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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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20 Math Critical Thinking Questions to Ask in Class Tomorrow

• November 20, 2023

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

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

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

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

Table of Contents

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

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

What skills do we actually want to teach our students?

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

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

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

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

When should I ask these math critical thinking questions?

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

Start small

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

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

Add critical thinking questions to word problems

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

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

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

When you want your students to defend their answers.

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

When you want your students to justify their opinions

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

When you want your students to think outside of the box

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

Let’s Recap:

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

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7 thoughts on “20 math critical thinking questions to ask in class tomorrow”.

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

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

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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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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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Critical Thinking in Math: A Focus on Mathematical Reasoning Competencies

TC²’s approach to math embraces the idea that sustained quality mathematical thinking, or reasoning, is the key to the success of current and future generations of math students.

What Is Mathematical Reasoning?

A mathematical reasoning approach optimizes the learning opportunities for every student in the classroom. It empowers students with the capacity to independently detect the need for, and to use, a wide range of math reasoning abilities.

• a strong understanding of foundational math concepts and content, and
• the capacity to reason soundly about and with these concepts and content.
• deeply understand
• appropriately act on, and
• effectively communicate using those concepts.

What Are Mathematical Reasoning Competencies?

Eight key mathematical reasoning competencies underpin all math learning and are needed for student success in math. These are presented in A Math Pedagogy Designed to Empower Learners [PDF].

"To provide each and every student equitable opportunities to improve their learning success in math, students need to learn how to reason soundly in a variety of ways through the application of critical thinking. —Laura Gini Newman (2020) A Math Pedagogy Designed to Empower Learners

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Describing Trends in Data: Which data set should be considered linear in the trends it presents? In this lesson, students learn how to use lines (curves) of best fit to help them effectively describe mathematical trends in data. Most suitable for grades 8–10.

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Grade 9 Student Lessons These lessons were developed in partnership with the Matawa Education and Care Centre.

What is the best way to represent information to help you make financial decisions: a table or a graph? [PDF] In this lesson, students compare different ways of organizing information to create a budget that will help them make the best financial decision.

Which best describes the trend in the data: a line of best fit or a curve of best fit? [PDF] In this lesson, students consider different patterns in the data that describes the relationship between fish and seafood consumption and the year. They then make the most accurate prediction about fish and seafood consumption for the year 2030.

How well does an equation match a line of best fit, a table, and a description in words? [PDF] In this lesson, students explain how well an equation given to them describes the line of best fit and the trend in the data. They then use the equation to make a prediction.

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NEW! Assessing Mathematical Thinking: A Focus on Reasoning Competencies This new title in our Quick Guides to Thinking Classrooms series presents a framework for effectively assessing and evaluating thinking in math. It shows how building math assessment practices on a foundation of essential mathematical reasoning competencies provides a clearly defined, manageable, and consistent way to focus assessments.

A Math Pedagogy Designed to Empower Learners [PDF] Laura Gini-Newman outlines a new pedagogical approach to the teaching and learning of mathematics that is focused on building student capacity to reason mathematically through critical inquiry.

Critical Inquiry in Math Class During TC²’s 25 th Anniversary celebration, each month explored a different focus. April focussed on how we can bring critical thinking into math. This webpage introduces the focus and explores ways to enrich your classroom with critical inquiry in math.

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An Introduction to the Why, What, How, When, and Who of Assessing Mathematical Thinking Listen to TC² math consultant Laura Gini-Newman as she explains why your assessments should focus on mathematical reasoning and offers a few tips on the why, what, how, when, and who of doing so.

OAME Talks Listen to TC² math consultant Laura Gini-Newman as she shares her thoughts on Assessing Mathematical Thinking: Who, What, When, and How on the OAME Talks podcast (Season 5, Talk 41)

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How Thinking Mathematically Changed My Teaching Jocelynn Foxon talks about her experience completing the Math Lead Teacher Certification Program (MLTCP) offered by TC², and the work she did with teachers and students in supporting the implementation of this approach in the math classroom.

