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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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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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Encyclopedia of Mathematics Education pp 159–163 Cite as

Critical Thinking in Mathematics Education

• Eva Jablonka 2  
• Reference work entry
• First Online: 01 January 2020

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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...

• Logical thinking
• Argumentation
• Deductive reasoning
• Mathematical problem solving
• Mathematical literacy
• Critical judgment
• Goals of mathematics education

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Course Contents

What is critical thinking, critical thinking.

Thinking comes naturally. You don’t have to make it happen—it just does. But you can make it happen in different ways. For example, you can think positively or negatively. You can think with “heart” and you can think with rational judgment. You can also think strategically and analytically, and mathematically and scientifically. These are a few of multiple ways in which the mind can process thought.

What are some forms of thinking you use? When do you use them, and why?

As a college student, you are tasked with engaging and expanding your thinking skills. One of the most important of these skills is critical thinking. Critical thinking is important because it relates to nearly all tasks, situations, topics, careers, environments, challenges, and opportunities. It’s a “domain-general” thinking skill—not a thinking skill that’s reserved for a one subject alone or restricted to a particular subject area.

Great leaders have highly attuned critical thinking skills, and you can, too. In fact, you probably have a lot of these skills already. Of all your thinking skills, critical thinking may have the greatest value.

What Is Critical Thinking?

Critical thinking is clear, reasonable, reflective thinking focused on deciding what to believe or do. It means asking probing questions like, “How do we know?” or “Is this true in every case or just in this instance?” It involves being skeptical and challenging assumptions, rather than simply memorizing facts or blindly accepting what you hear or read.

Who are critical thinkers, and what characteristics do they have in common? Critical thinkers are usually curious and reflective people. They like to explore and probe new areas and seek knowledge, clarification, and new solutions. They ask pertinent questions, evaluate statements and arguments, and they distinguish between facts and opinion. They are also willing to examine their own beliefs, possessing a manner of humility that allows them to admit lack of knowledge or understanding when needed. They are open to changing their mind. Perhaps most of all, they actively enjoy learning, and seeking new knowledge is a lifelong pursuit.

This may well be you!

The following video, from Lawrence Bland, presents the major concepts and benefits of critical thinking.

Critical Thinking and Logic

Critical thinking is fundamentally a process of questioning information and data. You may question the information you read in a textbook, or you may question what a politician or a professor or a classmate says. You can also question a commonly-held belief or a new idea. With critical thinking, anything and everything is subject to question and examination for the purpose of logically constructing reasoned perspectives.

Questions of Logic in Critical Thinking

Let’s use a simple example of applying logic to a critical-thinking situation. In this hypothetical scenario, a man has a PhD in political science, and he works as a professor at a local college. His wife works at the college, too. They have three young children in the local school system, and their family is well known in the community. The man is now running for political office. Are his credentials and experience sufficient for entering public office? Will he be effective in the political office? Some voters might believe that his personal life and current job, on the surface, suggest he will do well in the position, and they will vote for him. In truth, the characteristics described don’t guarantee that the man will do a good job. The information is somewhat irrelevant. What else might you want to know? How about whether the man had already held a political office and done a good job? In this case, we want to ask, How much information is adequate in order to make a decision based on logic instead of assumptions?

The following questions are ones you may apply to formulating a logical, reasoned perspective in the above scenario or any other situation:

• What’s happening? Gather the basic information and begin to think of questions.
• Why is it important? Ask yourself why it’s significant and whether or not you agree.
• What don’t I see? Is there anything important missing?
• How do I know? Ask yourself where the information came from and how it was constructed.
• Who is saying it? What’s the position of the speaker and what is influencing them?
• What else? What if? What other ideas exist and are there other possibilities?

Problem-Solving with Critical Thinking

For most people, a typical day is filled with critical thinking and problem-solving challenges. In fact, critical thinking and problem-solving go hand-in-hand. They both refer to using knowledge, facts, and data to solve problems effectively. But with problem-solving, you are specifically identifying, selecting, and defending your solution.

Problem-Solving Action Checklist

Problem-solving can be an efficient and rewarding process, especially if you are organized and mindful of critical steps and strategies. Remember, too, to assume the attributes of a good critical thinker. If you are curious, reflective, knowledge-seeking, open to change, probing, organized, and ethical, your challenge or problem will be less of a hurdle, and you’ll be in a good position to find intelligent solutions.

Critical Thinking, Problem Solving, and Math

In previous math courses, you’ve no doubt run into the infamous “word problems.” Unfortunately, these problems rarely resemble the type of problems we actually encounter in everyday life. In math books, you usually are told exactly which formula or procedure to use, and are given exactly the information you need to answer the question. In real life, problem solving requires identifying an appropriate formula or procedure, and determining what information you will need (and won’t need) to answer the question.

• "Student Success-Thinking Critically In Class and Online."  Critical Thinking Gateway . St Petersburg College, n.d. Web. 16 Feb 2016. ↵
• Critical Thinking Skills. Authored by : Linda Bruce. Provided by : Lumen Learning. Located at : https://courses.lumenlearning.com/collegesuccess-lumen/chapter/critical-thinking-skills/ . Project : College Success. License : CC BY: Attribution
• Critical Thinking. Authored by : Critical and Creative Thinking Program. Located at : http://cct.wikispaces.umb.edu/Critical+Thinking . License : CC BY: Attribution
• Thinking Critically. Authored by : UBC Learning Commons. Provided by : The University of British Columbia, Vancouver Campus. Located at : http://www.oercommons.org/courses/learning-toolkit-critical-thinking/view . License : CC BY: Attribution
• Problem Solving. Authored by : David Lippman. Located at : http://www.opentextbookstore.com/mathinsociety/ . Project : Math in Society. License : CC BY-SA: Attribution-ShareAlike
• Critical Thinking.wmv. . Authored by : Lawrence Bland. Located at : https://youtu.be/WiSklIGUblo . License : All Rights Reserved . License Terms : Standard YouTube License

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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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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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7 Concrete Strategies to Teach Conceptual Understanding in Math

These strategies will give you a head start on getting rid of math tips and tricks.

