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open ended problem solving questions

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Open ended math questions and problems for elementary students.

Does your current math instruction involve only situations where there is one answer? Are students expected to solve problems following rigid procedures that do not require critical or creative thinking? These components are important; however, it’s time to take your math instruction to the next level! The answer: open ended math questions!

Open ended math questions, also known as open ended math problems, help learners grow into true mathematicians who use diverse problem solving strategies to explore mathematical situations where there isn’t necessarily one “right” answer. It equips them with the critical thinking skills they need to solve real world problems in the twenty-first century.

a girl looking up and thinking about what math strategy to use

This blog post will answer the following questions:

  • What is an open ended math question?
  • What are the differences between open-ended and closed-ended problems in math?
  • Why should I implement open ended questions in my classroom?
  • What are the disadvantages of using open-ended math problems?
  • How do I implement open ended math questions in my classroom?
  • How do I create open ended math questions?
  • What are some examples of open ended math problems?
  • How do I grade open ended math tasks?

math journal cover and data tracker on a teacher's desk

What is an Open Ended Math Question?

An open ended math question (which is known as an open ended math problem or open ended math task) is a real world math situation presented to students in a word problem format where there is more than one solution, approach, and representation. This instructional strategy is more than reciting a fact or repeating a procedure. It requires students to apply what they have learned while using their problem solving, reasoning, critical thinking, and communication skills to solve a given problem.

This strategy naturally allows for differentiation because of its open-ended nature. In addition, it is a valuable formative assessment tool that allows teachers to assess accuracy in computation and abilities to think of and flexibly apply more than one strategy. In addition to the teacher being able to learn about their students from this tool, the students can thoughtfully extend their learning and reflect on their own thinking through whole group discussions or partner talks.

math prompts for guided math workshop with a pencil

What are the Differences Between Open-Ended and Closed-Ended Problems in Math?

The major difference between open-ended math problems and closed-ended math problems is that close-ended ones have one answer and open-ended ones have more than one answer. This simple difference creates a very different learning experience for elementary students when they work on solving the problem.

elementary students solving open ended math questions

What are the Advantages and Disadvantages of Open Ended Math Problems?

Advantages of open ended math problems.

There are many benefits to using open ended math questions in your classroom. This list of advantages of open ended questions will help you understand their ability to transform your math block! Here are 8 advantages to using open ended math tasks:

  • Provides valuable and specific information to the teacher about student understanding and application of learning
  • Allows the teacher to assess accuracy in computation and abilities to think of and flexibly apply more than one strategy
  • Permits the teacher to see flexibility in student thinking
  • Gives students the opportunity to practice and fine tune their problems solving, reasoning, critical thinking, and communication skills
  • Creates opportunities for real-world application of math
  • Empowers students to extend their learning and reflect on their thinking
  • Fosters creativity, collaboration, and engagement in students
  • Facilitates a differentiated learning experience where all students can access the task

math printables on a clipboard and desk

Disadvantages of Open Ended Math Problems

Although there are tons of benefits to using open ended math problems in your classroom, it is important to note that there are some disadvantages. Here are 3 disadvantages using open ended math tasks:

  • Increases time in collecting data
  • Provides a higher complexity of data
  • Requires the implementation and practice of routines

3 Ways to Implement Open Ended Problems During Your Math Block

Here are 3 ways you can implement open ended problems in your elementary classroom:

  • Start a lesson with an open-ended math problems for students to solve independently. Invite them to share their work and reasoning with a partner. Ask a few students to share their ideas with the whole class.
  • Use the open-ended math problems for fast finishers . If a student or a group of students tend to finish independent work before the rest of the class, invite them to work on an open-ended math problem.
  • Utilize open-ended math problems as a center during math workshop . You will not have to worry about students finishing that math center before it is time to switch to the next center.

open ended math questions in an elementary classroom

3 Ways to Write Open Ended Math Questions with Examples

Here are 3 ways to create open ended math questions accompanied with easy-to-understand open ended math problems examples:

  • Start with a Closed-Ended Question. For example, a closed-ended question could be: What is the sum of 10 plus 10? The related open-ended question would be: The sum is 20. What could the addends be? There are an infinite number of responses because students could use negative numbers.
  • Ask Students to Explain, Prove, or Justify their Thinking. An example of this is, “Prove 5 + 6 = 11.” One possible student response could be that they know the sum is 11 because of the doubles + 1 rule. Another student may take out counters, while another draws a picture.
  • Invite Students to Compare 2 Concepts. For example, ask students to identify the similarities and differences between 2D and 3D shapes. Some possible responses for similarities are that they are both geometry concepts and classifications of shapes. A difference they could say is that 2D shapes are flat, while a 3D shape is solid.

3rd grader drawing quadrilaterals on a geometry math worksheet

How do you Grade Open-Ended Math Questions?

Grading open-ended math tasks is not as clear cut as closed-ended questions. If you are using the task as a formative assessment for your own planning purposes, then you have flexibility on how you choose to evaluate students’ work. However, I recommend you use a rubric if you plan to use it as a summative assessment. Remember to share the rubric with students so that the expectations are clear.

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We would love for you to try these open ended question resources with your students. They offer students daily opportunities to practice solving open ended problems. You can download worksheets specific to your grade level (along with lots of other math freebies) in our free printable math resources bundle using this link: free printable math worksheets for elementary teachers .

Check out my daily open ended problem resources !

  • 1st Grade Open Ended Math Question Problems
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Model Teaching

Opening Minds with Open-Ended Math Problems in the Primary Classroom

by Model Teaching | December 21, 2018.

Wait, so, what is the RIGHT answer?” “Sarah got a different answer than I did…how can we BOTH be right?” You will most likely hear all kinds of responses like this when you start to incorporate open-ended math activities into your classroom. At first, they’ll probably make your students look at you as if you have two heads. But, these kinds of reactions will begin to subside once your students have been exposed to the idea that there are many ways to solve problems, even math problems! Encouraging this kind of “endless possibility” thinking is an effective way to teach your students to challenge themselves and think outside of the “normal” problem solving thinking.

Open ended math stations

What are open-ended math problems?

Open-ended math problems are problems that have more than one possible answer. These problems might present an end result and then ask students to work backward to figure out how that end result might have been achieved or they might ask students to compare two concepts that can be compared in a variety of different ways. But whatever way they are presented, the purpose of open-ended math problems is always to encourage students to use higher order thinking skills to solve problems and understand that some problems can be solved in many ways, with many outcomes.

Examples of Open-ended Math Problems

If you teach pre-k or kindergarten, an open-ended math problem might be: “You have 2 shapes that have a different amount of sides. What 2 shapes could you have? Show and name the shapes.” You would provide them with crayons, paper, pattern blocks, or whatever other manipulative they might be used to using when discussing shapes and students would use these manipulatives to come up with as many answers as they can. Your little ones may answer with a variety of answers based on their current skill level. You may get answers like “triangle and square”, “hexagon and parallelogram”, or “a circle is a shape” depending on what each student knows about shapes. This is a great way to reinforce what students already know and to quickly assess where they are in their knowledge.

If you teach first grade, an open-ended math problem might be: “I’m thinking of the number 8. What two numbers work could work together to make the number 8?” Again, you would provide them with manipulatives they normally use for composing and decomposing numbers, like counters, small erasers, counting bears, unifix cubes, or even playdoh balls. The extra bonus about this kind of problem is that it’s extremely easy for students to show their math skills. Some might use addition, others will use subtraction, and you may even run into a kiddo or two who can use multiplication to find the number. However students choose to explore all the possibilities for answers, be sure to give them a few options for how to show their thinking. This might include simply writing equations, drawing pictures with the equations, or even building the number with a manipulative and then taking a picture of it with an iPad.

As students get older and move onto more abstract thinking in second and third grade, you might incorporate more word problems like: “The difference between the temperature on Monday and Tuesday was 13 degrees. What could the temperature have been on each day? Find and explain at least 5 different answers.” Or “Penelope sees 37 children playing in a corn maze. If those children split into four groups, how many children could be in each group? Find and explain at least 5 different answers.” As always, be sure to provide students with manipulatives, paper and pencils, dry erase markers and whiteboards, or whatever you normally use to help them solve problems and then let them go to work! By presenting these kinds of word problems, you’ll expose students to a variety of math concepts (such as division in this example) just by allowing them to think about how to solve the problem on their own. Then, when these concepts are formally introduced, they will hopefully feel more familiar to some students.

Why should I use open-ended math problems with my students?

