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Differentiated Instruction

7 Strategies for Differentiated Math Instruction

Math classrooms are mosaics of strengths and experiences. When we have students with diverse backgrounds—with various languages, achievements, and interests—in the same space, everyone learns from each other and broadens their world.

On the flip side, though, teaching math to a broad array of students can be challenging. Do you struggle to reach all of your students? Are you a newer teacher who is looking to improve your practice? The strategies for differentiated instruction provided here might help you out.

What Is Differentiated Math Instruction?

Differentiated math instruction refers to the collection of techniques, strategies, and adaptations you can use to reach your diverse group of learners and make mathematics accessible to every single one. Dr. Timothy Kanold , former president of the National Council of Supervisors of Mathematics (NCSM)—and HMH author— clarifies that differentiation in a math lesson is “differentiation on the entry points into the task for support or the exit point to advance student thinking.”

By applying various tools and strategies, such as incorporating technology, assigning hands-on projects, and teaching in math small-group formats, you can help every student meet expectations. We know that there are different schools of thought regarding what differentiation means. When we use the term, we are talking about providing student choice, voice, and agency. Differentiating instruction isn’t meant to add more work to your day. Quite the opposite, in fact; it’s meant as a teaching approach that will help you to reach more students in terms of accessibility and equity, making your job both easier and more effective in the long run.

Why Is Differentiating Math Instruction Important?

Some people think that math, more than any other subject, is the best fit for differentiation. Even though a 2018 survey by Texas Instruments found that 46% of kids said they really liked math, there are hundreds of books, websites, and memes discussing the difficulty of the subject. From the anxiety caused by there being only one correct answer to the cultural buy-in to the myth of being—or not being—a “math person” to the fear of solving a word problem, many students struggle with math. In addition, many students and educators alike find it hard to make the connection between math and the real world, which only increases disillusionment with the subject. That’s why it’s especially important to be open to new ways of providing instruction .

The National Council of Teachers of Mathematics (NCTM) promotes differentiating math instruction for differences in learning as well as differences in achievement, interest, and confidence. NCTM advises that the need is greater in middle and high school, as higher-level math relies on more complex reasoning. When you differentiate your math instruction, you support all learners by targeting and addressing specific needs of groups and individual students.

Examples of Differentiated Instruction in Math

Do you need ideas for how to differentiate your teaching to be sure your math students are progressing? Below are seven differentiation strategies for math instruction, along with ways that you can use them in your math classroom. They serve as examples of differentiated instruction in math and may work better for some classrooms and math topics than others. Customize these ideas however you need to serve you and your students.

Strategy 1: Math Centers

For this, you’ll need to come up with a few activities your students can rotate through (be sure to browse our library of free activities and resources !), such as watching a video, reading an article, or solving a word problem. We spoke with Kristy McFarlane, an instructional supervisor at Sandshore Elementary School in New Jersey, about differentiation. She says math teachers at her school spend about 10 minutes on a mini-lesson for the whole class and then students spend about 15 minutes at various math centers. “They might meet with the teacher in a small group for extra help, use math software, do a game or project at the hands-on station, or do seat work based on the day’s mini-lesson,” she says.

Math centers are a powerful way to facilitate independent and small group learning within your classroom. Our Go Math! program, for example, is known for embedding resources and instructional time to math centers. If a select group of your students are all struggling to, say, add fractions, they may benefit from an activity that has them practice finding least common denominators. Think about ways to customize the groupings and centers so they’re perfect for your students’ strengths, misconceptions, and interests, and make use of tools that strategically group students and recommend activities for you.

Strategy 2: Activity Cards

Choice is an important part of differentiation, and letting students decide how they want to spend their time is a great way to appeal to various learning preferences. You’ll need to come up with math problems, tasks, or questions. As much as possible, use or create cards that span several lessons and offer options to work independently, with a partner, or in a small group. Ask for feedback so you can adjust future learning accordingly. Many of HMH’s math programs , including Into Math , Go Math! , and Into AGA include inquiry-based task and project cards that help teachers differentiate.

Strategy 3: Choice Boards

As we just mentioned, giving students the ability to make decisions about their learning is an important part of differentiation. A choice board is a graphic organizer that gives students activities to choose from. There are different types of choice boards, but they need to focus on specific learning needs, interests, and skills. Choice boards increase student ownership; students pace themselves and get to decide how to engage with information, along with how to demonstrate their learning. Some teachers create different versions of the same choice board; others will color-code options to signify topic, activity type, or expected level of challenge. Check out the choice board we developed for remote learning. This board covers all subjects but also includes a free template to get you started on a math-only version.

differentiate problem solving from drills

Strategy 4: Math Journals

Having students write about math is a great way for them to reflect on what they’ve learned and incorporate ELA instruction into the math classroom. Encourage your kids to summarize key points, answer open-ended questions, tie math into everyday experiences, or write about the most interesting or challenging math lesson. It’s also a way to provide an entry point for all students, including multilingual learners , as they can write a little or a lot in English or in their native language. Those who need extra support might be given sentence starters. Students might also be given the choice to illustrate their ideas instead of writing them. Similar to activity cards, math journals are included in many of HMH’s math programs, including Into Math and Into AGA .

Strategy 5: Learning Contracts

If metacognition is the ability to think about thinking—including about how you learn—we owe it to students to help them develop and expand their metacognitive skills. One way to do this is to work on learning contracts. Throughout the year, ask students to reflect on important lessons and set learning goals, including skills to learn or improve as well as new areas to explore. Use these learning contracts to help students learn to organize their thoughts. “One of our district’s goals is to have personalized learning opportunities for all students,” says McFarlane. “Each student creates a personalized success plan at the beginning of the year and does regular check-ins.” More broadly, metacognition is an idea that can be taught and practiced in the classroom and applies broadly to any subject.

$Strategies math differentiation girl thinking metacognition inline$

Strategy 6: Math Games

Games are fun, motivational, and can help students deepen their mathematical reasoning. Some games encourage students to develop strategic and problem-solving skills or improve computational fluency. Seek out games where the math learning objective matches the game objective as a way for students to find joy in learning. Go Math! was designed to include both ready-made games for math centers and recommended games for differentiation within the teacher’s edition.

You can also use non-math games to provide a short mental break or a context for having math discussions. Look for ways to turn the game into mathematical discourse. How could you have scored more points? How much time did it take? What strategies did you use?

Strategy 7: Digital Math Practice

There are also lots of math apps and online tools that are designed to reinforce foundational understanding by allowing students to practice arithmetic and other math standards. In particular, seek out apps that are not simply timed drills with fun graphics, which are likely to make math anxiety worse for students who are not yet fluent in math facts. For digital math practice that extends far beyond just practicing arithmetic, our newest Go Math! for Grades K–6 has adaptive and personalized practice that aligns to our supplemental practice program, Waggle .

to you, consider giving your students problem-solving tasks with open-ended solutions. A single math problem can reveal different ways that students think about mathematics, which might be a less time-consuming way to assess student progress and determine an effective way to differentiate.

When you think critically about how to transform math instruction into differentiated math instruction, students will be more engaged because the content will be more relevant. They will achieve more success because they’ll be experiencing different types of activities, using various modalities, and contributing to the best of their abilities as they continue to grow.

HMH offers a variety of math classroom solutions to help you reach every student. Just looking for more articles and resources to help you differentiate math instructions? Try one of these to keep reading!

This blog post, originally published in 2021, has been updated for 2022.

Wf926906 Hmh Diffin Classroom Middle School 2019

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Cultivating mathematical skills: from drill-and-practice to deliberate practice

Original Article
Published: 29 March 2017
Volume 49 , pages 625–636, ( 2017 )

Cite this article

Erno Lehtinen ORCID: orcid.org/0000-0001-6188-862X 1 ,
Minna Hannula-Sormunen 1 ,
Jake McMullen 1 &
Hans Gruber 1 , 2

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Contemporary theories of expertise development highlight the crucial role of deliberate practice in the development of high level performance. Deliberate practice is practice that intentionally aims at improving one’s skills and competencies. It is not a mechanical or repetitive process of making performance more fluid. Instead, it involves a great deal of thinking, problem solving, and reflection for analyzing, conceptualizing, and cultivating developing performance. This includes directing and guiding future training efforts that are then fine-tuned to dynamically evolving levels of performance. Expertise studies, particularly in music and sport, have described early forms of deliberate practice among children. These findings are made use of in our analysis of the various forms of practice in school mathematics. It is widely accepted that mathematics learning requires practice that results in effortless conducting of lower level processes (such as quick and accurate whole number arithmetic with small numbers), which relieve cognitive capacity for more complex tasks. However, the typical training of mathematical skills in educational contexts can be characterized as drill-and-practice that helps automatize basic skills, but often leads to inert routine skills instead of adaptive and flexible number knowledge. In this article we summarize findings of studies which describe students’ self-initiated, deliberate practice in learning number knowledge and intervention studies applying deliberate practice in mathematics teaching, including technology-based learning environments aimed at triggering practice that goes beyond mechanical repeating of number skills.

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Promoting Adaptive Number Knowledge Through Deliberate Practice in the Number Navigation Game

Given that the detailed original criteria for deliberate practice have not changed, could the understanding of this complex concept have improved over time? A response to Macnamara and Hambrick (2020)

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This research was supported by the Academy of Finland Grant 274163 to the first author.

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Lehtinen, E., Hannula-Sormunen, M., McMullen, J. et al. Cultivating mathematical skills: from drill-and-practice to deliberate practice. ZDM Mathematics Education 49 , 625–636 (2017). https://doi.org/10.1007/s11858-017-0856-6

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Differentiated Problem Solving: A New Approach

I know so many of you have been looking for a way to build deep math thinking with your intermediate students–I know this because I get questions about it all the time!

problem solving, differentiation, addition, subtraction, word problems, math enrichment, math workshop, math stations, guided math, third grade, fourth grade, fifth grade, tiered math, tiered problem solving, teaching resources

You want your students to be challenged in new and interesting ways—and be easily able to differentiate so that ALL your students can benefit, right? Here’s the problem…what’s challenging for some is way too tricky for others, right? Or you want to give a task–and you do–and then 1/3 of your class is finished in minutes and asking for more while others have barely gotten started. (Tell me that I’m not the only one this happens to!)