Multiplication is repeated addition.

Keep, switch, flip. 

The butterfly method.

These are all examples of math shortcuts, tips, or tricks that many students learn to rely on from an early age. I taught many students throughout my 16 years in the classroom who quickly pulled out these strategies!

But my students couldn’t explain why these tips and tricks work and would often stumble and become frustrated when they encountered situations where the tricks didn’t work or they forgot exactly what to do.

That’s why math education has transitioned in recent years to focus on teaching deep conceptual understanding rather than encouraging students to rely on shortcuts. Educators know that teaching children to deeply understand math leads to the development of problem-solvers and critical thinkers. 

But how can we ease away from teaching tips and tricks so our students have the opportunity to become true mathematicians?

Don’t worry; we’ve got a few ideas for you! Check out these seven tips for getting rid of the shortcuts and teaching true conceptual understanding in math.

1. Spiral Practice Through a Well-Thought-Out Scope and Sequence

Mathematics is a body of conceptual knowledge made up of interrelated concepts—it isn’t just a list of disconnected topics to check off a list as students move from grade to grade. Using a carefully considered scope and sequence to structure your school year is the first step in avoiding the pitfalls of math tips and tricks.

During my last few years of teaching, my district used the Carnegie Learning High School Math Solution for our Algebra 1 and Geometry classes. For the first time, I saw how much the scope and sequence really matter. Watching my Algebra 1 students pull from their Module 1 experiences in Module 5 to make sense of quadratic functions was a lightbulb moment–for all of us!

A thoughtful scope and sequence incorporating spiral review is key to teaching deep conceptual understanding in math. If we rely on teaching the “easy” shortcuts instead of giving students the time and space to master grade-level skills and see the connections between concepts, they’ll struggle to develop a body of conceptual knowledge that will help them understand more complex ideas in the future.

2. Use High-Order Tasks to Build Critical Thinking Skills

Although many students (and teachers!) love math shortcuts because they lead to quick “success,” having a toolbox packed with critical thinking skills and problem-solving strategies for students to pull from is so much more valuable. These skills will serve your students in various situations, whether they’re in advanced math classes or have to think critically about real-world problems.

One way to help students develop their critical thinking and problem-solving skills is to assign high-order math tasks in your classroom. When working on these rich tasks, they can think about what they already know and test out different ways to complete the task until they identify one that works. In the process, your students fill their toolbox with problem-solving strategies and critical thinking skills, eliminating the need for tips and tricks.

Some of my favorite high-order tasks to use with my Algebra 1 students were in a lesson titled, “Do You Mean: Recursion ?” This lesson is filled with activities that encourage students to think critically about arithmetic and geometric sequences and how to develop and deeply understand explicit and recursive formulas. They’re even asked to compare the pros and cons of using explicit or recursive formulas, using evidence developed over the last series of lessons!

The fact that there’s no “plug and chug” in this series of high-order tasks meant that my students were constantly using and developing their critical thinking skills and problem-solving strategies. 

I was always amazed at the deep conversations I heard around the room as they completed tables of cell divisions and eventually used those observations to understand why explicit and recursive formulas work.

3. Visual Representations for Better Retrieval

Visual aids are powerful tools for helping students to develop a deep, conceptual understanding of mathematical concepts. I loved supplementing as many lessons as possible with diagrams, graphs, anchor charts, manipulatives, and even high-quality math videos . In doing so, every learner had an entry point into even the most upper-level mathematic concepts.

When students visualize math concepts, they can more easily see patterns and make connections that might not be immediately apparent from written or verbal explanations. And when they have a visual cue stored in their brain, it makes retrieving information much more manageable. 

For example, suppose a student can recall that a quadratic function looks like a parabola because they’ve interacted with graphs illustrating a pumpkin catapult or diving into a swimming pool. In that case, they’re more likely to be able to interpret the formula of a quadratic function and apply that conceptual knowledge to different scenarios.

4. Manipulatives and Hands-On Learning

Another way to eliminate the need for tips and tricks (“A negative times a negative is a positive,” anyone?) is with manipulatives such as algebra tiles, counting chips, and even interactive number lines.

And I promise those hands-on materials aren’t just for the younger kids—your high schoolers won’t mind abandoning the paper and pencil note-taking in favor of digging into algebra tiles occasionally! 

I’ll never forget using algebra tiles for various purposes with my high schoolers. From watching a student with complex special needs finally understand the meaning and applications of a zero pair to seeing upper-level students suddenly “get” factoring trinomials, each visual and hands-on learning experience was pure magic!

5. Connect Concepts Instead of Teaching Math Shortcuts

Teaching is all about making connections. And while, yes, connecting with your students is one of the best ways to increase engagement, here we’re talking about making mathematical connections.

Teach your students to look for the interconnectedness of mathematical concepts, so they see how ideas fit together and build on one another, and watch as they develop a deeper understanding of the underlying concepts. Then, it’s time to kiss the shortcuts goodbye!

For example, the scope and sequence I used encouraged my students to apply their foundational knowledge of concrete geometric investigations and reasoning with shapes to formalize their understanding. Circles were also integrated throughout the course, rather than treating them as isolated geometric figures (as many other curriculums do). 

