critical thinking skills in maths

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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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Engaging Maths

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

• by cattard2017
• Posted on June 25, 2017

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

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

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

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

How and when do we embed critical and creative thinking?

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

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

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

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

Who should be engaging in critical and creative thinking?

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

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

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

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

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20 Math Critical Thinking Questions to Ask in Class Tomorrow

• November 20, 2023

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

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

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

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

Table of Contents

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

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

What skills do we actually want to teach our students?

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

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

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

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

When should I ask these math critical thinking questions?

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

Start small

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

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

Add critical thinking questions to word problems

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

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

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

When you want your students to defend their answers.

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

When you want your students to justify their opinions

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

When you want your students to think outside of the box

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

Let’s Recap:

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

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

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

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

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Khan Academy Blog

Unlocking the Power of Math Learning: Strategies and Tools for Success

posted on September 20, 2023

Mathematics, the foundation of all sciences and technology, plays a fundamental role in our everyday lives. Yet many students find the subject challenging, causing them to shy away from it altogether. This reluctance is often due to a lack of confidence, a misunderstanding of unclear concepts, a move ahead to more advanced skills before they are ready, and ineffective learning methods. However, with the right approach, math learning can be both rewarding and empowering. This post will explore different approaches to learning math, strategies for success, and cutting-edge tools to help you achieve your goals.

Math Learning

Math learning can take many forms, including traditional classroom instruction, online courses, and self-directed learning. A multifaceted approach to math learning can improve understanding, engage students, and promote subject mastery. A 2014 study by the National Council of Teachers of Mathematics found that the use of multiple representations, such as visual aids, graphs, and real-world examples, supports the development of mathematical connections, reasoning, and problem-solving skills.

Moreover, the importance of math learning goes beyond solving equations and formulas. Advanced math skills are essential for success in many fields, including science, engineering, finance, health care, and technology. In fact, a report by Burning Glass Technologies found that 71% of high-salary, entry-level positions require advanced math skills.

Benefits of Math Learning

In today’s 21st-century world, having a broad knowledge base and strong reading and math skills is essential. Mathematical literacy plays a crucial role in this success. It empowers individuals to comprehend the world around them and make well-informed decisions based on data-driven understanding. More than just earning good grades in math, mathematical literacy is a vital life skill that can open doors to economic opportunities, improve financial management, and foster critical thinking. We’re not the only ones who say so:

• Math learning enhances problem-solving skills, critical thinking, and logical reasoning abilities. (Source: National Council of Teachers of Mathematics )
• It improves analytical skills that can be applied in various real-life situations, such as budgeting or analyzing data. (Source: Southern New Hampshire University )
• Math learning promotes creativity and innovation by fostering a deep understanding of patterns and relationships. (Source: Purdue University )
• It provides a strong foundation for careers in fields such as engineering, finance, computer science, and more. These careers generally correlate to high wages. (Source: U.S. Bureau of Labor Statistics )
• Math skills are transferable and can be applied across different academic disciplines. (Source: Sydney School of Education and Social Work )

How to Know What Math You Need to Learn

Often students will find gaps in their math knowledge; this can occur at any age or skill level. As math learning is generally iterative, a solid foundation and understanding of the math skills that preceded current learning are key to success. The solution to these gaps is called mastery learning, the philosophy that underpins Khan Academy’s approach to education .

Mastery learning is an educational philosophy that emphasizes the importance of a student fully understanding a concept before moving on to the next one. Rather than rushing students through a curriculum, mastery learning asks educators to ensure that learners have “mastered” a topic or skill, showing a high level of proficiency and understanding, before progressing. This approach is rooted in the belief that all students can learn given the appropriate learning conditions and enough time, making it a markedly student-centered method. It promotes thoroughness over speed and encourages individualized learning paths, thus catering to the unique learning needs of each student.

Students will encounter mastery learning passively as they go through Khan Academy coursework, as our platform identifies gaps and systematically adjusts to support student learning outcomes. More details can be found in our Educators Hub . 

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How to learn math.

Learning at School

One of the most common methods of math instruction is classroom learning. In-class instruction provides students with real-time feedback, practical application, and a peer-learning environment. Teachers can personalize instruction by assessing students’ strengths and weaknesses, providing remediation when necessary, and offering advanced instruction to students who need it.

Learning at Home

Supplemental learning at home can complement traditional classroom instruction. For example, using online resources that provide additional practice opportunities, interactive games, and demonstrations, can help students consolidate learning outside of class. E-learning has become increasingly popular, with a wealth of online resources available to learners of all ages. The benefits of online learning include flexibility, customization, and the ability to work at one’s own pace. One excellent online learning platform is Khan Academy, which offers free video tutorials, interactive practice exercises, and a wealth of resources across a range of mathematical topics.