There are many benefits to incorporating these kinds of problems into your students’ daily routine, but here are a few of the most obvious and effective ones:

  • Open-ended problems encourage higher order thinking skills. Students will not only be “recognizing”, “identifying”, or “describing” their thinking; they’ll be “justifying”, “defending”, and “evaluating” their problem solving skills and how they arrived at their answers.
  • Open-ended problems build confidence in your students. Once students recognize that there are many possibilities for correct answers and thinking, they begin to participate more readily because they can bring to the table. Students who normally struggle with math might solve the problem on a very basic level, using a basic strategy, but they’ll be correct! And your advanced students can solve it on their advanced level and be just as correct as the student who struggles. Simply knowing that the way that they specifically thought and solved the problem was considered correct builds confidence for students.
  • Open-ended problems are engaging! Students are immediately engaged in these kinds of problems because they recognize that there are so many different ways to solve it. Whether students are working in small groups or independently, there is possibility for so many different ideas and answers to be correct that everyone wants in on it. This engagement, in turn, encourages collaboration among students and soon, they’re sharing their thinking and learning from each other to solve problems.
  • Open-ended problems encourage creativity. Students are capable of using so many strategies that they’ve learned over the years to solve problems and, given the space and time, they can even come up with some of their own strategies for solving problems. Open-ended problems give students permission to be creative in their thinking and problem solving.
  • Open-ended problems make it easy for teachers to see what levels students are working at. Simply by walking around the room while students are working to solve an open-ended math problem, you’ll be able to informally assess what kind of level they are independently working on. This can be extremely beneficial as you are collecting data, forming groups, or simply getting a feel for what kind of skills each student is working with.

For more information about the benefits of using open-ended math problems, read:

https://nzmaths.co.nz/benefits-problem-solving

How do I incorporate open-ended math problems into my math instructional time?

Some of the simplest ways to incorporate open-ended math problems into your math instructional time is to include them in math stations, use them in small groups, and use them as a warm up.

  • Math Stations:  You can implement open-ended problems into your math stations a number of ways, including thinking mats, task cards, or interactive math journals. The simplest way to implement them into your math stations is by using task cards. Task cards are pre-made cards that you can create or purchase to cut and laminate for students to use repeatedly. Task cards usually include words, pictures, diagrams, or a combination of all to present a problem to students. To use task cards in a math station, simply create or purchase the cards you want with open-ended word problems or picture problems. Then, simply print them out and cut/laminate them to make them durable and easily reused. (TEACHER TIP: Most dry erase markers wipe off of lamination pretty easily if it’s wiped off within a reasonable amount of time. Your students may want to mark the important parts of the problem on the actual task card with dry erase marker if you want them to. Just wipe if off after use!) I would suggest storing cards in a labeled plastic container or ziplock bag to keep them organized. It is suggested that you always allow students to use manipulatives as needed, as this can help students feel allowed to express their creative problem-solving thoughts. So, be sure your task card station provides anything students might use to solve problems in their own way: whiteboards, markers, papers, crayons, counters, manipulatives, thinking mats, laminated task cards, etc. For example, if you give students a task card with this problem on it: “Marcy finds 47 apples on the ground. What 3 addends could create this sum? Find and explain at least 5 answers.”, I would provide them with small apple erasers or counters, a whiteboard and dry erase marker, and an iPad to take picture evidence of their five (or more!) answers when they’re finished.  Please refer to pages 10-16 in the resource provided to you below this article for some sample open-ended word problem task cards that you can use with your students immediately!
  • Small Groups: To implement open-ended problems in your small groups, using thinking mats, manipulatives, and prepared open-ended problems is a great way to ease students into working on open-ended problems independently. This is a great way to model your own thinking and problem-solving to allow students to see how they can begin their own ways of solving the problems. Take a moment to download and look at the thinking mat activity in the downloadable resource below. You can incorporate these mats into your small group activity by providing each student with a laminated copy of the mat you want to use and manipulatives for them to work with to follow the mat’s directions. For example, the thinking mat that says “Make patterns out of these shapes and name them.” would be an excellent open-ended activity for a group of kindergartners who are working on shapes OR patterns. Give each student a few of each of the pattern blocks shown on the mat and a dry erase marker. Explain and model how YOU would complete the activity by creating a pattern with the pattern blocks, tracing the blocks or drawing your pattern, and then naming it with letters (ie.: rhombus, rhombus, circle would be named an AAB pattern). This will give your students an idea of what’s expected and their little brains can get started coming up with their own patterns!
  • Warm Up: Using your warm-up time to practice with open-ended problems is a great way to model your own thinking to the whole. Modeling how to solve these problems step-by-step along with the whole class can help give reluctant participants the courage and understanding to participate and ready participants the reassurance that they’re on the right track. As an example, look at “Activity 3: Creating and Solving Problems” in the downloadable resource. You will notice a few thinking mats included, along with cards that correspond to the mats. For a warm-up activity before you begin your lesson for the day, you could give each student a laminated thinking mat and a corresponding manipulative (like, pass out the table and basket cards and give every student some small apple erasers). Then, project a corresponding task card so that everyone can see it. Read the card together, model one way you could solve the problem using your own thinking mat and manipulatives, and then allow students to solve it their own way to find one or two other answers. I would ask students to record their thinking and answers in a math journal or something similar so I could look back on their skills from early in the year and compare them to the end of the year. This is a quick, great way to collect data on student’s skills without a lot of involvement from you!

These are just a few ways to incorporate open-ended problems into your math time. I encourage you to try one way for a week or two and then experiment with another way once your students are showing they feel confident in the first implementation.

How do I make sure to provide students with open ended math problems during math each day?

In order to provide your students with activities and resources that encourage deep thinking and allow every student to participate, detailed planning is required. Deciding what standards and concept you want to focus on and choosing the best way to practice skills related to that concept before having students complete an activity is crucial to creating an effective learning time. If you peek at page 37 of the resource below, you’ll see a planning page that you can use to plan out the open-ended activities you want to use in your classroom. By editing this page with your own information, you can plan for a week of open-ended activities quickly and efficiently. This is also a great way to hold yourself accountable for how often you’re giving your students the opportunity to work on open-ended activities.

Any way you choose to implement open-ended problems in your classroom, your students are sure to grow in their problem-solving abilities and confidence. Creating a space that is safe for your students to take chances and risks with their learning is one of the greatest gifts you can give them. By incorporating ways for your students to express their individual ways of thinking, like open-ended math problems, you’ll foster a love of creative thinking and confidence in problem-solving skills.

Notes About the Included Resources:

The resources included in this blog post are for you to use in your classroom with your students. More detailed explanations for how to incorporate the activities are included in the resources themselves. There are also a few blank templates within the resource so that you can create your own task cards, thinking mats, and activity plans.

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open ended problem solving questions

Open Ended Math Activities for the Primary Classroom

Use these templates and graphic organizers for students who may need additional support. Feel free to download and modify the editable version, including the Frayer model template and word bank template.

IMPLEMENTATION GOAL

Choose or create an open-ended math activity to incorporate into your math instructional time. Plan to introduce the activity to your students at the beginning of the week, model and practice how to complete the activity together, and then allow them to work on the activity for 10-15 minutes per day throughout the rest of the week. Take anecdotal notes about growth you notice and how your students react to these kinds of problems. Do they enjoy them? Dread them? Are you seeing improvement in their thinking and willingness to participate? Take note of these kinds of things as the week goes on. Then, decide what open-ended problems you’ll implement the following week.

  • What is Open-Ended Problem Solving? – https://mste.illinois.edu/users/aki/open_ended/WhatIsOpen-ended.html
  • The Effect of Open-ended Tasks- http://journals.yu.edu.jo/jjes/Issues/2013/Vol9No3/8.pdf
  • Clip art generated by Creative Clips Clipart by Krista Walden, http://www.teacherspayteachers/store/krista-walden

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open ended problem solving questions

Open-Ended Math Questions Reveal Student Thinking

Careful, intentional, and mindful questioning is one of the most powerful tools a skillful teacher possesses (Costa & Kallick, 2000).  Teachers can use open-ended questions during math instruction or assessments to learn how students are problem-solving.

A question is considered open-ended when it is framed in such a way that a variety of responses or approaches are possible (Small, 2009).  As shown in Figure 1, open-ended math questions are designed to uncover student understanding and misunderstandings. The responses are used to inform instruction rather than to make evaluative decisions (Rose & Arline, 2009).

Teachers analyze students’ responses to questions in order to learn how they think.  The responses reveal what students know and how they apply that knowledge.  Teachers then use this information to design instruction that supports student learning.  Additionally, open-ended questions provide opportunities for students to respond and contribute at their respective levels.  This is especially important for struggling students, since they are likely to be passive learners (Lovin, Kyger, & Allsopp, 2004).

Figure 1 illustrates how asking an open-ended question identifies students’ understanding along a continuum.  Teachers use this information to determine where to begin new instruction.

openedendedchart

Figure 1. Continuum of student responses and instructional decisions.

The examples below from Instructional Consultation and Assessment Team (ICAT) Manual Book 3 (Gravois, Gickling, & Rosenfield, 2011, pp. 82-84) illustrates examples of open-ended questions what the responses reveal about students’ understanding of a variety of math concepts.

Specific Assessment: Mathematical Thinking

Good Math Questions Share the Following Features

  • Begin with what the student knows
  • Engage students in the math skills and thinking that you are trying to assess
  • Require more than facts to resolve
  • Are open-ended and have more than one answer

Consider the following types of mathematical open-ended questions:

Number Sense

  • Can the student count 12 items?
  • Does the student know how to begin?
  • Does the student start with 12 and count back, or does she start with a number and add on to 12? (i.e. 10 or 5)
  • Does the student need counters or can he solve it mentally?
  • Does the student demonstrate knowledge of an operational algorithm?
  • What factors of 6 does the student use?
  • Does the student know how to begin the problem?
  • Does the student use counters or can she solve the problem mentally?

