how can i improve my critical thinking skills in math

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How To Encourage Critical Thinking in Math

By Mary Montero

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

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

What Does Critical Thinking in Math Look Like?

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

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

Mathematician Posters

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

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

A Marvelous Mathematician:

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

Critical Thinking Math Activities

Here are a few of my favorite critical thinking activities. 

Square Of Numbers

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

This square of numbers challenge was a hit!

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

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

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

Fifteen Cards

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

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

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

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

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

Logic Puzzles

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

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

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

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

Math Projects

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

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

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

Error Analysis

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

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

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

My FREE addition error analysis is a good place to start, no matter the grade level. I show them the process of walking through the problem and how best to complete an error analysis task.

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

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

Related Critical Thinking Posts

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

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

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

Mary Montero

I’m so glad you are here. I’m a current gifted and talented teacher in a small town in Colorado, and I’ve been in education since 2009. My passion (other than my family and cookies) is for making teachers’ lives easier and classrooms more engaging.

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

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20 Math Critical Thinking Questions to Ask in Class Tomorrow

• November 20, 2023

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

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

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

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

Table of Contents

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

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

What skills do we actually want to teach our students?

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

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

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

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

When should I ask these math critical thinking questions?

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

Start small

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

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

Add critical thinking questions to word problems

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

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

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

When you want your students to defend their answers.

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

When you want your students to justify their opinions

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

When you want your students to think outside of the box

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

Let’s Recap:

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

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

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

What is Problem-Solving?

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

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

How to Develop Critical Thinking Skills in Math

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

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

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

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

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

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

Making Math Fun and Relevant

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

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

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

The Underlying Beauty of Mathematics

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

Why Mathematics is the Ideal Playground for Problem-Solving

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

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

Enhancing the Learning Environment

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

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

Incorporating Technology

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

More than Numbers 

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

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

FAQ: Mathematics and Critical Thinking

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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5 Ways to Stop Thinking for Your Students

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

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

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

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

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

5 Ways to Get Your Students to Think

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

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

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

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

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

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

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

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

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

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

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

Promoting Critical Thinking in Math

Critical thinking is more than just an educational term. Experts have shown that it is an essential skill needed for students to be able to logically process concepts and become effective problem-solvers. While it is important for students to learn mathematical skills, facts, and vocabulary, critical thinking skills provide the avenue for students to use that knowledge in various real-life situations. Also, if you consciously follow Bloom’s Taxonomy (the original or revised versions) in your classroom, you’ll see how critical thinking plays a key role throughout learning. Here are a few ideas to promote critical thinking within your classroom: 

• Use language that supports critical thinking: “Why,” “how,” and “if” are keywords that teachers should use to support critical thinking in their math lessons. Instead of allowing students to accept an answer at face value, question them about why that is a logical answer. How do they know their answer is correct? Initially, this might cause some of your kids to become a little anxious, and they’ll probably think their answer is wrong. But as you teach them the importance of explaining/defending their answer, this can help them become more confident and be more apt to use critical thinking skills. Need ideas of what questions to ask? Check out this article from Cambridge University.
• Teach how to solve word problems and practice the skill regularly: If you were to poll your students on their least favorite part of a math assignment, the majority would probably express their dislike of word problems the most. Why? Because word problems make you think harder (and how dare we ask our kids to think, right?) Most students prefer factual problems where they know exactly which numbers they’re computing and the sign tells them what to do with those numbers. However, word problems require critical thinking skills that aren’t always natural for students. So teaching them how to solve word problems is a crucial skill. What is the problem asking? What are the keywords students need to look for that direct them to add, subtract, multiply, or divide? Is there extra information they need to overlook because it is not necessary to solve the problem? These are all critical thinking skills that have to be taught and practiced for students to become competent at solving word problems. Word walls and vocabulary notebooks are a great way to continuously expose children to those keywords they’ll need to be on the lookout for when solving word problems.
• Determine where a mistake occurred and how to fix it: We all make mistakes, especially in math. That’s why our pencils have erasers! It is important to be able to review a problem, find where the mistake occurred, and then take the steps to fix the mistake. Many times our students (and ourselves, if we’re honest) see a wrong answer and think, “It’s wrong. Oh well, move on.” Sometimes as teachers, we might take the time to analyze why the answer is incorrect to be able to help our students learn from their mistakes. But it is important to teach them those same skills to be able to fix their own mistakes. One way to practice this is to have students check their own or a peer’s answers using reverse computation skills (addition/subtraction and multiplication/division). While not every answer needs to be checked, it would be beneficial to practice regularly enough that students become effective and comfortable in using necessary critical thinking skills to analyze and correct their answers. Want to remove the personal emotional aspect of wrong answers? Have them correct pre-made problems with errors as a part of activities like warm-up or exit questions.
• Analyze graphs and interpret data: Another part of math that requires critical thinking skills is analyzing graphs and interpreting data. What do those dots, lines, and pictures mean? How can graphs and data be used to visually present and compare information? These are critical thinking skills students need to be taught and practice regularly. 
• Use peer tutors, student teachers, or “turn and talk”: This helps students present their knowledge one-on-one to a peer instead of in front of the whole class. It promotes discussion on a student level. One method is to have a partner pretend to be the teacher and the other partner pretend to be a student in a lower grade. The “teacher” can then explain the math skill or word problem to their “student.” Another option is to train your students using the “turn and talk” method. This means they turn to an assigned partner sitting near them to either solve a word problem together or work on it separately and then explain their answers to each other. 

