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How To Encourage Critical Thinking in Math

By Mary Montero

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Critical thinking is more than just a buzzword… It’s an essential skill that helps students develop problem-solving abilities and make logical connections between different concepts. By encouraging critical thinking in math, students learn to approach problems more thoughtfully, they learn to analyze and evaluate math concepts, identify patterns and relationships, and explore different strategies for finding the solution. Critical thinking also involves a great deal of persistence. Those are critical life skills!

When you think about it, students are typically asked to solve math problems and find the answer. Showing their work is frequently stressed too, which is important, but not the end. Instead, students need to be able to look at math in different ways in order to truly grasp a complete understanding of math concepts. Mathematics requires logical reasoning, problem-solving, and abstract thinking.

What Does Critical Thinking in Math Look Like?

When I think about critical thinking in math, I focus on:

• Solving problems through logical thinking . Students learn how to break down complex problems, analyze the different parts, and understand how they fit together logically.
• Identifying patterns and making connections. Students learn how to identify patterns across different math concepts, make connections between seemingly unrelated topics, and develop a more in-depth understanding of how math works.
• Evaluating and comparing solutions. Students learn to evaluate which solution is best for a given problem and identify any flaws in their reasoning or others’ reasoning when looking at different solutions

Mathematician Posters

These FREE Marvelous Mathematician posters have been a staple in my classroom for the last 8+ years! I first started using a version from MissMathDork and adapted them for my classroom over the years. 

I print, laminate, and add magnetic stickers on the back. At the beginning of the year, I only put one or two up at a time depending on our area of focus. Now, they are all hanging on my board, and I’ll pull out different ones depending on our area of focus. They are so empowering to my mathematicians and help them stay on track!

A Marvelous Mathematician:

• knows that quicker doesn’t mean better
• looks for patterns
• knows mistakes happen and keeps going
• makes sense of the most important details
• embraces challenges and works through frustrations
• uses proper math vocabulary to explain their thinking
• shows their work and models their thinking
• discusses solutions and evaluates reasonableness
• gives context by labeling answers
• applies mathematical knowledge to similar situations
• checks for errors (computational and conceptual)

Critical Thinking Math Activities

Here are a few of my favorite critical thinking activities. 

Square Of Numbers

I love to incorporate challenge problems (use Nrich and Openmiddle to get started) because they teach my students so much more than how to solve a math problem. They learn important lessons in teamwork, persistence, resiliency, and growth mindset. We talk about strategies for tackling difficult problems and the importance of not giving up when things get hard.

This square of numbers challenge was a hit!

ALL kids need to feel and learn to embrace challenge. Oftentimes, kids I see have rarely faced an academic challenge. Things have just come easy to them, so when it doesn’t, they can lack strategies that will help them. In fact, they will often give up before they even get started.

I tell them it’s my job to make sure I’m helping them stretch and grow their brain by giving them challenges. They don’t love it at first, but they eventually do! 

This domino challenge was another one from Nrich . I’m always on the hunt for problems like this!!  How would you guide students toward an answer??

Fifteen Cards

This is a well-loved math puzzle with my students, and it’s amazing for encouraging students to consider all options when solving a math problem.

We have number cards 1-15 (one of each number) and only seven are laid out. With the given clues, students need to figure out which seven cards should be put out and in what order. My students love these, and after they’ve done a few, they enjoy creating their own, too! Use products, differences, and quotients to increase the challenge.

This is also adapted from Nrich, which is an AMAZING resource for math enrichment!

This is one of my favorite fraction lessons that I’ve done for years! Huge shout out to Meg from The Teacher Studio for this one. I give each child a slip of paper with this figure and they have to silently write their answer and justification. Then I tally up the answers and have students take a side and DEBATE with their reasoning! It’s an AMAZING conversation, and I highly recommend trying it with your students. 

Sometimes we leave it hanging overnight and work on visual models to make some proofs. 

Logic Puzzles

Logic puzzles are always a hit too! You can enrich and extend your math lessons with these ‘Math Mystery’ logic puzzles that are the perfect challenge for 4th, 5th, and 6th grades. The puzzles are skills-based, so they integrate well with almost ANY math lesson. You can use them to supplement instruction or challenge your fast-finishers and gifted students… all while encouraging critical thinking about important math skills!

Three levels are included, so they’re perfect to use for differentiation.

• Introductory logic puzzles are great for beginners (4th grade and up!)
• Advanced logic puzzles are great for students needing an extra challenge
• Extra Advanced logic puzzles are perfect for expert solvers… we dare you to figure these puzzles out! 

Do you have a group of students who are ready for more of a fraction challenge? My well-loved fraction puzzlers are absolutely perfect for fraction enrichment. They’ll motivate your students to excel at even the most challenging tasks! 