Helping My Students Take Ownership of Their Own Learning Shamima Basrai talks about her experience embedding critical thinking in mathematics in her Grade 3/4 classroom.

The Perfect Fit for the Meandering but Wonderful Thinking Process of the Grade 4 Student David Markus talks about his experience shifting to a critical thinking framework with his Grade 4 math students.

Generating Enthusiasm in the Math Classroom Nina Perreault-Primeau talks about the transformation in her math classroom when she planned lessons focussing on student interest, creating authentic learning opportunities accompanied by sound critical inquiry questions.

But will it also work in math? [PDF] Sarah Sommers describes what happened when she and her teaching partner took a critical inquiry approach with their Grade 5 students during a math patterning unit.

Basics vs Inquiry in Math? A critical inquiry approach can achieve both Chris Achong talks about his experience with his Grade 9 math team implementing this approach to math learning—a comprehensive and balanced approach that improves the quality of every student’s capacity to think mathematically.

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Critical Thinking in Mathematics Education

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

• Eva Jablonka 2  

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

Eva Jablonka

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

Stephen Lerman

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

Bharath Sriraman

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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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Using Questioning to Stimulate Mathematical Thinking

Types of questions, levels of mathematical thinking, combining the categories.

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5. Teaching Mathematical Reasoning: Critical Math Thinking Through Problem-Solving and Modeling

• Mathematical problem-solving : This approach makes students think conceptually about problems before applying tools they’ve learned.
• Mathematical modeling : Modeling projects give students experience in weighing several factors against one another and using mathematical knowledge to make decisions.

I. Mathematical Problem-Solving

An emphasis on open-ended mathematical problem-solving can help develop mathematical reasoning skills and address a problem teachers have long been concerned about: too much “rote” learning in math. 

Too often students spend time in math class memorizing procedures and applying them mindlessly to problems. This leads to errors when students are confronted with unfamiliar problems. It also contributes to a widespread misperception of math as boring and lacking relevance to everyday life. 

On the other hand, attempting to remedy this problem by giving students open-ended problems has its own drawbacks. Without the conceptual and methodological tools to solve these problems students become frustrated and disengaged. It can end up being an inefficient way to spend class time.  

Although learning fundamental math skills like algorithms for adding, subtracting, multiplying, and dividing is absolutely critical for students in the early grades, the deeper mathematical problem-solving skills are the ones we really want students to graduate with. How can we ensure they do?

The deeper mathematical problem-solving skills are the ones we really want students to graduate with.

Evidence suggests that skills in mathematical problem-solving lead to more general improvements in outcomes related to math. They help students acquire a deeper understanding of mathematical reasoning and concepts. 

For instance, the commutative property, which most students learn applies to addition and multiplication problems (changing the order of the operations doesn’t affect the outcome), also applies to other logical and practical situations. A familiarity with some of these situations fosters deeper conceptual understanding, and deeper conceptual understanding leads to better critical thinking.

And learning these skills helps students improve outcomes related to critical thinking more generally. For example, students who become skilled in mathematical problem-solving tend to also:

• Create beneficial habits of mind — persistence, thoroughness, creativity in solution-finding, and improved self-monitoring.
• Break down hard problems into easier parts or reframing problems so that they can think about them more clearly. 
• Some problem solving tactics are applicable to situations well beyond math: making a visualization of a situation to understand it more clearly; creating a simplified version of the problem to more easily address the essence of the problem; creating branches of possibilities to solve the problem; creating “what if” example cases to test key assumptions, etc.
• Elevate the value of discussion and argumentation over simple appeals to authority.

Small-group mathematical problem solving targets skills that traditional mathematics instruction doesn’t. Instead of just finding a match between an algorithm and a question, students must: adapt or create an algorithm; evaluate and debate the merits of different solution paths; and verify their solution through additional evidence.