[showhide type=”post2″ more_text=”Addition and Subtraction Examples… show more” less_text=”Addition and Subtraction Examples… show less”]

  • How does the student approach the problem?
  • Does the student understand that something is being given away?
  • Can the student count to 8?
  • Does the student need manipulatives?[/showhide]

[showhide type=”post3″ more_text=”Multiplication Examples… show more” less_text=”Multiplication Examples… show less”]

  • Does the student add each leg?
  • Did the student multiply a number or group of number by 2?

[showhide type=”post4″ more_text=”Division Examples… show more” less_text=”Division Examples… show less”]

  • Does the student use flexible strategies to solve and organize the problem?
  • Does the student understand division?
  • Does the student recognize the different steps involved in the problem?
  • Does the student draw a picture or use manipulatives?[/showhide]

[showhide type=”post5″ more_text=”Fraction Examples… show more” less_text=”Fraction Examples… show less”]

  • Does the student understand ¼?
  • Does the student choose a set easily divided by ¼?
  • Does the student understand parts of a set as opposed to parts of a whole?[/showhide]

[showhide type=”post6″ more_text=”Probability Examples… show more” less_text=”Probability Examples… show less”]

  • Does the student know what likely means?
  • Does the student see the problem as a fraction, percent, or ratio?[/showhide]

[showhide type=”post7″ more_text=”Algebra Examples… show more” less_text=”Algebra Examples… show less”]

  • Jacob wrote this statement on his homework paper: 8 + 6 = 7 + 3 + 4 Is this statement true?  How do you know?
  • Does the student work from right to left?
  • Does the student see the equal sign as meaning the same, or does he see it as an action?
  • Does the student add mentally or use another strategy to add the numbers?[/showhide]

[showhide type=”post8″ more_text=”Measurement Examples… show more” less_text=”Measurement Examples… show less”]

  • Does the student recognize and understand perimeter ?
  • Does the student use a number of sides that represents a multiple of 36?[/showhide]

[showhide type=”post9″ more_text=”Geometry Examples… show more” less_text=”Geometry Examples… show less”]

  • Does the student recognize a shape within shapes?
  • Does the student recognize that the triangles keep repeating?
  • Can the student explain how he found the triangles?
  • Does the student understand parallel ?
  • Can the student make more than two items parallel?[/showhide]

[showhide type=”post10″ more_text=”Data Analysis Examples… show more” less_text=”Data Analysis Examples… show less”]

  • What type of graph does the student use to display this data?
  • How does the student interpret the data on the graph?
  • Does the student understand time as intervals?
  • Does the student understand that the data changes over time?
  • How does the student interpret the data on the graph?[/showhide]

[youtube]https://www.youtube.com/watch?v=_ofQ_WnQiZ4[/youtube] (Burn, 1993a)

[youtube]https://www.youtube.com/watch?v=1puQxclB2aw#t=10[/youtube] (Burn, 1993b)

Note to Readers: Please share how you are using open-ended questions in your classes or ask a question related to this topic by posting a comment at the end of this article.

Burn. M. (1993a).   Mathematics: Assessing, understanding Cena. Retrieved from  https://www.youtube.com/watch?v=_ofQ_WnQiZ4

Burn, M. (1993b). Mathematics: Assessing, understanding Jonathan . Retrieved from https://www.youtube.com/watch?v=1puQxclB2aw#t=10

Costa, A. L., & Kallick, B. (Eds.). (2000). Activating and engaging habits of mind. Alexandria, VA:  Association for Supervision and Curriculum Development.

Gravois, T., Gickling, E., & Rosenfield, S. (2011).   ICAT manual book 3 . Baltimore, MD: ICAT Publishing.

Lovin, A., Kyger, M., & Allsopp, D. (2004). Differentiation for special needs learners.      Teaching Children Mathematics , 11 , 158-167.

Rose, C., & Arline, C. (2009). Uncovering student thinking in mathematics, Grades 6-12. Thousand Oaks, CA: Corwin Press.

Small, M., (2009). Good questions: Great ways to differentiate mathematics i nstruction. New York, NY: Teachers College Press.

Categories: Mathematics

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Problem Solving

 A selection of resources containing a wide range of open-ended tasks, practical tasks, investigations and real life problems, to support investigative work and problem solving in primary mathematics.

Problem Solving in Primary Maths - the Session

Quality Assured Category: Mathematics Publisher: Teachers TV

In this programme shows a group of four upper Key Stage Two children working on a challenging problem; looking at the interior and exterior angles of polygons and how they relate to the number of sides. The problem requires the children to listen to each other and to work together co-operatively. The two boys and two girls are closely observed as they consider how to tackle the problem, make mistakes, get stuck and arrive at the "eureka" moment. They organise the data they collect and are then able to spot patterns and relate them to the original problem to find a formula to work out the exterior angle of any polygon. At the end of the session the children report back to Mark, explaining how they arrived at the solution, an important part of the problem solving process.

In a  second video  two maths experts discuss some of the challenges of teaching problem solving. This includes how and at what stage to introduce problem solving strategies and the appropriate moment to intervene when children find tasks difficult. They also discuss how problem solving in the curriculum also helps to develop life skills.

Cards for Cubes: Problem Solving Activities for Young Children

Quality Assured Category: Mathematics Publisher: Claire Publications

This book provides a series of problem solving activities involving cubes. The tasks start simply and progress to more complicated activities so could be used for different ages within Key Stages One and Two depending on ability. The first task is a challenge to create a camel with 50 cubes that doesn't fall over. Different characters are introduced throughout the book and challenges set to create various animals, monsters and structures using different numbers of cubes. Problems are set to incorporate different areas of mathematical problem solving they are: using maths, number, algebra and measure.

open ended problem solving questions

Problem solving with EYFS, Key Stage One and Key Stage Two children

Quality Assured Category: Computing Publisher: Department for Education

These three resources, from the National Strategies, focus on solving problems.

  Logic problems and puzzles  identifies the strategies children may use and the learning approaches teachers can plan to teach problem solving. There are two lessons for each age group.

Finding all possibilities focuses on one particular strategy, finding all possibilities. Other resources that would enhance the problem solving process are listed, these include practical apparatus, the use of ICT and in particular Interactive Teaching Programs .

Finding rules and describing patterns focuses on problems that fall into the category 'patterns and relationships'. There are seven activities across the year groups. Each activity includes objectives, learning outcomes, resources, vocabulary and prior knowledge required. Each lesson is structured with a main teaching activity, drawing together and a plenary, including probing questions.

open ended problem solving questions

Primary mathematics classroom resources

Quality Assured Collection Category: Mathematics Publisher: Association of Teachers of Mathematics

This selection of 5 resources is a mixture of problem-solving tasks, open-ended tasks, games and puzzles designed to develop students' understanding and application of mathematics.

Thinking for Ourselves: These activities, from the Association of Teachers of Mathematics (ATM) publication 'Thinking for Ourselves’, provide a variety of contexts in which students are encouraged to think for themselves. Activity 1: In the bag – More or less requires students to record how many more or less cubes in total...

8 Days a Week: The resource consists of eight questions, one for each day of the week and one extra. The questions explore odd numbers, sequences, prime numbers, fractions, multiplication and division.

Number Picnic: The problems make ideal starter activities

Matchstick Problems: Contains two activities concentrating upon the process of counting and spotting patterns. Uses id eas about the properties of number and the use of knowledge and reasoning to work out the rules.

Colours: Use logic, thinking skills and organisational skills to decide which information is useful and which is irrelevant in order to find the solution.

open ended problem solving questions

GAIM Activities: Practical Problems

Quality Assured Category: Mathematics Publisher: Nelson Thornes

Designed for secondary learners, but could also be used to enrich the learning of upper primary children, looking for a challenge. These are open-ended tasks encourage children to apply and develop mathematical knowledge, skills and understanding and to integrate these in order to make decisions and draw conclusions.

Examples include:

*Every Second Counts - Using transport timetables, maps and knowledge of speeds to plan a route leading as far away from school as possible in one hour.

*Beach Guest House - Booking guests into appropriate rooms in a hotel.

*Cemetery Maths - Collecting relevant data from a visit to a local graveyard or a cemetery for testing a hypothesis.

*Design a Table - Involving diagrams, measurements, scale.

open ended problem solving questions

Go Further with Investigations

Quality Assured Category: Mathematics Publisher: Collins Educational

A collection of 40 investigations designed for use with the whole class or smaller groups. It is aimed at upper KS2 but some activities may be adapted for use with more able children in lower KS2. It covers different curriculum areas of mathematics.

open ended problem solving questions

Starting Investigations

The forty student investigations in this book are non-sequential and focus mainly on the mathematical topics of addition, subtraction, number, shape and colour patterns, and money.

The apparatus required for each investigation is given on the student sheets and generally include items such as dice, counters, number cards and rods. The sheets are written using as few words as possible in order to enable students to begin working with the minimum of reading.

NRICH Primary Activities

Explore the NRICH primary tasks which aim to enrich the mathematical experiences of all learners. Lots of whole class open ended investigations and problem solving tasks. These tasks really get children thinking!

