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## How To Encourage Critical Thinking in Math

By Mary Montero

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

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

## What Does Critical Thinking in Math Look Like?

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

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

## Mathematician Posters

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

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

A Marvelous Mathematician:

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

## Critical Thinking Math Activities

Here are a few of my favorite critical thinking activities.

## Square Of Numbers

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

This square of numbers challenge was a hit!

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

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

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

## Fifteen Cards

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

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

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

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

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

## Logic Puzzles

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

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

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

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

## Math Projects

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

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

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

## Error Analysis

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

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

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

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

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

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

## Related Critical Thinking Posts

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

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

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

## Mary Montero

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

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## One Comment

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

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## 20 Math Critical Thinking Questions to Ask in Class Tomorrow

- November 20, 2023

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

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

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

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

Table of Contents

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

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

## What skills do we actually want to teach our students?

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

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

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

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

## When should I ask these math critical thinking questions?

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

## Start small

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

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

## Add critical thinking questions to word problems

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

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

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

When you want your students to defend their answers.

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

## When you want your students to justify their opinions

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

## When you want your students to think outside of the box

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

## Let’s Recap:

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

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

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

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

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## 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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- Math & Science

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

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

## What is Problem-Solving?

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

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

## How to Develop Critical Thinking Skills in Math

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

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

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

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

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

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

## Making Math Fun and Relevant

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

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

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

## The Underlying Beauty of Mathematics

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

## Why Mathematics is the Ideal Playground for Problem-Solving

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

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

## Enhancing the Learning Environment

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

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

## Incorporating Technology

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

## More than Numbers

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

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

## FAQ: Mathematics and Critical Thinking

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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## 5 Ways to Stop Thinking for Your Students

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

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

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

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

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

## 5 Ways to Get Your Students to Think

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

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

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

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

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

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

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

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

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

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

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

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## Stanford Online

Introduction to mathematical thinking.

HSTAR-Y0001

Stanford Graduate School of Education

The goal of the course is to help you develop a valuable mental ability – a powerful way of thinking that our ancestors have developed over three thousand years.

Mathematical thinking is not the same as doing mathematics – at least not as mathematics is typically presented in our school system. School math typically focuses on learning procedures to solve highly stereotyped problems. Professional mathematicians think a certain way to solve real problems, problems that can arise from the everyday world, or from science, or from within mathematics itself. The key to success in school math is to learn to think inside-the-box. In contrast, a key feature of mathematical thinking is thinking outside-the-box – a valuable ability in today's world. This course helps to develop that crucial way of thinking.

The course is offered in two versions. The eight-week-long Basic Course is designed for people who want to develop or improve mathematics-based, analytic thinking for professional or general life purposes. The ten-week-long Extended Course is aimed primarily at first-year students at college or university who are thinking of majoring in mathematics or a mathematically-dependent subject, or high school seniors who have such a college career in mind. The final two weeks are more intensive and require more mathematical background than the Basic Course. There is no need to make a formal election between the two. Simply skip or drop out of the final two weeks if you decide you want to complete only the Basic Course.

Subtitles for all video lectures available in: Portuguese (provided by The Lemann Foundation ), English

## Course Syllabus

Instructor's welcome and introduction

- Introductory material
- Analysis of language – the logical combinators
- Analysis of language – implication
- Analysis of language – equivalence
- Analysis of language – quantifiers
- Working with quantifiers
- Proofs
- Proofs involving quantifiers
- Elements of number theory
- Beginning real analysis

## Recommended Background

High school mathematics. Specific requirements are familiarity with elementary symbolic algebra, the concept of a number system (in particular, the characteristics of, and distinctions between, the natural numbers, the integers, the rational numbers, and the real numbers), and some elementary set theory (including inequalities and intervals of the real line). Students whose familiarity with these topics is somewhat rusty typically find that with a little extra effort they can pick up what is required along the way. The only heavy use of these topics is in the (optional) final two weeks of the Extended Course.

A good way to assess if your basic school background is adequate (even if currently rusty) is to glance at the topics in the book Adding It Up: Helping Children Learn Mathematics (free download), published by the US National Academies Press in 2001. Though aimed at K-8 mathematics teachers and teacher educators, it provides an excellent coverage of what constitutes a good basic mathematics education for life in the Twenty-First Century (which was the National Academies' aim in producing it).

