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Published 2011 Revised 2019

The Mathematical Problems Faced by Advanced STEM Students

Mathematics is critical to the study of any STEM subject; indeed, historically the development of science, technology, engineering and mathematics has often gone hand in hand. The scientist or engineer needs to embrace mathematics in order to get the most from their studies. Unfortunately, students often struggle with the mathematical aspects of their scientific degree courses. In this article we explore some of the main mathematical problems arising. Far from simply a lack of content knowledge, we believe that the main area of concern is in mathematical process skills. Problem: Students don't know enough maths! Whilst preparing stemNRICH it was clear that sometimes certain content knowledge was lacking: those teaching biology, chemistry, physics and engineering courses often claimed that students didn't know enough about various topics in mathematics. Sometimes this lack of content knowledge was obvious: students in engineering need to know about complex numbers; other times it was graded or more subtle: biologists needed to know more about graphs and equations. Whilst these various topics obviously varied across universities and courses, interestingly, there was a surprising large overlap between the mathematical needs. The following core topics seemed to emerge across many disciplines:

Problem: Students can't apply their knowledge! Beneath any issues which might arise in knowledge of content, many students with good grades in mathematics seem to find it difficult to apply the mathematical knowledge that they might have. Why would this be the case? It seems that there are several main reasons, common to all disciplines:

• Overly Procedural thinking Mathematics exams can often be passed by learning the content procedurally. This means that students can answer certain types of question by following a recipe. The problems in scientific mathematics arise because even minor deviations from the precise recipe cause the student to fail to know what to do.  
• Lack of ability to translate mathematical meaning to real-world meaning Students who are very skilled at mathematics might have trouble seeing how to relate the mathematical process to a real-world context; this hampers the use of common sense, so valuable in quantitative science.  
• Lack of ability to make approximations or estimations Real scientific contexts are rarely simple. In order to apply mathematics predictively, approximations or estimations will need to be made. To make approximations or estimations requires the student to really understand the meaning and structure of the mathematics, along with the underlying scientific meaning.  
• Lack of multi-step problem solving skills Scientific mathematics problems are not usually clearly 'signposted' from a mathematical point of view. The student must assess the physical situation, decide how to represent it mathematically, decide what needs to be solved and then solve the problem. Students who are not well versed in solving 'multi-step' problems in mathematics are very likely to struggle with the application of their mathematical knowledge.  
• Lack of practice There are two ways in which lack of practice can impact mathematical activity in the sciences. First is a lack of skill at basic numerical or symbolic manipulation. This leads to errors and hold-ups regardless of whether the student understands what they are trying to do. Second is a lack of practice at thinking mathematically in a scientific context.  
• Lack of confidence Lack of confidence builds with uncertainty and failure, leading to more problems. Students who freeze at the sight of numbers or equations will most certainly underperform.  
• Lack of mathematical interest Students are hopefully strongly driven by their interest in science. If mathematics is studied in an environment independent of this then mathematics often never finds meaning and remains abstract, dull and difficult.

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Why do students struggle with math word problems? (And What to Try)

Word problems can be a real challenge for students of all ages. While some learners quickly grasp the concepts and transfer these skills to multi-step word problems, others struggle with even the most straightforward, basic word problems. As teachers, we must understand why this is so to help students succeed.

In this blog post, we'll explore common issues that cause difficulty when solving word problems and potential solutions that can assist learners in becoming more proficient problem-solvers. So, let's dive into what makes word problems so tricky and how you can help your students master them!

How to Help Learners Conquer Word Problems: Common Challenges & Solutions

Problem #1: students have difficulty reading & understanding the problems..

Word problems can be a daunting task for students of all ages. Solving math problems demands students to comprehend mathematical terms and have solid decoding abilities. If either of these skills is lacking, students may need help understanding the meaning behind certain words and phrases.

Considering that only a few sentences can determine the solution to a problem, it is essential to comprehend the language used in word problems. Yet, only  32% of 4th graders are proficient readers, according to the National Assessment of Educational Progress .

The challenge of comprehending the language in word problems is not only difficult for students who struggle with reading but can also be an obstacle for high-achieving math students. Often, these students know how to solve a problem but need help understanding what the problem is asking.

Word problems further complicate matters due to their use of language that's different from how we communicate.

For example, students may read a problem that says, “Sarah is baking a pie for her grandmother's birthday. She needs 7 apples for the recipe. At the store, apples are $2 a piece. If she has $11, will she have enough money to make the pie?”

Students must decode the words and phrases used to understand what the problem is asking them.

Solution: Provide word problems in audio formats & consider how you can incorporate explicit teaching into your math problem-solving routine.

One common strategy for addressing this is to read the problems aloud. Technology can help with this. Recording and storing problems where students can listen to them repeatedly can be helpful. However, you will need to teach your students to use this technology purposefully to help them better understand the word problems they are tackling. Without proper instruction, these recorded problems are no more helpful than reading the problems themselves.

However, this only addresses issues with decoding. It is essential to explain to students the meaning of words and math terms used in questions. A  Problem of the Day format  offers an excellent opportunity to deeply discuss a single problem with students without taking over your entire math lesson.

Explaining these concepts helps students build a stronger foundation for understanding word problems and increases their math comprehension.

Problem #2 :  Students have gaps in vocabulary that would help with math word problems.

Sometimes, story problems require students to have an understanding of math vocabulary. When students don't wholly understand math vocabulary, they struggle to understand what the problems are asking.

This is more than just decoding!

Even if they can read these words, they may need help understanding how to solve the problems. A strong foundation in math vocabulary is integral to any math classroom.

Solution: Explicitly teach and review math vocabulary regularly.

Ensure that students have a strong foundation in math vocabulary by explicitly teaching terms and concepts. This can be done through direct instruction, visual representations, and activities reinforcing the concepts.

Review these terms regularly throughout the year to ensure they stay fresh in students' minds.

Problem #3:  Students lack efficient & effective strategies.

Often, students are taught to use keywords early on. However, as problems become more complex, this quickly becomes an ineffective and inefficient strategy For addressing multi-step word problems.

Research has shown keywords often misdirect students' efforts and derail problem-solving with math word problems.

As a result, many state tests now purposefully include tricky problems designed to fool students who have been taught keywords as a problem-solving strategy.

Solution :  Teach a problem-solving strategy, like CUBES, that helps students break the problem down efficiently.

While keywords are ineffective, giving students a framework for breaking down word problems and identifying the information that CAN help them is a great way to support their problem-solving efforts.

The CUBES strategy (Circle, Underline, Box, Evaluate, Solve) can help older students with math word problems . This strategy helps them break down problems into manageable steps that make sense to them.

Problem #4:  Difficulty mapping out and visualizing the story behind each problem can lead to confusion in solving for an answer.

Another familiar struggle students face when solving word problems is difficulty mapping out and visualizing the story behind each problem. This can lead to confusion in solving for an answer because students may be unable to see how all the pieces fit together. In other words, they don't have a complete understanding of the context of the problem.

Solution:  Give students an active way to create a picture of what the problem is asking them.

Diagrams with labels, breaking the problem into simpler parts, and making a step-by-step plan with math word problems can help students understand the situation. Having them explain the story in their own words helps them clarify what they're trying to solve.

Encourage students who automatically add all the numbers to slow down and process the question with numberless word problems.

A numberless word problem is a story problem that does not include numbers . Instead, students are asked to analyze the problem without numbers before they are given the numbers to solve. This can help students notice patterns in the problem and determine what operations will be necessary for solving it. Adding these types of word problems to your instructional routine can be a great way to help students slow down and focus on understanding the scenario being presented in the problem. 

By providing students with different ways to visualize word problems, we can increase their chances of success and provide meaningful math instruction. Equipping them with the right tools and strategies gives them a better chance of tackling any difficult word problem they may encounter.

Problem #5:  Those with poor numeracy skills are disadvantaged when attempting to solve math word problems.

Computational fluency  is a common buzzword in math circles these days. We often discuss whether students know their math facts. However, math fact fluency becomes even more critical when students dive into more challenging word problems.

According to cognitive load theory, students focusing on rote processes such as basic facts have fewer mental resources left for higher-level thinking and processing.

In other words, the more mental energy it takes to work through the first step of a two-step problem, the less likely the student will have the resources to persist in accurately making it through the rest of the problem.

Solution: Build fact fluency practice into your routine in fun, engaging ways. 

Fact fluency practice doesn't have to be boring, but it is integral to being an effective mathematician. Therefore, finding ways to build it into your math class is essential.

Here are some of my favorite online games that students love:  30+ Awesome Online Games for Math Fact Practice .

Problem #6:  Students lack experience or are only provided with structured word problem practice.

Some curricula only include problems that follow a specific pattern or directly connect to the skill learned in a given lesson. However, formulaic word problems, where students follow a specific set of steps repeatedly, promote complacency.

Students begin to approach every word problem with the same steps. Soon they are grabbing numbers instead of taking the time to comprehend the problem and how best to address it.

Additionally, many word problems require students to apply knowledge from multiple different units to solve the problem. This can be challenging for students still working on mastering previously taught skills. It overwhelms those who have missed chunks of their instruction due to illness or being pulled from instruction.

As a result, these word problems often begin to feel impossible. 

Solution:  Incorporate variety into your problem-solving and allow for productive struggle.

Students need to be provided with an opportunity to approach a variety of different problems across time. They need to see problems that come in various formats. They need uniquely worded problems. This novelty prevents them from sticking with a rote set of strategies. The goal is to get them critically thinking about the problem at hand .

Offering variety builds confidence, competence, and the ability to address any problem they are given. Many students lack confidence in word problems. Varied experience reduces fears and helps students develop a bank of strategies to overcome barriers when complex problems arise.

To help foster independence, you can also support students through the gradual release process. Provide learners with a step-by-step guide to ensure they have completed the problem-solving process's critical steps when you aren't doing problems with them. This can help boost their confidence and reduce the risk of careless mistakes.

I've created a free mini-book for students with guiding questions and steps to help them independently complete word problems.

Get it here.

Building the math problem solver's toolbox

Word problems can be difficult for learners, but with the right strategies and resources, teachers can help their students learn to approach word problems confidently. By providing a variety of word problems that come in various formats and require different steps to solve, teachers can allow their students to develop problem-solving skills and build confidence when addressing any problem they are given.

Don't forget to grab the free problem solver's guide!

I hope you found this post helpful. Problem-solving is an essential skill for learners. Learn more about word problems or check out my Daily Problem Solving for engaging and meaningful word problem practice. 

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Struggling to overcome challenges in your life? We all face problems, big and small, on a regular basis.

So how do you tackle them effectively? What are some key problem-solving strategies and skills that can guide you?

Effective problem-solving requires breaking issues down logically, generating solutions creatively, weighing choices critically, and adapting plans flexibly based on outcomes. Useful strategies range from leveraging past solutions that have worked to visualizing problems through diagrams. Core skills include analytical abilities, innovative thinking, and collaboration.

Want to improve your problem-solving skills? Keep reading to find out 17 effective problem-solving strategies, key skills, common obstacles to watch for, and tips on improving your overall problem-solving skills.

Key Takeaways:

• Effective problem-solving requires breaking down issues logically, generating multiple solutions creatively, weighing choices critically, and adapting plans based on outcomes.
• Useful problem-solving strategies range from leveraging past solutions to brainstorming with groups to visualizing problems through diagrams and models.
• Core skills include analytical abilities, innovative thinking, decision-making, and team collaboration to solve problems.
• Common obstacles include fear of failure, information gaps, fixed mindsets, confirmation bias, and groupthink.
• Boosting problem-solving skills involves learning from experts, actively practicing, soliciting feedback, and analyzing others’ success.
• Onethread’s project management capabilities align with effective problem-solving tenets – facilitating structured solutions, tracking progress, and capturing lessons learned.

What Is Problem-Solving?

Problem-solving is the process of understanding an issue, situation, or challenge that needs to be addressed and then systematically working through possible solutions to arrive at the best outcome.

It involves critical thinking, analysis, logic, creativity, research, planning, reflection, and patience in order to overcome obstacles and find effective answers to complex questions or problems.

The ultimate goal is to implement the chosen solution successfully.

What Are Problem-Solving Strategies?

Problem-solving strategies are like frameworks or methodologies that help us solve tricky puzzles or problems we face in the workplace, at home, or with friends.

Imagine you have a big jigsaw puzzle. One strategy might be to start with the corner pieces. Another could be looking for pieces with the same colors. 

Just like in puzzles, in real life, we use different plans or steps to find solutions to problems. These strategies help us think clearly, make good choices, and find the best answers without getting too stressed or giving up.

Why Is It Important To Know Different Problem-Solving Strategies?

Knowing different problem-solving strategies is important because different types of problems often require different approaches to solve them effectively. Having a variety of strategies to choose from allows you to select the best method for the specific problem you are trying to solve.

This improves your ability to analyze issues thoroughly, develop solutions creatively, and tackle problems from multiple angles. Knowing multiple strategies also aids in overcoming roadblocks if your initial approach is not working.

Here are some reasons why you need to know different problem-solving strategies:

• Different Problems Require Different Tools: Just like you can’t use a hammer to fix everything, some problems need specific strategies to solve them.
• Improves Creativity: Knowing various strategies helps you think outside the box and come up with creative solutions.
• Saves Time: With the right strategy, you can solve problems faster instead of trying things that don’t work.
• Reduces Stress: When you know how to tackle a problem, it feels less scary and you feel more confident.
• Better Outcomes: Using the right strategy can lead to better solutions, making things work out better in the end.
• Learning and Growth: Each time you solve a problem, you learn something new, which makes you smarter and better at solving future problems.

Knowing different ways to solve problems helps you tackle anything that comes your way, making life a bit easier and more fun!

17 Effective Problem-Solving Strategies

Effective problem-solving strategies include breaking the problem into smaller parts, brainstorming multiple solutions, evaluating the pros and cons of each, and choosing the most viable option. 

Critical thinking and creativity are essential in developing innovative solutions. Collaboration with others can also provide diverse perspectives and ideas. 

By applying these strategies, you can tackle complex issues more effectively.

Now, consider a challenge you’re dealing with. Which strategy could help you find a solution? Here we will discuss key problem strategies in detail.

1. Use a Past Solution That Worked

This strategy involves looking back at previous similar problems you have faced and the solutions that were effective in solving them.

It is useful when you are facing a problem that is very similar to something you have already solved. The main benefit is that you don’t have to come up with a brand new solution – you already know the method that worked before will likely work again.

However, the limitation is that the current problem may have some unique aspects or differences that mean your old solution is not fully applicable.

The ideal process is to thoroughly analyze the new challenge, identify the key similarities and differences versus the past case, adapt the old solution as needed to align with the current context, and then pilot it carefully before full implementation.

An example is using the same negotiation tactics from purchasing your previous home when putting in an offer on a new house. Key terms would be adjusted but overall it can save significant time versus developing a brand new strategy.

2. Brainstorm Solutions

This involves gathering a group of people together to generate as many potential solutions to a problem as possible.

It is effective when you need creative ideas to solve a complex or challenging issue. By getting input from multiple people with diverse perspectives, you increase the likelihood of finding an innovative solution.

The main limitation is that brainstorming sessions can sometimes turn into unproductive gripe sessions or discussions rather than focusing on productive ideation —so they need to be properly facilitated.

The key to an effective brainstorming session is setting some basic ground rules upfront and having an experienced facilitator guide the discussion. Rules often include encouraging wild ideas, avoiding criticism of ideas during the ideation phase, and building on others’ ideas.

For instance, a struggling startup might hold a session where ideas for turnaround plans are generated and then formalized with financials and metrics.

3. Work Backward from the Solution

This technique involves envisioning that the problem has already been solved and then working step-by-step backward toward the current state.

This strategy is particularly helpful for long-term, multi-step problems. By starting from the imagined solution and identifying all the steps required to reach it, you can systematically determine the actions needed. It lets you tackle a big hairy problem through smaller, reversible steps.

A limitation is that this approach may not be possible if you cannot accurately envision the solution state to start with.

The approach helps drive logical systematic thinking for complex problem-solving, but should still be combined with creative brainstorming of alternative scenarios and solutions.

An example is planning for an event – you would imagine the successful event occurring, then determine the tasks needed the week before, two weeks before, etc. all the way back to the present.

4. Use the Kipling Method

This method, named after author Rudyard Kipling, provides a framework for thoroughly analyzing a problem before jumping into solutions.

It consists of answering six fundamental questions: What, Where, When, How, Who, and Why about the challenge. Clearly defining these core elements of the problem sets the stage for generating targeted solutions.

The Kipling method enables a deep understanding of problem parameters and root causes before solution identification. By jumping to brainstorm solutions too early, critical information can be missed or the problem is loosely defined, reducing solution quality.

Answering the six fundamental questions illuminates all angles of the issue. This takes time but pays dividends in generating optimal solutions later tuned precisely to the true underlying problem.

The limitation is that meticulously working through numerous questions before addressing solutions can slow progress.

The best approach blends structured problem decomposition techniques like the Kipling method with spurring innovative solution ideation from a diverse team. 

An example is using this technique after a technical process failure – the team would systematically detail What failed, Where/When did it fail, How it failed (sequence of events), Who was involved, and Why it likely failed before exploring preventative solutions.

5. Try Different Solutions Until One Works (Trial and Error)

This technique involves attempting various potential solutions sequentially until finding one that successfully solves the problem.

Trial and error works best when facing a concrete, bounded challenge with clear solution criteria and a small number of discrete options to try. By methodically testing solutions, you can determine the faulty component.

A limitation is that it can be time-intensive if the working solution set is large.

The key is limiting the variable set first. For technical problems, this boundary is inherent and each element can be iteratively tested. But for business issues, artificial constraints may be required – setting decision rules upfront to reduce options before testing.

Furthermore, hypothesis-driven experimentation is far superior to blind trial and error – have logic for why Option A may outperform Option B.

Examples include fixing printer jams by testing different paper tray and cable configurations or resolving website errors by tweaking CSS/HTML line-by-line until the code functions properly.

6. Use Proven Formulas or Frameworks (Heuristics)

Heuristics refers to applying existing problem-solving formulas or frameworks rather than addressing issues completely from scratch.

This allows leveraging established best practices rather than reinventing the wheel each time.

It is effective when facing recurrent, common challenges where proven structured approaches exist.

However, heuristics may force-fit solutions to non-standard problems.

For example, a cost-benefit analysis can be used instead of custom weighting schemes to analyze potential process improvements.

Onethread allows teams to define, save, and replicate configurable project templates so proven workflows can be reliably applied across problems with some consistency rather than fully custom one-off approaches each time.

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7. Trust Your Instincts (Insight Problem-Solving)

Insight is a problem-solving technique that involves waiting patiently for an unexpected “aha moment” when the solution pops into your mind.

It works well for personal challenges that require intuitive realizations over calculated logic. The unconscious mind makes connections leading to flashes of insight when relaxing or doing mundane tasks unrelated to the actual problem.

Benefits include out-of-the-box creative solutions. However, the limitations are that insights can’t be forced and may never come at all if too complex. Critical analysis is still required after initial insights.

A real-life example would be a writer struggling with how to end a novel. Despite extensive brainstorming, they feel stuck. Eventually while gardening one day, a perfect unexpected plot twist sparks an ideal conclusion. However, once written they still carefully review if the ending flows logically from the rest of the story.

8. Reverse Engineer the Problem

This approach involves deconstructing a problem in reverse sequential order from the current undesirable outcome back to the initial root causes.

By mapping the chain of events backward, you can identify the origin of where things went wrong and establish the critical junctures for solving it moving ahead. Reverse engineering provides diagnostic clarity on multi-step problems.

However, the limitation is that it focuses heavily on autopsying the past versus innovating improved future solutions.

An example is tracing back from a server outage, through the cascade of infrastructure failures that led to it finally terminating at the initial script error that triggered the crisis. This root cause would then inform the preventative measure.

9. Break Down Obstacles Between Current and Goal State (Means-End Analysis)

This technique defines the current problem state and the desired end goal state, then systematically identifies obstacles in the way of getting from one to the other.

By mapping the barriers or gaps, you can then develop solutions to address each one. This methodically connects the problem to solutions.

A limitation is that some obstacles may be unknown upfront and only emerge later.

For example, you can list down all the steps required for a new product launch – current state through production, marketing, sales, distribution, etc. to full launch (goal state) – to highlight where resource constraints or other blocks exist so they can be addressed.

Onethread allows dividing big-picture projects into discrete, manageable phases, milestones, and tasks to simplify execution just as problems can be decomposed into more achievable components. Features like dependency mapping further reinforce interconnections.

Using Onethread’s issues and subtasks feature, messy problems can be decomposed into manageable chunks.

10. Ask “Why” Five Times to Identify the Root Cause (The 5 Whys)

This technique involves asking “Why did this problem occur?” and then responding with an answer that is again met with asking “Why?” This process repeats five times until the root cause is revealed.

Continually asking why digs deeper from surface symptoms to underlying systemic issues.

It is effective for getting to the source of problems originating from human error or process breakdowns.

However, some complex issues may have multiple tangled root causes not solvable through this approach alone.

An example is a retail store experiencing a sudden decline in customers. Successively asking why five times may trace an initial drop to parking challenges, stemming from a city construction project – the true starting point to address.

11. Evaluate Strengths, Weaknesses, Opportunities, and Threats (SWOT Analysis)

This involves analyzing a problem or proposed solution by categorizing internal and external factors into a 2×2 matrix: Strengths, Weaknesses as the internal rows; Opportunities and Threats as the external columns.

Systematically identifying these elements provides balanced insight to evaluate options and risks. It is impactful when evaluating alternative solutions or developing strategy amid complexity or uncertainty.

The key benefit of SWOT analysis is enabling multi-dimensional thinking when rationally evaluating options. Rather than getting anchored on just the upsides or the existing way of operating, it urges a systematic assessment through four different lenses:

• Internal Strengths: Our core competencies/advantages able to deliver success
• Internal Weaknesses: Gaps/vulnerabilities we need to manage
• External Opportunities: Ways we can differentiate/drive additional value
• External Threats: Risks we must navigate or mitigate

Multiperspective analysis provides the needed holistic view of the balanced risk vs. reward equation for strategic decision making amid uncertainty.

However, SWOT can feel restrictive if not tailored and evolved for different issue types.

Teams should view SWOT analysis as a starting point, augmenting it further for distinct scenarios.

An example is performing a SWOT analysis on whether a small business should expand into a new market – evaluating internal capabilities to execute vs. risks in the external competitive and demand environment to inform the growth decision with eyes wide open.

12. Compare Current vs Expected Performance (Gap Analysis)

This technique involves comparing the current state of performance, output, or results to the desired or expected levels to highlight shortfalls.

By quantifying the gaps, you can identify problem areas and prioritize address solutions.

Gap analysis is based on the simple principle – “you can’t improve what you don’t measure.” It enables facts-driven problem diagnosis by highlighting delta to goals, not just vague dissatisfaction that something seems wrong. And measurement immediately suggests improvement opportunities – address the biggest gaps first.

This data orientation also supports ROI analysis on fixing issues – the return from closing larger gaps outweighs narrowly targeting smaller performance deficiencies.

However, the approach is only effective if robust standards and metrics exist as the benchmark to evaluate against. Organizations should invest upfront in establishing performance frameworks.

Furthermore, while numbers are invaluable, the human context behind problems should not be ignored – quantitative versus qualitative gap assessment is optimally blended.

For example, if usage declines are noted during software gap analysis, this could be used as a signal to improve user experience through design.

13. Observe Processes from the Frontline (Gemba Walk)

A Gemba walk involves going to the actual place where work is done, directly observing the process, engaging with employees, and finding areas for improvement.

By experiencing firsthand rather than solely reviewing abstract reports, practical problems and ideas emerge.

The limitation is Gemba walks provide anecdotes not statistically significant data. It complements but does not replace comprehensive performance measurement.

An example is a factory manager inspecting the production line to spot jam areas based on direct reality rather than relying on throughput dashboards alone back in her office. Frontline insights prove invaluable.

14. Analyze Competitive Forces (Porter’s Five Forces)

This involves assessing the marketplace around a problem or business situation via five key factors: competitors, new entrants, substitute offerings, suppliers, and customer power.

Evaluating these forces illuminates risks and opportunities for strategy development and issue resolution. It is effective for understanding dynamic external threats and opportunities when operating in a contested space.

However, over-indexing on only external factors can overlook the internal capabilities needed to execute solutions.

A startup CEO, for example, may analyze market entry barriers, whitespace opportunities, and disruption risks across these five forces to shape new product rollout strategies and marketing approaches.

15. Think from Different Perspectives (Six Thinking Hats)

The Six Thinking Hats is a technique developed by Edward de Bono that encourages people to think about a problem from six different perspectives, each represented by a colored “thinking hat.”

The key benefit of this strategy is that it pushes team members to move outside their usual thinking style and consider new angles. This brings more diverse ideas and solutions to the table.

It works best for complex problems that require innovative solutions and when a team is stuck in an unproductive debate. The structured framework keeps the conversation flowing in a positive direction.

Limitations are that it requires training on the method itself and may feel unnatural at first. Team dynamics can also influence success – some members may dominate certain “hats” while others remain quiet.

A real-life example is a software company debating whether to build a new feature. The white hat focuses on facts, red on gut feelings, black on potential risks, yellow on benefits, green on new ideas, and blue on process. This exposes more balanced perspectives before deciding.

Onethread centralizes diverse stakeholder communication onto one platform, ensuring all voices are incorporated when evaluating project tradeoffs, just as problem-solving should consider multifaceted solutions.

16. Visualize the Problem (Draw it Out)

Drawing out a problem involves creating visual representations like diagrams, flowcharts, and maps to work through challenging issues.

This strategy is helpful when dealing with complex situations with lots of interconnected components. The visuals simplify the complexity so you can thoroughly understand the problem and all its nuances.

Key benefits are that it allows more stakeholders to get on the same page regarding root causes and it sparks new creative solutions as connections are made visually.

However, simple problems with few variables don’t require extensive diagrams. Additionally, some challenges are so multidimensional that fully capturing every aspect is difficult.

A real-life example would be mapping out all the possible causes leading to decreased client satisfaction at a law firm. An intricate fishbone diagram with branches for issues like service delivery, technology, facilities, culture, and vendor partnerships allows the team to trace problems back to their origins and brainstorm targeted fixes.

17. Follow a Step-by-Step Procedure (Algorithms)

An algorithm is a predefined step-by-step process that is guaranteed to produce the correct solution if implemented properly.

Using algorithms is effective when facing problems that have clear, binary right and wrong answers. Algorithms work for mathematical calculations, computer code, manufacturing assembly lines, and scientific experiments.

Key benefits are consistency, accuracy, and efficiency. However, they require extensive upfront development and only apply to scenarios with strict parameters. Additionally, human error can lead to mistakes.

For example, crew members of fast food chains like McDonald’s follow specific algorithms for food prep – from grill times to ingredient amounts in sandwiches, to order fulfillment procedures. This ensures uniform quality and service across all locations. However, if a step is missed, errors occur.

The Problem-Solving Process

The problem-solving process typically includes defining the issue, analyzing details, creating solutions, weighing choices, acting, and reviewing results.

In the above, we have discussed several problem-solving strategies. For every problem-solving strategy, you have to follow these processes. Here’s a detailed step-by-step process of effective problem-solving:

Step 1: Identify the Problem

The problem-solving process starts with identifying the problem. This step involves understanding the issue’s nature, its scope, and its impact. Once the problem is clearly defined, it sets the foundation for finding effective solutions.

Identifying the problem is crucial. It means figuring out exactly what needs fixing. This involves looking at the situation closely, understanding what’s wrong, and knowing how it affects things. It’s about asking the right questions to get a clear picture of the issue. 

This step is important because it guides the rest of the problem-solving process. Without a clear understanding of the problem, finding a solution is much harder. It’s like diagnosing an illness before treating it. Once the problem is identified accurately, you can move on to exploring possible solutions and deciding on the best course of action.

Step 2: Break Down the Problem

Breaking down the problem is a key step in the problem-solving process. It involves dividing the main issue into smaller, more manageable parts. This makes it easier to understand and tackle each component one by one.

After identifying the problem, the next step is to break it down. This means splitting the big issue into smaller pieces. It’s like solving a puzzle by handling one piece at a time. 

By doing this, you can focus on each part without feeling overwhelmed. It also helps in identifying the root causes of the problem. Breaking down the problem allows for a clearer analysis and makes finding solutions more straightforward. 

Each smaller problem can be addressed individually, leading to an effective resolution of the overall issue. This approach not only simplifies complex problems but also aids in developing a systematic plan to solve them.

Step 3: Come up with potential solutions

Coming up with potential solutions is the third step in the problem-solving process. It involves brainstorming various options to address the problem, considering creativity and feasibility to find the best approach.

After breaking down the problem, it’s time to think of ways to solve it. This stage is about brainstorming different solutions. You look at the smaller issues you’ve identified and start thinking of ways to fix them. This is where creativity comes in. 

You want to come up with as many ideas as possible, no matter how out-of-the-box they seem. It’s important to consider all options and evaluate their pros and cons. This process allows you to gather a range of possible solutions. 

Later, you can narrow these down to the most practical and effective ones. This step is crucial because it sets the stage for deciding on the best solution to implement. It’s about being open-minded and innovative to tackle the problem effectively.

Step 4: Analyze the possible solutions

Analyzing the possible solutions is the fourth step in the problem-solving process. It involves evaluating each proposed solution’s advantages and disadvantages to determine the most effective and feasible option.

