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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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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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7 Strategies to Teach Conceptual Understanding in Math

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

Multiplication is repeated addition.

Keep, switch, flip. 

The butterfly method.

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

But my students couldn’t explain why these tips and tricks work. Sometimes they would struggle and get upset. This happened when they faced situations where the tricks didn't work or they forgot what to do.

That’s why math education has changed recently to focus on teaching students a deep understanding of concepts instead of relying on shortcuts.  Educators know that teaching children to deeply understand math leads to the development of problem-solvers and critical thinkers. 

But how can we stop focusing on teaching shortcuts and instead help students become real mathematicians?

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

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

Mathematics is a body of conceptual knowledge made up of interrelated concepts. It isn’t just a list of disconnected topics to check off a list as students move from grade to grade. Plan your school year carefully to avoid math pitfalls by following a structured scope and sequence.

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

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

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

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

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

One way to help students develop their critical thinking and problem-solving skills is to assign high-order math tasks in your classroom. Rich tasks help students think about what they already know and test out different methods until they identify one that works. In the process, your students gain skills and strategies that eliminate the need for tips and tricks.

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

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

I was amazed by the intelligent conversations happening in the room. Students were discussing cell division tables and explaining why explicit and recursive formulas worked!

3. Visual Representations for Better Retrieval

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

Visualizing math concepts helps students see patterns and make connections that they may not immediately understand from written or verbal explanations. And when they have a visual cue stored in their brain, it makes retrieving information much more manageable. 

For example, suppose a student can recall that a quadratic function looks like a parabola because they’ve interacted with graphs illustrating a pumpkin catapult or diving into a swimming pool. If that happens, they're more likely to understand and use the formula of a quadratic function in various situations.

4. Manipulatives and Hands-On Learning

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

And I promise those hands-on materials aren’t just for the younger kids. Your high schoolers won’t mind abandoning note-taking in favor of digging into algebra tiles! 

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

5. Connect Concepts Instead of Teaching Math Shortcuts

Teaching is all about making connections. In this case, we're talking about mathematical connections.

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

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

Watching my Geometry students make connections between circles and angle relationships and complete constructions using arcs was a game changer! They remembered more when they understood how concepts were connected and could use their knowledge in unexpected ways.

6. Help Your Students Make Real-World Connections

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

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

Your students will stay interested and learn concepts that can be applied in different situations. Here are some examples using real-world scenarios to model integer subtraction that could be used in a 7th-grade class.

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

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

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

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

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

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

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

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

• Karen Sloan
• Content Marketing Specialist
• Carnegie Learning

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

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• April 17, 2023

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

Karen Sloan, Math and Special Education Teacher of 16 Years

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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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Spirit of Mathematics Critical Thinking Skills (CTS)

S Syafril 1 , N R Aini 1 , Netriwati 1 , A Pahrudin 1 , N E Yaumas 1 and Engkizar 2

Published under licence by IOP Publishing Ltd Journal of Physics: Conference Series , Volume 1467 , Young Scholar Symposium on Science Education and Environment 2019 4-5 November 2019, Lampung, Indonesia Citation S Syafril et al 2020 J. Phys.: Conf. Ser. 1467 012069 DOI 10.1088/1742-6596/1467/1/012069

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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 learning. Based on the analysis of several theories and research findings from various countries in the world, it can be concluded that the mathematical critical-thinking skill is very important for students, too; (i) help rational thinking in making decisions to express an idea, (ii) dare to make conclusions with alternative logical thinking, and (iii) able to examine and disregard various complex problems in learning Mathematics. Indeed, mathematics learning does not occur, if the learning process has not demonstrated the spirit of developing mathematical critical thinking skills.

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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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Engaging Maths

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

• by cattard2017
• Posted on June 25, 2017

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

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

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

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

How and when do we embed critical and creative thinking?

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

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

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

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

Who should be engaging in critical and creative thinking?

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

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

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

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

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20 Math Critical Thinking Questions to Ask in Class Tomorrow

• November 20, 2023

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

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

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

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

Table of Contents

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

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

What skills do we actually want to teach our students?

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

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

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

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

When should I ask these math critical thinking questions?

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

Start small

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

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

Add critical thinking questions to word problems

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

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

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

When you want your students to defend their answers.

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

When you want your students to justify their opinions

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

When you want your students to think outside of the box

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

Let’s Recap:

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

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How Math Can Help Children Develop Critical Thinking Skills

• 1.1 Thinking Deeply About Questions
• 1.2 Breaking Down Problems
• 1.3 Developing Logical Thinking
• 1.4 Encouraging Patience and Persistence
• 2.1 1. Mathematician Posters
• 2.2 2. Critical Thinking Math Activities
• 2.3 3. Fraction Lessons
• 2.4 4. Logic Puzzles
• 2.5 5. Math Projects
• 2.6 6. Error Analysis
• 3 Conclusion
• 4.1 1: What is critical thinking for math?
• 4.2 2: Can math improve critical thinking skills?
• 4.3 3: What are some activities that promote critical thinking skills in math?
• 4.4 4: How can parents support critical thinking for math at home?
• 4.5 5: Does math help with critical thinking in other areas?
• 4.6 6: What are some real-world applications of critical thinking skills in math?