Moreover, parents can encourage and monitor progress, answer questions, and demonstrate practical applications of math in everyday life. For example, when at the grocery store, parents can ask their children to help calculate the price per ounce of two items to discover which one is the better deal. Cooking and baking with your children also provides a lot of opportunities to use math skills, like dividing a recipe in half or doubling the ingredients. 

Learning Math with the Help of Artificial Intelligence (AI) 

AI-powered tools are changing the way students learn math. Personalized feedback and adaptive practice help target individual needs. Virtual tutors offer real-time help with math concepts while AI algorithms identify areas for improvement. Custom math problems provide tailored practice, and natural language processing allows for instant question-and-answer sessions. 

Using Khan Academy’s AI Tutor, Khanmigo

Transform your child’s grasp of mathematics with Khanmigo , the 24/7 AI-powered tutor that specializes in tailored, one-on-one math instruction. Available at any time, Khanmigo provides personalized support that goes beyond mere answers to nurture genuine mathematical understanding and critical thinking. Khanmigo can track progress, identify strengths and weaknesses, and offer real-time feedback to help students stay on the right track. Within a secure and ethical AI framework, your child can tackle everything from basic arithmetic to complex calculus, all while you maintain oversight using robust parental controls.

Get Math Help with Khanmigo Right Now

You can learn anything .

Math learning is essential for success in the modern world, and with the right approach, it can also be enjoyable and rewarding. Learning math requires curiosity, diligence, and the ability to connect abstract concepts with real-world applications. Strategies for effective math learning include a multifaceted approach, including classroom instruction, online courses, homework, tutoring, and personalized AI support. 

So, don’t let math anxiety hold you back; take advantage of available resources and technology to enhance your knowledge base and enjoy the benefits of math learning.

National Council of Teachers of Mathematics, “Principles to Actions: Ensuring Mathematical Success for All” , April 2014

Project Lead The Way Research Report, “The Power of Transportable Skills: Assessing the Demand and Value of the Skills of the Future” , 2020

Page. M, “Why Develop Quantitative and Qualitative Data Analysis Skills?” , 2016

Mann. EL, Creativity: The Essence of Mathematics, Journal for the Education of the Gifted. Vol. 30, No. 2, 2006, pp. 236–260, http://www.prufrock.com ’

Nakakoji Y, Wilson R.” Interdisciplinary Learning in Mathematics and Science: Transfer of Learning for 21st Century Problem Solving at University ”. J Intell. 2020 Sep 1;8(3):32. doi: 10.3390/jintelligence8030032. PMID: 32882908; PMCID: PMC7555771.

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Jun 30 Critical Thinking in Mathematics: Designing High Cognitive Demand Math Tasks

Oh hey, here we are again, talking about my favorite topic: making kids THINK. Let’s get into it.

At the heart of thinking tasks lie the mathematical process skills, including reasoning, problem solving, communicating, conjecturing, and representing. These skills empower students to think critically, analyze information, communicate their ideas effectively, and explore the intricacies of mathematical concepts. Moreover, they go hand in hand with general good thinking skills, such as collaborating, listening actively, sharing ideas, and reflecting on one's own thought processes.

One effective strategy for designing thinking tasks is to take existing activities and transform them into opportunities for deep thinking. By flipping traditional activities, we can infuse them with elements of open-ended exploration, encouraging students to delve into rich problem-solving experiences.

Let's consider a traditional numeracy task where students practice multiplication using flashcards or worksheets. To design a thinking task, we can flip it by presenting students with a numerical pattern and asking them to identify the rule or equation that generates the pattern. This task encourages students to think critically, make observations, and identify mathematical relationships. It goes beyond simple memorization of multiplication facts and invites students to actively engage in problem-solving and pattern recognition. By providing students with opportunities to uncover patterns and think flexibly, we foster their mathematical reasoning skills and promote a deeper understanding of multiplication concepts.

My first year of teaching was with kindergartners. During small group time, I decided to try out a new activity with them. I drew an "open equation" on the table and had them use blocks to make the equation true. Some students had ___ + ___ = 6, others had ___ + ___ + ___ = 10, or even ___ + ___ - ___ = 10. To my surprise, this activity quickly became their favorite during small group sessions. It not only engaged them but also showcased their remarkable ability to think deeply about mathematical concepts. It was a powerful reminder that even our youngest students are capable of high cognitive thinking and can thrive when given the opportunity to engage in meaningful tasks.