Small-group mathematical problem solving targets skills that traditional mathematics instruction doesn’t.

This process continues until the class has thoroughly explored the problem space, revealing multiple solution paths and exploring variations on the problem or contrasting problem-types.

Of course, the usefulness of a question like this depends on what students already know. If students don’t already know that chickens have two legs and pigs have four, they’re just going to be confused by the problem (and the explanation of the solution). It also requires some other basic skills—for instance, that if one chicken has two legs, four chickens would have eight.

As a way of evaluating student growth, teachers could also include some of these open-ended problems in homework assignments or as extra credit assignments.

Lesson Plan Outline

An example that might be appropriate for fifth grade is something like the following: A farmer has some pigs and some chickens. He finds that together they have 70 heads and 200 legs. How many pigs and how many chickens does he have? Divide the class into student groups of three to four. Have students spend a few minutes reading over the problem individually. Then let student groups discuss possible solution paths. The teacher walks around the classroom, monitoring the groups. Then the teacher leads a whole-class discussion about the problem.

• So how did you go about thinking about the problem?
• Show us how you got your answer and why you think it’s right. This might mean that a student goes up to the board to illustrate something if a verbal explanation is inadequate.
• And what was the answer you got?
• Does anyone else have a different way of thinking about the problem? If there are other ways of solving the problem that students didn’t come up with, teachers can introduce these other ways themselves.

Developing Math Problem-Solving Skills

Teachers should keep in mind the following as they bring mathematical problem-solving activities into their classrooms:

• Problem selection . Teachers have to select grade-appropriate problems. A question like “John is taller than Mary. Mary is taller than Peter. Who is the shortest of the three children?” may be considered an exercise to older students — that is, a question where the solutions steps are already known — but a genuine problem to younger students. It’s also helpful when problems can be extended in various ways. Adding variation and complexity to a problem lets students explore a class of related problems in greater depth.
• Managing student expectations . Introducing open-ended math problems to students who haven’t experienced them before can also be confusing for the students. Students who are used to applying algorithms to problems can be confused about what teachers expect them to do with open-ended problems, because no algorithm is available.
• Asking why . Asking students to explain the rationale behind their answer is critical to improving their thinking. Teachers need to make clear that these rationales or justifications are even more important than the answer itself. These justifications give us confidence that an answer is right. That is, if the student can’t justify her answer, it almost doesn’t matter if it’s correct, because there’s no way of verifying it.

II. Mathematical Modeling

Another approach is mathematical modeling. Usually used for students in middle or high school, mathematical modeling brings math tools to bear on real-world problems, keeping students engaged and helping them to develop deeper mathematical reasoning and critical thinking skills.

Math modeling is an extremely common practice in the professional world. Investors model returns and the effects of various events on the market; business owners model revenue and expenses, buying behavior, and more; ecologists model population growth, rainfall, water levels, and soil composition, among many other things. 

But, despite these many applications and the contributions it can make to general mathematical reasoning and critical thinking skills, mathematical modeling is rarely a main component of the math curriculum. Although textbook examples occasionally refer to real-world phenomena, the modeling process is not commonly practiced in the classroom.

Modeling involves engaging students in a big, messy real-world problem. The goals are for students to:

• refine their understanding of the situation by asking questions and making assumptions,
• leverage mathematical tools to solve the problem,
• make their own decisions about how to go about solving the problem,
• explain whether and how their methods and solutions make sense,
• and test or revise their solutions if necessary.

Mathematical modeling typically takes place over the course of several class sessions and involves working collaboratively with other students in small groups.

Modeling is not just about getting to a “right” answer — it’s about considering factors beyond mathematics as well.

Modeling also offers the opportunity to integrate other material across the curriculum and to “think mathematically” in several different contexts. Modeling is not just about getting to a “right” answer — it’s about considering factors beyond mathematics as well. For example, students deal with questions like:

• What is a “fair” split? 
• What level of risk should someone tolerate?
• What tradeoffs should a society make?