I often get people asking me what a “typical” math class looks like in my room–and I have to be honest. THere is no such thing. I feel like I operate on a menu system…I have all these “tasty” things and I serve them up when I think it makes the most sense! That being said, here are a few suggestions for working these quality tasks into your day. *Use as a “bell ringer” or warm up task. The goal of this should not be getting a correct answer, but the actual WORK of doing the problem solving! Each question has a starting point which can be used whole-class (you choose how much modeling/help) and then parts 2 and 3 can be used for everyone—or just for students who are ready! The colored slides are perfect to project from your computer…you can click to the next slides to show parts 2 and 3…but if students aren’t ready, no big deal! The original problem appears on every slide! *Print and laminate and use as task cards at a problem solving station. These half-page cards are low ink and are perfect for math rotations, math workshop, guided math, or for fast finishers. Whether you do rotations or organize your stations differently, having quality problems ready to go really saves your time. *Use as a reproducible problem-solving journal. I typed these problems up in a journal format with a full page of work space for the first part of the problem–with parts 2 and 3 copied on the next page. Copying the entire journal only takes 12 pieces of paper (without the cover) and is full of the 36 tasks. These can be used in so many ways—and even flexibly within a given classroom. I have some students who only do part of the collections–and others who might have the time and motivation to do much, much more. *Consider using open-ended tasks as enrichment opportunities for students needing just a bit more. This is perfect–whether you have one student or a handful. They can work together, practice that accountable math talk, and push each other.

Problem solving is not easy!

Like many of my resources, this set of problems is certainly not meant to be a time filler! It is meant to be a rich and meaningful problem solving experience for you to use with your students. HOW you use it is up to you! I know we are all busy…but the time we invest in modeling some of the thinking and strategies needed with this type of problem REALLY pays off in the long run as students become more and more independent.

When I use tasks like this with MY students, there are a few things I like to make very clear and I think really contribute to building a culture for problem solving. One of the most important things that I think teachers need to keep in mind is that we often “overteach”. We TELL too much. We push our own strategies and ideas onto them–even if they aren’t quite ready for them. Before I used this problem with my students, I thought about how I would solve this as an adult–and then thought about what might get students off track. In this case…I did NOT want to show students my “boxes” (although for a few students I did coach them in this direction after they had worked awhile), but I DID want to make sure students understood the task–and the terms (like digits) so that they didn’t waste time. I didn’t TELL them what the task was…but students worked in pairs to make sense of it, then we came back as a whole group to discuss it and get on the same page.

Push Yourself!

Interested in checking out this set of problems? Just click HERE or the image below.

Want to pin this for later? Here you go!

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The Differences between Problem-Based and Drill and Practice Games on Motivations to Learn

Problem-Based education has been put forward as the most fruitful approach when it comes to serious game design (Aldrich, 2009; Gee, 2005). In Problem-Based learning, students start with a problem. This problem is rather loosely defined as something ‘for which an individual lacks a ready response’ (Hallinger, 1992, p. 27). Problem-Based education distinguishes between well- and ill-defined problems. Ill-defined problems are those ‘in which one or several aspects of the situation is not well specified, the goals are unclear, and there is insufficient information to solve them’ (Ge & Land, p5 in Ertmer et al., 2008). Shaffer’s (Shaffer, Squire, Halverson, & Gee, 2005; Shaffer, 2008) suggestion for epistemic games, in which players adopt the perspective of a professional to confront complex problems in simulation-like game, aligns with Problem-Based learning approach.

Drill & Practice learning teaches the ‘what’ and the ‘when’, but not the ‘why’ and the ‘how’. Ke (2008) suggests that students in Drill & Practice Learning merely memorize facts. As a result, this kind of learning may not facilitate creative thought or stimulate problem-solving skills. Or, as Reeve et al. (2004) state, it may not present students with the opportunity to experiment, explore and struggle with the learning content to find the truth for themselves. Games such as Math Gran Prix (Atari Inc., 1982), Math Blaster (Davidson & Associates, 1994), and Dr. Kawashima’s Brain Training (Nintendo SDD, 2005) align with the Drill & Practice learning. In these games, there is only one solution to a mathematical challenge, and players are prompted to input the correct one.

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1.2: Problem or Exercise?

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Michelle Manes
University of Hawaii

The main activity of mathematics is solving problems. However, what most people experience in most mathematics classrooms is practice exercises. An exercise is different from a problem.

In a problem , you probably don’t know at first how to approach solving it. You don’t know what mathematical ideas might be used in the solution. Part of solving a problem is understanding what is being asked, and knowing what a solution should look like. Problems often involve false starts, making mistakes, and lots of scratch paper!

In an exercise , you are often practicing a skill. You may have seen a teacher demonstrate a technique, or you may have read a worked example in the book. You then practice on very similar assignments, with the goal of mastering that skill.

What is a problem for some people may be an exercise for other people who have more background knowledge! For a young student just learning addition, this might be a problem:

\[\textit{Fill in the blank to make a true statement} \: \_\_\_ + 4 = 7 \ldotp \nonumber \]

But for you, that is an exercise!

Both problems and exercises are important in mathematics learning. But we should never forget that the ultimate goal is to develop more and better skills (through exercises) so that we can solve harder and more interesting problems.

Learning math is a bit like learning to play a sport. You can practice a lot of skills:

hitting hundreds of forehands in tennis so that you can place them in a particular spot in the court,
breaking down strokes into the component pieces in swimming so that each part of the stroke is more efficient,
keeping control of the ball while making quick turns in soccer,
shooting free throws in basketball,
catching high fly balls in baseball,

But the point of the sport is to play the game. You practice the skills so that you are better at playing the game. In mathematics, solving problems is playing the game!

On Your Own

For each question below, decide if it is a problem or an exercise . (You do not need to solve the problems! Just decide which category it fits for you.) After you have labeled each one, compare your answers with a partner.

This clock has been broken into three pieces. If you add the numbers in each piece, the sums are consecutive numbers.(Note: Consecutive numbers are whole numbers that appear one after the other, such as 1, 2, 3, 4 or 13, 14, 15. )

$index-12_1.png$

Can you break another clock into a different number of pieces so that the sums are consecutive numbers? Assume that each piece has at least two numbers and that no number is damaged (e.g. 12 isn’t split into two digits 1 and 2).

A soccer coach began the year with a $500 budget. By the end of December, the coach spent $450. How much money in the budget was not spent?
What is the product of 4,500 and 27?
Arrange the digits 1–6 into a “difference triangle” where each number in the row below is the difference of the two numbers above it.
Simplify the following expression: $$\frac{2 + 2(5^{3} - 4^{2})^{5} - 2^{2}}{2(5^{3} - 4^{2})} \ldotp$$
What is the sum of $\frac{5}{2}$ and $\frac{3}{13}$?
You have eight coins and a balance scale. The coins look alike, but one of them is a counterfeit. The counterfeit coin is lighter than the others. You may only use the balance scale two times. How can you find the counterfeit coin?

$index-19_1-300x275.png$

How many squares, of any possible size, are on a standard 8 × 8 chess board?
What number is 3 more than half of 20?
Find the largest eight-digit number made up of the digits 1, 1, 2, 2, 3, 3, 4, and 4 such that the 1’s are separated by one digit, the 2’s are separated by two digits, the 3’s by three digits, and the 4’s by four digits.

How Drill And Practice Teaching Method Helps In Learning Math?

Last Updated on October 3, 2023 by Editorial Team

Prepare to embark on an adventure filled with numerical wonders and problem-solving prowess. Imagine stepping into a vibrant math circus, where students become mathematical acrobats, defying the limits of their skills and soaring to new heights of knowledge.

In this realm, the drill and practice method emerges as a magical tool, helping students master mathematical concepts. With each repetition, numbers come alive, formulas dance, and equations find their harmonious balance. It’s a symphony of drills, where practice becomes the key to unlocking mathematical brilliance.

But wait, this method is more than just a routine—it’s a catalyst for growth. It transforms math into an exhilarating adventure, where students build fluency, sharpen problem-solving skills, and unlock the secrets of numerical wizardry. Through targeted practice, mathematical doors swing open, revealing the beauty and logic that lie within.

So, join us under the big top of mathematics, where the drill and practice teaching method reigns supreme. Let’s embark on this extraordinary journey, where numbers become our allies, and mathematical mastery becomes a thrilling feat. Get ready to unleash the magic of drill and practice, as we soar to mathematical greatness, one equation at a time!

Drill and practice teaching method: Navigating through the meaning

The drill and practice teaching method is an instructional approach that focuses on the repetitive practice of specific skills or knowledge. It involves presenting students with targeted exercises, problems, or tasks that require them to repeatedly apply and reinforce what they have learned. This method aims to strengthen and solidify understanding, enhance retention, and promote automaticity in the mastery of concepts.

The purpose of drill and practice is to provide students with ample opportunities to practice and internalize essential skills or knowledge. By engaging in repetitive exercises, students develop fluency, accuracy, and efficiency in applying the learned content. This effectiveness of drill and practice has proven successful in building foundational skills, such as basic math operations, vocabulary acquisition, spelling, grammar rules, and procedural knowledge.

“Unlocking the secrets of math: The power of an effective teaching method”

The drill and practice teaching method finds extensive use in mathematics education due to its ability to reinforce foundational skills and promote mastery of mathematical concepts. The main goal of this method is to help students develop proficiency and fluency with a particular skill or concept. Here are ten ways in which the drill and practice teaching method can help students in math:

1. Improves mastery and fluency:

Repetitive practice can help students develop automaticity and fluency with math skills and concepts, allowing them to apply them more quickly and accurately in real-world situations. When students have to think less about how to perform a math operation and can just do it automatically, they are more likely to make fewer mistakes and solve problems more efficiently.

2. Provides immediate feedback:

With the drill and practice method, students can see their progress as they work through practice problems and quizzes, which can help them identify areas where they need more practice.

3. Allows for individualized instruction:

As students become more proficient and fluent in math skills and concepts, they may feel more confident in their ability to solve problems and tackle new challenges. This can help them approach math with a positive attitude and feel more motivated to engage with the material.

4. Builds confidence:

Drill and practice activities can help students develop a strong understanding of number sense, including basic arithmetic operations (addition, subtraction, multiplication, and division). Regular practice with number facts and calculations improves fluency and efficiency.

5. Encourages self-directed learning:

Drill and practice exercises are effective for memorizing essential math facts, such as multiplication tables, division facts, and number patterns. Repetition aids in automatic recall, allowing students to solve problems more quickly and accurately.

6. Promotes efficient problem-solving:

Drill and practice can reinforce problem-solving strategies, such as using algorithms, applying formulas, or using logical reasoning. Regular practice helps students become more proficient in selecting appropriate strategies and applying them effectively.

7. Facilitates retention:

Drill and practice activities encourage mental math skills by challenging students to perform calculations mentally and make quick estimations. Regular practice enhances mental agility and computational fluency.

8. Enhances problem-solving skills:

Drill and practice provide students with opportunities to practice mathematical procedures, such as long division, fraction operations, decimal conversions, and geometric formulas. Repetitive practice helps students internalize these procedures and become more proficient in their application.

9. Increases motivation:

By engaging in drill and practice exercises, students can improve their accuracy in mathematical calculations, reducing errors and promoting precision in their work.

10. Can be used in a variety of settings:

Drill and practice activities can reinforce mathematical vocabulary and terminology, ensuring students have a solid understanding of mathematical language and can effectively communicate their ideas.

What does the research state?

Research on drill and practice as a teaching method has generally shown that it can be effective for improving performance on specific skills or concepts. Furthermore, research [ 1 ] has shown that drills and practice can be effective ways to teach basic math facts, spelling words, and other skills that require automaticity or quick recall.