When first introducing thinking tasks, it is essential to start with tasks that are slightly lower in complexity and not directly tied to specific content or math standards. This approach allows students to experience success, build confidence, and develop their thinking skills without feeling overwhelmed. By emphasizing the importance of the thinking process rather than the final answer, we create a supportive environment that nurtures students' intellectual growth and fosters a love for learning.

Here are some examples of good thinking tasks that aren't necessarily math-content related:

Provide students with a list of unrelated words or objects (e.g., banana, shoe, cloud, clock). Challenge them to come up with as many unique and imaginative ways these items could be connected or combined.

Present students with logic puzzles or tasks that require them to deduce information based on given clues. For example:

Mary is older than Jane. Jane is older than Sarah. Who is the youngest among them?

There are five houses in a row, each painted a different color. The red house is to the left of the blue house. The green house is next to the yellow house. Which color house is in the middle?

Offer students puzzles that involve visual patterns, rotations, or transformations. For example, ask students to determine the missing piece in a sequence of shapes or to identify the next shape in a pattern.

These tasks are designed to facilitate that mindset shift for students. They allow time for students to start working on those thinking skills as well as collaboration.

After students have started shifting their mindset and are comfortable with the routine, you can incorporate tasks related to your current concept. Take a look at your curriculum or unit plans and ask yourself, "What is the concept here?" and "What is the math students need to be doing?" Then you can look for or create tasks that address the mathematical skills alongside sense-making.

Let's say you are working on a unit on area and perimeter. Traditionally, you might want students to solve the area and perimeter of many different shapes. You may even let them work in groups or play a game. But how could we still have them working on the concepts of area and perimeter and using deep thinking skills?

Consider providing tasks like these:

You have a length of fencing material that is 30 feet long. Your task is to create as many different rectangular enclosures as possible using the entire length of the fencing material.

Imagine you are designing the floor plan for your dream house or an amusement park. Create a floor plan that includes different rooms or attractions. Calculate the area of each room or attraction and consider how you can optimize the use of space. Be creative and make sure to label your dimensions.

Crystal decides that she wants fringe put all the way around the rug. If the rug maker puts fringe around the rugs, how much fringe will he need for each rug? Use pictures, equations, or words to model your thinking.

Jana is designing a daycare center for small dogs. She wants to design different rectangular pens for her dogs to play. Each pen must have a total area between 49 and 100 square feet.

What is the biggest perimeter you can make with a rectangle that has an area of 24?

You can see that some of these tasks are perfect for the beginning of the unit, and some are great for after students have been taught some vocabulary. The beauty is that the point of these tasks isn't just the mathematical thinking but the process skills they practice.

Similarly, when teaching a unit on place value, you could have students work on problems where they fill in the hundreds, tens, and ones of certain numbers. But what if you had them try these tasks:

Dylan has 634 cubes in a pile. How many groups of a hundred could he make? Any left over? How many groups of ten could he make? Any left over?

Pencils come in cases of 100, packs of 10, or as single pencils. Show me what an order of 283 would look like.

Designing tasks doesn't always have to be laborious or difficult. It can be as simple as flipping your task from asking for a solution to asking for a question. For example, in a unit on rounding, instead of asking what 83 rounds to, try asking students to make a list of numbers that are close to 83 and far from 83. Be sure to have them defend and explain their thinking.

AUBREETEACHES.COM

By incorporating these different types of thinking tasks into your teaching, you can create a dynamic and engaging learning environment that promotes critical thinking, problem-solving, and mathematical reasoning. These tasks not only help students develop their thinking skills but also foster a love for learning and a deeper understanding of mathematical concepts. So, embrace the power of thinking tasks and unlock the potential of your students' minds!

Happy math-ing!

Jul 5 Lions, Tigers, and WORD PROBLEMS - OH MY!

Jun 26 Supercharge Your Math Stations with Choice Boards

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

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

What is Problem-Solving?

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

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

How to Develop Critical Thinking Skills in Math

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

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

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

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

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

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

Making Math Fun and Relevant

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

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

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

The Underlying Beauty of Mathematics

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

Why Mathematics is the Ideal Playground for Problem-Solving

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

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

Enhancing the Learning Environment

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

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

Incorporating Technology

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

More than Numbers 

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

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

FAQ: Mathematics and Critical Thinking

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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6.1: Introduction on Mathematical Reasoning

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Mathematical reasoning plays a crucial role in problem-solving by providing a structured approach to analyzing and solving complex problems. It allows individuals to break down a problem into smaller, more manageable parts, identify patterns and relationships, and apply logical reasoning to reach a solution. This process helps in developing a systematic problem-solving strategy that can be applied to a wide range of problems, from simple arithmetic calculations to advanced mathematical concepts. Moreover, mathematical reasoning enhances decision-making skills by enabling individuals to evaluate options, consider different scenarios, and choose the most effective course of action based on logical reasoning and evidence.

