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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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• Critical Thinking

How To Encourage Critical Thinking in Math

By Mary Montero

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

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

What Does Critical Thinking in Math Look Like?

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

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

Mathematician Posters

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

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

A Marvelous Mathematician:

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

Critical Thinking Math Activities

Here are a few of my favorite critical thinking activities. 

Square Of Numbers

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

This square of numbers challenge was a hit!

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

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

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

Fifteen Cards

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

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

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

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

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

Logic Puzzles

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

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

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

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

Math Projects

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

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

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

Error Analysis

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

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

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

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

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

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

Related Critical Thinking Posts

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

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

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

Mary Montero

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

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

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Engaging Maths

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

• by cattard2017
• Posted on June 25, 2017

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

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

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

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

How and when do we embed critical and creative thinking?

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

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

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

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

Who should be engaging in critical and creative thinking?

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

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

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

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

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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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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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Rich Problems – Part 1

Rich problems – part 1, by marvin cohen and karen rothschild.

One of the underlying beliefs that guides Math for All is that in order to learn mathematics well, students must engage with rich problems. Rich problems allow ALL students, with a variety of neurodevelopmental strengths and challenges, to engage in mathematical reasoning and become flexible and creative thinkers about mathematical ideas. In this Math for All Updates, we review what rich problems are, why they are important, and where to find some ready to use. In a later Math for All Updates we will discuss how to create your own rich problems customized for your curriculum.

What are Rich Problems?

At Math for All, we believe that all rich problems provide:

• opportunities to engage the problem solver in thinking about mathematical ideas in a variety of non-routine ways.
• an appropriate level of productive struggle.
• an opportunity for students to communicate their thinking about mathematical ideas.

Rich problems increase both the problem solver’s reasoning skills and the depth of their mathematical understanding. Rich problems are rich because they are not amenable to the application of a known algorithm, but require non-routine use of the student’s knowledge, skills, and ingenuity. They usually offer multiple entry pathways and methods of representation. This provides students with diverse abilities and challenges the opportunity to create solution strategies that leverage their particular strengths.

Rich problems usually have one or more of the following characteristics:

• Several correct answers. For example, “Find four numbers whose sum is 20.”
• A single answer but with many pathways to a solution. For example, “There are 10 animals in the barnyard, some chickens, some pigs. Altogether there are 24 legs. How many of the animals are chickens and how many are pigs?”
• A level of complexity that may require an entire class period or more to solve.
• An opportunity to look for patterns and make connections to previous problems, other students’ strategies, and other areas of mathematics. For example, see the staircase problem below.
• A “low floor and high ceiling,” meaning both that all your students will be able to engage with the mathematics of the problem in some way, and that the problem has sufficient complexity to challenge all your students. NRICH summarizes this approach as “everyone can get started, and everyone can get stuck” (2013). For example, a problem could have a variety of questions related to the following sequence, such as: How many squares are in the next staircase? How many in the 20th staircase? What is the rule for finding the number of squares in any staircase?

• An expectation that the student be able to communicate their ideas and defend their approach.
• An opportunity for students to choose from a range of tools and strategies to solve the problem based on their own neurodevelopmental strengths.
• An opportunity to learn some new mathematics (a mathematical residue) through working on the problem.
• An opportunity to practice routine skills in the service of engaging with a complex problem.
• An opportunity for a teacher to deepen their understanding of their students as learners and to build new lessons based on what students know, their developmental level, and their neurodevelopmental strengths and challenges.

Why Rich Problems?

All adults need mathematical understanding to solve problems in their daily lives. Most adults use calculators and computers to perform routine computation beyond what they can do mentally. They must, however, understand enough mathematics to know what to enter into the machines and how to evaluate what comes out. Our personal financial situations are deeply affected by our understanding of pricing schemes for the things we buy, the mortgages we hold, and fees we pay. As citizens, understanding mathematics can help us evaluate government policies, understand political polls, and make decisions. Building and designing our homes, and scaling up recipes for crowds also require math. Now especially, mathematical understanding is crucial for making sense of policies related to the pandemic. Decisions about shutdowns, medical treatments, and vaccines are all grounded in mathematics. For all these reasons, it is important students develop their capacities to reason about mathematics. Research has demonstrated that experience with rich problems improves children’s mathematical reasoning (Hattie, Fisher, & Frey, 2017).

Where to Find Rich Problems

Several types of rich problems are available online, ready to use or adapt. The sites below are some of many places where rich problems can be found:

• Which One Doesn’t Belong – These problems consist of squares divided into 4 quadrants with numbers, shapes, or graphs. In every problem there is at least one way that each of the quadrants “doesn’t belong.” Thus, any quadrant can be argued to be different from the others.
• “Open Middle” Problems – These are problems with a single answer but with many ways to reach the answer. They are organized by both topic and grade level.
• NRICH Maths – This is a multifaceted site from the University of Cambridge in Great Britain. It has both articles and ready-made problems. The site includes  problems for grades 1–5 (scroll down to the “Collections” section) and problems for younger children . We encourage you to explore NRICH more fully as well. There are many informative articles and discussions on the site.
• Rich tasks from Virginia – These are tasks published by the Virginia Department of education. They come with complete lesson plans as well as example anticipated student responses.
• Rich tasks from Georgia – This site contains a complete framework of tasks designed to address all standards at all grades. They include 3-Act Tasks , YouCubed Tasks , and many other tasks that are open ended or feature an open middle approach.

The problems can be used “as is” or adapted to the specific neurodevelopmental strengths and challenges of your students. Carefully adapted, they can engage ALL your students in thinking about mathematical ideas in a variety of ways, thereby not only increasing their skills but also their abilities to think flexibly and deeply.

Hattie, J., Fisher, D., & Frey, N. (2017). Visible learning for mathematics, grades K-12: What works best to optimize student learning. Thousand Oaks, CA: Corwin Mathematics.

NRICH Team. (2013). Low Threshold High Ceiling – an Introduction . Cambridge University, United Kingdom: NRICH Maths.

The contents of this blog post were developed under a grant from the Department of Education. However, those contents do not necessarily represent the policy of the Department of Education, and you should not assume endorsement by the Federal Government.

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Critical Thinking in Mathematics Education

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

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

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Building a Thinking Classroom in Math

Over more than a decade, the author has developed a 14-point plan for encouraging students to engage deeply with math content.

One day in 2003, I was invited to help June implement problem solving in her grade 8 classroom. She had never done problem solving with her students before, but with its prominence in the recently revised British Columbia curriculum, she felt it was time.

June, as it turned out, was interested in neither co-planning nor co-teaching. What she wanted from me was simply a collection of problems she could try with her students. The first one I gave her was a Lewis Carroll problem that I’d had much success with, with students of different grade levels: If 6 cats can kill 6 rats in 6 minutes, how many will be needed to kill 100 rats in 50 minutes?