Math Projects

Math projects are another way to differentiation while building critical thinking skills. Math projects hold so much learning power with their real-world connections, differentiation options, collaborative learning opportunities, and numerous avenues for cross curricular learning too. 

If you’re new to math projects, I shared my best tips and tricks for using math projects in this blog post . They’re perfect for cumulative review, seasonal practice, centers, early finisher work, and more.

I use both concept-based math projects to focus on specific standards and seasonal math projects that integrate several skills.

Error Analysis

Finally, error analysis is always a challenging way to encourage critical thinking. When we use error analysis, we encourage students to analyze their own mistakes to prevent making the same mistakes in the future.

For my gifted students, I use error analysis tasks as an assessment when they have shown mastery of a unit during other tasks. For students in the regular classroom needing enrichment, I usually have them complete the tasks in a center or with a partner.

For students needing extra support, we complete error analysis in small groups.  We go step-by-step through the concept and they are always able to eventually identify what the error is. It is so empowering to students when they finally figure out the error AND it helps prevent them from making the same error in the future!

My 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.

When you’re ready for more, this bundle of error analysis tasks contains more than 240 tasks to engage and enrich your students in critical thinking practice.

If you want to dig even deeper, visit this conceptual vs computational error analysis post to learn more about using error analysis in the classroom. 

Related Critical Thinking Posts

• How to Increase Critical Thinking and Creativity in Your “Spare” Time
• More Tips to Increase Critical Thinking

Critical thinking is essential for students to develop a deeper understanding of math concepts, problem-solving skills, and a stronger ability to reason logically. When you learn how to encourage critical thinking in math, you’re setting your students up for success not only in more advanced math subjects they’ll encounter, but also in life. 

How do you integrate critical thinking in your classroom? Come share your ideas with us in our FREE Inspired In Upper Elementary Facebook group .

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.

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Mary Thankyou for your inspirational activities. I have just read and loved the morning talk activities. I do have meetings with my students but usually at end of day. What time do you

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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 .

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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 .

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Engaging Maths

Professor catherine attard, promoting creative and critical thinking in mathematics and numeracy.

• by cattard2017
• Posted on June 25, 2017

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?

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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.

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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.

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5 Ways to Stop Thinking for Your Students

Too often math students lean on teachers to think for them, but there are some simple ways to guide them to think for themselves.

Who is doing the thinking in your classroom? If you asked me that question a few years ago, I would have replied, “My kids are doing the thinking, of course!” But I was wrong. As I reflect back to my teaching style before I read Building Thinking Classrooms by Peter Liljedahl (an era in my career I like to call “pre-thinking classroom”), I now see that I was encouraging my students to mimic rather than think .

My lessons followed a formula that I knew from my own school experience as a student and what I had learned in college as a pre-service teacher. It looked like this: Students faced me stationed at the board; I demonstrated a few problems while students copied what I wrote in their notes. I would throw out a few questions to the class to assess understanding. If a few kids answered correctly, I felt confident that the lesson had gone well. Some educators might call this “ I do, we do, you do .”

What’s wrong with this formula? When it was time for them to work independently, which usually meant a homework assignment because I used most of class time for direct instruction, the students would come back to class and say, “The homework was so hard. I don’t get it. Can you go over questions 1–20?” Exhausted and frustrated, I would wonder, “But I taught it—why didn’t they get it?”

Now in the “peri-thinking classroom” era of my career, my students are often working at the whiteboards in random groups as outlined in Liljedahl’s book. The pendulum has shifted from the teacher doing the thinking to the students doing the thinking. Do they still say, “I don’t get it!”? Yes, of course! But I use the following strategies to put the thinking back onto them.

5 Ways to Get Your Students to Think

1. Answer questions with a refocus on the students’ point of view. Liljedahl found in his research that students ask three types of questions: “(1) proximity questions—asked when the teacher is close; (2) stop thinking questions—most often of the form ‘is this right’ or ‘will this be on the test’; and (3) keep thinking questions—questions that students ask so they can get back to work.” He suggests that teachers acknowledge “proximity” and “stop thinking questions” but not answer them.

Try these responses to questions that students ask to keep working:

• “What have you done so far?” 
• “Where did you get that number?” 
• “What information is given in the problem?” 
• “Does that number seem reasonable in this situation?”  

2. Don’t carry a pencil or marker. This is a hard rule to follow; however, if you hold the writing utensil, you’ll be tempted to write for them . Use verbal nudges and hints, but avoid writing out an explanation. If you need to refer to a visual, find a group that has worked out the problem, and point out their steps. Hearing and viewing other students’ work is more powerful .