Do your students need extra problem-solving and application practice? ORIGO’S Think Tanks are the perfect resource for all students. Think Tanks save teachers time by providing 12 imaginative sets of 20 task cards that increase in difficulty for scaffolded independence. Think Tanks complement any curriculum and are suitable for any setting: whole group, small group, or independent work.

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20 Effective Math Strategies To Approach Problem-Solving 

Katie Keeton

Math strategies for problem-solving help students use a range of approaches to solve many different types of problems. It involves identifying the problem and carrying out a plan of action to find the answer to mathematical problems.  

Problem-solving skills are essential to math in the general classroom and real-life. They require logical reasoning and critical thinking skills. Students must be equipped with strategies to help them find solutions to problems.

This article explores mathematical problem solving strategies, logical reasoning and critical thinking skills to help learners with solving math word problems independently in real-life situations. 

What are problem-solving strategies?

Problem-solving strategies in math are methods students can use to figure out solutions to math problems. Some problem-solving strategies: 

• Draw a model
• Use different approaches
• Check the inverse to make sure the answer is correct

Students need to have a toolkit of math problem-solving strategies at their disposal to provide different ways to approach math problems. This makes it easier to find solutions and understand math better. 

Strategies can help guide students to the solution when it is difficult ot know when to start.

The ultimate guide to problem solving techniques

Download these ready-to-go problem solving techniques that every student should know. Includes printable tasks for students including challenges, short explanations for teachers with questioning prompts.

20 Math Strategies For Problem-Solving

Different problem-solving math strategies are required for different parts of the problem. It is unlikely that students will use the same strategy to understand and solve the problem. 

Here are 20 strategies to help students develop their problem-solving skills. 

Strategies to understand the problem

Strategies that help students understand the problem before solving it helps ensure they understand: 

• The context
• What the key information is
• How to form a plan to solve it

Following these steps leads students to the correct solution and makes the math word problem easier .

Here are five strategies to help students understand the content of the problem and identify key information. 

1. Read the problem aloud

Read a word problem aloud to help understand it. Hearing the words engages auditory processing. This can make it easier to process and comprehend the context of the situation.

2. Highlight keywords 

When keywords are highlighted in a word problem, it helps the student focus on the essential information needed to solve it. Some important keywords help determine which operation is needed.  For example, if the word problem asks how many are left, the problem likely requires subtraction.  Ensure students highlight the keywords carefully and do not highlight every number or keyword. There is likely irrelevant information in the word problem.

3. Summarize the information

Read the problem aloud, highlight the key information and then summarize the information. Students can do this in their heads or write down a quick summary.  Summaries should include only the important information and be in simple terms that help contextualize the problem.

4. Determine the unknown

A common problem that students have when solving a word problem is misunderstanding what they are solving. Determine what the unknown information is before finding the answer.  Often, a word problem contains a question where you can find the unknown information you need to solve. For example, in the question ‘How many apples are left?’ students need to find the number of apples left over.

5. Make a plan

Once students understand the context of the word problem, have dentified the important information and determined the unknown, they can make a plan to solve it.  The plan will depend on the type of problem. Some problems involve more than one step to solve them as some require more than one answer.  Encourage students to make a list of each step they need to take to solve the problem before getting started.

Strategies for solving the problem 

1. draw a model or diagram.

Students may find it useful to draw a model, picture, diagram, or other visual aid to help with the problem solving process.  It can help to visualize the problem to understand the relationships between the numbers in the problem. In turn, this helps students see the solution.

Similarly, you could draw a model to represent the objects in the problem:

2. Act it out

This particular strategy is applicable at any grade level but is especially helpful in math investigation in elementary school . It involves a physical demonstration or students acting out the problem using movements, concrete resources and math manipulatives .  When students act out a problem, they can visualize and contectualize the word problem in another way and secure an understanding of the math concepts.  The examples below show how 1st-grade students could “act out” an addition and subtraction problem:

3. Work backwards

Working backwards is a popular problem-solving strategy. It involves starting with a possible solution and deciding what steps to take to arrive at that solution.  This strategy can be particularly helpful when students solve math word problems involving multiple steps. They can start at the end and think carefully about each step taken as opposed to jumping to the end of the problem and missing steps in between.