June used it the next day. It did not go well. A forest of arms immediately shot up, and June moved frantically around the room answering questions. Many students gave up quickly, so June also spent much effort trying to motivate them to keep going. In general, there was some work attempted when June was close by and encouraging the students, but as soon as she left the trying stopped. This continued for the whole period.

The following day I was back with a new problem. The results were as abysmal as they had been on the first day. The same was true the third day. Over the course of three 40-minute classes, we had seen little improvement in the students’ efforts to solve the problems, and no improvements in their abilities to do so. So June decided it was time to give up.

I wanted to understand why the results had been so poor, so I stayed to observe June and her students in their normal routines. After three full days of observation, I began to discern a pattern. That the students were lacking in effort was immediately obvious, but what took time for me to realize was that the students were not thinking. More alarming was the realization that June’s teaching was predicated on an assumption that the students either could not or would not think.

Once I realized this, I proceeded to visit 40 other mathematics classes in a number of schools. In each class, I saw the same thing—an assumption, implicit in the teaching, that the students either could not or would not think. Under such conditions it was unreasonable to expect that students were going to be able to spontaneously engage in problem solving.

This motivated me to find a way to build, within these same classrooms, a culture of thinking. I wanted to build what I now call a thinking classroom—one that’s not only conducive to thinking but also occasions thinking, a space inhabited by thinking individuals as well as individuals thinking collectively, learning together, and constructing knowledge and understanding through activity and discussion.

Over 14 years, and with the help of over 400 K–12 teachers, I’ve been engaged in a massive design-based research project to identify the variables that determine the degree to which a classroom is a thinking or non-thinking one, and to identify the pedagogies that maximize the effect of each of these variables in building thinking classrooms. From this research emerged a collection of 14 variables and corresponding optimal pedagogies that offer a prescriptive framework for teachers to build a thinking classroom.

1. The type of tasks used: Lessons should begin with good problem solving tasks. In the beginning of the school year, these tasks need to be highly engaging, non-curricular tasks. Later these are gradually replaced with curricular problem solving tasks that then permeate the entirety of the lesson.

2. How tasks are given to students: As much as possible, tasks should be given verbally. If there are data, diagrams, or long expressions in the task, these can be written or projected on a wall, but instructions should still be given verbally.

3. How groups are formed: At the beginning of every class, a visibly random method should be used to create groups of three students who will work together for the duration of the class.

4. Student work space: Groups should stand and work on vertical non-permanent surfaces such as whiteboards, blackboards, or windows. This makes the work visible to the teacher and other groups.

5. Room organization: The classroom should be de-fronted, with desks placed in a random configuration around the room—away from the walls—and the teacher addressing the class from a variety of locations within the room.

6. How questions are answered: Students ask only three types of questions: proximity questions, asked when the teacher is close; “stop thinking” questions—like “Is this right?” or “Will this be on the test?”; and “keep thinking” questions—ones that students ask in order to be able to get back to work. The teacher should answer only the third type of question.

7. How hints and extensions are used: The teacher should maintain student engagement through a judicious and timely use of hints and extensions to maintain a balance between the challenge of the task and the abilities of the students working on it.

8. Student autonomy: Students should interact with other groups frequently, for the purposes of both extending their work and getting help. As much as possible, the teacher should encourage this interaction by directing students toward other groups when they’re stuck or need an extension.

9. When and how a teacher levels their classroom: When every group has passed a minimum threshold, the teacher should pull the students together to debrief what they have been doing. This should begin at a level that every student in the room can participate in.

10. Student notes: Students should write thoughtful notes to their future selves. They should have autonomy as to what goes in the notes and how they’re formatted. The notes should be based on the work already on the boards done by their own group, another group, or a combination.

11. Practice questions: Students should be assigned four to six questions to check their understanding. They should have freedom to work on these questions in self-selected groups or on their own, and on the vertical non-permanent surfaces or at their desks. The questions should not be marked or checked for completeness—they’re for the students’ self-evaluation.

12. Formative assessment: Formative assessment should be focused primarily on informing students about where they are and where they’re going in their learning. This will require a number of different activities, from observation to check-your-understanding questions to unmarked quizzes where the teacher helps students decode their demonstrated understandings.

13. Summative assessment: Summative assessment should focus more on the processes of learning than on the products, and should include the evaluation of both group and individual work. Summative assessment should not in any way have a focus on ranking students.

14. Reporting out: Reporting out of students’ performance should be based not on the counting of points but on the analysis of the data collected for each student within a reporting cycle. The data need to be analyzed on a differentiated basis and focused on discerning the learning a student has demonstrated.

My research also shows that the variables and accompanying pedagogical tools are not all equally impactful in building thinking classrooms. And there is an optimal sequence for both teachers and students when first introducing these pedagogies. This sequence is presented as a set of four distinct toolkits that are meant to be enacted in sequence from top to bottom, as shown in the chart. When these toolkits are enacted in their entirety, an optimal transformation of the learning environment has been achieved in the vast majority of classrooms.

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Inspiring Minds: The Role of Mathematics in Critical Thinking

Join us for an enlightening conversation with Dr. Igor Subbotin, an esteemed mathematician and educator, as we explore the essential role mathematics plays in our world. Throughout our discussion, we uncover the profound impact that mathematics has on developing critical thinking and problem-solving skills, vital for the 21st-century landscape. Dr. Subbotin, with his extensive background in algebra and passion for the subject, shares his insights on how mathematics serves as both the queen and servant of the sciences, simplifying complex ideas and fostering analytical minds.

Listen in as we delve into the significance of mathematics within the educational sphere, particularly at National University. We emphasize the necessity for inspiring teachers who can ignite a lifelong appreciation for mathematics, crucial in dispelling the common apprehension surrounding the subject. Our journey through the history of algebra reveals its rich tapestry, from ancient civilizations to the Islamic Golden Age, demonstrating the subject’s evolution and the collaborative nature of its growth, transcending cultural and geographic divides.

Wrapping up our discussion, Dr. Subbotin shares personal anecdotes from his academic path, influenced by renowned mathematicians like Sergei Chernikov. He highlights the emergence of braces theory, a fascinating new branch of algebra, illustrating the interconnectedness of mathematics and physics. This narrative not only showcases the collaborative spirit within the mathematical community but also reinforces the notion that abstract mathematical theories can significantly influence various scientific fields. Tune in to discover the boundless universe of mathematics, where equations speak the language of nature, and every human activity is interwoven with numerical threads.