3. We instead of I . When I assign a handful of problems for groups to work on at the whiteboards, they are tempted to divvy up the task. “You do #30, and I’ll do #31.” This becomes an issue when they get stuck. I inevitably hear, “Can you help me with #30? I forgot how to start.”

I now require questions to use “we” instead of “I.” This works wonders. As soon as they start to ask a question with “I,” they pause and ask their group mates. Then they can legitimately say, “ We tried #30, and we are stumped.” But, in reality, once they loop in their group mates, the struggling student becomes unstuck, and everyone in the group has to engage with the problem.

4. Stall your answer. If I hear a basic computation question such as, “What is 3 divided by 5?” I act like I am busy helping another student: “Hold on, I need to help Marisela. I’ll be right back.” By the time I return to them, they are way past their question. They will ask a classmate, work it out, or look it up. If the teacher is not available to think for them, they learn to find alternative resources.

5. Set boundaries. As mentioned before, students ask “proximity” questions because I am close to them. I might reply with “Are you asking me a thinking question? I’m glad to give you a hint or nudge, but I cannot take away your opportunity to think.” This type of response acknowledges that you are there to help them but not to do their thinking for them.

When you set boundaries of what questions will be answered, the students begin to more carefully craft their questions. At this point of the year, I am starting to hear questions such as, “We have tried solving this system by substitution, but we are getting an unreasonable solution. Can you look at our steps?” Yes!

Shifting the focus to students doing the thinking not only enhances their learning but can also have the effect of less frustration and fatigue for the teacher. As the class becomes student-centered, the teacher role shifts to guide or facilitator and away from “sage on the stage.”

As another added benefit, when you serve as guide or facilitator, the students are getting differentiated instruction and assessment. Maybe only a few students need assistance with adding fractions, while a few students need assistance on an entirely different concept. At first, you might feel like your head is spinning trying to address so many different requests; however, as you carefully sift through the types of questions you hear, you will soon be comfortable only answering the “keep thinking” questions.

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Higher Order Thinking Math in 1st Grade

susanjones March 22, 2015 10 Comments

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But why are they not excelling on our standardized tests?!?

Over the years I created and collected tasks that are both challenging and fun for my first graders. Each task also has a challenge question that can be used when a student solves the problem and is looking to challenge themselves a little further.

Here are some in use in my classroom:

What’s the equation: The sum is 20, what is the equation? These two students worked together to come up with many different ways to make 20, including 3 and 4 addend equations.

Test time! To see if these two had progressed with subtraction within 20, their task was to create a math test (with an answer key) to give to a friend! They loved getting to be the teacher and I could tell right away that they knew how to subtract within 20 easily!

 Dress Teddy! This was a fun and tricky number sense problem for my students. They had to look at the clothes and try to figure out how many different outfits they could make for the teddy bear. At first, this group was all sorts of confused. Until one of my kids said, “let’s start with the pants… there’s only one pair of pants.” From there they could mix and match and record the different outfits. It is SO hard to bite my tongue and not guide them, but if you can hold back long enough it is amazing to see them persevere to get the answers. They feed off one another and it is pretty cool to watch!

I created 8 different tasks (which each have an additional *challenge* task) for each of the following domains:

Number sense Addition Subtraction Place Value Geometry & Measurement Time & Money

Each task comes in 3 different forms as well. There is a printable version that you can see above with the question on it. There is a guided printable version which is the same, but has guiding questions to help your students complete the problem, and a task card version to print, laminate and pass out to groups.

If you think these would be great for you students, head on over and check them out:

You can download the preview for more examples of the tasks 🙂

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Reader Interactions

10 comments.

March 22, 2015 at 11:34 pm

This looks great. I am adding it to my wishlist! Marcy [email protected]

March 23, 2015 at 1:20 am

This product would be awesome for incorporating those skills into higher level thinking but how do you incorporate this when you have a scripted math program as well? We use Saxon and have found it to do just what you said – teaches the skills but the kids still don't have the critical thinking aspect they need to be successful.

Thanks! you can email me at [email protected]

The Weekly Sprinkle

March 25, 2015 at 2:17 am

Love it! I'm also a first grade teacher who is passionate about high level math thinking skills! I always love finding fellow math junkies!

Whitney @ The First Grade Roundup

March 25, 2015 at 7:25 am

I just picked up this packet! Pinning your post for reading & rereading -I'm really excited about incorporating these ideas! Thank you, Jen

July 28, 2015 at 3:57 am

I LOVE this and am sharing it in my blog post tonight on math discussion strategies! Thank you! 🙂

September 26, 2020 at 5:05 pm

Thank you! I’m looking at setting up my Numeracy block for my K/1 students. I’m finding it challenging to spread myself between both groups! Your activities are super helpful!