Mathematical reasoning: activities for developing thinking skills

Quality Assured Category: Mathematics Publisher: SMILE

open ended problem solving questions

Problem Solving 2

Reasoning about numbers, with challenges and simplifications.

Quality Assured Category: Mathematics Publisher: Department for Education

Cambridge University Faculty of Mathematics

Or search by topic

Number and algebra

  • The Number System and Place Value
  • Calculations and Numerical Methods
  • Fractions, Decimals, Percentages, Ratio and Proportion
  • Properties of Numbers
  • Patterns, Sequences and Structure
  • Algebraic expressions, equations and formulae
  • Coordinates, Functions and Graphs

Geometry and measure

  • Angles, Polygons, and Geometrical Proof
  • 3D Geometry, Shape and Space
  • Measuring and calculating with units
  • Transformations and constructions
  • Pythagoras and Trigonometry
  • Vectors and Matrices

Probability and statistics

  • Handling, Processing and Representing Data
  • Probability

Working mathematically

  • Thinking mathematically
  • Mathematical mindsets
  • Cross-curricular contexts
  • Physical and digital manipulatives

For younger learners

  • Early Years Foundation Stage

Advanced mathematics

  • Decision Mathematics and Combinatorics
  • Advanced Probability and Statistics

Published 2005 Revised 2017

Problem Solving: Opening up Problems

Types of problems, benefits of open problems and investigations, opening up problems.

  • Remove the restriction: How many coins does it take to make 45p?
  • Remove known: I have a closed handful of 5p coins. How much do I have?
  • Swap the known/unknown, change the restriction: I have 5 coins. Three are the same. How much money do I have?
  • Swap the known/unknown, remove restriction: I have 5 coins. How much money could I have?
  • Remove known and restriction, change unknown: I have some coins in my hand. How much money do I have?
  • Change known, unknown, and restriction: What is the shortest/longest line that can be made with 5 coins?

Cristina Milos

Open-Ended Tasks and Questions in Mathematics

by CristinaM. | Sep 13, 2014 | inquiry , math , thinking | 5 comments

One way to differentiate in math class is creating open-ended tasks and questions (I talked about several differentiation strategies I use here – Mathematically Speaking ).

I think it is useful to clarify the scheme of mathematical problems – below I used Foong Pui’s research paper:

Problem types

“Problems in this classification scheme have their different roles in mathematics instruction as in teaching for problem solving, teaching about problem solving, or teaching via problem solving.”

1.   CLOSED problems   are well-structured problems in terms of clearly formulated tasks where the one correct answer can always be determined in some fixed ways from the necessary data given in the problem situation.

 A. Routine closed problems – are usually multi-step challenging problems that require the use of a specific procedure to arrive to the correct, unique, answer.

B. Non-routine closed problems – imply the use of heuristics strategies * in order to determine, again, a single correct answer.

*Problem-solving heuristics: work systematically, tabulate the data, try simpler examples, look for a pattern, generalize a rule etc.

Routine problem : Minah had a bag of rice. Her family ate an equal amount of rice each day. After 3 days, she had 1/3 of the rice left. After another 7 days, she had 24 kg of rice left. How much rice was in the bag at first?

Non-routine problem : How many squares are there in a chess board?

2. OPEN –ENDED problems – are often named “ill-structured” problems as they involve a higher degree of ambiguity and may allow for several correct solutions. Real-life mathematical problems or mathematical investigations are of this type – e.g. “How much water can our school save on a period of four months?” or “Design a better gym room considering the amount of money we can spend.”

FEATURES of open-ended problems :

  • There is no fixed answer (many possible answers)
  • Solved in different ways and on different levels (accessible to mixed abilities)
  • Empower students to make their own mathematical decisions and make room for own mathematical thinking
  • Develop reasoning and communication skills

HOW do you create open-ended tasks?

Usually, in order to create open-ended questions or problems, the teacher has to work backwards :

  • Indentify a mathematical topic or concept.
  • Think of a closed question and write down the answer.
  • Make up a new question that includes (or addresses) the answer.

STRATEGIES to convert closed problems/questions

  • Turning around a question

CLOSED: What is half of 20?

OPEN: 10 is the fraction of a number. What could the fraction and the number be? Explain.

CLOSED:  Find the difference between 23 and 7.

OPEN: The difference between two numbers is 16. What might the numbers be? Explain your thinking.

CLOSED: Round this decimal to the decimal place 5.7347

OPEN: A number has been rounded to 5.8. What might the number be?

CLOSED: There are 12 apples on the table and some in a basket. In all there are 50 apples. How many apples are in the basket?

OPEN: There are some apples on the table and some in a basket. In all there are 50 apples. How many apples might be on the table? Explain your thinking.

  • Asking for similarities and differences.

Choose two numbers, shapes, graphs, probabilities, measurements etc. and ask students how they are alike and how they are different.

Example: How are 95 and 100 alike? How are they different?

Possible answers:

They are alike because you can skip count by 5s, both are less than 200, both are greater than 90 etc.

They are different because one is a three-digit number, only one ends in 5, only one is greater than 99 etc.

Example: How are the numbers 6.001 and 1.006 alike? How are they different?

  • Asking for explanations.

Example: Compare two fractions with different denominators. Tell how you compare them.

Example: 4 is a factor for two different numbers. What else might be true about both numbers?

  • Creating a sentence

Students are asked to create a mathematical sentence that includes certain numbers and words.

Example: Create a sentence that includes numbers 3 and 4 along with the words “more” and “and”.

  • 3 and 4 are more than 2
  • 3 and 4 together are more than 6
  • 34 and 26 are more than 34 and 20 etc.

Example: Create a question involving multiplication or division of decimals where the digits 4, 9, and 2 appear somewhere.

Example: Create a sentence involving ½  and 64 and the words “less” and “twice as much”.

  • Using “soft” words.

Using the word “close” (or other equivalents) allows for a richer, more interesting mathematical discussion.

Example: You multiply two numbers and the product is almost 600. What could the numbers have been? Explain.

Example: Add two numbers whose sum is close to 750. What can the numbers be? Explain.

Example: Create two triangles with different but close areas. (*instead of, “Create a triangle with an area of 20 square inches.”)

……………………………………………………………………………………………………………………………………………………………………………………………………

A few important considerations are to be made when creating open-ended problems or questions.

  • Know your mathematical focus .
  • Develop questions with the right degree of ambiguity (vague enough to be interesting and to allow for different responses, but not too vague so as students get frustrated).
  • Plan for two types of prompts :
  • enabling prompts (for students who seem unable to start working)
  • extension prompts (for students who finish quickly)

High quality responses from students have the following features:

  • Are systematic (e.g. may record responses in a table or pattern).
  • If the solutions are finite, all solutions are found.
  • If patterns can be found, then they are evident in the response.
  • Where a student has challenged themselves and shown complex examples which satisfy the constraints.
  • Make connections to other content areas.

……………………………………………………………………………………………………………………………………………………………………………………………………………….

References:

Designing Quality Open-Ended Tasks in Mathematics , Louise Hodgson, 2012

Using Short Open-ended Mathematics Questions to Promote Thinking and Understanding , Foong Pui Yee, National Institute of Education, Singapore

Good Questions – Great Ways to Differentiate Mathematics Instruction , Marian Small, 2012

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Jessica

Thank you for posting this. I appreciate how there is a comparison between the two (closed and open ended) types of questions and the considerations that go along with each. Thanks!

CristinaM.

You are welcome!

Jen

Wow, well-written, thank you. I’m excited that my teaching is getting great, clear, and organized at the level that I’m at. But this article reminds me there are many higher levels I can get to, including this area of more open-endedness. Thank you!

I am happy to have helped even in a small way!

drmuneer

Thank you so much

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75 Open-Ended Questions Examples

open-ended questions examples definition and benefits, explained below

Open-ended questions are inquiries that cannot be answered with a simple “yes” or “no” and require elaboration.

These questions encourage respondents to provide more detailed answers, express opinions, and share experiences.

They can be useful in multiple contexts:

  • In conversation , it elicits more information about someone and can help break the ice or deepen your relationship with them.
  • In education , open-ended questions are used as prompts to encourage people to express themselves, demonstrate their knowledge, or think more deeply about other people.
  • In research , they are used to gather detailed responses from research participants who, if not asked open-ended questions, may not give valuable detailed or in-depth responses.

An example of an open-ended question is:

“What did you enjoy most about your recent vacation?”

Open-Ended Questions Examples

Examples of open-ended questions for students.