• 0:04:04 – The Importance of Mathematics in Science (105 Seconds)
• 0:08:56 – Discovering Mathematics (104 Seconds)
• 0:11:50 – Universal Application of Mathematical Concepts (85 Seconds)
• 0:21:12 – Global Influence of Mathematics (173 Seconds)
• 0:33:38 – Tragic End of Evariste Galois (55 Seconds)
• 0:38:29 – Otto Schmidt (165 Seconds)
• 0:42:52 – The Impact of Group Theory (108 Seconds)
• 0:51:47 – Studying Properties of Young-Baxter Equations (100 Seconds)
• 0:56:16 – Language and Mathematics Similarities and Differences (102 Seconds)
• 1:00:49 – Mathematics and Language Connection (117 Seconds)

Dr. Igor Subbotin

0:00:01 – Announcer

You are listening to the National University Podcast. 

0:00:10 – Kimberly King

Hello, I’m Kimberly King. Welcome to the National University Podcast, where we offer a holistic approach to student support, well-being and success- the whole human education. We put passion into practice by offering accessible, achievable higher education to lifelong learners. Today we are talking about the power of mathematics and, according to a recent article in the New York Times, learning mathematics is both crucial to the learning development of the 21st century students. So as not to be imposed upon learners too heavily. So, learning mathematics develops problem-solving skills which combine logic and reasoning in students as they grow. 

We’re going to be having a great conversation about the power of mathematics coming up on today’s show. On today’s episode, we’re talking about the power of mathematics, and joining us is National University’s Dr. Igor Subbotin, and he earned his PhD in mathematics at the Mathematics Institute of the National Academy of Sciences of Ukraine. Before joining National University, he taught mathematics at the most prestigious university in Ukraine, Kiev Polytech Institute. At National University, Dr. Subbotin regularly teaches different mathematics classes and supervises mathematics courses. Dr. Subbotin’s main area of research is algebra. His list of publications include more than 170 research articles in algebra published in major mathematics journals around the globe, and he has had the privilege to collaborate with several world-class mathematicians from different countries. He also authored more than 50 articles in mathematics education, dedicated mostly to the theoretical base of some topics of high school and college mathematics, and he’s published several books. We welcome you to the podcast, Dr. Subbotin. How are you? 

0:02:07 – Igor Subbotin

I’m fine. Thank you very much for inviting me. I’m really happy to be with you, thank you. 

0:02:13 – Kimberly King

Thank you. Why don’t you fill our audience in a little bit on your mission and your work before we get to today’s show topic? 

0:02:22 – Igor Subbotin

It’s very easy to talk about things that you love it. I love mathematics and felt a love in mathematics a long time ago. I continue loving it, the same kind of, let’s say, powers that used to be when I was very young and I love my students and the love to my students even grows. Comparison that I was young, because I got experience and understand people better. I love my university and I have been working for National University for 30 years already and university growth- I looked, I was part of the university development and growth. It was small at that time when I came and now it’s a big university with some traditions, prestige, some kind of place in the American higher education and I’m happy to work with this and I’m happy to continue development of the mathematics education in our country, in different countries, and Europe, that I participated in many different collaborations with many different scientists and promote some mathematics- new ideas and also disseminate these ideas, which is extremely important. Thank you for inviting me. 

0:03:51 – Kimberly King

Absolutely. I love more than anything, I can hear your passion for teaching and really helping your students understand the joy that you have for mathematics, and so today we’re talking about the power of mathematics. And so, Doctor, why is mathematics so useful? 

0:04:10 – Igor Subbotin

I will answer for this just bringing the citation from Joseph Louis LaGrange, one of the bright stars on the mathematics horizon. He was born in the middle of the 18th century, I mean, started to work in the middle of the 18th century in Turin, Italy, but he is a French mathematician actually, after all, and he was a key figure in many different mathematics development of that time- special calculus, differential equations. It was a time very in, and he said like that, mathematics as the queen and the servant of all sciences. Mathematics is a queen and the servant. 

I know that some other people say that mathematics, that science, became a science only if the science used mathematics. Start to use mathematics, it’s number one. And also I would like to repeat the word attributed to Galileo Galilei, who said that mathematics is a language in which God speaks to us. God could be just changed to nature, but the meaning is the same. Our nature, our God, will speak to us through the mathematics language, mathematics language. And why mathematics is so remarkably useful in every single human activity, not only in science, not only in physics, everywhere, everywhere. Number one, philosophically talking about everything. Every single event has some kind of qualities. How can we assess this quality? How we will talk about that? 

First of all, we try to measure this in some way- qualitative measurement- also appeals to quantitative measurement. There is no quality without quantity. There is no quantity without quality. There are two structures that inter-influence each other. It’s number one. Why mathematics has so much power and why it’s so useful? Because the main idea of mathematics is to strip off the second line, details, to look at the stems, not on the leaf and the stems ignoring by some details. When you start to study some physics or chemistry, some kind of events or what happened with them in this specific event, you just miss so many details, so many details- you don’t know to what you need to concentrate your attention. In this case, mathematics helps you to strip off of this mail by focusing on the most essential aspects. 

Mathematic enables a comprehensive understanding of complex natural processes. It distills vast amounts of information, stripping away irrelevant details to emphasize what truly matters, what truly matters. So mathematics is not a calculation. Mathematics is not only geometry. Mathematics is a way of thinking. This is exactly what we call critical thinking of the highest level of development. That’s why mathematics is so powerful. And this mathematics is powerful not only in science. It’s powerful in any kind of human activity. And I will tell you what is mathematics’ role in this human activity. This is the main thing. 

0:08:20 – Kimberly King

Well, I love that you explain mathematics as a language and again your passion comes through. In fact, last time I interviewed you I kept saying, as soon as we were through- I wish you were my mathematics professor, because you share such a passion and you make it easy for others to understand, and that is truly a gift. So thank you. Why is mathematics so universal? 

0:08:50 – Igor Subbotin

The idea of universality of mathematics- it belongs to its own structure. What is mathematics study? What is mathematics study? The main process in every single thing, event, and you know, sometime it always amazed me and not only me, maybe, it’s amazed many different people for sure, that when some science or some human activity thing face some needs of mathematical analysis, which means qualitative or quantitative analysis, the appropriate corresponding series already exists. You don’t have to create something new, it’s already there. What does it mean philosophically? For me, it means that this is some kind of answer for the very deep questions that everybody who is doing mathematics asks themselves. What we are doing? Creating new mathematical ideas? Or we are discovering this mathematics world, like a known country? So maybe all this idea exists already and we are just discovering them, like in physics, like in chemistry, like in any other thing, or we are creating these ideas. I believe it’s my opinion and not everybody shares this opinion that everything already exists. We are just not inventing, we are exploring these ideas. So what does it mean? So, for example, when we are talking we’ll talk today a lot about my favorite area of mathematic- algebra. This is not the same algebra that we are talking about during high school mathematics. No, it’s a totally different subject. I will talk about this today. 