In others words, students come to see mathematics as the socially indispensable tool that it is, rather than an abstract (and sometimes frustrating) school subject.

Mathematical Modeling and Critical Thinking

Research suggests that the ability to solve abstractly framed academic math problems is not necessarily related to mathematical reasoning more broadly: that is, the ability to use math well in everyday life or to integrate mathematical thinking into one’s decision-making. Students may be able to follow procedures when given certain cues, but unable to reason about underlying concepts. 

It’s also very common to hear complaints from students about math — that either they aren’t “ math people ,” that math is irrelevant, or that math is simply boring.

Mathematical modeling is one approach to resolving both these problems. It asks students to move between the concreteness of real — or at least relatively realistic — situations and the abstraction of mathematical models. Well-chosen problems can engage student interest. And the practice emphasizes revision, step-by-step improvement, and tradeoffs over single solution paths and single right-or-wrong answers.

Mathematical modeling often begins with a general question, one that may initially seem only loosely related to mathematics:

• how to design an efficient elevator system, given certain constraints;
• what the best gas station is to visit in our local area;
• how to distinguish between two kinds of flies, given some data about their physical attributes.

Then, over the course of the modeling process, students develop more specific questions or cases, adding constraints or assumptions to simplify the problem. Along the way, students identify the important variables — what’s changing, and what’s not changing? Which variables are playing the biggest role in the desired outcomes?

Students with little experience in modeling can leap too quickly into looking for a generalized solution, before they have developed a feel for the problem. They may also need assistance in developing those specific cases. During this part of the process, it can be easiest to use well-defined values for some variables. These values may then become variables later on.

After students explore some simplifying cases, then they work on extensions of these cases to reach ever more general solutions.

A key part of this activity is letting students be creative — students will often come up with unusual or especially innovative solutions.

Throughout the modeling process, the teacher may need to point out missing assumptions or constraints, or offer other ways of reframing the problem. For any given modeling problem, some solutions are usually more obvious than others, which leads to common stages students may reach as they solve the problem. But a key part of this activity is letting students be creative — students will often come up with unusual or especially innovative solutions.

A sample problem, from the Guidelines for Assessment and Instruction in Mathematical Modeling Education is below:

This problem involves variables that aren’t necessarily immediately apparent to students. For instance, the size of the gas tank, and how much gas you fill up on per trip. As students manage this specific case, they can take other hypothetical scenarios to generalize their solution: if it’s 10 miles away, how cheap would the gas have to be to make it worth it? What about the time spent in the car — is there a value to put on that?

Many modeling problems can be arbitrarily extended in various directions. Instead of just considering the best gas station to go to for a single car, for instance, students can explore the behavior of a fleet of trucks on set routes or seasonal changes to gas prices.

It’s also possible to include shorter modeling activities, where students work together in pairs or small groups to extend a problem or interpret the meaning of a solution.

These kinds of modeling activities are not reserved solely for older students. One example of a modeling problem for students in elementary school might be something like: what should go in a lunchbox? Students can talk about what kinds of things are important to them for lunch, “mathematize” the problem by counting student preferences or coming up with an equation (e.g., lunch = sandwich + vegetable + dessert + drink); and even explore geometrically how to fit such items into a lunchbox of a certain size.

Teaching Mathematical Modeling: Further Key Factors

Mathematical modeling activities can be challenging for both teachers and students. 

Often, mathematical modeling activities stretch over several class periods. Fitting modeling activities in, especially if standardized tests are focused on mathematical content, can be challenging. One approach is to design modeling activities that support the overall content goals.

The teacher’s role during mathematical modeling is more like a facilitator than a lecturer. Mathematical modeling activities are considerably more open-ended than typical math activities, and require active organization, monitoring, and regrouping by the teacher. Deciding when to let students persevere on a problem for a bit longer and when to stop the class to provide additional guidance is a key skill that only comes with practice.