After coming up with potential solutions, the next step is to analyze them. This means looking closely at each idea to see how well it solves the problem. You weigh the pros and cons of every solution.

Consider factors like cost, time, resources, and potential outcomes. This analysis helps in understanding the implications of each option. It’s about being critical and objective, ensuring that the chosen solution is not only effective but also practical.

This step is vital because it guides you towards making an informed decision. It involves comparing the solutions against each other and selecting the one that best addresses the problem.

By thoroughly analyzing the options, you can move forward with confidence, knowing you’ve chosen the best path to solve the issue.

Step 5: Implement and Monitor the Solutions

Implementing and monitoring the solutions is the final step in the problem-solving process. It involves putting the chosen solution into action and observing its effectiveness, making adjustments as necessary.

Once you’ve selected the best solution, it’s time to put it into practice. This step is about action. You implement the chosen solution and then keep an eye on how it works. Monitoring is crucial because it tells you if the solution is solving the problem as expected. 

If things don’t go as planned, you may need to make some changes. This could mean tweaking the current solution or trying a different one. The goal is to ensure the problem is fully resolved. 

This step is critical because it involves real-world application. It’s not just about planning; it’s about doing and adjusting based on results. By effectively implementing and monitoring the solutions, you can achieve the desired outcome and solve the problem successfully.

Why This Process is Important

Following a defined process to solve problems is important because it provides a systematic, structured approach instead of a haphazard one. Having clear steps guides logical thinking, analysis, and decision-making to increase effectiveness. Key reasons it helps are:

• Clear Direction: This process gives you a clear path to follow, which can make solving problems less overwhelming.
• Better Solutions: Thoughtful analysis of root causes, iterative testing of solutions, and learning orientation lead to addressing the heart of issues rather than just symptoms.
• Saves Time and Energy: Instead of guessing or trying random things, this process helps you find a solution more efficiently.
• Improves Skills: The more you use this process, the better you get at solving problems. It’s like practicing a sport. The more you practice, the better you play.
• Maximizes collaboration: Involving various stakeholders in the process enables broader inputs. Their communication and coordination are streamlined through organized brainstorming and evaluation.
• Provides consistency: Standard methodology across problems enables building institutional problem-solving capabilities over time. Patterns emerge on effective techniques to apply to different situations.

The problem-solving process is a powerful tool that can help us tackle any challenge we face. By following these steps, we can find solutions that work and learn important skills along the way.

Key Skills for Efficient Problem Solving

Efficient problem-solving requires breaking down issues logically, evaluating options, and implementing practical solutions.

Key skills include critical thinking to understand root causes, creativity to brainstorm innovative ideas, communication abilities to collaborate with others, and decision-making to select the best way forward. Staying adaptable, reflecting on outcomes, and applying lessons learned are also essential.

With practice, these capacities will lead to increased personal and team effectiveness in systematically addressing any problem.

 Let’s explore the powers you need to become a problem-solving hero!

Critical Thinking and Analytical Skills

Critical thinking and analytical skills are vital for efficient problem-solving as they enable individuals to objectively evaluate information, identify key issues, and generate effective solutions. 

These skills facilitate a deeper understanding of problems, leading to logical, well-reasoned decisions. By systematically breaking down complex issues and considering various perspectives, individuals can develop more innovative and practical solutions, enhancing their problem-solving effectiveness.

Communication Skills

Effective communication skills are essential for efficient problem-solving as they facilitate clear sharing of information, ensuring all team members understand the problem and proposed solutions. 

These skills enable individuals to articulate issues, listen actively, and collaborate effectively, fostering a productive environment where diverse ideas can be exchanged and refined. By enhancing mutual understanding, communication skills contribute significantly to identifying and implementing the most viable solutions.

Decision-Making

Strong decision-making skills are crucial for efficient problem-solving, as they enable individuals to choose the best course of action from multiple alternatives. 

These skills involve evaluating the potential outcomes of different solutions, considering the risks and benefits, and making informed choices. Effective decision-making leads to the implementation of solutions that are likely to resolve problems effectively, ensuring resources are used efficiently and goals are achieved.

Planning and Prioritization

Planning and prioritization are key for efficient problem-solving, ensuring resources are allocated effectively to address the most critical issues first. This approach helps in organizing tasks according to their urgency and impact, streamlining efforts towards achieving the desired outcome efficiently.

Emotional Intelligence

Emotional intelligence enhances problem-solving by allowing individuals to manage emotions, understand others, and navigate social complexities. It fosters a positive, collaborative environment, essential for generating creative solutions and making informed, empathetic decisions.

Leadership skills drive efficient problem-solving by inspiring and guiding teams toward common goals. Effective leaders motivate their teams, foster innovation, and navigate challenges, ensuring collective efforts are focused and productive in addressing problems.

Time Management

Time management is crucial in problem-solving, enabling individuals to allocate appropriate time to each task. By efficiently managing time, one can ensure that critical problems are addressed promptly without neglecting other responsibilities.

Data Analysis

Data analysis skills are essential for problem-solving, as they enable individuals to sift through data, identify trends, and extract actionable insights. This analytical approach supports evidence-based decision-making, leading to more accurate and effective solutions.

Research Skills

Research skills are vital for efficient problem-solving, allowing individuals to gather relevant information, explore various solutions, and understand the problem’s context. This thorough exploration aids in developing well-informed, innovative solutions.

Becoming a great problem solver takes practice, but with these skills, you’re on your way to becoming a problem-solving hero. 

How to Improve Your Problem-Solving Skills?

Improving your problem-solving skills can make you a master at overcoming challenges. Learn from experts, practice regularly, welcome feedback, try new methods, experiment, and study others’ success to become better.

Learning from Experts

Improving problem-solving skills by learning from experts involves seeking mentorship, attending workshops, and studying case studies. Experts provide insights and techniques that refine your approach, enhancing your ability to tackle complex problems effectively.

To enhance your problem-solving skills, learning from experts can be incredibly beneficial. Engaging with mentors, participating in specialized workshops, and analyzing case studies from seasoned professionals can offer valuable perspectives and strategies. 

Experts share their experiences, mistakes, and successes, providing practical knowledge that can be applied to your own problem-solving process. This exposure not only broadens your understanding but also introduces you to diverse methods and approaches, enabling you to tackle challenges more efficiently and creatively.

Improving problem-solving skills through practice involves tackling a variety of challenges regularly. This hands-on approach helps in refining techniques and strategies, making you more adept at identifying and solving problems efficiently.

One of the most effective ways to enhance your problem-solving skills is through consistent practice. By engaging with different types of problems on a regular basis, you develop a deeper understanding of various strategies and how they can be applied. 

This hands-on experience allows you to experiment with different approaches, learn from mistakes, and build confidence in your ability to tackle challenges.

Regular practice not only sharpens your analytical and critical thinking skills but also encourages adaptability and innovation, key components of effective problem-solving.

Openness to Feedback

Being open to feedback is like unlocking a secret level in a game. It helps you boost your problem-solving skills. Improving problem-solving skills through openness to feedback involves actively seeking and constructively responding to critiques. 

This receptivity enables you to refine your strategies and approaches based on insights from others, leading to more effective solutions. 

Learning New Approaches and Methodologies

Learning new approaches and methodologies is like adding new tools to your toolbox. It makes you a smarter problem-solver. Enhancing problem-solving skills by learning new approaches and methodologies involves staying updated with the latest trends and techniques in your field. 

This continuous learning expands your toolkit, enabling innovative solutions and a fresh perspective on challenges.

Experimentation

Experimentation is like being a scientist of your own problems. It’s a powerful way to improve your problem-solving skills. Boosting problem-solving skills through experimentation means trying out different solutions to see what works best. This trial-and-error approach fosters creativity and can lead to unique solutions that wouldn’t have been considered otherwise.

Analyzing Competitors’ Success

Analyzing competitors’ success is like being a detective. It’s a smart way to boost your problem-solving skills. Improving problem-solving skills by analyzing competitors’ success involves studying their strategies and outcomes. Understanding what worked for them can provide valuable insights and inspire effective solutions for your own challenges. 

Challenges in Problem-Solving

Facing obstacles when solving problems is common. Recognizing these barriers, like fear of failure or lack of information, helps us find ways around them for better solutions.

Fear of Failure

Fear of failure is like a big, scary monster that stops us from solving problems. It’s a challenge many face. Because being afraid of making mistakes can make us too scared to try new solutions. 

How can we overcome this? First, understand that it’s okay to fail. Failure is not the opposite of success; it’s part of learning. Every time we fail, we discover one more way not to solve a problem, getting us closer to the right solution. Treat each attempt like an experiment. It’s not about failing; it’s about testing and learning.

Lack of Information

Lack of information is like trying to solve a puzzle with missing pieces. It’s a big challenge in problem-solving. Because without all the necessary details, finding a solution is much harder. 

How can we fix this? Start by gathering as much information as you can. Ask questions, do research, or talk to experts. Think of yourself as a detective looking for clues. The more information you collect, the clearer the picture becomes. Then, use what you’ve learned to think of solutions. 

Fixed Mindset

A fixed mindset is like being stuck in quicksand; it makes solving problems harder. It means thinking you can’t improve or learn new ways to solve issues. 

How can we change this? First, believe that you can grow and learn from challenges. Think of your brain as a muscle that gets stronger every time you use it. When you face a problem, instead of saying “I can’t do this,” try thinking, “I can’t do this yet.” Look for lessons in every challenge and celebrate small wins. 

Everyone starts somewhere, and mistakes are just steps on the path to getting better. By shifting to a growth mindset, you’ll see problems as opportunities to grow. Keep trying, keep learning, and your problem-solving skills will soar!

Jumping to Conclusions

Jumping to conclusions is like trying to finish a race before it starts. It’s a challenge in problem-solving. That means making a decision too quickly without looking at all the facts. 

How can we avoid this? First, take a deep breath and slow down. Think about the problem like a puzzle. You need to see all the pieces before you know where they go. Ask questions, gather information, and consider different possibilities. Don’t choose the first solution that comes to mind. Instead, compare a few options. 

Feeling Overwhelmed

Feeling overwhelmed is like being buried under a mountain of puzzles. It’s a big challenge in problem-solving. When we’re overwhelmed, everything seems too hard to handle. 

How can we deal with this? Start by taking a step back. Breathe deeply and focus on one thing at a time. Break the big problem into smaller pieces, like sorting puzzle pieces by color. Tackle each small piece one by one. It’s also okay to ask for help. Sometimes, talking to someone else can give you a new perspective. 

Confirmation Bias

Confirmation bias is like wearing glasses that only let you see what you want to see. It’s a challenge in problem-solving. Because it makes us focus only on information that agrees with what we already believe, ignoring anything that doesn’t. 

How can we overcome this? First, be aware that you might be doing it. It’s like checking if your glasses are on right. Then, purposely look for information that challenges your views. It’s like trying on a different pair of glasses to see a new perspective. Ask questions and listen to answers, even if they don’t fit what you thought before.

Groupthink is like everyone in a group deciding to wear the same outfit without asking why. It’s a challenge in problem-solving. It means making decisions just because everyone else agrees, without really thinking it through. 

How can we avoid this? First, encourage everyone in the group to share their ideas, even if they’re different. It’s like inviting everyone to show their unique style of clothes. 

Listen to all opinions and discuss them. It’s okay to disagree; it helps us think of better solutions. Also, sometimes, ask someone outside the group for their thoughts. They might see something everyone in the group missed.

Overcoming obstacles in problem-solving requires patience, openness, and a willingness to learn from mistakes. By recognizing these barriers, we can develop strategies to navigate around them, leading to more effective and creative solutions.

What are the most common problem-solving techniques?

The most common techniques include brainstorming, the 5 Whys, mind mapping, SWOT analysis, and using algorithms or heuristics. Each approach has its strengths, suitable for different types of problems.

What’s the best problem-solving strategy for every situation?

There’s no one-size-fits-all strategy. The best approach depends on the problem’s complexity, available resources, and time constraints. Combining multiple techniques often yields the best results.

How can I improve my problem-solving skills?

Improve your problem-solving skills by practicing regularly, learning from experts, staying open to feedback, and continuously updating your knowledge on new approaches and methodologies.

Are there any tools or resources to help with problem-solving?

Yes, tools like mind mapping software, online courses on critical thinking, and books on problem-solving techniques can be very helpful. Joining forums or groups focused on problem-solving can also provide support and insights.

What are some common mistakes people make when solving problems?

Common mistakes include jumping to conclusions without fully understanding the problem, ignoring valuable feedback, sticking to familiar solutions without considering alternatives, and not breaking down complex problems into manageable parts.

Final Words

Mastering problem-solving strategies equips us with the tools to tackle challenges across all areas of life. By understanding and applying these techniques, embracing a growth mindset, and learning from both successes and obstacles, we can transform problems into opportunities for growth. Continuously improving these skills ensures we’re prepared to face and solve future challenges more effectively.

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Initial Thoughts

Perspectives & resources, what is high-quality mathematics instruction and why is it important, page 1: the importance of high-quality mathematics instruction.

• Page 2: A Standards-Based Mathematics Curriculum
• Page 3: Evidence-Based Mathematics Practices

What evidence-based mathematics practices can teachers employ?

• Page 4: Explicit, Systematic Instruction
• Page 5: Visual Representations
• Page 6: Schema Instruction
• Page 7: Metacognitive Strategies
• Page 8: Effective Classroom Practices
• Page 9: References & Additional Resources
• Page 10: Credits

Research Shows

Studies, such as those below, have raised concerns about the adequacy of mathematics instruction in the United States.

• In 2015, the Program for International Student Assessment (PISA) compared the mathematical literacy—that is, the ability to express, apply, and interpret mathematics in different contexts—of 15-year-old students in the United States to those in other industrialized countries. Out of 69 countries, the United States ranked 36th. (Kastberg, Chan, & Murray, 2016)
• In 2015, the Trends in International Mathematics and Science Study (TIMSS) indicated that 8th-grade students in the United States were outperformed in math proficiency by students in Singapore, the Republic of Korea, China, Japan, Kazakhstan, the Russian Federation, Canada, and Ireland. (Provasnik, Malley, Stephens, Landeros, Perkins, & Tang, 2016)

Description

This bar graph illustrates the results of the 2022 National Assessment of Educational Progress (NAEP) mathematics achievement test for 4th and 8th grades and the 2019 data for 12th grade. The table is divided into three columns—one for 4th-grade results, one for 8th-grade results, and the last for 12th-grade results—and each column is divided into two rows. The top row is labeled “Students Proficient & Advanced,” while the lower is labeled “Students Basic & Below Basic.”

The test results are displayed for three categories of test takers: “All Students,” “Students with Disabilities,” and “English Learners (ELs).” The “All Students” bars are colored aqua. The “Students with Disabilities” bars are peach. Finally, the “English Learners” bars are yellow.

In the 4th-grade column, “All Students” are 36% in the Proficient & Advanced range and 64% in the Basic & Below Basic range. “Students with Disabilities” are 16% in Proficient & Advanced and 84% in Basic & Below. Likewise, “English Learners” are 14% in Proficient & Advanced and 86% in Basic & Below.

In the 8th-grade column, “All Students” are 26% in the Proficient & Advanced range and 74% in the Basic & Below Basic range. “Students with Disabilities” are 7% in Proficient & Advanced and 93% in Basic & Below. Finally, “English Learners” are 4% in Proficient & Advanced and 96% in Basic & Below.

In the 12th-grade column, “All Students” are 24% in the Proficient & Advanced range and 76% in the Basic & Below Basic range. “Students with Disabilities” are 7% in Proficient & Advanced and 93% in Basic & Below. Finally, “English Learners” are 3% in Proficient & Advanced and 97% in Basic & Below .

What Do These Data Indicate?

The NAEP test is designed so that students who have learned the knowledge and skills deemed important for their grade level should score at the “Proficient” level at a minimum. However, the NAEP data show that a majority of students perform at the “Below Basic” and “Basic” levels, indicating little to partial mastery. And, of course, this lack of mastery will likely create new hurdles in subsequent grade levels, where a foundational understanding of mathematical concepts and procedures is required for success. Students who do not master foundational mathematics skills in early grades are likely to fall further behind as they progress through school.

With all of this in mind, take another look at the data in the table above. Note that many 4th-grade students (60%) have yet to master the expected mathematics knowledge and skills, a percentage that is even greater (75%) for students in the 12th grade. And the results are even worse for students with disabilities as well as for English language learners (ELLs).

The purpose of this module is to explore why some students struggle with mathematics and what steps teachers can take to give their students a significantly improved chance to succeed in the classroom.

Consider This

In recent years, a greater emphasis on college- and career-readiness has led to higher standards for student mathematics proficiency, as well as to more rigorous graduation requirements. Whereas in the past high school students in most states were required to complete Algebra 1 prior to graduation, those students are now often required to complete Algebra 1, Geometry, and Algebra II. Once more, reflect on the NAEP data presented above and consider the number of students who will struggle to meet these new requirements.

Why Do Some Students Struggle More with Math?

Students with learning disabilities.

Compared to the overall population, a much smaller percentage of students with disabilities demonstrate proficiency in mathematics. Although every learner is unique, students with a mathematics learning disability (MLD) tend to display any of a number of characteristics that affect their mathematics performance, including:

mathematics learning disability (MLD)

Condition characterized by significant difficulty in mathematics calculation and/or problem solving; a specific learning disability (SLD) in the area of mathematics, sometimes referred to as dyscalculia.

• Difficulty processing information
• Difficulty identifying relevant information in mathematics problems, especially in word problems
• Difficulty translating information into a mathematical expression or equation
• Problem maintaining attention
• Difficulty selecting an effective problem-solving strategy
• Poor reasoning and problem-solving skills
• Working through a problem without making sure all steps are completed or that the answer makes sense
• Deficits in the areas of mathematics facts and computational skills
• Memory and vocabulary difficulties
• Difficulty solving multi-step problems
• Weak visual/spatial representational skills
• Difficulty reading about mathematics
• Difficulty understanding the language, or vocabulary, of mathematics
• Difficulty understanding mathematics concepts and how concepts relate to procedures
• Mathematics anxiety
• Learned helplessness—that is, having low motivation, being a passive learner, and attributing both successes and failures to external, uncontrollable factors (e.g., luck)

Diane Bryant, who conducts research on mathematics interventions for students with LD, describes why students with mathematics learning disabilities and those who struggle with mathematics are often grouped together in research in this area (time: 1:00).

Diane Pedrotty Bryant, PhD Project Director, Mathematics Institute for Learning Disabilities and Difficulties University of Texas at Austin

View Transcript

Transcript: Diane Pedrotty Bryant, PhD

Students who struggle with mathematics and those who have been identified as having mathematics learning disabilities share similar characteristics. For example, both groups of students may have difficulty with what’s called number sense , just understanding numbers and the meaning of numbers and certainly calculation. Problem solving is another area. So I think this is one reason that these two groups are often included in studies. Another reason has to do with ensuring that you have a large enough sample size so that you can look at the treatment effects of practices that may become evidence-based practices. Therefore, it’s not uncommon to see both students with mathematics difficulties and students with identified mathematics learning disabilities included in the sample.

English Language Learners

ELLs, too, have difficulty with mathematics, though their struggles are more likely to be a result of linguistic issues. It is important for teachers to understand that mathematics consists of more than just numbers; it includes a significant amount of content-specific vocabulary. When teachers discuss mathematical procedures and concepts, they typically use academic language , which is composed of content-specific vocabulary—for example, words like factor , estimate , and sum —vocabulary with multiple meanings—for example, table —and complex language structures. Not surprisingly, then, many ELLs struggle to solve mathematical word problems.

academic language

The language used in academic settings to communicate information orally and in writing about discipline-specific content.

What Can Teachers Do?

To improve student mathematics performance, more and more school and districts have implemented high-quality mathematics instruction. This instruction involves the implementation of both:

• A standards-based curriculum — The concepts and skills believed to be important for students to learn
• Evidence-based practices (EBP) — Strategies or practices proven through research to be effective for teaching mathematical concepts and procedures
• When they provide effective mathematics instruction, teachers can reduce the performance gap between students who are at risk for mathematics difficulty and their average and high-performing peers. (Clarke, Smollkowski, Baker, Fien, Doabler, & Chard, 2011)
• Students with learning disabilities who receive effective mathematics intervention can develop critical skills like problem-solving and abstract reasoning, which are necessary to achieve mathematics proficiency. (Gersten, Chard, Jayanthi, Baker, Morphy, & Flojo, 2008; Allsopp, Lovin, & van Ingen, 2017)

Among the factors that sometimes influence effective high-quality mathematics instruction are teacher and student perceptions and beliefs about mathematics itself.

Perceptions and Beliefs Regarding Math

Are the statements about mathematics ability in the boxes below True or False? Read each and click “True” or “False” to find out the answer.

(Close this panel)

• If you are completing this activity in class, with a small group, or with a partner, discuss how your attitudes and beliefs about mathematics might influence or have already influenced your instructional practices. Next, brainstorm ideas for improving your mathematics instruction.

ORIGINAL RESEARCH article

Challenges of teachers when teaching sentence-based mathematics problem-solving skills.

• Faculty of Education, Universiti Kebangsaan Malaysia, Bangi, Malaysia

Sentence-based mathematics problem-solving skills are essential as the skills can improve the ability to deal with various mathematical problems in daily life, increase the imagination, develop creativity, and develop an individual’s comprehension skills. However, mastery of these skills among students is still unsatisfactory because students often find it difficult to understand mathematical problems in verse, are weak at planning the correct solution strategy, and often make mistakes in their calculations. This study was conducted to identify the challenges that mathematics teachers face when teaching sentence-based mathematics problem-solving skills and the approaches used to address these challenges. This study was conducted qualitatively in the form of a case study. The data were collected through observations and interviews with two respondents who teach mathematics to year four students in a Chinese national primary school in Kuala Lumpur. This study shows that the teachers have faced three challenges, specifically low mastery skills among the students, insufficient teaching time, and a lack of ICT infrastructure. The teachers addressed these challenges with creativity and enthusiasm to diversify the teaching approaches to face the challenges and develop interest and skills as part of solving sentence-based mathematics problems among year four students. These findings allow mathematics teachers to understand the challenges faced while teaching sentence-based mathematics problem solving in depth as part of delivering quality education for every student. Nevertheless, further studies involving many respondents are needed to understand the problems and challenges of different situations and approaches that can be used when teaching sentence-based mathematics problem-solving skills.

1. Introduction

To keep track of the development of the current world, education has changed over time to create a more robust and effective system for producing a competent and competitive generation ( Hashim and Wan, 2020 ). The education system of a country is a significant determinant of the growth and development of the said country ( Ministry of Education Malaysia, 2013 ). In the Malaysian context, the education system has undergone repeated changes alongside the latest curriculum, namely the revised Primary School Standard Curriculum (KSSR) and the revised Secondary School Standard Curriculum (KSSM). These changes have been implemented to ensure that Malaysian education is improving continually so then the students can guide the country to compete globally ( Adam and Halim, 2019 ). However, Malaysian students have shown limited skills in international assessments such as Trends in International Mathematics and Science Study (TIMSS) and the Program for International Student Assessment (PISA).

According to the PISA 2018 results, the students’ performance in mathematics is still below the average level of the Organization for Economic Co-operation and Development (OECD; Avvisati et al., 2019 ). The results show that almost half of the students in Malaysia have still not mastered mathematical skills fully. Meanwhile, the TIMSS results in 2019 have shown there to be a descent in the achievements of Malaysian students compared to the results in 2015 ( Ministry of Education Malaysia, 2020b ). This situation is worrying as most students from other countries such as China, Singapore, Korea, Japan, and others have a higher level of mathematical skills than Malaysian students. According to Mullis et al. (2016) , these two international assessments have in common that both assessments test the level of the students’ skills when solving real-world problems. In short, PISA and TIMSS have proven that Malaysian students are still weak when it comes to solving sentence-based mathematics problems.

According to Hassan et al. (2019) , teachers must emphasize the mastery of sentence-based mathematics problem-solving skills and apply it in mathematics teaching in primary school. Sentence-based mathematics problem-solving skills can improve the students’ skills when dealing with various mathematical problems in daily life ( Gurat, 2018 ), increase the students’ imagination ( Wibowo et al., 2017 ), develop the students’ creativity ( Suastika, 2017 ), and develop the students’ comprehension skills ( Mulyati et al., 2017 ). The importance of sentence-based mathematic problem-solving skills is also supported by Ismail et al. (2021) . They stated that mathematics problem-solving skills are similar to high-level thinking skills when it comes to guiding students with how to deal with problems creatively and critically. Moreover, problem-solving skills are also an activity that requires an individual to select an appropriate strategy to be performed by the individual to ensure that movement occurs between the current state to the expected state ( Sudarmo and Mariyati, 2017 ). There are various strategies that can be used by teachers to guide students when developing their problem-solving skills such as problem-solving strategies based on Polya’s Problem-Solving Model (1957). Various research studies have used problem-solving models to solve specific problems to improve the students’ mathematical skills. Polya (1957) , Lester (1980) , Gick (1986) , and DeMuth (2007) are examples. One of the oldest problem-solving models is the George Polya model (1957). The model is divided into four major stages: (i) understanding the problem; (ii) devising a plan that will lead to the solution; (iii) Carrying out the plan; and (iv) looking back. In contrast to traditional mathematics classroom environments, Polya’s Problem-Solving Process allows the students to practice adapting and changing strategies to match new scenarios. As a result, the teachers must assist the students to help them recognize whether the strategy is appropriate, including where and how to apply the technique.

In addition, problem-solving skills are one of the 21st-century skills that need to be mastered by students through education now so then they are prepared to face the challenges of daily life ( Khoiriyah and Husamah., 2018 ). This statement is also supported by Widodo et al. (2018) who put forward four main reasons why students need to master problem-solving skills through mathematics learning. One reason is that sentence-based mathematics problem-solving skills are closely related to daily life ( Wong, 2015 ). Such skills can be used to formulate concepts and develop mathematical ideas, a skill that needs to be conveyed according to the school’s content standards. The younger generation is expected to develop critical, logical, systematic, accurate, and efficient thinking when solving a problem. Accordingly, problem solving has become an element that current employers emphasize when looking to acquire new energy sources ( Zainuddin et al., 2018 ). This clearly shows that problem-solving skills are essential skills that must be mastered by students and taken care of by mathematics teachers in primary school.

In the context of mathematics learning in Malaysia, students are required to solve sentence-based mathematics problems by applying mathematical concepts learned at the end of each topic. Two types of sentence-based mathematics problems are presented when teaching mathematics: routine and non-routine ( Wong and Matore, 2020 ). According to Nurkaeti (2018) , routine sentence-based math problems are questions that require the students to solve problems using algorithmic calculations to obtain answers. For non-routine sentence-based math problems, thinking skills and the ability to apply more than one method or solution step are needed by the student to solve the problem ( Shawan et al., 2021 ). According to Rohmah and Sutiarso (2018) , problem-solving skills when solving a non-routine sentence-based mathematical problem is a high-level intellectual skill where the students need to use logical thinking and reasoning. This statement also aligns with Wilson's (1997) opinion that solving non-routine sentence-based mathematics always involves high-order thinking skills (HOTS). To solve non-routine and HOTS fundamental sentence-based math problems, a student is required to know various problem-solving strategies for solving the problems ( Wong and Matore, 2020 ). This situation has indirectly made the mastery of sentence-based mathematics problem-solving skills among students more challenging ( Mahmud, 2019 ).

According to Alkhawaldeh and Khasawneh’s findings (2021) , the failure of students stems from the teachers’ inability to perform their role effectively in the classroom. This statement is also supported by Abdullah (2020) . He argues that the failure of students in mastering non-routine sentence-based mathematics problem-solving skills is due to the teachers rarely supplying these types of questions during the process of learning mathematics in class. A mathematics teacher should consider this issue because the quality of their teaching will affect the students’ mastery level of sentence-based mathematics problem-solving skills.

In addition, the teachers’ efforts to encourage the students to engage in social interactions with the teachers ( Jatisunda, 2017 ) and the teachers’ method of teaching and assessing the level of sentence-based mathematics problem-solving skills ( Buschman, 2004 ) are also challenges that the teachers must face. Strategies that are not appropriate for the students will affect the quality of delivery of the sentence-based mathematics problem-solving skills as well as cause one-way interactions to exist in the classroom. According to Rusdin and Ali (2019) , a practical teaching approach plays a vital role in developing the students’ skills when mastering specific knowledge. However, based on previous studies, the main challenges that mathematics teachers face when teaching sentence-based mathematics problem solving are due to the students. These challenges include the students having difficulty understanding sentence-based math problems, lacking knowledge about basic mathematical concepts, not calculating accurately, and not transforming the sentence-based mathematics problems into an operational form ( Yoong and Nasri, 2021 ). This also means that they cannot transform the sentence-based math problems into an operational form ( Yoong and Nasri, 2021 ). As a result, the teacher should diversify his or her teaching strategy by emphasizing understanding the mathematical concepts rather than procedural teaching to reinforce basic mathematical concepts, to encourage the students to work on any practice problems assigned by the teacher before completing any assignments to help them do the calculation correctly, and engaging in the use of effective oral questioning to stimulate student thinking related to the operational need when problem solving. All of these strategies actually help the teachers facilitate and lessen the students’ difficulty understanding sentence-based math problems ( Subramaniam et al., 2022 ).

Meanwhile, Dirgantoro et al. (2019) stated several challenges that the students posed while solving the sentence-based problem. For example, students do not read the questions carefully, the students lack mastery of mathematical concepts, the students solve problems in a hurry due to poor time management, the students are not used to making hypotheses and conclusions, as well as the students, being less skilled at using a scientific calculator. These factors have caused the students to have difficulty mastering sentence-based mathematics problem-solving skills, which goes on to become an inevitable challenge in maths classes. Therefore, teachers need to study these challenges to self-reflect so then their self-professionalism can be further developed ( Dirgantoro et al., 2019 ).