But the power of algebra based on the idea of isomorphism. Isomorphism, what is that? It means if you have two different structures consisting of two of different subjects, objects of different objects, and you find out some kind of one-to-one relations between these two structures, in which it doesn’t matter what you’re doing in one subject, in one object, work with the same kind of result for the second object. It’s in the isomorphism. You can start to study one area and after that, all the rules that you will come to will work for another way. This is the power of mathematics. Simple examples- you can use the same kind of linear equation to describe many different things, many different things. In mechanic, in accounting, in, let’s say, the different disciplines that are directed to the, for example, structural things, some like concrete structures and so on. 

It’s a very simple example. Idea of isomorphism this is algebraic ideas that just came to our attention, I believe, not so long time ago, maybe 200 years ago, no more. But mathematics, mathematics use it for long period of time, many, many, many years, without understanding what is that. I believe that algebra, which is the most abstract subject in mathematics, could be a wonderful illustration how this idea works. The idea of isomorphism is crucial in explaining how mathematical concepts can be applied across diverse fields. For instance, in algebra, the same equation can be used to solve problem various areas, showcasing the universality of mathematical principles. So we will talk about this today. 

0:13:13 – Kimberly King

So interesting- yeah, go ahead. 

0:13:16 – Igor Subbotin

Let me add a little bit about your remark about the study of mathematics, how the teacher role is important in that. I would like to assure our future students, or some people, that we have right now at National University, that main idea for selecting faculty for mathematics department for classes, staffing them to the classes, is the idea how these people really feel about the subject, if they’re really motivated to bring their knowledge and their passion in the classroom and they really understand with whom they’re dealing with. Because it’s a very different approach in our study when we come to the class of the elementary teacher future elementary teacher or to some art designers, all of us are very passionate about the subject. All of us understand our role at the university and how to treat students in the right way. 

I believe not only me, it’s statistical knowledge that most of the hate to mathematics born in the elementary school classrooms, where some teachers hate it and don’t understand it enough. That’s why our mission, our mission- and we teach elementary teachers also- to bring the light of mathematics understanding to them, to build up the respect to the subject, respect to the teaching. And that’s why I believe that major, I would say almost all our students are successful, not because we are not keeping rigor. We are keeping high rigor in our classes, our classes. But we are doing our best not only to fill out students like a job but to light them as a torch in mathematics. Sorry for interruption, but it’s an important point that I would like to mention, answering for your remark.

0:15:38 – Kimberly King

I’m glad you did, because it is true that when we’re learning, I mean it’s almost like now you’re playing catch up to get these students to love and have that affair- a love affair- with mathematics and that understanding, and it really does need to start at a younger level, just so that you know we can continue to move forward and grow. So thank you for taking that moment out to explain that, because it really does truly show, and I think we’re doing a disservice, you know, for those teachers that are in place and either don’t have that love, that understanding, that passion, and then they’re, you know, not necessarily bringing up our kids, our children, to love it like you do. So it’s good, thank you. Can you discuss abstract algebra and how it’s stated and its applications? 

0:16:31 – Igor Subbotin

Most of our students who are not math majors, they will not study abstract algebra in the university course. They will just, I believe, will be starting studying, some of them, calculus. Some of them will study just college algebra. Some of them will study linear algebra at most, like computer science people. Abstract algebra, this is only for math majors and this is very interesting to trace the genesis of algebra, how it became totally different from other areas of mathematic language, developed language, develop understanding and what is the most important- at the end I will show this- how this may be one of the most abstract, without any, sometimes, visualization ability, subject became the most useful and most applied. It’s interesting. So if I will start about talking about algebra, you immediately just come to the original. 

Algebra can be found in the mathematics of ancient civilizations, particularly Babylonian people and ancient Greece, of course, with Euclid, with his famous book Elements. Do you know that the book Elements of Euclid was it’s about 2,300 years ago published? by Euclid, and Euclid is a very mystical figure in mathematics because there is not any portrait of Euclid that exists. Most of the mathematician portraits we have it came to us from the anthropology but not Euclid’s picture and according to his, let’s say his in quotation input, in mathematics it’s too much for one person to be so educated and so powerful. So there is some hypothesis that Euclid this is just, let’s say, like nickname for the group of mathematicians of that time they put together their knowledge in the group of elements, elements. It’s had in the group of elements. You now that the book elements is the second book by the amount of publications after Bible, only one book that was published more than Euclid. Why? Because during 2000 years, it has been the maybe only textbook for mathematics for our civilization, for 2000 years almost. So, Euclid, ancient Greece, so who started developing equation solving procedure and manipulating symbols to depict mathematical relationships. 

After that we jump to the golden age of Islamic, that next significant advancement which was made possible by the writing of academic-like Al-Khwarizmi. Al-Khwarizmi – listen, algorithm. Al-Khwarizmi- [laughs] it’s the same, it’s the same. Algorithm come from Al-Khwarizmi name. This is some golden age of the Islamic age, something 15th, 14th, 15th century, when this very famous author wrote the book. I will read the book- “al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābalah”- translating is going to be the Comprehensive Book on Calculation, Completion and Balancing- Al-Jabr. Al-Jabr, this is a book we named. It was brought to Europe by mathematicians from Middle Eastern countries. So what is interesting? Definitely, this book is not absolutely just created by people from Islamic countries. It was a lot of roots in China, a lot of roots come to India and so on. 

Let me remind you my favorite words from brilliant mathematician David Gilbert, who was one of the most prominent, if not most prominent mathematicians in the world in the end of the 19th, beginning of 20th century up to the middle of 20th century. He said that for mathematics, there is no boundary in culture and race. Mathematics considered the entire intellectual world as one country. If you take mathematics, you will not find any other science that would be so internationally developed, internationally developed. And now we have many different countries that work in mathematic development. It was maybe something like 19th century. 

The most influential was French mathematics. Before it was Newton, English, England mathematics. At the same time it was German mathematics. After that, again, German mathematics became prevalent. After that, Soviet Union mathematical school became a golden, when it was golden age. It’s the most powerful and most developed school. 

American mathematics. American mathematics became very, very influential and powerful, but it’s mostly after 40 years of the previous century, 20th century, and I will bring you many other examples like that. Every Chinese, Chinese school. Look at China now. How many prominent fields, medal holders and mathematicians we have in China. What a genius was born in India. What a great development. 

In my own experience and seeing how the Middle East developed mathematics, about 30, 40 years ago they didn’t talk about, for example, abstract algebra. Now they have a few, a few journals in the Middle East, especially in Iranian people. It doesn’t matter what kind of relation we have with Iran. Mathematics is unity, it’s all people work together and we develop the same subject and we work on the same field, which is extremely important. That’s why mathematics is so influential. Why we are doing it? This is the reason, because everybody needs it. Everybody needs it, not only for developing some kind of new technical idea, but also for understanding the world. 