For example,

To solve this problem working backwards, start with the final condition, which is Sam’s grandmother’s age (71) and work backwards to find Sam’s age. Subtract 20 from the grandmother’s age, which is 71.  Then, divide the result by 3 to get Sam’s age. 71 – 20 = 51 51 ÷ 3 = 17 Sam is 17 years old.

4. Write a number sentence

When faced with a word problem, encourage students to write a number sentence based on the information. This helps translate the information in the word problem into a math equation or expression, which is more easily solved.  It is important to fully understand the context of the word problem and what students need to solve before writing an equation to represent it.

5. Use a formula

Specific formulas help solve many math problems. For example, if a problem asks students to find the area of a rug, they would use the area formula (area = length × width) to solve.   Make sure students know the important mathematical formulas they will need in tests and real-life. It can help to display these around the classroom or, for those who need more support, on students’ desks.

Strategies for checking the solution 

Once the problem is solved using an appropriate strategy, it is equally important to check the solution to ensure it is correct and makes sense. 

There are many strategies to check the solution. The strategy for a specific problem is dependent on the problem type and math content involved.

Here are five strategies to help students check their solutions. 

1. Use the Inverse Operation

For simpler problems, a quick and easy problem solving strategy is to use the inverse operation. For example, if the operation to solve a word problem is 56 ÷ 8 = 7 students can check the answer is correct by multiplying 8 × 7. As good practice, encourage students to use the inverse operation routinely to check their work. 

2. Estimate to check for reasonableness

Once students reach an answer, they can use estimation or rounding to see if the answer is reasonable.  Round each number in the equation to a number that’s close and easy to work with, usually a multiple of ten.  For example, if the question was 216 ÷ 18 and the quotient was 12, students might round 216 to 200 and round 18 to 20. Then use mental math to solve 200 ÷ 20, which is 10.  When the estimate is clear the two numbers are close. This means your answer is reasonable. 

3. Plug-In Method

This method is particularly useful for algebraic equations. Specifically when working with variables.  To use the plug-in method, students solve the problem as asked and arrive at an answer. They can then plug the answer into the original equation to see if it works. If it does, the answer is correct.

If students use the equation 20m+80=300 to solve this problem and find that m = 11, they can plug that value back into the equation to see if it is correct. 20m + 80 = 300 20 (11) + 80 = 300 220 + 80 = 300 300 = 300 ✓

4. Peer Review

Peer review is a great tool to use at any grade level as it promotes critical thinking and collaboration between students. The reviewers can look at the problem from a different view as they check to see if the problem was solved correctly.   Problem solvers receive immediate feedback and the opportunity to discuss their thinking with their peers. This strategy is effective with mixed-ability partners or similar-ability partners. In mixed-ability groups, the partner with stronger skills provides guidance and support to the partner with weaker skills, while reinforcing their own understanding of the content and communication skills.  If partners have comparable ability levels and problem-solving skills, they may find that they approach problems differently or have unique insights to offer each other about the problem-solving process.

5. Use a Calculator

A calculator can be introduced at any grade level but may be best for older students who already have a foundational understanding of basic math operations. Provide students with a calculator to allow them to check their solutions independently, accurately, and quickly. Since calculators are so readily available on smartphones and tablets, they allow students to develop practical skills that apply to real-world situations.  

Step-by-step problem-solving processes for your classroom

In his book, How to Solve It , published in 1945, mathematician George Polya introduced a 4-step process to solve problems. 

Polya’s 4 steps include:

• Understand the problem
• Devise a plan
• Carry out the plan

Today, in the style of George Polya, many problem-solving strategies use various acronyms and steps to help students recall. 

Many teachers create posters and anchor charts of their chosen process to display in their classrooms. They can be implemented in any elementary, middle school or high school classroom. 

Here are 5 problem-solving strategies to introduce to students and use in the classroom.

How Third Space Learning improves problem-solving 

Resources .

Third Space Learning offers a free resource library is filled with hundreds of high-quality resources. A team of experienced math experts carefully created each resource to develop students mental arithmetic, problem solving and critical thinking. 

Explore the range of problem solving resources for 2nd to 8th grade students. 

One-on-one tutoring 

Third Space Learning offers one-on-one math tutoring to help students improve their math skills. Highly qualified tutors deliver high-quality lessons aligned to state standards. 

Former teachers and math experts write all of Third Space Learning’s tutoring lessons. Expertly designed lessons follow a “my turn, follow me, your turn” pedagogy to help students move from guided instruction and problem-solving to independent practice. 