As for the school factor, challenges such as limited teaching resources, a lack of infrastructure facilities, and a large number of students in a class ( Rusdin and Ali, 2019 ) have meant that a conducive learning environment for learning sentence-based mathematics problem-solving skills cannot be created. According to Ersoy (2016) , problem-solving skills can be learned if an appropriate learning environment is provided for the students to help them undergo a continuous and systematic problem-solving process.

To develop sentence-based mathematics problem-solving skills among students, various models, pedagogies, activities, etc. have been introduced to assist mathematics teachers in delivering sentence-based mathematics problem-solving skills more effectively ( Gurat, 2018 ; Khoiriyah and Husamah., 2018 ; Özreçberoğlu and Çağanağa, 2018 ; Hasibuan et al., 2019 ). However, students nowadays still face difficulties when trying to master sentence-based mathematics problem-solving skills. This situation occurs due to the lack of studies examining the challenges faced by these mathematics teachers and how teachers use teaching approaches to overcome said challenges. This has led to various issues during the teaching and facilitation of sentence-based mathematics problem-solving skills in mathematics classes. According to Rusdin and Ali (2019) , these issues need to be addressed by a teacher wisely so then the quality of teaching can reach the best level. Therefore, mathematics teachers must understand and address these challenges to improve their teaching.

However, so far, not much is known about how primary school mathematics teachers face the challenges encountered when teaching sentence-based mathematics problem-solving skills and what approaches are used to address the challenges in the context of education in Malaysia. Therefore, this study needs to be carried out to help understand the teaching of sentence-based mathematics problem-solving skills in primary schools ( Pazin et al., 2022 ). Due to the challenges when teaching mathematics as stipulated in the Mathematics Curriculum and Assessment Standard Document ( Ministry of Education Malaysia, 2020a ) which emphasizes mathematical problem-solving skills as one of the main skills that students need to master in comprehensive mathematics learning, this study focuses on identifying the challenges faced by mathematics teachers when teaching sentence-based mathematics problem-solving skills and the approaches that mathematics teachers have used to overcome those challenges. The results of this study can provide information to mathematics teachers to help them understand the challenges when teaching sentence-based mathematics problem-solving skills and the approaches that can be applied to overcome the challenges faced. Therefore, it is very important for this study to be carried out so then all the visions set within the framework of the Malaysian National Mathematics Curriculum can be successfully achieved.

2. Conceptual framework

The issue of students lacking mastery of sentence-based mathematics problem-solving skills is closely related to the challenges that teachers face and the teaching approach used. Based on the overall findings of the previous studies, the factors that pose a challenge to teachers when delivering sentence-based mathematics problem-solving skills include challenges from the teacher ( Buschman, 2004 ; Jatisunda, 2017 ; Abdullah, 2020 ), challenges from the pupils ( Dirgantoro et al., 2019 ), and challenges from the school ( Rusdin and Ali, 2019 ). As for the teaching approach, previous studies have suggested teaching approaches such as mastery learning, contextual learning, project-based learning, problem-based learning, simulation, discovery inquiry, the modular approach, the STEM approach ( Curriculum Development Division, 2019 ), game-based teaching which uses digital games ( Muhamad et al., 2018 ), and where a combination of the modular approach especially the flipped classroom is applied alongside the problem-based learning approach when teaching sentence-based mathematics problem solving ( Alias et al., 2020 ). This is as well as the constructivism approach ( Jatisunda, 2017 ). The conceptual framework in Figure 1 illustrates that the teachers will face various challenges during the ongoing teaching and facilitation of sentence-based mathematics problem-solving skills.

Figure 1 . Conceptual framework of the study.

3. Methodology

The objective of this study was to determine the challenges that teachers face while teaching sentence-based mathematics problem-solving skills and the approaches used when teaching those skills. Therefore, a qualitative research approach in the form of a case study was used to collect data from the participants in a Chinese national type of school (SJKC) in Bangsar and Pudu, in the Federal Territory of Kuala Lumpur. The school, SJKC, in the districts of Bangsar and Pudu, was chosen as the location of this study because the school is implementing the School Transformation Program 2025 (TS25). One of the main objectives of the TS25 program is to apply the best teaching concepts and practices so then the quality of the learning and teaching in the classes is improved. Thus, schools that go through the program are believed to be able to diversify their teachers’ teaching and supply more of the data needed to answer the questions of this study. This is because case studies can develop an in-depth description and analysis of the case to be studied ( Creswell and Poth, 2018 ). All data collected through the observations, interviews, audio-visual materials, documents, and reports can be reported on in terms of both depth and detail based on the theme of the case. Therefore, this study collected data related to the challenges and approaches of SJKC mathematics teachers through observations, interviews, and document analysis.

Two primary school mathematics teachers who teach year four mathematics were selected to be the participants of this research using the purposive sampling technique to identify the challenges faced and the approaches used to overcome those challenges. The number of research participants in this study was sufficient enough to allow the researcher to explore the real picture of the challenges found when teaching sentence-based mathematics problem-solving skills and the approaches that can be applied when teaching to overcome the challenges faced. According to Creswell and Creswell (2018) , the small number of study participants is sufficient when considering that the main purpose of the study is to obtain findings that can give a holistic and meaningful picture of the teaching and learning process in the classroom. However, based on the data analysis for both study participants, the researcher considered repeated information until it reached a saturation point. The characteristics of the study participants required when they were supplying the information for this study were as follows:

i. New or experienced teachers.

ii. Year four math teacher.

iii. Teachers teach in primary schools.

iv. The teacher teaches the topic of sentence-based mathematics problem-solving skills.

The types of instruments used in the study were the observation protocol, field notes, interview protocol, and participants’ documents. In this study, the researcher used participatory type observations to observe the teaching style of the teachers when engaged in sentence-based mathematics problem-solving skills lessons. Before conducting the study, the researcher obtained consent to conduct the study from the school as well as informed consent from the study participants to observe their teaching. During the observation, the teacher’s teaching process was recorded and transcribed using the field notes provided. Then, the study participants submitted and validated the field notes to avoid biased data. After that, the field notes were analyzed based on the observation protocol to identify the teachers’ challenges and teaching approaches in relation to sentence-based mathematics problem-solving skills. Throughout the observation process of this study, the researcher observed the teaching of mathematics teachers online at least four times during the 2 months of the data collection at the research location.

Semi-structured interviews were used to identify the teachers’ perspectives and views on teaching sentence-based mathematics problem-solving skills in terms of the challenges faced when teaching sentence-based mathematics problem-solving skills and the approaches used by the teachers to overcome those challenges. To ensure that the interview data collected could answer the research questions, an interview protocol was prepared so then the required data could be collected from the study participants ( Cohen et al., 2007 ). Two experts validated the interview protocol, and a pilot study was conducted to ensure that the questions were easy to understand and would obtain the necessary data. Before the interview sessions began, the participants were informed of their rights and of the related research ethics. Throughout the interview sessions, the participants were asked two questions, namely:

1. What are the challenges faced when teaching mathematical problem-solving skills earlier?

2. What teaching approaches are used by teachers when facing these challenges? Why?

Semi-structured interviews were used to interview the study participants for 30 min every interview session. The timing ensured sufficient time for both parties to complete the question-and-answer process. Finally, the entire interview process was recorded in audio form. The audio recordings were then transcribed into text form and verified by the study participants.

The types of document collected in this study included informal documents, namely the daily lesson plan documents of the study participants, the work of the students of the study participants, and any teaching aids used. All of the documents were analyzed and used to ensure that the triangulation of the data occurred between the data collected from observations, interviews, and document analysis.

All data collected through the observations, interviews, and documentary analyses were entered into the NVIVO 11 software to ensure that the coding process took place simultaneously. The data in this study were analyzed using the constant comparative analysis method including open coding, axial coding, and selective coding to obtain the themes and subthemes related to the focus of the study ( Kolb, 2012 ). The NVIVO 11 software was also used to manage the data stack obtained from the interviews, observations, and document analysis during the data analysis process itself. In order to ensure that the themes generated from all of the data were accurate, the researcher carried out a repetitive reading process. The process of theme development involved numerous steps. First, the researcher examined the verbatim instruction data several times while looking for statements or paragraphs that could summarize a theme in a nutshell. This process had already been completed during the verbatim formation process of the teaching, while preparing the transcription. Second, the researcher kept reading (either from the same or different data), and if the researcher found a sentence that painted a similar picture to the theme that had been developed, the sentence was added to the same theme. This process is called “pattern matching” because the coding of the sentences refers to the existing categories ( Yin, 2003 ). Third, if the identified sentence was incompatible with an existing theme, a new theme was created. Fourth, this coding procedure continued throughout each data set’s theme analysis. The repeated reading process was used to select sentences able to explain the theme or help establish a new one. In short, the researcher conducted the data analysis process based on the data analysis steps proposed by Creswell and Creswell (2018) , as shown in Figure 2 .

Figure 2 . Data analysis steps ( Creswell and Creswell, 2018 ).

4. Findings

The findings of this study are presented based on the objective of the study, which was to identify the challenges faced by teachers and the approaches used to addressing those challenges when imparting sentence-based mathematics problem-solving skills to students in year four. Several themes were formed based on the analysis of the field notes, interview transcripts, and daily lesson plans of the study participants. This study found that teachers will face challenges that stem from the readiness of students to master sentence-based mathematics problem-solving skills, the teachers’ teaching style, and the equipment used for delivering sentence-based mathematics problem-solving skills. Due to facing these challenges, teachers have diversified their teaching approaches ( Figure 3 ; Table 1 ).

Figure 3 . Sentence-based mathematics problem-solving teaching approaches.

Table 1 . Teacher challenges when teaching sentence-based mathematics problem-solving skills.

5. Discussion

5.1. challenges for teachers when imparting sentence-based mathematics problem-solving skills.

A mathematics teacher will face three challenges when teaching sentence-based mathematics problem-solving skills. The first challenge stems from the low mastery skills held by a student. Pupils can fail to solve sentence-based mathematics problems because they have poor reading skills, there is a poor medium of instruction used, or they have a poor mastery of mathematical concepts ( Johari et al., 2022 ). This indicates that students who are not ready or reach a minimum level of proficiency in a language, comprehension, mathematical concepts, and calculations will result in them not being able to solve sentence-based mathematics problems smoothly.

These findings are consistent with the findings of the studies by Raifana et al. (2016) and Dirgantoro et al. (2019) who showed that students who are unprepared in terms of language skills, comprehension, mathematical concepts, and calculations are likely to make mistakes when solving sentence-based mathematics problems. If these challenges are not faced well, the students will become passive and not interact when learning sentence-based mathematics problem-solving skills. This situation occurs because students who frequently make mistakes will incur low self-confidence in mathematics ( Jailani et al., 2017 ). This situation should be avoided by teachers and social interaction should be encouraged during the learning process because the interaction between students and teachers can ensure that the learning outcomes are achieved by the students optimally ( Jatisunda, 2017 ).

The next challenge stems from the teacher-teaching factor. This study found that how teachers convey problem-solving skills has been challenging in terms of ensuring that their students master sentence-based mathematics problem-solving skills ( Nang et al., 2022 ). The mastery teaching approach has caused the teaching time spent on mathematical content to be insufficient. Based on the findings of this study, the allocation of time spent ensuring that the students master the skills of solving sentence-based mathematics problems through a mastery approach has caused the teaching process not to follow the rate set in the annual lesson plan.

In this study, the participants spent a long time correcting the students’ mathematical concepts and allowing students to apply the skills learned. The actions of the participants of this study are in line with the statement of Adam and Halim (2019) that teachers need more time to arouse their students’ curiosity and ensure that students understand the correct ideas and concepts before doing more challenging activities. However, this approach has indirectly posed challenges regarding time allocation and ensuring that the students master the skills of sentence-based mathematics problem-solving. Aside from ensuring that the students’ master problem-solving skills, the participants must also complete the syllabus set in the annual lesson plan.

Finally, teachers also face challenges in terms of the lack of information and communication technology (ICT) infrastructure when implementing the teaching and facilitation of sentence-based mathematics problem-solving skills. In this study, mathematics teachers were found to face challenges caused by an unstable internet connection such as the problem of their students dropping out of class activities and whiteboard links not working. These problems have caused one mathematics class to run poorly ( Mahmud and Law, 2022 ). Throughout the implementation of teaching and its facilitation, ICT infrastructure equipment in terms of hardware, software, and internet services has become an element that will affect the effectiveness of virtual teaching ( Saifudin and Hamzah, 2021 ). In this regard, a mathematics teacher must be wise when selecting a teaching approach and diversifying the learning activities to implement a suitable mathematics class for students such as systematically using tables, charts, or lists, creating digital simulations, using analogies, working back over the work, involving reasoning activities and logic, and using various new applications such as Geogebra and Kahoot to help enable their students’ understanding.

5.2. Teaching sentence-based mathematics problem-solving skills—Approaches

In this study, various approaches have been used by the teachers facing challenges while imparting sentence-based mathematics problem-solving skills. Among the approaches that the mathematics teachers have used when teaching problem-solving skills are the oral questioning approach, mastery learning approach, contextual learning approach, game approach, and modular approach. This situation has shown that mathematics teachers have diversified their teaching approaches when facing the challenges associated with teaching sentence-based mathematics problem-solving skills. This action is also in line with the excellent teaching and facilitation of mathematics proposal in the Curriculum and Assessment Standards Document revised KSSR Mathematics Year 4 ( Curriculum Development Division, 2019 ), stating that teaching activities should be carefully planned by the teachers and combine a variety of approaches that allow the students not only to understand the content in depth but also to think at a higher level. Therefore, a teacher needs to ensure that this teaching approach is applied when teaching sentence-based mathematics problem-solving skills so then the students can learn sentence-based mathematics problem-solving teaching skills in a more fun, meaningful, and challenging environment ( Mahmud et al., 2022 ).

Through the findings of this study, the teaching approach used by mathematics teachers was found to have a specific purpose, namely facing the challenges associated with teaching sentence-based mathematics problem-solving skills in the classroom. First of all, the oral questioning approach has been used by teachers facing the challenge of students having a poor understanding of the medium of instruction. The participants stated that questioning the students in stages can guide them to understanding the question and helping them plan appropriate problem-solving strategies. This opinion is also supported by Maat (2015) who stated that low-level oral questions could help the students achieve a minimum level of understanding, in particular remembering, and strengthening abstract mathematical concepts. The teacher’s action of guiding the students when solving sentence-based mathematics problems through oral questioning has ensured that the learning takes place in a student-centered manner, providing opportunities for the students to think and solve problems independently ( Mahmud and Yunus, 2018 ). This action is highly encouraged because teaching mathematics through the conventional approach is only effective for a short period, as the students can lack an understanding or fail to remember the mathematical concepts presented by the teacher ( Ali et al., 2021 ).

In addition, this study also found that the participants used the mastery approach to overcome the challenges of poor reading skills and poor mastery of mathematical concepts among the students. The mastery approach was used because it can provide more opportunities and time for the students to improve their reading skills and mastery of mathematical concepts ( Shawan et al., 2021 ). This approach has ensured that all students achieve the teaching objectives and that the teachers have time to provide enrichment and rehabilitation to the students as part of mastering the basic skills needed to solve sentence-based mathematics problems. This approach is very effective at adapting students to solving sentence-based mathematics problems according to the solution steps of the Polya model as well as the mathematical concepts learned in relation to a particular topic. The finding is in line with Ranggoana et al. (2018) and Mahmud (2019) study, which has shown that teaching through a mastery approach can enhance the student’s learning activities. This situation clearly shows that the mastery approach has ensured that the students have time to learn at their own pace, where they often try to emulate the solution shown by the teacher to solve a sentence-based mathematics problem.

Besides that, this study also found that mathematics teachers apply contextual learning approaches when teaching and facilitating sentence-based mathematics problem-solving skills. In this study, mathematics teachers have linked non-routine problems with examples from everyday life to guide the students with poor language literacy to help them understand non-routine problems and plan appropriate solution strategies. Such relationships can help the students process non-routine problems or mathematical concepts in a more meaningful context where the problem is relevant to real situations ( Siew et al., 2016 ). This situation can develop the students’ skill of solving sentence-based math problems where they can choose the right solution strategy to solve a non-routine problem. This finding is consistent with the results of Afni and Hartono (2020) . They showed that the contextual approach applied in learning could guide the students in determining appropriate strategies for solving sentence-based math problems. These findings are also supported by Seliaman and Dollah (2018) who stated that the practice of teachers giving examples that exist around the students and in real situations could make teaching and the subject facilitation easier to understand and fun.

Furthermore, the game approach was also used by the participants when imparting sentence-based mathematics problem-solving skills. According to Sari et al. (2018) , the game approach to teaching mathematics can improve the student learning outcomes because the game approach facilitates the learning process and provides a more enjoyable learning environment for achieving the learning objectives. In this study, the game approach was used by the teachers to overcome the challenge of mastering the concept of unit conversion, which was not strong among the students ( Tobias et al., 2015 ; Hui and Mahmud, 2022 ). The participants used the game approach to teach induction sets that guided the students in recalling mathematical concepts. The action provided a fun learning environment and attracted the students to learning mathematical concepts, especially in the beginning of the class. This situation is consistent with the findings of Muhamad et al. (2018) . They showed that the game approach improved the students’ problem-solving skills, interests, and motivation to find a solution to the problem.

Regarding the challenge of insufficient teaching time and a lack of ICT infrastructure, modular approaches such as flipped classrooms have been used to encourage students to learn in a situation that focuses on self-development ( UNESCO, 2020 ). In this study, the participants used instructional videos with related content, clear instructions, and worksheets as part of the Google classroom learning platform. The students can follow the instructions to engage in revision or self-paced learning in their spare time. This modular approach has ensured that teachers can deliver mathematical content and increase the effectiveness of learning a skill ( Alias et al., 2020 ). For students with unstable internet connections, the participants have used a modular approach to ensure that the students continue learning and send work through other channels such as WhatsApp, by email, or as a hand-in hardcopy. In short, an appropriate teaching approach needs to be planned and implemented by the mathematics teachers to help students master sentence-based mathematics problem-solving skills.

6. Conclusion

Overall, this study has expanded the literature related to the challenges when teaching sentence-based mathematics problem-solving skills and the approaches that can be applied while teaching to overcome the challenges faced. This study has shown that students have difficulty mastering sentence-based mathematics problem-solving skills because they do not achieve the minimum mastery of factual knowledge, procedural skills, conceptual understanding, and the ability to choose appropriate strategies ( Collins and Stevens, 1983 ). This situation needs to be taken into account because sentence-based mathematics problem-solving skills train the students to always be prepared to deal with problems that they will be faced with in their daily life. Through this study, teachers were found to play an essential role in overcoming the challenges faced by choosing the most appropriate teaching approach ( Baul and Mahmud, 2021 ). An appropriate teaching approach can improve the students’ sentence-based mathematics problem-solving skills ( Wulandari et al., 2020 ). Teachers need to work hard to equip themselves with varied knowledge and skills to ensure that sentence-based mathematics problem-solving skills can be delivered to the students more effectively. Finally, the findings of this study were part of obtaining extensive data regarding the challenges that mathematics teachers face when teaching sentence-based mathematics problem-solving skills and the approaches used to address those challenges in the process of teaching mathematics. It is suggested that a quantitative study be conducted to find out whether the findings obtained can be generalized to other populations. This is because this study is a qualitative one, and the findings of this study cannot be generalized to other populations.

The findings of this study can be used as a reference to develop the professionalism of mathematics teachers when teaching mathematical problem-solving skills. However, the study’s findings, due to being formulated from a small sample size, cannot be generalized to all mathematics teachers in Malaysia. Further studies are proposed to involve more respondents to better understand the different challenges and approaches used when teaching sentence-based mathematics problem-solving skills.

Data availability statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

Ethics statement

This study was reviewed and approved by The Malaysian Ministry of Education. The participants provided their written informed consent to participate in this study.

Author contributions

AL conceived and designed the study, collected and organized the database, and performed the analysis. AL and MM co-wrote the manuscript and contributed to manuscript revision. All authors read and approved the final submitted version.

The publication of this article is fully sponsored by the Faculty of Education Universiti Kebangsaan Malaysia and University Research Grant: GUP-2022-030, GGPM-2021-014, and GG-2022-022.

Acknowledgments

The authors appreciate the commitment from the respondent. Thank you to the Faculty of Education, Universiti Kebangsaan Malaysia, and University Research Grant: GUP-2022-030, GGPM-2021-014, and GG-2022-022 for sponsoring the publication of this article. Thanks also to all parties directly involved in helping the publication of this article to success.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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Keywords: mathematics, problem solving, teaching challenges, teaching approaches, primary school

Citation: Ling ANB and Mahmud MS (2023) Challenges of teachers when teaching sentence-based mathematics problem-solving skills. Front. Psychol . 13:1074202. doi: 10.3389/fpsyg.2022.1074202

Received: 19 October 2022; Accepted: 21 December 2022; Published: 01 February 2023.

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Copyright © 2023 Ling and Mahmud. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY) . The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Muhammad Sofwan Mahmud, ✉ [email protected]

Disclaimer: All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.

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Solving Multi-Step Equations

Digging into 8th-grade standards 8.EE.C.7 and 8.EE.C.8

Welcome back to the Unfinished Learning in Middle School Math series, in which math educators Chrissy Allison and Becca Varon illustrate how to make some of the trickiest standards in grades 6-8 accessible for all students. Over the course of six blog posts, we’ll provide concrete examples of how math educators can address unfinished learning within the context of grade-level lessons, which in the long term will help prevent an entrenched pattern of over-remediation and below-grade-level teaching. You can read our introductory post here . In this fifth post, we will explore ways to “bridge the gap” with the 8th-grade standards 8.EE.C.7 and 8.EE.C.8.

Let’s take a look at the standards:

Ah, solving equations—an essential skill students need to be successful in Algebra I and beyond. Whether students are working with inequalities, quadratics, or exponential functions in high school, they’ll leverage skills and understanding they developed in the elementary and middle grades.

8th grade is not students’ first encounter with finding missing values in a number sentence. In fact, with the introduction of college- and career-readiness standards, many students have been solving word problems “with unknowns in all positions” using “a symbol for the unknown number to represent the problem” since 1st grade (1.OA.A.1).

Despite the ongoing focus on algebraic thinking in grades K-5, and a continued emphasis in grades 6-7, it’s not uncommon for 8th-grade students to enter the year with a shaky understanding of how to solve equations. In fact, it’s likely the number one concern we hear from 8th-grade teachers who ask, “How can I teach students to solve multi-step equations and systems of equations when they can’t solve simple one-step equations?”

Before we share ideas to help address students’ unfinished learning with equations, let’s take a moment to study and understand these two 8th-grade standards:

To anchor the discussion about how to address students’ unfinished learning with equations, we encourage you to take a minute or two to solve the problem below, which is aligned to standard 8.EE.C.7. As you do so, think metacognitively about the steps you’re taking, the purpose of each step, and how you decided what to do next.

Solving this problem involves the distributive property, combining like terms, and operations with rational numbers. Not only that, but there are myriad decisions students must make in order to isolate the variable and find a solution—and more than one way to do so.

After solving the equation above, can you see why a teacher might hesitate to put multi-step equations in front of students who have struggled to solve simpler, one-step equations involving positive whole numbers?

In fact, when teachers diagnose unfinished learning in this area, many assume they must start back at square one, helping students solve the most basic equations before working their way up. However, much of the conceptual understanding and procedural skills that are necessary to solve complex equations overlap with those necessary to solve simpler equations. 

As a result, when teachers identify students needing additional support, they have the opportunity to strengthen prerequisite skills and concepts alongside the 8th-grade content. 

Below are some ideas of how to take action to address students’ unfinished learning while moving forward with grade-level content:

• Reinforce important concepts, such as the meaning of “solution.” Rather than solving a series of 6th-grade tasks, reinforce important ideas such as the meaning of solutions and the difference between “constant” and “variable.” This can be done within the context of grade-level aligned problems such as two-step equations, multi-step equations involving the distributive property, and system of equations. The Solving Equations task and The Sign of Solutions task , both from Illustrative Mathematics, provide opportunities to deepen understanding of these concepts.
• Emphasize the properties of operations instead of step-by-step procedures. Teaching students to solve one-step equations by identifying the “inverse operation,” doing the “same thing to both sides,” and “canceling terms” is commonplace in 6th-grade classrooms because they work for the majority of simple equations students are solving. However, they do not set students up for success in future grade levels when students solve increasingly complex equations. By focusing more time and attention on the why behind each step, students use reasoning to make decisions, think flexibility while solving problems, and deepen their understanding of concepts like equivalence and equality.  Check out our previous blog post about addressing unfinished learning when teaching the properties of operations here .
• Engage students in tasks with multiple entry points and solution pathways . Assign problems that are set in a real-life or mathematical context. Allow and encourage students to find solutions using strategies other than writing and solving an equation . Then, invite students to share their different approaches with the class. Ask questions to probe students’ thinking and help them make connections between strategies. Want to try it out? Kimi and Jordan and Fixing the Furnace from Illustrative Mathematics are tasks that provide access for students no matter where they are in the learning progression. At times, you may want to modify a task to increase accessibility for students. For example, the Summer Swimming task from Illustrative Mathematics can become more open-ended by reordering pieces of the task. Try assigning Part E first, then having a brief class discussion about ways students approached the problem. This will prepare students for Parts A-D by helping them make sense of the problem context first without the expectation to represent it using an equation.

We hope the ideas shared here will help you create an on-ramp for students with unfinished learning in the expressions and equations domain. We’d love to keep the conversation going in the comments by hearing from you! Please share:

• What approaches have you used to “bridge the gap” to multi-step equations?
• How have you helped students find success with systems of equations?

Finally, don’t forget to check out other posts in the Unfinished Learning in Middle School Math series for additional strategies, examples, and ideas.

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About the Author: Chrissy Allison is the founder and CEO of Mindful Math Coach whose core purpose is to support secondary math educators in providing positive and equitable learning experience for students of color. A former middle school math teacher, Chrissy became an instructional leader where she led the school's math team to increase summative assessment scores by 30% within three years. Her 5+ years experience serving as Director of Math Professional Learning & Content Design at educational nonprofits gives Chrissy unique insight and expertise about what it takes to shift teacher practice and move the needle with student learning. As host of The Mindful Math Podcast, Chrissy interviews experienced educators and shares her own advice and lessons learned to help teachers reach every learner while finding balance in their own lives. Chrissy lives in Chicago with her husband, Dan, and two children, Liviana and Otto. Learn more about Chrissy, access the podcast, and download free resources at www.mindfulmathcoach.com.

About the Author: Becca Varon is the Director of Math Content Learning at ANet. Prior to joining ANet, Becca taught 6th grade math and science in Glendale, Arizona and 5th grade math in Boston, Massachusetts. Becca now lives in Oakland, CA, where she continues to tutor local students in math.

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How to master the seven-step problem-solving process

In this episode of the McKinsey Podcast , Simon London speaks with Charles Conn, CEO of venture-capital firm Oxford Sciences Innovation, and McKinsey senior partner Hugo Sarrazin about the complexities of different problem-solving strategies.

Podcast transcript

Simon London: Hello, and welcome to this episode of the McKinsey Podcast , with me, Simon London. What’s the number-one skill you need to succeed professionally? Salesmanship, perhaps? Or a facility with statistics? Or maybe the ability to communicate crisply and clearly? Many would argue that at the very top of the list comes problem solving: that is, the ability to think through and come up with an optimal course of action to address any complex challenge—in business, in public policy, or indeed in life.

Looked at this way, it’s no surprise that McKinsey takes problem solving very seriously, testing for it during the recruiting process and then honing it, in McKinsey consultants, through immersion in a structured seven-step method. To discuss the art of problem solving, I sat down in California with McKinsey senior partner Hugo Sarrazin and also with Charles Conn. Charles is a former McKinsey partner, entrepreneur, executive, and coauthor of the book Bulletproof Problem Solving: The One Skill That Changes Everything [John Wiley & Sons, 2018].

Charles and Hugo, welcome to the podcast. Thank you for being here.

Hugo Sarrazin: Our pleasure.

Charles Conn: It’s terrific to be here.

Simon London: Problem solving is a really interesting piece of terminology. It could mean so many different things. I have a son who’s a teenage climber. They talk about solving problems. Climbing is problem solving. Charles, when you talk about problem solving, what are you talking about?

Charles Conn: For me, problem solving is the answer to the question “What should I do?” It’s interesting when there’s uncertainty and complexity, and when it’s meaningful because there are consequences. Your son’s climbing is a perfect example. There are consequences, and it’s complicated, and there’s uncertainty—can he make that grab? I think we can apply that same frame almost at any level. You can think about questions like “What town would I like to live in?” or “Should I put solar panels on my roof?”

You might think that’s a funny thing to apply problem solving to, but in my mind it’s not fundamentally different from business problem solving, which answers the question “What should my strategy be?” Or problem solving at the policy level: “How do we combat climate change?” “Should I support the local school bond?” I think these are all part and parcel of the same type of question, “What should I do?”

I’m a big fan of structured problem solving. By following steps, we can more clearly understand what problem it is we’re solving, what are the components of the problem that we’re solving, which components are the most important ones for us to pay attention to, which analytic techniques we should apply to those, and how we can synthesize what we’ve learned back into a compelling story. That’s all it is, at its heart.