So after this Islamic time, symbolic notation was developed by a mathematician during the Renaissance time, European mathematicians during the Renaissance time. What happens at that time? People can solve quadratic equations. Long time ago, linear equation was not a problem at all, but quadratic equation long time ago. But when people face equation of the power 3, power 3 with one variable, it was a big problem. Sooner or later it was solved. Also, what does it mean solving equation? It means to get a formula that will express the solution, the roots of the equation, expressing them through the main algebraic operation addition, subtraction, multiplication, division and so on, and radicals and so on, in one formulas through the coefficient, coefficient- number of this equation, using this equation. So for cube roots it could be huge formula. After that, for four roots, people were successful. They got it. 

After they got it, I believe that the creator of this was a mathematician who was nicknamed Tartaglia. Tartaglia mean people with some kind of empire of speaking, Tartaglia in Italy. This, it was a very strange person. He was very- He was not, he didn’t have really nice personality, he was, let’s say, some kind of gloomy and always not happy guy. But he, according to what I read, he was the person who solved the equation of the power of four. At the same time. Cardano, Federico Cardano, who is a physicist, great physician, great engineer and great mathematician of the time, very big person at the time, star number one in Italy, have heard that Tartaglia has some formula for solving equation of the force power, certain force power. 

Why it was so important? At that time, to get position to have some money for doing mathematics was a very difficult thing. It was a very difficult thing. It was very limited opportunity, for example, to be some kind of the mathematician in the court of some kind of prince or king or duke or something like that. So people just applying for this position, they’re supposed to go through the competition. Competition looks like that that the persons who would like to apply for the position, two weeks before the meeting, send to each other the list of the problems that they would like that their counterpart solved. A counterpart solved Okay, if you have a formula that nobody knows, you are a winner. It’s a big chance that you solve the problems that this person sent to you. But there is no chance that somebody will solve the problem if they don’t know the formula. So it was a huge privilege, it was a huge, huge benefit to having the formula. It was a big secret. Nobody knows. 

Cardano, who was a huge guy, a huge star at that time, came to this unknown guy, Tartaglia, and asked him tell me please about the formula I have heard you have it. Tell me please. And Tartaglia said okay, but don’t tell anybody. For Tartaglia it was a big, big how to say honor to meet a guy like that and he told him the formula. What Cardano did- from the point of view of nowadays, he did a very honest and great thing. He published a book and the end of the book published the appendix and he said this formula was given to me by my esteemed great colleagues Tartaglia. So he didn’t try to cheat, he didn’t try to get his Tartaglia let’s say, copyright, like we said right now. But from point of view of nowadays you can say it was a huge support to the people, to the Tartaglia from the big, big star. 

No, guys, that time it was totally different. Tartaglia lost his power to win in competition by publishing this formula. So, change time, change the vision, change the vision. So this time now the formula calls like usually called, usually called something that if you create in mathematics it’s always a different person name. Coming to the history, it’s Cardano formula, but definitely it was Tertaglia. In mathematics there are some kind of buyer rule that most of the invention in mathematics are called by the name of the people who didn’t do this. It’s even said like that Bayer rules, okay. 

So what happened after that? People started to study the equation of power of five. Next fifth power no success, no success. This way, that way, no success. Nobody can do this, anything. And until October 25, 1811, a brilliant, very unusual, mathematic star was born- Evariste Galois. Evariste Galois, French mathematician. He was born, again, in 1811. He created his main idea and write his main idea in written when he was 16 years old. 16 years old and he was killed when he was 20. Only 70 years after his death, Camille Jordan published his work in his book about matrices, and this is the beginning of modern algebra, beginning of modern algebra. 

Do you know what the Evariste Galois finds out? He finds out that any equation, polynomial equation with one variable, with a power greater than four, like five, six, 7, and so on, has no formula at all in general, cannot be found. But partial cases, please. To find some approximate solution, please, but not general. There is no, and never going to happen. It’s mathematical power. You see, he proves that it and never going to happen, it’s mathematical power. You see, he proves that it’s never going to happen that anybody with the hugest star in the world there is no formula in general, case like that. Moreover, he finds out when, in some partial cases, this formula exists and when not. What’s the condition for this. This is what. 16 years old how to say, teenager, 16 years old. 

He of course tried to find out the opinion of the people who was in power in mathematics that time. He sent his manuscript to Augustin-Louis Cauchy. If you will study calculus- I’m talking to our future audience- you will find Cauchy’s name every I mean tens, dozens of times in different theorems and calculus. He was a huge star at that time it was the first quarter maybe, of the 19th century right and he published a lot of different works. He was a genius and he is a genius. And Galois sent him a manuscript and after some time asked what do you think about that? And Augustin-Louis Cauchy said all right to him, I lost it. I don’t know if it was true or not. Some people said that after that, the publication of Augustin-Louis Cauchy has some influence on this paper. So it’s again, humans are humans everywhere, not only in the economics and history, but also in science, even such straight science like mathematics. Human is human. 

So what can we say about this situation? And Evariste Galois continued to promote his idea. He organized special seminars for people who wanted to come there. But he also participated in the revolutionary activity and he was very active in this, and the police decided to say too active, too much active, the police in French, and they sent to him the killer. He just sent him invitation to duel because of some woman. Sorry but it’s true and he just came to the duel and before the night of the duel he continued to work on mathematics and he was killed in the duel. It was a main idea. It was political kill, definitely, but he was 20. He was 20 at that time. So you know, teenagers create a new huge area of mathematics. 

It started with a name, like Galois theory and we studied Galois theory. But this is a partial idea. The idea was to study, not the numbers, not the equation- operations, operations. This is the power of abstract algebra. This is the power when the abstract algebra was born. Study operation. 

I will give you a very simple example that everybody will understand about operations and so on. You know that entire world knows chess game. In different countries we have different names, different names for the checkboard names. We have different names for the figures. We have different names for the different language, for the combinations that we consider. But we have great masters from different countries. They play the same game. They cannot speak common language, but they know the rules and the rules are the same rules. It’s operations, how to operate with this special figure in this special situation. This is a rule. So algebra, abstract algebra, they don’t deal with the equations, specifically. They don’t deal with the numbers. They can deal with the matrices, which are big tables. They can deal with the transformation of the space of the plane, doesn’t matter the idea how this object behaves under the operations. 

Under the operations. 