However, the effectiveness of drill and practice may depend on the specific learning goals and objectives, as well as the needs and characteristics of the students. In some cases, drill and practice may be less effective for more complex skills or for students who are struggling to learn new material.

Research [ 2 ] has also shown that drill methods along with other methods such as concept-based learning, prove effective in enhancing students’ comprehension of concepts and fostering high-level thinking, promoting a deeper understanding of the subject matter, and facilitating critical analysis and synthesis of information. Thus, the drill and practice method is a great method when used with other approaches to learning maths.

The drill and practice teaching method can be helpful for improving math skills because it allows students to repeatedly practice specific math concepts or techniques. This can help students to become more proficient and accurate in their math skills, as they are able to develop muscle memory and automaticity through repetition.

Additionally, the drill and practice method can be useful for reinforcing previously learned material, helping students retain important math concepts over time. At the same time, educators and parents can go through certain examples , that can explain the idea better how to employ this teaching method.

Lehtinen, Erno & Hannula-Sormunen, Minna & McMullen, Jake & Gruber, Hans. (2017). Cultivating mathematical skills: from drill-and-practice to deliberate practice. ZDM. 49. 10.1007/s11858-017-0856-6.
Lufri, Fitri, R., & Yogica, R. (2018). Effectiveness of concept-based learning model, drawing and drill methods to improve student’s ability to understand concepts and high-level thinking in animal development course. Journal of Physics: Conference Series , 1116 , 052040. https://doi.org/10.1088/1742-6596/1116/5/052040

An engineer, Maths expert, Online Tutor and animal rights activist. In more than 5+ years of my online teaching experience, I closely worked with many students struggling with dyscalculia and dyslexia. With the years passing, I learned that not much effort being put into the awareness of this learning disorder. Students with dyscalculia often misunderstood for having just a simple math fear. This is still an underresearched and understudied subject. I am also the founder of Smartynote -‘The notepad app for dyslexia’,

Problem or Exercise?

The main activity of mathematics is solving problems. However, what most people experience in most mathematics classrooms is practice exercises. An exercise is different from a problem.

Note: What is a problem for some people may be an exercise for other people who have more background knowledge! For a young student just learning addition, this might be a problem:

$\underline{\qquad} + 4 = 7$

But for you, that is an exercise!

Learning math is a bit like learning to play a sport. You can practice a lot of skills:

hitting hundreds of forehands in tennis so that you can place them in a particular spot in the court,
breaking down strokes into the component pieces in swimming so that each part of the stroke is more efficient,
keeping control of the ball while making quick turns in soccer,
shooting free throws in basketball,
catching high fly balls in baseball,

But the point of the sport is to play the game. You practice the skills so that you are better at playing the game. In mathematics, solving problems is playing the game!

On Your Own

1. This clock has been broken into three pieces. If you add the numbers in each piece, the sums are consecutive numbers.(Note: Consecutive numbers are whole numbers that appear one after the other, such as 1, 2, 3, 4 or 13, 14, 15. )

2. A soccer coach began the year with a $500 budget. By the end of December, the coach spent $450. How much money in the budget was not spent?

3. What is the product of 4,500 and 27?

4. Arrange the digits 1–6 into a “difference triangle” where each number in the row below is the difference of the two numbers above it.

5. Simplify the following expression:

$\displaystyle \frac{2 + 2(5^3-4^2)^5 - 2^2}{2(5^3-4^2)}.$

7. You have eight coins and a balance scale. The coins look alike, but one of them is a counterfeit. The counterfeit coin is lighter than the others. You may only use the balance scale two times. How can you find the counterfeit coin?

8. How many squares, of any possible size, are on a standard 8 × 8 chess board?

9. What number is 3 more than half of 20?

10. Find the largest eight-digit number made up of the digits 1, 1, 2, 2, 3, 3, 4, and 4 such that the 1’s are separated by one digit, the 2’s are separated by two digits, the 3’s by three digits, and the 4’s by four digits.

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Site navigation, problem-solving vs. exercise solving.

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Problem-solving

Involves a process used to obtain a best answer to an unknown, subject to some constraints.
The situation is ill defined. There is no problem statement and there is some ambiguity in the information given. Students must define the problem themselves. Assumptions must be made regarding what is known and what needs to be found.
The context of the problem is brand new (i.e., the student has never encountered this situation before).
There is no explicit statement in the problem that tells the student what knowledge / technique / skill to use in order to solve the problem.
There may be more than one valid approach.
The algorithm for solving the problem is unclear.
Integration of knowledge from a variety of subjects may be necessary to address all aspects of the problem.
Requires strong oral / written communication skills to convey the essence of the problem and present the results.

Exercise Solving

Involves a process to obtain the one and only right answer for the data given.
The situation is well defined. There is an explicit problem statement with all the necessary information (known and unknown).
The student has encountered similar exercises in books, in class or in homework.
Exercises often prescribe assumptions to be made, principles to be used and sometimes they even give hints.
There is usually one approach that gives the right answer.
A usual method is to recall familiar solutions from previously solved exercises.
Exercises involve one subject and in many cases only one topic from this subject.
Communication skills are not essential, as most of the solution involves math and sketches.

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NASP: The National Association of School Psychologists

Conducting Crisis Exercises & Drills: Guidelines for Schools

In this section.

Responding to Civil Unrest in Schools: Prevention to Response
A Framework for Effective School Discipline
A Framework for Safe and Successful Schools
Guidance for Measuring and Using School Climate Data
A Framework for School-Wide Bullying Prevention and Safety (PDF)
Threat Assessment at School
Conducting Crisis Exercises and Drills
Best Practice Considerations for Armed Assailant Drills in Schools
Mitigating Psychological Effects of Lockdowns
Reunification
Promoting Compassion and Acceptance in Crisis
Recovery From Large-Scale Crises: Guidelines for Crisis Teams and Administrators

While high profile crisis events and instances of violent crimes at school are extremely rare (e.g., the odds of a student being the victim of a school-associated homicide are about 1 in 2.5 million), it is essential that all schools be prepared to respond to emergency situations as part of their school safety and crisis planning and preparation. Current state laws already require certain types of drills (e.g., fire drills) and many schools have begun to conduct a much broader range of crisis exercises and drills. Which type of drills are conducted and how is critical to both their effectiveness and minimizing the potential to cause trauma or harm unintentionally. Members of the National Association of School Psychologists’ PREP a RE Workgroup offer the following guidelines to help schools understand what might be considered best practices in the development and implementation of a variety of exercises and drills.

Start With Simple Exercises

School crisis response training and exercises can be discussion-based (orientation seminars, workshops, or tabletop drills) or operations-based (a variety of specific emergency drills, functional exercise drills, or full-scale exercises), each of which can be useful in preparing school staff, crisis team members, students and other agencies for a wide variety of crises. However, it is recommended that districts start with simple, low-cost discussion-based exercises (e.g., orientations) and work their way toward more complex and expensive, operations-based exercises (e.g., full scale drills U.S. Department of Education, [USDE], 2006; Freeman & Taylor, 2010). Although there is little empirical research about drills, existing research suggests that drills implemented according to best practice can increase students’ knowledge and skills of how to respond in an emergency, without elevating their anxiety or perceived safety (Zhe & Nickerson, 2007).

Select an Appropriate Exercise Scenario

There is a difference between crises that are possible and those that are more probable, and exercises are most useful when based on a vulnerability assessment that identifies the types of risks or potential hazards that have a probability of occurring in a specific community. Vulnerability assessments that address both physical and psychological safety help schools identify areas wherein they are most vulnerable (e.g., responding to wild animals, trespassers, food contamination, chemical spills, angry students or parents; Reeves et al., 2011). Building administration and crisis response teams should consider training on how to respond to different emergency protocols within their crisis plans (e.g., fire drills, lockdowns, shelter-in-place drills, and evacuation procedures). Districts that are vulnerable to certain types of natural disasters (e.g., tornadoes, hurricanes, floods, wildfires, and earthquakes) should consider drills related to hazards associated with specific events. Exercises should be planned to address multiple hazards and consider unexpected occurrences (e.g., crisis occurring during a passing period or recess). When conducting crisis exercises and drills, schools also need to consider how they will respond to individuals with special needs. This includes students and staff members with physical handicaps (including temporary ones), medical needs, and emotional concerns (Reeves et al., 2011).

Discussion-Based Exercises

Discussion-based exercises are used to familiarize students and school staff members with crisis plans, policies, agency agreements, and emergency procedures. Orientation seminars and workshops can be an efficient way to introduce school staff members, first responders, and volunteers to the school’s crisis plans and procedures, and tabletop drills can be an effective first step in testing crisis response protocols (USDE, 2006).

Orientations. These relatively brief seminars (which can be a part of regularly scheduled staff meetings) are discussions facilitated by a school crisis team leader (e.g., school principal). This is often the first step in ensuring that all school staff members understand a recently developed (or revised) school safety or crisis preparedness plan. These meetings review the school’s emergency response procedures; and they provide the opportunity to discuss crisis response coordination, roles, responsibilities, procedures, and the equipment that might be needed to respond to a school emergency. Orientations can be facilitated by the use of a PowerPoint presentation, handouts, or videos illustrating the correct response to an emergency situation (Freeman & Taylor, 2010).

Workshops. Relative to an orientation, crisis response workshops typically last longer (up to 3 hours), involve more participant interaction, and may focus on a specific issue. They include sharing information; obtaining different perspectives; testing new ideas, policies, or procedures; training groups to perform specific coordinated crisis response activities; problem-solving; obtaining consensus; and building teams through lecture, discussion, and break-outs (U.S. Department of Homeland Security [USDHS], 2007).

Tabletop drills. These drills involve presenting crisis response teams with a crisis scenario and asking them to then discuss what their crisis response roles would require them to do in the given situation. These drills help participants better understand their crisis response roles and responsibilities, and can last from 1-4 hours (Freeman & Taylor, 2011). Tabletop drills are designed to prompt an in-depth, constructive, problem-solving discussion about existing emergency response plans as participants identify, investigate, and resolve issues (Reeves et al., 2012; USDE, 2006). When conducting this kind of drill a specific individual facilitates the drill, another person records each step the team suggests, and another is responsible for facilitating an evaluative discussion covering what the team did well and what areas are in need of improvement. Schools can develop written or video scenarios for the crisis team to follow during the tabletop drill. Effective tabletop scenarios should inject unexpected events into the discussion. Crisis events typically do not occur predictably. Thus, injecting “new” pieces of information into the tabletop drill makes it more realistic.

Operations-Based Exercises

Operations-based exercises serve to validate plans, policies, and procedures; clarify roles and responsibilities; and identify gaps in resources. They involve school staff and students reacting to a simulated crisis; practicing the response to specific emergency conditions. They may include the mobilization of emergency equipment, resources and networks. When planning operations-based exercises, it is important that schools start with less intense emergency drills and work their way up to functional exercises and full-scale drills. Practicing different types of emergency drills can help a school prepare for a more involved emergency response (Freeman & Taylor, 2010; USDHS, 2007).