Furthermore, mathematical reasoning is essential for understanding complex concepts by providing a framework for organizing and synthesizing information. It helps individuals make sense of abstract ideas and relationships that may initially seem confusing or overwhelming. By applying mathematical reasoning, individuals can identify underlying principles and patterns that govern complex systems, leading to a deeper understanding of the world around them. Additionally, mathematical reasoning fosters creativity by encouraging individuals to think outside the box and explore unconventional solutions to problems, which can lead to innovative breakthroughs and advancements in various fields.

What is Mathematical Reasoning?

Definition: mathematical reasoning.

Mathematical reasoning is the process of thinking logically about mathematics. It involves analyzing mathematical concepts, making connections between them, and applying logical principles to solve problems and draw conclusions. Mathematical reasoning is not just about performing calculations; it is about understanding the underlying principles and relationships that govern mathematical systems. It requires careful thought and attention to detail, as well as the ability to construct logical arguments and justify conclusions based on evidence. Mathematical reasoning is essential for understanding advanced mathematical concepts and theories, as well as for solving complex problems in a wide range of fields, from science and engineering to economics and finance.

Mathematical reasoning plays a significant role in various aspects of everyday life, including cooking, sports, and shopping. In cooking, mathematical reasoning is used when measuring ingredients and adjusting recipes. When following a recipe, cooks must carefully measure ingredients to ensure the right balance of flavors and textures. This involves understanding fractions and proportions to scale ingredients up or down based on the desired serving size. Additionally, cooks often need to adjust recipes based on personal preferences or ingredient availability, requiring them to use mathematical reasoning to make accurate adjustments.

In sports, mathematical reasoning is used in calculating scores and analyzing statistics. For example, in basketball, players and coaches use statistics such as field goal percentage, free throw percentage, and points per game to evaluate performance and make strategic decisions. These statistics are calculated using mathematical formulas, requiring players and coaches to apply mathematical reasoning to interpret the data effectively. Mathematical reasoning is also used in sports betting, where individuals use probabilities and statistics to make informed decisions about which teams or players to bet on.

In shopping, mathematical reasoning is used when comparing prices and calculating discounts. When comparing prices, shoppers need to calculate the unit price of items to determine which offers the best value for money. This involves dividing the price of an item by its quantity to find the cost per unit. Similarly, when calculating discounts, shoppers need to understand percentages and apply mathematical reasoning to determine the final price of an item after the discount is applied. Overall, mathematical reasoning is an essential skill in everyday life, helping individuals make informed decisions and solve practical problems in a variety of contexts.

Why is Mathematical Reasoning Important?

Mathematical reasoning is a powerful tool for developing critical thinking skills because it requires individuals to analyze, evaluate, and solve complex problems using logical reasoning. When faced with a mathematical problem, individuals must first understand the problem, identify relevant information, and formulate a plan to solve it. This process involves critical thinking skills such as analysis, interpretation, and inference. Additionally, mathematical reasoning requires individuals to evaluate different strategies for solving a problem and choose the most effective one based on logical reasoning and evidence. By practicing mathematical reasoning, individuals can enhance their critical thinking skills, which are essential for success in academics, careers, and everyday life.

Furthermore, mathematical reasoning encourages individuals to think creatively and explore multiple approaches to problem-solving. It teaches individuals to look beyond the obvious solution and consider alternative methods and perspectives. This ability to think outside the box is a key aspect of critical thinking, as it allows individuals to approach problems from different angles and develop innovative solutions. Additionally, mathematical reasoning teaches individuals to evaluate the validity of their conclusions and consider the implications of their decisions. This aspect of critical thinking is essential for making informed choices and solving complex problems in a wide range of contexts. Overall, mathematical reasoning is a valuable tool for developing critical thinking skills, as it challenges individuals to think logically, creatively, and analytically about mathematical concepts and problems.

Something to think about...

1st Grade: Mathematical reasoning is important when students learning to count and understand number relationships. For example, they use reasoning to understand that if they have three apples and give one away, they will have two left.

2nd Grade: Mathematical reasoning helps students understand place value and how numbers can be decomposed into tens and ones. For example, they use reasoning to understand that 47 is composed of 4 tens and 7 ones.

3rd Grade: Mathematical reasoning is crucial for understanding multiplication and division. Students use reasoning to figure out that if they know 5 x 3 = 15, then they also know that 15 ÷ 5 = 3.

4th Grade: Mathematical reasoning is important for understanding fractions and decimals. They use reasoning to compare fractions and understand that 0.25 is the same as 1/4.