0:36:20 – Igor Subbotin

It’s a huge step. It became a very abstract subject, a very abstract subject and it’s very interesting to say the algebra started with the ideals of Galois and the idea of Galois leads us to the group theory. This group, the algebra subject, without operation group is the most possible. Group theory. But group theory was long time stay long time as a group theory of permutations. It’s a special object in algebra. Only in 1920s, 1920s, great mathematician, Otto Yulyevich Schmidt. Otto Schmidt, it’s a Soviet Union mathematician, Russian but definitely with the German roots. Schmidt, it’s German name. Otto Yulyevich Otto, also German name. It’s also another kind of brilliant guy and I’m his scientific grandson. 

0:37:26 – Kimberly King

Perfect yes. 

0:37:30 – Igor Subbotin

Why you will be amazed in a different way. In our department we have biologists, people who study biology. We have now a department called Mathematics and Natural Science. One of these professors is a prominent researcher in biology who uses a lot of statistics, and his dissertation also was supervised by the statistician mathematician statistician, because to study biology you need statistics, you need to watch. Okay, so what is interesting? When we together came four steps back, we will find out that our roots both of us, came to Carl Friedrich Gauss. I am grand-grand-grandson and he is grand-grand-grandson to Carl Friedrich Gauss. 

0:38:23 – Kimberly King

Wow, that is fascinating. My goodness Wow. 

0:38:27 – Igor Subbotin

The world is unique, it’s one. So Otto Yulyevich Schmidt, Otto Schmidt was not only mathematician, with the he, by the way, first wrote the book which called Abstract Group Theory. When this object, in the group object, elements of this set, absolutely abstract, doesn’t matter what the nature, only operation is important. Otto Schmitt was the one and is the one of the most famous geodesists about the science of the Earth, and he is well known also as a creator of the first scientific theory how our solar system was born. This is his name. Also, he was a guy who was the, who was the how to say director of the expedition to the North Pole on the ship Chelyuskin, and that time, in 1930s, it was the most like today, let’s say, a trip to the moon. It was the same kind of importance for the entire human race. And so, by the way, some of them article in group theory and algebra written in the Chelyuskin during this expedition, written in the Chelyuskin during this expedition and signed up like ships Chelyuskin, whereas it was written Chips Chelyuskin. So he was there in the North Pole and write the book, and write the book and write the article in mathematics, people like that is a brilliant our human civilization topics. Okay. So algebra became very abstract. Nobody expected that algebra became really, really applicable, right. So because I have been working in algebra for some time, I see it in my own eye when the group theory subject just transformed to the new abstract algebra. Of course I was not born in 1920s when it came, but in 1970s I see the most peak of development on the infinite group theory, and now I see that some other subject was developed like that. Let me continue with the history and I will tell you a very exciting thing about how algebra, so abstract, became so useful and became so applicable. 

Next step was German mathematicians, like we are supposed to mention David Hilbert and his school, and also the biggest star in mathematics for all times, Emmy Noether. Emmy Noether, this is not only you know, of course. Everybody knows about Sofia Kovalevskaya. Sofia Kovalevskaya or Gepardia Alexandriyevskaya, some other woman who brings their huge input in mathematics, but Emmy Noether is number one. She was a German mathematician. In 1938, she immigrated, like many other people, from Nazi Germany to the United States and I believe she was a professor in Bryn Maur College after her death. She developed the main idea of investigating some algebraic structure like rings, fields, groups and so on, the idea of chains. I’m not going to proceed with this too far, but she gave us the instruments, the tools to open these fields of investigation. Everybody up to now work on that, everybody up to now work on this and will continue, because this is only one, let’s say by now, useful tool to study infinite structure, algebraic structures. So it’s very important to mention that again. 

That algebra abstract algebra, I would like to underline this was created by many different people but having their definitely first step and the most important input by the genius Evariste Galois and in particular, group theory, has made great progress in the sphere of new ideas and theorems. Many different mathematicians brought their attention and now this is a very well-developed part of algebra that have their application in physics, in cosmology, in painting, in art, in crystallography, everywhere where we have symmetry, symmetry. You know everybody what now? What’s the tool to study symmetry, group theory. So it was created for solving equations, but find their application in any science that dealing with the symmetry, any kind of symmetry, geometrical symmetry or some other kind of symmetry, elementary particle symmetry, cosmology, and so on and so on. But I would like to finish with the development of how it was developed and I’m ready for your next question because I will continue this forever, definitely. 

0:44:26 – Kimberly King

It’s really fascinating, though, to hear the history and to hear all of the countries that have been involved with the very beginning, the establishment and the beginning of mathematics. So this is quite fascinating. We do have to take a quick break, doctor, if you don’t mind, and we’ll be right back with you in just a moment. Don’t go away and hold onto those thoughts, stay with us. You Thank you. And now back to our interview with National University’s math professor, Dr. Igor Subbotin, and we’re discussing the power of mathematics and, doctor, this has been so interesting, just hearing the history of it and how it is all the nations working together, as we were just discussing, without a particular agenda other than the love of mathematics. The answers, we say numbers don’t lie, and so, with everybody working together, it is universal. Can you talk about the bridge between high abstraction and realm? 

0:46:14 – Igor Subbotin

I would be happy to do it. I will give you some kind of examples that I faced myself lately. 

Okay, this is again for our future students. It’s very important that the people who will teach you the subject be active in this subject, be professional in this subject, not just read the book and explain book to you, but do something by their hands in order to develop and to bring very modest input, but input in the object. I will give you some interesting story about that that based on my own experience lately, I have a very old friend not old person, old friend to me which we are friends, we have our friendship, for you cannot believe more than 50 years and we keep our collaboration, maybe the same amount of years because we are from the same school. Our supervisor was a brilliant star who was one of the founders of Infinite Group Theory, Sergei Chernikov. He is a huge international star and we are proud to be his students. I was lucky to work under his supervision and in his seminar since 1967 to 1987 when he passed away. I was lucky and my friends that I’m talking about, I can give you his name. Somebody can go to Google to look and find out who is it. It’s Leonid Kurdachenko. It’s Ukrainian mathematician, very, very, very famous mathematician in our area, abstract algebra, distinguished professor, and so on and so on and my close friends. We worked together for many years and we both have our roots in working in abstract algebra, in group theory, and we’ve been witnesses and we are witnesses about the time when the group theory and pheningo theory was very powerful in new development field and so on. But lately we found out, by some different reasons, some people came to work in so-called braces theory. Braces, not the stomatological braces, not the braces that you use like parentheses, some kind of parentheses. Algebra brace is totally different. It’s algebraic structure, new algebraic structure. What is it about? Taking example from physical algebraic equation, we have behavior of particles and waves which make possible to predict what will happen next, to explain nature of phenomena. Again, this is about symmetry, for example, biological sign, complete genetic interaction, dynamics and so on. Algebra is resonant also to artistic work, but lately it’s happened like that. Let me give you the exact what I would like to say. The theory of braces, the theory of braces is very young, very young, it’s just right now. 

