critical thinking question in math

• Math Resources Links
• Math in the Real World
• Differentiated Math Unlocked
• Math in the Real World Workshop

20 Math Critical Thinking Questions to Ask in Class Tomorrow

• November 20, 2023

The level of apathy towards math is only increasing as each year passes and it’s up to us as teachers to make math class more meaningful . This list of math critical thinking questions will give you a quick starting point for getting your students to think deeper about any concept or problem. 

Since artificial intelligence has basically changed schooling as we once knew it, I’ve seen a lot of districts and teachers looking for ways to lean into AI rather than run from it.

The idea of memorizing formulas and regurgitating information for a test is becoming more obsolete. We can now teach our students how to use their resources to make educated decisions and solve more complex problems.

With that in mind, teachers have more opportunities to get their students thinking about the why rather than the how.

Table of Contents

Looking for more about critical thinking skills? Check out these blog posts:

• Why You Need to Be Teaching Writing in Math Class Today
• How to Teach Problem Solving for Mathematics
• Turn the Bloom’s Taxonomy Verbs into Engaging Math Activities

What skills do we actually want to teach our students?

As professionals, we talk a lot about transferable skills that can be valuable in multiple jobs, such as leadership, event planning, or effective communication. The same can be said for high school students. 

It’s important to think about the skills that we want them to have before they are catapulted into the adult world. 

Do you want them to be able to collaborate and communicate effectively with their peers? Maybe you would prefer that they can articulate their thoughts in a way that makes sense to someone who knows nothing about the topic.

Whatever you decide are the most essential skills your students should learn, make sure to add them into your lesson objectives.

When should I ask these math critical thinking questions?

Critical thinking doesn’t have to be complex or fill an entire lesson. There are simple ways that you can start adding these types of questions into your lessons daily!

Start small

Add specific math critical thinking questions to your warm up or exit ticket routine. This is a great way to start or end your class because your students will be able to quickly show you what they understand. 

Asking deeper questions at the beginning of your class can end up leading to really great discussions and get your students talking about math.

Add critical thinking questions to word problems

Word problems and real-life applications are the perfect place to add in critical thinking questions. Real-world applications offer a more choose-your-own-adventure style assignment where your students can expand on their thought processes. 

They also allow your students to get creative and think outside of the box. These problem-solving skills play a critical role in helping your students develop critical thinking abilities.

Keep reading for math critical thinking questions that can be applied to any subject or topic!

When you want your students to defend their answers.

• Explain the steps you took to solve this problem
• How do you know that your answer is correct?
• Draw a diagram to prove your solution.
• Is there a different way to solve this problem besides the one you used?
• How would you explain _______________ to a student in the grade below you?
• Why does this strategy work?
• Use evidence from the problem/data to defend your answer in complete sentences.

When you want your students to justify their opinions

• What do you think will happen when ______?
• Do you agree/disagree with _______?
• What are the similarities and differences between ________ and __________?
• What suggestions would you give to this student?
• What is the most efficient way to solve this problem?
• How did you decide on your first step for solving this problem?

When you want your students to think outside of the box

• How can ______________ be used in the real world?
• What might be a common error that a student could make when solving this problem?
• How is _____________ topic similar to _______________ (previous topic)?
• What examples can you think of that would not work with this problem solving method?
• What would happen if __________ changed?
• Create your own problem that would give a solution of ______________.
• What other math skills did you need to use to solve this problem?

Let’s Recap:

• Rather than running from AI, help your students use it as a tool to expand their thinking.
• Identify a few transferable skills that you want your students to learn and make a goal for how you can help them develop these skills.
• Add critical thinking questions to your daily warm ups or exit tickets.
• Ask your students to explain their thinking when solving a word problem.
• Get a free sample of my Algebra 1 critical thinking questions ↓

8 thoughts on “20 Math Critical Thinking Questions to Ask in Class Tomorrow”

I would love to see your free math writing prompts, but there is no place for me to sign up. thank you

Ahh sorry about that! I just updated the button link!

Pingback:  How to Teach Problem Solving for Mathematics -

Pingback:  5 Ways Teaching Collaboration Can Transform Your Math Classroom

Pingback:  3 Ways Math Rubrics Will Revitalize Your Summative Assessments

Pingback:  How to Use Math Stations to Teach Problem Solving Skills

Pingback:  How to Seamlessly Add Critical Thinking Questions to Any Math Assessment

Pingback:  13 Math Posters and Math Classroom Ideas for High School

Leave a Reply Cancel reply

Your email address will not be published. Required fields are marked *

Publications

On-demand strategy, speaking & workshops, latest articles, write for us, library/publications.

• Competency-Based Education
• Early Learning
• Equity & Access
• Personalized Learning
• Place-Based Education
• Post-Secondary
• Project-Based Learning
• SEL & Mindset
• STEM & Maker
• The Future of Tech and Work

Kyle Wagner on 12 Shifts for Student-Centered Environments

Town hall: back to school with ai, rachel davison humphries on the bill of rights institute and the importance of civics projects, betsy revell and marie mackintosh on employindy and modern apprenticeships, recent releases.

Health Science Pathways Guide

New Pathways Handbook: Getting Started with Pathways

Unfulfilled Promise: The Forty-Year Shift from Print to Digital and Why It Failed to Transform Learning

The Portrait Model: Building Coherence in School and System Redesign

Green Pathways: New Jobs Mean New Skills and New Pathways

Support & Guidance For All New Pathways Journeys

Unbundled: Designing Personalized Pathways for Every Learner

Credentialed Learning for All

AI in Education

For more, see Publications |  Books |  Toolkits

Microschools

New learning models, tools, and strategies have made it easier to open small, nimble schooling models.

Horizon Three (H3) Learning

Focused on charting and defining the future-ready education system we need to serve and prepare all learners.

Difference Making

Focusing on how making a difference has emerged as one of the most powerful learning experiences.

New Pathways

This campaign will serve as a road map to the new architecture for American schools. Pathways to citizenship, employment, economic mobility, and a purpose-driven life.

Green Schools

The climate crisis is the most complex challenge mankind has ever faced . We’re covering what edleaders and educators can do about it. 

Schools Worth Visiting

We share stories that highlight best practices, lessons learned and next-gen teaching practice.

View more series…

About Getting Smart

Getting smart collective, impact update, talking math: 100 questions that help promote mathematical discourse.

Update 2022: For a recent article on how to talk about math, click here. 

Think about the questions that you ask in your math classroom. Can they be answered with a simple “yes” or “no,” or do they open a door for students to really share their knowledge in a way that highlights their true understanding and uncovers their misunderstandings? Asking better questions can open new doors for students, helping to promote mathematical thinking and encouraging classroom discourse. Such questions help students:

• Work together to make sense of mathematics.
• Rely more on themselves to determine whether something is mathematically correct.
• Learn to reason mathematically.
• Evaluate their own processes and engage in productive peer interaction.
• Discover and seek help with problems in their comprehension.
• Learn to conjecture, invent and solve problems.
• Learn to connect mathematics, its ideas and its applications.
• Focus on the mathematical skills embedded within activities.

Dr. Gladis Kersaint

Help students work together to make sense of mathematics

• What strategy did you use?
• Do you agree?
• Do you disagree?
• Would you ask the rest of the class that question?
• Could you share your method with the class?
• What part of what he said do you understand?
• Would someone like to share ___?
• Can you convince the rest of us that that makes sense?
• What do others think about what [student] said?
• Can someone retell or restate [student]’s explanation?
• Did you work together? In what way?
• Would anyone like to add to this?
• Have you discussed this with your group? With others?
• Did anyone get a different answer?
• Where would you go for help?
• Did everybody get a fair chance to talk, to use the manipulatives, or to be recorded?
• How could you help another student without telling the answer?
• How would you explain ___ to someone who missed class today?

Refer questions raised by students back to the class.

Help students rely more on themselves to determine whether something is mathematically correct

• Is this a reasonable answer?
• Does that make sense?
• Why do you think that? Why is that true?
• Can you draw a picture or make a model to show that?
• How did you reach that conclusion?
• Does anyone want to revise his or her answer?
• How were you sure your answer was right?

Help students learn to reason mathematically

• How did you begin to think about this problem?
• What is another way you could solve this problem?
• How could you prove that?
• Can you explain how your answer is different from or the same as [student]’s?
• Let’s see if we can break it down. What would the parts be?
• Can you explain this part more specifically?
• Does that always work?
• Is that true for all cases?
• How did you organize your information? Your thinking?

Help students evaluate their own processes and engage in productive peer interaction

• What do you need to do next?
• What have you accomplished?
• What are your strengths and weaknesses?
• Was your group participation appropriate and helpful?

Help students with problem comprehension

• What is this problem about? What can you tell me about it?
• Do you need to define or set limits for the problem?
• How would you interpret that?
• Would you please reword that in simpler terms?
• Is there something that can be eliminated or that is missing?
• Would you please explain that in your own words?
• What assumptions do you have to make?
• What do you know about this part?
• Which words were most important? Why?