5th Grade: Mathematical reasoning is essential for understanding more complex concepts like ratios and proportions. Students use reasoning to solve problems like if it takes 3 hours to mow 2 lawns, then it will take 6 hours to mow 4 lawns.

Helping Students Develop Mathematical Reasoning Skills

Elementary school teachers play a crucial role in helping students develop mathematical reasoning skills. Here are some strategies they can use:

• Problem-solving : Engage students in solving a variety of math problems that require reasoning and critical thinking. Encourage them to explain their thought process and justify their solutions.
• Real-world applications : Use real-world examples to illustrate mathematical concepts and their relevance. This helps students see the practical applications of math and how it is used in everyday life.
• Mathematical discussions : Encourage students to discuss their strategies and reasoning with their peers. This helps them gain new perspectives and refine their own reasoning skills.
• Multiple representations : Present mathematical concepts using different representations such as visual models, manipulatives, and symbolic notation. This helps students develop a deeper understanding of the concepts and strengthens their reasoning skills.
• Open-ended tasks : Assign open-ended tasks that allow for multiple approaches and solutions. This encourages students to think creatively and develop their own problem-solving strategies.
• Scaffolding : Provide support and guidance as students work on more complex problems. Gradually remove scaffolding as they become more confident in their reasoning abilities.
• Reflective journaling : Ask students to keep a journal where they reflect on their problem-solving processes, challenges they faced, and strategies they used. This helps them become more aware of their reasoning skills and how to improve them.
• Promote a growth mindset: Encourage students to see mistakes as opportunities for learning and to persevere when faced with challenging problems. Emphasize that mathematical reasoning is a skill that can be developed with practice and effort.
• Use manipulatives and visual aids: Use concrete objects and visual representations to help students understand abstract mathematical concepts. This can help them see the patterns and relationships that underlie mathematical ideas.
• Encourage mathematical discussions: Foster a classroom environment where students feel comfortable sharing their ideas and reasoning processes. Encourage them to explain their thinking and listen to and learn from their peers.
• Assign open-ended tasks: Give students tasks that have multiple solutions or require them to explain their reasoning. This encourages them to think creatively and develop their own problem-solving strategies.
• Provide feedback: Give students feedback on their reasoning processes, highlighting both correct reasoning and areas for improvement. Encourage them to reflect on their thinking and consider different approaches.
• Relate math to real life: Help students see the relevance of mathematical reasoning by connecting it to real-world situations. This can help them understand why mathematical reasoning is important and how it can be applied outside the classroom.
• Differentiate instruction: Recognize that students may be at different levels of mathematical reasoning development and tailor instruction to meet their individual needs. Provide additional support or challenges as needed.
• Model mathematical reasoning: Demonstrate your own reasoning processes when solving problems or explaining concepts. This can help students see how mathematical reasoning is applied in practice and inspire them to develop their own reasoning skills.

Mathematical reasoning is a foundational skill that is essential for success in mathematics and many other areas of life. It enables individuals to think critically, solve complex problems, and make informed decisions based on evidence and logic. By developing their mathematical reasoning skills, students not only become better mathematicians but also better equipped to navigate the challenges of the modern world. It is therefore crucial for educators to foster the development of mathematical reasoning skills in their students through engaging and challenging activities that encourage them to think deeply and creatively about mathematical concepts. With strong mathematical reasoning skills, students can approach any problem with confidence and find innovative solutions that benefit themselves and society as a whole.

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 Mathematics Education

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Characteristics

Educational psychologists frame critical thinking (CT) as a set of generic thinking and reasoning skills, including a disposition for using them, as well as a commitment to using the outcomes of CT as a basis for decision-making and problem solving. In such descriptions, CT is established as a general standard for making judgments and decisions. Some descriptions of CT activities and skills include a sense for fairness and the assessment of practical consequences of decisions as characteristics of CT (e.g., Paul and Elder 2001 ). This assumes 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 also 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 and shared system of skills and criteria for...

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A Powerful Rethinking of Your Math Classroom

We look at strategies you can reset this year—adjusting your testing regimen, tackling math anxiety, encouraging critical thinking, and fostering a mistake-friendly environment.

The beginning of school is a great time for teachers—both veteran and early career—to consider ways they can improve upon their classroom practices over the next year. This may be especially important for math teachers, who often spend the early days of the school year confronting math anxiety , convincing students that they are indeed “math people,” and coming up with engaging practices to guide students to find pleasure in math challenges. Innovating in these areas has the potential to yield big benefits for students all year long.