August 27, 2022 at 5:49 am

This is awesome i loved it. I am a parent and gonna try this for sure. Thank you for creating amazing sheets.

September 17, 2022 at 1:42 am

I’ve been searching for ways to increase higher order thinking skills in math. I love these ideas!

September 17, 2022 at 1:43 am

I’ve been searching for ways to increase higher order thinking skills in math. I love these ideas! Are these items for sale in a bundle?

September 17, 2022 at 1:44 am

I love the ways that you have incorporated higher order thinking skills in fun and engaging activities.

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Math Games for 1st Grade: Print, Play, LEARN!

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Hello friends.

Welcome to Susan Jones Teaching. When it comes to the primary grades, learning *All Things* in the K-2 world has been my passion for many years! I just finished my M.Ed. in Curriculum and Instruction and love sharing all the latest and greatest strategies I learn with you through this blog and my YouTube channel! I hope you'll enjoy learning along with me :)

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Increasing Critical Thinking Skills in Math

• Math , Planning

It’s important that we are building critical thinking skills in math. Too often these are overlooked or assumed that students do it because they have to problem solve sometimes. While that does help build the all-important critical thinking skills, we need to make sure we are also finding ways to purposely bring it into instruction.

One such way that I like to implement critical thinking skills in my math class is through a game called Puzzlers. Recently I discussed why you should use games in the classroom and this one is no exception. Games go beyond just having fun and “entertaining” students. They aren’t just fillers.

Building Critical Thinking Skills with the Puzzler Game

The puzzler game is a game that not only increases critical thinking skills, but it also practices both fact fluency and the order of operations!

In the puzzler game, students are given a target number. This happens by rolling a die or dice, but it can also be any chosen number between 1 and 36. For instance, I have randomly chosen the date before.

Next, students are provided with a 3×3 grid of the numbers 1 through 9 mixed up. (See the image below.)

Once students have their target number and a mixed up grid of the numbers 1-9, they are ready to begin. This is where the critical thinking skills will come in.

Now, students will need to come up with a way to use ONLY three numbers (in a row, diagonally, or in a column) to get that target number. They will do this by creating equations that total the target number. They can add, subtract, multiply, divide, or even come up with a combination of them. If needed, they can use parentheses. This is where knowing the order of operations is necessary!

For instance, let’s take the example above with the 9 numbers on the sticky notes. Let’s say that the target number was 18. The student could create these two equations to come up with the solution of the target number 18:

• (9 x 6) ÷ 3
• (9 + 8) – 1

Here’s an example of a puzzler card with multiple solutions:

What I love about this puzzler game is the variety of ways it can be used to help build critical thinking skills! For instance, students could list all of the equations, or solutions, to get the target number:

or go through multiple cards trying to list as many solutions as they can:

Or they could skip rolling the dice altogether and see how many solutions they can find for the target numbers one through ten. Why not even through in zero?!

Students love this game and it’s perfect for independent work, early finishers, small groups, and even enrichment. It’s differentiated and there are cards that are strictly for adding and subtracting for students who can’t multiply yet.

You don’t have to purchase my puzzler resource to play this critical thinking skills builder! You can easily create it in your classroom as a bulletin board and change out the numbers each day!

If you want to save some time, grab the extra differentiated materials, and the specifics, head to my store now to purchase it! It’s definitely worth it!

Click here to purchase this Puzzler Game.

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How to Seamlessly Add Critical Thinking Questions to Any Math Assessment

• March 5, 2024

I’ve said it once and I’ll say it again. quality over quantity. I am on a mission to help other teachers make their math assessments more meaningful and insightful by reducing the number of questions that they’re asking students to complete and adding in critical thinking questions.

I’ll always keep it real with you. Your students are not going to need 90% of the math that they learn in your class. This is what they are going to need: how to problem solve and justify their answers based on evidence. 

Colleges and employers do not care about a student’s GPA. They want to know that a student can work hard and persevere through any problem that they face in their job or career. We cannot continue to teach using the plug-and-chug method because that is diminishing the importance of math class and the life skills that students have the potential to learn in high school.

There are a few different ways to add critical thinking questions to any of your formative or summative assessments. 

Table of Contents

Let's start with formative assessments…

You can easily add higher order thinking questions to exit tickets , classwork, or guided notes.

How to level up your exit tickets

You can make any question become a higher order thinking question just by asking your students to list out the steps. 

Let me give you an example. If you were to ask your students how to solve a multi step equation, start by asking them to simply solve the problem. Then, on the side have them list out their step by step process and explain how they know that the answer that they got was correct. 