Help students learn to conjecture, invent and solve problems

• What would happen if ___? What if not?
• Do you see a pattern?
• What are some possibilities here?
• Where could you find the information you need?
• How would you check your steps or your answer?
• What did not work?
• How is your solution method the same as or different from [student]’s?
• Other than retracing your steps, how can you determine if your answers are appropriate?
• What decision do you think he or she should make?
• How did you organize the information? Do you have a record?
• How could you solve this using (tables, trees, lists, diagrams, etc.)?
• What have you tried? What steps did you take?
• How would it look if you used these materials?
• How would you draw a diagram or make a sketch to solve the problem?
• Is there another possible answer? If so, explain.
• How would you research that?
• Is there anything you’ve overlooked?
• How did you think about the problem?
• What was your estimate or prediction?
• How confident are you in your answer?
• What else would you like to know?
• What do you think comes next?
• Is the solution reasonable, considering the context?
• Did you have a system? Explain it.
• Did you have a strategy? Explain it.
• Did you have a design? Explain it.

Help students learn to connect mathematics, its ideas and its application

• What is the relationship of this to that?
• Have we ever solved a problem like this before?
• What uses of mathematics did you find in the newspaper last night?
• What is the same?
• What is different?
• Did you use skills or build on concepts that were not necessarily mathematical?
• Which skills or concepts did you use?
• What ideas have we explored before that were useful in solving this problem?
• Is there a pattern?
• Where else would this strategy be useful?
• How does this relate to ___?
• Is there a general rule?
• Is there a real-life situation where this could be used?
• How would your method work with other problems?
• What other problem does this seem to lead to?

Help students persevere

• Have you tried making a guess?
• What else have you tried?
• Would another recording method work as well or better?
• Is there another way to (draw, explain, say) that?
• Give me another related problem. Is there an easier problem?
• How would you explain what you know right now?

Help students focus on the mathematics from activities

• What was one thing you learned (or two, or more)?
• Where would this problem fit on our mathematics chart?
• How many kinds of mathematics were used in this investigation?
• What were the mathematical ideas in this problem?
• What is the mathematically different about these two situations?
• What are the variables in this problem? What stays constant?

Facilitating student engagement in mathematical discourse begins with the decisions teachers make when they plan classroom instruction. In the next and final blog in this series, we will dive into the specific strategies that teachers can use to foster meaningful conversations about what students are thinking, doing and learning.

This blog is part of a three-post series on the importance of mathematical discourse from Curriculum Associates   and Dr. Gladis Kersaint, the author of the recently published whitepaper Orchestrating Mathematical Discourse to Enhance Student Learning . Download your free copy here . For more on mathematical discourse and Curriculum Associates, check out:

• Talking Math: How to Engage Students in Mathematical Discourse
• Talking Math: 6 Strategies for Getting Students to Engage in Mathematical Discourse
• Curriculum Associates: Leveraging For-profit Power With a Nonprofit Purpose

Dr. Gladis Kersaint is a Professor of Mathematics Education at the University of Connecticut.

Stay in-the-know with all things EdTech and innovations in learning by signing up to receive the weekly Smart Update .  This post includes mentions of a Getting Smart partner. For a full list of partners, affiliate organizations and all other disclosures please see our Partner page .

Guest Author

Discover the latest in learning innovations.

Sign up for our weekly newsletter.

Related Reading

Book Review: The Math(s) Fix

12 Shifts to Move from Teacher-Led to Student-Centered Environments

Empowering Math Teachers to be Culturally Responsive

The portrait model, melissa long.

I love how you have the questions categorized by outcome goal. The infographic is one that I will be printing and using very often next year in my middle school classroom.

Joan Arumemi

It's an amazing application and approach in addressing math issues.Shall use them in my class .

Leave a Comment

Your email address will not be published. All fields are required.

Nominate a School, Program or Community

Stay on the cutting edge of learning innovation.

Subscribe to our weekly Smart Update!

Smart Update

What is pbe (spanish), designing microschools download, download quick start guide to implementing place-based education, download quick start guide to place-based professional learning, download what is place-based education and why does it matter, download 20 invention opportunities in learning & development.

Check Out the New Website Shop!

Novels & Picture Books

Anchor Charts

6 Strategies for Increasing Critical Thinking with Problem Solving

By Mary Montero

Share This Post:

• Facebook Share
• Twitter Share
• Pinterest Share
• Email Share

For many teachers, problem-solving feels synonymous with word problems, but it is so much more. That’s why I’m sharing my absolute favorite lessons and strategies for increasing critical thinking through problem solving below. You’ll learn six strategies for increasing critical thinking through mathematical word problems, the importance of incorporating error analysis into your weekly routines,  and several resources I use for improving critical thinking – almost all of which are free! I’ll also briefly touch on teaching students to dissect word problems in a way that enables them to truly understand what steps to take to solve the problem.

This post is based on my short and sweet (and FREE!) Increasing Critical Thinking with problem Solving math mini-course . When you enroll in the free course you’ll get access to everything you need to get started:

• Problem Solving Essentials
• Six lessons to implement into your classroom
• How to Implement Error Analysis
• FREE Error Analysis Starter Kit
• FREE Mathematician Posters
• FREE Multi-Step Problem Solving Starter Kit
• FREE Task Card Starter Kit

Introduction to Critical Thinking and Problem Solving

According to the National Council of Teachers of Mathematics, “The term “problem solving” refers to mathematical tasks that have the potential to provide intellectual challenges for enhancing students’ mathematical understanding and development .)”

That’s a lot of words, but I’d like to focus in on the word POTENTIAL. I’m going to share with you strategies that move these tasks from having the potential to provide a challenge to actually providing that challenge that will enrich their mathematical understanding and development. 

If you’re looking for an introduction to multi-step problem solving, I have a free multi-step problem solving starter kit for that! 

I also highly encourage you to download and use my free Mathematician posters that help students see what their “jobs” are as mathematicians. Giving students this title of mathematician not only holds them accountable, but it gives them greater confidence and gives me very specific verbiage to use when discussing math with my students. 

The impacts of Incorporating Problem Solving

When I made the shift to incorporate problem solving into my everyday instruction intentionally, I saw a distinct increase in student understanding and application of mathematical concepts, more authentic connections to real-world mathematics scenarios, greater student achievement, and notably increased engagement. There are also ripple effects observed in other areas, as students learn grit and a growth mindset after tackling some more challenging problem-solving situations. I hope that by implementing some of these ideas, you see the very same shift.

Here’s an overview of some problem solving essentials I use to teach students to solve problems.

Routine vs. Non-Routine Problem Solving

Routine problems comprise the vast majority of the word problems we pose to students. They require using an algorithm through one or more of the four major operations, have relevance to real-world situations, and often have a distinct answer. They are solvable, and students can use several concrete strategies for solving, like “make a table” or “draw a picture” to solve.

Conversely, non-routine problem-solving focuses on mathematical reasoning. These are often more open-ended and allow students to make generalizations about math and numbers. There isn’t usually a straight path leading to the answer, there isn’t an algorithm readily available for finding the solution (or students are going to have to come up with the algorithm), and it IS going to require some level of experimentation and manipulation of numbers in order to solve it. In non-routine problems, students learn to look for patterns, work backwards, build models, etc. 

Incorporating both routine and non-routine problems into your instruction for EVERY student is critical. When solving non-routine problems, students can use some of the strategies they’ve learned for solving routine problems, and when solving routine problems, students benefit from a deeper understanding of the complexity of numbers that they gained from non-routine problems. For this training, we will focus heavily on routine problems, though the impacts of these practices will transition into non-routine problem solving.

Increasing Critical Thinking in Problem Solving

When tackling a problem, students need to be able to determine WHAT to do and HOW to do it.  Knowing the HOW is what you likely teach every day – your students know how to add, subtract, multiply, and divide. But knowing WHAT to do is arguably the most essential part of solving problems – once students know what needs to be done, then they can apply the conceptual skills – the algorithms and strategies – they’ve learned and will know how to solve. While dissecting word problems is an excellent starting point, exposing students to various ways to examine problems can help them figure out the WHAT. 

Being faced with a lengthy, complex word problem can be intimidating to even your most adept students. Having a toolbox of strategies to use when you tackle problems and seeing problems in various ways can enable students to get to the point where they feel comfortable knowing where to begin.

Shifting away from keywords

While it isn’t best practice to rely solely on operation “keywords” to determine what operation needs to occur when solving a problem, I’m not ready to fully ditch keyword-based instruction in math. I think there’s a huge difference between teaching students to blindly rely on keywords to determine which operation to use for a solution and using words found in the text to guide students in figuring out what to do. For that reason, I place heavy emphasis on using precise mathematical vocabulary , including specific operation keywords, and when students become accustomed to using that precise mathematical vocabulary every day, it really helps them to identify that language in word problems as well.

I also allow my students to dissect math word problems using strategies like CUBES , but in a way that is more aligned with best practice. 