Throughout each lesson, tutors ask higher-level thinking questions to promote critical thinking and ensure students are developing a deep understanding of the content and problem-solving skills.

Problem-solving

Educators can use many different strategies to teach problem-solving and help students develop and carry out a plan when solving math problems. Incorporate these math strategies into any math program and use them with a variety of math concepts, from whole numbers and fractions to algebra. 

Teaching students how to choose and implement problem-solving strategies helps them develop mathematical reasoning skills and critical thinking they can apply to real-life problem-solving.

READ MORE :

• 8 Common Core math examples
• Tier 3 Interventions: A School Leaders Guide
• Tier 2 Interventions: A School Leaders Guide
• Tier 1 Interventions: A School Leaders Guide

There are many different strategies for problem-solving; Here are 5 problem-solving strategies: • draw a model  • act it out  • work backwards  • write a number sentence • use a formula

Here are 10 strategies for problem-solving: • Read the problem aloud • Highlight keywords • Summarize the information • Determine the unknown • Make a plan • Draw a model  • Act it out  • Work backwards  • Write a number sentence • Use a formula

1. Understand the problem 2. Devise a plan 3. Carry out the plan 4. Look back

Some strategies you can use to solve challenging math problems are: breaking the problem into smaller parts, using diagrams or models, applying logical reasoning, and trying different approaches.

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Ultimate Guide to Metacognition [FREE]

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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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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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Welcome to AubreeTeaches. I share my tips, tricks, and resources for being an educator that does things differently!

Happy Teaching!

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.

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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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Aug 8 high cognitive demand math tasks

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7 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. Sometimes they would struggle and get upset. This happened when they faced situations where the tricks didn't work or they forgot what to do.

That’s why math education has changed recently to focus on teaching students a deep understanding of concepts instead of relying 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 stop focusing on teaching shortcuts and instead help students become real 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. Plan your school year carefully to avoid math pitfalls by following a structured scope and sequence.

I used the   Carnegie Learning High School Math Solution  for Algebra 1 and Geometry in my last years of teaching. For the first time, I saw how much the scope and sequence really matter. My Algebra 1 students used what they learned in Module 1 to understand quadratic functions in Module 5. It was a lightbulb moment for all of us!

My Algebra 1 students used their prior knowledge and noticed recurring concepts. This helped them avoid relying on shortcuts or tricks.

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

Many students (and teachers!) love math shortcuts for quick “success.” But having a toolbox packed with critical thinking skills and problem-solving strategies is so much more valuable. These skills will serve your students not only in class, but in the real world.

One way to help students develop their critical thinking and problem-solving skills is to assign high-order math tasks in your classroom. Rich tasks help students think about what they already know and test out different methods until they identify one that works. In the process, your students gain skills and strategies that eliminate 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 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 amazed by the intelligent conversations happening in the room. Students were discussing cell division tables and explaining why explicit and recursive formulas worked!

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.

Visualizing math concepts helps students see patterns and make connections that they may not immediately understand 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. If that happens, they're more likely to understand and use the formula of a quadratic function in various situations.

4. Manipulatives and Hands-On Learning

Another way to eliminate tips and tricks (“A negative times a negative is a positive,” anyone?) is with manipulatives. I love 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 note-taking in favor of digging into algebra tiles! 

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. In this case, we're talking about mathematical connections.

Teach your students to look for the interconnectedness of mathematical concepts. Show them how ideas fit together and build on one another. 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 remembered more when they understood how concepts were connected and could use their knowledge in unexpected ways.

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.

Your students will stay interested and learn concepts that can be applied in different 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. I 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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How to Get Better at Math: 14 Effective Steps

Mathematics is an essential skill that affects every aspect of our lives, from basic daily activities to complex problem-solving in various fields. Whether it's the intimidating numbers or simply a lack of confidence, struggling with math problems is a common challenge for both kids and adults.

Fear no more! This blog post will equip you with effective strategies to learning how to get better at math. As a bonus, you'll also discover some parenting tips to support your kid’s journey on how to get better at math fast.

Simple Tips & Tricks for How to Get Better at Math

Most people believe that practicing math is the only way to become better at it. However, in reality, there are many simple yet effective methods to improve your math skills and support your kids in learning math faster through day-to-day practice.

Here are the most effective tips to learn how to get better at math if you struggle.

1. Brush Up on Basic Math Problems

Excelling on basic math problems is a fundamental step toward improving math skills. By revisiting concepts like addition, subtraction, multiplication, and division, individuals can strengthen their foundation and enhance their problem-solving abilities.

So if you are wondering how to get better at basic math, try solving simple math problems consistently, both on paper and mentally, to increase speed and accuracy. You can dedicate a few minutes to integrating these basic problems into your everyday activities, like shopping or calculating expenses.