Watching my Geometry students make connections between circles and angle relationships and complete constructions using arcs was a game changer! They retained much more information when they saw the connections between concepts and were able to apply their knowledge and skills in new situations that I never expected.

6. Help Your Students Make Real-World Connections

Another vital connection that will lead to the elimination of shortcuts, tips, and tricks is between the mathematics your students learn in the classroom and the real-world applications of the concepts.

When you help your students discover these links to the real world, math suddenly loses its abstract nature. It becomes relevant, practical, and motivating.

Now your students will be more likely to remain engaged and acquire conceptual knowledge that can be generalized across various situations. Here are some examples using real-world scenarios to model integer subtraction that could be used in a 7th-grade class.

7. Don’t Use Math Tips and Tricks—Collaborate!

Most kids love to work in groups, right? It enhances the social aspect of school that many students value, and when structured correctly, these collaborative learning experiences can be the perfect setting for developing deep mathematical understanding.

Collaborating to create their conceptual knowledge is a powerful experience for your students. They may productively struggle , disagree, and even argue a bit, but these experiences are where the magic happens. 

“Allow students to experience and play and notice and wonder,” writes Tina Cardone, author of Nix the Tricks: A Guide to Avoiding Shortcuts That Cut Out Math Concept Development . “They will surprise you! Being a mathematician is not limited to rote memorization…Being a mathematician is about critical thinking, justification, and using tools from past experiences to solve new problems.”

And I can think of no better opportunity to notice, wonder, think critically, and justify those thoughts than when collaborating with peers. It may be hard to give up that “sage on the stage” lecture style (I definitely struggled!), but hearing your students engage in rich, mathematical conversations and watching them abandon the shortcuts in favor of deeply understanding the math is worth it. The feeling is second to none!

Don’t Let Tips and Tricks Take Away the Beauty of Math

Math is a beautiful, creative, and thought-provoking subject that sets the perfect stage for your students to become critical thinkers, problem solvers, and leaders of tomorrow. Don’t let a reliance on math shortcuts, tips, and tricks rob them of that experience!

I hope you’re ready to ditch the tips and tricks in your classroom, but if you need more convincing, check out this case study from Muleshoe Independent School District in Texas. They were able to teach their students deep conceptual understanding in math and get rid of the shortcuts—with some great results to show for it!

• Karen Sloan
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• Carnegie Learning

Before joining Carnegie Learning's marketing team in 2022, Karen spent 16 years teaching mathematics and social studies in Ohio classrooms. She has a passion for inclusive education and believes that all learners can be meaningfully included in academic settings from day one. As a former math and special education teacher, she is excited to provide educators with the latest in best-practices content so that they can set all students on the path to becoming confident "math people."

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• April 17, 2023

Math is a beautiful, creative, and thought-provoking subject that sets the perfect stage for your students to become critical thinkers, problem solvers, and leaders of tomorrow. Don’t let a reliance on math shortcuts, tips, and tricks rob them of that experience!

Karen Sloan, Math and Special Education Teacher of 16 Years

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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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Let us know how we can best support the implementation of a mathematical reasoning approach in your classroom(s) or school(s). Complete our short survey and receive 10% off our publications!

To learn more about critical thinking in math with TC² check out: Classroom Ready Materials Professional Learning Resources What Teachers Are Saying Professional Learning

Classroom Ready Materials

Online learning.

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.

Intermediate (7–9)

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.

Elementary (K–6)

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Secondary (10–12)

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Professional Learning Resources

Books, articles, & discussions.

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.

Videos and Presentations

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)

What Teachers Are Saying

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.

Professional Learning

Math Lead Teacher Certification This program supports teacher development of a rich understanding of the diverse role of critical thinking in math classrooms through 18 hours of personalized coaching and 10 hours of implementation support. Available as face-to-face and online sessions or a combination of both.

Book Professional Learning in Math Learn how we can help you plan affordable professional learning in math facilitated by our experienced team. We consult and collaborate with you to develop custom, focused, and engaging face-to-face and online sessions to meet your math specific needs and grade level requirements.

Contact Victoria Campoli to learn more.

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Common Core Math Explained: 8 Common Core Math Examples To Use In The Classroom 

Samantha dock.

Common Core math examples can be a tricky world to navigate for teachers trying to meet the individual needs of their students. Having a bank of Common Core math examples to hand can be helpful when planning your lessons. 

Embraced by the majority of states in the U.S., the Common Core math standards help to develop students’ conceptual understanding, problem-solving skills, and real-world applications.  In this article, we explore what Common Core math is, 8 Common Core math examples and top tips for educators teaching Common Core math.

What is Common Core math?

How is common core math different from traditional math, common core standards and 8 common core math examples, 1. make sense of problems and persevere in solving them, 2. reason abstractly and quantitatively, 3. construct viable arguments and critique the reasoning of others, 4. model mathematics, 5. use appropriate tools strategically, 6. attend to precision, 7. look for and make use of structure, 8. look for and express regularity in repeated reasoning, tips for teaching common core math, common core math and the wider world.

Common Core math standards are a set of educational standards for mathematics adopted by forty states in the United States. Each standard outlines the math knowledge students should know and be able to do at each grade level, from kindergarten through to high school. 

These standards aim to provide a more focused and coherent set of learning goals for students, emphasizing conceptual understanding, problem-solving, and critical thinking skills.  

Often, Common Core math involves multiple strategies and approaches to solving problems. In turn, this encourages students to understand the underlying concepts rather than simply memorizing algorithms.  