The teacher’s role during math modeling is more like a facilitator than a lecturer.

Students — especially students who have traditionally been successful in previous math classes — may also experience frustration when encountering modeling activities for the first time. Traditional math problems involve applying the right procedure to a well-defined problem. But expertise at this kind of mathematical reasoning differs markedly from tackling yet-to-be-defined problems with many possible solutions, each of which has tradeoffs and assumptions. Students might feel unprepared or even that they’re being treated unfairly.

Students also have to have some knowledge about the situation to reason mathematically about it. If the question is about elevators, for example, they need to know that elevators in tall buildings might go to different sets of floors; that elevators have a maximum capacity; that elevators occasionally break and need to be repaired. 

Finally, the mathematical question needs to be tailored to students’ experience and interests. Asking a group of students who don’t drive about how to efficiently purchase gas won’t garner student interest. Teachers should use their familiarity with their students to find and design compelling modeling projects. This is chance for both students and teachers to be creative. 

To download the PDF of the Teachers’ Guide

(please click here)

Sources and Resources

O’Connell, S. (2000). Introduction to Problem Solving: Strategies for The Elementary Classroom . Heinemann. A recent handbook for teachers with tips on how to implement small-group problem solving.

Youcubed.org , managed by Jo Boaler.  A community with lots of resources for small-group problem solving instruction.

Yackel, E., Cobb, P., & Wood, T. (1991). Small group interactions as a source of learning opportunities in second-grade mathematics . Journal for research in mathematics education , 390-408. Education research that illustrates how small-group problem solving leads to different kinds of learning opportunities than traditional instruction.

Guidelines for Assessment and Instruction in Mathematical Modeling Education , 2nd ed. (2019). Consortium for Mathematics and its Applications & Society for Industrial and Applied Mathematics.  An extensive guide for teaching mathematical modeling at all grade levels.

Hernández, M. L., Levy, R., Felton-Koestler, M. D., & Zbiek, R. M. (March/April 2017). Mathematical modeling in the high school curriculum . The variable , 2(2). A discussion of the advantages of mathematical modeling at the high school level.

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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.

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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.

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Critical Thinking: A Simple Guide and Why It’s Important

• Share This: Share Critical Thinking: A Simple Guide and Why It’s Important on Facebook Share Critical Thinking: A Simple Guide and Why It’s Important on LinkedIn Share Critical Thinking: A Simple Guide and Why It’s Important on X

Critical Thinking: A Simple Guide and Why It’s Important was originally published on Ivy Exec .

Strong critical thinking skills are crucial for career success, regardless of educational background. It embodies the ability to engage in astute and effective decision-making, lending invaluable dimensions to professional growth.

At its essence, critical thinking is the ability to analyze, evaluate, and synthesize information in a logical and reasoned manner. It’s not merely about accumulating knowledge but harnessing it effectively to make informed decisions and solve complex problems. In the dynamic landscape of modern careers, honing this skill is paramount.

The Impact of Critical Thinking on Your Career

☑ problem-solving mastery.

Visualize critical thinking as the Sherlock Holmes of your career journey. It facilitates swift problem resolution akin to a detective unraveling a mystery. By methodically analyzing situations and deconstructing complexities, critical thinkers emerge as adept problem solvers, rendering them invaluable assets in the workplace.

☑ Refined Decision-Making

Navigating dilemmas in your career path resembles traversing uncertain terrain. Critical thinking acts as a dependable GPS, steering you toward informed decisions. It involves weighing options, evaluating potential outcomes, and confidently choosing the most favorable path forward.

☑ Enhanced Teamwork Dynamics

Within collaborative settings, critical thinkers stand out as proactive contributors. They engage in scrutinizing ideas, proposing enhancements, and fostering meaningful contributions. Consequently, the team evolves into a dynamic hub of ideas, with the critical thinker recognized as the architect behind its success.