Six Lessons for Easy Implementation

Here are six super quick “outside the box” word problem, problem solving lessons to begin implementing into your classroom. These lessons shouldn’t replace your everyday problem solving, but are instead extensions that will help students tackle those tricky problems they encounter everyday. As a reminder, we look at all of these lessons in the FREE Increasing Critical Thinking with problem Solving math mini-course .

Lesson #1: What’s the Question?

In this lesson, we’ll encourage students to see. just how many different questions can be asked about the same statements or information. We start with a typical, one-step, one-operation problem. Then we cross out or cover up the answer and ask students to generate possible questions.

After students have come up with a variety of questions, ask them to determine HOW they would solve for each one.

Reveal the question and ask students how they would solve this one and see if any of the questions they came up with match.

This activity is important because it demonstrates to students just how many different questions can be asked about the same statement or information. It’s perfect for your students who automatically pick out numbers and start “operating” on them blindly. I’ve had students come up with 5-8 questions with a single statement!

I like to do this throughout the year using different word problems based on the skill we’re focused on at the time AND skills we’ve previously mastered, but be careful not to only use examples based on the skill you’re teaching right then so their brains don’t automatically go to the same place.

These 32 What’s My Operation? task cards will help your student learn and review which operations to use for different types of word problems! They’re perfect to use as a quick assessment, game of SCOOT, math center activity, or homework.

Lesson 2: Similar Scenarios

In this lesson, students will evaluate similar scenarios to determine the appropriate operations. Start with three similar scenarios requiring different operations and identify what situation is happening in each scenario (finding total, determining an amount, splitting or combining, etc.).

Read all three-word problems on a similar topic. Determine the similarity of all of them and determine which operation would be used to solve them. How does the situation/action of the problem help you determine what step to take?

I also created these differentiated word problem task cards after noticing my students struggling with which operation to choose, especially when given multiple problems from a similar scenario. They encourage students to select the appropriate operation for each word problem.

Lesson 3: Opposing Operations

In this lesson, students will determine relevant information from a set of facts, which requires a great deal of critical thinking to determine which operation to use. Give students a scenario and a variety of facts/information relating to the scenario as well as several questions to answer based on the facts . Students will focus on determining HOW they will solve each question using only the relevant information. 

These Operation Fascination task c ards engage students in critical thinking about operations. Each card has a scenario, multiple clues and facts to support the scenario, and four questions to accompany each scenario. The questions are a variety of operations so that students can see how using the same information can solve multiple problems.

Lesson 4: Next Level Numberless

In this lesson, we’ll take numberless word problems to the next level by developing a strong conceptual understanding of word problems. Give students scenarios without numbers and have them write a question and/or insert numbers using a specific operation and purpose . This requires a great deal of thinking to not only determine the situation, but to also figure out numbers that fit into the situation in a way that makes sense.

By integrating these types of math problems into your daily lessons, you can significantly enhance your students’ comprehension of word problems and problem-solving. These numberless word problem task cards are the ideal to improve your students’ critical thinking and problem-solving skills. They offer a variety of numbered and numberless word problems.

Lesson 5: Story Situations

In this lesson, we’ll discuss the importance of students generating their own word problems with a given set of information. This requires a great deal of quantitative reasoning as students determine how they would use a given set of numbers to create a realistic situation. Present students with two predetermined numbers and a theme. Then have students write a word problem, including a question, using the given information. 

Engage your students in additional practice with these differentiated division task cards that require your students to write their OWN word problems (and create real-world relevance in their learning!). Each task card has numbers and a theme that students use to guide their thinking and creation of a word problem.

Lesson 6: No Scenario Solving

In this lesson, we’ll decontextualize problem solving and require students to create the situation, represent it numerically, and solve. It’s a cognitively demanding task! Give students an operation and a purpose (joining, separating, comparing, etc.) with no other context, numbers, numbers, or theme. Then have students generate a word problem.

For additional practice, have students swap problems to identify the operation, purpose, and solution.

Implementing Error Analysis

Error analysis is an exceptional way to promote thinking and learning, but how do we teach students to figure out which type of math error they’ve made? This error analysis starter kit can help!

First, it is very rare that I will tell my students what error they have made in their work. I want to challenge them to figure it out on their own. So, when I see that they have a wrong answer, I ask them to go back and figure out where something went wrong. Because I resist the urge to tell them right away where their error is, my students tend to get a lot more practice identifying them!

Second, when I introduce a concept, I always, always, always create anchor charts with students and complete interactive notebook activities with them so that they have step-by-step procedures for completing tasks right at their fingertips. I have them go back and reference their notebooks while they are looking at their errors.  Usually, they can follow the anchor chart step-by-step to make sure they haven’t made a conceptual error, and if they have, they can identify it.

Third, I let them use a calculator. When worst comes to worst, and they are fairly certain they haven’t made a conceptual mistake to identify, I let them get out a calculator and start computing, step-by-step to see where they’ve made a mistake.

IF, after taking these steps, a student can’t figure out their mistake (especially if I find that it’s a conceptual mistake), I know I need to go back and do some individual reteaching with them because they don’t have a solid understanding of the concept.

This FREE addition error analysis is a good place to start, no matter the grade level. I show them the process of walking through the problem and how best to complete an error analysis task.

Digging Deeper into Error Analysis

Once students show proficiency in the standard algorithm (or strategies), I take it a step further and have them dive into error analysis where they can show a “reverse” understanding as they evaluate mistakes made and fix them. Being able to identify an error in someone else’s work requires higher order thinking not found in most other projects or activities and certainly not found in basic math fact completion.

First, teach students the difference between a computational error and a conceptual error. 

• Computational is when they make a mistake in basic math facts. This might look as simple as  64/8 does not equal 7. Oops!
• A Conceptual or Procedural Error is when they make a mistake in the procedure or concept. 
• I can’t tell you how many times students show as not proficient on a topic when the mistakes they are making are COMPUTATIONAL and not conceptual or procedural. They don’t need more review in how to use a strategy… they need to slow down and pay closer attention to their math facts!

Once we’ve introduced the types of errors they should be looking out for, we move on to actually analyzing these errors in someone else’s work and fixing the mistake.

I have created error analysis tasks for you to use with you students so they can identify the errors, types of errors, rework the problem, and create their own version of the problem and solve it. I have seen great success with incorporating these tasks into ALL of my math units. I even have kids beg to take their error analysis tasks out to recess to finish! These are great resources to start:

• Error Analysis Bundle
• 3rd Grade Word Problem of the Day
• 4th Grade Word Problem of the Day
• 5th Grade Word Problem of the Day

The final step in using error analysis is actually having students correct their OWN mistakes. Once I have instructed on types of errors, I will start by simply telling them, Oops! You’ve made a computational error here! That way they aren’t furiously looking through the procedure for a mistake, instead they are looking to see where they computed wrong. Conversely, I’ll tell them if they’ve made a procedural mistake, and that can guide them in figuring out what they need to look for.

Looking at the different types of errors students are making is essential to guiding my instruction as well, so even though it takes a bit longer to grade things like this, it is immensely helpful to me as I make adjustments to my instruction.

Resources and Ideas for Critical Thinking

I’ve compiled a collection of websites for complex tasks with multiple, open-ended answers and scenarios. The majority of these tasks are non-routine and so easy to implement. I often post these tasks and allow students short bursts of time to strategize and plan for a solution. Consider using the tasks and problems from these sites as warm-ups, extensions of your morning meeting, during enrichment groups, or on a Problem of the Week board. I also highly encourage you to incorporate these non-routine problems into your core instruction time for all students at least once or twice a month.

• NRICH provides thousands of FREE online mathematics resources for ages 3 to 18. The tasks focus on developing problem-solving skills, perseverance, mathematical reasoning, the ability to apply knowledge creatively in unfamiliar contexts, and confidence in tackling new challenges..
• Open Middle offers challenging math word problems that require a higher depth of knowledge than most problems that assess procedural and conceptual understanding. They support the Common Core State Standards and provide students with opportunities for discussing their thinking. All problems have a “closed beginning,” meaning that they all start with the same initial problem, a “closed-end” meaning that they all end with the same answer, and an “open middle” meaning that there are multiple ways to approach and ultimately solve the problem.
• Mathcurious offers interactive digital puzzles. Each adventure is dedicated to exploring the world of math and sharing experiences, knowledge, and ideas.
• Robert Kaplinsky shares math strategies, lessons, and resources designed to create problem solvers. The lessons are detailed and challenging!
• Mathigon “The mathematical playground” offers free manipulatives, activities, and lessons to make online learning interactive and engaging. The digital manipulates are a must-use!
• Fractal Foundation uses fractals to inspire interest in science, math and art. It has numerous fractal activities, software to help your students create their own fractals, and more.
• Greg Fletcher 3 Act Tasks contain engaging math videos with guiding questions. You can also download recording sheets to go with each video.

Mary Montero

I’m so glad you are here. I’m a current gifted and talented teacher in a small town in Colorado, and I’ve been in education since 2009. My passion (other than my family and cookies) is for making teachers’ lives easier and classrooms more engaging.