One aim of the Common Core State Standards is to move away from traditional memorization of procedures and algorithms towards a deeper understanding of connections between mathematical concepts.  

Common Core math standards are organized by grade level and cover a wide range of mathematical topics, including:  

• Arithmetic 
• Probability

Each standard is divided into domains, which represent broad categories of mathematical content such as: 

• Counting and cardinality
• Operations and algebraic thinking
• Numbers and operations in base ten
• Measurement and data

3rd to 6th grade Common Core math test

Help your students prepare for their state math test with these Common Core practice math tests for 3rd - 6th grade.

Common Core math standards have been controversial in some areas due to concerns about curriculum changes, standardized testing, and complexity. But some argue that these standards provide a more coherent approach to mathematics education and better prepare students for higher education than traditional math. 

Read more: Why is Math Important?

Common Core math and traditional math represent two different approaches to teaching mathematics.

Traditional math typically refers to methods of teaching mathematics that were used before the adoption of the Common Core standards. These methods often focused on rote memorization of formulas and procedures, with less emphasis on understanding the concepts or on real-life application of mathematical skills.

Here are some key differences between the two:

Focus on Conceptual Understanding vs. Memorization

A strong emphasis is placed on developing students’ conceptual understanding of math concepts under the common core. It aims to help learners understand the “why” behind mathematical procedures rather than just memorizing algorithms. Traditional math often focuses more on rote memorization of formulas and procedures without necessarily understanding the underlying concepts needed to approach math questions.

Problem-Solving and Critical Thinking vs. Rote Practice

Common Core math problems encourage critical thinking skills. They promote multiple approaches to solving new math problems and require students to justify their reasoning. Often, traditional math involves repetitive practice of standard procedures with less emphasis on problem-solving and critical thinking.

Real-World Applications vs. Abstract Exercises

Connections between mathematical concepts to real-world situations is valued under the common core. This helps students see the relevance of the math skills they are learning. Tasks and problems require the application of mathematical skills in practical contexts. Traditional math lessons focus more on abstract exercises and textbook problems that may not always have clear real-world connections.

Depth of Understanding vs. Breadth of Coverage

Rather than covering a wide range of topics, Common Core math aims for depth of understanding and maths mastery. Fewer topics at each grade level allow for deeper exploration and mastery of key concepts. In contrast, traditional math tends to cover a broader range of topics in less depth.

Flexibility and Multiple Strategies vs. One Correct Method

Students are encouraged to use multiple strategies and approaches to solve problems through the Common Core math standards. Flexibility and creativity are valued when approaching problem-solving. Emphasis on a single “correct” method or algorithm for solving problems is the general approach in traditional math. Overall, Common Core State Standards aim to develop students’ mathematical proficiency in alignment with the demands of the modern world. This includes the need for critical thinking, problem-solving, and application of mathematical concepts to real-world situations

Overall, Common Core State Standards aim to develop students’ mathematical proficiency in alignment with the demands of the modern world. This includes the need for critical thinking, problem-solving, and application of mathematical concepts to real-world situations. 

Third Space Learning provides one-on-one math instruction for students who need it most. Personalized one-on-one math lessons are designed by math experts and aligned to your state’s math standards — including the Common Core State Standards. 

Students should not only be able to understand problems and make sense of them, but persevere in finding solutions. 

Finding solutions may involve math skills such as: 

• Analyzing problems
• Making conjectures
• Planning approaches to solving math problems 

Common Core math example 1

A student is faced with a word problem about finding the area of a garden. They must take the time to carefully read and understand the problem before attempting to solve it. 

This problem may require several approaches to answer the math question. Small group work and discussion can encourage students to persevere through the challenge and try different strategies until they find a solution.

In order to reason abstractly, students need to be able to make sense of quantities and their relationships in mathematical situations. 

This will be easier for students if they can take abstract information from context and quantify information. Being able to decontextualize and contextualize mathematical ideas will benefit students.

Common Core math example 2

A graph shows the relationship between the number of hours worked and the amount earned. Students can analyze the graph to determine patterns and make predictions about future earnings based on proportional relationships between hours worked and money earned. 

For example, if John worked for 13 hours, how much money would he earn?

Introducing math vocabulary in the classroom helps students construct viable arguments and critique the mathematical reasoning of others. Exposure to mathematics language and sentence stems will help students to reason mathematically, construct arguments, and justify their thinking, without creating cognitive overload. 

Common Core math example 3

During a class discussion about strategies for solving a particular math problem, you might ask students to present their solutions — justifying and explaining their reasoning. 

They can also be encouraged to critique each other’s approaches, identify strengths and weaknesses in their arguments and offer alternative methods. 

Math lessons should prepare students to use math to solve real-world problems. It may help students to do this if you represent mathematical concepts with visual models and math manipulatives . 

Common Core math example 4

Subtraction of fractions is a skill that many students struggle with. Using a visual model to describe and analyze the word problem can release cognitive load for students. 

For example, Paul had 11 ⅔ yards of twine. He used 6 ½ yards to make macrame wall hangings, how many yards of twine does Paul have left?

To solve math problems effectively and efficiently, students must be able to select and use appropriate tools. This includes recognizing when and how to use tools, as well as evaluating effectiveness and efficiency.

Common Core math example 5

When solving a complex geometry problem, students should recognise the effectiveness of using a protractor and ruler to accurately measure angles and lengths. 

For example, Given an angle ABC where point B is the vertex of the angle, construct an angle bisector of angle ABC using a ruler and a protractor. Then, using the angle bisector you have constructed, draw a line segment from point B to the bisected angle’s line that is exactly 5 cm long. Measure and report the angle sizes of the two new angles created by the angle bisector.