This just takes your original problem a bit further so that you know your students actually understand without having to give them a full assessment. It’s also just a really quick and easy way to get your students thinking more critically. 

If you want to go a little bit deeper with your exit ticket and your students are still solving equations, you can ask them to explain solving multi-step equations to a student in 5th grade. 

We know that fifth grade students can add, subtract, multiply, and divide but how would they explain solving multi-step equations to a student that has really never seen an equation before. What would that look like? 

It is always really interesting to see what your students will say about the steps that they take to solve the problem and how they know that they are confident that their answer is correct.

Slim down your homework assignments

You can do the same thing with regular class work, homework, or quizzes. I really believe in less is more and quality over quantity. Instead of assigning your students 20 problems for multi step equations you can assign them five and then add one or two  critical thinking questions . 

It’s important for your students to know that these questions are low stakes at first because you want them to just share their ideas and thought processes. It doesn’t necessarily have to be right. This is especially true for class work that you might not be grading for accuracy. 

If you’re able to give your students the opportunity to think through a problem then you will be able to see what they do and don’t understand, and it will get them thinking about math in a deeper way. 

Adding critical thinking questions to summative assessments

Before adding a lot of challenging critical thinking questions to your assessments, it’s really important to prepare your students so that they know what to expect. Of course, you never want to catch your students off guard with the assessment. 

Adding these types of questions to your daily routine (or even weekly routine) is really helpful for having your students practice that deeper thinking and letting them know that the expectation is that you want them to go beyond just getting the right answer.

If problem solving and having a deeper understanding of math is a priority for you and your classroom, then they should know that that’s the case.

The same thing that I said about formative assessments and homework applies to summative assessments as well. If you are going to add critical thinking questions, do you really need to have 20 to 30 questions on a test? Probably not. 

Instead, you can have 8-10 questions, depending on how long your unit is, and add in 2 to 3 critical thinking questions that combine multiple types of questions from the test.

For example, if you are teaching polynomial operations , your critical thinking question could be compare and contrast adding polynomial expressions and multiplying polynomial expressions. Multiple subtopics within a larger unit are included while you are cutting down on the amount of questions. 

Another critical thinking question that I love to have in my summative assessments is error analysis . I recommend taking common errors from questions that your students have shown in the past and replicating those as error analysis problems. 

It’s very evident from error analysis problems which students actually understand and which students are just going through the motion of solving the problems.

I’ve always found that my students really struggle with identifying the errors because they aren’t really analyzing the problems. They see the work on the paper, think that it looks right, and assume that there is no error.

However, the process of finding an error, fixing it to show the correct work, and explaining how they know that their work is correct is an invaluable skill. 

My final thoughts about adding critical thinking questions to your assessments.

Adding critical thinking questions to your summative assessments doesn’t need to be challenging or overcomplicated. You don’t have to create a whole new assessment. Take out some of the repetitive problems and replace it with a critical thinking question. Some really good examples of this would be compare and contrast, error analysis, explaining your thinking, and explaining a concept to someone in a previous grade level.

If you're looking for more ideas for how to add critical thinking questions seamlessly into your daily routine check out these blog posts.

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How to use questions to inspire mathematical thinking

• Powerful teaching strategies
• May 30, 2023
• Michaela Epstein

Blog > How to use questions to inspire mathematical thinking

Take a look at this puzzle. What do you think it might be about?

Only the numbers 1 to 4 can be used. Now, look at the puzzle again and consider: does that change what might be going on?

We'll come back to this puzzle – and what it's all about – in a bit.

The difference between instructions and questions

In the puzzle you've been looking, what you weren't given, was a list of step-by-step instructions on how to solve it:

Instead, you were given a series of prompts designed to get you focusing on the information in front of you, and thinking about what might be going on:

If you step into the shoes of a learner, consider what these questions offerered you. For example:

• Time to take in complex information
• An opportunity to form your own ideas, before ‘getting started' with the maths
• A space for being creative
• A chance to make observations without needing to be right or wrong.

But, how do you know what questions to ask? What kinds of questions can encourage thoughtful, reflective mathematical thinking rather than mindless rule following?

Helpfully, there's a categorisation that can make this easier for you.

Introducing funnelling and focusing questions

Questions are the bread and butter of a teacher's day. They're used for:

• teaching new skills and concepts,
• help students getting unstuck
• extending and challenging thinking,
• affirming and confirming ideas,
• redirecting attention.

While questions are obviously present in classroom conversation, what's less obvious are the types of questions that get asked.

Enter: funnelling and focusing questions.

