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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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The Power of One Good Math Question

By focusing on authentic moments of inquiry, elementary students engage in critical thinking in math class.

Most of the math problems we present to elementary students are scripted. They might come from the curriculum we are teaching, or maybe they are a three-act math task we learned at a workshop or found in a blog. But often the best problems are the ones that we had not planned and are instead based in the real life of the classroom. There is a lot of power in a single, authentic question.

Moments of Inquiry in Math Class

Everyday life presents lots of interesting math problems. The best of these inquiry opportunities come from questions the students themselves pose if we stop and listen to what they are asking. No one works on problems like these in class every day, and they shouldn’t. But an openness to tackling something that comes from actual experience connects the classroom to the underlying power of mathematics to help us make sense of the world. 

One day, Maggie, who was new to teaching first grade, went to retrieve a box of dominoes she had set aside to use for the next unit. Dominoes are a great manipulative for first grade because they encourage subitizing (recognizing small numbers without counting), and they help students see fact families. For instance, a single two-dot/three-dot domino can show all of these: 2+3=5, 3+2=5, 5–2=3, 5–3=2.

Instead of finding neatly organized sets, she found a box full of jumbled dominoes. When she met with Steven, the math specialist who was coaching her, they both looked at the plastic container and had the same sinking feeling: It was going to take a while to sort them back into sets. Neither of them had a good sense of how many sets were even in the box or how many dominoes constituted a set. But this felt like a good question to think about, so instead of doing the sorting themselves, they brought the problem to the students. 

presenting students with the task

Maggie presented the bin of dominoes to her class and explained that there were many sets mixed together. Students had some prior knowledge of what individual dominoes looked like, but for this exercise, the only additional information that students had was that there would not be any duplicates in a single set. There was no explanation of how many dominoes were in a full set or clues about how to organize them.

The class split into small groups and set to work. The students had the opportunity to approach the task in any way that made sense to them. Some students sorted the dominoes based on one half of the dominoes’ pips. Some students sought out patterns within the full collection of dominoes. For example, one group looked at how many had one pip on one side and six on the other. Some students focused primarily on casting aside duplicates or bringing them to another group that could use them. 

The first completed set of dominoes cast a hush over the room—the excitement was palpable. Now, with a visual cue, students noticed patterns that they had been grappling with. Students observed how the group had arranged them: a complete row of dominoes with six pips on the bottom, moving up the ladder to finally one domino without any pips on either side. Students asked all sorts of questions about where their peers had started and how they thought about the sequence. They noticed that the total number of pips in each row increased as they worked their way down from top to bottom. Students shared how organizing it this way made it easier to notice which dominoes were duplicates within the set. 

making one big question work

First of all, the task was clear and compelling. It was a single question that was easy to understand, but there was not a single path toward a solution. There were curricular connections—pattern making, subitizing, and number properties (a nice physical model for the commutative property)—but the students had the freedom to figure out their own approach. The students had to do the thinking, the organizing, and the planning. They had ownership over the problem . Moreover, it was a question that caught their curiosity, and they wanted to know the answer.

These aren’t the only questions that Maggie’s class has considered this year. A picture of a sundial led to looking at Roman numerals and other number systems. A poll about favorite shapes led to a lively discussion about whether a heart was a shape or not (and introduced the word cardioid !). In order to choose questions that guide conversation, here are some guidelines: 

1. Make it real. Kids can tell if the question is a setup or something you really want to know. 

2. Keep it simple. Frame the question in the simplest way possible. Simple questions leave the most room for creativity and thinking. 

3. Stay out of their way. Often we rush in with too much structure and over-scaffold the problem. When we do that, we reduce the thinking, and it becomes about doing what is expected rather than solving the problem. When we hand students a list of questions to guide them or a form for their answers, it becomes another worksheet to fill out, and filling out the worksheet correctly becomes the goal. Let the answer to the question be the goal. 

4. Let them struggle. Don’t jump in to help. Don’t make suggestions. Limit yourself to asking questions and only answering questions with questions. 