Emergency drills. These drills involve practicing a single specific emergency procedure or protocol and can last from 30 minutes to 2 hours (Freeman &Taylor, 2011). Many schools already conduct a variety of these drills (e.g., lockdown, fire, evacuation, reverse evacuation, duck-cover-hold, and shelter-in-place) with students and staff, which allow them to practice the steps they should take in emergency situations. These exercises may include local public safety agencies (USDE, 2006). Each state requires a different number and type of annual emergency drills. Some states also require that local public safety agency representatives be present when schools conduct these drills.

Functional exercises. These exercises are simulations of emergency situations with realistic timelines that can last from 3 to 8 hours, and that test one or more functions of a school’s emergency response plan during an interactive, time-pressured, simulated event. Functional exercises are often conducted in a school district’s emergency operations center, but do not involve the movement of emergency personnel and equipment. Participants are given directions by exercise controllers and simulators via telephones, radios, and televisions, and they must respond appropriately to the incidents as they arise. Evaluators candidly critique the exercise and the team’s performance. Roles in a functional drill include (a) an exercise controller who manages and directs the exercise, (b) players who respond as they would in a real emergency, (c) crisis simulators who assume external roles and deliver planned messages to the players, and (d) evaluators who assess performance through observations. Functional exercises are also 3 less expensive than a full-scale drill due to the lack of movement of emergency personnel or equipment (Freeman & Taylor, 2010; 2011; USDE, 2006).

Full-scale drills. As a school considers a full-scale drill, it is essential that it be carefully planned and that it does not cause harm (e.g., unnecessarily frighten participants). The local community and parents must be informed about the drill starting at least one month in advance, and the school should work with the media and its local municipality to inform all members of the surrounding community . These drills are the most elaborate, expensive, and time consuming, lasting from a half-day to multiple days, and often have a significant effect on instructional time. The full-scale drill is a simulation of emergency situations in real time with all necessary resources deployed, allowing for the evaluation of operational capabilities of emergency management systems in a highly stressful environment that simulates actual conditions. This type of drill will test multiple emergency protocols at once (e.g., reverse evacuation and lockdown to an off-site evacuation). To design and conduct a full-scale drill, districts collaborate with multiple agencies (including but not limited to police, fire, health departments, mental health agencies, transportation, local utilities, hospitals, and emergency management agencies). Full-scale drills also may involve multiple municipalities and jurisdictions (Freeman & Taylor, 2010; 2011; USDHS, 2007). Additional considerations for the full-scale drill are offered in Table 1.

When conducting full-scale drills, schools should choose a scenario that is most likely to occur in their communities and thereby increase the likelihood of involving all community stakeholders. A vulnerability assessment that includes a local hazard analysis can assist schools in determining what scenario should be chosen for the full-scale drill (Reeves et al., 2011). Additional considerations for the full-scale drill are offered in Table 1.

Table 1: Essential Considerations When Developing/Conducting Full-Scale Drill

May require as long as a year to 18 months to develop.
Should be proceeded by orientation sessions, emergency drills, and functional exercises.
Should be part of a long-term emergency exercise plan that begins with basic drills and culminates with the full-scale drill.
May require collaboration with an outside expert or consultant to provide guidance in conducting crisis exercises.
Must not be mistaken for a real crisis event.
Likely will not require exposing students and staff to potentially traumatic stimuli (e.g., shooting blanks, fake blood) to meet drill objectives, as this exposure may lead to increased threat perceptions, serve as a reminder of prior trauma, and generate distressing reactions. b
Does not need to involve the entire student body. c
Should have participating agencies follow the National Incident Management System’s Incident Command System and activate an Emergency Operations Center.
Should require participants to sign in before the drill begins, receive an initial briefing, and wear identification on who they are and what their roles are during the drill.
Should generate a postincident critique to identify issues to correct. d
Establishes a no-fault/no-fail expectation and emphasizes that mistakes or inconsistencies are learning opportunities to improve future crisis response.

Notes. a Sources include Freeman & Taylor (2010). b Schools must consider the costs and benefits of using certain types of props during a full-scale drill and the developmental level of the children that are involved. The district’s risk manager and a mental health staff member must be involved in the process of determining what types of dramatizations will occur. In addition, it should be recognized that some props may damage school or community facilities (e.g., blanks can leave nicks in the wall). c The drill can be conducted on the weekend with a small group of student actors and other adults playing students. All drill participants should be carefully selected and screened for past trauma history, trained about what the drill will involve, and supported afterward. d This report summarizes the findings of the drill and analyzes its outcomes relative to drill goals and objectives. Any areas that need improvement are identified and provided to the crisis team to determine further training needs or changes to the crisis response plan.

The guidance in this document is not a substitute for crisis team training, planning and more in-depth knowledge of the school crisis prevention and intervention process. For more extensive school crisis prevention and intervention information please refer to Brock et al. (2009) or visit www.nasponline.org/prepare for details the PREP a RE School Crisis Prevention and Intervention Training Curriculum. For detailed guidance on planning, conducting, and evaluating crisis exercises and drills specifically, review FEMA’s Homeland Security Exercise and Evaluation Program (HSEEP) at https://hseep.dhs.gov/pages/1001_HSEEP7.aspx .

Brock, S. E., Nickerson, A. B., Reeves, M. A., Jimerson, S. R., Lieberman, R. A., & Feinberg, T. A. (2009). School crisis prevention and intervention: The PREP a RE model. Bethesda, MD: National Association of School Psychologists.

Freeman, W., & Taylor, M. (2010, July). Conducting effective tabletops, drills and other exercises. Workshop presented at the U.S. Department of Education, Office of Safe and Drug-Free Schools Readiness and Emergency Management for Schools (REMS) Final Grantee Meeting, Boston, MA. Retrieved from http://rems.ed.gov/docs/Training_FY09REMS_BOMA_TableTopsDrills.pdf

Freeman, W., & Taylor, M. (2011, August). Overview of emergency management exercises. Workshop presented at the U.S. Department of Education, Office of Safe and Drug-Free Schools Readiness and Emergency Management for Schools (REMS) Final Grantee Meeting, National Harbor, MD. Retrieved from http://rems.ed.gov/docs/FY10REMS_FGM_NHMD_EMExercises.pdf

Reeves, M. A., Conolly-Wilson, C. N., Pesce, R. C., Lazzaro, B. R., Nickerson, A. B., Feinberg, T., . . . Brock, S. E. (2012). Providing the comprehensive school crisis response. In S. E. Brock & S. R.Jimerson (Eds.) Best practices in school crisis prevention and intervention (pp. 305-316; 2nd ed.). Bethesda, MD: National Association of School Psychologists.

Reeves, M. A., Nickerson, A. B., Connolly-Wilson, C. N., Susan, M. K., Lazzaro, B. R., Jimerson, S. R., & Pesce, R. C. (2011). Crisis prevention and preparedness: Comprehensive school safety planning (2nd ed.). Bethesda, MD: National Association of School Psychologists.

U.S. Department of Education, REMS Technical Assistance Center. (USDE; 2006). Emergency exercises: An effective way to validate school safety plans. ERCMExpress, 2(3), 1–4. Retrieved from http://rems.ed.gov/docs/Emergency_NewsletterV2I3.pdf

U.S. Department of Homeland Security. (USDHS; 2007, February). Homeland security exercise and evaluation program. Vol. I: HSEEP overview and exercise program management. Washington, DC: Author. Retrieved from https://hseep.dhs.gov/pages/1001_HSEEP7.aspx

Zhe, E. J., & Nickerson, A. B. (2007). The effects of an intruder crisis drill on children’s self-perceptions of anxiety, school safety, and knowledge. School Psychology Review , 36, 501-508. Retrieved from http://www.nasponline.org/publications/spr/abstract.aspx?ID=1850

Related Resources

Best Practice Considerations for Schools in Active Shooter and Other Armed Assailant Drills Guidance from NASP and the National Association of School Resource Officers on key considerations and guidelines to safely conduct drills around armed assailant scenarios.

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Team Building Exercises – Problem Solving and Decision Making

Fun ways to turn problems into opportunities.

By the Mind Tools Content Team

Whether there's a complex project looming or your team members just want to get better at dealing with day-to-day issues, your people can achieve much more when they solve problems and make decisions together.

By developing their problem-solving skills, you can improve their ability to get to the bottom of complex situations. And by refining their decision-making skills, you can help them work together maturely, use different thinking styles, and commit collectively to decisions.

In this article, we'll look at three team-building exercises that you can use to improve problem solving and decision making in a new or established team.

Exercises to Build Decision-Making and Problem-Solving Skills

Use the following exercises to help your team members solve problems and make decisions together more effectively.

Exercise 1: Lost at Sea*

In this activity, participants must pretend that they've been shipwrecked and are stranded in a lifeboat. Each team has a box of matches, and a number of items that they've salvaged from the sinking ship. Members must agree which items are most important for their survival.

Download and print our team-building exercises worksheet to help you with this exercise.

This activity builds problem-solving skills as team members analyze information, negotiate and cooperate with one another. It also encourages them to listen and to think about the way they make decisions.

What You'll Need

Up to five people in each group.
A large, private room.
A "lost at sea" ranking chart for each team member. This should comprise six columns. The first simply lists each item (see below). The second is empty so that each team member can rank the items. The third is for group rankings. The fourth is for the "correct" rankings, which are revealed at the end of the exercise. And the fifth and sixth are for the team to enter the difference between their individual and correct score, and the team and correct rankings, respectively.
The items to be ranked are: a mosquito net, a can of petrol, a water container, a shaving mirror, a sextant, emergency rations, a sea chart, a floating seat or cushion, a rope, some chocolate bars, a waterproof sheet, a fishing rod, shark repellent, a bottle of rum, and a VHF radio. These can be listed in the ranking chart or displayed on a whiteboard, or both.
The experience can be made more fun by having some lost-at-sea props in the room.

Flexible, but normally between 25 and 40 minutes.

Instructions

Divide participants into their teams, and provide everyone with a ranking sheet.
Ask team members to take 10 minutes on their own to rank the items in order of importance. They should do this in the second column of their sheet.
Give the teams a further 10 minutes to confer and decide on their group rankings. Once agreed, they should list them in the third column of their sheets.
Ask each group to compare their individual rankings with their collective ones, and consider why any scores differ. Did anyone change their mind about their own rankings during the team discussions? How much were people influenced by the group conversation?
Now read out the "correct" order, collated by the experts at the US Coast Guard (from most to least important): - Shaving mirror. (One of your most powerful tools, because you can use it to signal your location by reflecting the sun.) - Can of petrol. (Again, potentially vital for signaling as petrol floats on water and can be lit by your matches.) - Water container. (Essential for collecting water to restore your lost fluids.) -Emergency rations. (Valuable for basic food intake.) - Plastic sheet. (Could be used for shelter, or to collect rainwater.) -Chocolate bars. (A handy food supply.) - Fishing rod. (Potentially useful, but there is no guarantee that you're able to catch fish. Could also feasibly double as a tent pole.) - Rope. (Handy for tying equipment together, but not necessarily vital for survival.) - Floating seat or cushion. (Useful as a life preserver.) - Shark repellent. (Potentially important when in the water.) - Bottle of rum. (Could be useful as an antiseptic for treating injuries, but will only dehydrate you if you drink it.) - Radio. (Chances are that you're out of range of any signal, anyway.) - Sea chart. (Worthless without navigational equipment.) - Mosquito net. (Assuming that you've been shipwrecked in the Atlantic, where there are no mosquitoes, this is pretty much useless.) - Sextant. (Impractical without relevant tables or a chronometer.)