☑ Communication Prowess

Effective communication is the cornerstone of professional interactions. Critical thinking enriches communication skills, enabling the clear and logical articulation of ideas. Whether in emails, presentations, or casual conversations, individuals adept in critical thinking exude clarity, earning appreciation for their ability to convey thoughts seamlessly.

☑ Adaptability and Resilience

Perceptive individuals adept in critical thinking display resilience in the face of unforeseen challenges. Instead of succumbing to panic, they assess situations, recalibrate their approaches, and persist in moving forward despite adversity.

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So, how can one cultivate and harness this invaluable skill?

✅ developing curiosity and inquisitiveness:.

Embrace a curious mindset by questioning the status quo and exploring topics beyond your immediate scope. Cultivate an inquisitive approach to everyday situations. Encourage a habit of asking “why” and “how” to deepen understanding. Curiosity fuels the desire to seek information and alternative perspectives.

✅ Practice Reflection and Self-Awareness:

Engage in reflective thinking by assessing your thoughts, actions, and decisions. Regularly introspect to understand your biases, assumptions, and cognitive processes. Cultivate self-awareness to recognize personal prejudices or cognitive biases that might influence your thinking. This allows for a more objective analysis of situations.

✅ Strengthening Analytical Skills:

Practice breaking down complex problems into manageable components. Analyze each part systematically to understand the whole picture. Develop skills in data analysis, statistics, and logical reasoning. This includes understanding correlation versus causation, interpreting graphs, and evaluating statistical significance.

✅ Engaging in Active Listening and Observation:

Actively listen to diverse viewpoints without immediately forming judgments. Allow others to express their ideas fully before responding. Observe situations attentively, noticing details that others might overlook. This habit enhances your ability to analyze problems more comprehensively.

✅ Encouraging Intellectual Humility and Open-Mindedness:

Foster intellectual humility by acknowledging that you don’t know everything. Be open to learning from others, regardless of their position or expertise. Cultivate open-mindedness by actively seeking out perspectives different from your own. Engage in discussions with people holding diverse opinions to broaden your understanding.

✅ Practicing Problem-Solving and Decision-Making:

Engage in regular problem-solving exercises that challenge you to think creatively and analytically. This can include puzzles, riddles, or real-world scenarios. When making decisions, consciously evaluate available information, consider various alternatives, and anticipate potential outcomes before reaching a conclusion.

✅ Continuous Learning and Exposure to Varied Content:

Read extensively across diverse subjects and formats, exposing yourself to different viewpoints, cultures, and ways of thinking. Engage in courses, workshops, or seminars that stimulate critical thinking skills. Seek out opportunities for learning that challenge your existing beliefs.

✅ Engage in Constructive Disagreement and Debate:

Encourage healthy debates and discussions where differing opinions are respectfully debated.

This practice fosters the ability to defend your viewpoints logically while also being open to changing your perspective based on valid arguments. Embrace disagreement as an opportunity to learn rather than a conflict to win. Engaging in constructive debate sharpens your ability to evaluate and counter-arguments effectively.

✅ Utilize Problem-Based Learning and Real-World Applications:

Engage in problem-based learning activities that simulate real-world challenges. Work on projects or scenarios that require critical thinking skills to develop practical problem-solving approaches. Apply critical thinking in real-life situations whenever possible.

This could involve analyzing news articles, evaluating product reviews, or dissecting marketing strategies to understand their underlying rationale.

In conclusion, critical thinking is the linchpin of a successful career journey. It empowers individuals to navigate complexities, make informed decisions, and innovate in their respective domains. Embracing and honing this skill isn’t just an advantage; it’s a necessity in a world where adaptability and sound judgment reign supreme.

So, as you traverse your career path, remember that the ability to think critically is not just an asset but the differentiator that propels you toward excellence.