5. Celebrate the solution. Or, if there are no solutions that emerge, celebrate that we couldn’t figure it out. That happens too. 

6. If there is a natural audience, share the results. Neither Maggie nor Steven knew how many sets of dominoes there were until the students told them. That in itself was worth celebrating. 

Improving secondary school student’s mathematics critical thinking in geometry transformation through GeoGebra-Based flipped learning: An experimental study

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Millenia Hanifah , Naufal Ishartono , Siti Hajar Binti Halili , Rafiza Binti Abdul Razak; Improving secondary school student’s mathematics critical thinking in geometry transformation through GeoGebra-Based flipped learning: An experimental study. AIP Conf. Proc. 7 June 2023; 2727 (1): 020024. https://doi.org/10.1063/5.0141401

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The current situation of the Covid-19 pandemic is a challenge for education practitioners, especially in the online learning process, including in maths learning which plays an essential role in improving student’s critical thinking skills. Of course, this encourages various efforts to optimize student’s critical thinking skills during online learning. However, there are currently few research results that examine further efforts to improve student’s critical thinking skills through the GeoGebra-assisted Flipped Learning learning approach on geometric transformation during online learning. The study aims to test the effectiveness of GeoGebra integration on the Flipped Learning approach in improving student’s critical thinking skills during online learning. The type of research used is quantitative with experimental methods using the draft pre-test post-test control group design and involves 102 students of grade XI from one of the high schools in Indonesia who were selected by simple random sampling. Based on the pre-test and post-test results using statistical prerequisite tests, One-Way ANOVA, and post-hoc tests, the GeoGebra-Based Flipped Learning approach is more effective to improve student’s critical thinking skills during online learning when compared to flipped learning or direct learning approaches. Therefore, the results of this study are expected to be an alternative for education practitioners to develop student’s critical thinking skills so that online learning runs more effectively and efficiently.

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Critical Thinking in Mathematics Education

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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...

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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

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Junior High Students' Critical Thinking in Geometry Problem Solving

2020, Universal Journal of Educational Research

In this study, students' critical thinking in geometry problem solving is referred to the five stages namely Identifying, Visualizing, Analyzing, Applying and Evaluating. Some aspects were evaluated at every stage. Identifying included elaborating information, associating problems with prior knowledge and stating the root of the problem; Visualizing included using images or sketches to represent the problem; Analyzing encompassed simplifying problem situations and making an initial prediction or conjecture; Applying involved determining the stages for solving the problem and finding the solution to the problem; and Evaluating included clarifying the key terms used and rechecking the problem-solving result. This qualitative study was conducted on 60 students from SMP Negeri 1 Pangkajene. The data were collected using a geometry problem-solving test. The results of the analysis showed that at the Visualizing stage, three subjects used a garden drawing or sketched a garden to represent the problem, while one subject solved the problem without representation. At the Applying stage, all of the subjects used different strategies to solve the geometry problem.

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This study aimed to describe the students’ critical thinking processes in solving plane geometry problems based on cognitive styles and gender styles. This research used the descriptive qualitative method. The subjects of this study were four junior high school students selected based on differences in cognitive styles and gender styles. The instrument consisted of the main instruments are researchers and supporting instruments, namely problem-solving instruments, cognitive style instruments, and interview guidelines. The data were collected by means of tests and interviews. The results showed there is a significant difference in critical thinking based on students’ cognitive styles. Students with field independent cognitive style tend to go through critical thinking stages more fully than students with field dependent cognitive style. Besides that, gender differences also have an impact, although not as significant as the cognitive style.

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Engaging Maths

Dr 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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Understanding Geometry

Understanding Geometry is part of the Mathematical Reasoning series, but it differs significantly enough from the rest of the series to warrant its own review.

The Mathematical Reasoning series (for preK through sixth grade), along with some other elementary math programs, such as Singapore Math/Primary Mathematics, is quite advanced by the end of sixth grade. Rather than push students into algebra, the option selected here is to offer serious geometry for students who might be in grades seven through nine.