You might also like…

Leave a Reply Cancel reply

Your email address will not be published. Required fields are marked *

©2023 Teaching With a Mountain View . All Rights Reserved | Designed by Ashley Hughes

Username or Email Address

Remember Me

Lost your password?

Review Cart

No products in the cart.

Rich Problems – Part 1

Rich problems – part 1, by marvin cohen and karen rothschild.

One of the underlying beliefs that guides Math for All is that in order to learn mathematics well, students must engage with rich problems. Rich problems allow ALL students, with a variety of neurodevelopmental strengths and challenges, to engage in mathematical reasoning and become flexible and creative thinkers about mathematical ideas. In this Math for All Updates, we review what rich problems are, why they are important, and where to find some ready to use. In a later Math for All Updates we will discuss how to create your own rich problems customized for your curriculum.

What are Rich Problems?

At Math for All, we believe that all rich problems provide:

• opportunities to engage the problem solver in thinking about mathematical ideas in a variety of non-routine ways.
• an appropriate level of productive struggle.
• an opportunity for students to communicate their thinking about mathematical ideas.

Rich problems increase both the problem solver’s reasoning skills and the depth of their mathematical understanding. Rich problems are rich because they are not amenable to the application of a known algorithm, but require non-routine use of the student’s knowledge, skills, and ingenuity. They usually offer multiple entry pathways and methods of representation. This provides students with diverse abilities and challenges the opportunity to create solution strategies that leverage their particular strengths.

Rich problems usually have one or more of the following characteristics:

• Several correct answers. For example, “Find four numbers whose sum is 20.”
• A single answer but with many pathways to a solution. For example, “There are 10 animals in the barnyard, some chickens, some pigs. Altogether there are 24 legs. How many of the animals are chickens and how many are pigs?”
• A level of complexity that may require an entire class period or more to solve.
• An opportunity to look for patterns and make connections to previous problems, other students’ strategies, and other areas of mathematics. For example, see the staircase problem below.
• A “low floor and high ceiling,” meaning both that all your students will be able to engage with the mathematics of the problem in some way, and that the problem has sufficient complexity to challenge all your students. NRICH summarizes this approach as “everyone can get started, and everyone can get stuck” (2013). For example, a problem could have a variety of questions related to the following sequence, such as: How many squares are in the next staircase? How many in the 20th staircase? What is the rule for finding the number of squares in any staircase?

• An expectation that the student be able to communicate their ideas and defend their approach.
• An opportunity for students to choose from a range of tools and strategies to solve the problem based on their own neurodevelopmental strengths.
• An opportunity to learn some new mathematics (a mathematical residue) through working on the problem.
• An opportunity to practice routine skills in the service of engaging with a complex problem.
• An opportunity for a teacher to deepen their understanding of their students as learners and to build new lessons based on what students know, their developmental level, and their neurodevelopmental strengths and challenges.

Why Rich Problems?

All adults need mathematical understanding to solve problems in their daily lives. Most adults use calculators and computers to perform routine computation beyond what they can do mentally. They must, however, understand enough mathematics to know what to enter into the machines and how to evaluate what comes out. Our personal financial situations are deeply affected by our understanding of pricing schemes for the things we buy, the mortgages we hold, and fees we pay. As citizens, understanding mathematics can help us evaluate government policies, understand political polls, and make decisions. Building and designing our homes, and scaling up recipes for crowds also require math. Now especially, mathematical understanding is crucial for making sense of policies related to the pandemic. Decisions about shutdowns, medical treatments, and vaccines are all grounded in mathematics. For all these reasons, it is important students develop their capacities to reason about mathematics. Research has demonstrated that experience with rich problems improves children’s mathematical reasoning (Hattie, Fisher, & Frey, 2017).

Where to Find Rich Problems

Several types of rich problems are available online, ready to use or adapt. The sites below are some of many places where rich problems can be found:

• Which One Doesn’t Belong – These problems consist of squares divided into 4 quadrants with numbers, shapes, or graphs. In every problem there is at least one way that each of the quadrants “doesn’t belong.” Thus, any quadrant can be argued to be different from the others.
• “Open Middle” Problems – These are problems with a single answer but with many ways to reach the answer. They are organized by both topic and grade level.
• NRICH Maths – This is a multifaceted site from the University of Cambridge in Great Britain. It has both articles and ready-made problems. The site includes  problems for grades 1–5 (scroll down to the “Collections” section) and problems for younger children . We encourage you to explore NRICH more fully as well. There are many informative articles and discussions on the site.
• Rich tasks from Virginia – These are tasks published by the Virginia Department of education. They come with complete lesson plans as well as example anticipated student responses.
• Rich tasks from Georgia – This site contains a complete framework of tasks designed to address all standards at all grades. They include 3-Act Tasks , YouCubed Tasks , and many other tasks that are open ended or feature an open middle approach.

The problems can be used “as is” or adapted to the specific neurodevelopmental strengths and challenges of your students. Carefully adapted, they can engage ALL your students in thinking about mathematical ideas in a variety of ways, thereby not only increasing their skills but also their abilities to think flexibly and deeply.

Hattie, J., Fisher, D., & Frey, N. (2017). Visible learning for mathematics, grades K-12: What works best to optimize student learning. Thousand Oaks, CA: Corwin Mathematics.

NRICH Team. (2013). Low Threshold High Ceiling – an Introduction . Cambridge University, United Kingdom: NRICH Maths.

The contents of this blog post were developed under a grant from the Department of Education. However, those contents do not necessarily represent the policy of the Department of Education, and you should not assume endorsement by the Federal Government.

Math for All is a professional development program that brings general and special education teachers together to enhance their skills in planning and adapting mathematics lessons to ensure that all students achieve high-quality learning outcomes in mathematics.

Our Newsletter Provides Ideas for Making High-Quality Mathematics Instruction Accessible to All Students

Sign up for our newsletter, recent blogs.

• Social Justice in the Math Classroom May 30, 2024
• Parents and Teachers as Co-Constructors of Children’s Success as Mathematical Learners May 1, 2024
• The Problem with Word Problems March 11, 2024
• Honoring Diversity: What, why, and how? February 22, 2024
• Looking at a Student at Work January 3, 2024

Team & Partners

Testimonials

Newsletter Signup

Engaging Maths

Promoting creative and critical thinking in mathematics and numeracy.

What is critical and creative thinking, and why is it so important in mathematics and numeracy education?

Numeracy is often defined as the ability to apply mathematics in the context of day to day life. However, the term ‘critical numeracy’ implies much more. One of the most basic reasons for learning mathematics is to be able to apply mathematical skills and knowledge to solve both simple and complex problems, and, more than just allowing us to navigate our lives through a mathematical lens, being numerate allows us to make our world a better place.

The mathematics curriculum in Australia provides teachers with the perfect opportunity to teach mathematics through critical and creative thinking. In fact, it’s mandated. Consider the core processes of the curriculum. The Australian Curriculum (ACARA, 2017), requires teachers to address four proficiencies : Problem Solving, Reasoning, Fluency, and Understanding. Problem solving and reasoning require critical and creative thinking (). This requirement is emphasised more heavily in New South wales, through the graphical representation of the mathematics syllabus content , which strategically places Working Mathematically (the proficiencies in NSW) and problem solving, at its core. Alongside the mathematics curriculum, we also have the General Capabilities , one of which is Critical and Creative Thinking – there’s no excuse!

Critical and creative thinking need to be embedded in every mathematics lesson . Why? When we embed critical and creative thinking, we transform learning from disjointed, memorisation of facts, to sense-making mathematics. Learning becomes more meaningful and purposeful for students.

How and when do we embed critical and creative thinking?

There are many tools and many methods of promoting thinking. Using a range of problem solving activities is a good place to start, but you might want to also use some shorter activities and some extended activities. Open-ended tasks are easy to implement, allow all learners the opportunity to achieve success, and allow for critical thinking and creativity. Tools such as Bloom’s Taxonomy and Thinkers Keys  are also very worthwhile tasks. For good mathematical problems go to the nrich website . For more extended mathematical investigations and a wonderful array of rich tasks, my favourite resource is Maths300   (this is subscription based, but well worth the money). All of the above activities can be used in class and/or for homework, as lesson starters or within the body of a lesson.

Will critical and creative thinking take time away from teaching basic concepts?

No, we need to teach mathematics in a way that has meaning and relevance, rather than through isolated topics. Therefore, teaching through problem-solving rather than for problem-solving. A classroom that promotes and critical and creative thinking provides opportunities for:

• higher-level thinking within authentic and meaningful contexts;
• complex problem solving;
• open-ended responses; and
• substantive dialogue and interaction.

Who should be engaging in critical and creative thinking?

Is it just for students? No! There are lots of reasons that teachers should be engaged with critical and creative thinking. First, it’s important that we model this type of thinking for our students. Often students see mathematics as black or white, right or wrong. They need to learn to question, to be critical, and to be creative. They need to feel they have permission to engage in exploration and investigation. They need to move from consumers to producers of mathematics.