I think sometimes when people think about seven steps, they assume that there’s a rigidity to this. That’s not it at all. It’s actually to give you the scope for creativity, which often doesn’t exist when your problem solving is muddled.

Simon London: You were just talking about the seven-step process. That’s what’s written down in the book, but it’s a very McKinsey process as well. Without getting too deep into the weeds, let’s go through the steps, one by one. You were just talking about problem definition as being a particularly important thing to get right first. That’s the first step. Hugo, tell us about that.

Hugo Sarrazin: It is surprising how often people jump past this step and make a bunch of assumptions. The most powerful thing is to step back and ask the basic questions—“What are we trying to solve? What are the constraints that exist? What are the dependencies?” Let’s make those explicit and really push the thinking and defining. At McKinsey, we spend an enormous amount of time in writing that little statement, and the statement, if you’re a logic purist, is great. You debate. “Is it an ‘or’? Is it an ‘and’? What’s the action verb?” Because all these specific words help you get to the heart of what matters.

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Simon London: So this is a concise problem statement.

Hugo Sarrazin: Yeah. It’s not like “Can we grow in Japan?” That’s interesting, but it is “What, specifically, are we trying to uncover in the growth of a product in Japan? Or a segment in Japan? Or a channel in Japan?” When you spend an enormous amount of time, in the first meeting of the different stakeholders, debating this and having different people put forward what they think the problem definition is, you realize that people have completely different views of why they’re here. That, to me, is the most important step.

Charles Conn: I would agree with that. For me, the problem context is critical. When we understand “What are the forces acting upon your decision maker? How quickly is the answer needed? With what precision is the answer needed? Are there areas that are off limits or areas where we would particularly like to find our solution? Is the decision maker open to exploring other areas?” then you not only become more efficient, and move toward what we call the critical path in problem solving, but you also make it so much more likely that you’re not going to waste your time or your decision maker’s time.

How often do especially bright young people run off with half of the idea about what the problem is and start collecting data and start building models—only to discover that they’ve really gone off half-cocked.

Hugo Sarrazin: Yeah.

Charles Conn: And in the wrong direction.

Simon London: OK. So step one—and there is a real art and a structure to it—is define the problem. Step two, Charles?

Charles Conn: My favorite step is step two, which is to use logic trees to disaggregate the problem. Every problem we’re solving has some complexity and some uncertainty in it. The only way that we can really get our team working on the problem is to take the problem apart into logical pieces.

What we find, of course, is that the way to disaggregate the problem often gives you an insight into the answer to the problem quite quickly. I love to do two or three different cuts at it, each one giving a bit of a different insight into what might be going wrong. By doing sensible disaggregations, using logic trees, we can figure out which parts of the problem we should be looking at, and we can assign those different parts to team members.

Simon London: What’s a good example of a logic tree on a sort of ratable problem?

Charles Conn: Maybe the easiest one is the classic profit tree. Almost in every business that I would take a look at, I would start with a profit or return-on-assets tree. In its simplest form, you have the components of revenue, which are price and quantity, and the components of cost, which are cost and quantity. Each of those can be broken out. Cost can be broken into variable cost and fixed cost. The components of price can be broken into what your pricing scheme is. That simple tree often provides insight into what’s going on in a business or what the difference is between that business and the competitors.

If we add the leg, which is “What’s the asset base or investment element?”—so profit divided by assets—then we can ask the question “Is the business using its investments sensibly?” whether that’s in stores or in manufacturing or in transportation assets. I hope we can see just how simple this is, even though we’re describing it in words.

When I went to work with Gordon Moore at the Moore Foundation, the problem that he asked us to look at was “How can we save Pacific salmon?” Now, that sounds like an impossible question, but it was amenable to precisely the same type of disaggregation and allowed us to organize what became a 15-year effort to improve the likelihood of good outcomes for Pacific salmon.

Simon London: Now, is there a danger that your logic tree can be impossibly large? This, I think, brings us onto the third step in the process, which is that you have to prioritize.

Charles Conn: Absolutely. The third step, which we also emphasize, along with good problem definition, is rigorous prioritization—we ask the questions “How important is this lever or this branch of the tree in the overall outcome that we seek to achieve? How much can I move that lever?” Obviously, we try and focus our efforts on ones that have a big impact on the problem and the ones that we have the ability to change. With salmon, ocean conditions turned out to be a big lever, but not one that we could adjust. We focused our attention on fish habitats and fish-harvesting practices, which were big levers that we could affect.

People spend a lot of time arguing about branches that are either not important or that none of us can change. We see it in the public square. When we deal with questions at the policy level—“Should you support the death penalty?” “How do we affect climate change?” “How can we uncover the causes and address homelessness?”—it’s even more important that we’re focusing on levers that are big and movable.

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Simon London: Let’s move swiftly on to step four. You’ve defined your problem, you disaggregate it, you prioritize where you want to analyze—what you want to really look at hard. Then you got to the work plan. Now, what does that mean in practice?

Hugo Sarrazin: Depending on what you’ve prioritized, there are many things you could do. It could be breaking the work among the team members so that people have a clear piece of the work to do. It could be defining the specific analyses that need to get done and executed, and being clear on time lines. There’s always a level-one answer, there’s a level-two answer, there’s a level-three answer. Without being too flippant, I can solve any problem during a good dinner with wine. It won’t have a whole lot of backing.

Simon London: Not going to have a lot of depth to it.

Hugo Sarrazin: No, but it may be useful as a starting point. If the stakes are not that high, that could be OK. If it’s really high stakes, you may need level three and have the whole model validated in three different ways. You need to find a work plan that reflects the level of precision, the time frame you have, and the stakeholders you need to bring along in the exercise.

Charles Conn: I love the way you’ve described that, because, again, some people think of problem solving as a linear thing, but of course what’s critical is that it’s iterative. As you say, you can solve the problem in one day or even one hour.

Charles Conn: We encourage our teams everywhere to do that. We call it the one-day answer or the one-hour answer. In work planning, we’re always iterating. Every time you see a 50-page work plan that stretches out to three months, you know it’s wrong. It will be outmoded very quickly by that learning process that you described. Iterative problem solving is a critical part of this. Sometimes, people think work planning sounds dull, but it isn’t. It’s how we know what’s expected of us and when we need to deliver it and how we’re progressing toward the answer. It’s also the place where we can deal with biases. Bias is a feature of every human decision-making process. If we design our team interactions intelligently, we can avoid the worst sort of biases.

Simon London: Here we’re talking about cognitive biases primarily, right? It’s not that I’m biased against you because of your accent or something. These are the cognitive biases that behavioral sciences have shown we all carry around, things like anchoring, overoptimism—these kinds of things.

Both: Yeah.

Charles Conn: Availability bias is the one that I’m always alert to. You think you’ve seen the problem before, and therefore what’s available is your previous conception of it—and we have to be most careful about that. In any human setting, we also have to be careful about biases that are based on hierarchies, sometimes called sunflower bias. I’m sure, Hugo, with your teams, you make sure that the youngest team members speak first. Not the oldest team members, because it’s easy for people to look at who’s senior and alter their own creative approaches.

Hugo Sarrazin: It’s helpful, at that moment—if someone is asserting a point of view—to ask the question “This was true in what context?” You’re trying to apply something that worked in one context to a different one. That can be deadly if the context has changed, and that’s why organizations struggle to change. You promote all these people because they did something that worked well in the past, and then there’s a disruption in the industry, and they keep doing what got them promoted even though the context has changed.

Simon London: Right. Right.

Hugo Sarrazin: So it’s the same thing in problem solving.

Charles Conn: And it’s why diversity in our teams is so important. It’s one of the best things about the world that we’re in now. We’re likely to have people from different socioeconomic, ethnic, and national backgrounds, each of whom sees problems from a slightly different perspective. It is therefore much more likely that the team will uncover a truly creative and clever approach to problem solving.

Simon London: Let’s move on to step five. You’ve done your work plan. Now you’ve actually got to do the analysis. The thing that strikes me here is that the range of tools that we have at our disposal now, of course, is just huge, particularly with advances in computation, advanced analytics. There’s so many things that you can apply here. Just talk about the analysis stage. How do you pick the right tools?

Charles Conn: For me, the most important thing is that we start with simple heuristics and explanatory statistics before we go off and use the big-gun tools. We need to understand the shape and scope of our problem before we start applying these massive and complex analytical approaches.

Simon London: Would you agree with that?

Hugo Sarrazin: I agree. I think there are so many wonderful heuristics. You need to start there before you go deep into the modeling exercise. There’s an interesting dynamic that’s happening, though. In some cases, for some types of problems, it is even better to set yourself up to maximize your learning. Your problem-solving methodology is test and learn, test and learn, test and learn, and iterate. That is a heuristic in itself, the A/B testing that is used in many parts of the world. So that’s a problem-solving methodology. It’s nothing different. It just uses technology and feedback loops in a fast way. The other one is exploratory data analysis. When you’re dealing with a large-scale problem, and there’s so much data, I can get to the heuristics that Charles was talking about through very clever visualization of data.

You test with your data. You need to set up an environment to do so, but don’t get caught up in neural-network modeling immediately. You’re testing, you’re checking—“Is the data right? Is it sound? Does it make sense?”—before you launch too far.

Simon London: You do hear these ideas—that if you have a big enough data set and enough algorithms, they’re going to find things that you just wouldn’t have spotted, find solutions that maybe you wouldn’t have thought of. Does machine learning sort of revolutionize the problem-solving process? Or are these actually just other tools in the toolbox for structured problem solving?

Charles Conn: It can be revolutionary. There are some areas in which the pattern recognition of large data sets and good algorithms can help us see things that we otherwise couldn’t see. But I do think it’s terribly important we don’t think that this particular technique is a substitute for superb problem solving, starting with good problem definition. Many people use machine learning without understanding algorithms that themselves can have biases built into them. Just as 20 years ago, when we were doing statistical analysis, we knew that we needed good model definition, we still need a good understanding of our algorithms and really good problem definition before we launch off into big data sets and unknown algorithms.

Simon London: Step six. You’ve done your analysis.

Charles Conn: I take six and seven together, and this is the place where young problem solvers often make a mistake. They’ve got their analysis, and they assume that’s the answer, and of course it isn’t the answer. The ability to synthesize the pieces that came out of the analysis and begin to weave those into a story that helps people answer the question “What should I do?” This is back to where we started. If we can’t synthesize, and we can’t tell a story, then our decision maker can’t find the answer to “What should I do?”

Simon London: But, again, these final steps are about motivating people to action, right?

Charles Conn: Yeah.

Simon London: I am slightly torn about the nomenclature of problem solving because it’s on paper, right? Until you motivate people to action, you actually haven’t solved anything.

Charles Conn: I love this question because I think decision-making theory, without a bias to action, is a waste of time. Everything in how I approach this is to help people take action that makes the world better.

Simon London: Hence, these are absolutely critical steps. If you don’t do this well, you’ve just got a bunch of analysis.

Charles Conn: We end up in exactly the same place where we started, which is people speaking across each other, past each other in the public square, rather than actually working together, shoulder to shoulder, to crack these important problems.

Simon London: In the real world, we have a lot of uncertainty—arguably, increasing uncertainty. How do good problem solvers deal with that?

Hugo Sarrazin: At every step of the process. In the problem definition, when you’re defining the context, you need to understand those sources of uncertainty and whether they’re important or not important. It becomes important in the definition of the tree.

You need to think carefully about the branches of the tree that are more certain and less certain as you define them. They don’t have equal weight just because they’ve got equal space on the page. Then, when you’re prioritizing, your prioritization approach may put more emphasis on things that have low probability but huge impact—or, vice versa, may put a lot of priority on things that are very likely and, hopefully, have a reasonable impact. You can introduce that along the way. When you come back to the synthesis, you just need to be nuanced about what you’re understanding, the likelihood.

Often, people lack humility in the way they make their recommendations: “This is the answer.” They’re very precise, and I think we would all be well-served to say, “This is a likely answer under the following sets of conditions” and then make the level of uncertainty clearer, if that is appropriate. It doesn’t mean you’re always in the gray zone; it doesn’t mean you don’t have a point of view. It just means that you can be explicit about the certainty of your answer when you make that recommendation.

Simon London: So it sounds like there is an underlying principle: “Acknowledge and embrace the uncertainty. Don’t pretend that it isn’t there. Be very clear about what the uncertainties are up front, and then build that into every step of the process.”

Hugo Sarrazin: Every step of the process.

Simon London: Yeah. We have just walked through a particular structured methodology for problem solving. But, of course, this is not the only structured methodology for problem solving. One that is also very well-known is design thinking, which comes at things very differently. So, Hugo, I know you have worked with a lot of designers. Just give us a very quick summary. Design thinking—what is it, and how does it relate?

Hugo Sarrazin: It starts with an incredible amount of empathy for the user and uses that to define the problem. It does pause and go out in the wild and spend an enormous amount of time seeing how people interact with objects, seeing the experience they’re getting, seeing the pain points or joy—and uses that to infer and define the problem.

Simon London: Problem definition, but out in the world.

Hugo Sarrazin: With an enormous amount of empathy. There’s a huge emphasis on empathy. Traditional, more classic problem solving is you define the problem based on an understanding of the situation. This one almost presupposes that we don’t know the problem until we go see it. The second thing is you need to come up with multiple scenarios or answers or ideas or concepts, and there’s a lot of divergent thinking initially. That’s slightly different, versus the prioritization, but not for long. Eventually, you need to kind of say, “OK, I’m going to converge again.” Then you go and you bring things back to the customer and get feedback and iterate. Then you rinse and repeat, rinse and repeat. There’s a lot of tactile building, along the way, of prototypes and things like that. It’s very iterative.

Simon London: So, Charles, are these complements or are these alternatives?

Charles Conn: I think they’re entirely complementary, and I think Hugo’s description is perfect. When we do problem definition well in classic problem solving, we are demonstrating the kind of empathy, at the very beginning of our problem, that design thinking asks us to approach. When we ideate—and that’s very similar to the disaggregation, prioritization, and work-planning steps—we do precisely the same thing, and often we use contrasting teams, so that we do have divergent thinking. The best teams allow divergent thinking to bump them off whatever their initial biases in problem solving are. For me, design thinking gives us a constant reminder of creativity, empathy, and the tactile nature of problem solving, but it’s absolutely complementary, not alternative.

Simon London: I think, in a world of cross-functional teams, an interesting question is do people with design-thinking backgrounds really work well together with classical problem solvers? How do you make that chemistry happen?

Hugo Sarrazin: Yeah, it is not easy when people have spent an enormous amount of time seeped in design thinking or user-centric design, whichever word you want to use. If the person who’s applying classic problem-solving methodology is very rigid and mechanical in the way they’re doing it, there could be an enormous amount of tension. If there’s not clarity in the role and not clarity in the process, I think having the two together can be, sometimes, problematic.

The second thing that happens often is that the artifacts the two methodologies try to gravitate toward can be different. Classic problem solving often gravitates toward a model; design thinking migrates toward a prototype. Rather than writing a big deck with all my supporting evidence, they’ll bring an example, a thing, and that feels different. Then you spend your time differently to achieve those two end products, so that’s another source of friction.

Now, I still think it can be an incredibly powerful thing to have the two—if there are the right people with the right mind-set, if there is a team that is explicit about the roles, if we’re clear about the kind of outcomes we are attempting to bring forward. There’s an enormous amount of collaborativeness and respect.

Simon London: But they have to respect each other’s methodology and be prepared to flex, maybe, a little bit, in how this process is going to work.

Hugo Sarrazin: Absolutely.

Simon London: The other area where, it strikes me, there could be a little bit of a different sort of friction is this whole concept of the day-one answer, which is what we were just talking about in classical problem solving. Now, you know that this is probably not going to be your final answer, but that’s how you begin to structure the problem. Whereas I would imagine your design thinkers—no, they’re going off to do their ethnographic research and get out into the field, potentially for a long time, before they come back with at least an initial hypothesis.

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Hugo Sarrazin: That is a great callout, and that’s another difference. Designers typically will like to soak into the situation and avoid converging too quickly. There’s optionality and exploring different options. There’s a strong belief that keeps the solution space wide enough that you can come up with more radical ideas. If there’s a large design team or many designers on the team, and you come on Friday and say, “What’s our week-one answer?” they’re going to struggle. They’re not going to be comfortable, naturally, to give that answer. It doesn’t mean they don’t have an answer; it’s just not where they are in their thinking process.

Simon London: I think we are, sadly, out of time for today. But Charles and Hugo, thank you so much.

Charles Conn: It was a pleasure to be here, Simon.

Hugo Sarrazin: It was a pleasure. Thank you.

Simon London: And thanks, as always, to you, our listeners, for tuning into this episode of the McKinsey Podcast . If you want to learn more about problem solving, you can find the book, Bulletproof Problem Solving: The One Skill That Changes Everything , online or order it through your local bookstore. To learn more about McKinsey, you can of course find us at McKinsey.com.

Charles Conn is CEO of Oxford Sciences Innovation and an alumnus of McKinsey’s Sydney office. Hugo Sarrazin is a senior partner in the Silicon Valley office, where Simon London, a member of McKinsey Publishing, is also based.

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Mastering Multi-Step Word Problems - Lesson Plan

This interactive math lesson focuses on teaching strategies for solving multi-step word problems involving multiplication. students will learn how to estimate products, compare multiplication expressions, and solve problems in multiple steps. the lesson includes warm-up activities, guided practice, and opportunities for independent practice. by the end of the lesson, students will have developed problem-solving skills and gained confidence in tackling complex word problems..

Know more about Mastering Multi-Step Word Problems - Lesson Plan

The key objectives of this lesson are to teach students how to estimate products, compare multiplication expressions, and solve multi-step word problems involving multiplication.

This lesson helps develop problem-solving skills by guiding students through each step of solving multi-step word problems. It encourages critical thinking and strategic planning as students identify the expressions involved at each step and apply appropriate mathematical operations.

Yes! The optional sections provide additional practice problems and challenges for students who want to further reinforce their understanding or take on more advanced problem-solving tasks.

Your one stop solution for all grade learning needs.

What Are They?

Students who experience significant problems learning and applying mathematics manifest their math learning problems in a variety of ways. Research indicates that there are a number of reasons these students experience difficulty learning mathematics (Mercer, Jordan, & Miller, 1996; Mercer, Lane, Jordan, Allsopp, & Eisele, 1996; Mercer & Mercer, 1998; Miller & Mercer, 1997.) The following list includes these research-based math disability characteristics.

Characteristics of Students Who Have Learning Problems

Learned Helplessness - Students who experience continuous failure in math expect to fail. Their lack of confidence compels them to rely on assistance from others to complete tasks such as worksheets. Assistance that only helps the student "get through" the current set of problems or tasks and does not include re-teaching the concept/skill, only reinforces the student's belief that he cannot learn math.

Passive Learners - Students who have learning problems often are not "active" learners. They do not actively make connections between what they already know and what they are presently learning. When presented with a problem-solving situation, they do not employ strategies or activate prior knowledge to solve the problem. For example, students may learn that 8 x 4 = 32, but when presented with 8 x 5 = ___, they do not actively connect the process of multiplication to that of repeated addition. They do not think to add eight more to thirty-two in order to solve the problem. Students that have learning problems often believe that students who are successful in math just know the answers. They do not understand that students who are successful in math are good at employing strategies to solve problems.

Memory Problems - Memory deficits play a significant role in these students' math learning problems. Memory problems are most evident when students demonstrate difficulty remembering their basic addition, subtraction, multiplication, & division facts. Memory deficits also play a significant role when students are solving multi-step problems and when problem-solving situations require the use of particular problem solving strategies. A common misconception about the memory problems of these students is that it is an information storage problem; that somehow, these students just never get it stored properly. This belief probably arises because one day the student can do a math task but then the next day they can't. Teachers then re-teach the skill only to have the same experience repeated. In contrast to an information storage problem, these memory deficits are often a result of an information retrieval problem. For these students, instruction should include teaching students strategies for accessing and retrieving the information they have stored.

Attention Problems - Math requires a great deal of attention, particularly when multiple steps are involved in the problem solving process. During instruction, students who have attention problems often "miss" important pieces of information. Without these important pieces of information, students have difficulty trying to implement the problem solving process they have just learned. For example, when learning long division, students may miss the "subtract" step in the "divide, multiply, subtract, bring down" long division process. Without subtracting in the proper place, the student will be unable to solve long division problems accurately. Additionally, these students may be unable to focus on the important features that make a mathematical concept distinct. For example, when teaching geometric shapes, these students may attend to features not relevant to identifying shapes. Instead of counting the number of sides to distinguish triangles from rectangles, the student may focus on size or color. Using visual, auditory, tactile (touch), and kinesthetic (movement) cues to highlight the relevant features of a concept is helpful for these students.

Cognitive/Metacognitive Thinking Deficits - Metacognition has to do with students' ability to monitor their learning: 1.) Evaluating whether they are learning; 2.) Employing strategies when needed; 3.) Knowing whether a strategy is successful; and, 4.) Making changes when needed. These are essential skills for any problem solving situation. Because math is problem solving, students who are not metacognitively adept will have great difficulty being successful with mathematics. These students need to be explicitly taught how to be metacognitive learners. Teachers who model this process, who teach students problem solving strategies, who reinforce students' use of these strategies, and who teach students to organize themselves so they can access strategies, will help students who have metacognitive deficits become metacognitive learners.

Low Level of Academic Achievement - Students who experience math failure often lack basic math skills. Because it takes students with math disabilities a longer time to process visual and auditory information than typical learners, they often never have enough time or opportunity to master the foundational concepts/skills that make learning higher level mathematics possible. Providing these students many opportunities to respond to math tasks and providing these practice opportunities in a variety of ways is essential if these students are to ever master the math concepts/skills we teach. Additionally, teachers need to plan periodic review and practice of concepts/skills that students have previously mastered.

Math Anxiety - These students often approach math with trepidation. Because math is difficult for them, "math time" is often an anxiety-ridden experience. The best cure for math anxiety is success. Providing success starts first with the teacher. By understanding why students are having the difficulties they are having, we are less inclined to place "blame" on the students for their lack of math success. These students already feel they are not capable. The attitude with which we approach these students can be a crucial first step in rectifying the math problems they are having. Providing these students with non-threatening, risk-free opportunities to learn and practice math skills is greatly encouraged. Celebrating both small and great advances is also important. Last, if we provide instruction that is effective for these students, we will help them learn math, thereby helping them to experience the success they deserve.

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Math Instruction Issues That Impact Students Who Have Math Learning Problems

Although it is very important to understand the learning characteristics of students with math learning problems, it is also important to understand how math instruction/curriculum issues negatively affect these students (Mercer, Jordan, & Miller, 1996; Mercer, Lane, Jordan, Allsopp, & Eisele, 1996; Mercer & Mercer, 1998; Miller & Mercer, 1997). The following list includes these instruction/curriculum issues as well as how they impact the students described above.

Spiraling Curriculum - Within a spiraling curriculum, students are exposed to a number of important math concepts the first year. The next year, students return to those math concepts, expanding on the foundation established the year before. This cycle continues with each successive year. While the purpose of this approach is logical and may be appropriate for students who are average to above average achievers, the spiraling curriculum can be a significant impediment for students who have math learning problems. The primary problem for these students is the limited time that is devoted to each concept. Students who have math learning problems are never able to truly master the concept/skill being taught. For these students, "exposure" to foundational skills is not enough. Without an appropriate number of practice opportunities, these students will only partially acquire the skill. When the concept/skill is revisited the next year, the student is at a great disadvantage because the foundation they are expected to have is incomplete. After several years, the student has not only "not mastered" basic skills, but has also not been able to make the important connections between those basic skills and the higher level math skills being taught as the students moves through the elementary, middle, & secondary grades.

Teaching Understanding/Algorithm Driven Instruction - Although the National Council on Teaching Mathematics (NCTM) strongly encourages teaching mathematical understanding and reasoning, the reality for students with math learning problems is that they spend most of their math time learning and practicing computation procedures. Because of their memory problems, attention problems, and metacognitive deficits, these students have difficulty accurately performing multi-step computations. Therefore, instructional emphasis for these students is often placed on procedural accuracy rather than on conceptual understanding. This emphasis on algorithm (procedure) proficiency supersedes emphasis on conceptual understanding. An example of this is the process of multiplication. Students who only are taught the procedure of multiplication through drill and practice often do not really understand what the process represents. For example, consider the relationship of the following two multiplication problems: 2 x 4 = 8 and ½ x ¼ = 1/8. When students are asked why the answer in the first problem is greater than its multipliers but the answer to the second problem is less than each of its multipliers, the students are unable to answer why. They have never really understood that the multiplication sign really means "of" and that "2 x 4 = 8" means two groups of fours objects, while " ½ x ¼ = 1/8" means one-half of one-fourth. Teaching understanding of the math processes as well as teaching the algorithms (procedures) for computing solutions is critical for students with math learning problems.

Teaching to Mastery - As described under "Spiraling Curriculum," students with learning problems need many opportunities to respond to specific math tasks in order to master them. Teaching to mastery requires that both the teacher and the student monitor the student's learning progress on a daily basis. Mastery is indicated only when the student is able to perform a math task at 100% accuracy for at least three consecutive days. In situations where student progress is assessed only by unit tests, it is very difficult to determine whether a student has really mastered the skills covered in that unit. Even if the student performs well on the unit test, a teacher cannot be certain that the student actually has reached mastery. Because of the learning characteristics common for these students, it is possible that the student would not score as well if given the same test the next day. Mastery can only be inferred when the student demonstrates consistent mastery performance over time. Such continuous assessment is rare in math classrooms. When evaluation of student progress occurs only by unit tests and the students with learning problems do not perform well, the teacher is left with a difficult dilemma. Does the teacher take additional class time to re-teach the skill, thereby falling behind the mandated curriculum's instructional pace? Conversely, does the teacher instead move on to new material, knowing that these students have not mastered the preceding skills, making it less likely the student will have the prerequisite skills to learn the new information? This no-win situation can be avoided if continuous daily assessment is implemented for these students. It is easier and more time efficient to re-teach an individual math skill the same day of initial instruction, or on the following day. Attempting to re-teach multiple math skills many days after initial instruction is much more difficult and time consuming. Due to the hierarchical nature of mathematics, if students do not master prerequisite skills, it is likely that they will not master future skills.

Reforms That Are Cyclical in Nature - The cyclical nature of mathematics curriculum/instruction reforms creates changing instructional practices that confuse students with learning problems. Like reforms for reading instruction, math instruction swings from primarily skills-based emphasis to primarily meaning-based emphasis dependent on the philosophical and political trends of the day. Most students experience at least one of these shifts as they move through grades K to 12. While students who are average to above average achievers are able to manage these changes in instruction, students who have learning problems do not adjust well to such change.

Application of Effective Teaching Practices for Students who have Learning Problems - Research has identified math instructional practices that are effective for students who have learning problems, but these instructional practices are not always implemented in our schools. These instructional practices are described and modeled in this CD-ROM program. Descriptions also include how the particular characteristics of each instructional strategy complement the learning characteristics of students with learning problems. Guidance is also provided which will help you implement these instructional practices in an organized and systematic way.

How Does This Information Help Me?

Teachers who understand the learning needs of their students are more empowered to provide the kind of instruction their students need. Knowing why a student is struggling to learn math provides a basis for understanding why particular instructional strategies/approaches are effective for him/her. Each of the instructional strategies included in this program has unique characteristics that positively impact the learning characteristics of students who have math learning problems. As you learn about each strategy, you are encouraged to refer often to the learner characteristics described in this section. While reading about each instructional strategy and then watching a teacher model the strategy, note how the specific instructional characteristics of the strategy complement or "match" the learning characteristics of students with math learning problems. The text descriptions for each instructional strategy found in this manual clarify these relationships. The elaborated video clips in the CD-ROM also emphasize how the specific characteristics for each instructional strategy positively impact students who have math learning problems.

How to improve your problem solving skills and build effective problem solving strategies

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A step-by-step guide to planning a workshop, how to create an unforgettable training session in 8 simple steps, 47 useful online tools for workshop planning and meeting facilitation.

Effective problem solving is all about using the right process and following a plan tailored to the issue at hand. Recognizing your team or organization has an issue isn’t enough to come up with effective problem solving strategies. 

To truly understand a problem and develop appropriate solutions, you will want to follow a solid process, follow the necessary problem solving steps, and bring all of your problem solving skills to the table.  

We’ll first guide you through the seven step problem solving process you and your team can use to effectively solve complex business challenges. We’ll also look at what problem solving strategies you can employ with your team when looking for a way to approach the process. We’ll then discuss the problem solving skills you need to be more effective at solving problems, complete with an activity from the SessionLab library you can use to develop that skill in your team.

Let’s get to it! 

What is a problem solving process?

• What are the problem solving steps I need to follow?

Problem solving strategies

What skills do i need to be an effective problem solver, how can i improve my problem solving skills.

Solving problems is like baking a cake. You can go straight into the kitchen without a recipe or the right ingredients and do your best, but the end result is unlikely to be very tasty!

Using a process to bake a cake allows you to use the best ingredients without waste, collect the right tools, account for allergies, decide whether it is a birthday or wedding cake, and then bake efficiently and on time. The result is a better cake that is fit for purpose, tastes better and has created less mess in the kitchen. Also, it should have chocolate sprinkles. Having a step by step process to solve organizational problems allows you to go through each stage methodically and ensure you are trying to solve the right problems and select the most appropriate, effective solutions.

What are the problem solving steps I need to follow? 

All problem solving processes go through a number of steps in order to move from identifying a problem to resolving it.