Advice for the Facilitator

The ideal scenario is for teams to arrive at a consensus decision where everyone's opinion is heard. However, that doesn't always happen naturally: assertive people tend to get the most attention. Less forthright team members can often feel intimidated and don't always speak up, particularly when their ideas are different from the popular view. Where discussions are one-sided, draw quieter people in so that everyone is involved, but explain why you're doing this, so that people learn from it.

You can use the Stepladder Technique when team discussion is unbalanced. Here, ask each team member to think about the problem individually and, one at a time, introduce new ideas to an appointed group leader – without knowing what ideas have already been discussed. After the first two people present their ideas, they discuss them together. Then the leader adds a third person, who presents his or her ideas before hearing the previous input. This cycle of presentation and discussion continues until the whole team has had a chance to voice their opinions.

After everyone has finished the exercise, invite your teams to evaluate the process to draw out their experiences. For example, ask them what the main differences between individual, team and official rankings were, and why. This will provoke discussion about how teams arrive at decisions, which will make people think about the skills they must use in future team scenarios, such as listening , negotiating and decision-making skills, as well as creativity skills for thinking "outside the box."

A common issue that arises in team decision making is groupthink . This can happen when a group places a desire for mutual harmony above a desire to reach the right decision, which prevents people from fully exploring alternative solutions.

If there are frequent unanimous decisions in any of your exercises, groupthink may be an issue. Suggest that teams investigate new ways to encourage members to discuss their views, or to share them anonymously.

Exercise 2: The Great Egg Drop*

In this classic (though sometimes messy!) game, teams must work together to build a container to protect an egg, which is dropped from a height. Before the egg drop, groups must deliver presentations on their solutions, how they arrived at them, and why they believe they will succeed.

This fun game develops problem-solving and decision-making skills. Team members have to choose the best course of action through negotiation and creative thinking.

Ideally at least six people in each team.
Raw eggs – one for each group, plus some reserves in case of accidents!
Materials for creating the packaging, such as cardboard, tape, elastic bands, plastic bottles, plastic bags, straws, and scissors.
Aprons to protect clothes, paper towels for cleaning up, and paper table cloths, if necessary.
Somewhere – ideally outside – that you can drop the eggs from. (If there is nowhere appropriate, you could use a step ladder or equivalent.)
Around 15 to 30 minutes to create the packages.
Approximately 15 minutes to prepare a one-minute presentation.
Enough time for the presentations and feedback (this will depend on the number of teams).
Time to demonstrate the egg "flight."
Put people into teams, and ask each to build a package that can protect an egg dropped from a specified height (say, two-and-a-half meters) with the provided materials.
Each team must agree on a nominated speaker, or speakers, for their presentation.
Once all teams have presented, they must drop their eggs, assess whether the eggs have survived intact, and discuss what they have learned.

When teams are making their decisions, the more good options they consider, the more effective their final decision is likely to be. Encourage your groups to look at the situation from different angles, so that they make the best decision possible. If people are struggling, get them to brainstorm – this is probably the most popular method of generating ideas within a team.

Ask the teams to explore how they arrived at their decisions, to get them thinking about how to improve this process in the future. You can ask them questions such as:

Did the groups take a vote, or were members swayed by one dominant individual?
How did the teams decide to divide up responsibilities? Was it based on people's expertise or experience?
Did everyone do the job they volunteered for?
Was there a person who assumed the role of "leader"?
How did team members create and deliver the presentation, and was this an individual or group effort?

Exercise 3: Create Your Own*

In this exercise, teams must create their own, brand new, problem-solving activity.

This game encourages participants to think about the problem-solving process. It builds skills such as creativity, negotiation and decision making, as well as communication and time management. After the activity, teams should be better equipped to work together, and to think on their feet.

Ideally four or five people in each team.
Paper, pens and flip charts.

Around one hour.

As the participants arrive, you announce that, rather than spending an hour on a problem-solving team-building activity, they must design an original one of their own.
Divide participants into teams and tell them that they have to create a new problem-solving team-building activity that will work well in their organization. The activity must not be one that they have already participated in or heard of.
After an hour, each team must present their new activity to everyone else, and outline its key benefits.

There are four basic steps in problem solving : defining the problem, generating solutions, evaluating and selecting solutions, and implementing solutions. Help your team to think creatively at each stage by getting them to consider a wide range of options. If ideas run dry, introduce an alternative brainstorming technique, such as brainwriting . This allows your people to develop one others' ideas, while everyone has an equal chance to contribute.

After the presentations, encourage teams to discuss the different decision-making processes they followed. You might ask them how they communicated and managed their time . Another question could be about how they kept their discussion focused. And to round up, you might ask them whether they would have changed their approach after hearing the other teams' presentations.

Successful decision making and problem solving are at the heart of all effective teams. While teams are ultimately led by their managers, the most effective ones foster these skills at all levels.

The exercises in this article show how you can encourage teams to develop their creative thinking, leadership , and communication skills , while building group cooperation and consensus.

* Original source unknown. Please let us know if you know the original source.

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v.9(5); 2023 May
PMC10208825

Development and differences in mathematical problem-solving skills: A cross-sectional study of differences in demographic backgrounds

Ijtihadi kamilia amalina.

a Doctoral School of Education, University of Szeged, Hungary

Tibor Vidákovich

b Institute of Education, University of Szeged, Hungary

Associated Data

Data will be made available on request.

Problem-solving skills are the most applicable cognitive tool in mathematics, and improving the problem-solving skills of students is a primary aim of education. However, teachers need to know the best period of development and the differences among students to determine the best teaching and learning methods. This study aims to investigate the development and differences in mathematical problem-solving skills of students based on their grades, gender, and school locations. A scenario-based mathematical essay test was administered to 1067 students in grades 7–9 from schools in east Java, Indonesia, and their scores were converted into a logit scale for statistical analysis. The results of a one-way analysis of variance and an independent sample t -test showed that the students had an average level of mathematical problem-solving skills. The number of students who failed increased with the problem-solving phase. The students showed development of problem-solving skills from grade 7 to grade 8 but not in grade 9. A similar pattern of development was observed in the subsample of urban students, both male and female. The demographic background had a significant effect, as students from urban schools outperformed students from rural schools, and female students outperformed male students. The development of problem-solving skills in each phase as well as the effects of the demographic background of the participants were thoroughly examined. Further studies are needed with participants of more varied backgrounds.

1. Introduction

Problem-solving skills are a complex set of cognitive, behavioral, and attitudinal components that are situational and dependent on thorough knowledge and experience [ 1 , 2 ]. Problem-solving skills are acquired over time and are the most widely applicable cognitive tool [ 3 ]. Problem-solving skills are particularly important in mathematics education [ 3 , 4 ]. The development of mathematical problem-solving skills can differ based on age, gender stereotypes, and school locations [ [5] , [6] , [7] , [8] , [9] , [10] ]. Fostering the development of mathematical problem-solving skills is a major goal of educational systems because they provide a tool for success [ 3 , 11 ]. Mathematical problem-solving skills are developed through explicit training and enriching materials [ 12 ]. Teachers must understand how student profiles influence the development of mathematical problem-solving skills to optimize their teaching methods.

Various studies on the development of mathematical problem-solving skills have yielded mixed results. Grissom [ 13 ] concluded that problem-solving skills were fixed and immutable. Meanwhile, other researchers argued that problem-solving skills developed over time and were modifiable, providing an opportunity for their enhancement through targeted educational intervention when problem-solving skills developed quickly [ 3 , 4 , 12 ]. Tracing the development of mathematical problem-solving skills is crucial. Further, the results of previous studies are debatable, necessitating a comprehensive study in the development of students’ mathematical problem-solving skills.

Differences in mathematical problem-solving skills have been identified based on gender and school location [ [6] , [7] , [8] , [9] , [10] ]. School location affects school segregation and school quality [ 9 , 14 ]. The socioeconomic and sociocultural characteristics of a residential area where a school is located are the factors affecting academic achievement [ 14 ]. Studies in several countries have shown that students in urban schools demonstrated better performance and problem-solving skills in mathematics [ 9 , 10 , 15 ]. However, contradictory results have been obtained for other countries [ 6 , 10 ].

Studies on gender differences have shown that male students outperform female students in mathematics, which has piqued the interest of psychologists, sociologists, and educators [ 7 , 16 , 17 ]. The differences appear to be because of brain structure; however, sociologists argue that gender equality can be achieved by providing equal educational opportunities [ 8 , 16 , 18 , 19 ]. Because the results are debatable and no studies on gender differences across grades in schools have been conducted, it would be interesting to investigate gender differences in mathematical problem-solving skills.

Based on the previous explanations, teachers need to understand the best time for students to develop mathematical problem-solving skills because problem-solving is an obligatory mathematics skill to be mastered. However, no relevant studies focused on Indonesia have been conducted regarding the mathematical problem-solving skill development of students in middle school that can provide the necessary information for teachers. Further, middle school is the important first phase of developing critical thinking skills; thus relevant studies are required in this case [ 3 , 4 ]. In addition, a municipal policy-making system can raise differences in problem-solving skills based on different demographic backgrounds [ 10 ]. Moreover, the results of previous studies regarding the development and differences in mathematical problem-solving skills are debatable. Thus, the present study has been conducted to meet these gaps. This study investigated the development of mathematical problem-solving skills in students and the differences owing demographic backgrounds. Three aspects were considered: (1) student profiles of mathematical problem-solving skills, (2) development of their mathematical problem-solving skills across grades, and (3) significant differences in mathematical problem-solving skills based on gender and school location. The results of the present study will provide detailed information regarding the subsample that contributes to the development of mathematical problem-solving skills in students based on their demographic backgrounds. In addition, the description of the score is in the form of a logit scale from large-scale data providing consistent meaning and confident generalization. This study can be used to determine appropriate teaching and learning in the best period of students’ development in mathematical problem-solving skills as well as policies to achieve educational equality.

2. Theoretical background

2.1. mathematical problem-solving skills and their development.