Did you know that math can do more than just teach your child how to add and subtract? It can also help children develop critical thinking skills, which are super important for solving problems and making good decisions in life. Let’s understand how math can help your child become a better thinker and some fun activities you can try at home.

How Math Helps Children Develop Critical Thinking Skills

Math isn’t just about numbers and equations; it’s a way to help kids think more deeply and critically. Here’s how:

Thinking Deeply About Questions

When children tackle math problems, they have to think carefully about what the question is asking. This helps them understand the problem fully before jumping to a solution. By practicing this, they learn to approach all kinds of problems with a thoughtful mindset.

Breaking Down Problems

Math teaches kids to break down big problems into smaller, more manageable parts. For example, solving a complex equation often involves handling simpler steps first. This skill of breaking down problems is crucial for critical thinking, as it makes complicated issues easier to handle.

Developing Logical Thinking

Logical thinking is a big part of math. Kids learn to follow a series of steps to reach a conclusion, which helps them think logically about other types of problems too. 

Encouraging Patience and Persistence

Solving math problems can be challenging, and it often requires patience and persistence. Kids learn that it’s okay to make mistakes and that they can learn from them. 

Fun Math Activities to Boost Critical Thinking

Here are some easy and enjoyable activities to help your children develop critical thinking skills through math:

1. Mathematician Posters

What They Are: These are free posters with fun math facts and problems.

How to Use Them: Print them out, laminate them if you can, and put them up in your child’s room or study area. They’ll remind your child of important math concepts and encourage them to think critically.

2. Critical Thinking Math Activities

Square of Numbers: Give your child some tricky math problems that involve finding the square of numbers. This helps them learn persistence and teamwork if they do it with friends.

Domino Challenge: Use dominoes to create math problems. This game helps children develop problem-solving and logical thinking skills.

Fifteen Cards: This is a card game where your child has to make combinations that add up to fifteen. It encourages them to consider all options and think strategically.

3. Fraction Lessons

Activity: Have your child write down their answers to fraction problems and explain why they think they are correct. This promotes debate and a deeper understanding of fractions.

Benefits: Discussing and visualizing fractions helps your child understand and think critically about abstract concepts.

4. Logic Puzzles

What They Are: These are math puzzles that require logical thinking to solve.

How to Use Them: Give your child puzzles that match their skill level. These puzzles are fun and challenging, helping them think critically while having fun.

5. Math Projects

What They Are: These projects connect math to real-world situations and involve group work.

How to Use Them: Pick projects that are interesting to your child and do them together. These projects help children see how math applies to everyday life and develop their critical thinking skills.

6. Error Analysis

What It Is: This activity involves looking at mistakes to understand why they happened and how to avoid them in the future.

How to Use It: When your child makes a mistake, help them analyze what went wrong. This helps them learn from their errors and think more critically about their approach to problems.

Math is not just about numbers and equations; it’s a fantastic way to help your child develop critical thinking skills. By doing fun math activities and using simple resources, you can help your child learn to think deeply, solve problems, and make better decisions. You can consider Mathema for math resources for your children to score higher on their next test!

1: What is critical thinking for math?

Critical thinking for math involves using logic and reasoning to solve problems. It means not just finding the answer but understanding why it’s the right answer and how to approach similar problems in the future.

2: Can math improve critical thinking skills?

Yes, math can improve critical thinking skills. By working through math problems, children learn to think deeply, consider multiple solutions, and justify their answers.

3: What are some activities that promote critical thinking skills in math?

Activities like logic puzzles, fraction lessons, and error analysis are great for promoting critical thinking skills in math. 

4: How can parents support critical thinking for math at home?

As a parent, you can support critical thinking for math at home by engaging your children in fun math activities. You can encourage them to explain their reasoning and provide resources like mathematician posters and logic puzzles to stimulate their thinking.

5: Does math help with critical thinking in other areas?

Yes, math helps with critical thinking in other areas like science, reading, and even everyday decision-making.

6: What are some real-world applications of critical thinking skills in math?

Real-world applications of critical thinking skills in math include budgeting, planning trips, cooking (measuring ingredients), and solving everyday problems that require logical thinking and analysis.

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Developing Critical Thinking Skills from Dispositions to Abilities: Mathematics Education from Early Childhood to High School

• Einav Aizikovitsh-Udi , Diana S. Cheng
• Published 24 March 2015
• Mathematics, Education
• Creative Education

149 Citations

Developing students critical thinking ability through lesson study.