  • What did you find most interesting or surprising about today’s lesson?
  • How would you explain this concept to someone who has never encountered it before?
  • Can you think of a real-life example of what we are talking about today?
  • When doing the task, what did you find most challenging and why?
  • How does this topic connect to the topic we were discussing in last week’s lesson?
  • When you walk out of this lesson today, what is the most important insight you’ll take with you?
  • When you were solving this problem, what strategies did you draw upon? Can you show them to me?
  • If you could change one thing about how you did today’s task, what would it be and why?
  • How do you feel about the progress you have made in the unit so far, and what areas do you think you need to work on?
  • What questions do you still have about this topic that we can address in our next lesson?
  • How do you think this subject will be relevant to your life outside of the classroom, such as on the weekends or even in the workplace once you leave school?
  • We tried just one way to solve this problem. Can you think of any alternative approaches we could have taken to reach the same results?
  • What resources or strategies do you think were most useful when solving this problem?
  • What were the challenges you faced when completing this group work task and how would you work to resolve them next time?
  • What are some of the possible weaknesses of the theory we’ve been exploring today?
  • How has your understanding of this topic evolved throughout the course of this unit?
  • What are some real-world applications of what we’ve learned today?
  • If you were to design an experiment to test this hypothesis, what would be your approach?
  • Can you think of any counterarguments or alternative perspectives on this issue?
  • How would you rate your level of engagement with this topic, and what factors have influenced your level of interest?

Examples of Open-Ended Questions for Getting to Know People

  • So, can you tell me about the first time you met our mutual friend who introduced us?
  • How did you get interested in your favorite hobby?
  • How have your tastes in music changed over time?
  • Can you explain a memorable memory from your childhood?
  • Are there any books, movies, or TV shows that you’ve enjoyed recently that you could recommend? Why would you recommend them to me?
  • How do you usually spend your weekends or leisure time?
  • Can you tell me about a restaurant experience you had that you really enjoyed and why it was so memorable?
  • What’s your fondest memory of your childhood pet?
  • What first got you interested in your chosen career?
  • If you could learn a new skill or take up a new hobby, what would it be and why?
  • What’s the best piece of advice you’ve ever received from a parent or mentor?
  • If you were to pass on one piece of advice to your younger self, what would lit be?
  • Tell me about something fun you did in the area recently that you could recommend that I do this weekend on a budget of $100?
  • If you could have a think for a second, would you be able to tell me your short-term, medium-term, and long-term personal goals ?
  • If you could travel anywhere in the world, where would you go and why?

Examples of Open-Ended Questions for Interviews

  • Can you tell me about yourself and your background, and how you came to be in your current position/field?
  • How do you approach problem-solving, and what methods have you found to be most effective?
  • Can you describe a particularly challenging situation you faced, and how you were able to navigate it?
  • What do you consider to be your greatest strengths, and how have these played a role in your career or personal life?
  • Can you describe a moment of personal growth or transformation, and what led to this change?
  • What are some of your passions and interests outside of work, and how do these inform or influence your professional life?
  • Can you tell me about a time when you faced criticism or negative feedback, and how you were able to respond to it?
  • What do you think are some of the most important qualities for success in your field, and how have you worked to develop these qualities in yourself?
  • Can you describe a moment of failure or setback, and what you learned from this experience?
  • Looking to the future, what are some of your goals or aspirations, and how do you plan to work towards achieving them?

Examples of Open-Ended Questions for Customer Research

  • What factors influenced your decision to purchase this product or service?
  • How would you describe your overall experience with our customer support team?
  • What improvements or changes would you suggest to enhance the user experience of our website or app?
  • Can you provide an example of a time when our product or service exceeded your expectations?
  • What challenges or obstacles did you encounter while using our product or service, and how did you overcome them?
  • How has using our product or service impacted your daily life or work?
  • What features do you find most valuable in our product or service, and why?
  • Can you describe your decision-making process when choosing between competing products or services in the market?
  • What additional products or services would you be interested in seeing from our company?
  • How do you perceive our brand in comparison to our competitors, and what factors contribute to this perception?
  • What sources of information or communication channels did you rely on when researching our product or service?
  • How likely are you to recommend our product or service to others, and why?
  • Can you describe any barriers or concerns that might prevent potential customers from using our product or service?
  • What aspects of our marketing or advertising caught your attention or influenced your decision to engage with our company?
  • How do you envision our product or service evolving or expanding in the future to better meet your needs?

Examples of Open-Ended Questions for Preschoolers

  • Can you tell me about the picture you drew today?
  • What is your favorite thing to do at school, and why do you like it?
  • How do you feel when you play with your friends at school?
  • What do you think would happen if animals could talk like people?
  • Can you describe the story we read today? What was your favorite part?
  • If you could be any animal, which one would you choose to be and why?
  • What would you like to learn more about, and why does it interest you?
  • How do you help your friends when they’re feeling sad or upset?
  • Can you tell me about a time when you solved a problem all by yourself?
  • What is your favorite game to play, and how do you play it?
  • If you could create your own superhero, what powers would they have and why?
  • Can you describe a time when you were really brave? What happened?
  • What do you think it would be like to live on another planet?
  • If you could invent a new toy, what would it look like and what would it do?
  • Can you tell me about a dream you had recently? What happened in the dream?

Open-Ended vs Closed-Ended Questions

Benefits of open-ended questions.

Above all, open-ended questions require people to actively think. This engages them in higher-order thinking skills (rather than simply providing restricted answers) and forces them to expound on their thoughts.

The best thing about these questions is that they benefit both the questioner and the answerer:

  • Questioner: For the person asking the question, they benefit from hearing a full insight that can deepen their knowledge about their interlocutor.
  • Answerer: For the person answering the question, they benefit because the very process of answering the question helps them to sort their thoughts and clarify their insights.

To expound, below are four of the top benefits.

1. Encouraging critical thinking

When we have to give full answers, our minds have to analyze, evaluate, and synthesize information. We can’t get away with a simple yes or no.

This is why educators embrace open-ended questioning, and preferably questions that promote higher-order thinking .

Expounding on our thoughts enables us to do things like:

  • Thinking more deeply about a subject
  • Considering different perspectives
  • Identifying logical fallacies in our own conceptions
  • Developing coherent and reasoned responses
  • Reflecting on our previous actions
  • Clarifying our thoughts.

2. Facilitating self-expression

Open-ended questions allow us to express ourselves. Imagine only living life being able to say “yes” or “no” to questions. We’d struggle to get across our own personalities!

Only with fully-expressed sentences and monologues can we share our full thoughts, feelings, and experiences. It allows us to elaborate on nuances, express our hesitations, and explain caveats.

At the end of explaining our thoughts, we often feel like we’ve been more heard and we have had the chance to express our full authentic thoughts.

3. Building stronger relationships

Open-ended questioning creates good relationships. You need to ask open-ended questions if you want to have good conversations, get to know someone, and make friends.

These sorts of questions promote open communication, speed up the getting-to-know-you phase, and allow people to share more about themselves with each other.

This will make you more comfortable with each other and give the person you’re trying to get to know a sense that you’re interested in them and actively listen to what they have to say. When people feel heard and understood, they are more likely to trust and connect with others.

Tip: Avoid Loaded Questions

One mistake people make during unstructured and semi-structured interviews is to ask open-ended questions that have bias embedded in them.

For an example of a loaded question, imagine if you asked a question: “why did the shop lifter claim he didn’t take the television without paying?”

Here, you’ve made a premise that you’re asking the person to consent to (that the man was a shop lifter).

A more neutral wording might be “why did the man claim he didn’t take the television without paying?”

The second question doesn’t require the person to consent to the notion that the man actually did the shop lifting.

This might be very important, for example, in cross-examining witnesses in a police station!

When asking questions, use questions that encourage people to provide full-sentence responses, at a minimum. Use questions like “how” and “why” rather than questions that can be answered with a brief point. This will allow people the opportunity to provide more detailed responses that give them a chance to demonstrate their full understanding and nuanced thoughts about the topic. This helps students think more deeply and people in everyday conversation to feel like you’re actually interested in what they have to say.

Chris

Chris Drew (PhD)

Dr. Chris Drew is the founder of the Helpful Professor. He holds a PhD in education and has published over 20 articles in scholarly journals. He is the former editor of the Journal of Learning Development in Higher Education. [Image Descriptor: Photo of Chris]

  • Chris Drew (PhD) https://helpfulprofessor.com/author/chris-drew-phd/ 50 Durable Goods Examples
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Open-Ended Problems

  • Student Generated Media
  • Role-Playing-Activities

open ended problem solving questions

An Open-Ended problem is an activity designed to learn course content within the framework of a realistic problem. These are particularly well suited for courses whose main thrust is to help students develop the capacity for critical thinking and analysis. While Open-Ended problems are primarily preformed in groups, it is not to be confused with group work. The structure of the problem set and the indented outcomes are unique to this method.

Open-Ended problems challenge students by forcing them to identify what they know in relation to a problem, allowing the group to then focus on the aspects of the problem they do not understand. The group then decides which issues to consider in order of importance this is then delegated. Finally the groups integrate new knowledge in the context of the problem, making connections between previous and newly acquired knowledge; this process is repeated until the problem is satisfied.

Math Needs a Makeover

In this video professor Dan Meyer, illustrates an example of decoupling the formulaic method of traditional math education with real world examples. To achieve this, he provides his students with a video of a fish tank, slowly (very slowly) filling up with water. The students inevitably ask themselves, “how long will it take to fill?” That question triggers a though process connected to real world experience allowing non-traditional students the ability to participate at the same level as the students who excel at formulaic mathematics. One question leads to another, and by answering these in order of self-perceived importance the student empowers his or her self toward initiating the initiating question.