I first came across Funnelling and Focusing questions from maths educator, Mark Chubb . It was one of those lightbulb moments that suddenly helped me to look at the purpose of questioning and its value to student thinking in an entirely new way.

So, what are they?

Funneling questions guide students through a procedure or to a pre-determined endpoint. For example:

• What should you do first?
• What's the next step?
• So, what's the answer?

The message that's being communicated to students through funnelling questions is to correctly follow your steps.

In contrast, focusing questions guide students based on their ideas and responses. For example:

• What's this question asking you?
• What do you notice about this example?
• Can you explain your approach to me?
• Why do you think this symbol/word is used?
• Look at these solutions. How are they similar/different?

Focusing questions are a way of you understanding a student's thinking. These questions put you in a position to then provided meaningful, targeted support.

There's a subtle value to focusing questions, which I believe is worth surfacing here. These questions:

• help students to think for themselves,
• to become reflective and analytic, and
• they can inspire confidence and curiosity about what’s going on.

All of these things are absolutely worthwhile in the maths classroom.

Why should we care about focusing vs funnelling?

I believe that both focusing and funnelling questions can have a role in maths class, depending on a combination of factors, such as:

• the purpose of a lesson,
• where students are at in their understanding and confidence,
• student motivation,
• the culture and values that you want to promote around doing maths.

We all get into patterns of speaking, instructing and communicating with students – and many of these patterns are, unknowingly, built up over time. Reflecting on the extent to which you're using funnelling and focusing questions, is a way of making these patterns visible – and then putting yourself in a position to adjust your practice.

So…. what was that puzzle about?

The puzzle at the very start of this article is called a Skyscraper puzzle. It's one of my favourites!

The numbers inside the grid represent the heights of skyscraper buildings. Each row and column contains one building of each size. The clues outside the grid tell you how many skyscrapers are visible (in a row or column) as seen from that vantage point.

Sometimes it helps to look at the answer to make sense of what you're first given.

See if you can understand what's meant by the number of visible buildings from each vantage point that's outside the grid:

If you want to play around with Skyscraper puzzles, here are 3 great sites:

• Krazy Dad has loads of printable puzzles, starting easy and getting really tough
• The Art of Puzzles has standard versions, plus many fantastic variations
• Mark Chubb shows how you can make the puzzle concrete using linking cubes.

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Thinking Mathematically - Secondary Students

Successful mathematicians  understand and use mathematical ideas and methods, solve problems, explain and justify their thinking, and have a positive attitude towards learning mathematics. 

Exploring, questioning, working systematically, visualising, conjecturing, explaining, generalising, convincing, proving... are all at the heart of mathematical thinking. The activities below are designed to give you the opportunity to think and work as a mathematician.

For problems arranged by mathematical topics, see our  Topics in Secondary Mathematics  page. For problems arranged by mathematical mindsets, see our Mathematical Mindsets  page.

Exploring and Noticing - Secondary Students

What do you notice as you explore these problems?

Working Systematically - Secondary Students

Work on these problems to improve your ability to work systematically.

Conjecturing and Generalising - Secondary Students

Work on these problems to improve your conjecturing and generalising skills.

Visualising and Representing - Secondary Students

Work on these problems to improve your visualising and representing skills.

Reasoning, Convincing and Proving - Secondary Students

Work on these problems to improve your reasoning skills.

Thinking Mathematically - Short Problems

A collection of short problems which require students to think mathematically.

In this film  (available  here if you live outside the UK) the mathematician Andrew Wiles talks about his personal experience of seeking a proof of Fermat's Last Theorem.  He describes what it is like to do mathematics, to be creative, to have difficulties, to make mistakes, to persevere, to make progress, to have a dream and love what you are doing so much that you are willing to devote yourself to it for a long time.  Of course, each mathematician's experience is different, and most mathematicians do not work alone for such prolonged periods without discussing their work with others, but much of Andrew Wiles' experience is shared amongst mathematicians, and reminds us of the rewards of perseverance in the face of difficulty.  

We have compiled a list of books for young people who are interested in mathematics.  

• School Guide
• Mathematics
• Number System and Arithmetic
• Trigonometry
• Probability
• Mensuration
• Maths Formulas
• Class 8 Maths Notes
• Class 9 Maths Notes
• Class 10 Maths Notes
• Class 11 Maths Notes
• Class 12 Maths Notes

Critical Thinking Math Problems

Ability to make decisions by the application of logical, sceptic, and objective analyses and evaluations of data, arguments, and other evidence is known as critical thinking. Regarding mathematics, critical thinking is not only about making calculations but also involves logical reasoning, pattern recognition and problem-solving strategies.