Secondly, teachers need to think critically and creatively about their practice as teachers of mathematics. We need to be reflective practitioners who constantly evaluate our work, questioning curriculum and practice, including assessment, student grouping, the use of technology, and our beliefs of how children best learn mathematics.

Critical and creative thinking is something we cannot ignore if we want our students to be prepared for a workforce and world that is constantly changing. Not only does it equip then for the future, it promotes higher levels of student engagement, and makes mathematics more relevant and meaningful.

How will you and your students engage in critical and creative thinking?

Share this:

[…] of this recognition that the 1940s menu makes a perfect stimulus for mathematical investigation and critical thinking. The links between mathematics and children’s lives are not always obvious for students, so […]

[…] several of my previous posts I discussed the importance of promoting critical thinking in mathematics teaching and learning. I’ve also discussed at length various ways to contextualise […]

Leave a comment Cancel reply

• Already have a WordPress.com account? Log in now.
• Subscribe Subscribed
• Copy shortlink
• Report this content
• View post in Reader
• Manage subscriptions
• Collapse this bar

How to Improve Problem-Solving Skills: Mathematics and Critical Thinking

In today’s rapidly changing world, problem-solving has become a quintessential skill. When we discuss the topic, it’s natural to ask, “What is problem-solving?” and “How can we enhance this skill, particularly in children?” The discipline of mathematics offers a rich platform to explore these questions. Through math, not only do we delve into numbers and equations, but we also explore how to improve problem-solving skills and how to develop critical thinking skills in math. Let’s embark on this enlightening journey together.

What is Problem-Solving?

At its core, problem-solving involves identifying a challenge and finding a solution. But it’s not always as straightforward as it sounds. So, what is problem-solving? True problem-solving requires a combination of creative thinking and logical reasoning. Mathematics, in many ways, embodies this blend. When a student approaches a math problem, they must discern the issue at hand, consider various methods to tackle it, and then systematically execute their chosen strategy.

But what is problem-solving in a broader context? It’s a life skill. Whether we’re deciding the best route to a destination, determining how to save for a big purchase, or even figuring out how to fix a broken appliance, we’re using problem-solving.

How to Develop Critical Thinking Skills in Math

Critical thinking goes hand in hand with problem-solving. But exactly how to develop critical thinking skills in math might not be immediately obvious. Here are a few strategies:

• Contextual Learning: Teaching math within a story or real-life scenario makes it relevant. When students see math as a tool to navigate the world around them, they naturally begin to think critically about solutions.
• Open-ended Questions: Instead of merely seeking the “right” answer, encourage students to explain their thought processes. This nudges them to think deeply about their approach.
• Group Discussions: Collaborative learning can foster different perspectives, prompting students to consider multiple ways to solve a problem.
• Challenging Problems: Occasionally introducing problems that are a bit beyond a student’s current skill level can stimulate critical thinking. They will have to stretch their understanding and think outside the box.

What are the Six Basic Steps of the Problem-Solving Process?

Understanding how to improve problem-solving skills often comes down to familiarizing oneself with the systematic approach to challenges. So, what are the six basic steps of the problem-solving process?

• Identification: Recognize and define the problem.
• Analysis: Understand the problem’s intricacies and nuances.
• Generation of Alternatives: Think of different ways to approach the challenge.
• Decision Making: Choose the most suitable method to address the problem.
• Implementation: Put the chosen solution into action.
• Evaluation: Reflect on the solution’s effectiveness and learn from the outcome.

By embedding these steps into mathematical education, we provide students with a structured framework. When they wonder about how to improve problem-solving skills or how to develop critical thinking skills in math, they can revert to this process, refining their approach with each new challenge.

Making Math Fun and Relevant

At Wonder Math, we believe that the key to developing robust problem-solving skills lies in making math enjoyable and pertinent. When students see math not just as numbers on a page but as a captivating story or a real-world problem to be solved, their engagement skyrockets. And with heightened engagement comes enhanced understanding.

As educators and parents, it’s crucial to continuously ask ourselves: how can we demonstrate to our children what problem-solving is? How can we best teach them how to develop critical thinking skills in math? And how can we instill in them an understanding of the six basic steps of the problem-solving process?

The answer, we believe, lies in active learning, contextual teaching, and a genuine passion for the beauty of mathematics.

The Underlying Beauty of Mathematics

Often, people perceive mathematics as a rigid discipline confined to numbers and formulas. However, this is a limited view. Math, in essence, is a language that describes patterns, relationships, and structures. It’s a medium through which we can communicate complex ideas, describe our universe, and solve intricate problems. Understanding this deeper beauty of math can further emphasize how to develop critical thinking skills in math.

Why Mathematics is the Ideal Playground for Problem-Solving

Math provides endless opportunities for problem-solving. From basic arithmetic puzzles to advanced calculus challenges, every math problem offers a chance to hone our problem-solving skills. But why is mathematics so effective in this regard?

• Structured Challenges: Mathematics presents problems in a structured manner, allowing learners to systematically break them down. This format mimics real-world scenarios where understanding the structure of a challenge can be half the battle.
• Multiple Approaches: Most math problems can be approached in various ways . This teaches learners flexibility in thinking and the ability to view a single issue from multiple angles.
• Immediate Feedback: Unlike many real-world problems where solutions might take time to show results, in math, students often get immediate feedback. They can quickly gauge if their approach works or if they need to rethink their strategy.

Enhancing the Learning Environment

To genuinely harness the power of mathematics in developing problem-solving skills, the learning environment plays a crucial role. A student who is afraid of making mistakes will hesitate to try out different approaches, stunting their critical thinking growth.

However, in a nurturing, supportive environment where mistakes are seen as learning opportunities, students thrive. They become more willing to take risks, try unconventional solutions, and learn from missteps. This mindset, where failure is not feared but embraced as a part of the learning journey, is pivotal for developing robust problem-solving skills.

Incorporating Technology

In our digital age, technology offers innovative ways to explore math. Interactive apps and online platforms can provide dynamic problem-solving scenarios, making the process even more engaging. These tools can simulate real-world challenges, allowing students to apply their math skills in diverse contexts, further answering the question of how to improve problem-solving skills.

More than Numbers 

In summary, mathematics is more than just numbers and formulas—it’s a world filled with challenges, patterns, and beauty. By understanding its depth and leveraging its structured nature, we can provide learners with the perfect platform to develop critical thinking and problem-solving skills. The key lies in blending traditional techniques with modern tools, creating a holistic learning environment that fosters growth, curiosity, and a lifelong love for learning.

Join us on this transformative journey at Wonder Math. Let’s make math an adventure, teaching our children not just numbers and equations, but also how to improve problem-solving skills and navigate the world with confidence. Enroll your child today and witness the magic of mathematics unfold before your eyes!

FAQ: Mathematics and Critical Thinking

1. what is problem-solving in the context of mathematics.

Problem-solving in mathematics refers to the process of identifying a mathematical challenge and systematically working through methods and strategies to find a solution.

2. Why is math considered a good avenue for developing problem-solving skills?

Mathematics provides structured challenges and allows for multiple approaches to find solutions. This promotes flexibility in thinking and encourages learners to view problems from various angles.

3. How does contextual learning enhance problem-solving abilities?

By teaching math within a story or real-life scenario, it becomes more relevant for the learner. This helps them see math as a tool to navigate real-world challenges , thereby promoting critical thinking.

4. What are the six basic steps of the problem-solving process in math?

The six steps are: Identification, Analysis, Generation of Alternatives, Decision Making, Implementation, and Evaluation.

5. How can parents support their children in developing mathematical problem-solving skills?

Parents can provide real-life contexts for math problems , encourage open discussions about different methods, and ensure a supportive environment where mistakes are seen as learning opportunities.

6. Are there any tools or apps that can help in enhancing problem-solving skills in math?

Yes, there are various interactive apps and online platforms designed specifically for math learning. These tools provide dynamic problem-solving scenarios and simulate real-world challenges, making the learning process engaging.

7. How does group discussion foster critical thinking in math?

Group discussions allow students to hear different perspectives and approaches to a problem. This can challenge their own understanding and push them to think about alternative methods.

8. Is it necessary to always follow the six steps of the problem-solving process sequentially?

While the six steps provide a structured approach, real-life problem-solving can sometimes be more fluid. It’s beneficial to know the steps, but adaptability and responsiveness to the situation are also crucial.

9. How does Wonder Math incorporate active learning in teaching mathematics?

Wonder Math integrates mathematics within engaging stories and real-world scenarios, making it fun and relevant. This active learning approach ensures that students are not just passive recipients but active participants in the learning process.

10. What if my child finds a math problem too challenging and becomes demotivated?

It’s essential to create a supportive environment where challenges are seen as growth opportunities. Remind them that every problem is a chance to learn, and it’s okay to seek help or approach it differently.