Congressional Internships

From: The Practice of Problem Based Learning

You have applied for several writing-intensive internships through the University of Rhode Island’s Office of Experiential Education, and you have been offered a position in the Rhode Island office of Congressman James Langevin. Because House of Representative terms are only two years, you assume you will be writing campaign materials. You have also been told that you will be working on a team with three or four other writing interns.

Questions for Team Discussion

  • Who is James Langevin? What do members of your team already know about him?
  • How long has he been in Congress? Who does he represent?
  • What issues seem to concern him most?
  • What constituent groups and platform have gotten him elected and reelected?
  • What has he done during his time in Congress?
  • What do different interest groups think of him so far?
  • What more do you need to learn about him?
  • Where can you go to learn more about him?

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How to Use Open-Ended Questions with Preschoolers (with 50+ Examples)

Open-ended questions are an effective tool for teaching young children how to think deeper and express their thoughts.

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  • Child development

How to Use Open-Ended Questions with Preschoolers

Imagine one of your preschoolers enjoys playing with a plush dog toy. You ask, “ Do you like dogs? ” They respond with, “ Yes ,” and go back to playing. Conversation over? To better engage with your children, asking a question requiring a more detailed answer is best. “ Do you like dogs? ” becomes “ Why do you like dogs? ”

In this article, we’ll discuss open-ended questions—what they are, their importance in early childhood education, and examples you can use in the classroom.

group of children sitting at a table raising their hands

What is an open-ended question?

An open-ended question is a question that can’t be answered with a “ yes ” or “ no ” response. They’re alternatives to closed-ended questions, which are narrow in focus and can typically be answered with a limited or single-word response. It’s the difference between asking the closed-ended question, “ Do you like playing with this toy? ” versus an open-ended alternative, “ Why do you like playing with this toy? ”

Open-ended questions create a language-rich environment and are key to starting a long conversation. Some significant characteristics of open-ended questions are that they don’t have a right or wrong answer, encourage discussion, and give control to the child. By using questions that start with what if , why , or how , you’re giving your children the opportunity to use their knowledge, feelings, and understanding to answer questions. Additionally, you’re helping them develop and exercise their language skills, which are a strong foundation for their social-emotional development and reading, writing, and math skills.

Why are open-ended questions important?

Through conversation, children can practice and develop many skills, and open-ended questions are at the heart of guiding them. These questions allow children to speak and communicate, problem-solve, control conversations, and think creatively. 

Open-ended questions encourage young children to use their language skills because they require longer answers, which helps them strengthen and build their vocabulary. In answering open-ended questions, children are driven toward expanding their cognitive and problem-solving skills. They have to use their memory, search their mind for words, and form sentences. It requires them to think about their answer and give details to answer the question reasonably. 

An effective way to develop social-emotional skills is through conversation. These skills allow you to understand your thoughts and feelings while being able to relate to others. They also include the ability to express yourself and recognize emotions in others. When you ask young children open-ended questions, it creates a space where they can relate to you and build a relationship with you. It also gives them the freedom to be creative and use their imagination as they express their thoughts and feelings and offer their opinions.

How to ask open-ended questions

There are three components to open-ended questions—asking the question, actively listening, and responding. The purpose of asking open-ended questions is to get children talking, allowing them to use their language skills and participate in the conversation. While there is no exact approach to asking open-ended questions, there are several strategies you can use.

Combine open-ended and closed-ended questions

Closed-ended questions aren’t a great tool to keep conversations going; however, they are a useful way to start one. Use them to introduce a topic to your children. Follow up by asking open-ended questions to allow your children to use their words while expressing their thoughts, feelings, and opinions. For example, in a conversation about sports, you can start by asking the closed-ended question, “ What is your favorite sport? ” To keep the conversation going, follow up with open-ended questions like, “ Why is it your favorite sport? ,” “ How do you feel when you’re playing it? ,” and “ What’s your favorite thing about it? ”

Ask stimulating questions

You wouldn’t expect an adult to enjoy answering “boring” questions, so why would we expect the same of children? In conjunction with asking them stimulating questions, ask questions in a way that relates to them. Ask your children about their interests and experiences. You can ask them about what they’re doing and prompt them to make predictions or provide explanations. The open-ended questions you ask can also guide them to connect concepts back to their own lives and experiences.

Give them time to answer

Language and cognitive development continue past early childhood education. Young children are at the start of this process, so it may take some time for them to hear your question, process the information, think up a response, and express it. When you ask your children an open-ended question, give them time to formulate an answer. You should also allow them to respond without interruption.

Express interest in their answer

As their social-emotional skills develop, children begin to pick up on cues. They notice when someone isn’t interested or paying attention. Just imagine how often you’ve heard your name repeated by a child waiting for your attention. After asking your open-ended question, make sure you express interest. As they respond, use facial expressions and nods to confirm that you’re listening. You can also try repeating parts of their response in any follow-up questions.

Extend their language

To help children increase their vocabulary, it’s necessary to introduce them to new words. Language scaffolding is an effective way to develop and strengthen their skills. Extend their language by repeating concepts or actions with more complex vocabulary. Try restating their language using correct grammar. Summarize their thoughts while making the phrases more complex and adding new words. 

Keep the conversation going

A one-sided conversation isn’t a conversation; it’s practically a monologue. When asking your open-ended questions, keep the conversation going. Offer your experiences and insight into the discussion. This opens the door to asking more questions and allows your children to engage in back-and-forth conversations. 

Watch their expressions and behavior

Non-verbal communication is a key factor when engaging children in dialogue with open-ended questions. Watch their facial expressions and body language to make sure they’re engaged. If you realize they’re no longer interested, switch the topic or move on.

Woman helping child play with number toy

Examples of open-ended questions for preschoolers

Open-ended questions will help you guide a conversation with your children. In addition to general questions, below are several types of questions you can use to guide your preschoolers through experiences, feelings, problem-solving, predictions, and thinking.

  • How did you do that?
  • Why did you do that?
  • How can we find out?
  • What made you think of that?
  • What does that remind you of?
  • Why do you think that happened?
  • How does it work?
  • What could you change?
  • What is your plan?
  • What are you thinking about?

Experiences

  • What’s the funniest thing you’ve ever seen?
  • What sounds do you hear when you go to a park?
  • What do you see when you look outside at night time?
  • What’s the strangest thing you’ve ever eaten?
  • How did you learn to ride your bike?
  • What new things did you learn today?
  • What is your happiest memory?
  • What do you take to bed with you every night?
  • How does a TV work?

What is the first thing you do when you go to the beach?

  • What makes you feel like dancing?
  • What is the nicest thing a friend has ever done for you?
  • What are you excited about in the morning?
  • What makes someone a good friend?
  • What are you thankful for?
  • What makes you feel afraid?
  • Do you like dogs or cats better, and why?
  • What is your favorite toy and why?
  • What is your favorite part of the school day?

Why did you draw that picture?

Problem-solving

What is something you can do today that would make your day better?

  • How can we build a block tower so it’s really tall?
  • What are some different ways to draw a bird?
  • What do you need to make a sandwich?
  • How are dogs and cats the same and different?
  • How can you turn on a TV if you can’t find the remote?
  • What can we do if your backpack is too heavy for you to carry?
  • If your parent’s coffee isn’t sweet enough, how could we make it sweeter?
  • What could someone do if they want to be stronger?
  • What are some ways we can make clean-up time faster?

Predictions

  • What do you think this story is about?
  • What would happen if your parent(s) missed their alarm in the morning?
  • What do you think happens when a worm burrows itself into the dirt?
  • What would happen if you skipped breakfast and lunch?
  • How do you think Mary felt when you shared your toy?
  • What might happen if we built your block tower 10 feet tall?
  • How many books do you think we need in a library?
  • What could we do differently so your block tower doesn’t fall?
  • What happens after an egg hatches?
  • What do you think happens when a bee lands on a flower?
  • What do you think about when you wake up?
  • If you had a million dollars, what would you do with it?
  • What is the best thing about being a grown-up?
  • What do you think is the hardest job in the world and why?
  • If you could go on an adventure anywhere, where would you go?
  • If you could change the color of the ocean to any color you’d like, what would you change it to and why?
  • If you could be any animal, what would it be and why?
  • What do you think a giraffe says?
  • If your toys could talk, what would they say?
  • If you were a teacher, what would you teach your students?

What will you do next?

It’s exceedingly challenging to learn about someone and facilitate a conversation if you keep asking “yes” or “no” questions. Using open-ended questions with your preschoolers effectively builds their language skills and helps with social-emotional development. As you incorporate open-ended questions, ask about their interests, give them time to answer, and show interest. Children are like a sponge—constantly soaking in new information and social cues. The conversation and connections they learn and make with you help set the foundation for the ones they’ll make in the future.

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Daniel Levitin

How to Solve Google's Crazy Open-Ended Interview Questions

brain

One of the most important tools in critical thinking about numbers is to grant yourself permission to generate wrong answers to mathematical problems you encounter. Deliberately wrong answers!

Engineers and scientists do it all the time, so there’s no reason we shouldn’t all be let in on their little secret: the art of approximating, or the “back of the napkin” calculation. As the British writer Saki wrote, “a little bit of inaccuracy saves a great deal of explanation.”