In this article, we dive into various aspects of critical thinking in mathematics, including problems, strategies for solving them, which activities can increase critical thinking, and some examples with solutions. If you want to improve your critical thinking or even know more about getting better at math, then keep reading.

Table of Content

Types of Critical Thinking Math Problems

Strategies for solving critical thinking problems, activities to enhance critical thinking, benefits of critical thinking in math, examples and solutions.

There can be various types of problems. Some may be related to numbers, others to shapes and some even comprehensions. Let’s take a closer look at them:

• Word Problems : Students must convert verbal descriptions into mathematical equations to solve word problems. These exercises improve understanding abilities and encourage applying mathematical ideas to practical settings.
• Logic Puzzles : Logic puzzles, like logic grids or Sudoku, require students to apply deductive reasoning and pattern recognition skills. A methodical approach and multiple steps are frequently needed to solve these puzzles.
• Algebraic Problems : Variables and unknowns are part of algebra. Students must work with equations and inequalities. Working through these kinds of problems promotes the development of abstract thinking as well as proficiency with symbols, formulas, and identities.
• Geometric Problems : Geometric problems relate to the shapes, sizes and properties of figures such as circles, triangles and rhombuses. Solving these problems involves spatial reasoning and the application of theorems and postulates.
• Probability and Statistics : While statistics entails evaluating vast amounts of data, generating hypotheses, and concluding, probability involves determining the possibility that an event will occur. Students learn to deal with ambiguity and generate well-informed predictions from these problems.

The majority of people think they are terrible at math, and when they are asked to solve problems in math, they just don’t know how. The following are some methods that you can apply to resolve an issue:

• Understand Problem : The most important thing is to understand the problem statement. The question always provides some key information, such as relations, conditions or equations. If required, read the problem several times and carefully note the provided information and what is being asked.
• Break Down Problem : Some problems can be more complex than others and may require lengthy procedures and much thinking. Breaking down such problems into smaller parts can make them easier to solve. This step-by-step approach makes it easier to tackle each component individually.
• Visualize Problem : Try to draw diagrams, graphs or charts to simplify the problem and make abstract concepts more concrete. Visualization aids in understanding relationships and patterns.
• Use Logical Reasoning : Logical reasoning involves making connections between known information and new insights. This process includes making assumptions, drawing inferences, and evaluating the validity of conclusions.
• Check Your Work : After arriving at a solution, it is crucial to verify its accuracy. This can involve reworking the problem, checking calculations, and ensuring that the solution makes sense in the context of the problem.

Improving critical thinking is not only about working out problems; there are other ways in which you can boost your critical thinking:

• Math Games: What better way to learn than playing games? Chess or strategy board games can help develop strategic thinking and problem-solving. These games make learning fun and engaging.
• Group Work: Working in a group can be a great and fun way of learning. A problem can be solved in multiple ways. When you work together in a group, you get to know about how others perceive problems and can learn different ways to solve them.
• Real-World Projects: It is really important for students to learn to use their critical thinking skills in practical situation. When students solve real world problems, they get to use their textbook knowledge which can help to deepen their understanding of the subject matter. This also helps students build real-world skills that can benefit their careers.
• Technology Integration: Learning may become more interactive through the use of technology in the classroom. Students’ learning in the classroom has changed dramatically as a result of educational technology, including software, digital whiteboards, and, more recently, artificial intelligence.

Working on critical thinking in math has several benefits, such as:

• Improved Problem-Solving Skills: Critical thinking boosts students ability to solve problems. It equips students with the ability to approach problems with various methods come up with different solutions and choose the most effective one.
• Enhanced Creativity: Critical thinking exercises help students become more creative because they let them see problems from multiple perspectives and come up with unique solutions.
• Better Decision Making: Critical thinking equips students with the ability to take well-informed decisions. It allows them to take action based on logical reasoning and evidence.
• Greater Academic Success: Students with strong critical thinking skills often perform better in their academics as they can handle complex problems across multiple subjects.
• Lifelong Learning: Critical thinking fosters a curiosity and continuous learning mindset essential for personal and professional growth.

In conclusion, critical thinking is the ability to solve problem by analyzing available information, and observations. Enhancing one’s critical thinking abilities can be achieved through tackling an array of problems. Beyond the classroom, critical thinking helps students make better decisions and solve problems throughout their lives.

Let’s take some examples of different types of critical thinking questions to understand better:

Example 1: Students were scheduled for a field trip by a school. The field trip is being attended by 120 students in total. 28 students can fit on a school bus. Find out the number of buses required to hold every student and the number of students on the final bus.