Dr Keith Devlin, Co-founder and Executive Director H-STAR Institute

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## Mathematical Literacy and Critical Thinking

- First Online: 29 April 2020

## Cite this chapter

- Estela Rojas 2 &
- Nadia Benakli 2

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The development of mathematical literacy enables students to become skilled critical thinkers and problem-solvers who have a better understanding of the world they live in. However, very often students are unable to understand the mathematical principles and apply them to real-life situations. In many college mathematics classrooms, the lessons focus only on abstract concepts and routine exercises. Mathematics teachers need to transform the way they present and deliver the concepts, which should be contextualized in real-life applications, in order to motivate the students and enable them to acquire the necessary skills to understand and utilize the mathematical language. Reading is essential to access the mathematical language, but many beginning college students lack the literacy skills to navigate the abstract concepts to acquire a deeper understanding of how mathematics works. Reading in mathematics involves not just literal and linear comprehension; the process requires a broad range of thinking and reasoning skills. In each stage of the reading process, students need to engage themselves in understanding words, symbols, and concepts; analyze problems; and apply content knowledge and mathematical models to solve problems.

Since the application of the math content requires both general and specialized vocabulary knowledge, student success in math courses requires the mastery of general and discipline-specific literacy skills. The lack of these skills generates obstacles for students to learn math effectively. This chapter discusses the development of observation, generating questions, communication, listening skills, implementation of vocabulary strategies, metacognitive skills, cooperative learning, and emotional intelligence to develop students’ disciplinary literacy. These skills are fundamental to acquire a solid critical thinking process.

…Socrates: And it won’t be as a result of any teaching that he’ll have become knowledgeable: he’ll just have been asked questions, and he’ll recover the knowledge by himself, from within himself . —Meno dialogue by Plato

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Rojas, E., Benakli, N. (2020). Mathematical Literacy and Critical Thinking. In: But, J. (eds) Teaching College-Level Disciplinary Literacy. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-030-39804-0_8

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

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

## I. Mathematical Problem-Solving

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

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

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

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

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

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

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

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

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

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

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

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

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

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

## Lesson Plan Outline

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

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

## Developing Math Problem-Solving Skills

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

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

## II. Mathematical Modeling

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

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

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

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

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

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

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

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

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

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

## Mathematical Modeling and Critical Thinking

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

## Teaching Mathematical Modeling: Further Key Factors

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

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

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

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

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

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

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

To download the PDF of the Teachers’ Guide

(please click here)

## Sources and Resources

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

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

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

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

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

## Privacy Overview

## China nurse saves life of 820-gram baby in ‘critical danger’ born in toilet during flight

- Pregnant mother with 4-year-old goes into labour at 25 weeks, newborn cannot breathe inside fetal membrane

A nurse in China has won massive plaudits for her calm, swift actions as she saved the life of a premature baby during a flight.

The mother was only 25-week pregnant when she gave birth in the toilet of a Southern Airlines plane from Haikou, Hainan province in the south of the country to Beijing on August 3, reported state broadcaster CCTV.

The woman was travelling with her four-year-old daughter to meet her husband in Beijing.

Chen Shanshan, a nurse who works in the neonatal department at Hainan Provincial People’s Hospital, responded quickly after flight attendants asked around the aircraft for emergency medical aid for the newly-born baby.

Chen said she saw the mother, surnamed Zhang, holding a palm-sized infant in her hand, and the baby was still wrapped in the fetal membrane.

Assisted by two doctors from different departments of the same hospital, Chen, wearing gloves, tore off the fetal membrane so the baby could breathe.

They soon realised the newborn’s entire body looked far too pale, it was not crying or breathing and they could not feel a pulse. The baby did not respond to stimulation.

“This little weak life is in critical danger. Every single second is vital to her,” Chen said as she performed emergency CPR on the tiny girl.

She asked cabin crew for a warm water bag to keep the child’s temperature stable, which is especially vital for premature babies who are at the risk of hematosepsis and even death.

Chen and the two doctors let out a sigh of relief as the baby’s breath and heartbeat stabilised.