Depending on your problem solving model and who you ask, there can be anything between four and nine problem solving steps you should follow in order to find the right solution. Whatever framework you and your group use, there are some key items that should be addressed in order to have an effective process.

We’ve looked at problem solving processes from sources such as the American Society for Quality and their four step approach , and Mediate ‘s six step process. By reflecting on those and our own problem solving processes, we’ve come up with a sequence of seven problem solving steps we feel best covers everything you need in order to effectively solve problems.

1. Problem identification 

The first stage of any problem solving process is to identify the problem or problems you might want to solve. Effective problem solving strategies always begin by allowing a group scope to articulate what they believe the problem to be and then coming to some consensus over which problem they approach first. Problem solving activities used at this stage often have a focus on creating frank, open discussion so that potential problems can be brought to the surface.

2. Problem analysis 

Though this step is not a million miles from problem identification, problem analysis deserves to be considered separately. It can often be an overlooked part of the process and is instrumental when it comes to developing effective solutions.

The process of problem analysis means ensuring that the problem you are seeking to solve is the right problem . As part of this stage, you may look deeper and try to find the root cause of a specific problem at a team or organizational level.

Remember that problem solving strategies should not only be focused on putting out fires in the short term but developing long term solutions that deal with the root cause of organizational challenges. 

Whatever your approach, analyzing a problem is crucial in being able to select an appropriate solution and the problem solving skills deployed in this stage are beneficial for the rest of the process and ensuring the solutions you create are fit for purpose.

3. Solution generation

Once your group has nailed down the particulars of the problem you wish to solve, you want to encourage a free flow of ideas connecting to solving that problem. This can take the form of problem solving games that encourage creative thinking or problem solving activities designed to produce working prototypes of possible solutions. 

The key to ensuring the success of this stage of the problem solving process is to encourage quick, creative thinking and create an open space where all ideas are considered. The best solutions can come from unlikely places and by using problem solving techniques that celebrate invention, you might come up with solution gold. 

4. Solution development

No solution is likely to be perfect right out of the gate. It’s important to discuss and develop the solutions your group has come up with over the course of following the previous problem solving steps in order to arrive at the best possible solution. Problem solving games used in this stage involve lots of critical thinking, measuring potential effort and impact, and looking at possible solutions analytically. 

During this stage, you will often ask your team to iterate and improve upon your frontrunning solutions and develop them further. Remember that problem solving strategies always benefit from a multitude of voices and opinions, and not to let ego get involved when it comes to choosing which solutions to develop and take further.

Finding the best solution is the goal of all problem solving workshops and here is the place to ensure that your solution is well thought out, sufficiently robust and fit for purpose. 

5. Decision making 

Nearly there! Once your group has reached consensus and selected a solution that applies to the problem at hand you have some decisions to make. You will want to work on allocating ownership of the project, figure out who will do what, how the success of the solution will be measured and decide the next course of action.

The decision making stage is a part of the problem solving process that can get missed or taken as for granted. Fail to properly allocate roles and plan out how a solution will actually be implemented and it less likely to be successful in solving the problem.

Have clear accountabilities, actions, timeframes, and follow-ups. Make these decisions and set clear next-steps in the problem solving workshop so that everyone is aligned and you can move forward effectively as a group. 

Ensuring that you plan for the roll-out of a solution is one of the most important problem solving steps. Without adequate planning or oversight, it can prove impossible to measure success or iterate further if the problem was not solved. 

6. Solution implementation 

This is what we were waiting for! All problem solving strategies have the end goal of implementing a solution and solving a problem in mind. 

Remember that in order for any solution to be successful, you need to help your group through all of the previous problem solving steps thoughtfully. Only then can you ensure that you are solving the right problem but also that you have developed the correct solution and can then successfully implement and measure the impact of that solution.

Project management and communication skills are key here – your solution may need to adjust when out in the wild or you might discover new challenges along the way.

7. Solution evaluation 

So you and your team developed a great solution to a problem and have a gut feeling its been solved. Work done, right? Wrong. All problem solving strategies benefit from evaluation, consideration, and feedback. You might find that the solution does not work for everyone, might create new problems, or is potentially so successful that you will want to roll it out to larger teams or as part of other initiatives. 

None of that is possible without taking the time to evaluate the success of the solution you developed in your problem solving model and adjust if necessary.

Remember that the problem solving process is often iterative and it can be common to not solve complex issues on the first try. Even when this is the case, you and your team will have generated learning that will be important for future problem solving workshops or in other parts of the organization. 

It’s worth underlining how important record keeping is throughout the problem solving process. If a solution didn’t work, you need to have the data and records to see why that was the case. If you go back to the drawing board, notes from the previous workshop can help save time. Data and insight is invaluable at every stage of the problem solving process and this one is no different.

Problem solving workshops made easy

Problem solving strategies are methods of approaching and facilitating the process of problem-solving with a set of techniques , actions, and processes. Different strategies are more effective if you are trying to solve broad problems such as achieving higher growth versus more focused problems like, how do we improve our customer onboarding process?

Broadly, the problem solving steps outlined above should be included in any problem solving strategy though choosing where to focus your time and what approaches should be taken is where they begin to differ. You might find that some strategies ask for the problem identification to be done prior to the session or that everything happens in the course of a one day workshop.

The key similarity is that all good problem solving strategies are structured and designed. Four hours of open discussion is never going to be as productive as a four-hour workshop designed to lead a group through a problem solving process.

Good problem solving strategies are tailored to the team, organization and problem you will be attempting to solve. Here are some example problem solving strategies you can learn from or use to get started.

Use a workshop to lead a team through a group process

Often, the first step to solving problems or organizational challenges is bringing a group together effectively. Most teams have the tools, knowledge, and expertise necessary to solve their challenges – they just need some guidance in how to use leverage those skills and a structure and format that allows people to focus their energies.

Facilitated workshops are one of the most effective ways of solving problems of any scale. By designing and planning your workshop carefully, you can tailor the approach and scope to best fit the needs of your team and organization. 

Problem solving workshop

• Creating a bespoke, tailored process
• Tackling problems of any size
• Building in-house workshop ability and encouraging their use

Workshops are an effective strategy for solving problems. By using tried and test facilitation techniques and methods, you can design and deliver a workshop that is perfectly suited to the unique variables of your organization. You may only have the capacity for a half-day workshop and so need a problem solving process to match. 

By using our session planner tool and importing methods from our library of 700+ facilitation techniques, you can create the right problem solving workshop for your team. It might be that you want to encourage creative thinking or look at things from a new angle to unblock your groups approach to problem solving. By tailoring your workshop design to the purpose, you can help ensure great results.

One of the main benefits of a workshop is the structured approach to problem solving. Not only does this mean that the workshop itself will be successful, but many of the methods and techniques will help your team improve their working processes outside of the workshop. 

We believe that workshops are one of the best tools you can use to improve the way your team works together. Start with a problem solving workshop and then see what team building, culture or design workshops can do for your organization!

Run a design sprint

Great for: 

• aligning large, multi-discipline teams
• quickly designing and testing solutions
• tackling large, complex organizational challenges and breaking them down into smaller tasks

By using design thinking principles and methods, a design sprint is a great way of identifying, prioritizing and prototyping solutions to long term challenges that can help solve major organizational problems with quick action and measurable results.

Some familiarity with design thinking is useful, though not integral, and this strategy can really help a team align if there is some discussion around which problems should be approached first. 

The stage-based structure of the design sprint is also very useful for teams new to design thinking.  The inspiration phase, where you look to competitors that have solved your problem, and the rapid prototyping and testing phases are great for introducing new concepts that will benefit a team in all their future work. 

It can be common for teams to look inward for solutions and so looking to the market for solutions you can iterate on can be very productive. Instilling an agile prototyping and testing mindset can also be great when helping teams move forwards – generating and testing solutions quickly can help save time in the long run and is also pretty exciting!

Break problems down into smaller issues

Organizational challenges and problems are often complicated and large scale in nature. Sometimes, trying to resolve such an issue in one swoop is simply unachievable or overwhelming. Try breaking down such problems into smaller issues that you can work on step by step. You may not be able to solve the problem of churning customers off the bat, but you can work with your team to identify smaller effort but high impact elements and work on those first.

This problem solving strategy can help a team generate momentum, prioritize and get some easy wins. It’s also a great strategy to employ with teams who are just beginning to learn how to approach the problem solving process. If you want some insight into a way to employ this strategy, we recommend looking at our design sprint template below!

Use guiding frameworks or try new methodologies

Some problems are best solved by introducing a major shift in perspective or by using new methodologies that encourage your team to think differently.

Props and tools such as Methodkit , which uses a card-based toolkit for facilitation, or Lego Serious Play can be great ways to engage your team and find an inclusive, democratic problem solving strategy. Remember that play and creativity are great tools for achieving change and whatever the challenge, engaging your participants can be very effective where other strategies may have failed.

LEGO Serious Play

• Improving core problem solving skills
• Thinking outside of the box
• Encouraging creative solutions

LEGO Serious Play is a problem solving methodology designed to get participants thinking differently by using 3D models and kinesthetic learning styles. By physically building LEGO models based on questions and exercises, participants are encouraged to think outside of the box and create their own responses. 

Collaborate LEGO Serious Play exercises are also used to encourage communication and build problem solving skills in a group. By using this problem solving process, you can often help different kinds of learners and personality types contribute and unblock organizational problems with creative thinking. 

Problem solving strategies like LEGO Serious Play are super effective at helping a team solve more skills-based problems such as communication between teams or a lack of creative thinking. Some problems are not suited to LEGO Serious Play and require a different problem solving strategy.

Card Decks and Method Kits

• New facilitators or non-facilitators 
• Approaching difficult subjects with a simple, creative framework
• Engaging those with varied learning styles

Card decks and method kids are great tools for those new to facilitation or for whom facilitation is not the primary role. Card decks such as the emotional culture deck can be used for complete workshops and in many cases, can be used right out of the box. Methodkit has a variety of kits designed for scenarios ranging from personal development through to personas and global challenges so you can find the right deck for your particular needs.

Having an easy to use framework that encourages creativity or a new approach can take some of the friction or planning difficulties out of the workshop process and energize a team in any setting. Simplicity is the key with these methods. By ensuring everyone on your team can get involved and engage with the process as quickly as possible can really contribute to the success of your problem solving strategy.

Source external advice

Looking to peers, experts and external facilitators can be a great way of approaching the problem solving process. Your team may not have the necessary expertise, insights of experience to tackle some issues, or you might simply benefit from a fresh perspective. Some problems may require bringing together an entire team, and coaching managers or team members individually might be the right approach. Remember that not all problems are best resolved in the same manner.

If you’re a solo entrepreneur, peer groups, coaches and mentors can also be invaluable at not only solving specific business problems, but in providing a support network for resolving future challenges. One great approach is to join a Mastermind Group and link up with like-minded individuals and all grow together. Remember that however you approach the sourcing of external advice, do so thoughtfully, respectfully and honestly. Reciprocate where you can and prepare to be surprised by just how kind and helpful your peers can be!

Mastermind Group

• Solo entrepreneurs or small teams with low capacity
• Peer learning and gaining outside expertise
• Getting multiple external points of view quickly

Problem solving in large organizations with lots of skilled team members is one thing, but how about if you work for yourself or in a very small team without the capacity to get the most from a design sprint or LEGO Serious Play session? 

A mastermind group – sometimes known as a peer advisory board – is where a group of people come together to support one another in their own goals, challenges, and businesses. Each participant comes to the group with their own purpose and the other members of the group will help them create solutions, brainstorm ideas, and support one another. 

Mastermind groups are very effective in creating an energized, supportive atmosphere that can deliver meaningful results. Learning from peers from outside of your organization or industry can really help unlock new ways of thinking and drive growth. Access to the experience and skills of your peers can be invaluable in helping fill the gaps in your own ability, particularly in young companies.

A mastermind group is a great solution for solo entrepreneurs, small teams, or for organizations that feel that external expertise or fresh perspectives will be beneficial for them. It is worth noting that Mastermind groups are often only as good as the participants and what they can bring to the group. Participants need to be committed, engaged and understand how to work in this context. 

Coaching and mentoring

• Focused learning and development
• Filling skills gaps
• Working on a range of challenges over time

Receiving advice from a business coach or building a mentor/mentee relationship can be an effective way of resolving certain challenges. The one-to-one format of most coaching and mentor relationships can really help solve the challenges those individuals are having and benefit the organization as a result.

A great mentor can be invaluable when it comes to spotting potential problems before they arise and coming to understand a mentee very well has a host of other business benefits. You might run an internal mentorship program to help develop your team’s problem solving skills and strategies or as part of a large learning and development program. External coaches can also be an important part of your problem solving strategy, filling skills gaps for your management team or helping with specific business issues. 

Now we’ve explored the problem solving process and the steps you will want to go through in order to have an effective session, let’s look at the skills you and your team need to be more effective problem solvers.

Problem solving skills are highly sought after, whatever industry or team you work in. Organizations are keen to employ people who are able to approach problems thoughtfully and find strong, realistic solutions. Whether you are a facilitator , a team leader or a developer, being an effective problem solver is a skill you’ll want to develop.

Problem solving skills form a whole suite of techniques and approaches that an individual uses to not only identify problems but to discuss them productively before then developing appropriate solutions.

Here are some of the most important problem solving skills everyone from executives to junior staff members should learn. We’ve also included an activity or exercise from the SessionLab library that can help you and your team develop that skill. 

If you’re running a workshop or training session to try and improve problem solving skills in your team, try using these methods to supercharge your process!

Active listening

Active listening is one of the most important skills anyone who works with people can possess. In short, active listening is a technique used to not only better understand what is being said by an individual, but also to be more aware of the underlying message the speaker is trying to convey. When it comes to problem solving, active listening is integral for understanding the position of every participant and to clarify the challenges, ideas and solutions they bring to the table.

Some active listening skills include:

• Paying complete attention to the speaker.
• Removing distractions.
• Avoid interruption.
• Taking the time to fully understand before preparing a rebuttal.
• Responding respectfully and appropriately.
• Demonstrate attentiveness and positivity with an open posture, making eye contact with the speaker, smiling and nodding if appropriate. Show that you are listening and encourage them to continue.
• Be aware of and respectful of feelings. Judge the situation and respond appropriately. You can disagree without being disrespectful.   
• Observe body language. 
• Paraphrase what was said in your own words, either mentally or verbally.
• Remain neutral. 
• Reflect and take a moment before responding.
• Ask deeper questions based on what is said and clarify points where necessary.   

Active Listening   #hyperisland   #skills   #active listening   #remote-friendly   This activity supports participants to reflect on a question and generate their own solutions using simple principles of active listening and peer coaching. It’s an excellent introduction to active listening but can also be used with groups that are already familiar with it. Participants work in groups of three and take turns being: “the subject”, the listener, and the observer.

Analytical skills

All problem solving models require strong analytical skills, particularly during the beginning of the process and when it comes to analyzing how solutions have performed.

Analytical skills are primarily focused on performing an effective analysis by collecting, studying and parsing data related to a problem or opportunity. 

It often involves spotting patterns, being able to see things from different perspectives and using observable facts and data to make suggestions or produce insight. 

Analytical skills are also important at every stage of the problem solving process and by having these skills, you can ensure that any ideas or solutions you create or backed up analytically and have been sufficiently thought out.

Nine Whys   #innovation   #issue analysis   #liberating structures   With breathtaking simplicity, you can rapidly clarify for individuals and a group what is essentially important in their work. You can quickly reveal when a compelling purpose is missing in a gathering and avoid moving forward without clarity. When a group discovers an unambiguous shared purpose, more freedom and more responsibility are unleashed. You have laid the foundation for spreading and scaling innovations with fidelity.

Collaboration

Trying to solve problems on your own is difficult. Being able to collaborate effectively, with a free exchange of ideas, to delegate and be a productive member of a team is hugely important to all problem solving strategies.

Remember that whatever your role, collaboration is integral, and in a problem solving process, you are all working together to find the best solution for everyone. 

Marshmallow challenge with debriefing   #teamwork   #team   #leadership   #collaboration   In eighteen minutes, teams must build the tallest free-standing structure out of 20 sticks of spaghetti, one yard of tape, one yard of string, and one marshmallow. The marshmallow needs to be on top. The Marshmallow Challenge was developed by Tom Wujec, who has done the activity with hundreds of groups around the world. Visit the Marshmallow Challenge website for more information. This version has an extra debriefing question added with sample questions focusing on roles within the team.

Communication  

Being an effective communicator means being empathetic, clear and succinct, asking the right questions, and demonstrating active listening skills throughout any discussion or meeting. 

In a problem solving setting, you need to communicate well in order to progress through each stage of the process effectively. As a team leader, it may also fall to you to facilitate communication between parties who may not see eye to eye. Effective communication also means helping others to express themselves and be heard in a group.

Bus Trip   #feedback   #communication   #appreciation   #closing   #thiagi   #team   This is one of my favourite feedback games. I use Bus Trip at the end of a training session or a meeting, and I use it all the time. The game creates a massive amount of energy with lots of smiles, laughs, and sometimes even a teardrop or two.

Creative problem solving skills can be some of the best tools in your arsenal. Thinking creatively, being able to generate lots of ideas and come up with out of the box solutions is useful at every step of the process. 

The kinds of problems you will likely discuss in a problem solving workshop are often difficult to solve, and by approaching things in a fresh, creative manner, you can often create more innovative solutions.

Having practical creative skills is also a boon when it comes to problem solving. If you can help create quality design sketches and prototypes in record time, it can help bring a team to alignment more quickly or provide a base for further iteration.

The paper clip method   #sharing   #creativity   #warm up   #idea generation   #brainstorming   The power of brainstorming. A training for project leaders, creativity training, and to catalyse getting new solutions.

Critical thinking

Critical thinking is one of the fundamental problem solving skills you’ll want to develop when working on developing solutions. Critical thinking is the ability to analyze, rationalize and evaluate while being aware of personal bias, outlying factors and remaining open-minded.

Defining and analyzing problems without deploying critical thinking skills can mean you and your team go down the wrong path. Developing solutions to complex issues requires critical thinking too – ensuring your team considers all possibilities and rationally evaluating them. 

Agreement-Certainty Matrix   #issue analysis   #liberating structures   #problem solving   You can help individuals or groups avoid the frequent mistake of trying to solve a problem with methods that are not adapted to the nature of their challenge. The combination of two questions makes it possible to easily sort challenges into four categories: simple, complicated, complex , and chaotic .  A problem is simple when it can be solved reliably with practices that are easy to duplicate.  It is complicated when experts are required to devise a sophisticated solution that will yield the desired results predictably.  A problem is complex when there are several valid ways to proceed but outcomes are not predictable in detail.  Chaotic is when the context is too turbulent to identify a path forward.  A loose analogy may be used to describe these differences: simple is like following a recipe, complicated like sending a rocket to the moon, complex like raising a child, and chaotic is like the game “Pin the Tail on the Donkey.”  The Liberating Structures Matching Matrix in Chapter 5 can be used as the first step to clarify the nature of a challenge and avoid the mismatches between problems and solutions that are frequently at the root of chronic, recurring problems.

Data analysis 

Though it shares lots of space with general analytical skills, data analysis skills are something you want to cultivate in their own right in order to be an effective problem solver.

Being good at data analysis doesn’t just mean being able to find insights from data, but also selecting the appropriate data for a given issue, interpreting it effectively and knowing how to model and present that data. Depending on the problem at hand, it might also include a working knowledge of specific data analysis tools and procedures. 

Having a solid grasp of data analysis techniques is useful if you’re leading a problem solving workshop but if you’re not an expert, don’t worry. Bring people into the group who has this skill set and help your team be more effective as a result.

Decision making

All problems need a solution and all solutions require that someone make the decision to implement them. Without strong decision making skills, teams can become bogged down in discussion and less effective as a result. 

Making decisions is a key part of the problem solving process. It’s important to remember that decision making is not restricted to the leadership team. Every staff member makes decisions every day and developing these skills ensures that your team is able to solve problems at any scale. Remember that making decisions does not mean leaping to the first solution but weighing up the options and coming to an informed, well thought out solution to any given problem that works for the whole team.

Lightning Decision Jam (LDJ)   #action   #decision making   #problem solving   #issue analysis   #innovation   #design   #remote-friendly   The problem with anything that requires creative thinking is that it’s easy to get lost—lose focus and fall into the trap of having useless, open-ended, unstructured discussions. Here’s the most effective solution I’ve found: Replace all open, unstructured discussion with a clear process. What to use this exercise for: Anything which requires a group of people to make decisions, solve problems or discuss challenges. It’s always good to frame an LDJ session with a broad topic, here are some examples: The conversion flow of our checkout Our internal design process How we organise events Keeping up with our competition Improving sales flow

Dependability

Most complex organizational problems require multiple people to be involved in delivering the solution. Ensuring that the team and organization can depend on you to take the necessary actions and communicate where necessary is key to ensuring problems are solved effectively.

Being dependable also means working to deadlines and to brief. It is often a matter of creating trust in a team so that everyone can depend on one another to complete the agreed actions in the agreed time frame so that the team can move forward together. Being undependable can create problems of friction and can limit the effectiveness of your solutions so be sure to bear this in mind throughout a project. 

Team Purpose & Culture   #team   #hyperisland   #culture   #remote-friendly   This is an essential process designed to help teams define their purpose (why they exist) and their culture (how they work together to achieve that purpose). Defining these two things will help any team to be more focused and aligned. With support of tangible examples from other companies, the team members work as individuals and a group to codify the way they work together. The goal is a visual manifestation of both the purpose and culture that can be put up in the team’s work space.

Emotional intelligence

Emotional intelligence is an important skill for any successful team member, whether communicating internally or with clients or users. In the problem solving process, emotional intelligence means being attuned to how people are feeling and thinking, communicating effectively and being self-aware of what you bring to a room. 

There are often differences of opinion when working through problem solving processes, and it can be easy to let things become impassioned or combative. Developing your emotional intelligence means being empathetic to your colleagues and managing your own emotions throughout the problem and solution process. Be kind, be thoughtful and put your points across care and attention. 

Being emotionally intelligent is a skill for life and by deploying it at work, you can not only work efficiently but empathetically. Check out the emotional culture workshop template for more!

Facilitation

As we’ve clarified in our facilitation skills post, facilitation is the art of leading people through processes towards agreed-upon objectives in a manner that encourages participation, ownership, and creativity by all those involved. While facilitation is a set of interrelated skills in itself, the broad definition of facilitation can be invaluable when it comes to problem solving. Leading a team through a problem solving process is made more effective if you improve and utilize facilitation skills – whether you’re a manager, team leader or external stakeholder.

The Six Thinking Hats   #creative thinking   #meeting facilitation   #problem solving   #issue resolution   #idea generation   #conflict resolution   The Six Thinking Hats are used by individuals and groups to separate out conflicting styles of thinking. They enable and encourage a group of people to think constructively together in exploring and implementing change, rather than using argument to fight over who is right and who is wrong.

Flexibility 

Being flexible is a vital skill when it comes to problem solving. This does not mean immediately bowing to pressure or changing your opinion quickly: instead, being flexible is all about seeing things from new perspectives, receiving new information and factoring it into your thought process.

Flexibility is also important when it comes to rolling out solutions. It might be that other organizational projects have greater priority or require the same resources as your chosen solution. Being flexible means understanding needs and challenges across the team and being open to shifting or arranging your own schedule as necessary. Again, this does not mean immediately making way for other projects. It’s about articulating your own needs, understanding the needs of others and being able to come to a meaningful compromise.

The Creativity Dice   #creativity   #problem solving   #thiagi   #issue analysis   Too much linear thinking is hazardous to creative problem solving. To be creative, you should approach the problem (or the opportunity) from different points of view. You should leave a thought hanging in mid-air and move to another. This skipping around prevents premature closure and lets your brain incubate one line of thought while you consciously pursue another.

Working in any group can lead to unconscious elements of groupthink or situations in which you may not wish to be entirely honest. Disagreeing with the opinions of the executive team or wishing to save the feelings of a coworker can be tricky to navigate, but being honest is absolutely vital when to comes to developing effective solutions and ensuring your voice is heard. 

Remember that being honest does not mean being brutally candid. You can deliver your honest feedback and opinions thoughtfully and without creating friction by using other skills such as emotional intelligence. 

Explore your Values   #hyperisland   #skills   #values   #remote-friendly   Your Values is an exercise for participants to explore what their most important values are. It’s done in an intuitive and rapid way to encourage participants to follow their intuitive feeling rather than over-thinking and finding the “correct” values. It is a good exercise to use to initiate reflection and dialogue around personal values.

Initiative 

The problem solving process is multi-faceted and requires different approaches at certain points of the process. Taking initiative to bring problems to the attention of the team, collect data or lead the solution creating process is always valuable. You might even roadtest your own small scale solutions or brainstorm before a session. Taking initiative is particularly effective if you have good deal of knowledge in that area or have ownership of a particular project and want to get things kickstarted.

That said, be sure to remember to honor the process and work in service of the team. If you are asked to own one part of the problem solving process and you don’t complete that task because your initiative leads you to work on something else, that’s not an effective method of solving business challenges.

15% Solutions   #action   #liberating structures   #remote-friendly   You can reveal the actions, however small, that everyone can do immediately. At a minimum, these will create momentum, and that may make a BIG difference.  15% Solutions show that there is no reason to wait around, feel powerless, or fearful. They help people pick it up a level. They get individuals and the group to focus on what is within their discretion instead of what they cannot change.  With a very simple question, you can flip the conversation to what can be done and find solutions to big problems that are often distributed widely in places not known in advance. Shifting a few grains of sand may trigger a landslide and change the whole landscape.

Impartiality

A particularly useful problem solving skill for product owners or managers is the ability to remain impartial throughout much of the process. In practice, this means treating all points of view and ideas brought forward in a meeting equally and ensuring that your own areas of interest or ownership are not favored over others. 

There may be a stage in the process where a decision maker has to weigh the cost and ROI of possible solutions against the company roadmap though even then, ensuring that the decision made is based on merit and not personal opinion. 

Empathy map   #frame insights   #create   #design   #issue analysis   An empathy map is a tool to help a design team to empathize with the people they are designing for. You can make an empathy map for a group of people or for a persona. To be used after doing personas when more insights are needed.

Being a good leader means getting a team aligned, energized and focused around a common goal. In the problem solving process, strong leadership helps ensure that the process is efficient, that any conflicts are resolved and that a team is managed in the direction of success.

It’s common for managers or executives to assume this role in a problem solving workshop, though it’s important that the leader maintains impartiality and does not bulldoze the group in a particular direction. Remember that good leadership means working in service of the purpose and team and ensuring the workshop is a safe space for employees of any level to contribute. Take a look at our leadership games and activities post for more exercises and methods to help improve leadership in your organization.

Leadership Pizza   #leadership   #team   #remote-friendly   This leadership development activity offers a self-assessment framework for people to first identify what skills, attributes and attitudes they find important for effective leadership, and then assess their own development and initiate goal setting.

In the context of problem solving, mediation is important in keeping a team engaged, happy and free of conflict. When leading or facilitating a problem solving workshop, you are likely to run into differences of opinion. Depending on the nature of the problem, certain issues may be brought up that are emotive in nature. 

Being an effective mediator means helping those people on either side of such a divide are heard, listen to one another and encouraged to find common ground and a resolution. Mediating skills are useful for leaders and managers in many situations and the problem solving process is no different.

Conflict Responses   #hyperisland   #team   #issue resolution   A workshop for a team to reflect on past conflicts, and use them to generate guidelines for effective conflict handling. The workshop uses the Thomas-Killman model of conflict responses to frame a reflective discussion. Use it to open up a discussion around conflict with a team.

Planning 

Solving organizational problems is much more effective when following a process or problem solving model. Planning skills are vital in order to structure, deliver and follow-through on a problem solving workshop and ensure your solutions are intelligently deployed.

Planning skills include the ability to organize tasks and a team, plan and design the process and take into account any potential challenges. Taking the time to plan carefully can save time and frustration later in the process and is valuable for ensuring a team is positioned for success.

3 Action Steps   #hyperisland   #action   #remote-friendly   This is a small-scale strategic planning session that helps groups and individuals to take action toward a desired change. It is often used at the end of a workshop or programme. The group discusses and agrees on a vision, then creates some action steps that will lead them towards that vision. The scope of the challenge is also defined, through discussion of the helpful and harmful factors influencing the group.

Prioritization

As organisations grow, the scale and variation of problems they face multiplies. Your team or is likely to face numerous challenges in different areas and so having the skills to analyze and prioritize becomes very important, particularly for those in leadership roles.

A thorough problem solving process is likely to deliver multiple solutions and you may have several different problems you wish to solve simultaneously. Prioritization is the ability to measure the importance, value, and effectiveness of those possible solutions and choose which to enact and in what order. The process of prioritization is integral in ensuring the biggest challenges are addressed with the most impactful solutions.

Impact and Effort Matrix   #gamestorming   #decision making   #action   #remote-friendly   In this decision-making exercise, possible actions are mapped based on two factors: effort required to implement and potential impact. Categorizing ideas along these lines is a useful technique in decision making, as it obliges contributors to balance and evaluate suggested actions before committing to them.

Project management

Some problem solving skills are utilized in a workshop or ideation phases, while others come in useful when it comes to decision making. Overseeing an entire problem solving process and ensuring its success requires strong project management skills. 

While project management incorporates many of the other skills listed here, it is important to note the distinction of considering all of the factors of a project and managing them successfully. Being able to negotiate with stakeholders, manage tasks, time and people, consider costs and ROI, and tie everything together is massively helpful when going through the problem solving process. 