It has its roots in addressing the Young-Baxter equation, a fundamental concept with profound amplification in both pure mathematics and physics. It originated from the groundbreaking work of the Nobel Prize winner, physicist Young, of the Nobel Prize winner, physicist Young, in the realm of statistical mechanics and, independently, in the contribution of Baxter to the 8-vertex model. It came from the knot theory and came from the statistical mechanics, its quantum theory. This theory holds substantial significance across diverse domains such as knot theory, braid theory, operator theory, hoppe, algebra, quantum groups, Tremont Foyle and monodromy of the differential equation. This is a fundamental equation in mathematics and physics that arises in the study of central algebraic structure, arises the study of central algebraic structure and it came to our attention thanks to the works in some first time in 1960s. But about the theory itself is thanks to the work of other mathematician, is became popular since 2008,9. 

Many people just came to study this because it has huge application. What is it about? It’s about to find the properties of the solution of this equation. Solution of this equation is not numbers, it’s matrices. It’s matrices, different kinds of matrices, special subjects in algebra. We don’t have a general formula or a general approach to solving this equation. We don’t have it, but it’s very important to us not waiting until maybe we’ll never have it, who knows? 

I gave you today an example of a various Galois equation. There is no formula for the solution of the fifth power equation. There is no formula. We are doing this approximately. We have many methods to solve approximately, which is enough for us. For the Young-Baxter equation. We have some situation, but study the properties of solutions as very, very important for many different disciplines and the people start to study this kind of equation, study the properties of this equation, even though we don’t have these solutions. We don’t have solutions, but we study properties of the solution and they find out that this just could be studied with the approach of abstract algebra. The properties of this solution could be described with the help of the new algebraic structure, old algebraic structure groups, fields, rings and so on. It’s classical structures already. This is absolutely young, absolutely young and new. This is braces. Braces like a fusion of two groups together. It’s very difficult to describe on the fingers but it’s very important. 

Solution of the Young-Baxter equation, known as the Young-Baxter matrices or R matrices, have found numerous applications beyond their original context. People start to study this. My friend, Leonid Kordachenko, just tried to apply the ideas that we had developed in the group theory, not we. Saying we, I mean all mathematics community, not myself, separate, okay. And he was so kind, he involved me to this and we together started to work on some specific points of it me to this and we together start to work on some specific points of it and find out that, you know, the idea of group theory works there. Works there, not different results different, the same approach, the same approach, the same idea how to mine this, but totally different results. Totally different results. But its results are natural and ideas natural. So we start to work on this and let’s say we work with success and we are doing this for the last two years and we published already a few articles about that and we also developed some and delivered some talks in different conferences and was very welcome in the community of the people who work in this area and very famous and algebraic for there. Why? Because of applications. 

But what is interesting, even though this is absolutely new structure in algebra, not like we used to study and we will study in our algebra course in our university- New structure, absolutely new. The approaches that we use, the approaches work there in the same way. So the ideas work there. So what I would like to say- It exists. When physicists need it, mathematicians said welcome, we have it. We have it. We were just a little bit adjusted and the lock will be open. It’s the power of mathematics. That’s why we need to study that. That’s why everybody likes to study mathematics around the globe. That’s why there’s no difference to us, to our colleagues, whoever, wherever they live, whether it’s race, whether it’s nationality, whether it’s language, we don’t care. We are one community. We united the globe, we united the human nation together like nobody else. 

0:56:03 – Kimberly King

Well, I love that and I wish that we can continue to just be united as a nation and not get politics involved in everything. Mathematics, there is a universal answer and that’s just beautiful. And speaking of the language, how are language and mathematics alike and what are the differences? I know you’ve been talking a little bit about this, but I just I really do love that you’ve talked about the history of it in a universal manner, but talk about the reasons why mathematics and language are alike and different. 

0:56:36 – Igor Subbotin

Mathematics and language. You know, English is not my native language, as you may have already seen, and my third language that I use, and analyzing my experience in writing mathematics and analyzing my experience in writing mathematics in Russian, in Ukrainian, in English, I can find out some very interesting things of the cross influence of the language and mathematics. As I told you before, mathematics is a language, Mathematics is a language. So, in my my opinion, it looks like that we have a box which called mathematics. On the input, we have a regular language. We translate this language in mathematics language using symbols, place this in the box and forget about everything. Forget about the, what we are dealing with. We are just using the rules automatically, like in algebra, solving equation. We don’t care what the A, what the B, what the C. We have a formula. We substitute number to the formula, get the result, go back, output and translate the solution to the common language. This is how it works, right In reality. But it’s very interesting that English, in my opinion. 

I am not a polyglot. I don’t have too many languages in my, let’s say, possession, but I’m really good at Russian and I’m not bad at Ukrainian. I can express myself in English, but English is a very interesting language. It’s totally different than the language that I started to use before. English is a beautiful language because it’s close to mathematics nature. It’s a very straightforward language. Everything is structural. For example, a Russian can say something like this is a beautiful girl and this girl is beautiful. I can say beautiful, this girl in Russian. It will be the same meaning In English. No, in the order in the sentence. It’s very, very important. So it became English, close to mathematics. 

Also, I find out when I translate my articles. For example, I need to write an abstract. For some international journal, it’s going to be in two languages, for example Russian and English. I look in English. I have, let’s say, 75% of the amount of sentences written in Russian words. It’s shorter, straightforward. Also, in English we prefer something like short sentences. In Russia, for example, in Leo Tolstoy, War and Peace, you will find War and Peace. You will find something like two pages languages by the same Charles Dickens. I found out that the same in Charles Dickens’ writing huge sentences. But for mathematics, English- English maybe the most, in my experience, the most close to express their idea, knowledge, very easy to understand, very easy to write, mathematics, much easier to write in any other languages, it’s number one. So language and mathematics, while seemingly distant, share common futures and serve as conveyors of thoughts, ideas and concepts. 

Symbolic system, use words and structure to represent ideas, allowing for the exchange of information without requiring a deep understanding of the subject matter, which never happened in mathematics. Language, much more rich structure. You can explain something that you don’t understand for yourself. For example, I talked today about Young-Baxter equation. I don’t want to pretend that I understand this equation absolutely clearly like physicists no way. I look at this from one part, from the algebraic approach, and also I’m not far to be a full understanding of this. Okay, but I can express my opinion about, I can express my opinion about, I can express my approach and so on. 