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2 Ways to Encourage Reflection on Math Concepts

Open-ended questions guide students to participate and to think mathematically, which cements their learning.

For many students, math is a subject where every question has one (and only one) correct answer. If a student is asked, “What is two plus two?” the only acceptable response is “Four.”

What if students were also asked, “Why does two plus two equal four?” Reflection questions like this, which are purposely open-ended, do not have a single correct answer. Instead, these questions remove the fear of being wrong and encourage mathematical thinking, participation, and growth.

“Reflection questions are important for students and help move the focus from performance to learning,” says Stanford professor Jo Boaler , who believes that “assessment plays a key role in the messages given to students about their potential, and many classrooms need to realign their assessment approach in order to encourage growth instead of fixed mindsets among students.”

In addition to performance-focused questions and assessments (“What is the total sum of the interior angles of a triangle?”), you can ask open-ended reflection questions that encourage mathematical thinking and participation (“Why do you think the total sum of the interior angles of a triangle always equals 180 degrees?”). The second question shifts the focus from performance toward thinking, learning, and engaging with mathematics without the fear of being wrong.

How can you incorporate reflection questions into your math lessons? Try these two useful strategies.

Which One Doesn’t Belong?

If you grew up watching Sesame Street , you probably remember the “One of These Things Is Not Like the Others” segment, where viewers had to identify one object out of a set of four that did not belong. This simple activity helps children to identify similarities and differences, and this type of thinking can be extended to learning math.

Which One Doesn’t Belong? (WODB) math activities present students with four different visual graphics that are all similar and different from each other in some way. This four-quadrant activity is my go-to for getting whole-class participation, as each option can be argued as the correct answer.

Observe the photo above of a WODB activity showing the numbers 22, 33, 44, and 50, and identify which choice does not belong and explain why. Since the graphics are purposely ambiguous and have overlapping similarities and differences, there is no single correct answer. One student might conclude that 50 doesn’t belong because it is the only number not divisible by 11. Another student may also believe that 50 doesn’t belong but for a different reason, namely that it is the only number with two different digits. A third student might conclude that 33 doesn’t belong because it is the only odd number. With this one graphic, you can easily spark a deep mathematical discussion where all students are eager to participate and share their thinking without any fear of being wrong.

WODB activities can be used for any math topic and can include images, numbers, charts, and graphs. They can also be used as formative assessments where students write their responses on sticky notes and stick them on the graphic that is projected at the front of the classroom.

Think-Notice-Wonder

Writing about math helps students organize their thoughts, use important vocabulary terms, and express their ideas in depth—which leads to deeper understanding.

Think-Notice-Wonder (TNW) activities are open-ended writing prompts where students are required to complete I think… , I notice… , I wonder… , based on a given graphic related to a math topic.

For example, students observe the soda and popcorn price graphic above and are prompted: What do you think? What do you notice? What do you wonder?

Encourage your students to think deeply for a minute or two before putting their thoughts into writing. They can share their ideas about the relationship between the price of a bag of popcorn and a soda based on size. They can verbalize how they perceive the proportional relationship to behave, wonder about which option provides the most value, and question how the prices were determined in the first place.

Since TNW writing activities are open-ended and do not have a correct answer, they encourage full group participation. Teachers often have students share their responses in a math journal notebook, but you can also use this free TNW student response template .

If you are looking for free images and graphics to use as TNW writing prompts, here are a few helpful resources:

• Find math-related graphics and images using Google Image Search and display them at the front of your classroom.
• Access and share teacher-created TNW activities on Twitter by searching the math education hashtags, including #ITeachMath, #MTBoS, and #NoticeWonder.
• Free stock photo websites such as Unsplash have an excellent collection of photos that relate to math topics, including estimation, three-dimensional figures, and mathematical patterns in nature.

When you add more reflection questions into your math lessons, students will have more opportunities to participate and engage in mathematical thinking without fear, which leads to a most-desired outcome—accessibility and growth.