Solving mathematical problems is a complex cognitive ability that requires students to understand the problem as well as apply mathematical concepts to them [ 20 ]. Researchers have described the phases of solving a mathematical problem as understanding the problem, devising a plan, conducting out the plan, and looking back [ [20] , [24] , [21] , [22] , [23] ]. Because mathematical problems are complex, students may struggle with several phases, including applying mathematical knowledge, determining the concepts to use, and stating mathematical sentences (e.g., arithmetic) [ 20 ]. Studies have concluded that more students fail at later stages of the solution process [ 25 , 26 ]. In other words, fewer students fail in the phase of understanding a problem than during the plan implementation phase. Different studies have stated that students face difficulties in understanding the problem, determining what to assume, and investigating relevant information [ 27 ]. This makes them unable to translate the problem into a mathematical form.

Age or grade is viewed as one factor that influences mathematical problem-solving skills because the skills of the students improve over time as a result of the teaching and learning processes [ 28 ]. Neuroscience research has shown that older students have fewer problems with arithmetic than younger students; however, the hemispheric asymmetry is reduced [ 29 ]. In other words, older students are more proficient, but their flexibility to switch among different strategies is less. Ameer & Sigh [ 28 ] obtained similar results and found a considerable difference in mathematical achievement; specifically, older students performed better than younger students in number sense and computation using one-way analysis of variance (ANOVA) ( F ) of F (2,411) = 4.82, p < 0.01. Molnár et al. [ 3 ] found that the student grade affects domain-specific and complex problem-solving skills. They observed that the development of problem-solving skills was noticeable across grades in elementary school but stopped in secondary school. The fastest development of domain-specific problem-solving occurred in grades 7 and 8 [ 3 ], but the fastest development of complex problem-solving occurred in grades 5–7 [ 3 ]. No development was detected between grades 4 and 5 as well as grades 6 and 7 for domain-specific and complex problem-solving skills, respectively. Similarly, Greiff et al. [ 4 ] concluded that students developed problem-solving skills across grades 5–11 with older students being more skilled. However, the grade 9 students deviated from the development pattern, and their problem-solving skills dropped. The theories from Molnár et al. [ 3 ] and Greiff et al. [ 4 ] are the benchmark cases herein.

The above studies showed that problem-solving skills mostly developed during compulsory schooling and developed most quickly in specific grades. This indicates that specific development times can be targeted to enhance the problem-solving skills [ 3 ]. However, Jabor et al. [ 30 ] observed contradictory results showing statistically significant differences with small effects in mathematical performance between age groups: those under the age of 19 outperformed those over the age of 19 years old. Grissom [ 13 ] observed a negative correlation between age and school achievement that remained constant over time.

2.2. Effects of school location and gender on mathematical problem-solving skills

School location has been shown to affect mathematical achievement [ 9 , 14 ]. In 15 countries, students in rural schools performed considerably worse than students in urban schools in mathematics [ 9 , 10 ], science and reading [ 9 ]. In addition, Nepal [ 15 ] discovered that urban students significantly outperformed rural students in mathematical problem-solving skills ( t = −5.11, p < 0.001) and achievement ( t = −4.45, p < 0.001) using the results of an independent sample t -test (t). However, other countries have found that rural students outperformed urban students in mathematics [ 6 , 10 ]. These variations may be attributed to a lack of instructional resources (e.g., facilities, materials, and programs), professional training (e.g., poorly trained teachers), and progressive instruction [ 6 ]. The results of Williams's study [ 10 ] serve as the basis for the current study.

Gender differences in mathematics have received attention because studies show that male students outperform female students on higher-level cognitive tasks [ 31 ]. This is a shift from a meta-analysis study that found gender differences in mathematics to be insignificant and favored female students [ 32 ]. At the college level, female students slightly outperform male students in computation while male students outperform female students in problem solving. However, no gender differences have been observed among elementary and middle school students. This result was strengthened by other meta-analysis studies [ 7 , 8 ], which concluded that there was no gender difference in mathematical performance and problem-solving skills [ 15 , [33] , [35] , [34] ]. Gender similarity in mathematics is achieved when equal learning opportunities and educational choices are provided and the curriculum is expanded to include the needs and interests of the students [ 16 , 18 , 31 ].

From a sociological perspective, gender similarity in mathematics makes sense. If there is a gender difference in mathematics, this has been attributed to science, technology, engineering, and mathematics (STEM) being stereotyped as a male domain [ 8 ]. Stereotypes influence beliefs and self-efficacy of students and perceptions of their own abilities [ 8 , 19 ]. This is the reason for the low interest of female students in advanced mathematics courses [ 18 , 19 ]. However, Halpern et al. [ 16 ] found that more female students are entering many occupations that require a high level of mathematical knowledge. Moreover, Anjum [ 36 ] found that female students outperformed male students in mathematics. This may be because female students prepared better than the male students before the test and were more thorough [ 36 , 37 ]. The study of Anjum [ 36 ] is one of the basis cases of the current study.

Differences in brain structure support the argument that there are gender differences in mathematical performance [ 16 , 17 ]. Females have less brain lateralization (i.e., symmetric left and right hemispheres), which helps them perform better verbally. Meanwhile, males have more brain lateralization, which is important for spatial tasks [ 17 ]. In addition, the male hormone testosterone slows the development of the left hemisphere [ 16 ], which improves the performance of right brain-dominant mathematical reasoning and spatial tasks.

3.1. Instrumentation

In this study, a science-related mathematical problem-solving test was used. This is a mathematics essay test where the problems are in the form of scenarios related to environmental management. Problems are solved by using technology as a tool (e.g., calculator, grid paper). The test was developed in an interdisciplinary STEM framework, and it is targeted toward grades 7–9. There were six scenarios in total: some were given to multiple grades, and others were specific to a grade. They included ecofriendly packaging (grade 7), school park (grade 7), calorie vs. greenhouse gas emissions (grades 7–9), floodwater reservoir (grade 8), city park (grades 8–9), and infiltration well (grade 9). These scenarios cover topics such as number and measurement, ratio and proportion, geometry, and statistics. Every scenario had a challenge, and students were provided with eight metacognitive prompt items to help them explore their problem-solving skills.

The test was administered by using paper and pencils for a 3-h period with a break every hour. At the end of the test, students were asked to fill in their demographic information. Each prompt item had a maximum score of 5 points: a complete and correct answer (5 points), a complete answer with a minor error (4 points), an incomplete answer with a minor error (3 points), an incomplete answer with a major error (2 points), and a completely wrong and irrelevant answer (1 point). Each scenario had a maximum total score of 40 points.

The test was validated to determine whether it contained good and acceptable psychometric evidence. It had an acceptable content validity index (CVI >0.67), moderate intraclass correlation coefficient (ICC) (rxx = 0.63), and acceptable Cronbach's alpha (α = 0.84). The construct validity indicated all scenarios and prompt items were fit (0.77 ≤ weighted mean square ≤1.59) with an acceptable discrimination value (0.48 ≤ discrimination value ≤ 0.93), acceptable behavior of the rating score, and good reliability (scenario reliability = 0.86; prompt item reliability = 0.94).

3.2. Participants

The test was administered to grades 7–9 students in east Java, Indonesia (n = 1067). The students were selected from A-accreditation schools in urban and rural areas; random classes were selected for each grade. The majority of the students were Javanese (95.01%), with the remainder being Madurese (3.3%) and other ethnicities. Table 1 describes the demographics of the participants.

Demographic characteristics of participants.

3.3. Data analysis

Data were collected between July and September 2022. Prior to data collection, ethical approval was sought from the institutional review board (IRB) of the Doctoral School of Education, University of Szeged and was granted with the ethical approval number of 7/2022. In addition, permission letters were sent to several schools to request permission and confirm their participation. The test answers of the students were scored by two raters – the first author of this study and a rater with master's degree in mathematics education – to ensure that the rating scale was consistently implemented. The results showed good consistency with an ICC of 0.992 and Cronbach's alpha of 0.996.

The scores from one of the raters were converted to a logit scale by weighted likelihood estimation (WLE) using the ConQuest software. A logit scale provides a consistent value or meaning in the form of intervals. The logit scale represents the unit interval between locations on the person–item map. WLE was chosen rather than maximum likelihood estimation (MLE) because WLE is more central than MLE, which helps to correct for bias [ 38 ]. The WLE scale was represented by using descriptive statistics to profile the students' mathematical problem-solving skills in terms of the percentage, mean score ( M ) and standard deviation ( SD ) for each phase. The WLE scale was also used to describe common difficulties for each phase. The development of students’ mathematical problem-solving skills across grades was presented by a pirate plot, which is used in R to visualize the relationship between 1 and 3 categorical independent variables and 1 continuous dependent variable. It was chosen because it displays raw data, descriptive statistics, and inferential statistics at the same time. The data analysis was performed using R studio version 4.1.3 software with the YaRrr package. A one-way ANOVA was performed to find significant differences across grades. An independent sample t -test was used to analyze significant differences based on gender and school location. The descriptive statistics, one-way ANOVA test, and independent sample t -test were performed using the IBM SPSS Statistics 25 software.

4.1. Student profiles

The scores of students were converted to the WLE scale, where a score of zero represented a student with average ability, a positive score indicated above-average ability, and a negative score indicated below-average ability. A higher score indicated higher ability. The mean score represented a student with average mathematical problem-solving skills ( M = 0.001, SD = 0.39). Overall, 52.1% of students had a score below zero. The distribution of scores among students was predominantly in the interval between −1 and 0. When the problem-solving process was analyzed by phase, the results showed that exploring and understanding were the most mastered problem-solving skills ( M = 0.24, SD = 0.51). Only 27.9% of students had below-average scores for the exploring and understanding phases, which indicates that they mostly understood the given problem and recognized the important information. However, the problem-solving skills decreased with higher phases. The students had below-average abilities in the phases of representing and formulating ( M = −0.01, SD = 0.36), planning and executing ( M = −0.15, SD = 0.41), and monitoring and reflecting ( M = −0.16, SD = 0.36). About 57.9% of the students had below-average scores for the representing and formulating phase, which indicates that they had problems making hypotheses regarding science phenomena, representing problems in mathematical form, and designing a prototype. The obvious reason for their difficulty with making hypotheses was that they did not understand simple concepts of science (e.g., CO 2 vs. O 2 ). In the planning and executing phase, 66.8% of the students failed to achieve a score greater than zero. This happened because they failed to apply mathematical concepts and procedures. Because they were unable to plan and execute a strategy, this affected the next phase of the problem-solving process. In the monitoring and reflecting phase, 68.0% of the students had a below-average score.

4.2. Development of mathematical problem-solving skills across grades

The development of the mathematical problem-solving skills of the students across grades was observed based on the increase in the mean score. The problem-solving skills developed from grade 7 to grade 8. The students of grade 7 had a mean score of −0.04 while grade 8 students had the highest mean score of 0.03. The students in grades 7 and 8 also showed more varied problem-solving skills than the grade 9 students did. In contrast, the grade 9 students showed a different pattern of development, and their mean score dropped to 0.01. Although the difference was not large, further analysis was needed to determine its significance.