Record keeping

Working out meaningful solutions to organizational challenges is only one part of the process.  Thoughtfully documenting and keeping records of each problem solving step for future consultation is important in ensuring efficiency and meaningful change. 

For example, some problems may be lower priority than others but can be revisited in the future. If the team has ideated on solutions and found some are not up to the task, record those so you can rule them out and avoiding repeating work. Keeping records of the process also helps you improve and refine your problem solving model next time around!

Personal Kanban   #gamestorming   #action   #agile   #project planning   Personal Kanban is a tool for organizing your work to be more efficient and productive. It is based on agile methods and principles.

Research skills

Conducting research to support both the identification of problems and the development of appropriate solutions is important for an effective process. Knowing where to go to collect research, how to conduct research efficiently, and identifying pieces of research are relevant are all things a good researcher can do well. 

In larger groups, not everyone has to demonstrate this ability in order for a problem solving workshop to be effective. That said, having people with research skills involved in the process, particularly if they have existing area knowledge, can help ensure the solutions that are developed with data that supports their intention. Remember that being able to deliver the results of research efficiently and in a way the team can easily understand is also important. The best data in the world is only as effective as how it is delivered and interpreted.

Customer experience map   #ideation   #concepts   #research   #design   #issue analysis   #remote-friendly   Customer experience mapping is a method of documenting and visualizing the experience a customer has as they use the product or service. It also maps out their responses to their experiences. To be used when there is a solution (even in a conceptual stage) that can be analyzed.

Risk management

Managing risk is an often overlooked part of the problem solving process. Solutions are often developed with the intention of reducing exposure to risk or solving issues that create risk but sometimes, great solutions are more experimental in nature and as such, deploying them needs to be carefully considered. 

Managing risk means acknowledging that there may be risks associated with more out of the box solutions or trying new things, but that this must be measured against the possible benefits and other organizational factors. 

Be informed, get the right data and stakeholders in the room and you can appropriately factor risk into your decision making process. 

Decisions, Decisions…   #communication   #decision making   #thiagi   #action   #issue analysis   When it comes to decision-making, why are some of us more prone to take risks while others are risk-averse? One explanation might be the way the decision and options were presented.  This exercise, based on Kahneman and Tversky’s classic study , illustrates how the framing effect influences our judgement and our ability to make decisions . The participants are divided into two groups. Both groups are presented with the same problem and two alternative programs for solving them. The two programs both have the same consequences but are presented differently. The debriefing discussion examines how the framing of the program impacted the participant’s decision.

Team-building 

No single person is as good at problem solving as a team. Building an effective team and helping them come together around a common purpose is one of the most important problem solving skills, doubly so for leaders. By bringing a team together and helping them work efficiently, you pave the way for team ownership of a problem and the development of effective solutions. 

In a problem solving workshop, it can be tempting to jump right into the deep end, though taking the time to break the ice, energize the team and align them with a game or exercise will pay off over the course of the day.

Remember that you will likely go through the problem solving process multiple times over an organization’s lifespan and building a strong team culture will make future problem solving more effective. It’s also great to work with people you know, trust and have fun with. Working on team building in and out of the problem solving process is a hallmark of successful teams that can work together to solve business problems.

9 Dimensions Team Building Activity   #ice breaker   #teambuilding   #team   #remote-friendly   9 Dimensions is a powerful activity designed to build relationships and trust among team members. There are 2 variations of this icebreaker. The first version is for teams who want to get to know each other better. The second version is for teams who want to explore how they are working together as a team.

Time management 

The problem solving process is designed to lead a team from identifying a problem through to delivering a solution and evaluating its effectiveness. Without effective time management skills or timeboxing of tasks, it can be easy for a team to get bogged down or be inefficient.

By using a problem solving model and carefully designing your workshop, you can allocate time efficiently and trust that the process will deliver the results you need in a good timeframe.

Time management also comes into play when it comes to rolling out solutions, particularly those that are experimental in nature. Having a clear timeframe for implementing and evaluating solutions is vital for ensuring their success and being able to pivot if necessary.

Improving your skills at problem solving is often a career-long pursuit though there are methods you can use to make the learning process more efficient and to supercharge your problem solving skillset.

Remember that the skills you need to be a great problem solver have a large overlap with those skills you need to be effective in any role. Investing time and effort to develop your active listening or critical thinking skills is valuable in any context. Here are 7 ways to improve your problem solving skills.

Share best practices

Remember that your team is an excellent source of skills, wisdom, and techniques and that you should all take advantage of one another where possible. Best practices that one team has for solving problems, conducting research or making decisions should be shared across the organization. If you have in-house staff that have done active listening training or are data analysis pros, have them lead a training session. 

Your team is one of your best resources. Create space and internal processes for the sharing of skills so that you can all grow together. 

Ask for help and attend training

Once you’ve figured out you have a skills gap, the next step is to take action to fill that skills gap. That might be by asking your superior for training or coaching, or liaising with team members with that skill set. You might even attend specialized training for certain skills – active listening or critical thinking, for example, are business-critical skills that are regularly offered as part of a training scheme.

Whatever method you choose, remember that taking action of some description is necessary for growth. Whether that means practicing, getting help, attending training or doing some background reading, taking active steps to improve your skills is the way to go.

Learn a process 

Problem solving can be complicated, particularly when attempting to solve large problems for the first time. Using a problem solving process helps give structure to your problem solving efforts and focus on creating outcomes, rather than worrying about the format. 

Tools such as the seven-step problem solving process above are effective because not only do they feature steps that will help a team solve problems, they also develop skills along the way. Each step asks for people to engage with the process using different skills and in doing so, helps the team learn and grow together. Group processes of varying complexity and purpose can also be found in the SessionLab library of facilitation techniques . Using a tried and tested process and really help ease the learning curve for both those leading such a process, as well as those undergoing the purpose.

Effective teams make decisions about where they should and shouldn’t expend additional effort. By using a problem solving process, you can focus on the things that matter, rather than stumbling towards a solution haphazardly. 

Create a feedback loop

Some skills gaps are more obvious than others. It’s possible that your perception of your active listening skills differs from those of your colleagues. 

It’s valuable to create a system where team members can provide feedback in an ordered and friendly manner so they can all learn from one another. Only by identifying areas of improvement can you then work to improve them. 

Remember that feedback systems require oversight and consideration so that they don’t turn into a place to complain about colleagues. Design the system intelligently so that you encourage the creation of learning opportunities, rather than encouraging people to list their pet peeves.

While practice might not make perfect, it does make the problem solving process easier. If you are having trouble with critical thinking, don’t shy away from doing it. Get involved where you can and stretch those muscles as regularly as possible. 

Problem solving skills come more naturally to some than to others and that’s okay. Take opportunities to get involved and see where you can practice your skills in situations outside of a workshop context. Try collaborating in other circumstances at work or conduct data analysis on your own projects. You can often develop those skills you need for problem solving simply by doing them. Get involved!

Use expert exercises and methods

Learn from the best. Our library of 700+ facilitation techniques is full of activities and methods that help develop the skills you need to be an effective problem solver. Check out our templates to see how to approach problem solving and other organizational challenges in a structured and intelligent manner.

There is no single approach to improving problem solving skills, but by using the techniques employed by others you can learn from their example and develop processes that have seen proven results. 

Try new ways of thinking and change your mindset

Using tried and tested exercises that you know well can help deliver results, but you do run the risk of missing out on the learning opportunities offered by new approaches. As with the problem solving process, changing your mindset can remove blockages and be used to develop your problem solving skills.

Most teams have members with mixed skill sets and specialties. Mix people from different teams and share skills and different points of view. Teach your customer support team how to use design thinking methods or help your developers with conflict resolution techniques. Try switching perspectives with facilitation techniques like Flip It! or by using new problem solving methodologies or models. Give design thinking, liberating structures or lego serious play a try if you want to try a new approach. You will find that framing problems in new ways and using existing skills in new contexts can be hugely useful for personal development and improving your skillset. It’s also a lot of fun to try new things. Give it a go!

Encountering business challenges and needing to find appropriate solutions is not unique to your organization. Lots of very smart people have developed methods, theories and approaches to help develop problem solving skills and create effective solutions. Learn from them!

Books like The Art of Thinking Clearly , Think Smarter, or Thinking Fast, Thinking Slow are great places to start, though it’s also worth looking at blogs related to organizations facing similar problems to yours, or browsing for success stories. Seeing how Dropbox massively increased growth and working backward can help you see the skills or approach you might be lacking to solve that same problem. Learning from others by reading their stories or approaches can be time-consuming but ultimately rewarding.

A tired, distracted mind is not in the best position to learn new skills. It can be tempted to burn the candle at both ends and develop problem solving skills outside of work. Absolutely use your time effectively and take opportunities for self-improvement, though remember that rest is hugely important and that without letting your brain rest, you cannot be at your most effective. 

Creating distance between yourself and the problem you might be facing can also be useful. By letting an idea sit, you can find that a better one presents itself or you can develop it further. Take regular breaks when working and create a space for downtime. Remember that working smarter is preferable to working harder and that self-care is important for any effective learning or improvement process.

Want to design better group processes?

Over to you

Now we’ve explored some of the key problem solving skills and the problem solving steps necessary for an effective process, you’re ready to begin developing more effective solutions and leading problem solving workshops.

Need more inspiration? Check out our post on problem solving activities you can use when guiding a group towards a great solution in your next workshop or meeting. Have questions? Did you have a great problem solving technique you use with your team? Get in touch in the comments below. We’d love to chat!

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Enhancing computational thinking skills of students with disabilities

• Original Research
• Published: 29 April 2022
• Volume 50 , pages 625–651, ( 2022 )

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• Serhat Bahadır Kert   ORCID: orcid.org/0000-0002-1093-6326 1 ,
• Sabiha Yeni 1 &
• Mehmet Fatih Erkoç 1  

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Computational thinking (CT) and computer science (CS) are becoming more widely adopted in K-12 education. However, there is a lack of focus on CT and CS access for children with disabilities. This study investigates the effect of the robot development process at the secondary school level on the algorithmic thinking and mental rotation skills of students with learning disabilities (LD). The study was conducted with the single-subject model and as an A-B-A design. In the study, the CT skill development of four students with LD (1 female, 3 male) was monitored throughout 13 weeks with the pre-treatment sessions running from weeks 1–4, treatment sessions running from weeks 5–9, and post-treatment sessions running from weeks 10–13. During the treatment sessions, robot design and programming implementations were performed. During these 13 sessions, the observer scored participants’ both algorithmic problem-solving and mental rotation skills. These skills are also required to use some other cognitive sub-skills (i.e., selective attention, processing speed) which were defined by ten special education experts at the beginning of the study. All these skills were evaluated according to how well the students performed the following four criteria: (1) To start to perform the instructions quickly (processing speed), (2) to focus on the task by filtering out distractions (selective attention), (3) to fulfill the task without having to have the instructions repeated, (4) to perform algorithmic problem-solving/mental rotation tasks without any help. Considering the results on the participants’ algorithmic problem-solving skills, a significant improvement was obtained in their skills after the treatment process. The improvement obtained in the participants’ mental rotation skills is another important result of the study. Considering the study results from a holistic perspective, it can be concluded that the robot development implementation, as educational technology, can be used to support the cognitive development of students with learning disabilities.

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5 Strategies to Teach Multi-Step Word Problems: Teacher’s Guide

Hey there, Teacher! Are you ready to empower your students with the skills they need to conquer multi-step word problems? Look no further! This comprehensive blog post is your go-to resource for effective strategies that will transform your classroom into a problem-solving powerhouse. Let’s dive in and unleash your students’ problem-solving potential!

Table of Contents

Word Problems

Word problems are an essential part of math education, as they help develop critical thinking and problem-solving skills.

Multi-step word problems, in particular, provide an excellent opportunity for students to apply their math skills in real-life scenarios.

Teaching students how to solve one-step word problems or multi-step word problems can be a complex task as it can be a challenging concept for most students to grasp.

However, with the right strategies and guidance, teachers can help their students become proficient problem solvers.

Strategies to Teach Multi-Step Word Problems

Now, let’s delve into the 5 strategies that teachers can employ to effectively teach multi-step word problem-solving to their students.

Model the Problem-Solving Process

Provide clear problem-solving strategies.

• Provide Scaffolded Practice

Differentiate Instruction

• Practice Regularly for Proficiency

One of the most effective ways to help students understand and solve multi-step word problems is by modeling the problem-solving process.

When teachers model the steps involved in solving a problem, students can observe and learn from their thinking and approach.

This approach helps students develop a deeper understanding of the problem and the strategies required to solve it.

You can use the following approaches when modeling the problem-solving process.

Step-by-Step Approach

Storytelling approach, real-life examples.

When modeling the problem-solving process, it is crucial to take a step-by-step approach.

Break down the problem into smaller, manageable parts, and demonstrate how to tackle each part systematically.

By breaking down the problem, students can focus on one step at a time, reducing confusion and overwhelm.

Another effective strategy is to take a storytelling approach when presenting multi-step word problems.

Create narratives or scenarios around the problems, making them more interesting and captivating for students.

By presenting problems in a story format, students can visualize the context and relate to the characters or situations involved.

This approach enhances their problem-solving abilities and engages their imagination.

To make the modeling process more engaging and relatable, incorporate real-life examples into the multi-step word problems.

Relating the problems to situations that students encounter in their daily lives helps them connect with the content and see the practical applications of mathematical concepts.

For instance, you can present a word problem involving shopping or calculating distances during a trip.

As a teacher, your role is to equip your students with effective problem-solving strategies that will empower them to confidently tackle multi-step word problems.

Here are some key multi-step problem-solving strategies to share with your students:

Read the Problem Carefully

Identify the question, plan and break down the problem, choose the correct operations, solve step-by-step, check and reflect.

The first step in solving a multi-step word problem is to read the problem carefully.

Emphasize the importance of taking the time to understand the given information.

Encourage your students to identify the essential details, underline or highlight important numbers or quantities.

By comprehending the problem thoroughly, students will lay a strong foundation for their problem-solving journey.

After understanding the given information, students should identify the question they need to answer.

Guide your students to identify the question or the unknown they need to solve.

Assist them in recognizing what information is missing and what they are being asked to find.

Encourage them to clearly define the question to maintain focus and direction throughout the problem-solving process.

Teach your students to develop a plan and break down the problem into smaller, more manageable steps.

By doing so, they can avoid feeling overwhelmed by the complexity of multi-step word problems.

Discuss possible approaches, such as drawing diagrams, making tables, or using equations.

By having a plan in place or by creating a roadmap for the problem-solving journey, students can proceed with confidence and avoid making unnecessary mistakes.

Guide your students in choosing the correct operations or mathematical strategies for each step of the problem.

Help them consider the problem’s context and the relationships between different quantities.

By selecting the appropriate operations, such as addition, subtraction, multiplication, or division, students can accurately solve each component of the multi-step word problem.

Encourage your students to solve the problem step-by-step, following the plan they devised earlier.

By solving each step individually and linking them together, students can see the overall problem as a series of interconnected components, making it easier to navigate and solve.

Remind students to show their work neatly and clearly. This approach not only helps students organize their thoughts but also allows you, as their teacher, to identify any errors or misconceptions they may have.

Provide guidance and support as needed throughout their problem-solving process.

After obtaining a solution, it’s essential that students check their answer.

Teach your students the importance of checking their work and reflecting on their solution.

Encourage them to evaluate whether their answer makes sense within the context of the problem. They can do this by revisiting the original problem, reapplying the given information, and verifying if the solution satisfies the question asked.

This step promotes critical thinking and helps students identify and correct any mistakes they may have made.

Scaffolded Practice

As a teacher, it is crucial to provide scaffolded practice to support your students in mastering multi-step word problems.

Scaffolded practice involves breaking down the problem-solving process into smaller, manageable steps and gradually increasing the complexity of the problems.

This approach allows students to build their skills incrementally and gain confidence as they progress.

Here are some strategies to implement scaffolded practice:

Start with Simple Problems

Provide guided practice, increase complexity gradually, use visual aids and models, break down complex problems, provide independent practice opportunities.

Begin by introducing your students to simple, one-step word problems .

These problems will help them understand the basics of problem-solving and build their confidence.

Provide clear explanations and model the problem-solving process step-by-step.

Reinforce the importance of reading the problem carefully and identifying the key information.

Guide your students through practice problems by solving them together as a class.

Discuss the steps involved and explain the reasoning behind each step.

Encourage students to ask questions and engage in discussions about problem-solving strategies.

Offer support and guidance as needed, ensuring that students understand each concept before moving on.

Gradually increase the complexity of the problems as your students become more comfortable with one-step word problems.

Introduce multistep word problems by adding another step or operation at a time.

Provide clear explanations of each step and emphasize the relationships between different quantities in the problem.

Encourage students to apply the problem-solving strategies they have learned.

Utilize visual aids and models to help students visualize the problem and understand the relationships between different quantities.

Use diagrams, charts, or manipulatives to represent the information given in the problem. This visual representation can enhance students’ understanding and support their problem-solving process.

For complex multi-step word problems, break them down into smaller components.

Guide students through each step, explaining the rationale behind each operation and encouraging them to show their work.

This step-by-step approach helps students tackle complex problems more effectively and reduces feelings of overwhelm.

Offer opportunities for students to practice independently.

Provide worksheets or online resources that offer a range of multi-step word problems at various difficulty levels.

Encourage students to work through the problems on their own, applying the strategies they have learned. Offer feedback and support as they progress.

As a teacher, it is essential to differentiate your instruction to meet the diverse needs of your students.

Differentiation allows you to tailor your teaching methods, materials, and assessments to accommodate the individual learning styles, abilities, and interests of your students.

Here are some strategies for differentiating instruction in multi-step word problem-solving:

Assess Readiness and Grouping

Offer multiple problem-solving approaches, vary the level of difficulty, flexible grouping strategies, accommodations and supports, formative assessments and feedback.

Assess the readiness of your students by evaluating their prior knowledge and skills related to multi-step word problems.

Group students based on their readiness levels, whether they require additional support or enrichment.

Provide targeted instruction and resources to address the specific needs of each group.

Recognize that students have different learning styles and preferences.

Offer a variety of problem-solving approaches to cater to their individual needs.

Provide visual representations, manipulatives, or verbal explanations to support visual, kinesthetic, or auditory learners.

Allow students to choose the method that best suits their learning style.

Recognize that students have different levels of proficiency in problem-solving.

Differentiate the level of difficulty in the problems you assign.

Provide additional challenge problems for high-achieving students and offer extra support or modified problems for those who may struggle.

Adjust the complexity of the problems to ensure that each student is appropriately challenged.

Implement flexible grouping strategies to allow students to learn from and support each other.

Arrange students in pairs or small groups based on their strengths and weaknesses.

Encourage collaborative problem-solving activities where students can share their strategies, explain their thinking, and learn from their peers.

Offer opportunities for peer tutoring or mentoring.

Recognize the diverse needs of students with learning disabilities or English language learners.

Provide accommodations and support to ensure their understanding and success.

Adapt the materials, provide visual cues or graphic organizers, offer additional time, or provide translated resources when necessary.

Collaborate with special education or language support specialists to address individual needs effectively.

Use formative assessments to gather ongoing feedback on students’ progress in multi-step word problem-solving.

Monitor their understanding, identify misconceptions, and provide timely feedback. Offer specific guidance and support based on individual needs.

Encourage students to reflect on their problem-solving strategies and provide self-assessment opportunities.

Practice Multi-Step Word Problems Regularly for Proficiency

It is important to emphasize the value of regular practice to help your students achieve proficiency in solving multi-step word problems.

Consistent practice not only reinforces problem-solving strategies and concepts but also builds confidence and fluency in applying them.

Here are some strategies to promote regular practice:

Assign Regular Problem-Solving Exercises

• Integrate Multistep Word Problems in Daily Lessons
• Encourage Independent Practice

Offer Problem-Solving Discussions and Presentations

Assign regular problem-solving exercises as part of your students’ homework or in-class activities.

Provide a variety of multi-step word problems that cover different mathematical concepts and real-life scenarios.

Gradually increase the difficulty level of the problems to challenge your students and foster their growth.

Integrate Multi-Step Word Problems in Daily Lessons

Incorporate multi-step word problems into your daily math lessons. Introduce them as real-world applications of the mathematical concepts you are teaching.

Show your students how these problems relate to their everyday lives and the importance of problem-solving skills in various contexts.

Inspire active participation and engagement during problem-solving activities.

Encourage Independent Practice of Multi-Step Word Problems

Encourage your students to practice independently outside of class.

Recommend math resources, such as textbooks, workbooks, or online platforms, that provide additional multi-step word problems for practice.

Encourage them to set aside dedicated time each week to work on problem-solving skills.

Remind them of the benefits of consistent practice in building proficiency.

Create opportunities for students to present and discuss their problem-solving strategies with the class.

Encourage them to explain their thinking, justify their solutions, and engage in constructive discussions.

This not only enhances their communication skills but also exposes them to different problem-solving approaches and fosters a collaborative learning environment.

Recommended Materials for Multi-Step Word Problems Practice

To enhance your students’ practice in solving multi-step word problems, consider incorporating the following recommended materials:

Math Workbooks

Online problem-solving platforms, math games and activities, task cards and worksheets.

Math workbooks provide a structured approach to practicing multi-step word problems.

Look for workbooks that specifically focus on problem-solving skills and offer a variety of problem types and difficulty levels.

These workbooks typically include step-by-step explanations and examples, providing students with opportunities to apply their problem-solving strategies independently.

Encourage your students to work through the problems in the workbooks, highlighting the importance of showing their work and explaining their reasoning.

You can assign specific pages or problem sets from the workbook as part of their regular practice routine.

Leverage the power of technology by using online problem-solving platforms.

These platforms offer interactive multi-step word problems that engage students and provide immediate feedback.

Look for platforms that align with your curriculum and allow you to track students’ progress.

Online problem-solving platforms often provide a range of difficulty levels and adaptive features that adjust the complexity of the problems based on individual performance.

Encourage your students to explore these platforms during their independent practice time or assign specific problems for them to solve online.

Engage your students in learning through math games and activities.

These interactive and hands-on experiences make practice enjoyable and foster a positive attitude towards problem-solving.

Look for math games and activities that specifically focus on multi-step word problems.

You can create your own games using task cards or find existing games that align with your curriculum.

Set up problem-solving stations where students rotate and solve different multi-step word problems in a game format.

These activities promote collaboration, critical thinking, and a deeper understanding of problem-solving strategies.

Task cards and worksheets provide targeted practice opportunities for multi-step word problems.

While worksheets offer a collection of problems on a single sheet, task cards typically consist of individual problems on small cards.

These resources allow for flexibility in assigning practice and can be tailored to the specific needs of your students.

Select task cards or worksheets that align with the topics and skills you want your students to practice.

Consider using a variety of formats, such as multiple-choice, open-ended, or guided questions, to cater to different learning preferences.

Provide clear instructions and encourage students to work through the problems independently or in small groups.

If you’re looking for engaging worksheets and task cards for multi-step word problems to use with your students, then my differentiated worksheets and task cards will be a perfect fit for you. You can find them at my resources stores below:

• Website shop
• Made by Teachers store

Congratulations, teachers! You have now equipped yourself with a toolkit of effective strategies and resources to teach multi-step word problems with confidence.

By providing clear problem-solving strategies, implementing scaffolded practice, differentiating your instruction, and promoting regular practice, you are setting your students up for success.

Embrace the journey of teaching and learning multi-step word problems, celebrate the progress of your students, and watch as they develop into proficient problem solvers.

Remember, with your guidance and support, their potential knows no bounds!

Multistep Word Problems – FAQs

What are two-step and multi-step problems.

Two-step and multi-step problems are types of word problems that require multiple mathematical operations to be performed to find the solution. In two-step problems, students need to apply two operations, such as addition and subtraction or multiplication and division, to solve the problem. Multi-step problems may involve more than two operations and often require students to perform a series of steps to arrive at the final answer.

Why Are Multi-Step Word Problems So Important?

Multi-step word problems are important because they reflect real-world situations that students may encounter in their daily lives. By solving these problems, students develop critical thinking, analytical reasoning, and problem-solving skills. Multi-step word problems also help students apply mathematical concepts in context, promoting a deeper understanding of the subject matter.

Why do students struggle with multi-step word problems?

Students often struggle with multi-step word problems due to several reasons. First, these problems require higher-level thinking skills, such as analyzing and synthesizing information. Second, students may find it challenging to identify the relevant information and determine which operations to use. Additionally, students may struggle with organizing their thoughts and applying problem-solving strategies effectively.

What are the common errors in solving multi-step word problems?

Common errors in solving multi-step word problems include:

• Misinterpreting the problem: Failing to understand the problem correctly and misidentifying the key information.
• I ncorrect selection of operations: Choosing the wrong operations or using them in the wrong order.
• Calculation errors: Making mistakes in performing arithmetic calculations.
• Forgetting to check the answer: Neglecting to verify if the solution aligns with the question asked and the context of the problem.

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The Efficacy and Development of Students' Problem-Solving Strategies During Compulsory Schooling: Logfile Analyses

Gyöngyvér molnár.

1 Department of Learning and Instruction, University of Szeged, Szeged, Hungary

Benő Csapó

2 MTA-SZTE Research Group on the Development of Competencies, University of Szeged, Szeged, Hungary

The purpose of this study was to examine the role of exploration strategies students used in the first phase of problem solving. The sample for the study was drawn from 3 rd - to 12 th -grade students (aged 9–18) in Hungarian schools ( n = 4,371). Problems designed in the MicroDYN approach with different levels of complexity were administered to the students via the eDia online platform. Logfile analyses were performed to ascertain the impact of strategy use on the efficacy of problem solving. Students' exploration behavior was coded and clustered through Latent Class Analyses. Several theoretically effective strategies were identified, including the vary-one-thing-at-a-time (VOTAT) strategy and its sub-strategies. The results of the analyses indicate that the use of a theoretically effective strategy, which extract all information required to solve the problem, did not always lead to high performance. Conscious VOTAT strategy users proved to be the best problem solvers followed by non-conscious VOTAT strategy users and non-VOTAT strategy users. In the primary school sub-sample, six qualitatively different strategy class profiles were distinguished. The results shed new light on and provide a new interpretation of previous analyses of the processes involved in complex problem solving. They also highlight the importance of explicit enhancement of problem-solving skills and problem-solving strategies as a tool for knowledge acquisition in new contexts during and beyond school lessons.

Introduction

Computer-based assessment has presented new challenges and opportunities in educational research. A large number of studies have highlighted the importance and advantages of technology-based assessment over traditional paper-based testing (Csapó et al., 2012 ). Three main factors support and motivate the use of technology in educational assessment: (1) the improved efficiency and greater measurement precision in the already established assessment domains (e.g., Csapó et al., 2014 ); (2) the possibility of measuring constructs that would be impossible to measure by other means (e.g., Complex Problem Solving (CPS) 1 ; see Greiff et al., 2012 , 2013 ); and (3) the opportunity of logging and analyzing not only observed variables, but metadata as well (Lotz et al., 2017 ; Tóth et al., 2017 ; Zoanetti and Griffin, 2017 ). Analyzing logfiles may contribute to a deeper and better understanding of the phenomenon under examination. Logfile analyses can provide answers to research questions which could not be answered with traditional assessment techniques.

This study focuses on problem solving, especially on complex problem solving (CPS), which reflects higher-order cognitive processes. Previous research identified three different ways to measure CPS competencies: (1) Microworlds (e.g., Gardner and Berry, 1995 ), (2) formal frameworks (Funke, 2001 , 2010 ) and (3) minimal complex systems (Funke, 2014 ). In this paper, the focus is on the MicroDYN approach, which is a specific form of complex problem solving (CPS) in interactive situations using minimal complex systems (Funke, 2014 ). Recent analyses provide both a new theory and data-based evidence for a global understanding of different problem-solving strategies students employ or could employ in a complex problem-solving environment based on minimal complex systems.

The problem scenarios within the MicroDYN approach consist of a small number of variables and causal relations. From the perspective of the problem solver, solving a MicroDYN problem requires a sequence of continuous activities, in which the outcome of one activity is the input for the next. First, students interact with the simulated system, set values for the input variables, and observe the impacts of these settings on the target (dependent) variable. Then, they plot their conclusion about the causal relationships between the input and output variables on a graph (Phase 1). Next, they manipulate the independent variables again to set their values so that they result in the required values for the target variables (Phase 2).

When it comes to gathering information about a complex problem, as in the MicroDYN scenarios, there may be differences between the exploration strategies in terms of efficacy. Some of them may be more useful for generating knowledge about the system. Tschirgi ( 1980 ) identified different exploration strategies. When control of variables strategies (Greiff et al., 2014 ) were explored, findings showed that the vary-one-thing-at-a-time (VOTAT, Tschirgi, 1980 ; Funke, 2014 ) was the most effective strategy for identifying causal relations between the input and output variables in a minimal complex system (Fischer et al., 2012 ). Participants who employed this strategy tended to acquire more structural knowledge than those who used other strategies (Vollmeyer et al., 1996 ; Kröner et al., 2005 ). With the VOTAT strategy, the problem solver systematically varies only one input variable, while the others remain unchanged. This way, the effect of the variable that has just been changed can be observed directly by monitoring the changes in the output variables. There exist several types of VOTAT strategies.