2. MentalUP Math Games

Are you wondering how to train your brain to get better at math? Discover the ultimate app both kids and adults love and trust.

Engaging in MentalUP Math Games offers a fun and effective way to improve math skills by providing a variety of challenges that stimulate critical thinking, logical reasoning, and problem-solving.

As a learning app designed with an adaptive algorithm, MentalUP will be continuously testing your family members' math skills and providing them with personalized math exercises tailored to their strengths, weaknesses, and needs.

Thanks to its easy-to-navigate layout and colorfully designed games and activities, the combination of entertainment and learning will motivate all your family members to develop a positive attitude toward math.

MentalUP is full of brain training games and exercises that combine fun and learning to enhance individuals' cognitive skills, like number sense , critical thinking , working memory , and logical thinking . 🚀

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3. Use Mental Math

If you wonder what is mental math , here is the answer: Mastering mental math techniques is a valuable skill that entails performing calculations without relying heavily on paper or calculators. Practicing it can enhance individuals' computational skills, number sense, and overall efficiency in solving math problems.

If you are thinking about how to get better at simple math with mental math tricks, you can incorporate mental math exercises into daily activities, such as estimating expenses while grocery shopping or mentally calculating tips at restaurants.

4. Practice Math in Everyday Scenarios

One of the most effective ways to improve math skills is to apply mathematical concepts to everyday scenarios. So if you are thinking about how to get better at math as an adult, you can try recognizing math in day-to-day activities.

For example, tasks that require measurement and estimation, such as baking, DIY projects, or planning a budget, can help you and your kids think critically and solve problems in real-world situations. You can engage more in these activities to gain a deeper understanding of math.

5. Solve Puzzles and Riddles

Solving puzzles and riddles is one of the best ways to enhance problem-solving and critical thinking, that’s why many problem solving apps offer puzzles and riddles. These activities require logical reasoning and help individuals develop their mathematical thinking skills.

Both kids and adults that wonder how to get better at critical thinking in math need to practice regularly with different types of puzzles and riddles. This will be helpful for becoming more comfortable with mathematical patterns and improving analytical skills.

Looking for a safe and effective resource full of engaging puzzles and riddles ? MentalUP is here for you!

It offers a safe learning platform trusted by world-leading universities like Troy and Indiana. With hundreds of brain teasers , riddles , and puzzles , it functions as the ultimate resource for both kids and adults to sharpen their cognitive skills and mathematical knowledge. 🎉

6. Break Down Complex Problems Into Smaller Ones

Complex math problems can be intimidating, but breaking them down into smaller, more manageable parts can make them much more approachable. This trick can be also helpful when you are struggling with how to get better at discrete math.

You can start by identifying the key components or steps within the problem. Break it down into simpler sub-problems and solve each one systematically. This approach will allow you to focus on one step at a time, reducing stress and building confidence as you progress.

7. Join a Study Group

Joining a study group can greatly enhance your math learning experience. Collaborating with peers who share similar goals and challenges provides an opportunity for discussion, explanation, and problem-solving together.

So if you need to explore how to help yourself get better at math, you can join a study group that meets regularly. Through active engagement with others, you can gain different perspectives, uncover new problem-solving approaches, and clarify concepts that may be unclear.

8. Utilize Practice Tests

Practice tests are invaluable resources for improving math skills. They simulate the format and style of actual exams, allowing individuals to familiarize themselves with the types of questions they may encounter.

If your child is asking for advice on how to get better at competition math, you can encourage them to take more practice tests. These tests will help them identify areas where they need more practice and enhance their speed and accuracy.

9. Memorize Important Formulas and Facts

Memorizing key formulas and facts is an essential step toward becoming better at math. So if you are wondering how to get better at quick math, keep in mind that solidifying your knowledge of facts and formulas can help.

You should regularly review and quiz yourself on important formulas to reinforce your understanding and retention. Also, flashcards, mnemonic devices, and interactive online resources can be helpful tools in this process.

10. Establish a Consistent Study Routine

This tip is especially useful for students wondering how to get better at algebra in higher level math classes. All they need is to establish a regular study routine that creates a structured learning environment and allows for effective practice.

Students can set aside time each day or week to focus on math-related activities. This could include reviewing concepts, practicing problem-solving, or working on math exercises. Also, it will be better to benefit from math websites for kids during these routines.

Our tips are about to finish. Do you still have questions about how to get better at math or which strategies to utilize? 👀

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11. Improve Your Number Sense

Number sense refers to the ability to understand and work with numbers flexibly and fluently. Improving it involves developing a deep understanding of the relationship between numbers and properties.

If your kids are asking you about how to get better at problem-solving math, you can encourage them to engage in activities that promote number sense, such as estimating, comparing quantities, and exploring number patterns.