It’s a language, mathematics. You cannot do it. It’s only one subject that always answers for the question why, why? And this is very important. That’s why I love mathematics. But definitely language and mathematics have a lot in common, because there is no mathematics without language and I believe that we cannot express any of our thoughts without language. Mathematics is a kind of shortness and compact. It’s some kind of observations that they make, that when you have an information, some piece of information and you just really study this. It became very small in your brain and take only one small cell. When you need to go back, you just open it up again in the big structure. Mathematics does the same in language so-called word problems. It’s interesting. 

1:02:13 – Kimberly King

It is so interesting also the way you have explained that English has mathematical you know, and when you compare it to Russian or Ukrainian, I mean I can’t even imagine. I have heard that English is one of the hardest languages to speak and to learn, which I don’t believe. That, because I think Russian would be just off the top. I can’t even imagine. So kudos to you for being so proficient and putting this all together. I think it’s so fascinating and I love interviewing you every time we have you on. So thank you for your time today. This has been wonderful that you’ve shared your knowledge today. This has been wonderful that you’ve shared your knowledge, and if you want more information, you can visit National University’s website. It’s nu.edu. Thank you, doctor, so very much for your time. 

1:03:01 – Igor Subbotin

Thank you, Kimberly. I really appreciate it. I’m always happy to meet with you and I was happy to work with you as a one team to promote my favorite subject mathematics. 

1:03:15 – Kimberly King

We need mathematicians. Thank you so much. 

1:03:17 – Igor Subbotin

Thank you, thank you. 

1:03:22 – Kimberly King

You’ve been listening to the National University Podcast. For updates on future or past guests, visit us at nu.edu. You can also follow us on social media. Thanks for listening.

Show Quotables

“Mathematics enables a comprehensive understanding of complex natural processes. It distills vast amounts of information, stripping away irrelevant details to emphasize what truly matters.” – Igor Subbotin, https://shorturl.at/jORW4

“Mathematics is a way of thinking. This is exactly what we call critical thinking of the highest level of development. That’s why mathematics is so powerful.” – Igor Subbotin, https://shorturl.at/jORW4

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Understanding Pre-Algebra This book teaches and develops the math concepts and critical thinking skills necessary for success in Algebra I and future mathematics courses at the high school level.  It was written with the premise that students cannot problem solve or take leaps of reasoning without understanding the concepts and elements that lead to discovery.  The author—with 35 years of experience teaching mathematics—is a firm believer that understanding leads to confidence and confidence gives students the resolve to succeed in higher level mathematics rather than fear it.   It is standards-based, but what makes it different from other pre-algebra books is that it organizes concepts in a logical fashion, stressing practice and critical thinking. It avoids the mistakes—found in many other math books—of trying to teach new concepts before students receive the prerequisite skills and practice necessary for success. The concepts are presented clearly and in connection to other concepts. Math vocabulary is very important to success in higher mathematics, so this book includes easy-to-follow explanations and a user-friendly glossary.

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Understanding Geometry The successful completion of this colorful 272-page book will prepare middle schoolers for high school geometry. It covers more than 50% of the concepts taught in high school geometry using a step-by-step approach and teaches the reasoning behind the properties taught in geometry–instead of merely asking them to memorize them. Students are also taught the basics of geometric proofs and coordinate geometry in a way middle school students can understand. Students who struggle with high school geometry usually have lower standardized test scores because it is a fundamental subject in high school standardized testing. A glossary of terms that every student should master is included. This book can be used as a classroom textbook in Grades 7, 8, or 9 (usually over a two-year period) or as a reference for high school students. This book covers more than the National Math Standards for middle school mathematics.

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NOTE:  It is our recommendation that students complete Understanding Pre-Algebra (see description above) before attempting Understanding Geometry .

Understanding Algebra I This is a one-year Algebra I course for Grades 7-9. Students who have a solid algebra background will have no trouble with the algebra problems from SAT and even the GRE.  This 384-page book highlights vocabulary and notation, and has examples from the history of math. What makes this book unique and different from other algebra textbooks is that it is built from the experiences of an award-winning algebra teacher with more than 30 years of teaching experience. Many textbooks are written by a committee of authors, and many of those authors have little experience teaching beginning algebra students in middle school or high school. Understanding Algebra I presents the most essential concepts and skills needed to fully understand and gain confidence in algebra in a step-by-step fashion, teaching students that algebra is generalized arithmetic. It helps students see the connection between mathematics that they already know and algebra, so that learning algebra becomes easier and less abstract. This book provides students with real strategies to succeed in solving word problems by using charts and translating strategies that guarantee success.

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Essential Algebra for Advanced High School and SAT

Discover Essential Algebra for Advanced High School and SAT , a 241-page math book in the esteemed Mathematical Reasoning series written by award-winning author and teacher with 30 years of expertise in secondary mathematics. This powerful resource teaches the ‘essential’ connection of arithmetic and geometric concepts with algebraic concepts. Without this understanding, students tend to memorize Algebra I problem-solving steps—which is sufficient to pass Algebra I—but leaves them unprepared for math courses beyond Algebra I and the SATs. Algebra, the essential language of all advanced mathematics, lies at the core of this book's teachings. By delving into the generalized arithmetic that underpins algebra, students develop a solid foundation in the rules governing number and fraction operations, including factors and multiples. This vital knowledge empowers students to move beyond mere memorization of Algebra I problem-solving steps and confidently tackle the complexities of math courses beyond Algebra I. Without the knowledge and skills taught in this book, students often struggle or even fail in advanced mathematics courses and on the SATs. Imagine a good high school student who sees a problem like 3•x•y•4 and hesitates to write 12xy due to uncertainty about the rules governing multiplication. Or not understanding how to add 2x to 1/4y to combine it into a single fraction. Or why –6 2 is different than (–6) 2 . It is easy to see that not having a strong understanding of the foundational rules of algebra can stop even the smartest students from succeeding in advanced high school math courses. Essential Algebra for Advanced High School and SAT serves as a companion to an Algebra I course or aids in post-Algebra I readiness. To ensure students’ long-term success in advanced math beyond Algebra I, this book teaches the following 'essential' mathematics skills and concepts:

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Mathematical Reasoning™ Supplements These supplemental books reinforce grade math concepts and skills by asking students to apply these skills and concepts to non-routine problems. Applying mathematical knowledge to new problems is the ultimate test of concept mastery and mathematical reasoning. These user-friendly, engaging books are made up of 50 theme-based collections of problems, conveniently grouped in self-contained, double-sided activity sheets that provide space for student work. Each collection contains relevant math facts at the end of the worksheet in case students need hints to solve the problems. Calculators are allowed on activity sets that have a calculator icon at the top of the front side of the set. Each activity set is accompanied by a single-sided answer sheet containing strategy tips and detailed solutions. Teachers and parents will appreciate the easy-to-understand, comprehensive solutions. These books are a wonderful enrichment tool, but also can be used to assess how well students have learned their grade level's math concepts.

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