For over a decade, when Google conducted job interviews, they’d ask their applicants questions that have no answers. Google is a company whose very existence depends on innovation—on inventing things that are new and didn’t exist before, and on refining existing ideas and technologies to allow consumers to do things they couldn’t do before.

Contrast this with how most companies conduct job interviews: In the skills portion of the interview, the company wants to know if you can actually do the things that they need doing.

But Google doesn’t even know what skills they need new employees to have. What they need to know is whether an employee can think his way through a problem.

Consider the following question that has been asked at actual Google job interviews: How much does the Empire State Building weigh?

Now, there is no correct answer to this question in any practical sense because no one knows the answer. Google isn’t interested in the answer, though; they’re interested in the process. They want to see a reasoned, rational way of approaching the problem to give them insight into how an applicant’s mind works, how organized a thinker she is.

Excerpted from The Organized Mind Thinking Straight in the Age of Information Overload. By Daniel J Levitin

Excerpted from

There are four common responses to the problem. People throw up their hands and say “that’s impossible” or they try to look up the answer somewhere.

The third response? Asking for more information. By “weight of the Empire State Building,” do you mean with or without furniture? Do I count the people in it? But questions like this are a distraction. They don’t bring you any closer to solving the problem; they only postpone being able to start it.

The fourth response is the correct one, using approximating, or what some people call guesstimating. These types of problems are also called estimation problems or Fermi problems, after the physicist Enrico Fermi, who was famous for being able to make estimates with little or no actual data, for questions that seemed impossible to answer. Approximating involves making a series of educated guesses systematically by partitioning the problem into manageable chunks, identifying assumptions, and then using your general knowledge of the world to fill in the blanks.

How would you solve the Fermi problem of “How many piano tuners are there in Chicago?”

Where to begin? As with many Fermi problems, it’s often helpful to estimate some intermediate quantity, not the one you’re being asked to estimate, but something that will help you get where you want to go. In this case, it might be easier to start with the number of pianos that you think are in Chicago and then figure out how many tuners it would take to keep them in tune.

>There is an infinity of ways one might solve the problem, but the final number is not the point—the thought process, the set of assumptions and deliberations, is the answer.

In any Fermi problem, we first lay out what it is we need to know, then list some assumptions:

1. How often pianos are tuned 2. How long it takes to tune a piano 3. How many hours a year the average piano tuner works 4. The number of pianos in Chicago

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Knowing these will help you arrive at an answer. If you know how often pianos are tuned and how long it takes to tune a piano, you know how many hours are spent tuning one piano. Then you multiply that by the number of pianos in Chicago to find out how many hours are spent every year tuning Chicago’s pianos. Divide this by the number of hours each tuner works, and you have the number of tuners.

Assumption 1: The average piano owner tunes his piano once a year.

Where did this number come from? I made it up! But that’s what you do when you’re approximating. It’s certainly within an order of magnitude: The average piano owner isn’t tuning only one time every ten years, nor ten times a year. One time a year seems like a reasonable guesstimate.

Assumption 2: It takes 2 hours to tune a piano. A guess. Maybe it’s only 1 hour, but 2 is within an order of magnitude, so it’s good enough.

Assumption 3: How many hours a year does the average piano tuner work? Let’s assume 40 hours a week, and that the tuner takes 2 weeks’ vacation every year: 40 hours a week x 50 weeks is a 2,000-hour work year. Piano tuners travel to their jobs—people don’t bring their pianos in—so the piano tuner may spend 10 percent–20 percent of his or her time getting from house to house. Keep this in mind and take it off the estimate at the end.

Assumption 4: To estimate the number of pianos in Chicago, you might guess that 1 out of 100 people have a piano—again, a wild guess, but probably within an order of magnitude. In addition, there are schools and other institutions with pianos, many of them with multiple pianos. This estimate is trickier to base on facts, but assume that when these are factored in, they roughly equal the number of private pianos, for a total of 2 pianos for every 100 people.

Now to estimate the number of people in Chicago. If you don’t know the answer to this, you might know that it is the third-largest city in the United States after New York (8 million) and Los Angeles (4 million). You might guess 2.5 million, meaning that 25,000 people have pianos. We decided to double this number to account for institutional pianos, so the result is 50,000 pianos.

So, here are the various estimates: 1. There are 2.5 million people in Chicago. 2. There are 2 pianos for every 100 people. 3. There are 50,000 pianos in Chicago. 4. Pianos are tuned once a year. 5. It takes 2 hours to tune a piano. 6. Piano tuners work 2,000 hours a year. 7. In one year, a piano tuner can tune 1,000 pianos (2,000 hours per year ÷ 2 hours per piano). 8. It would take 50 tuners to tune 50,000 pianos (50,000 pianos ÷ 1,000 pianos tuned by each piano tuner). 9. Add 15 percent to that number to account for travel time, meaning that there are approximately 58 piano tuners in Chicago.

What is the real answer? The Yellow Pages for Chicago lists 83. This includes some duplicates (businesses with more than one phone number are listed twice), and the category includes piano and organ technicians who are not tuners. Deduct 25 for these anomalies, and an estimate of 58 appears to be very close.

Back to the Google interview and the Empire State Building question. If you were sitting in that interview chair, your interviewer would ask you to think out loud and walk her through your reasoning. There is an infinity of ways one might solve the problem, but to give you a flavor of how a bright, creative, and systematic thinker might do it, here is one possible “answer.” And remember, the final number is not the point—the thought process, the set of assumptions and deliberations, is the answer.

Let’s see. One way to start would be to estimate its size, and then estimate the weight based on that. I’ll begin with some assumptions. I’m going to calculate the weight of the building empty—with no human occupants, no furnishings, appliances, or fixtures. I’m going to assume that the building has a square base and straight sides with no taper at the top, just to simplify the calculations.

For size I need to know height, length, and width. I don’t know how tall the Empire State Building is, but I know that it is definitely more than 20 stories tall and probably less than 200 stories.

I don’t know how tall one story is, but I know from other office buildings I’ve been in that the ceiling is at least 8 feet inside each floor and that there are typically false ceilings to hide electrical wires, conduits, heating ducts, and so on. I’ll guess that these are probably 2 feet. So I’ll approximate 10–15 feet per story.

I’m going to refine my height estimate to say that the building is probably more than 50 stories high. I’ve been in lots of buildings that are 30–35 stories high. My boundary conditions are that it is between 50 and 100 stories; 50 stories work out to being 500–750 feet tall (10–15 feet per story), and 100 stories work out to be 1,000–1,500 feet tall. So my height estimate is between 500 and 1,500 feet. To make the calculations easier, I’ll take the average, 1,000 feet.

Now for its footprint. I don’t know how large its base is, but it isn’t larger than a city block, and I remember learning once that there are typically 10 city blocks to a mile.

>How many uses can you come up with for a broomstick? A lemon? These are skills that can be nurtured beginning at a young age. Most jobs require some degree of creativity and flexible thinking.

A mile is 5,280 feet, so a city block is 1/10 of that, or 528 feet. I’ll call it 500 to make calculating easier. I’m going to guess that the Empire State Building is about half of a city block, or about 265 feet on each side. If the building is square, it is 265 x 265 feet in its length x width. I can’t do that in my head, but I know how to calculate 250 x 250 (that is, 25 x 25 = 625, and I add two zeros to get 62,500). I’ll round this total to 60,000, an easier number to work with moving forward.

Now we’ve got the size. There are several ways to go from here. All rely on the fact that most of the building is empty—that is, it is hollow. The weight of the building is mostly in the walls and floors and ceilings. I imagine that the building is made of steel (for the walls) and some combination of steel and concrete for the floors.

The volume of the building is its footprint times its height. My footprint estimate above was 60,000 square feet. My height estimate was 1,000 feet. So 60,000 x 1,000 = 60,000,000 cubic feet. I’m not accounting for the fact that it tapers as it goes up.

I could estimate the thickness of the walls and floors and estimate how much a cubic foot of the materials weighs and come up then with an estimate of the weight per story. Alternatively, I could set boundary conditions for the volume of the building. That is, I can say that it weighs more than an equivalent volume of solid air and less than an equivalent volume of solid steel (because it is mostly empty). The former seems like a lot of work. The latter isn’t satisfying because it generates numbers that are likely to be very far apart. Here’s a hybrid option: I’ll assume that on any given floor, 95 percent of the volume is air, and 5 percent is steel.

I’m just pulling this estimate out of the air, really, but it seems reasonable. If the width of a floor is about 265 feet, 5 percent of 265 ≈ 13 feet. That means that the walls on each side, and any interior supporting walls, total 13 feet. As an order of magnitude estimate, that checks out—the total walls can’t be a mere 1.3 feet (one order of magnitude smaller) and they’re not 130 feet (one order of magnitude larger).

I happen to remember from school that a cubic foot of air weights 0.08 pounds. I’ll round that up to 0.1. Obviously, the building is not all air, but a lot of it is—virtually the entire interior space—and so this sets minimum boundary for the weight. The volume times the weight of air gives an estimate of 60,000,000 cubic feet x 0.1 pounds = 6,000,000 pounds.