Calculations need to be carried out precisely. To do this, students need to be aware of key mathematical terminology for the Common Core Standards they are studying. This involves using appropriate units and labels and stating mathematical results clearly.

Common Core math example 6

A student ensures that their work is clear and organized. They pay attention to detail, avoiding errors and inaccuracies in their calculations. Below is a worked example of a student showing how to solve a word problem involving multiple percentages.

Solving math problems accurately means students need to recognize and use mathematical patterns and structure. They should be able to identify relationships between mathematical ideas and make connections between different mathematical representations.

Common Core math example 7

When solving a multiplication of decimals problem, a student recognizes that breaking down the whole numbers and decimal parts into their factors makes the problem easier to solve. They identify the underlying structure of the problem and use it to their advantage.

Identifying and generalizing patterns and regularities in mathematical situations is key to proficiency in problem soving and reasoning . Students should be able to notice repeated reasoning and use it to solve math problems efficiently. 

Common Core math example 8

Example: A student identifies similarities between a problem they’re working on and a previous math problem. They utilize the patterns in the prior example to complete the new problem. This also helps them to solve similar problems in the future.

Teachers need to understand Common Core math standards to recognize the appropriate instructional strategies and promote a growth mindset in the classroom .  

Here are 8 tips for maximising student progress when teaching the Common Core State Standards:

1. Understand the Common Core State Standards

Familiarize yourself with the math Common Core State Standards for your specific grade level. Take the time to understand the mathematical practices, domains and teaching strategies required for your grade.

2. Focus on conceptual understanding

Prioritize conceptual understanding over rote memorization. You can achieve this by helping students understand the “why” behind math concepts and skills. Always encourage them to explain and justify their reasoning.

3. Promote multiple approaches

Offer your students a range of math strategies and approaches to problem-solving. The more methods in their math bank, the better equipped they are to find a solution. Asking students to share their thinking process helps those who are not grasping the content from the math instruction.

4. Real-world connections

Connecting mathematical concepts to real-world situations makes learning more meaningful and relevant. You can do this by implementing math problems where students work collaboratively to solve complex, open-ended word problems with real-world relevance. 

5. Use visual representations

Diagrams, models, and manipulatives support students’ understanding of mathematical concepts by making abstract concepts more concrete and accessible. For example, you could use algebra tiles when students are first learning how to solve algebraic equations and inequalities to help them contexutalize the abstract nature of algebra.

6. Encourage discourse and collaboration

Promote a classroom environment where students feel comfortable sharing their ideas, asking questions and engaging in mathematical discourse. 

Encourage discourse by using techniques such as turn and talk, or the 3 reads method for word problems.

7. Assess progress

Use formative and summative assessments to monitor students’ progress and understanding of mathematical concepts and adjust instruction accordingly based on assessment data.  

Some examples of a formative assessment are: 

• Exit tickets
• Rating scales
• Thumbs up or thumbs down 

Summative assessments include: 

• Check for understanding quizzes
• End-of-topic quizzes

Assessment resources:

• Practice state assessments  

8. Professional development

Continuously seek professional development opportunities to deepen your understanding of Common Core math and improve your teaching practices. Collaborate with colleagues and participate in workshops, conferences, and online courses.

Embracing Common Core principles can help equip students for future challenges

Educators’ commitment to teaching Common Core math goes beyond math instruction. It’s about nurturing critical thinking and problem solving, ensuring students are prepared for the wider world. 

Math lessons are no longer simply giving students math worksheets and grading them on the correct answer. The American education system has developed a math curriculum that anchors mathematical concepts in real-world relevance, promotes diverse problem-solving strategies, and encourages a collaborative learning environment.  

Educators have a responsibility to ensure students have the tools and mathematical literacy they need to succeed.

Common Core math examples FAQ

1. Make sense of problems and persevere in solving them 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning

1. Focus: Emphasizes focusing deeply on a smaller number of key topics at each grade level. This is done to ensure students develop a deep understanding of foundational mathematical ideas. 2. Coherence: Emphasizes the importance of coherence in mathematical instruction. This is done to support students in making meaningful connections between different mathematical ideas, helping them see how concepts are related and reinforcing their understanding over time 3. Rigor: Focuses on increasing the rigor of mathematical instruction by demanding that students engage in conceptual understanding, procedural fluency, and application of mathematical concepts in real-world contexts. In this context, rigor means ensuring that students develop a deep understanding of mathematical concepts, are able to apply their knowledge in various contexts, and can solve complex problems through reasoning and critical thinking.

Forty states have fully adopted Common Core math, while Minnesota partially embraces it. South Carolina, Oklahoma, Indiana, Florida, and Arizona initially adopted but later repealed Common Core. Alaska, Nebraska, Texas, and Virginia never adopted it.

Do you have students who need extra support in math? Give your students more opportunities to consolidate learning and practice skills through personalized math tutoring with their own dedicated online math tutor. Each student receives differentiated instruction designed to close their individual learning gaps, and scaffolded learning ensures every student learns at the right pace. Lessons are aligned with your state’s standards and assessments, plus you’ll receive regular reports every step of the way. Personalized one-on-one math tutoring programs are available for: – 2nd grade tutoring – 3rd grade tutoring – 4th grade tutoring – 5th grade tutoring – 6th grade tutoring – 7th grade tutoring – 8th grade tutoring Why not learn more about how it works ?

Ultimate Guide to Problem Solving Techniques [FREE]

Are you trying to build problem solving and reasoning skills in the classroom?