To help you re-imagine some of your teaching practices this year, we’ve pulled together a collection of big and small strategies aimed at:

• helping students adopt a more productive, curious mindset when they approach math,
• engaging students as soon as they walk through the door,
• rethinking how you handle testing,
• fostering a mistake-friendly environment,
• incorporating humanities-style discussions into math, and
• building what Peter Liljedahl calls a “ thinking classroom .”

Tackle Negative Math Mindsets

“Most of my work as a math teacher isn’t even math ,” former middle school math teacher José Vilson has said. “It’s helping students believe that they can also do math.”

Striking a similar note, middle school math coordinator Alessandra King writes that supporting students—particularly students from marginalized backgrounds—to develop a “ positive mathematical identity ” is crucial to fostering a sense of belonging in a larger math community, boosting “their willingness and ability to engage” in challenging work.

King recommends spending the first few days of the school year spotlighting mathematicians from the past who reflect the makeup of your classroom. In her all-girls school, for example, she has students read and respond to a play about the achievements and struggles of Maria Agnesi, Sofya Kovalevskaya, and Emmy Noether, three historical female mathematicians. You can also hang up posters of famous Black or Latino mathematicians—such as Euphemia Haynes, the first African American woman to earn a PhD in math, or Robert Luis Santos, a Latino statistician and director of the U.S. Census Bureau—and devote lessons to discussing their achievements and backgrounds.

To get students thinking about—and challenging—their own math identity, educator Rolanda Baldwin suggests asking students early in the year to write a “ math autobiography .” They might respond to questions like: “How do you feel about math? How did your relationship with math change over time?” To draw students closer together—and make them realize that many students, of all backgrounds, struggle—have them share their responses in groups, or with the entire class.

Sometimes, our best intentions can go awry. Rachel Fuhrman, a former special education math teacher, notes that teachers can sometimes dampen a student’s math identity by wheeling out phrases that might seem helpful but can actually demotivate students , like: “This is so easy.” Framing something as “easy,” she writes, can leave students feeling uncomfortable or afraid of asking crucial, clarifying questions.

Engage Students the Moment They Enter the Classroom

To set a playful tone and lower the stakes of the work so students can effectively experiment and collaborate, high school math teacher Lorenzo Robinson suggests starting off class with fun, challenging brain teasers .

For example, ask students to draw a cross on a sheet of paper (you should draw one on the board as a point of reference). Ask students to draw two straight lines that will segment or cut the cross into pieces. The goal is to cut the cross in a way that produces the most pieces .

You could also try math riddles: Ask students to imagine they have two coins that total 30 cents. Tell them that one of the coins is not a nickel, and ask them to figure out what the two coins are.

Robinson finds that brain teasers like these can get students primed for problem solving and critical thinking, without even realizing it.

Lower the Stakes of Testing

Big tests—centered around units or near the end of a grading period—are staples of many math classrooms. But that doesn’t mean they have to be the only opportunity for students to show what they know. One way to lower the stakes for students and give them more opportunities to practice, while providing yourself more of an opportunity to both teach content and check for understanding is to give short assessments on a regular schedule. 

Math coordinator Steven Goldman’s school switched from tests to checkpoints . These short assessments include a mix of current and past topics—retrieval practice is a research-backed way to support greater learning , after all—with some repetition for the most important skills students need to know. The checkpoints are given every two weeks and are not formally graded, but mistakes are noted and teachers leave feedback to guide student revisions. The change, Goldman writes, resulted in a big reduction in student stress levels—something research shows is a benefit of frequent practice tests, alongside a boost in long-term retention. Because the checkpoints happen all the time, Goldman and his colleagues don’t have to “go through all kinds of contortions to finish a unit before a break or on a Thursday so that we could give the test at the right time.”

Fresno State math instructor Howie Hua suggests lowering the stake in your classroom by allowing students to discuss a test before starting to work on it . Hua recommends having students put their pencils on the floor so they can focus on their discussion. Hand out the test and give students five minutes to discuss strategies they can use to solve the problems. 

Good Mistakes, Better Mistakes

Mistakes are bound to happen in any math classroom, and how you respond to them throughout the year can make a world of difference. Making light of your own mistakes is a good first gambit, but research suggests a more advanced approach: Giving students space to make mistakes—and opportunities to analyze and discuss them with peers—can better encode information in their brains than simply providing them with the correct answer.

To use mistakes as building blocks to better solutions, math teacher Emma Chiapetta uses an ingenious, small-group activity that asks students to identify and reflect on common mistakes , and then explain the rationale behind them to their peers. Here’s how it works: Randomly separate students into groups and assign each group to a board to generate a problem and solve it incorrectly. Groups rotate so they’re looking at a problem and an incorrect solution, and have to identify the error and solve the problem correctly. After another rotation—so students are looking at a problem and both the incorrect and correct solutions—they explain to the class the mistake made by the first group and the correct solution provided by the second. The activity, Chiapetta writes, helps students think about the same content from various perspectives, which can lead to deeper understanding.