12. Celebrate Your Achievements

Whether it's acknowledging your own progress or that of your kids, such as mastering a new concept, solving a challenging problem, or improving accuracy, it's important to celebrate these milestones.

By celebrating small achievements along the way, you provide positive reinforcement and foster a sense of accomplishment. This recognition inspires both you and your kids to continue working towards improving math skills with enthusiasm and determination.

13. Expand Your Problem-Solving Skills

Problem-solving lies at the heart of mathematics. By expanding your problem-solving skills, you can approach math challenges with confidence and efficiency. To improve your problem-solving abilities, begin by taking the time to thoroughly understand the problem.

Next, try breaking down complex problems into smaller steps and visualize them using diagrams and graphs. Additionally, practice different solution strategies, such as working backward, making an organized list, and using logical reasoning.

14. Learn Math Shortcuts and Tricks

Math shortcuts can speed up calculations, increase accuracy, and improve overall efficiency in solving math problems. To benefit from these techniques in your daily math practice, start by memorizing key formulas for geometric shapes, trigonometry, or algebraic operations.

You can also set timers and challenge yourself to complete calculations within a specified time frame to gradually excel at these tricks and tips. Remember that as you maintain your curiosity and seek out new math shortcuts, you'll become more proficient.

Parenting Strategies to Improve Your Kid’s Math Success & Grades

When it comes to fostering kids' interest in learning math, many parents face challenges with encouraging their children to practice math concepts. Here are some insightful strategies that can help you transform your kids into math whizzes:

• Encourage a positive mindset by emphasizing the importance and relevance of math in daily life.
• Create a math-friendly environment at home with math-related books, puzzles, games, and toys.
• Show interest in your child's math progress and engage in conversations about math topics.
• Incorporate math into daily routines, such as counting objects, calculating prices during shopping, or solving math problems together.
• Help your child develop problem-solving skills by teaching them various strategies, such as breaking down problems.
• Explore educational apps that can provide extra support and practice opportunities.
• Encourage your child to take ownership of their math learning by setting goals, managing their time, and seeking help when needed.
• Make math fun and relatable with activities and games.
• Celebrate achievements to boost confidence and motivation.

Experience the award-winning learning app , MentalUP , and unlock the shortcut to igniting your child's passion for math! 🏆

MentalUP offers an engaging and safe learning environment . Its adaptive algorithm continuously assesses your child's skills and tailors personalized learning plans to cultivate their interest. 🎮

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Frequently Asked Questions

How to get better at math as an adult.

Practice with the best math apps for adults regularly and consistently, working through math problems and seeking additional resources if needed.

What are the most effective ways to use for how to get better at critical thinking in math?

You can try engaging in problem-solving activities, analyzing mathematical concepts deeply, and questioning assumptions to enhance critical thinking in math.

How to train your brain to get better at math?

To improve your math skills, you need to constantly challenge yourself with new problems, utilize mnemonic techniques, and engage in a variety of math-related activities.

Which are the best solutions for how to get better at math if you struggle?

Seek personalized tutoring or study groups, use online resources and apps, and maintain a positive mindset to overcome math struggles and improve proficiency. Don’t forget to use an online dyscalculia test for detailed information about your struggle.

Whether you need to improve your math skills or proficiency in any other subject, the key to success lies in enhancing your cognitive skills and practicing regularly with effective resources.

As one of the most trusted and downloaded learning apps, MentalUP takes both your and your kids' skills to a whole new level! 📊

With 150+ brain-training games and 240+ physical exercises , MentalUP challenges and improves its users' cognitive and physical abilities in memory, logic, visual, and verbal intelligence skills. Plus, with its performance-tracking tools , it provides insights into your learning journey. 👏

Don't miss this opportunity to boost your brainpower and have fun at the same time!

PLAY & LEARN WITH MENTALUP

Online Math Games for Kids

15 Best Math Apps for Kids (Teacher-Recommended)

The Best 24 Math Jokes With Answers to Laugh

97 Fun and Challenging Math Riddles With Answers

• For Parents

6 Effective Math Problem-Solving Strategies

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Author: BYJU’S Math Companion Tutor

How do you develop critical thinking in math?

• Encourage exploration: Provide opportunities for your child to explore math concepts beyond textbooks. Engage them in activities that challenge their thinking, such as puzzles, brain teasers, and logic games. 
• Ask open-ended questions: Pose questions that require more than a simple yes-or-no answer. Encourage your child to explain their reasoning and provide evidence for their solutions, fostering analytical thinking. 
• Promote active discussions : Engage in conversations with your child about real-world scenarios where math comes into play. Discuss how math concepts apply to everyday situations, encouraging them to think critically about problem-solving approaches.
• Provide diverse problem sets : Present a variety of math problems that require different strategies to solve. This broadens your child’s problem-solving toolkit and enhances their ability to adapt their skills to various challenges.