Related posts

Summer Math Programs: How They Can Prevent Learning Loss in Young Students

As summer approaches, parents and educators alike turn their attention to how they can support young learners during the break. Summer is a time for relaxation, fun, and travel, yet it’s also a critical period when learning loss can occur. This phenomenon, often referred to as the “summer slide,” impacts students’ progress, especially in foundational subjects like mathematics. It’s reported…

Math Programs 101: What Every Parent Should Know When Looking For A Math Program

  As a parent, you know that a solid foundation in mathematics is crucial for your child’s success, both in school and in life. But with so many math programs and math help services out there, how do you choose the right one? Whether you’re considering Outschool classes, searching for “math tutoring near me,” or exploring tutoring services online, understanding…

Critical Thinking in Mathematics Education

• Reference work entry
• First Online: 01 January 2020
• Cite this reference work entry

• Eva Jablonka 2  

972 Accesses

13 Citations

Mainstream educational psychologists view critical thinking (CT) as the strategic use of a set of reasoning skills for developing a form of reflective thinking that ultimately optimizes itself, including a commitment to using its outcomes as a basis for decision-making and problem solving. In such descriptions, CT is established as a general methodological standard for making judgments and decisions. Accordingly, some authors also include a sense for fairness and the assessment of practical consequences of decisions as characteristics (e.g., Paul and Elder 2001 ). This conception assumes rational, autonomous subjects who share a common frame of reference for representation of facts and ideas, for their communication, as well as for appropriate (morally “good”) action. Important is the difference as to what extent a critical examination of the criteria for CT is included in the definition: If education for CT is conceptualized as instilling a belief in a more or less fixed...

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save.

• Get 10 units per month
• Download Article/Chapter or eBook
• 1 Unit = 1 Article or 1 Chapter
• Cancel anytime
• Available as PDF
• Read on any device
• Instant download
• Own it forever
• Available as EPUB and PDF
• Durable hardcover edition
• Dispatched in 3 to 5 business days
• Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Appelbaum P, Davila E (2009) Math education and social justice: gatekeepers, politics and teacher agency. In: Ernest P, Greer B, Sriraman B (eds) Critical issues in mathematics education. Information Age, Charlotte, pp 375–394

Google Scholar  

Applebaum M, Leikin R (2007) Looking back at the beginning: critical thinking in solving unrealistic problems. Mont Math Enthus 4(2):258–265

Bacon F (1605) Of the proficience and advancement of learning, divine and human. Second Book (transcribed from the 1893 Cassell & Company edition by David Price. Available at: http://www.gutenberg.org/dirs/etext04/adlr10h.htm

Common Core State Standards Initiative (2010) Mathematics standards. http://www.corestandards.org/Math . Accessed 20 July 2013

Ernest P (2010) The scope and limits of critical mathematics education. In: Alrø H, Ravn O, Valero P (eds) Critical mathematics education: past, present and future. Sense Publishers, Rotterdam, pp 65–87

Fawcett HP (1938) The nature of proof. Bureau of Publications, Columbia/New York City. University (Re-printed by the National Council of Teachers of Mathematics in 1995)

Fenner P (1994) Spiritual inquiry in Buddhism. ReVision 17(2):13–24

Fish M, Persaud A (2012) (Re)presenting critical mathematical thinking through sociopolitical narratives as mathematics texts. In: Hickman H, Porfilio BJ (eds) The new politics of the textbook. Sense Publishers, Rotterdam, pp 89–110

Chapter   Google Scholar  

Garfield JL (1990) Epoche and śūnyatā: skepticism east and west. Philos East West 40(3):285–307

Article   Google Scholar  

Jablonka E (1997) What makes a model effective and useful (or not)? In: Blum W, Huntley I, Houston SK, Neill N (eds) Teaching and learning mathematical modelling: innovation, investigation and applications. Albion Publishing, Chichester, pp 39–50

Keitel C, Kotzmann E, Skovsmose O (1993) Beyond the tunnel vision: analyzing the relationship between mathematics, society and technology. In: Keitel C, Ruthven K (eds) Learning from computers: mathematics education and technology. Springer, New York, pp 243–279

Legrand M (2001) Scientific debate in mathematics courses. In: Holton D (ed) The teaching and learning of mathematics at university level: an ICMI study. Kluwer, Dordrect, pp 127–137

National Council of Teachers of Mathematics (NCTM) (1989) Curriculum and evaluation standards for school mathematics. National Council of Teachers of Mathematics (NCTM), Reston

O’Daffer PG, Thomquist B (1993) Critical thinking, mathematical reasoning, and proof. In: Wilson PS (ed) Research ideas for the classroom: high school mathematics. MacMillan/National Council of Teachers of Mathematics, New York, pp 31–40

Paul R, Elder L (2001) The miniature guide to critical thinking concepts and tools. Foundation for Critical Thinking Press, Dillon Beach

Pimm D (1990) Mathematical versus political awareness: some political dangers inherent in the teaching of mathematics. In: Noss R, Brown A, Dowling P, Drake P, Harris M, Hoyles C et al (eds) Political dimensions of mathematics education: action and critique. Institute of Education, University of London, London

Skovsmose O (1989) Models and reflective knowledge. Zentralblatt für Didaktik der Mathematik 89(1):3–8

Stallman J (2003) John Dewey’s new humanism and liberal education for the 21st century. Educ Cult 20(2):18–22

Steiner H-G (1988) Theory of mathematics education and implications for scholarship. In: Steiner H-G, Vermandel A (eds) Foundations and methodology of the discipline mathematics education, didactics of mathematics. In: Proceedings of the second tme conference, Bielefeld-Antwerpen, pp 5–20

Straehler-Pohl H, Bohlmann N, Pais A (eds) (2017) The disorder of mathematics education: challenging the socio-political dimensions of research. Springer, Berlin

Walkerdine V (1988) The mastery of reason: cognitive development and the production of rationality. Routledge, London

Walshaw M (2003) Democratic education under scrutiny: connections between mathematics education and feminist political discourses. Philos Math Educ J 17. http://people.exeter.ac.uk/PErnest/pome17/contents.htm

Download references

Author information

Authors and affiliations.

Department of Education and Psychology, Freie Universität Berlin, Berlin, Germany

Eva Jablonka

You can also search for this author in PubMed   Google Scholar

Corresponding author

Correspondence to Eva Jablonka .

Editor information

Editors and affiliations.

Department of Education, Centre for Mathematics Education, London South Bank University, London, UK

Stephen Lerman

Section Editor information

Department of Mathematical Sciences, The University of Montana, Missoula, MT, USA

Bharath Sriraman

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this entry

Cite this entry.

Jablonka, E. (2020). Critical Thinking in Mathematics Education. In: Lerman, S. (eds) Encyclopedia of Mathematics Education. Springer, Cham. https://doi.org/10.1007/978-3-030-15789-0_35

Download citation

DOI : https://doi.org/10.1007/978-3-030-15789-0_35

Published : 23 February 2020

Publisher Name : Springer, Cham

Print ISBN : 978-3-030-15788-3

Online ISBN : 978-3-030-15789-0

eBook Packages : Education Reference Module Humanities and Social Sciences Reference Module Education

Share this entry

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

• Publish with us

Policies and ethics

• Find a journal
• Track your research

• Our Mission

10 Brilliant Math Brain Teasers

Tap into rigorous problem-solving and critical thinking with these playful math brain teasers for middle and high school students.  

To break the ice during the first few awkward moments of class in the new school year, high school math teacher Lorenzo Robinson uses an unusual strategy: He reads his students’ minds.

Here’s how the mystical feat unfolds: Each student picks a number between 1 and 100. Next, they use calculators to add, subtract, multiply, and divide their way through a set of predetermined numbers. At the end, everyone ends up with the same answer: 427. Robinson isn’t psychic, of course, but because the mechanics of the teaser are always the same, regardless of which initial numbers his students select, he’s able to correctly “guess” their final result—much to their amazement. 

Starting off the lesson with a math brain teaser sets a playful tone and lowers the stakes for kids, “generating a buzz around my class,” Robinson says. “It makes students feel as though this class is not going to be scary, it’s going to be interesting. ‘We’re going to be learning, but we’re also going to have some fun.’” 

Robinson thinks of math brain teasers as a variation on brain breaks —a brief respite from dense curricular material that gives kids time to pause and process. They can also provide an opportunity to build relationships and community as kids put their heads together to find solutions. Good brain teasers can be sneaky: They get kids developing problem-solving and critical thinking skills. 

Meanwhile, before introducing a new one, Robinson works through the problem himself, identifying questions that students might have along the way and making sure his class has the background knowledge to understand how the teaser works. It’s important, he says, to provide a few minutes for kids to examine and discuss the teaser. Ask them to observe, highlight, and share things that stand out. 

“The most powerful reaction is when a kid doesn’t get the correct answer,” Robinson says, and they ask to try the problem again. “They want to feel what the other kids are feeling, that educational euphoria. They want to do it again because they want to be right.” That organic intellectual curiosity is hugely helpful in high school math, Robinson says, because it can be “parlayed into the other stuff that we do.”