Not all mistakes are created equal, and they often conceal thoughtful, underlying work. Former math teacher Colin Seale asks students to reflect on which wrong answer to a problem is “ more right .” He suggests offering students two equations that are both incorrect—perhaps one is wrong conceptually and the other computationally. Seale notes that this exercise gets students to tease out nuances around the skills and concepts they’re learning, while also correcting their approach to similar problems moving forward.

Bring Humanities Strategies to Math Class

The discussion, analysis of reasoning, and argumentation that happen in humanities classrooms can be extremely useful in math classrooms to help students slow down and think through the work they’re doing , says middle school math teacher Connell Cloyd. 

He does this in his classroom by posting four incorrectly solved math problems around the room and having students rotate around each problem in groups to discuss the error and write down (in complete sentences) a claim and supporting evidence to show why they believe the error occurred. As they rotate around, students read the arguments of peers and either support an argument or refute it with new evidence. This is like adding techniques from debate to Chiappetta’s strategy.

Math journals can also inject more writing and reasoning into your classroom. Former math teacher Nell McAnelly prompts students to reflect on concepts they’re having trouble with : They work through a problem and then write about the strategy they used to solve it, or informally journal about other approaches they could have used. Doing so, she writes, gives students a chance to “synthesize learning and address unanswered questions.”

Driving Deep, Critical Thinking

A traditional math classroom, where the teacher demonstrates a skill numerous times before students take the reins, can inhibit higher order thinking and result in students who “mimic” teachers rather than develop their own strategies to solve complex problems , says Peter Liljedahl, a researcher and professor of mathematics at Simon Fraser University. 

Instead of starting lessons with direct instruction, Liljedahl says, give students novel “ thinking tasks ” to work on in groups. These are problem-solving activities and mental puzzles that should get students in the mindset of challenging themselves. Work groups should not be chosen based on ability or students’ preferences—Liljedahl’s research shows that students are more likely to contribute in randomized groups. Having students stand while they engage in this collaborative, messy thinking—as Chiappetta does in her mistake-analysis exercise—is another way to engage them.

Instead of using notebooks to compute, Liljedahl calls for groups to work at vertical, non-permanent surfaces, such as whiteboards, blackboards, or windows—surfaces that he says promote more risk-taking because students have the freedom to quickly erase false starts without feeling committed. As students work in groups, teachers bounce around the room and avoid directly answering questions such as “Is this right?” that circumvent student thinking and instead make suggestions that lead to further independent thinking. 

Because the goal of this approach is to get students to develop perseverance, curiosity, and collaboration, Liljedahl suggests evaluating them in a way that prioritizes these competencies. He developed formative assessments that focus on informing students “about where they are and where they’re going in their learning.” These include observations, check-for-understanding questions, and unmarked quizzes. Summative assessments, meanwhile, should focus less on end products and more on the process of learning through both group and individual work.

Shifting to this thinking classroom model requires a fundamental shift in how you run your classroom, but it could result in large rewards throughout the year.

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Hidden power of Math: Here are parenting tips to boost your child’s brain and problem-solving abilities

Math skills may be the missing piece in your child’s life. here's the parenting key to boost kids' brain, critical thinking and problem-solving abilities.

Math skills are essential for academic success, problem-solving, logical thinking, financial literacy and cognitive development in children however, gaps in guidance, both at school and at home , often leave this crucial piece missing. By mastering Maths skills, children sharpen their cognitive faculties, enabling them to navigate complex challenges with clarity and precision but embracing Math education isn't merely about numbers; it is about nurturing a robust mental framework that empowers children to excel academically and thrive in various facets of their lives.

In an interview with HT Lifestyle, Sripurna Sarma, Principal at Ryan International Academy in Sarjapur, addressed the factors vital to nurturing well-rounded individuals:

1. Attention difficulties – Complex mathematics procedures can overwhelm students, leading to distraction during lessons and difficulties with self-evaluation. The intricate nature of these concepts often causes students to lose focus, making it challenging for them to assess their understanding independently.

2. Math is always built on previous concepts- Mathematics relies on sequential learning; difficulties with earlier concepts can hinder understanding of new ones. For instance, mastering division is crucial to reducing fractions. Similarly, algebra builds upon arithmetic skills. Students often feel hesitant to ask questions when they're already struggling, contributing to the cycle of misunderstanding.