Using this approach—defining the effectiveness of a strategy on a conceptual level, independently of empirical effectiveness—we developed a labeling system and a mathematical model based on all theoretically effective strategies. Thus, effectiveness was defined and linked to the amount of information extracted. An exploration strategy was defined as theoretically effective if the problem solver was able to extract all the information needed to solve the problem, independently of the application level of the information extracted and of the final achievement. We split the effectiveness of the exploration strategy and the usage and application of the information extracted to be able to solve the problem and control the system with respect to the target values based on the causal knowledge acquired. Systematicity was defined on the level of effectiveness based on the amount of information extracted and on the level of awareness based on the implementation of systematicity in time.

Students' actions were logged and coded according to our input behavior model and then clustered for comparison. We were able to distinguish three different VOTAT strategies and two successful non-VOTAT ones. We empirically tested awareness of the input behavior used in time. Awareness of strategy usage was analyzed by the sequence of the trials used, that is, by the systematicity of the trials used in time. We investigated the effectiveness of and differences in problem-solving behavior between three age groups by conducting latent class analyses to explore and define patterns in qualitatively different VOTAT strategy uses.

Although the assessment of problem solving within the MicroDYN approach is a relatively new area of research, its processes have already been studied in a number of different contexts, including a variety of educational settings with several age groups. Our cross-sectional design allows us to describe differences between age groups and outline the developmental tendencies of input behavior and strategy use among children in the age range covered by our data collection.

Reasoning strategies in complex problem solving

Problem-solving skills have been among the most extensively studied transversal skills over the last decade; they have been investigated in the most prominent comprehensive international large-scale assessments today (e.g., OECD, 2014 ). The common aspects in the different theoretical models are that a problem is characterized by a gap between the current state and the goal state with no immediate solution available (Mayer and Wittrock, 1996 ).

Parallel to the definition of the so-called twenty first-century skills (Griffin et al., 2012 ), recent research on problem solving disregards content knowledge and domain-specific processes. The reason for this is that understanding the structure of unfamiliar problems is more effective when it relies on abstract representation schemas and metacognitive strategies than on specifically relevant example problems (Klahr et al., 2007 ). That is, the focus is more on assessing domain-general problem-solving strategies (Molnár et al., 2017 ), such as complex problem solving, which can be used to solve novel problems, even those arising in interactive situations (Molnár et al., 2013 ).

Logfile analyses make it possible to divide the continuum of a problem-solving process into several scoreable phases by extracting information from the logfile that documents students' problem-solving behavior. In our case, latent class analysis extracts information from the file that logs students' interaction with the simulated system at the beginning of the problem-solving process. The way students manipulate the input (independent) variables represents their reasoning strategy. Log data, on the one hand, make it possible to analyze qualitative differences in these strategies and then their efficiency in terms of how they generate knowledge resulting in the correct plotting of the causal relationship in Phase 1 and then the proper setting to reach the required target value in Phase 2. On the other hand, qualitative strategy data can be quantified, and an alternative scoring system can be devised.

From the perspective of the traditional psychometric approach and method of scoring, these problems form a test task consisting of two scoreable items. The first phase is a knowledge acquisition process, where scores are assigned based on how accurately the causal relationship was plotted. The second phase is knowledge application, where the correctness of the value for the target variable is scored. Such scoring based on two phases of solving MicroDYN problems has been used in a number of previous studies (e.g., Greiff et al., 2013 , 2015 ; Wüstenberg et al., 2014 ; Csapó and Molnár, 2017 ; Greiff and Funke, 2017 ).

To sum up, there is great potential to investigate and cluster the problem-solving behavior and exploration strategy usage of the participants at the beginning of the problem-solving process and correlate the use of a successful exploration strategy with the model-building solution (achievement in Phase 1) observed directly in these simulated problem scenarios. Using logfile analyses (Greiff et al., 2015 ), the current article wishes to contribute insights into students' approaches to explore and solve problems related to minimal complex systems. By addressing research questions on the problem-solving strategies used, the study aims to understand students' exploration behavior in a complex problem-solving environment and the underlying causal relations. In this study, we show that such scoring can be developed through latent class analysis and that this alternative method of scoring may produce more reliable tests. Furthermore, such scoring can be automated and then employed in a large-scale assessment.

There are two major theoretical approaches to cognition relevant to our study; both offer general principles to interpret cognitive development beyond the narrower domain of problem solving. Piaget proposed the first comprehensive theory to explain the development of children's thinking as a sequence of four qualitatively different stages, the formal operational stage being the last one (Inhelder and Piaget, 1958 ), while the information processing approach describes human cognition by using terms and analogies borrowed from computer science. The information processing paradigm was not developed into an original developmental theory; it was rather aimed at reinterpreting and extending Piaget's theory (creating several Neo-Piagetian models) and synthesizing the main ideas of the two theoretical frameworks (Demetriou et al., 1993 ; Siegler, 1999 ). One of the focal points of these models is to explain the development of children's scientific reasoning, or, more closely, the way children understand how scientific experiments can be designed and how causal relationships can be explored by systematically changing the values of (independent) variables and observing their impact on other (target) variables.

From the perspective of the present study, the essential common element of cognitive developmental research is the control of variables strategy. Klahr and Dunbar ( 1988 ) distinguished two related skills in scientific thinking, hypothesis formation and experimental design, and they integrated these skills into a coherent model for a process of scientific discovery. The underlying assumption is that knowledge acquisition requires an iterative process involving both. System control as knowledge application tends to include both processes, especially when acquired knowledge turns out to be insufficient or dysfunctional (J. F. Beckmann, personal communication, August 16, 2017). Furthermore, they separated the processes of rule induction and problem solving, defining the latter as a search in a space of rules (Klahr and Dunbar, 1988 , p. 5).

de Jong and van Joolingen ( 1998 ) provided an overview of studies in scientific discovery learning with computer simulations. They concluded that a number of specific skills are needed for successful discovery, like systematic variation of variable values, which is in a focus of the present paper, and the use of high-quality heuristics for experimentation. They identified several characteristic problems in the discovery process and stressed that learners often have trouble interpreting data.

In one of the earliest systematic studies of students' problem-solving strategies, Vollmeyer et al. ( 1996 ) explored the impact of strategy systematicity and effectiveness on complex problem-solving performance. Based on previous studies, they distinguished the VOTAT strategy from other possible strategies [Change All (CA) and Heterogeneous (HT) other strategies], as VOTAT allows systematic exploration of the behavior of a system and a disconfirmation of hypotheses. In one of their experiments, they examined the hypothesis that VOTAT was more effective for acquiring knowledge than less systematic strategies. According to the results, the 36 undergraduate students had clearly shown strategy development. After interacting with the simulated system in several rounds, they tended to use the VOTAT strategy more frequently. In a second experiment, it was also demonstrated that goal specificity influences strategy use as well (Vollmeyer et al., 1996 ).

Beckmann and Goode ( 2014 ) analyzed the systematicity in exploration behavior in a study involving 80 first-year psychology students and focusing on the semantic context of a problem and its effect on the problem solvers' behavior in complex and dynamic systems. According to the results, a semantically familiar problem context invited a high number of a priori assumptions on the interdependency of system variables. These assumptions were less likely tested during the knowledge acquisition phase, this proving to be the main barrier to the acquisition of new knowledge. Unsystematic exploration behavior tended to produce non-informative system states that complicated the extraction of knowledge. A lack of knowledge ultimately led to poor control competency.

Beckmann et al. ( 2017 ) confirmed research results by Beckmann and Goode ( 2014 ) and demonstrated how a differentiation between complexity and difficulty leads to a better understanding of the cognitive mechanism behind CPS. According to findings from a study with 240 university students, the performance differences observed in the context of the semantic effect were associated with differences in the systematicity of the exploration behavior, and the systematicity of the exploration behavior was reflected in a specific sequence of interventions. They argued that it is only the VOTAT strategy—supplemented with the vary- none -at-a-time strategy in the case of noting autonomous changes—that creates informative system state transitions which enable problem solvers to derive knowledge of the causal structure of a complex, dynamic system.

Schoppek and Fischer ( 2017 ) also investigated VOTAT and the related “PULSE” strategy (all input variables to zero), which enables the problem solver to observe the eigendynamics of the system in a transfer experiment. They proposed that besides VOTAT and PULSE, other comprehensive knowledge elements and strategies, which contribute to successful CPS, should be investigated.

In a study with 2 nd - to 4 th -grade students, Chen and Klahr found little spontaneous development when children interacted with physical objects (in situations similar to that of Piaget's experiments), while more direct teaching of the control of variables strategy resulted in good effect sizes and older children were able to transfer the knowledge they had acquired (improved control of variable strategy) to remote contexts (Chen and Klahr, 1999 ). In a more recent study, Kuhn et al. ( 2008 ) further extended the scope of studies on scientific thinking, identifying three further aspects beyond the control of variables strategy, including coordinating effects of multiple influences, understanding the epistemological foundations of science and engaging in argumentation. In their experiment with 91 6th-grade students, they explored how students were able to estimate the impact of five independent variables simultaneously on a particular phenomenon, and they found that most students considered only one or two variables as possible causes.

In this paper, we explore several research questions on effective and less effective problem-solving strategies used in a complex problem-solving environment and detected by logfile analyses. We use logfile analyses to empirically test the success of different input behavior and strategy usage in CPS tasks within the MicroDYN framework. After constructing a mathematical model based on all theoretically effective strategies, which provide the problem solver with all the information needed to solve the problem, and defining several sub-strategies within the VOTAT strategy based on the amount of effort expended to extract the necessary information, we empirically distinguish different VOTAT and non-VOTAT strategies, which can result in good CPS performance and which go beyond the isolated variation strategy as an effective strategy for rule induction (Vollmeyer et al., 1996 ). We highlight the most and least effective VOTAT strategies used in solving MicroDYN problems and empirically investigate the awareness of the strategy used based on the sequence of the sub-strategies used. Based on these results, we conduct latent class analyses to explore and define patterns in qualitatively different VOTAT strategy uses.

We thus intend to answer five research questions:

• RQ1: Does the use of a theoretically effective strategy occur prior to high performance? In other words, does the use of a theoretically effective strategy result in high performance?
• RQ2: Do all VOTAT strategies result in a high CPS performance? What is the most effective VOTAT strategy?
• RQ3: How does awareness of the exploration strategy used influence overall performance on CPS tasks?
• RQ4: What profiles characterize the various problem solvers and explorers?
• RQ5: Do exploration strategy profiles differ across grade levels, which represent different educational stages during compulsory schooling?

In this study, we investigated qualitatively different classes of students' exploration behavior in CPS environments. We used latent class analysis (LCA) to study effective and non-effective input behavior and strategy use, especially the principle of isolated variation, across several CPS tasks. We compared the effectiveness of students' exploration behavior based on the amount of information they extracted with their problem-solving achievement. We posed five separate hypotheses.

Hypothesis 1: We expect that high problem-solving achievement is not closely related to expert exploration behavior.

Vollmeyer et al. ( 1996 ) explored the impact of strategy effectiveness on problem-solving performance and reported that effectiveness correlated negatively and weakly to moderately with solution error ( r = −0.32 and r = −0.54, p < 0.05). They reported that “most participants eventually adopted the most systematic strategy, VOTAT, and the more they used it, the better they tended to perform. However, even those using the VOTAT strategy generally did not solve the problem completely” (p. 88). Greiff et al. ( 2015 ) confirmed that different exploration behaviors are relevant to CPS and that the number of sub-strategies implemented was related to overall problem-solving achievement.

Hypothesis 2: We expect that students who use the isolated variation strategy in exploring CPS problems have a significantly better overall performance than those who use a theoretically effective, but different strategy.

Sonnleiter et al. ( 2017 ) noted that “A more effective exploration strategy leads to a higher system knowledge score and the higher the gathered knowledge, the better the ability to achieve the target values. Thus, system knowledge can be seen as a reliable and valid measure of students' mental problem representations” (p. 169). According to Wüstenberg et al. ( 2012 ), students who consistently apply the principle of isolated variation—the most systematic VOTAT strategy—in CPS environments show better overall CPS performance, compared to those who use different exploration strategies. Kröner et al. ( 2005 ) reported a positive correlation between using the principle of isolated variation and the likelihood of solving the overall problem.

Hypothesis 3: We expected that more aware CPS exploration behavior would be more effective than exploration behavior that generally results in extracting all the necessary information from the system to solve the problem, but within which the steps have no logically built structure and no systematicity in time.

Vollmeyer et al. ( 1996 ) explored the impact of strategy systematicity on problem-solving performance. They emphasized that “the systematicity of participants' spontaneous hypothesis-testing strategies predicted their success on learning the structure of the biology lab problem space” (p. 88). Vollmeyer and her colleagues restricted systematic strategy users to isolated variation strategy users; this corresponds to our terminology usage of aware isolated variation strategy users.

Hypothesis 4: We expected to find a distinct number of classes with statistically distinguishable profiles of CPS exploration behavior. Specifically, we expected to find classes of proficient, intermediate and low-performing explorers.

Several studies (Osman and Speekenbrink, 2011 ; Wüstenberg et al., 2012 ; Greiff et al., 2015 ) have indicated that there exist quantitative differences between different exploration strategies, which are relevant to a CPS environment. The current study is the first to investigate whether a relatively small number of qualitatively different profiles of students' exploration proficiency can be derived from their behavior detected in a CPS environment in a broad age range.

Hypothesis 5: We expected that more proficient CPS exploration behavior would be more dominant at later grade levels as an indication of cognitive maturation and of increasing abilities to explore CPS environments.

The cognitive development in children between Grades 3 and 12 is immense. According to Piaget's stage theory, they move from concrete operations to formal operations and they will be able to think logically and abstractly. According to Galotti ( 2011 ) and Molnár et al. ( 2013 ), the ability to solve problems effectively and to make decisions in CPS environments increases in this period of time; Grades 6–8 seem especially crucial for development. Thus, we expect that cognitive maturation will also be reflected in more proficient exploration behavior.

Participants

The sample was drawn from 3 rd - to 12 th -grade students (aged 9–18) in Hungarian primary and secondary schools ( N = 4,371; Table ​ Table1). 1 ). School classes formed the sampling unit. 180 classes from 50 schools in different regions were involved in the study, resulting in a wide-ranging distribution of students' background variables. The proportion of boys and girls was about the same.

Composition of samples.

The MicroDYN approach was employed to develop a measurement device for CPS. CPS tasks within the MicroDYN approach are based on linear structural equations (Funke, 2001 ), in which up to three input variables and up to three output variables are related (Greiff et al., 2013 ). Because of the small set of input and output variables, the MicroDYN problems could be understood completely with precise causal analyses (Funke, 2014 ). The relations are not presented to the problem solver in the scenario. To explore these relations, the problem solver must interact directly with the problem situation by manipulating the input variables (Greiff and Funke, 2010 ), an action that can influence the output variables (direct effects), and they must use the feedback provided by the computer to acquire and employ new knowledge (Fischer et al., 2012 ). Output variables can change spontaneously and can consist of internal dynamics, meaning they can change without changing the input variables (indirect effects; Greiff et al., 2013 ). Both direct and indirect effects can be detected with an adequate problem-solving strategy (Greiff et al., 2012 ). The interactions between the problem situation and the test taker play an important role, but they can only be identified in a computerized environment based on log data collected during test administration.

In this study, different versions with different levels of item complexity were used (Greiff et al., 2013 ), which varied by school grade (Table ​ (Table2; 2 ; six MicroDYN scenarios were administered in total in Grades 3–4; eight in Grade 5: nine in Grades 6–8; and twelve in Grades 9–12); however, we only involved those six tasks where the principle of isolated variation was the optimal exploration strategy. That is, we excluded problems with an external manipulation-independent, internal dynamic effect or multiple dependence effect from the analyses, and there were no delayed or accumulating effects used in the problem environments created. Complexity was defined by the number of input and output variables and the number of relations based on Cognitive Load Theory (Sweller, 1994 ). “Findings show that increases in the number of relations that must be processed in parallel in reasoning tasks consistently lead to increases in task difficulty” (Beckmann and Goode, 2017 ).

The design of the whole study: the complexity of the systems administered and the structure and anchoring of the tests applied in different grades.

The tasks were designed so that all causal relations could be identified with systematic manipulation of the inputs. The tasks contained up to three input variables and up to three output variables with different fictitious cover stories. The values of the input variables were changed by clicking on a button with a + or – sign or by using a slider connected to the respective variable (see Figure ​ Figure1). 1 ). The controllers of the input variables range from “– –” (value = −2) to “++” (value = +2). The history of the values of the input variables within the same scenario was presented on a graph connected to each input variable. Beyond the input and output variables, each scenario contained a Help, Reset, Apply and Next button. The Reset button set the system back to its original status. The Apply button made it possible to test the effect of the currently set values of the input variables on the output variables, which appeared in the form of a diagram of each output variable. According to the user interface, within the same phase of each of the problem scenarios, the input values remained at the level at which they were set for the previous input until the Reset button was pressed or they were changed manually. The Next button implemented the navigation between the different MicroDYN scenarios and the different phases within a MicroDYN scenario.

Exploration in phase 1 of the MicroDYN problems (two input variables and two output variables).

In the knowledge acquisition phase, participants were freely able to change the values of the input variables and attempt as many trials for each MicroDYN scenario as they liked within 180 s. During this 180 s, they had to draw the concept map (or causal diagram; Beckmann et al., 2017 ); that is, they had to draw the arrows between the variables presented on the concept map under the MicroDYN scenario on screen. In the knowledge application phase, students had to check their respective system using the right concept map presented on screen by reaching the given target values within a given time frame (90 s) in no more than four trials, that is, with a maximum of four clicks on the Apply button. This applied equally to all participants.

All of the CPS problems were administered online via the eDia platform. At the beginning, participants were provided with instructions about the usage of the user interface, including a warm-up task. Subsequently, participants had to explore, describe and operate unfamiliar systems. The assessment took place in the schools' ICT labs using the available school infrastructure. The whole CPS test took approximately 45 min to complete. Testing sessions were supervised by teachers who had been thoroughly trained in test administration. Students' problem-solving performance in the knowledge acquisition and application phases was automatically scored as CPS performance indicators; thus, problem solvers received immediate performance feedback at the end of the testing session. We split the sample into three age groups, whose achievement differed significantly (Grades 3–5, N = 1,871; Grades 6–7, N = 1,284; Grades 8–12, N = 1,216; F = 122.56, p < 0.001; t level_1_2 = −6.22, p < 0.001; t level_2_3 = −8.92, p < 0.001). This grouping corresponds to the changes in the developmental curve relevant to complex problem solving. The most intensive development takes place in Grades 6–7 (see Molnár et al., 2013 ). Measurement invariance, that is, the issue of structural stability, has been demonstrated with regard to complex problem solving in the MicroDYN approach already (e.g., Greiff et al., 2013 ) and was confirmed in the present study (Table ​ (Table3). 3 ). Between group differences can be interpreted as true and not as psychometric differences in latent ability. The comparisons across grade levels are valid.

Goodness of fit indices for measurement invariance of MicroDYN problems.

χ 2 and df were estimated by the weighted least squares mean and variance adjusted estimator (WLSMV). Δχ 2 and Δdf were estimated by the Difference Test procedure in MPlus. Chi-square differences between models cannot be compared by subtracting χ 2 s and dfs if WLSMV estimators are used. CFI, comparative fit index; TLI, Tucker Lewis index; RMSEA, root mean square error of approximation .

The latent class analysis (Collins and Lanza, 2010 ) employed in this study seeks students whose problem-solving strategies show similar patterns. It is a probabilistic or model-based technique, which is a variant of the traditional cluster analysis (Tein et al., 2013 ). The indicator variables observed were re-coded strategy scores. Robust maximum likelihood estimation was used and two to seven cluster solutions were examined. The process of latent class analysis is similar to that of cluster analysis. Information theory methods, likelihood ratio statistical test methods and entropy-based criteria were used in reducing the number of latent classes. As a measure of the relative model fit, AIC (Akaike Information Criterion), which considers the number of model parameters, and BIC (Bayesian Information Criterion), which considers the number of parameters and the number of observations, are the two original and most commonly used information theory methods for model selection. The adjusted Bayesian Information Criterion (aBIC) is the sample size-adjusted BIC. Lower values indicated a better model fit for each criterion (see Dziak et al., 2012 ). Entropy represents the precision of the classification for individual cases. MPlus reports the relative entropy index of the model, which is a re-scaled version of entropy on a [0,1] scale. Values near one, indicating high certainty in classification, and values near zero, indicating low certainty, both point to a low level of homogeneity of the clusters. Finally, the Lo–Mendell–Rubin Adjusted Likelihood Ratio Test (Lo et al., 2001 ) was employed to compare the model containing n latent classes with that containing n −1 latent classes. A significant p -value ( p < 0.05) indicates that the n −1 model is rejected in favor of a model with n classes, as it fits better than the previous one (Muthén and Muthén, 2012 ).

As previous research has found (Greiff et al., 2013 ), achievement in the first and second phases of the problem-solving process can be directly linked to the concept of knowledge acquisition (representation) and knowledge application (generating a solution) and was scored dichotomously. For knowledge acquisition, students' responses were scored as correct (“1”) if the connections between the variables were accurately indicated on the concept map (students' drawings fully matched the underlying problem structure); otherwise, the response was scored as incorrect (“0”). For knowledge application, students' responses were scored as correct (“1”) if students reached the given target values within a given time frame and in no more than four steps, that is, with a maximum of four clicks on the Apply button; otherwise, the response was scored as incorrect (“0”).

We developed a labeling procedure to divide the continuum of the problem-solving process into more scoreable phases and to score students' activity and behavior in the exploration phase at the beginning of the problem-solving process. For the different analyses and the most effective clustering, we applied a categorization, distinguishing students' use of the full, basic and minimal input behavior within a single CPS task (detailed description see later). The unit of this labeling process was a trial, a setting of the input variables, which was tested by clicking on the Apply button during the exploration phase of a problem, thus between receiving the problem and clicking on the Next button to reach the second part, the application part of the problem. The sum of these trials, within the same problem environment is called the input behavior. The input behavior was called a strategy if it followed meaningful regularities.

By our definition, the full input behavior model describes what exactly was done throughout the exploration phase and what kinds of trials were employed in the problem-solving process. It consists of all the activities with the sliders and Apply buttons in the order they were executed during the first phase, the exploration phase of the problem-solving process. The basic input behavior is part of the full input behavior model by definition, when the order of the trials attempted was still being taken into account, but it only consists of activities where students were able to acquire new information on the system. This means that the following activities and trials were not included in the basic input behavior model (they were deleted from the full input behavior model to obtain the basic behavior model):

• - where the same scenario, the same slider adjustment, was employed earlier within the task (that is, we excluded the role of ad hoc control behavior from the analyses),
• - where the value (position) of more than one input variable (slider) was changed and where the effect of the input variable on the operation of the system was still theoretically unknown to the problem solver,
• - where a new setting or new slider adjustment was employed, though the effect of the input variables used was known from previous settings.
• - As the basic input behavior involves timing, that is, the order of the trials used, it is suitable for the analyses with regard to the awareness of the input behavior employed.

Finally, we generated the students' minimal input behavior model from the full input behavior model. By our definition, the minimal input behavior focuses on those untimed activities (a simple list, without the real order of the trials), where students were able to obtain new information from the system and were able to do so by employing the most effective trials.

Each of the activities in which the students engaged and each of the trials which they used were labeled according to the following labeling system to be able to define students' full input behavior in a systematic format (please note that the numerical labels are neither scores nor ordinal or metric information):

• Only one single input variable was manipulated, whose relationship to the output variables was unknown (we considered a relationship unknown if its effect cannot be known from previous settings), while the other variables were set at a neutral value like zero. We labeled this trial +1.
• One single input variable was changed, whose relationship to the output variables was unknown. The others were not at zero, but at a setting used earlier. We labeled this trial +2.
• One single input variable was changed, whose relationship to the output variables was unknown, and the others were not at zero; however, the effect of the other input variable(s) was known from earlier settings. Even so, this combination was not attempted earlier. We labeled this trial +3.
• Everything was maintained in a neutral (zero) position. This trial is especially important for CPS problems with their own internal dynamics. We labeled this +A.
• The value of more than one input variable, whose relationship to the output variables was unknown, was changed at the same time, resulting in no additional information on the system. It was labeled –X.
• The same trial, the slider adjustment, had already been employed earlier within the task, resulting in no additional information on the system. It was labeled −0.
• A new slider adjustment was employed; however, the effect of the manipulated input variables was known from previous settings. This trial offered no additional information on the system and was labeled +0.

Although several input variables were changed by the scenario, it was theoretically possible to count the effect of the input variables on the output variables based on the information from the previous and present settings by using and solving linear equations. It was labeled +4.

An extra code (+5) was employed in the labeling process, but only for the basic input behavior, when the problem solver was able to figure out the structure of the problem based on the information obtained in the last trial used. This labeling has no meaning in the case of the minimal input behavior.

The full, basic and minimal input behavior models as well as the labeling procedure can be employed by analyzing problem solvers' exploration behavior and strategies for problems that are based on minimal complex systems. The user interface can preserve previous input values, and the values are not reset to zero after each exploration input. According to Fischer et al. ( 2012 ), VOTAT strategies are best for identifying causal relations between variables and they maximize the successful strategic behavior in minimal complex systems, such as CPS. By using a VOTAT strategy, the problem solver systematically varies only one input variable, while the others remain unchanged. This way, the effect of the changed variable can be found in the system by monitoring the changes in the output variables. There exist several types of VOTAT strategies based on the different combinations of VOTAT-centered trials +1, +2, and +3. The most obvious systematic strategy is when only one input variable is different from the neutral level in each trial and all the other input variables are systematically maintained at the neutral level. Thus, the strategy is a combination of so-called +1 trials, where it is employed for every input variable. Known as the isolated variation strategy (Müller et al., 2013 ), this strategy has been covered extensively in the literature. It must be noted that the isolated variation strategy is not appropriate to detect multiple dependence effects within the MicroDYN approach. We hypothesize that there are more and less successful input behaviors and strategies. We expect that theoretically effective, non-VOTAT strategies do not work as successfully as VOTAT strategies and that the most effective VOTAT strategy will be the isolated variation strategy.

We will illustrate the labeling and coding process and the course of generating a minimal input behavior out of a basic or full input behavior through the following two examples.

Figure ​ Figure1 1 shows an example with two input variables and two output variables. (The word problem reads as follows: “When you get home in the evening, there is a cat lying on your doorstep. It is exhausted and can barely move. You decide to feed it, and a neighbor gives you two kinds of cat food, Miaow and Catnip. Figure out how Miaow and Catnip impact activity and purring.”). The student who mapped the operation of the system as demonstrated in the figure pressed the Apply button six times in all, using the various settings for the Miaow and Catnip input variables.

In mapping the system, the problem solver kept the value of both the input variables at 0 in the first two steps (making no changes to the base values of the input variables), as a result of which the values of the output variables remained unchanged. In steps 3 and 4, he set the value of the Miaow input variable at 2, while the value of the Catnip variable remained at 0 (the bar chart by the name of each variable shows the history of these settings). Even making this change had no effect on the values of the output variables; that is, the values in each graph by the purring and activity variables are constantly horizontal. In steps 5 and 6, the student left the value of the Miaow input variable at 2, but a value of 2 was added to this for the Catnip input variable. As a result, the values of both output variables (purring and activity) began to grow by the same amount. The coding containing all the information (the full input behavior) for this sequence of steps was as follows: +A, −0, +1, −0, +2, −0. The reason for this is since steps 2, 4, and 6 were repetitions of previous combinations, we coded them as −0. Step 3 involved the purest use of a VOTAT strategy [changing the value of one input variable at a time, while keeping the values of the other input values at a neutral level (+1)], while the trial used in step 5 was also a VOTAT strategy. After all, only the value of one input variable changed compared to step 4. This is therefore not the same trial as we described in step 3 (+2). After step 5, all the necessary information was available to the problem solver. The basic input behavior for the same sequence of steps was +A, +1, +2, since the rest of the steps did not lead the problem solver to acquire unknown information. Independently of the time factor, the minimal input behavior in this case was also +A, +1, +2. The test taker was able to access new information on the operation of the system through these steps. From the point of view of awareness, this +1+2 strategy falls under aware strategy usage, as the +1 and +2 sub-strategies were not applied far apart (excluding the simple repetition of the executed trials next to each other) from each other in time. A good indicator of aware strategy usage is if there is no difference between minimal and basic input behavior.

In the second example (Figure ​ (Figure2), 2 ), we demonstrate the sequence of steps taken in mapping another problem as well as the coding we used. Here the students needed to solve a problem consisting of two input variables and one output variable. The word problem reads as follows: “Your mother has bought two new kinds of fruit drink mix. You want to make yourself a fruit drink with them. Figure out how the green and blue powders impact the sweetness of the drink. Plot your assumptions in the model.” The test taker attempted eight different trials in solving this problem, which were coded as follows: +1, +2, +0, +0, +0, +0, −0, −0. After step 2, the student had access to practically all the information required to plot the causal diagram. (In step 1, the problem solver checked the impact of one scoop of green powder and left the quantity of blue powder at zero. Once mixed, the resultant fruit drink became sweeter. In step 2, the problem solver likewise measured out one scoop of green powder for the drink but also added a scoop of blue powder. The sweetness of the drink changed as much as it had in step 1. After that, the student measured out various quantities of blue and then green powder, and looked at the impact.) The basic input behavior coded from the full input behavior used by the problem solver was +1+2, and the minimal input behavior was +1+1 because the purest VOTAT strategy was used in steps 1 and 6. (Thus, both variables separately confirmed the effects of the blue and the green powder on the sweetness of the drink.) From the point of view of awareness, this +1+1 strategy falls under non-aware strategy usage, as the two applications of the +1 trial occurred far apart from each other in time.