Understanding Geometry can be used in junior high as a one-year course or over two years if supplemented with other math learning. It might be used as a high school course but it does not go as far in depth as do most high school courses, and it does not include trigonometry. Nevertheless, it does teach all of the other key topics of a high school geometry course, including proofs—sufficient coverage for a high-school credit. An answer key is included at the back.

The author acknowledges that students might need “more exercises per topic.” Instruction is thorough but there are significantly fewer practice problems, no review problems, and no quizzes or tests. You can create additional worksheets for free at sites such as http://www.math-aids.com/Geometry/ , but these worksheets and most I could find online for free are suitable only up through eighth grade. They do not provide practice for all concepts taught in this book. Books such as Mark Twain Media’s Geometry Practice workbook might be more useful as supplements. The limited number of practice problems and lack of assessment tools are the only negative issues I see with this course.

Before starting Understanding Geometry , students need to understand enough algebra to understand how to write and work with algebraic equations for lines. This is a math-based approach to geometry as is used in other high school level geometry courses (and in contrast to Key to Geometry which teaches concepts without mathematical computation.) Students use symbolic language and equations. If students have been adequately prepared with prior courses such as those in the Mathematical Reasoning series, they should have no problem. Coming from some other programs, students might need to have completed a seventh or eighth-grade level program before tackling Understanding Geometry . Sample pages can be viewed on the publisher’s website so you can judge whether or not a student is ready for the course. (click here)

Understanding Geometry sometimes uses inductive methods where students “discover” mathematical concepts through explorations, but it also uses direct instruction. Students are presented with some standard problems such as determining the volume of a cylinder. However, many problems ask students to explain their reasoning or describe a problem-solving strategy. Critical thinking is stressed throughout the course. In addition, students learn how to do geometric constructions with a compass and straightedge, a strategy that I think is especially helpful for understanding some geometry concepts.

Like the rest of the Mathematical Reasoning series, this worktext is printed in full color with many illustrations. A large font for the text and the limited amount of information per page make this much more manageable and unintimidating than most geometry courses.

I would recommend it for junior high students who want to get a jump on high school geometry and who will follow up with more intensive geometry later on. I would also recommend it for high school students who need a less-challenging option or for those who might even use it alongside another course. It might be a particularly good choice as a supplement for those using the classic Saxon series for high school math who want to cover more geometry earlier in the high school years.

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Understanding Geometry

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Critical thinking definition

Critical thinking, as described by Oxford Languages, is the objective analysis and evaluation of an issue in order to form a judgement.

Active and skillful approach, evaluation, assessment, synthesis, and/or evaluation of information obtained from, or made by, observation, knowledge, reflection, acumen or conversation, as a guide to belief and action, requires the critical thinking process, which is why it's often used in education and academics.

Some even may view it as a backbone of modern thought.

However, it's a skill, and skills must be trained and encouraged to be used at its full potential.

People turn up to various approaches in improving their critical thinking, like:

• Developing technical and problem-solving skills
• Engaging in more active listening
• Actively questioning their assumptions and beliefs
• Seeking out more diversity of thought
• Opening up their curiosity in an intellectual way etc.

Is critical thinking useful in writing?

Critical thinking can help in planning your paper and making it more concise, but it's not obvious at first. We carefully pinpointed some the questions you should ask yourself when boosting critical thinking in writing:

• What information should be included?
• Which information resources should the author look to?
• What degree of technical knowledge should the report assume its audience has?
• What is the most effective way to show information?
• How should the report be organized?
• How should it be designed?
• What tone and level of language difficulty should the document have?

Usage of critical thinking comes down not only to the outline of your paper, it also begs the question: How can we use critical thinking solving problems in our writing's topic?

Let's say, you have a Powerpoint on how critical thinking can reduce poverty in the United States. You'll primarily have to define critical thinking for the viewers, as well as use a lot of critical thinking questions and synonyms to get them to be familiar with your methods and start the thinking process behind it.

Are there any services that can help me use more critical thinking?

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