3. Concepts are learnt but not understood – Students frequently learn operations through repetition without grasping their underlying meaning. For instance, memorizing times tables like "4 x 4 = 16" may not signify true comprehension of multiplication; rather, it's rote memorization lacking deeper understanding of the concept.

4. Lack of practice of patience – Sometimes a student will feel like they have understood the concept but when attempting to do the problem themselves, they do not know how to begin or end, it results in struggling throughout the process.

5. “When am I ever going to use this?’- Many students struggle to see the practical application of math skills beyond the classroom. Difficulty with fractions, like converting 1/2 to 0.50, highlights this disconnect. However, they easily grasp concepts like "half of ₹ 100" equating to ₹ 50, illustrating the challenge of relating math to real-life situations.

Math is accessible to all; success relies on practice, not just natural ability. A positive attitude is crucial for achievement.

Empowering Young Minds: Embracing Mathematics

Bringing her expertise to the same, Shobha Sivaramakrishnan, Head at Ekya Learning Center of Ekya Schools, shared, “In the pursuit of enhancing children's mental focus and overall cognitive abilities, one cannot overlook the pivotal role that math skills play. Embedded within the framework of the Singapore Math methodology lies a potent approach that not only teaches mathematical concepts but also cultivates critical thinking, problem solving, and mental discipline.”

She added, “As children engage with mathematical concepts, they exercise their minds in logical reasoning, pattern recognition, and analytical thinking, all of which are fundamental components of mental focus. The practical application of these skills in real-life situations further underscores their centrality to mental focus, demonstrating the tangible benefits of mathematical proficiency.”

When students develop strong math skills, they gain the ability to:

1. Analyse and Solve Complex Problems: Children learn to break down complex issues into manageable parts, applying logical reasoning and mathematical principles to find solutions. This skill is invaluable across all areas of life, from navigating personal finances to understanding scientific challenges.

2. Think Critically and Make Informed Decisions: Strong Maths skills enable children to evaluate information critically, assess the validity of arguments, and make decisions based on quantitative data, preparing them for leadership and responsible citizenship in a data-driven world.

3. Innovate and Create: By understanding the foundational principles of Mathematics, children learn to innovate and create new solutions to real-world problems. This ability is essential for future scientists, engineers, technologists, and entrepreneurs in a ever evolving global economy.

4. Communicate Effectively Using Quantitative Information: By embracing Maths, students learn to present and interpret data clearly and persuasively, a skill that is increasingly important in a world where information is abundant and data literacy is a key ompetency in many professions.

5. Adapt to New Challenges with Flexibility : A strong foundation in Maths teaches children how to learn and adapt to new situations. They become more flexible thinkers, able to apply their knowledge in varied contexts and embrace challenges with confidence.

The Power of Math in Your Child's Development

Neelakantha Bhanu, CEO and Founder at Bhanzu, revealed, “Many parents are surprised to learn that as many as 75% of students struggle with math anxiety. One can overcome math anxiety by experiencing math which is different than the traditional way of learning.”

There is a constant question as to why 3 out of 4 students dislike math as a subject. Neelakantha Bhanu said, “My years of experience teaching math and creating curriculum has proven that it can improve your child's confidence and problem-solving skills. It is a mental workout that engages multiple areas of the brain and improves cognitive abilities. We can take steps as parents to Incorporate math games, puzzles, and hands-on activities into their child's routine for cognitive development and mental well-being.”

By developing strong mathematical skills, students unlock the ability to

1. Unlock Speed Math: This skill comes with practice and develops the ability of students not only to make them faster in calculations but also to boost their confidence in math. Students can calculate up to 4 times faster and have shown immense improvement. Speed math has also helped them academically.

2. Multitask with fluidity : Not everyone excels at multitasking with fluidity. Some find it overwhelming and see a decrease in productivity when juggling multiple things. However, getting yourself involved in early learning with math can train your mind to focus on the present moment and can improve your ability to switch between tasks with less mental strain giving you a competitive edge over other students.

3. Paradigm Adapt: In a world driven by data, math is the language of the future. By equipping your child with strong math skills, you're not just giving them a leg up in school, you're empowering them to become architects of tomorrow's world. Math fosters thought-leaders who are adaptive to change, it not only equips you with specific skills but also strengthens the cognitive foundation for paradigm adaptability.

4. Build Problem-Solving Muscles: Solving math problems is like training for a mental sport. Like athletes train their bodies, math problems become a workout for your brain. Each problem you solve strengthens your mental muscles – logic, problem solving and critical thinking.

5. Real-life applications: Strong math skills can encourage students to explore the math around them. With the right direction, students can apply the learnings and understand how the world works through the lens of mathematics.

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