How can critical thinking help in problem-solving?

• Analyzing information: Critical thinking enables children to break down complex problems into manageable components, making it easier to understand and address the underlying issues.
• Developing creative solutions: Encouraging critical thinking sparks creativity. Children learn to think outside the box, exploring unconventional approaches to problem-solving.
• Enhancing decision-making: Critical thinkers are equipped to evaluate various options before making decisions. They consider the potential outcomes and choose the most effective course of action.
• Building confidence: As children successfully tackle challenging problems, their confidence in their problem-solving abilities grows. This newfound confidence extends to other areas of learning too.
• Applying learning to real life: Critical thinking extends beyond the classroom. Children become adept at applying problem-solving skills to real-world scenarios, preparing them for future challenges.

What are the Strategies for Effective Math Problem-Solving ?

Break it down, visualize the answer, trial and error, collaborative problem-solving, real-life applications, celebrate progress.

How do online math tutors help children develop critical thinking and problem-solving skills through active learning?

• Online math tutors use active learning strategies, presenting various math problems that demand analytical thought.
• They encourage children to explain problem-solving strategies, promoting self-reflection and deeper understanding.
• Tutors facilitate discussions, asking open-ended questions to stimulate reasoning and logical thinking.
• Interactive quizzes and simulations engage children in hands-on problem-solving.
• Tutors connect math concepts to practical situations through real-world scenarios.
• They challenge children with puzzles, riddles, and brainteasers, encouraging creative thinking and diverse approaches.
• Immediate feedback through online platforms helps children learn from mistakes and refine strategies.
• Tutors employ Socratic questioning, guiding children to analyze and justify solutions.
• Collaborative problem-solving is encouraged through group activities, fostering learning from peers.
• Online math tutors customize strategies to suit individual learning styles, promoting optimal critical thinking growth.

10 Practical Tips for Creating a Math-Friendly Environment at Home

10 Ways to Build a Strong Math Foundation via Encouragement

3-Step Math Practice Routine for Your Child's Success (A Parent's Guide)

5 Games with a Math Twist to Spark Joy (and Math Skills) in Your Child!

6 Strategies to Boost Math Problem-Solving Skills in Your Child

6 Ways Parents can Help Children Improve Math Skills at Home

7 Effective Techniques to Overcome Math Frustration: A Guide for Parents & Tutors

7 Proven Strategies for Mastering Math Fluency

8 Essential Qualities to Seek in Your Child's Online Math Tutor

8 Fun Math Games for Homeschoolers: Enhancing Learning With Online Math Tutoring

A Math Tutor's Impact: Empowering Students via Online Tutoring

Addressing Dyscalculia: How Online Math Tutoring can Make a Difference

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CRITICAL THINKING IN MATHEMATICS: WHAT, WHY, AND HOW CAN BE DEVELOPED IN MATHEMATICS LEARNING

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Mainstream educational psychologists view critical thinking (CT) as the strategic use of a set of reasoning skills for developing a form of reflective thinking that ultimately optimizes itself, including a commitment to using its outcomes as a basis for decision-making and problem solving. In such descriptions, CT is established as a general methodological standard for making judgments and decisions. Accordingly, some authors also include a sense for fairness and the assessment of practical consequences of decisions as characteristics (e.g., Paul and Elder 2001 ). This conception assumes rational, autonomous subjects who share a common frame of reference for representation of facts and ideas, for their communication, as well as for appropriate (morally “good”) action. Important is the difference as to what extent a critical examination of the criteria for CT is included in the definition: If education for CT is conceptualized as instilling a belief in a more or less fixed...

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What book would you recommend to significantly improve my problem solving skills? [duplicate]

I am a straight-A student (going to the ninth grade) and do nearly perfectly in math, the problem is that my school (like many other schools I suppose) makes you memorize the steps, the formulas, etc. and just apply that to similar questions, and since I’ll probably be opting for IGCSE and eventually A levels, I find it necessary to significantly improve my critical thinking skills because I never had adequate practice for that. I’m sorry for all that background, but I would like your recommendation of books/websites that would help me.

Do you think these books are appropriate:

• How to Solve It by Polya
• The Art of Problem Solving Vol. 1: The Basics & How to Solve It Vol. 2: And Beyond
• Problem Solving Strategies by Engel
• Problem-Solving through Problems
• Mathematical Discovery: On Understanding, Learning and Teaching Problem Solving

Do you think that Solving Mathematical Problems by Tao is enough for me to significantly improve?