Here are 9 ready-to-go printable problem solving techniques that all your students should know, including challenges, short explanations and questioning prompts.

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11 Real World Math Activities That Engage Students

Bridging the gap between abstract math concepts and real life experiences can make the subject accessible and relevant for kids.

During a unit on slope, José Vilson’s students just weren’t getting it, and their frustration was growing. The former middle school math teacher began brainstorming creative ways to illustrate the concept. “I kept thinking, ‘My students already understand how this works—they just don’t know that they know,’” Vilson writes in a recent article for Teacher2Teacher . “How can I activate knowledge they don’t believe they have?”

Then he thought about a hill a couple of blocks from school that his students “walk up every day to get to the subway.” He tacked up paper and began sketching stick figures on the hill. “One was at the top of the hill, one was halfway up, one was near the bottom skating on flat ground, and one was on a cliff,” writes Vilson, now the executive director of EduColor. “Which of these figures will go faster and why?” he asked his students. “That got my kids laughing because, of course, my stick figures weren’t going to hang in the MoMA.” Still, his sketch got them thinking and talking, and it provided a simple stepping stone that “gave that math relevance and belonging in their own lives,” Vilson concludes. 

“It’s not unusual for students to walk into our classrooms thinking that math belongs to people who are smarter, who are older, or who aren’t in their immediate circle,” Vilson writes. “But every time I teach math in a way that’s accessible and real for my students, I’m teaching them: ‘The math is yours.’”

To build on Vilson’s idea, we posted on our social channels asking teachers to share their favorite strategies for connecting math to students’ experiences and lives outside of school. We received hundreds of responses from math educators across grade levels. Here are 11 teacher-tested ideas that get students seeing and interacting with the math that surrounds them each day.

Hunt for clues

Coordinate systems can feel abstract to some students—but using coordinates to navigate a familiar space can solidify the concept in a relevant and fun way. “Before starting a unit on coordinates, I make gridded maps of the school—I make them look old using tea staining —and send my students off on a treasure hunt using the grid references to locate clues,” says Kolbe Burgoyne, an educator in Australia. “It’s meaningful, it’s fun, and definitely gets them engaged.”

Budget a trip

Students enjoy planning and budgeting for imaginary trips, teachers tell us, offering ample opportunities to practice adding, subtracting, and multiplying large numbers. In Miranda Henry’s resource classroom, for example, students are assigned a budget for a fictional spring break trip; then they find flights, hotels, food, and whatever else they’ll need, while staying within budget.

Math teacher Alicia Wimberley has her Texas students plan and budget a hypothetical trip to the Grand Canyon. “They love the real world context of it and start to see the relevance of the digits after the decimal—including how the .00 at the end of a price was relevant when adding.” One of Wimberley’s students, she writes, mixed up his decimals and nearly planned a $25,000 trip, but found his mistake and dialed back his expenses to under $3,000.

Tap into pizza love

Educators in our audience are big fans of “pizza math”—that is, any kind of math problem that involves pizza. “Pizza math was always a favorite when teaching area of a circle,” notes Shane Capps. If a store is selling a 10-inch pizza, for example, and we know that’s referring to its diameter, what is its total area? “Pizza math is a great tool for addition, subtraction, multiplication, word problems, fractions, and geometry,” another educator writes on our Instagram. There are endless pizza-based word problems online. Here’s a simple one to start, from Jump2Math : “The medium pizza had six slices. Mom and Dad each ate one slice. How much pizza is left?”

Break out the measuring cups

Lindsey Allan has her third-grade students break into pairs, find a recipe they like online, and use multiplication to calculate how much of each ingredient they’d need in order to feed the whole class. The class then votes on a favorite recipe, and they write up a shopping list—“which involves more math, because we have to decide, ‘OK, if we need this much butter for the doubled recipe, will we need three or four sticks, and then how much will be left over?’” Allan writes. “And then it turns out students were also doing division without even realizing!” 

Sometimes, a cooking mistake teaches students about proportions the hard way. “Nobody wants a sad chocolate chip cookie where you doubled the dough but not the chocolate chips,” adds teacher Holly Satter.

Heading outdoors is good for kids’ bodies , of course, but it can also be a rich mathematical experience. In second grade, kids can head out to measure perimeters, teacher Jenna McCann suggests—perhaps of the flower boxes in the school garden. If outdoors isn’t an option, there’s plenty of math to be found by walking around inside school—like measuring the perimeter of the tables in the cafeteria or the diameters of circles taped off on the gym floor.

In Maricris Lamigo’s eighth-grade geometry class, “I let [students] roam around the school and take photos of things where congruent triangles were applied,” says Lamigo. “I have students find distances in our indoor courtyard between two stickers that I place on the floor using the Pythagorean theorem,” adds Christopher Morrone, another eighth-grade teacher. In trigonometry, Cathee Cullison sends students outside “with tape measures and homemade clinometers to find heights, lengths, and areas using learned formulas for right and non-right triangles.” Students can make their own clinometers , devices that measure angles of elevation, using protractors and a few other household items.

Plan for adult life

To keep her math lessons both rigorous and engaging, Pamela Kranz runs a monthlong project-based learning activity where her middle school students choose an occupation and receive a salary based on government data. Then they have to budget their earnings to “pay rent, figure out transportation, buy groceries,” and navigate any number of unexpected financial dilemmas, such as medical expenses or car repairs. While learning about personal finance, they develop their mathematical understanding of fractions, decimals, and percents, Kranz writes.