We combed through dozens of math brain teasers to find 10 good ones—including several of Robinson’s tried-and-true favorites.  

Number Magic: I’ll Bet Your Number is… 427

• Start by having students pick any number between 1 and 100. 
• Add 28. 
• Multiply that number by 6. 
• Subtract 3. 
• Divide that number by 3. 
• Subtract 3 more than your original number. 
• Add 8. 
• Subtract 1 less than your original number. 
• Multiply that number by 7. 

And voilà, you’ll correctly identify each student’s final result as 427. Courtesy of: Lorenzo Robinson  

Can Your Shoe Size Tell Your Age? 

• Start with your shoe size. If you are a half size—for example, size 8.5—round up to 9.
• Multiply your shoe size by 5. 
• Add 50. 
• Multiply that number by 20. 
• Subtract the year you were born—for example, 1991. Add 1 if you already had your birthday this year. 

The first digit(s) are your shoe size, and the last two digits are your age.  Courtesy of: Lorenzo Robinson.  

Cutting Across a Cross

Ask students to draw a cross on a sheet of paper. Drawing one on the board as a point of reference is helpful. Next, ask students to draw two straight lines that will segment or cut the cross into pieces. The goal is to produce the most pieces.

The solution can be found here .  Sourced from: MathisFun.com .

Number Magic: I’ll Bet Your Final Number is… 5  

• Start with a positive number. Students shouldn’t say the number out loud.
• Square that number. 
• Add 10x the original number to what you have now. 
• Add 25 to the result of the previous step. 
• Now take the square root of that number, rounding to the nearest whole number. 
• Subtract your original number.
• Before students share their final figure, reveal that you guess their collective result is 5. 

Courtesy of: Lorenzo Robinson.

Birthday Math 

Have students work in pairs and share the following instructions with their partner:

• Start with the number 7.
• Multiply that by the month of your birth. For example, if you were born in September, you’d use the number 9 to represent your birth month. 
• Subtract 1. 
• Multiply that number by 13. 
• Add the day of your birth. 
• Add 3. 
• Multiply that number by 11. 
• Subtract the month of your birth. 
• Subtract the day of your birth. 
• Divide by 10.
• Add 11 to that number. 
• Divide by 100. 

The result on the calculator screen should be their partner’s birthday.  Sourced from: Dr. Mike’s Math Games for Kids .

Coin Conundrum

Ask students to imagine that they have two coins that total 30 cents in value. Have them try to figure out what the two coins are, only providing them with a singular piece of information: One of the coins is not a nickel. The answer: A quarter and a nickel. (Only one of the coins is not a nickel.)

Sourced from: WeAreTeachers .

The Phone Number Trick  

• Ignoring your area code, type the first three digits of your phone number into a calculator.
• Multiply that number by 80. 
• Add 1. 
• Multiply that number by 250. 
• Add the last four digits of your phone number. 
• Add the last four digits of your phone number again. 
• Subtract 250. 
• Divide this number by 2. Do you recognize your phone number?

Courtesy of: Lorenzo Robinson. 

A Number Challenge

For a slightly more independent teaser, challenge students to produce a math equation that works using these four numbers—2, 3, 4, and 5—as well as a plus (+) and equal sign (=). Students can work individually, in pairs, or in small groups as they try to create a valid equation. The answer: 2 + 5 = 3 + 4.

Sourced from: WeAreTeachers .  

Math Mind Reader 

Students can work in pairs with this teaser. One person will start off by holding the calculator so their partner cannot see it; the other person can read the steps aloud to the partner with the calculator. 

• The student with the calculator starts by choosing a whole number from 1 to 20 and writing it down on a piece of paper without letting their partner see it. 
• Next, the student with the calculator enters their secret number into the calculator. 
• Multiply that number by 3. 
• Add the secret number, then subtract 5. 
• Multiply by 3, then multiply by 3 again. 
• Add the secret number, then subtract the number of their favorite month (you don’t have to know what month it is). For example, 9 represents the ninth month of the year, September. 
• Multiply by 3, then multiply by 3 again, and then again a third time. 
• Add the secret number, then subtract their favorite day of the month (again, you don’t have to know what it is on your end). 
• Ask them to show the non-calculator partner the result. At this stage, the non-calculator partner can guess the original secret number, even though what appears on the screen may be a very large number. 

If the result is negative, their secret number is 1.

If the result has only three digits, their secret number is 2.

In all other cases, ignore the last three digits, and then add 2 to get the secret number! 

Sourced from: Dr. Mike’s Math Games for Kids .

What’s Unique About This Number?

After writing the number 8,549,176,320 on the board, ask students to observe the number and tell you everything they think is unique about the number.

Answer: It is the digits 0 to 9 in alphabetical order (eight, five, four, nine, one, seven, six, three, two, zero), but it’s surprising and fun to see what students come up with. This number can also be evenly divided by the digits 1 through 9 except for the number 7, for example. 

Sourced from: MathisFun.com . 

JavaScript seems to be disabled in your browser. For the best experience on our site, be sure to turn on Javascript in your browser.

• Order Tracking
• Create an Account

200+ Award-Winning Educational Textbooks, Activity Books, & Printable eBooks!

• Compare Products

Reading, Writing, Math, Science, Social Studies

• Search by Book Series
• Algebra I & II  Gr. 7-12+
• Algebra Magic Tricks  Gr. 2-12+
• Algebra Word Problems  Gr. 7-12+
• Balance Benders  Gr. 2-12+
• Balance Math & More!  Gr. 2-12+
• Basics of Critical Thinking  Gr. 4-7
• Brain Stretchers  Gr. 5-12+
• Building Thinking Skills  Gr. Toddler-12+
• Building Writing Skills  Gr. 3-7
• Bundles - Critical Thinking  Gr. PreK-9
• Bundles - Language Arts  Gr. K-8
• Bundles - Mathematics  Gr. PreK-9
• Bundles - Multi-Subject Curriculum  Gr. PreK-12+
• Bundles - Test Prep  Gr. Toddler-12+
• Can You Find Me?  Gr. PreK-1
• Complete the Picture Math  Gr. 1-3
• Cornell Critical Thinking Tests  Gr. 5-12+
• Cranium Crackers  Gr. 3-12+
• Creative Problem Solving  Gr. PreK-2
• Critical Thinking Activities to Improve Writing  Gr. 4-12+
• Critical Thinking Coloring  Gr. PreK-2
• Critical Thinking Detective  Gr. 3-12+
• Critical Thinking Tests  Gr. PreK-6
• Critical Thinking for Reading Comprehension  Gr. 1-5
• Critical Thinking in United States History  Gr. 6-12+
• CrossNumber Math Puzzles  Gr. 4-10
• Crypt-O-Words  Gr. 2-7
• Crypto Mind Benders  Gr. 3-12+
• Daily Mind Builders  Gr. 5-12+
• Dare to Compare Math  Gr. 2-7
• Developing Critical Thinking through Science  Gr. 1-8
• Dr. DooRiddles  Gr. PreK-12+
• Dr. Funster's  Gr. 2-12+
• Editor in Chief  Gr. 2-12+
• Fun-Time Phonics!  Gr. PreK-2
• Half 'n Half Animals  Gr. K-4
• Hands-On Thinking Skills  Gr. K-1
• Inference Jones  Gr. 1-6
• James Madison  Gr. 10-12+
• Jumbles  Gr. 3-5
• Language Mechanic  Gr. 4-7
• Language Smarts  Gr. 1-4
• Mastering Logic & Math Problem Solving  Gr. 6-9
• Math Analogies  Gr. K-9
• Math Detective  Gr. 3-8
• Math Games  Gr. 3-8
• Math Mind Benders  Gr. 5-12+
• Math Ties  Gr. 4-8
• Math Word Problems  Gr. 4-10
• Mathematical Reasoning  Gr. Toddler-11
• Middle School Science  Gr. 6-8
• Mind Benders  Gr. PreK-12+
• Mind Building Math  Gr. K-1
• Mind Building Reading  Gr. K-1
• Novel Thinking  Gr. 3-6
• OLSAT® Test Prep  Gr. PreK-K
• Organizing Thinking  Gr. 2-8
• Pattern Explorer  Gr. 3-9
• Practical Critical Thinking  Gr. 8-12+
• Punctuation Puzzler  Gr. 3-8
• Reading Detective  Gr. 3-12+
• Red Herring Mysteries  Gr. 4-12+
• Red Herrings Science Mysteries  Gr. 4-9
• Science Detective  Gr. 3-6
• Science Mind Benders  Gr. PreK-3
• Science Vocabulary Crossword Puzzles  Gr. 4-6
• Sciencewise  Gr. 4-12+
• Scratch Your Brain  Gr. 2-12+
• Sentence Diagramming  Gr. 3-12+
• Smarty Pants Puzzles  Gr. 3-12+
• Snailopolis  Gr. K-4
• Something's Fishy at Lake Iwannafisha  Gr. 5-9
• Teaching Technology  Gr. 3-12+
• Tell Me a Story  Gr. PreK-1
• Think Analogies  Gr. 3-12+
• Think and Write  Gr. 3-8
• Think-A-Grams  Gr. 4-12+
• Thinking About Time  Gr. 3-6
• Thinking Connections  Gr. 4-12+
• Thinking Directionally  Gr. 2-6
• Thinking Skills & Key Concepts  Gr. PreK-2
• Thinking Skills for Tests  Gr. PreK-5
• U.S. History Detective  Gr. 8-12+
• Understanding Fractions  Gr. 2-6
• Visual Perceptual Skill Building  Gr. PreK-3
• Vocabulary Riddles  Gr. 4-8
• Vocabulary Smarts  Gr. 2-5
• Vocabulary Virtuoso  Gr. 2-12+
• What Would You Do?  Gr. 2-12+
• Who Is This Kid? Colleges Want to Know!  Gr. 9-12+
• Word Explorer  Gr. 4-8
• Word Roots  Gr. 3-12+
• World History Detective  Gr. 6-12+
• Writing Detective  Gr. 3-6
• You Decide!  Gr. 6-12+

Mathematical Reasoning™

Bridging the gap between computation and math reasoning.