• algebra-precalculus
• reference-request
• problem-solving
• book-recommendation

• $\begingroup$ Probably I will get a personal tutor once I actually begin A-levels, but for now I just wanted to acquire methods and techniques on problem-solving and do some practice. Do you have any book recommendations? $\endgroup$ –  Tala Sadaqa Commented May 27, 2020 at 20:35
• $\begingroup$ If you want to really make a name for yourself, talk to your math teacher and ace the Putnam exam while in high school: amazon.com/William-Lowell-Putnam-Mathematical-Competition/dp/… $\endgroup$ –  David G. Stork Commented May 27, 2020 at 20:43
• $\begingroup$ Thank you for your comment. I am not that of a math lover, so I don't really want to make a name for myself or ace the Putnam Exams. However, I like math and now how essential mastering it is, so I am ready to set time aside and seriously work on it. I checked the book you suggested, and it seems too advanced, don't you think? $\endgroup$ –  Tala Sadaqa Commented May 27, 2020 at 21:22
• $\begingroup$ Hello Tala. I understand that you've learnt the material up to about Year 8 of a curriculum similar to that of England. Tao's and Engel's books are certainly too advanced for that. I'm not a big fan of the Art of Problem Solving series, though they might be at your level. I would suggest that there are two (non-mutually exclusive) kinds of reading that could be beneficial. Firstly, there are some good problem books for roughly your year level. The best I know of is Mathematical Circles by Fomin and Itenberg. Secondly, you could bring your knowledge of algebra up to about GCSE level (though.. $\endgroup$ –  Anonymous Commented May 27, 2020 at 22:36
• $\begingroup$ not necessarily following a specific curriculum), after which your options for collateral reading on interesting mathematical subjects will expand considerably. Suitable textbooks for this might be Basic Mathematics by Serge Lang or Algebra the Easy Way by Douglas Downing, though which one you choose isn't terribly important. If I've misunderstood and you are in fact already at GCSE level in your knowledge, then please say so and I'll make different recommendations. $\endgroup$ –  Anonymous Commented May 27, 2020 at 22:40

3 Answers 3

The ruling class doesn't want critical thinkers. It wants trained workers. Apparently you've outstripped your school's ability to educate you. So move on from the Dummy Down Dunce Dance school you're currently attending. Learn math on your own. Choose the topics that most interest you and study them. That is where you'll excel. Skip that I've got to curriculum, it's just baggage. If you bump into something that you like but lack the background, then you'll be motivated to study the background material. Skip that high school stuff and look for a community college that'll accept you. You'll likely be happier there. Visit a community college, meet with a math teacher, and find what opportunities be there for you. I have a friend who got so bored with high school, she attended a community college instead where she was happier. I also know a self tutored math student who was accepted into a liberal arts college based upon his success in a college math exam. PS. Skip the most modern text books and look for books written for mathematicians where the cook book method is ignored in preference to concepts, theorems and proofs. Good luck with your struggle living in a country where being of above average intelligence is a handicap and with your new adventure beyond high school into adult education. PSS. If you want a tutor, I'll give you a throwaway email address by which you can contact me. I have successful experience tutoring both elbow to elbow and by phone. Being retired, I've no need of payment. I also suggest you use this web cite for questions and when you get stuck understanding a concept or a formula.

• $\begingroup$ "If you bump into something that you like but lack the background, then you'll be motivated to study the background material." Can't agree with this more. $\endgroup$ –  BLUC Commented May 28, 2020 at 5:20
• 3 $\begingroup$ Could you perhaps focus on the question asked and leave commentary and asides away. $\endgroup$ –  quid Commented May 28, 2020 at 10:46
• $\begingroup$ I would be very cautious when a student shows "math" talent. I would really want him/her to differentiate between math capability and the nature of rigorous math. In my opinion, Olympic math/class math is ruining most children's interest in going deeper. On the other hand, rigorous logic education enables everyone to learn math on their own, and allows them to move forward as he/she wishes. Some of them may become mathematicians, and some of them know when to stop and turn to other jobs (engineers, for example), while still keep a good sense of math. $\endgroup$ –  Ziqi Fan Commented Dec 31, 2020 at 5:01

Yes, I think the Art of Problem Solving is a good book series.

I have not heard of Solving Mathematical Problems by Tao, though, but I do know that you can practice your mathematics skills by completing AMC Tests from past years here.

https://artofproblemsolving.com/wiki/index.php/AMC_Problems_and_Solutions

I wrote a book that may be helpful, as it covers the mental process of problem solving, something rarely taught in school. It has a series of engineering problems solved with the top-down approach, as a boot camp for the reader to get the hang of it. It's titled "The Top-Down Approach to Problem Solving", ISBN 979-8464073296. You can find it on amazon as paperback and ebook. I hope you like it.

Not the answer you're looking for? Browse other questions tagged algebra-precalculus reference-request problem-solving book-recommendation .

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