Fig. 1 displays the development of the mathematical problem-solving skills of the students. The dots represent raw data or WLE scores. The middle line shows the mean score. The beans represent a smoothed density curve showing the full data distribution. The scores of the students in grades 7 and 9 were concentrated in the interval between −0.5 and 0. However, the scores of the grade 8 students were concentrated in the interval between 0 and 0.5. The scores of the students in grades 7 and 8 showed a wider distribution than those of the grade 9 students. The bands which overlap with the line representing the mean score, define the inference around the mean (i.e., 95% of the data are in this interval). The inference of the WLE score was close to the mean.

Differences in students' mathematical problem-solving skills across grades.

Note : PS: Problem-Solving Skills of Students.

The one-way ANOVA results indicated a significant difference among the problem-solving skills of the students of grades 7–9 ( F (1,066) = 3.01, p = 0.046). The students of grade 8 showed a significant difference in problem-solving skills and outperformed the other students. The students of grades 7 and 9 showed no significant difference in their mathematical problem-solving skills. Table 2 presents the one-way ANOVA results of the mathematical problem-solving skills across grades.

One-way ANOVA results of the mathematical problem-solving across grades.

Note. Post hoc test: Dunnett's T3. 7, 8, and 9: subsample grade. <: direction of significant difference ( p < 0.05).

Fig. 2 shows the development of the mathematical problem-solving skills of the students across grades based on school location and gender. The problem-solving skills of the urban students increased from a mean score of 0.07 in grade 7 to 0.14 in grade 8. However, the mean score of urban students in grade 9 dropped. In contrast, the mean scores of the rural students increased continuously with grade. The improvements were significant for both the rural ( F (426) = 10.10, p < 0.001) and urban ( F (639) = 6.10, p < 0.01) students. For the rural students, grade 9 students showed a significant difference in problem-solving skills. In contrast, urban students in grades 8 and 9 showed significant differences in problem-solving skills but not in grade 7.

Differences in students' mathematical problem-solving skills across grades and different demographic backgrounds.

(a) Differences in students grade 7 of mathematical problem-solving skills across grades and different demographic backgrounds

(b) Differences in students grade 8 of mathematical problem-solving skills across grades and different demographic backgrounds

(c) Differences in students grade 9 of mathematical problem-solving skills across grades and different demographic backgrounds

Note: WLE_PS: The students' problem-solving skills in WLE scale; F: Female; M: Male; ScLoc: School location; R: Rural; U: Urban.

When divided by gender, both female and male students showed improvements in their problem-solving skills from grades 7 and 8. However, female students in grade 9 showed a stable score while the scores of male students in grade 9 declined. Only male students in grade 7 showed a significant difference in the mean score. In urban schools, the scores of male and female students increased and decreased, respectively, from grade 7 to grade 8. Male students in rural schools showed an increase in score from grade 7 to grade 9. However, the scores of female students in rural schools decreased from grade 7 to grade 8. Table 3 presents the one-way ANOVA results for the mathematical problem-solving skills of the students considering gender and school location.

One-way ANOVA results for mathematical problem-solving skills across grades and different demographic backgrounds.

Fig. 2 shows that the distributions of the male and female scores of students were similar for every grade except rural grade 9 students. The scores of the rural female students were concentrated in the interval between 0 and 0.5 while the scores of the rural male students were mostly below 0. The scores of rural students in grade 7 and urban students in grade 9 (both male and female) were concentrated in the interval between −0.5 and 0. The scores of urban students in grades 7 and 8 were concentrated in the interval between −0.5 and 0.5.

Fig. 3 shows a detailed analysis of the development of mathematical problem-solving skills across grades for each phase of the problem-solving process. Similar patterns were observed in the exploring and understanding and the representing and formulating phases: the mean score increased from grade 7 to grade 8 but decreased from grade 8 to grade 9. Grade 8 students had the highest mean score and differed significantly from the scores of students in other grades.

Differences in students' mathematical problem-solving skills in every phase across grades: (1) Exploring & understanding, (2) Representing & formulating, (3) Planning & executing, (4) Monitoring & reflecting.

(a) Differences in students' mathematical problem-solving skills in exploring and understanding phase

(b) Differences in students' mathematical problem-solving skills in representing and formulating phase

(d) Differences in students' mathematical problem-solving skills in monitoring and reflecting phase

Note: WLE_Exp_Un: The WLE score in exploring and understanding; WLE_Rep_For: The WLE score in representing and formulating; WLE_Plan_Ex: The WLE score in planning and executing; WLE_Mon_Ref: The WLE score in monitoring and reflecting.

The scores of the students for the planning and executing phase increased with grade. However, the difference was only significant at grade 9. Grades 7 and 8 students showed an increase in score, but the improvement was not significant. There was no pattern detected in the monitoring and reflecting phase. The score was stable for grades 7 and 8 students but improved for grade 9 students. The mean score for each phase and the one-way ANOVA results are presented in Table 4 .

One-way ANOVA results for every phase of problem-solving across grades.

Fig. 3 shows that the distributions of the problem-solving skills of the students were similar across grades and phases. However, the distributions were different for grade 9 students in the representing and formulating, planning and executing, and monitoring and reflecting phases, where 95% of the data were in the interval between −0.5 and 0.5.

4.3. Effects of demographic background

4.3.1. school location.

The mathematical problem-solving skills of the students differed significantly based on school location. Urban students scored higher than rural students. The results of the t -test for mathematical problem-solving skills based on school location are presented in Table 5 .

T-test results for mathematical problem-solving skills based on school location.

The effects of the school's location on the performances of male and female students were analyzed. The results showed that the scores of the female students differed significantly based on school location ( t (613) = −6.09, p < 0.001). Female students in urban schools ( M = 0.18, SD = 0.39) outperformed female students in rural schools ( M = −0.08, SD = 0.37). Similar results were observed for male students with urban students ( M = −0.01, SD = 0.35) outperforming rural students ( M = −0.12, SD = 0.39) by a significant margin ( t (382.764) = −3.25, p < 0.01).

When analyzed by grade, grades 7 and 8 students contributed to the difference based on school location with t (377.952) = −6.34, p < 0.001 and t (300.070) = −5.04, p < 0.001, respectively. Urban students in grades 7 and 8 performed significantly better than their rural counterparts did. However, there was no significant difference between rural and urban students in grade 9 ( t (354) = 0.71, p = 0.447).

4.3.2. Gender

Male and female students showed a significant difference in their mathematical problem-solving skills. Overall, female students outperformed male students. The detailed results of the independent sample t -test for mathematical problem-solving skills based on gender are presented in Table 6 .

T-test results for mathematical problem-solving skills based on gender.

The results were analyzed to determine whether the school location contributed to the gender difference. The gender difference was most significant among urban students ( t (596.796) = −4.36, p < 0.001). Female students from urban schools ( M = 0.12, SD = 0.39) outperformed male students from urban schools ( M = −0.01, SD = 0.35). There was no significant difference between female and male students from rural schools ( t (425) = −1.31, p = 0.191).

Grades 7 and 9 students contributed to the gender difference with t (372.996) = −3.90, p < 0.001 and t (354) = −2.73, p < 0.01, respectively. Female students in grades 7 and 9 outperformed their male counterparts. However, there was no significant gender difference among grade 8 students ( t (329) = −0.10, p = 0.323).

5. Discussion

The mathematical problem-solving skills of the students were categorized as average. In addition, the difficulties of students increased in line with the problem-solving phase. Fewer students failed the exploring and understanding phase than the subsequent phases. This confirms the results of previous studies indicating that more students failed further along the problem-solving process [ 25 , 26 ]. Because the problem-solving process is sequential, students who have difficulty understanding a problem will fail the subsequent phases [ 27 ].

The development of mathematical problem-solving skills was evaluated according to the mean WLE score. The mathematical problem-solving skills of the students developed from grade 7 to grade 8 based on the increase in their mean scores. However, the development dropped in grade 9. This agrees with previous results that concluded that higher grades had the highest problem-solving skills, but the fastest skill development took place in grades 7–8 after which it dropped [ 3 , 4 ]. These results indicate that the mathematical problem-solving skills of the students should improve and be strengthened in grades 7–8, which will help them perform better in grade 9.

In this study, the effects of the demographic background of the students were analyzed in detail, which is an aspect missing from previous studies. The results showed that the mathematical problem-solving skills of urban students increased from grade 7 to grade 8 but decreased in grade 9. The same pattern was found among male and female students. However, a different pattern was observed for rural students, where the skills of grade 9 students continued to increase. The different patterns may be attributed to a structural reorganization of cognitive processes at a particular age [ 3 ]. However, more research is needed on the effects of the demographic backgrounds of students on mathematical problem-solving skills. These results were different from previous results because the previous studies only analyzed the development in general, without focusing on their demographic background. Hence, different patterns of development were observed when it was thoroughly examined.

Because solving problems is a cognitive process, the development of problem-solving skills for particular phases and processes needed to be analyzed. The students showed the same pattern for knowledge acquisition (i.e., exploring and understanding, and representing and formulating phases), with an increase in skill from grade 7 to grade 8 but a decrease in grade 9. However, the students showed increasing skill in knowledge application (i.e., planning and executing, as well as monitoring and reflecting phases) across grades. This means that the difference between the mean scores in grade 9 was not significant across phases. Grade 9 students had lower scores than grade 8 students for the knowledge acquisition phase but higher scores for the knowledge application phase. In contrast, the gap between the mean scores of grades 7 and 8 was large across phases.

These results proved that there is a significant difference in the mathematical problem-solving skills of students based on their demographic backgrounds. The urban students outperformed rural students, which confirms the results of previous studies [ 9 , 10 , 15 ]. The difference can be attributed to the availability of facilities, teacher quality, and interactive teaching and learning instruction [ 6 ]. In Indonesia, the policies for the public educational system for middle schools are set at the municipal level. This means that each city has its own policies for teacher training, teacher recruitment, teaching and learning processes, facilities, etc. Urban schools mostly have stricter policies as well as various programs to help students improve their knowledge and skills. In addition, they have supportive facilities for teaching and learning. This unequal environment is the strongest reason for the difference in mathematical problem-solving skills.

The results were analyzed in detail to observe which groups in the rural and urban schools contributed to the difference. Both male and female students in urban schools performed better than their counterparts in rural schools did. In addition, urban students in grades 7 and 8 outperformed their rural counterparts. There was no significant difference between urban and rural students in grade 9. This may be because grade 9 is the last grade in middle school, so students have to prepare for high school entrance requirements, including exam and/or grade point average scores. Hence, both rural and urban schools focus much effort on the teaching and learning process in this grade.

In this study, the female students surprisingly had better mathematical problem-solving skills than the male students did. This confirmed the results of the meta-analysis by Hyde et al. [ 32 ] and study by Anjum [ 36 ], which found that female students slightly outperformed male students in mathematics. This difference may be because of motivation and attitude [ 39 , 40 ]. Female Indonesian students are typically more diligent, thorough, responsible, persistent, and serious with their tasks.