To find the number of buses required, divide the total number of students by the number of students each bus can hold. 120 ÷ 28 ≈ 4.29 You need five buses because one cannot have a fraction of a bus. Multiply the number of full buses (4) by the bus capacity (28) to determine the number of students on the final bus, then deduct that amount from the total number of students: 120 – (4×28) = 120 – 112 = 8 students on the last bus.

Example 2: There are three crates labeled Apples, Oranges, and Apples and Oranges. Each label is incorrect. How can you determine the contents of each crate by picking one fruit from one crate?

Pick a fruit from the crate labeled Apples and Oranges. If you pick an apple, this crate must be Apples (since all labels are wrong). The crate labeled Apples must then be Oranges, and the crate labeled Oranges must be Apples and Oranges. If you pick an orange instead, the crate is Oranges, the one labeled Apples is Apples and Oranges, and the one labeled Oranges is Apples.

Example 3: Solve for x in the equation 3x – 7 = 2x + 5

Subtract 2x from both sides: 3x – 2x – 7 = 5. Simplify: x – 7 = 5. Add 7 to both sides: x = 12.

Example 4: Find the area of a trapezoid with bases of 5 cm and 7 cm, and a height of 4 cm.

Use formula for the area of a trapezoid: Area = 12×(Base 1​ + Base 2 ​)×Height = 12​×(5 + 7)×4 = 24 cm²

Example 5: What is the probability of drawing an ace or a king from a standard deck of 52 playing cards?

In a 52-card deck, there are 4 aces and 4 kings, which results in 8 positive outcomes. Probability is calculated by dividing the total number of outcomes by the number of positive outcomes. 8/52 = 2/13

Example 6: A survey was conducted to find out whether people prefer tea or coffee. 80 out of 200 people said they prefer tea over coffee. What percentage of the surveyed population prefers tea?

To calculate the percentage of people who preferred tea: = 80200 × 100 = 40% So, 40% of the surveyed population prefers tea.

Example 7: What is the next number in the sequence 2, 6, 12, 20, 30, …?

Addition of consecutive even numbers is the pattern: 2 + 4 = 6 6 + 6 = 12 12 + 8 = 20 20 + 10 = 30 Thus, 30 + 12 = 42 would be the following number.

Practice Problems with Solutions

Q1. A library has 400 books. 25% of them are fiction. How many fiction books are there?

0.25×400 = 100 There are 100 fiction books

Q2. There are five houses in a row. Each house is painted in a different color. It is known that the red house is not next to the blue house, and the green house is to the left of the yellow house. If the blue house is on one end and the white house is directly to the right of the red house. Find where is the green house?

Blue house should be the last one since it is the only possible solution to meet all the given criteria. The red house cannot be next to the blue house and must be placed such that the white house is directly to its right. So the possible arrangement is {R, W, G, Y, B}. Green house is the 3rd house.

Q3. Solve for z: 5z – 2 = 3z + 8.

5z – 3z – 2 = 8 2z – 2 = 8 2z = 10 z = 5

Q4. Determine the volume of an 8-cm-long, 5-cm-wide, and 10-cm-tall rectangular prism.

Volume = length×width×height = 8×5×10 = 400 cm³

Q5. What is the probability of drawing a heart from a deck of 52 cards?

There are 13 hearts in the deck. Probability is 13/52 = 1/4

Q6. The sales of a company increased from $150,000 to $180,000. What was the percentage increase in sale?

Percentage Increase = (180,000 – 150,000)/150,000×100 Percentage Increase = 20%

Q7. What is the next number in the sequence 3, 9, 27, 81, …?

The pattern is multiplication by 3: 81×3 = 243 The next number is 243.

Q8. If 4y + 6 = 3y – 2, determine the value of y?

4y + 6 = 3y – 2 4y-3y + 6 = – 2 y + 6 = – 2 y = – 8

Frequently Asked Questions

What is critical thinking in math problems.

In mathematics, critical thinking is a talent that enables pupils to examine, assess, and interpret data in order to come up with a solution. These issues frequently need the application of multiple steps and a profound comprehension of mathematical ideas.

How to improve mathematical critical thinking abilities?

Solve a variety of problems, including word puzzles, algebraic equations, and logic issues, to hone your critical thinking skills in mathematics. Engage in activities like games and puzzles that test your ability to think and analyze information.

Why are critical thinking skills important in math?

Mathematical critical thinking abilities are crucial because they enable students to comprehend difficult subjects, work through difficulties efficiently, and apply their knowledge of mathematical reasoning to practical situations.

Can critical thinking math problems be used in standardized tests?

Yes, analytical thinking Standardized examinations frequently utilize arithmetic questions to evaluate a student’s capacity for critical thought, analysis, and problem-solving. The purpose of these questions is to assess a student’s deeper comprehension of mathematical ideas rather than just rote memory.

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