Grades: Toddler-11

Mathematics

Full curriculum

•  Multiple Award Winner

Forget boring math lessons and dreaded drill sheets.  These fun, colorful books use engaging lessons with easy-to-follow explanations, examples, and charts to make mathematical concepts easy to understand.  They can be used as textbooks or comprehensive workbooks with your textbooks to teach the math skills and concepts that students are expected to know in each grade—and several concepts normally taught in the next grade. Every lesson is followed with a variety of fun, colorful activities to ensure concept mastery.  The lessons and activities spiral slowly, allowing students to become comfortable with concepts, but also challenging them to continue building their problem-solving skills.  These books teach more than mathematical concepts; they teach mathematical reasoning, so students learn to devise different strategies to solve a wide variety of math problems.  All books are written to the standards of the National Council of Teachers of Mathematics.

Beginning 1 , Beginning 2 , and Level A Contents

Understanding Pre-Algebra This book teaches and develops the math concepts and critical thinking skills necessary for success in Algebra I and future mathematics courses at the high school level.  It was written with the premise that students cannot problem solve or take leaps of reasoning without understanding the concepts and elements that lead to discovery.  The author—with 35 years of experience teaching mathematics—is a firm believer that understanding leads to confidence and confidence gives students the resolve to succeed in higher level mathematics rather than fear it.   It is standards-based, but what makes it different from other pre-algebra books is that it organizes concepts in a logical fashion, stressing practice and critical thinking. It avoids the mistakes—found in many other math books—of trying to teach new concepts before students receive the prerequisite skills and practice necessary for success. The concepts are presented clearly and in connection to other concepts. Math vocabulary is very important to success in higher mathematics, so this book includes easy-to-follow explanations and a user-friendly glossary.

Free Detailed Solutions are available!

Understanding Pre-Algebra Contents

Understanding Geometry The successful completion of this colorful 272-page book will prepare middle schoolers for high school geometry. It covers more than 50% of the concepts taught in high school geometry using a step-by-step approach and teaches the reasoning behind the properties taught in geometry–instead of merely asking them to memorize them. Students are also taught the basics of geometric proofs and coordinate geometry in a way middle school students can understand. Students who struggle with high school geometry usually have lower standardized test scores because it is a fundamental subject in high school standardized testing. A glossary of terms that every student should master is included. This book can be used as a classroom textbook in Grades 7, 8, or 9 (usually over a two-year period) or as a reference for high school students. This book covers more than the National Math Standards for middle school mathematics.

Understanding Geometry Contents

NOTE:  It is our recommendation that students complete Understanding Pre-Algebra (see description above) before attempting Understanding Geometry .

Understanding Algebra I This is a one-year Algebra I course for Grades 7-9. Students who have a solid algebra background will have no trouble with the algebra problems from SAT and even the GRE.  This 384-page book highlights vocabulary and notation, and has examples from the history of math. What makes this book unique and different from other algebra textbooks is that it is built from the experiences of an award-winning algebra teacher with more than 30 years of teaching experience. Many textbooks are written by a committee of authors, and many of those authors have little experience teaching beginning algebra students in middle school or high school. Understanding Algebra I presents the most essential concepts and skills needed to fully understand and gain confidence in algebra in a step-by-step fashion, teaching students that algebra is generalized arithmetic. It helps students see the connection between mathematics that they already know and algebra, so that learning algebra becomes easier and less abstract. This book provides students with real strategies to succeed in solving word problems by using charts and translating strategies that guarantee success.

Understanding Algebra I Contents

Essential Algebra for Advanced High School and SAT

Discover Essential Algebra for Advanced High School and SAT , a 241-page math book in the esteemed Mathematical Reasoning series written by award-winning author and teacher with 30 years of expertise in secondary mathematics. This powerful resource teaches the ‘essential’ connection of arithmetic and geometric concepts with algebraic concepts. Without this understanding, students tend to memorize Algebra I problem-solving steps—which is sufficient to pass Algebra I—but leaves them unprepared for math courses beyond Algebra I and the SATs. Algebra, the essential language of all advanced mathematics, lies at the core of this book's teachings. By delving into the generalized arithmetic that underpins algebra, students develop a solid foundation in the rules governing number and fraction operations, including factors and multiples. This vital knowledge empowers students to move beyond mere memorization of Algebra I problem-solving steps and confidently tackle the complexities of math courses beyond Algebra I. Without the knowledge and skills taught in this book, students often struggle or even fail in advanced mathematics courses and on the SATs. Imagine a good high school student who sees a problem like 3•x•y•4 and hesitates to write 12xy due to uncertainty about the rules governing multiplication. Or not understanding how to add 2x to 1/4y to combine it into a single fraction. Or why –6 2 is different than (–6) 2 . It is easy to see that not having a strong understanding of the foundational rules of algebra can stop even the smartest students from succeeding in advanced high school math courses. Essential Algebra for Advanced High School and SAT serves as a companion to an Algebra I course or aids in post-Algebra I readiness. To ensure students’ long-term success in advanced math beyond Algebra I, this book teaches the following 'essential' mathematics skills and concepts:

• Understanding Terms and Order of Operations
• Understanding the Family of Real Numbers
• Rationals and Irrationals
• Working with Terms and Polynomials
• Polynomial Division, Factoring, and Rational Expressions
• Solving Equations and Inequalities
• Ratio, Proportion, and Percent
• How Algebra is Used in Geometry
• Understanding Functions
• Working With Quadratic Equations and Functions

Mathematical Reasoning™ Supplements These supplemental books reinforce grade math concepts and skills by asking students to apply these skills and concepts to non-routine problems. Applying mathematical knowledge to new problems is the ultimate test of concept mastery and mathematical reasoning. These user-friendly, engaging books are made up of 50 theme-based collections of problems, conveniently grouped in self-contained, double-sided activity sheets that provide space for student work. Each collection contains relevant math facts at the end of the worksheet in case students need hints to solve the problems. Calculators are allowed on activity sets that have a calculator icon at the top of the front side of the set. Each activity set is accompanied by a single-sided answer sheet containing strategy tips and detailed solutions. Teachers and parents will appreciate the easy-to-understand, comprehensive solutions. These books are a wonderful enrichment tool, but also can be used to assess how well students have learned their grade level's math concepts.

Description and Features

All products in this series.

    • Our eBooks digital, electronic versions of the book pages that you may print to any paper printer.     • You can open the PDF eBook from any device or computer that has a PDF reader such as Adobe® Reader®.     • Licensee can legally keep a copy of this eBook on three different devices. View our eBook license agreement details here .     • You can immediately download your eBook from "My Account" under the "My Downloadable Product" section after you place your order.

• Add to Cart Add to Cart Remove This Item
• Special of the Month
• Sign Up for our Best Offers
• Bundles = Greatest Savings!
• Sign Up for Free Puzzles
• Sign Up for Free Activities
• Toddler (Ages 0-3)
• PreK (Ages 3-5)
• Kindergarten (Ages 5-6)
• 1st Grade (Ages 6-7)
• 2nd Grade (Ages 7-8)
• 3rd Grade (Ages 8-9)
• 4th Grade (Ages 9-10)
• 5th Grade (Ages 10-11)
• 6th Grade (Ages 11-12)
• 7th Grade (Ages 12-13)
• 8th Grade (Ages 13-14)
• 9th Grade (Ages 14-15)
• 10th Grade (Ages 15-16)
• 11th Grade (Ages 16-17)
• 12th Grade (Ages 17-18)
• 12th+ Grade (Ages 18+)
• Test Prep Directory
• Test Prep Bundles
• Test Prep Guides
• Preschool Academics
• Store Locator
• Submit Feedback/Request
• Sales Alerts Sign-Up
• Technical Support
• Mission & History
• Articles & Advice
• Testimonials
• Our Guarantee
• New Products
• Free Activities
• Libros en Español