Dig into sports stats

To help students learn how to draw conclusions from data and boost their comfort with decimals and percentages, fourth-grade teacher Kyle Pisselmyer has his students compare the win-loss ratio of the local sports team to that of Pisselmyer’s hometown team. While students can struggle to grasp the relevance of decimals—or to care about how 0.3 differs from 0.305—the details snap into place when they look at baseball players’ stats, educator Maggierose Bennion says.

March Madness is a great source of real world data for students to analyze in math class, says sixth-grade math teacher Jeff Norris. Last March, Norris decorated his classroom like a basketball court, then had his students do basic statistical analysis—like calculating mean, median, and mode—using March Madness data, including individual game scores and the total win rate of each team. “We also did some data collection through our own basketball stations to make it personally relevant,” Norris says; students lined up in teams to shoot paper balls into a basket in a set amount of time, recorded their scores in a worksheet, and then examined the scoring data of the entire class to answer questions about mean, median, mode, range, and outliers.

Go on a (pretend) shopping spree

“My students love any activities that include SHOPPING!” says Jessie, a sixth-grade teacher who creates shopping-related problems using fake (or sometimes real) store ads and receipts. Her students practice solving percentage problems, and the exercise includes opportunities to work with fractions and decimals.

To get students more engaged with the work, math educator Rachel Aleo-Cha zeroes in on objects she knows students are excited about. “I make questions that incorporate items like AirPods, Nike shoes, makeup, etc.,” Aleo-Cha says. She also has students calculate sales tax and prompts them to figure out “what a 50% off plus 20% off discount is—it’s not 70% off.”

Capture math on the fly

Math is everywhere, and whipping out a smartphone when opportunities arise can lead to excellent content for math class. At the foot of Mount Elbert in Colorado, for example, math teacher Ryan Walker recorded a short word problem for his fourth- and fifth-grade students. In the video, he revealed that it was 4:42 a.m., and it would probably take him 249 minutes to reach the summit. What time would he reach the summit, he asked his students—and, assuming it took two-thirds as long to descend, what time would he get back down?

Everyday examples can be especially relatable. At the gas station, “I record a video that tells the size of my gas tank, shows the current price of gas per gallon, and shows how empty my gas tank is,” says Walker. “Students then use a variety of skills (estimation, division, multiplying fractions, multiplying decimals, etc.) to make their estimate on how much money it will cost to fill my tank.”

Connect to social issues

It can be a powerful exercise to connect math to compelling social issues that students care about. In a unit on ratios and proportions, middle school teacher Jennifer Schmerler starts by having students design the “most unfair and unjust city”—where resources and public services like fire departments are distributed extremely unevenly. Using tables and graphs that reflect the distribution of the city’s population and the distribution of its resources, students then design a more equitable city.

Play entrepreneur

Each year, educator Karen Hanson has her fourth- and fifth-grade students brainstorm a list of potential business ideas and survey the school about which venture is most popular. Then the math begins: “We graph the survey results and explore all sorts of questions,” Hanson writes, like whether student preferences vary with age. Winning ideas in the past included selling T-shirts and wallets made of duct tape.

Next, students develop a resource list for the business, research prices, and tally everything up. They calculate a fair price point for the good they’re selling and the sales quantity needed to turn a profit. As a wrap-up, they generate financial statements examining how their profits stack up against the sales figures they had projected.

HELP OTHER TEACHERS OUT!

We’d love this article to be an evolving document of lesson ideas that make math relevant to kids. So, teachers, please tell us about your go-to activities that connect math to kids’ real world experiences.

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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.

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Open-Ended Questions for Students: How to Craft Them (20+ Examples)

As an educator, your main responsibility is to instill critical thinking skills in your students.

No student becomes successful in their career simply because they memorized and used facts in their jobs. Success comes from the ability to think deeply about the subject matter and interpret it in ways that let them apply it to real-world situations.

It’s hard for students to go beyond their textbooks unless you ask open-ended questions that allow them to engage in such thought processes.

To help you facilitate such discussions in the classroom, we’ll explore a list of open-ended question examples for students. In addition, we’ll also look at ways to craft thoughtful questions and analyze responses with ease.

Open-ended vs. closed-ended questions

The key difference between open-ended and closed-ended questions is that the former has no predefined answer. It encourages the individual to think about the question in their own way and respond accordingly.

As typical university assessments use close-ended questions with specific answers, students tend to think from a one-dimensional perspective. For example, yes/no types of questions .

While the method is sound to test their grasp of a topic’s theoretical roots, more lively discussions are needed to develop other skills.

Also, as open-ended questions require more cognitive effort , it results in a pool of answers that offer more insights compared with simple yes/no questions. This leads to more fruitful discussions in the classroom as students actively participate and put their questions and points of view forward.

Open-ended question examples for students in different subjects

Usually, open-ended questions start with a “what,” “who,” “which,” “why,” or “how.” They force the student to articulate their thoughts in a structured manner and go beyond single-word answers.

If you’re stuck trying to spin up questions for your next lecture, here are a few examples of open-ended questions to get you started:

Arts and Literature

• How does the cultural context of a novel influence its themes and characters?
• How do gender dynamics shape the narratives in classical literature versus contemporary works?
• How has digital media changed traditional forms of art and literature?
• In what ways does the [book you discussed] reflect the societal issues of its time?
• How does the “anti-hero” concept in literature challenge traditional notions of heroism?
• How do the narrative techniques in [specific book] affect your story interpretation?
• How do genetic mutations contribute to the evolution of antibiotic resistance in bacteria?
• Why do you think your environment plays a significant role in your mental health?
• What are the ethical implications of [new medical intervention] in humans?
• Which renewable energy technologies will prove to be sustainable 20 years down the line?
• What do you think about the correlation between psychology and neurobiology?
• How can we improve the method we discussed today to detect earthquakes?
• Which types of rovers do you think are best for a planet like Venus?