I don’t know what a cubic foot of steel weighs. But I can estimate that, based on some comparisons. It seems to me that 1 cubic foot of steel must certainly weigh more than a cubic foot of wood. I don’t know what a cubic foot of wood weighs either, but when I stack firewood, I know that an armful weighs about as much as a 50-pound bag of dog food. So I’m going to guess that a cubic foot of wood is about 50 pounds and that steel is about 10 times heavier than that. If the entire Empire State Building were steel, it would weigh 60,000,000 cubic feet x 500 pounds = 30,000,000,000 pounds.

This gives me two boundary conditions: 6 million pounds if the building were all air, and 30 billion pounds if it were solid steel. But as I said, I’m going to assume a mix of 5 percent steel and 95 percent air. 5% x 30 billion = 1,500,000,000 + 95% x 6 million = 5,700,000 __________________ 1,505,700,000 pounds or roughly 1.5 billion pounds. Converting to tons, 1 ton = 2,000 pounds, so 1.5 billion pounds/2,000 = 750,000 tons.

This hypothetical interviewee stated her assumptions at each stage, established boundary conditions, and then concluded with a point estimate at the end, of 750,000 tons. Nicely done!

Another job interviewee might approach the problem much more parsimoniously. Using the same assumptions about the size of the building, and assumptions about its being empty, a concise protocol might come down to this.

Skyscrapers are constructed from steel. Imagine that the Empire State Building is filled up with cars. Cars also have a lot of air in them, they’re also made of steel, so they could be a good proxy. I know that a car weighs about 2 tons and it is about 15 feet long, 5 feet wide, and 5 feet high. The floors, as estimated above, are about 265 x 265 feet each. If I stacked the cars side by side on the floor, I could get 265/15 = 18 cars in one row, which I’ll round to 20 (one of the beauties of guesstimating).

How many rows will fit? Cars are about 5 feet wide, and the building is 265 feet wide, so 265/5 = 53, which I’ll round to 50. That’s 20 cars x 50 rows = 1,000 cars on each floor. Each floor is 10 feet high and the cars are 5 feet high, so I can fit 2 cars up to the ceiling. 2 x 1,000 = 2,000 cars per floor. And 2,000 cars per floor x 100 floors = 200,000 cars. Add in their weight, 200,000 cars x 4,000 pounds = 800,000,000 pounds, or in tons, 400,000 tons.

These two methods produced estimates that are relatively close—one is a bit less than twice the other—so they help us to perform an important sanity check. Because this has become a somewhat famous problem (and a frequent Google search), the New York State Department of Transportation has taken to giving their estimate of the weight, and it comes in at 365,000 tons. So we find that both guesstimates brought us within an order of magnitude of the official estimate, which is just what was required.

These so-called back-of-the-envelope problems are just one window into assessing creativity. Another test that gets at both creativity and flexible thinking without relying on quantitative skills is the “name as many uses” test.

For example, how many uses can you come up with for a broomstick? A lemon? These are skills that can be nurtured beginning at a young age. Most jobs require some degree of creativity and flexible thinking.

As an admissions test for flight school for commercial airline pilots, the name-as-many-uses test was used because pilots need to be able to react quickly in an emergency, to be able to think of alternative approaches when systems fail. How would you put out a fire in the cabin if the fire extinguisher doesn’t work? How do you control the elevators if the hydraulic system fails?

Exercising this part of your brain involves harnessing the power of free association—the brain’s daydreaming mode—in the service of problem solving, and you want pilots who can do this in a pinch. This type of thinking can be taught and practiced, and can be nurtured in children as young as five years old. It is an increasingly important skill in a technology-driven world with untold unknowns.

There are no right answers, just opportunities to exercise ingenuity, find new connections, and to allow whimsy and experimentation to become a normal and habitual part of our thinking, which will lead to better problem solving.

Excerpt from THE ORGANIZED MIND: Thinking Straight in the Age of Information Overload . Copyright © 2014 by Daniel Levitin. Reprinted by arrangement with Dutton, a member of Penguin Group (USA) LLC, A Penguin Random House Company.

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General Scoring Rubric For Open-Ended Questions

Here is an example rubric, in this case a general scoring rubric for open-ended questions…

Sort papers first into three stacks:

  • good responses (5 or 6 points),
  • adequate responses (3 or 4 points), and
  • inadequate responses (1 or 2 points).

Each of those three stacks then can be re-sorted into 2 stacks…

  • Good Responses:
  • * Exemplary=6 points Gives a complete response with a clear, coherent, unambiguous, and elegant explanation; includes a clear and simplified diagram; communicates effectively to the identified audience. shows understanding of the open-ended problem’s..ideas and processes; identifies all the important elements of the problem; may include examples and counterexamples; provides strong supporting arguments.
  • * Competent=5 points Gives a fairly complete response with reasonably clear explanations; may include an appropriate diagram; communicates effectively to the identified audience; shows understanding of the problem’s..ideas and processes; identifies the most important elements of the problem; presents solid supporting arguments.
  • Adequate Responses:
  • * Satisfaction=4 but minor flaws Completes the problem satisfactorily but the explanation may be muddled; argumentation may be incomplete; diagram may be inapropriate or unclear; understands the underlying…ideas and uses them effectively.
  • * Nearly satisfactory=3 but serious flaws Begins the problem apropriately but may fail to complete or may omit significant parts of the problem; may fail to show full understanding of ideas and processes; may make major computational errors; may misuse or fail to use correct terms; response may reflect an inappropriate strategy for solving the problem.
  • Inadequate Responses:
  • * Begins but fails to complete problem=2 Explanation is not understandable; diagram may be unclear; shows no understanding of the problem situation; may make major computational errors;
  • * Unable to begin effectively=1 Words do not reflect the problem; drawings misrepresent the problem situation copies parts of the problem but without attempting a solution; fails to indicate which information is appropriate to problem.
  • * No attempt=0.

[ Originally published in Assessment of Authentic Performance in School Mathematics (p. 159) edited by Richard Lesh and Susan J. Lamon, 1992 ISBN 0-87168-5 ] 

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    These interactive activity cards are all about getting children to problem solve and present an argument. Designed specifically for Year 5-6 children, they will provide a great challenge by forcing them to think outside the box to come up with an answer. There are 27 individual open-ended maths activities included. Each is teacher-made, saving you time on lesson planning while ensuring that ...

  14. 75 Open-Ended Questions Examples (2024)

    Open-ended questions are inquiries that cannot be answered with a simple "yes" or "no" and require elaboration. These questions encourage respondents to provide more detailed answers, express opinions, and share experiences. They can be useful in multiple contexts:

  15. How to Ask Open-Ended Questions That Spark Good Conversation

    Open-ended questions are questions that are designed to encourage people to share more than a one-word response and typically start with words like "what," "how," or "why.". Open-ended questions help people expound on an idea or issue and carry the conversation forward without getting stunted in potentially awkward silence or little ...

  16. PDF Open-Ended Problem-Solving Projections

    specific "share" questions from the 1-10 on the Problem-solving Questions page. The process should be followed by students for all open-ended questions is to answer questions 1-3, then complete the solution to the problem, and then finally answer questions 4-10.

  17. Open-Ended Questions: Everything You Need to Know

    Additionally, open-ended questions can be used as a tool for problem-solving and critical thinking. By encouraging individuals to think outside the box and explore alternative perspectives, open-ended questions can help generate creative solutions.

  18. How Open-Ended Questions Boost Creative Problem Solving

    1 Why ask open-ended questions? Open-ended questions can help you understand the problem better, by revealing the needs, wants, and expectations of the people involved. They can also...

  19. Open-Ended Problems

    An Open-Ended problem is an activity designed to learn course content within the framework of a realistic problem. These are particularly well suited for courses whose main thrust is to help students develop the capacity for critical thinking and analysis. While Open-Ended problems are primarily preformed in groups, it is not to be confused ...

  20. 18 Tough Open-Ended Questions (And How To Answer Them)

    Updated November 30, 2023 Interviewers ask open-ended interview questions during the hiring process to learn more about a candidate's experience and relevant abilities. The ability to answer open-ended interview questions in a detailed and thoughtful manner can show your problem-solving and critical-thinking skills.

  21. How to Use Open-Ended Questions with Preschoolers (with 50+ Examples)

    In answering open-ended questions, children are driven toward expanding their cognitive and problem-solving skills. They have to use their memory, search their mind for words, and form sentences. It requires them to think about their answer and give details to answer the question reasonably.

  22. How to Solve Google's Crazy Open-Ended Interview Questions

    Consider the following question that has been asked at actual Google job interviews: How much does the Empire State Building weigh? Now, there is no correct answer to this question in any...

  23. General Scoring Rubric For Open-Ended Questions

    General Scoring Rubric For Open-Ended Questions. Here is an example rubric, in this case a general scoring rubric for open-ended questions…. Sort papers first into three stacks: good responses (5 or 6 points), adequate responses (3 or 4 points), and. inadequate responses (1 or 2 points). Each of those three stacks then can be re-sorted into 2 ...

  24. Brooke

    301 likes, 114 comments - facelessfreedommarketing on February 14, 2024: "Just my personal two cents 睊 Sales is just this: one's ability to communicate conviction..."