Exploration in phase 1 of the problems based on minimal complex systems (two input variables and one output variable).

Based on students' minimal input behavior we executed latent class analyses. We narrowed the focus to the principle of isolated variation, especially to the extent to which this special strategy was employed in the exploration phase as an indicator of students' ability to proficiently explore the problem environment. We added an extra variable to each of the problems, describing students' exploration behavior based on the following three categories: (1) no isolated variation at all (e.g., isolated variation was employed for none of the input variables – 0 points); (2) partially isolated variation (e.g., isolated variation was employed for some but not all the input variables – 1 point); and (3) fully isolated variation (e.g., isolated variation was employed for all the input variables – 2 points). Thus, depending on the level of optimal exploration strategy used, all the students received new categorical scores based on their input exploration behavior, one for each of the CPS tasks. Let us return to the example provided in Figures ​ Figures1, 1 , ​ ,2. 2 . In the first example, a partially isolated strategy was applied, since the problem solver only used this strategy to test the effect of the Miaow input variables (in trials 3 and 4). In the second example, a full isolated strategy was applied, as the problem solver used this isolated variation strategy for both the input variables during the exploration phase in the first and sixth trials.

The reliability of the test improved when scoring was based on the log data

The reliability of the MicroDYN problems as a measure of knowledge acquisition and knowledge application, the traditional CPS indicators for phases 1 and 2, were acceptable at α = 0.72–0.86 in all grades (Table ​ (Table4). 4 ). After we re-scored the problem solvers' behavior at the beginning of the problem-solving process, coded the log data and assigned new variables for the effectiveness of strategy usage during the exploration phase of the task for each task and person, the overall reliability of the test scores improved. This phenomenon was noted in all grades and in both coding procedures, when the amount of information obtained was examined (Cronbach's α ranged from 0.86 to 0.96) and when the level of optimal exploration strategy used was analyzed (Cronbach's α ranged from 0.83 to 0.98; the answers to the warm-up tasks were excluded from these analyses).

Internal consistencies in scoring the MicroDYN problems: analyses based on both traditional CPS indicators and re-coded log data based on student behavior at the beginning of the problem-solving process.

Use of a theoretically effective strategy does not result in high performance (RQ1)

Use of a theoretically effective strategy did not always result in high performance. The percentage of effective strategy use and high CPS performance varied from 20 to 80%, depending on the complexity of the CPS tasks and the age group. The percentage of theoretically effective strategy use in each cohort increased by 20% for age when problems with the same complexity were compared (Table ​ (Table5) 5 ) and decreased about 20% for the increasing number of input variables in the problems.

Percentage of theoretically effective and non-effective strategy use and high CPS performance.

The percentage of theoretically effective strategy use was the same for the less complex problems in Grades 3–5 and for the most complex tasks in Grades 8–12 (58%). More than 80% of these students solved the problem correctly in the first case, but only 60% had the correct solution in the second case. There was a 50% probability of effective and non-effective strategy use for problems with two input and two output variables in Grades 3–5 and for problems with three input and three output variables in Grades 6–7. In Grades 8–12, the use of a theoretically effective strategy was always higher than 50%, independently of the complexity of the problems (with no internal dynamic). The guessing factor, that is, the ad hoc optimization (use of a theoretically non-effective strategy with the correct solution) also changed, mostly based on the complexity and position of the tasks in the test. The results confirmed our hypothesis that the use of a theoretically effective strategy does not necessary represent the correct solution and that the correct solution does not always represent the use of an even theoretically effective problem-solving strategy.

Not all the VOTAT strategies result in high CPS performance (RQ2)

On average, only 15% of the theoretically effective strategy uses involved non-VOTAT strategies. The isolated variation strategy comprised 45% of the VOTAT strategies employed. It was the only theoretically effective strategy which always resulted in the correct solution to the problem with higher probability independently of problem complexity or the grade of the students. The real advantage of this strategy was most remarkable in the case of the third cohort, where an average of 80% of the students who employed this strategy solved the problems correctly (Figures ​ (Figures3, 3 , ​ ,4 4 ).

Efficacy of the most frequently employed VOTAT strategies on problems with two input variables and one or two output variables in Grades 3–5, 6–7, and 8–12.

Efficacy of the most frequently employed VOTAT strategies on problems with three input variables and one or two output variables in Grades 3–5, 6–7, and 8–12.

The second most frequently employed and successful VOTAT strategy was the +1+2 type or the +1+2+2 type, depending on the number of input variables. In the +1+2 type, only one single input variable was manipulated in the first step, while the other variable remained at a neutral value; in the second step, only the other input variable was changed and the first retained the setting used previously. This proved to be relatively successful on problems with a low level of complexity independently of age, but it generally resulted in a good solution with a low level of probability on more complex problems.

VOTAT strategies of the +1+3 type (in the case of two input variables) and of the +1+1+2 type (in the case of three input variables) were employed even less frequently and with a lower level of efficacy than all the other VOTAT strategies (+1+1+3, +1+2+1, +1+2+2, +1+2+3, +1+3+1, +1+3+2 and +1+3+3 in the case of three input variables) and theoretically effective, non-VOTAT strategies (e.g., +4 in the case of two input variables or +1+4, +4+2 and +4+3 in the case of three input variables). In the following, we provide an example of the +4+2 type, where the MicroDyn problem has three input variables (A, B, and C) and three output variables. In the first trial, the problem solver set the input variables to the following values: 0 (for variable A), 1 (for variable B), and 1 (for variable C); that is, he or she changed two input variables at the same time. In the second trial, he or she changed the value of two input variables at the same time again and applied the following setting: 0 (for variable A), −2 (for variable B), and −1 (for variable C). In the third trial, he set variable A to 1, and left variables B and C unchanged. That is, the problem solver's input behavior can be described with the following trials: –X +4 +2. Based on this strategy, it was possible to map the relationships between the input and output variables without using any VOTAT strategy in the exploration phase.

Aware explorers perform significantly higher on the CPS tasks (RQ3)

We compared the achievement of the aware, isolated strategy users with that of the non-aware explorers (Table ​ (Table6). 6 ). The percentage of high achievers among the non-aware explorers seemed to be almost independent of age, but strongly influenced by the complexity of the problem and the learning effect we noted in the testing procedure (see RQ5). Results for problems with two input variables and one output variable confirmed our previous results, which showed that the probability of providing the correct solution is very high even without aware use of a theoretically effective strategy (60–70%). With more complex problems, the difference between the percentages of aware and non-aware explorers was huge. Generally, 85% of the non-aware explorers failed on the problems, while at least 80% of the aware, isolated strategy users were able to solve the problems correctly.

Percentage of high achievers among aware and non-aware explorers by grade and problem complexity.

Six qualitatively different explorer class profiles can be distinguished at the end of the elementary level and five at the end of the secondary level (RQ4 and RQ5)

In all three cohorts, each of the information theory criteria used (AIC, BIC, and aBIC) indicated a continuous decrease in an increasing number of latent classes. The likelihood ratio statistical test (Lo–Mendell Rubin Adjusted Likelihood Ratio Test) showed the best model fit in Grades 3–5 for the 4-class model, in Grades 6–7 for the 6-class model and in Grades 8–12 for the 5-class model. The entropy-based criterion reached the maximum values for the 2- and 3-class solutions, but it was also high for the best-fitting models based on the information theory and likelihood ratio criteria. Thus, the entropy index for the 4-class model showed that 80% of the 3 rd - to 5 th -graders, 82% of the 6 th - to 7 th -graders and 85% of the 8 th - to 12 th -graders were accurately categorized based on their class membership (Table ​ (Table7 7 ).

Information theory, likelihood ratio and entropy-based fit indices for latent class analyses.

AIC, Akaike Information Criterion; BIC, Bayesian Information Criterion; aBIC, adjusted Bayesian Information Criterion; L–M–R test, Lo–Mendell–Rubin Adjusted Likelihood Ratio Test. The best fitting model solution is in italics .

We distinguished four latent classes in the lower grades based on the exploration strategy employed and the level of isolated variation strategy used (Table ​ (Table8): 8 ): 40.5% of the students proved to be non-performing explorers on the basis of their strategic patterns in the CPS environments. They did not use any isolated or partially isolated variation at all; 23.6% of the students were among the low-performing explorers who only rarely employed a fully or partially isolated variation strategy (with 0–20% probability on the less complex problems and 0–5% probability on the more complex problems). 24.7% of the 3 rd - to 5 th -graders were categorized as slow learners who were intermediate performers with regard to the efficiency of the exploration strategy they used on the easiest problems with a slow learning effect, but low-performing explorers on the complex ones. In addition, 11.1% of the students proved to be proficient explorers, who used the isolated or partially isolated variation strategy with 80–100% probability on all the proposed CPS problems.

Relative frequencies and average latent class probabilities across grade levels 3–5, 6–7, and 8–12.

In Grades 6–7, in which achievement proved to be significantly higher on average, 10% fewer students were observed in each of the first two classes (non-performing explorers and low-performing explorers). The percentage of intermediate explorers remained almost the same (26%), and we noted two more classes with the analyses: the class of rapid learners (4.4%) and that of slow learners, who are almost proficient explorers on the easiest problems, employing the fully or partially isolated variation strategy with 60–80% probability, but low-performing explorers on the complex ones (10.3%). The frequency of proficient strategy users was also increased (to 14.2%) compared to students in the lower grades. Finally, there was almost no change detected in the low-performing explorers' classes in Grades 8–12. We did not detect anyone in the class of intermediate explorers; they must have developed further and become (1) rapid learners (7.7%), (2) slow learners with almost high achievement with regard to the exploration strategy they used on the easiest problems, but low achievers on the complex ones (17.6%), or (3) proficient strategy users (26.3%), whose achievement was high both on the simplest and the most complex problems.

Based on these results, the percentage of non- and low explorers, who have very low exploration skills and do not learn during testing, decreased from almost 65 to 50% between the lower and higher primary school levels and then remained constant at the secondary level. There was a slight increase in respect of the percentage of students among the rapid learners. The students in that group used the fully or partially isolated strategy at very low levels at the beginning of the test, but they learned very quickly and detected these effective exploration strategies; thus, by the end of the test, their proficiency level with regard to exploration was equal to the top performers' achievement. However, we were unable to detect the class of rapid learners among 3 rd - to 5 th -graders.

Generally, students' level of exploration expertise with regard to fully and partially isolated variation improved significantly with age ( F = 70.376, p < 0.001). According to our expectations based on the achievement differences among students in Grades 3–5, 6–8 and 9–12, there were also significant differences in the level of expertise in fully or partially isolated strategy use during problem exploration between 3 rd - to 5 th - and 6 th - to 7 th -grade students ( t = −6.833, p < 0.001, d = 0.03) and between 6 th - to 7 th - and 8 th - to 12 th -grade students ( t = −6.993, p < 0.001, d = 0.03).

In this study, we examined 3 rd - to 12 th -grade (aged 9–18) students' problem-solving behavior by means of a logfile analysis to identify qualitatively different exploration strategies. Students' activity in the first phase of the problem-solving process was coded according to a mathematical model that was developed based on strategy effectiveness and then clustered for comparison. Reliability analyses of students' strategy use indicated that strategies used in the knowledge acquisition phase described students' development (ability level) better than traditional quantitative psychometric indicators, including the goodness of the model. The high reliability indices indicate that there are untapped possibilities in analyzing log data. Our analyses of logfiles extracted from a simulation-based assessment of problem solving have expanded the scope of previous studies and made it possible to identify a central component of children's scientific reasoning: the way students understand how scientific experiments can be designed and how causal relationships can be explored by systematically changing the values of (independent) variables and observing their impact on other (target) variables.

In this way, we have introduced a new labeling and scoring method that can be employed in addition to the two scores that have already been used in previous studies. We have found that using this scoring method (based on student strategy use) improves the reliability of the test. Further studies are needed to examine the validity of the scale based on this method and to determine what this scale really measures. We may assume that the general idea of varying the values of the independent variables and connecting them to the resultant changes in the target variable is the essence of scientific reasoning and that the systematic manipulation of variables is related to combinatorial reasoning, while summarizing one's observations and plotting a model is linked to rule induction. Such further studies have to place CPS testing in the context of other cognitive tests and may contribute to efforts to determine the place of CPS in a system of cognitive abilities (see e.g., Wüstenberg et al., 2012 ).

We have found that the use of a theoretically effective strategy does not always result in high performance. This is not surprising, and it confirms research results by de Jong and van Joolingen ( 1998 ), who argue that learners often have trouble interpreting data. As we observed earlier, using a systematic strategy requires combinatorial thinking, while drawing a conclusion from one's observations requires rule induction (inductive reasoning). Students showing systematic strategies but failing to solve the problem may possess combinatorial skills but lack the necessary level of inductive reasoning. It is more difficult to find an explanation for the other direction of discrepancy, when students actually solve the problem without an effective (complete) strategy. Thus, solving the problem does not require the use of a strategy which provides the problem solver with sufficient information about the problem environment to be able to form the correct solution. This finding is similar to results from previous research (e.g., Vollmeyer et al., 1996 ; Greiff et al., 2015 ). Goode and Beckmann ( 2010 ) reported two qualitatively different, but equally effective approaches: knowledge- based and ad hoc control.

In the present study, the contents of the problems were not based on real knowledge, and the causal relationships between the variables were artificial. Content knowledge was therefore no help to the students in filling the gap between the insufficient information acquired from interaction and the successful solution to the problem. We may assume that students guessed intuitively in such a case. Further studies may ascertain how students guess in such situations.

The percentage of success is influenced by the complexity of the CPS tasks, the type of theoretically effective strategy used, the age group and, finally, the degree to which the strategy was consciously employed.

The most frequently employed effective strategies fell within the class of VOTAT strategies. Almost half the VOTAT strategies were of the isolated variation strategy type, which resulted with higher probability in the correct solution independently of the complexity of the problem or the grade of the students. As noted earlier, not all the VOTAT strategies resulted in high CPS performance; moreover, all the other VOTAT strategies proved to be significantly less successful. Some of them worked with relative success on problems with a low level of complexity, but failed with a high level of probability on more complex problems independently of age group. Generally, the advantage of the isolated variation strategy (Wüstenberg et al., 2014 ) compared to the other VOTAT and non-VOTAT, theoretically effective strategies is clearly evident from the outcome. The use of the isolated variation strategy, where students examined the effect of the input variables on the output variables independently, resulted in a good solution with the highest probability and proved to be the most effective VOTAT strategy independently of student age or problem complexity.

Besides the type of strategy used, awareness also played an influential role. Aware VOTAT strategy users proved to be the most successful explorers. They were followed in effectiveness by non-aware VOTAT strategy users and theoretically effective, but non-VOTAT strategy users. They managed to represent the information that they had obtained from the system more effectively and made good decisions in the problem-solving process compared to their peers.

We noted both qualitative and quantitative changes of problem-solving behavior in the age range under examination. Using latent class analyses, we identified six qualitatively different class profiles during compulsory schooling. (1) Non-performing and (2) low-performing students who usually employed no fully or partially isolated variation strategy at all or, if so, then rarely. They basically demonstrated unsystematic exploration behavior. (3) Proficient strategy users who consistently employed optimal exploration strategies from the very first problem as well as the isolated variation strategy and the partially isolated variation, but only seldom. They must have more elaborated schemas available. (4) Slow learners who are intermediate performers on the easiest problems, but low performers on the complex ones or (5) high performers on the easiest problems, but low performers on the complex ones. Most members of this group managed to employ the principle of isolated or partially isolated variation and had an understanding of it, but they were only able to use it on the easiest task and then showed a rapid decline on the more complex CPS problems. They might have been cognitively overloaded by the increasingly difficult problem-solving environments they faced. (6) Rapid learners, a very small group from an educational point of view. These students started out as non-performers in their exploration behavior on the first CPS tasks, showed a rapid learning curve afterwards and began to use the partially isolated variation strategy increasingly and then the fully isolated variation strategy. By the end of the test, they reached the same high level of exploration behavior as the proficient explorers. We observed no so-called intermediate strategy users, i.e., those who used the partially isolated variation strategy almost exclusively on the test. As we expected, class membership increased significantly in the more proficient classes at the higher grade levels due to the effects of cognitive maturation and schooling, but this did not change noticeably in the two lowest-level classes.

Limitations of the study include the low sample size for secondary school students; further, repetition is required for validation. The generalizability of the results is also limited by the effects of semantic embedding (i.e., cover stories and variable labels), that is, the usage of different fictitious cover stories “with the intention of minimizing the uncontrollable effects of prior knowledge, beliefs or suppositions” (Beckmann and Goode, 2017 ). An assumption triggered by semantic contexts has an impact on exploration behavior (e.g., the range of interventions, or strategies employed by the problem solver; Beckmann and Goode, 2014 ), that is, how the problem solver interacts with the system. Limitations also include the characteristics of the interface used. In our view, analyses with regard to VOTAT strategies are only meaningful in systems with an interface where inputs do not automatically reset to zero from one input to the next (Beckmann and Goode, 2017 ). That is, we excluded problem environments from the study where the inputs automatically reset to zero from one input to the next. A further limitation of the generalizability of the results is that we have omitted problems with autonomic changes from the analyses.

The main reason why we have excluded systems that contain autoregressive dependencies from the analyses is that different strategy usage is required on problems which also involve the use of trial +A (according to our coding of sub-strategies), which is not among the effective sub-strategies for problems without autonomic changes. Analyses of students' behavior on problems with autonomic changes will form part of further studies, as well as a refinement of the definition of what makes a problem complex and difficult. We plan to adapt the Person, Task and Situation framework published by Beckmann and Goode ( 2017 ). The role of ad hoc control behavior was excluded from the analyses; further studies are required to ascertain the importance of the repetitive control behavior. Another limitation of the study could be the interpretation of the differences across age group clusters as indicators of development and not as a lack of stability of the model employed.

These results shed new light on and provide a new interpretation of previous analyses of complex problem solving in the MicroDYN approach. They also highlight the importance of explicit enhancement of problem-solving skills and problem-solving strategies as a tool for applying knowledge in a new context during school lessons.

Ethics statement

Ethical approval was not required for this study based on national and institutional guidelines. The assessments which provided data for this study were integrated parts of the educational processes of the participating schools. The coding system for the online platform masked students' identity; the data cannot be connected to the students. The results from the no-stakes diagnostic assessments were disclosed only to the participating students (as immediate feedback) and to their teachers. Because of the anonymity and no-stakes testing design of the assessment process, it was not required or possible to request and obtain written informed parental consent from the participants.

Author contributions

Both the authors, GM and BC, certify that they have participated sufficiently in the work to take responsibility for the content, including participation in the concept, design and analysis as well as the writing and final approval of the manuscript. Each author agrees to be accountable for all aspects of the work.

Conflict of interest statement

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

1 With regard to terminology, please note that different terms are used for the subject at hand (e.g., complex problem solving, dynamic problem solving, interactive problem solving and creative problem solving). In this paper, we use the modifier “complex” (see Csapó and Funke, 2017 ; Dörner and Funke, 2017 ).

Funding. This study was funded by OTKA K115497.

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Helping with Math

Multi-Step Math Word Problems

What to expect in this article.

After reading this article, you will be able to analyze, process, and solve multi-step word problems . This lesson will provide help and guidance in solving these types of problems as it includes tips on how to solve a multi-step problem . There are two given examples wherein you can practice and guide your children in honing their mathematical skills. You can also read the common errors and misconceptions of students in solving multi-step problems. Furthermore, this article consists of links directed to worksheets – which you can find at the bottom of the page. 

What is a multi-step word problem?

Math word problems are a critical component of the mathematics curriculum because they help students develop their mental abilities , improve logical analysis , and stimulate creative thinking . Word problems are fun and challenging to solve because they represent actual situations that happen in our world. More so, having the ability to solve math word problems significantly benefits one’s career and personal life.

To be able to solve any math word problem , children must be familiar with the mathematics language associated with the mathematical symbols they are accustomed in order to comprehend the word problem.

A multi-step math word problem is a type of problem wherein you need to solve one or more problems first in order to get the necessary information to solve the question being asked. It usually involves multiple operations and may also involve more than one strand of the curriculum. Say, for example, a multi-step word problem involving area and perimeter may also require the application of ratio and multiplication .

How to solve multi-step word problems?

In any word problem, the true challenge is deciding which mathematical operation to use. In solving multi-step word problems, there may be two or more operations that you need to work on, and you must solve them in the correct order to be able to get the correct answer. Since word problems describe a real situation in detail, the question being asked can get lost in all the information, especially in a multi-step problem.

To solve multi-step word problems, you may follow these strategy:

• Analyze and understand the problem. 
• Break down each sentence of the problem and identify the clues.
• List all the information.
• Identify the unknown in the problem.
• Devise a plan or identify the mathematical operations you are going to use.
• Carry out the plan.
• Label your final answer.

Multi-Step Word Problem #1

Step 1: Break down each sentence of the problem and identify the information needed to solve the problem.

• The first sentence states that “Steven is reading a book that has 260 pages.” Hence, the total number of pages of that particular book is 260 .
• The second statement says, “He read 35 pages on Monday night and 40 pages on Tuesday night.” 

Step 2: Analyze the question of the problem and find the keyword for the unknown. The last sentence of the problem, “How many pages does he has left to read?” asks us how many more pages Steven needs to read. Hence, we are going to find the number of pages he still needs to read.

Step 3: Based on the second statement, Steven read 35 pages on a Monday night and 45 pages on a Tuesday night. Hence, we will use addition in getting the total number of pages he read for 2 nights. Thus, 

35 + 40 = 75

Therefore, Steven read 75 pages in the span of two days. However, that is not the answer we are looking for. 

Step 4: Since we are asked to get the number of pages he still needs to read, the first sentence on our problem shows us that there are 260 pages in the book. Hence, we need to subtract the number of pages Steven has read from the total number of pages of the book. Thus,

260 – 75 = 185

Therefore, Steven has 185 pages left to read.

Multi-Step Word Problem #2

Jesy bought a dozen of boxes, each containing 24 highlighter pens inside. Each box costs \$8. Jesy repacked five of these boxes into packages of six highlighters each and sold them for \$3 per package. She sold the rest of the highlighters at the price of three pens for \$2. How much profit did Jesy make?

• The statement, “Jesy bought a dozen of boxes , each containing 24 highlighter pens inside,” tells us that there are a dozen of boxes that contains 24 highlighters. A dozen means there are 12 boxes . 
• The second sentence, “Each box costs \$8”, means Jesy bought 12 boxes at \$8 each . 
• “Jesy repacked five of these boxes into packages of six highlighters each and sold them for \ $3 per package ” means that Jesy separated 5 boxes from the original 12 boxes to be repacked at a package of six, which was sold at \$3 each. 
• “She sold the rest of the highlighters at the price of three pens for \$2 ” means that Jesy sold the remaining highlighters and bundled it for 3 pens for \$2.

Step 2: Analyze the question of the problem and find the keyword. The last sentence of the problem, “How much profit did Jesy make?” asks us how much profit Jesy earned after repacking the highlighter pens. Profit is defined as the amount earned minus the amount spent to buy the highlighters. 

Step 3: Based on the first statement, Jesy bought 12 boxes containing 24 highlighters. The follow-up statement that says, “Each box costs \$8” refers to the price of each box. In this particular statement, we can find the total expenditures of Jesy for the highlighter pens by simply multiplying the total number of boxes to \$8. Hence, 

12 x \$8 = \$96

This means that Jesy spent \$96 to buy all the highlighters. However, that is not the question being asked. Hence, we need to work on the follow-up statements and find more clues to get Jesy’s profit in selling highlighters. 

Step 4: The next statement says that “Jesy repacked five of these boxes into packages of six highlighters each and sold them for \$3 per package” means that Jesy separated 5 boxes from the dozen to be repacked at a package of six, which was sold at \$3 each. Based on this statement, we need to do three things:

• Find the total number of highlighters she got from separating 5 boxes;
• Find the total number of packages she made by repacking it by 6; and
• Find how much money she made by selling the sets of 6 at \$3.  

Step 5: To find the total number of highlighters she got from separating 5 boxes, we simply multiply 5 by the number of highlighters inside the box. Based on the first statement, each box contains 24 highlighters. Hence,

5 x 24 = 120

This means Jesy repacked a total of 120 highlighter pens.

Step 6: To find the total number of packages she made by repacking 120 highlighter pens by 6, we will divide 120 by 6. Thus,

So, she was able to make 20 sets of 6 highlighter pens. 

Step 7: The next thing we need to do is find how much money she made by selling the sets of 6 by \$3. This can be done by multiplying 20 sets by \$3. Hence,

20 x \$3 = \$60

Thus, Jesy made \$60 from the 20 sets of 6 highlighter pens.

Step 8: The third sentence, “She sold the rest of the highlighters at the price of three pens for \$2” means that Jesy sold the remaining highlighters and bundled it for 3 pens for \$2. From this statement, we need to work on four things first:

• Find the remaining number of boxes; 
• Find the total number of highlighter pens she repacked;
• Find the number of sets she repacked by making sets of 3; and
• Find how much money Jesy made by selling packs of 3 at \$2. 

Step 9: To find the remaining number of boxes, we need to go through some of the problem statements. Based on the first statement, we have 12 boxes, then 5 boxes were separated to make a highlighter set of 6. Hence, we will subtract 5 from 12. 

So, we still have 7 remaining boxes.

Step 10: To find the total number of highlighter pens she repacked, we need to multiply the remaining 7 boxes to the number of highlighter pens inside the box. Going back to the information we already have, we know that there are 24 highlighter pens inside a box. Thus, 

7 x 24 = 168

This means Jesy repacked a total of 168 highlighter pens. 

Step 11: Find the number of sets Jesy made by repacking 168 highlighter pens by 3. This can be done by dividing 168 by 3. Hence,

Thus, Jesy was able to make 56 sets of 3 highlighter pens. 

Step 12: Determine how much money Jesy made by selling each set for \$2. Hence, 

56 x \$2 = \$112

This means Jesy made \$112 by selling 3 highlighter pens for \$2. 

Step 13: The question asks us to determine the profit Jesy made by selling the highlighter pens. In order to find the profit, we need the information of:

• How much did Jesy spend on the highlighter. In Step 3, we found out that she paid \$96 on buying all the highlighter pens. 
• How much money does Jesy make on selling packs of 6 highlighters for \$3. In Step 7, we already know that she made \$60; and
• How much money does Jesy make on selling sets of 3 highlighter pens for \$2. In Step 12, we found out that she made \$112.

Step 14: Before getting the profit Jesy made, we need to know the total money Jesy made in selling the highlighters. Hence, we will simply add the money of \$60 and \$112. Thus,

\$60 + \$112 = \$172

However, \$172 is not the profit Jesy made. This is just the money she was able to make in selling the highlighter pens.

Step 15: Lastly, we will subtract the money Jesy spent on buying the highlighters from the money she made by selling it to find the profit. Thus, 

\ $172 – \$96 = \$76

Therefore, Jesy made a profit of \$76 by selling the highlighter pens.

You can tell that there are lots of things to remember with a multi-step word problem, even when the problem itself is relatively easy. But that’s what makes these problems challenging: you get to use both sides of your brain – your logical math skills and your verbal language skills. That’s why they are often more fun to do than problems that are just numbers without the details and context that word problems give you. The better you understand how to solve them, the more fun they are to solve. 

What are the common errors in solving multi-step word problems?

Mathematical word problems can be challenging to solve. To obtain the correct answer, children must read the words and carefully analyze the problem, determine the appropriate math operation, and then perform the calculations correctly. An error in working on one of the steps may result in a wrong answer. 

Here’s a list of some errors students make when solving multi-step word problems:

• The most common error of students is stopping at one process if they solve one problem. Consider the same word problem about Steven. 

“Steven is reading a book that has 260 pages. He read 35 pages on Monday night and 40 pages on Tuesday night. How many pages does he has left to read?” 

Most students recognize that they need to add 35 and 40 together to get the total number of pages Steven has read so far. Most errors occur when students stop at one process. Adding 35 + 40 will tell you that Steven has read 75 pages so far, but if we go back to the question you are being asked, you will notice that 75 pages are not the answer you are being asked. Thus, we still need to take another step to get there. Steven has read 75 pages so far, but the questions asked us to solve the number of pages he has left to read. Hence, subtracting 75 from the total number of pages of the book makes much more sense. 

• Students get confused with the mathematical operation to use. Even if children are strong readers, they may struggle to pick up on clues in word problems. These clues are phrases that instruct children on how to solve a problem, such as adding or subtracting. The children are then required to convert these phrases into a number sentence in order to solve word problems.

How to teach multi-step problems to children?

There are certain activities or practices that you can try with your child in order to develop their skills in solving multi-step problems. 

• The first and most important skill in working with multi-step is being able to understand the problem clearly. Hence, practicing your child in slowly reading and visualizing problems is the first step in implementing our effective reading comprehension strategies.
• Practice your child in recognizing mathematics terms and vocabulary that are used in word problems. There are keywords or clues that we can easily spot in a word problem if we familiarize ourselves with these mathematical terms. 

Let’s look at the sample words related to addition, subtraction , multiplication, and division.

However, some English words can sometimes be confusing as they may mean differently depending on the context. 

Let’s look at the table below:

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Shape Partitions (Rectangles and Circles) 2nd Grade Math Worksheets

Understanding the Metric System Unit of Measurement for Length 2nd Grade Math Worksheets

Naming Numbers Up To 1000 2nd Grade Math Worksheets

Understanding Picture Graph and Bar Graph 2nd Grade Math Worksheets