Mathematics

• How can mathematical models be used to predict financial markets?
• How can differential equations be applied to model and solve real-world problems in engineering?
• What role does probability theory play in everyday decision-making and risk assessment?
• How does game theory apply to economics?
• What are the mathematical foundations of AI?

Social Studies

• How has social media influenced political discourse and public opinion?
• How do historical events shape current foreign policy decisions in [country]?
• What are the ethical considerations for governments using surveillance technology?
• How do migration patterns affect urban development and demographic shifts?
• How has the concept of work changed with technological advances and the global economy?

Tips to ask open-ended questions and analyze responses

Here are a few tips for integrating this format of questioning within the classroom:

1. Integrate open-ended questions into discussions and homework

Typically, open-ended questions are limited to post-lecture discussions. But it’s essential to include them wherever you can—especially for homework assignments.

It gives learners time to think and even further research the topic to build on their understanding. This lets them develop an argument for their viewpoint, explore different perspectives, and clarify the concept.

For instance, ask them to write a short essay of 250 words or more on the lecture’s topic. As it requires them to articulate their thoughts and identify gaps in their knowledge, it’ll force them to take on further reading.

In the classroom, ask them similar questions and keep following up with statements like “Why do you think that?” or “How would that work?” to help them develop their arguments.

2. Craft questions that encourage the development of critical-thinking skills

Design a questionnaire of open-ended questions for the end of each lecture to stimulate critical thinking and problem-solving skills. This will push students to go beyond mere facts and “islands of knowledge” to make connections between seemingly disconnected issues.

Here are a few templates that can help encourage elaboration:

• “What is the impact of…”
• “How would you solve…”
• “Why do you think…”
• “Which factors do you think influence…”
• “What challenges do you foresee in…”
• “What potential solutions would work for…”

Students learn to question the status quo and assess the credibility of sources. This format also helps them get a well-rounded perspective on complex issues.

3. Make sure the question design is simple and effective

A well-crafted question is clear and straightforward. Also, it avoids overly complex language or any form of ambiguity.

The goal is to avoid confusion and keep students on the same topic. One way to do that is to prevent the use of double-barreled or leading questions.

NOT: “What do you think about renewable energy, and how will it impact our environment?”

BUT: “What do you think about renewable energy?”

When you connect the question to real-world scenarios or dilemmas, it’ll make the discussion relatable and engaging . This gives students the much-needed context required to offer meaningful answers.

4. Use technology to create an interactive classroom environment

These days, there are many types of tools available for creating engaging discussions, such as:

• Polling software
• Presentation software
• Digital whiteboards
• Discussion boards

For instance, Poll Everywhere lets you create open-ended questions or surveys that you can use within a remote or in-person session. Students can add their responses anonymously or with their names, and you can discuss the answers in real time.

Alternatively, students can also send in their questions, and you can pick the most common or interesting ones for discussion.

5. Give students a time limit to respond thoughtfully

When you set a time limit, students are compelled to think on their feet. However, it also encourages focused reflection and prioritization of ideas. This prevents overthinking or drifting off-topic, resulting in better answers.

Time constraints also give a sense of what making decisions in the real world is like, especially under pressure. This will ultimately prepare students to handle situations confidently outside the classroom.

6. Create a safe and inclusive environment to encourage responses

When you ask open-ended questions, ensure the classroom environment is safe. You can do that by allowing the students to anonymize their answers, mainly when dealing with sensitive topics.

Tools such as Poll Everywhere let you anonymize responses and moderate live discussions to create a healthy learning environment. Poll Everywhere has a feature that can automatically filter any profanity and censor abusive words. Also, students who have difficulty speaking up in class and sharing their ideas might be more likely to participate now.

When you’re asking follow-up questions, don’t use a judgmental tone. For instance, instead of saying, “Why?” use “Why do you think so?” or something similar. This fosters trust and openness in the classroom.

7. Develop a criteria for assessment of student responses

Create a rubric for assessment to ensure consistency while grading. For example, assign a weighted grade to a student mentioning a specific keyword or concept.

This grading system needs to align with the learning objectives of the course. If students miss out on specific concepts, for example, it shows that they don’t thoroughly understand the topic.

Also, communicate the criteria for grading before or after the assessment so students know what was expected in the assignment.

8. Provide real-time feedback on student responses

When students offer their responses, give them feedback. This is particularly useful in a discussion setting as other students also get to form their opinions.

Feedback should be constructive, focusing on the strengths of the response and offering specific suggestions for enhancement. It builds confidence and motivates students to engage more deeply with the material.

Feedback also provides valuable information by highlighting areas for improvement for both the student and the lecturer. The students know what went wrong with their answers, and the lecturer knows what needs to be improved or focused on in the next lecture.

Engage in lively discussions using Poll Everywhere

Using open-ended questions results in a more interactive and reflective learning environment. But a key part of that is how you implement this learning method in the classroom.

This is where digital tools can play a massive role. For instance, instead of asking open-ended questions in class directly, use a tool such as Poll Everywhere to ask them. Students can all send in their responses without needing to raise their hands and speak to the entire class. Additionally, educators can also moderate responses and grade them in one space, making it an effective tool for learning.

Poll Everywhere allows students to get real-time feedback, participate anonymously, and participate in an inclusive learning environment. This teaches them to think critically while experiencing an interactive mode of learning.

If you’re an educator looking to deploy open-ended questions, schedule a demo with Poll Everywhere today.

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