A detailed analysis was performed to evaluate which group of students contributed to the gender differences. The results showed that female students outperformed male students in urban schools. This may be because male students at urban schools typically display an unserious attitude toward low-stake tests. In addition, female students outperformed their male counterparts in grades 7 and 9. The reason for this difference requires further analysis.

6. Conclusion

Studying the problem-solving skills of students is crucial to facilitating their development. In this study, the conclusions are presented as follows:

• The mathematical problem-solving skills of the students were categorized as average. More students failed at higher phases of the problem-solving process.
• Students showed development of their mathematical problem-solving skills from grade 7 to grade 8 but a decline in grade 9. The same pattern was detected across grades for urban students, both female and male. However, the problem-solving skills of rural students increased with the grade.
• A similar development was observed for the individual problem-solving phases. In the knowledge acquisition phase, the problem-solving skills of the students developed from grade 7 to grade 8 but decreased in grade 9. However, problem-solving skills increased across grades in the knowledge application phase.
• The school location was shown to have a significant effect on the mathematical problem-solving skills of the students. Urban students generally outperform students in rural schools. However, gender and grade contributed to differences in mathematical problem-solving skills based on school location. Female and male urban students in grades 7 and 8 outperformed their rural counterparts.
• In general, female students outperformed male students in mathematical problem-solving skills, particularly those from urban schools and in grades 7 and 9.

The sampling method and the number of demographic backgrounds limited the scope of this study. Only students from A-accreditation schools were selected because higher-order problem-solving skills were considered assets. Moreover, the study only included three demographic factors: grade, gender, and school location. More demographic information, such as school type, can be added (public or private schools). Hence, future studies will need to broaden the sample size and consider more demographic factors. Despite these limitations, this study can help teachers determine the best period for enhancing the development of mathematical problem-solving skills. Moreover, the differences in mathematical problem-solving skills due to demographic background can be used as a basis for educational policymakers and teachers to provide equal opportunity and equitable education to students.

Author contribution statement

Ijtihadi Kamilia Amalina: Conceived and designed the experiments; Performed the experiments; Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data; Wrote the paper.

Tibor Vidákovich: Conceived and designed the experiments; Contributed reagents, materials, analysis tools or data.

Funding statement

This work was supported by University of Szeged Open Access Fund with the grant number of 6020.

Data availability statement

Additional information.

No additional information is available for this paper.

Declaration of competing interest

No potential conflict of interest was reported by the authors.

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Drill Down technique

Drill Down technique: this article explains the Drill Down technique in a practical way. After reading this article, you’ll understand the basics of this powerful tool for problem solving .

What is the Drill Down technique?

The Drill Down technique is a method for gaining insight into the root causes of a problem within a department or area. After the root causes are known, a larger plan can be devised to address the problem.

A Drill Down is not the same as a diagnosis, but rather a broad and deep general examination. Furthermore, the technique is not used to find out the causes of all problems, but only the 20 percent of the causes behind 80 percent of the effects. This is a principle from the Pareto-analysis .

Method of the Drill Down technique

The technique starts with a table describing the main problem in the leftmost column. The factors and causes that create this problem are then described right next to it in the second column. The idea is to “drill through” until the real causes of the problem are identified. Solutions are then built based on these causes.

The idea here is that it is easier to deal with poor time management than poor quality customer service in general. In addition, some other causes of poor customer service are also discussed.

Step 1: Note down the most important problem

The aim of the first step is to take inventory of all core problems. Be specific in this, and do not generalise or use plurals such as “we” and “they”. Also mention the names of people who are affected by the problem. This is the only way to work on solutions effectively.

Have every individual connected in any way to the problem at hand participate in this Drill Down. You will benefit from this because they each bring their own insight to the brainstorming table. Don’t focus on a rare event or trivial problems. Don’t focus on the pursuit of unrealistic perfection, either.

Leave the search for solutions to the following steps. In the first step it is especially important that the problems are summarised.

Step 2: Identify the causes of the problems

In the second step, the more deeply rooted reasons causing the problems are identified. Often problems arise in different departments because it is not clear who is responsible, or because someone does not account for his or her responsibilities. Direct causes must be distinguished from underlying causes.

To find out the root cause of a problem, the Five Times Why method can be used, for example. Below is an example:

Problem: The project team is working overtime too often and is in danger of burning out
Why? There isn’t enough capacity to meet the team’s demands
Why? Because new responsibilities have been added without extra resources
Why? Because the manager did not correctly estimate the amount of work before taking responsibility
Why? Because the manager is unable to anticipate problems and make plans

Relevant individuals should not be left out while performing this Drill Down technique. At the same time, remember that people tend to respond defensively to criticism. It is the manager’s job to find out the truth and to come up with a good solution. In practice, this can mean that people have to be trained, relocated, or even fired.

After this step, take a short break and then start developing a plan.

Step 3: Make a plan

The third step is to develop a plan that addresses the root causes of a problem. Such an implementation plan works like a script: everything that has to be done and by whom is visualised and recorded. Risk management also plays an important role in this. The likelihood of achieving goals is set against the costs and risks. The plan must consist of at least:

Specific tasks and responsibilities
Measuring variables

Step 4: Implement the plan

Execute the established implementation plan and be transparent in documenting progress. Report at least once a month on actual progress and expectations for the coming period.

Drill Down technique in combination with other methods

The Drill Down technique fits seamlessly with other forms and methods of problem solving. The closest method is the 5-Why analysis . Both methods aim to get to the heart of a problem instead of solving all sorts of other problems first.

Neither method provides a quick way to a solution, but that isn’t the solution that should be sought anyway. Instead, it makes much more sense to have a clear understanding of the situational aspects of doing business.

It is very important that everyone in a company is on the same page when it comes to using the Drill Down technique. The method will not be optimally effective within the company if only a small part of the team uses the method.

Take the time to teach everyone how to get to the root of a problem by zooming in with the Drill Down technique. As indicated, the Drill Down technique does not automatically solve problems, but when used properly it can certainly help to move forward.

Drill Down Technique: pitfalls in general problem solving

Problem solving is not achieved by simply employing methods and frameworks and following them blindly. It is a very broad discipline in which various effects occur that can hinder the way to the solution. In general problem solving and research, there are the following pitfalls to watch out for.

Confirmation bias

Confirmation bias is the tendency for people to seek or interpret information in a way that confirms a person’s previous knowledge, values or beliefs. It is an important type of bias that has a significant effect on the effective performance of problem-solving methods such as the Drill Down technique.

People show this bias when they collect or remember information and interpret it in a biased way. For example, a team member may choose information while preparing for a new task that supports their beliefs and ignore what is not supportive. This effect is strongest when people envision desired outcomes, when a problem is emotionally charged, and for deeply held beliefs.

Perceptual expectations

A perceptual expectation in psychology is also called a set. A set is a group of expectations that shape a specific experience by making people sensitive to certain types of information. It is the disposition or habit to perceive things in a certain way.

This was demonstrated in an experiment by Abraham Luchins in the 1940s. In this experiment, participants were asked to fill a pitcher with a specific amount of water with the aid of only three other pitchers of different capacities.

After Luchins gave the participants this problem that could be solved by a simple technique, he gave them new assignments for other pitchers. This new problem could be solved by the same method, or by a newer and simpler method.

Luchins found that many of his participants tended to use the same old technique, despite the possibility for a better method. Thus, the mental set describes a person’s tendency to solve problems in a way that has previously proven successful.

As in Luchins’ experiment, choosing a method that has worked in the past is sometimes no longer sufficient or optimal for the new problem. Therefore, it is necessary for people to transcend their mental set.

Functional fixation

Functional fixation is a cognitive bias that limits a person to using or accessing an object only as it is traditionally used. This fixation also occurs when solving a problem through the Drill Down Technique. The concept of functional fixation stems from the Gestalt psychological movement.

This movement emphasizes holistic processing. Karl Duncker defined functional fixation as a mental block against using an object in a new way that is necessary to solve a problem. This block limits an individual’s ability to complete a task or solve a problem, as it does not look beyond the original purpose of the components of the solution.

Functional fixation is the inability to see, for example, the use of a hammer as anything different than for hitting nails.

Unnecessary limitations

Unnecessary limitations- or constraints, is a barrier that occurs when people subconsciously set limits on the task at hand. A well-known example of this is the point problem. In this assignment nine points are arranged in a square of three by three.

The task is to draw no more than four lines, without removing the pen or pencil from the paper, to connect all the dots. In the minds of the people who have never seen this problem before, the thought probably arises that the line does not come out of the square of the points. Unnecessary restrictions in this case are about literally thinking ‘outside the box’.

The term group mindset is also linked to unnecessary restrictions. Group thinking, or adopting the mentality of the group members, occurs when team members start to think the same. This is common, but also ensures that people take longer to start thinking “outside the box”.

Irrelevant information

Irrelevant information is information presented within the context of a problem but unrelated to the specific problem. Within the context of the problem, irrelevant information has no influence on whether or not the problem is solved. In fact, irrelevant information is often detrimental to the problem-solving process. Irrelevant information is a common problem that people struggle with. This is mainly because people are not aware of the existence of irrelevant information.

One of the reasons that irrelevant information is so effective in keeping people from the solution is how it is presented. The way information is presented can make a big difference for the level of interpreted difficulty of the problem. Below is a well-known example of irrelevant information in the Buddhist monk problem.

A monk starts walking up a mountain at sunrise one day and reaches the temple at the top of the mountain at sunset. After a few days of meditation, he leaves at sunrise to descend from the mountain. He arrives at sunset. There is a spot along the path the monk takes both ways where he will pass at the same time of the day.

Now it is your turn

What do you think? Do you recognise yourself in the explanation of the Drill Down method? Is this tool used in your own working environment? If not, do you think this could be valuable in your work? What other helpful troubleshooting methods and tools do you know? What do you believe are pros and cons of the Drill Down technique? Do you have any tips or solutions?

Share your experience and knowledge in the comments box below.

More information

de Aguiar Ciferri, C. D., Ciferri, R. R., Forlani, D. T., Traina, A. J. M., & da Fonseca de Souza, F. (2007, March). Horizontal fragmentation as a technique to improve the performance of drill-down and roll-up queries . In Proceedings of the 2007 ACM symposium on Applied computing (pp. 494-499).
Joglekar, M., Garcia-Molina, H., & Parameswaran, A. (2017). Interactive data exploration with smart drill-down . IEEE Transactions on Knowledge and Data Engineering, 31(1), 46-60.
McDonald, A., & Leyhane, T. (2005). Drill down with root cause analysis . Nursing management, 36(10), 26-31.

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Published on: 22/11/2020 | Last update: 04/03/2022

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Ben Janse is a young professional working at ToolsHero as Content Manager. He is also an International Business student at Rotterdam Business School where he focusses on analyzing and developing management models. Thanks to his theoretical and practical knowledge, he knows how to distinguish main- and side issues and to make the essence of each article clearly visible.

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