The pilot made an emergency landing in Changsha, central Hunan province, southern China for the safety of the infant and her mother.

Chen had kept up the chest compressions for 90 minutes until the baby was sent to a hospital in Changsha.

“It was only as I saw the baby taken into the rescue room that I realised my arms were numb,” she said.

Zhang’s husband hurried to Changsha hospital when he heard about the birth of his younger daughter.

He said the infant weighed a mere 820 grams, gaining 50 grams during her two-week stay in hospital.

“Thank you for being there at the critical moment. We will tell our kid and she will remember you forever,” the father said in a video clip sent to Chen.

He said he would thank the nurse in person after the baby was discharged from hospital.

Chen said that she was not thinking about much at the time, she was so focused on rescuing the baby. She said the cabin crew and the two doctors also deserve thanks.

The story has triggered an outpouring of compliments for Chen on mainland social media.

## More From Forbes

What is an ai-first mindset.

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Developing an AI-first mindset

As AI becomes an increasing part of our daily lives, people are starting to change the way they perform their daily tasks. The idea of incorporating AI into everyday activities is gradually becoming less of a thing people have to do with careful forethought to something that is being done increasingly with default afterthought.

Of course, everybody's using generative AI tools to write documents, emails, business reports, homework, and help with work and tasks of many types. Similarly, image generation is now built into so many tools that it’s becoming a default way of doing things, from creating presentations to images for marketing materials. People are realizing that AI tools can really help speed things up, make things more efficient, and provide that extra level of augmented guidance that improves overall output.

## How AI Is Changing People’s Behavior

AI is changing the way people look for information online. In the past, and indeed still the present for most, when you wanted some information, you would just go onto your phone, laptop, or desktop machine and use your favorite search engine with a typed search to your query. You’d then find the answer after scrolling through the many entries listed and hopping across multiple websites. But now that conversational Large Language Models (LLMs) are accessible and easy to use, people are just asking their LLM for direct answers. This is changing people's behaviors in terms of how they seek answers to their needs. And as it changes people's behaviors, it's really changing people's mindsets.

This is the idea of the AI-first mindset. As AI increasingly provides people with advantages, people are starting to act differently with AI so they can incorporate those advantages into their everyday lives. At a high level, an AI first mindset represents an intentional approach to integrating AI into all aspects of your life, both personal and professional. The AI-first mindset involves prioritizing and using AI as your primary interaction for things like decision making, innovation, communication, creativity, and problem solving.

An individual with an AI first mindset not only embraces AI technologies, but also knows how to appropriately use them, what they're best suited for, and they continually seek out opportunities to leverage AI to enhance and optimize their existing skill sets. The idea of the AI-first mindset embraces the idea of augmented intelligence, in which you are not using AI to replace what you're doing, but rather to help do your job better.

## Floyd Mayweather Jr. Vs. John Gotti III Results: Winner And Reaction

Apple iphone 16 release date: new report hones in on precise date, it ‘ends now’—donald trump reveals surprise assault on ‘crooked’ wall street, what does it mean to have an ai-first mindset.

One of the ways that you know that someone is embracing an AI first mindset is if you observe that their home screen or default front page of their browser or laptop is a conversational chat interface to an LLM and not, say, a search engine or some default portal or informational home page. The first thing that these users see when they open up their laptop or they open up a browser is an LLM, which suggests a shift in the way they're thinking. These individuals consider starting their interactions first with an AI conversation rather than a search or a generic informational page. That small change is potent in that they're integrating AI into their daily life and work and thinking to use AI first before anything else.

This idea of putting AI as your default interaction helps you realize very quickly what AI is good for because you're using it all the time. You could see when AI is working well, and you could see when AI is not working well because it's part of your daily life. People who start their interactions with an AI conversation will know inherently where AI will them up, augment, or enhance something they’re doing. People with an AI-first mindset will inherently know to use AI systems for complex tasks where AI performs well, such as working with large data sets, summarizing or analyzing information quickly, or interactively diving deeper into data and content.

Those with an AI-first mindset also inherently will know when AI systems are not good for a task. They know that LLMs are not super smart entities, they're just tools. They also know that garbage in is garbage out when it comes to LLMs. This means people with an AI-first mindset will become much better at working with data going into and coming out of AI systems. That also drives people to be more data driven because then they can feel more confident about the outputs of the AI systems they are increasingly depending on.

## An AI-First Mindset Means a Growth Mindset

Since AI is constantly evolving, and the capabilities of the tools and sophistication of users grows every day, a growth mindset becomes increasingly important. With a growth mindset, it’s not that you don’t necessarily know something or can’t do something, but rather that you don’t know it yet or can’t do it yet. With AI, the expectation will be that you can know or do something in the future as long as you keep improving and iterating your skills. This forms the idea of being a life-long learner with an emphasis on being data literate and improving your prompt engineering skills.

The interesting irony is that the more we work with technical AI systems, the more that interpersonal, human, “soft” skills become more important. The more you use AI, the more you have to be better at the things that AI is not good at such as creativity, problem solving, critical thinking, collaborating with others and communication because AI is going to do the other things well. So being in that AI-first mindset means that you have to master these soft skills. If you're not good at the soft skills, you won't be able to truly gain all the advantages from AI systems because you'll just be cutting and pasting things from AI systems, or simply not leveraging them to their fullest potential.

If you want to have an AI first mindset then you need to interact and iterate with AI systems in a way that helps you develop that reflex for AI so that you are inherently using AI in a highly valuable way while knowing where AI provides real benefit, and where it is not a good fit.

Listen more on this topic in a recent AI Today podcast episode recorded on this subject.

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## IMAGES

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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 ...

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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 ...

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In mathematics, creative thinking occurs when students generalise. Generalising involves identifying common properties or patterns across more than one case and communicating a rule (conjecture) to describe the common property, pattern or relationship. In order to generalise students need to first analyse the problem to notice things that are ...

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1. Answer questions with a refocus on the students' point of view. Liljedahl found in his research that students ask three types of questions: " (1) proximity questions—asked when the teacher is close; (2) stop thinking questions—most often of the form 'is this right' or 'will this be on the test'; and (3) keep thinking ...

Critical thinking in mathematics education. The classroom dynamics and atmosphere determine students' interaction and the kind of thinking and learning that takes place. Engaging students in argumentation in the mathematics classroom means going beyond recalling prescribed procedures and providing the right answers.

The mathematical critical-thinking skill is a process of thinking systematically to develop logical and critical thinking on mathematical problems, which characterize and demand to learn in the 21st century. This conceptual paper aims to analyze the spirit of critical thinking skill, and various approaches that can be applied in mathematics ...

Critical Thinking in Math: A Focus on Mathematical Reasoning Competencies. TC²'s approach to math embraces the idea that sustained quality mathematical thinking, or reasoning, is the key to the success of current and future generations of math students. What Is Mathematical Reasoning?

Professional mathematicians think a certain way to solve real problems, problems that can arise from the everyday world, or from science, or from within mathematics itself. The key to success in school math is to learn to think inside-the-box. In contrast, a key feature of mathematical thinking is thinking outside-the-box - a valuable ability ...

The development of mathematical literacy enables students to become skilled critical thinkers and problem-solvers who have a better understanding of the world they live in. However, very often students are unable to understand the mathematical principles and apply them to real-life situations. In many college mathematics classrooms, the lessons ...

Mathematical thinking is not the same as doing mathematics - at least not as mathematics is typically presented in our school system. School math typically focuses on learning procedures to solve highly stereotyped problems. Professional mathematicians think a certain way to solve real problems, problems that can arise from the everyday world ...

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The developed critical thinking skills also affect student's performance differently, and high-achieving students can develop critical thinking skills in mathematics more easily than low-achieving students [54 - 57]. There are different components in measuring critical thinking skills in mathematics among high school students.

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

A study by Indonesian researchers published in University of Nigeria, Nsukka, UNN, scholar's Scopus indexed journal, Ianna Journal of Interdisciplinary Studies, shows that using Artificial ...

A pregnant woman went into labour early on a domestic flight in China and her premature baby, who needed CPR, was lucky a quick-thinking nurse was on board to get her out of danger.

The more you use AI, the more you have to be better at the things that AI is not good at such as creativity, problem solving, critical thinking, collaborating with others and communication because ...