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Cognitive Growth Learning Model to Improve the Students’ Critical Thinking Skills

The effect of a problem centered learning on student’s mathematical critical thinking, the relationship between prospective middle school mathematics teachers’ critical thinking skills and reflective thinking skills, student understanding of derivative: mathematics education in the senior high school, supporting activities for critical thinking skills development based on students' perspective, critical thinking skills based on mathematical dispositions in problem-based learning, the students’ mathematical critical thinking skill ability in solving mathematical problems, an innovative model to promote secondary students' critical thinking skills in algebra learning, developing critical thinking skills of students in mathematics learning, 32 references, the change in mathematics teachers' perceptions of critical thinking after 15 years of educational reform in jordan, higher order thinking skills and low-achieving students: are they mutually exclusive, the disposition of eleventh-grade science students toward critical thinking, infusing the teaching of critical and creative thinking into content instruction: a lesson design handbook for the elementary grades, ' needed : research in critical thinking, teaching the language of thinking., “you're going to want to find out which and prove it”: collective argumentation in a mathematics classroom, education for critical thinking, practical strategies for the teaching of thinking, critical thinking and subject specificity: clarification and needed research, related papers.

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Math enrichment for all: 3 ways to engage all learners in deep mathematical thinking

When I first started teaching, I thought of enrichment as the puzzles and activities that I had on hand for fast finishers. I used these frequently in math when I thought I needed something to challenge kids who clearly (or so I thought) understood the math. Both of those assumptions—that what I was giving those students was enrichment and that finishing quickly meant students understood math deeply—were among the many things that I misunderstood as a beginning teacher. So, what is a better model for enrichment in math?

Math enrichment for all

There are several issues with my original approach to enrichment. First, it provided students who might actually have been ready for more challenge with glorified busywork that didn’t deepen their understanding of math. Second, it created a delineation between regular (aka “boring”) math work and “fun” enrichment work. Finally—and most importantly—it reinforced the assumption that not all students were capable of deeper understanding of math or worthy of enrichment. If indeed there had been an activity of value, it would only have been offered to a select few students (and usually the same few), denying others the opportunity to explore math in a richer way. As Jennifer Piggott of the University of Cambridge’s NRICH Mathematics Project puts it, “Enrichment should pervade the curriculum as a whole and not simply be available to those who work fastest.”

Ironically, it wasn’t until I became an enrichment teacher that I was introduced to the concept of enrichment for all. My school had a two-tiered enrichment program. In addition to a traditional pull-out program, every class got multiple one-week enrichment sessions each year where I worked with the grade-level classroom teachers to create a unit that deepened on-grade content. The lessons were designed so that all students could participate in all activities but were purposefully open-ended enough to allow students to explore the content at different depths. In these whole-class sessions, I saw both high levels of engagement and an incredible depth of thinking by all students.

Setting things straight

Before digging further into enrichment, I want to clarify some terminology. First, let’s address the difference between “enrichment” and “acceleration.” Researchers Susanne Schnell and Susanne Prediger nicely distinguish the two concepts in their study “Mathematics enrichment for all—Noticing and enhancing mathematical potentials of underprivileged students as an issue of equity” : “Acceleration means learning mathematics in accelerated pace (mainly by taking special courses ahead of the normally scheduled year). Enrichment means to expand students’ experiences and skills by exposing them to rich learning processes.” They further distinguish two approaches to enrichment: enrichment by “broadening,” or the learning of extra topics or subjects, often via extracurricular programs, and enrichment by “deepening,” which focuses on greater depth and complexity with the current topic. I’m going to focus on enrichment by deepening.

Why math enrichment for all?

Numerous research studies have highlighted the fact that African American, Latino, and Native American students , English learners, students with disabilities , and students from lower socioeconomic backgrounds are significantly underrepresented in traditional gifted and talented programs. This can be attributed to multiple factors , ranging from biased or flawed entry criteria to lack of equal educational opportunities for all students. Couple this with inequitable access to on-grade instruction and high-quality instructional materials and you can see that some students face real barriers to accessing anything beyond low-level content.

NCTM’s position on Access and Equity in Mathematics Education calls out that “To increase opportunities to learn, educators at all levels must focus on ensuring that all students have access to high-quality instruction, challenging curriculum, innovative technology, exciting extracurricular offerings, and the differentiated supports and enrichment necessary to promote students’ success at continually advancing levels.” Schnell and Prediger propose that an enrichment for all approach is necessary to expose the potential of traditionally underserved students: “Only if the situation has a potential of becoming mathematically rich, then the student can show some potentials. And in longer-term perspectives: if the student experiences his or her mathematical potential in a mathematically rich learning situation, then the potential can become a stable characteristic of the student in the long run.”

Students who are perpetually underchallenged have little opportunity to showcase their potential. I also appreciate Schnell and Prediger’s use of the term “potential,” as it implies a more fluid and dynamic characteristic, which may appear situationally but can be nurtured and developed. By contrast, labels like “gifted” and “talented” tend to be approached as more static and “given by nature.”

Furthermore, studies have shown that access to enriching math content has a positive impact on students’ engagement and attitudes. One study examined the Secondary Mathematics Masterclass program in the UK, which is designed to “inspire and engage young people in the art and practice of mathematics.” Students in the program reported enjoying learning “through experimenting rather than just being told something.” Over 60% felt the program improved their attitude toward and confidence in math, as well as their mathematical ability. John Hattie’s Visible Learning project cites attitude toward content domain as a factor with the potential to accelerate learning. All students need access to opportunities that support such positive experiences in math.

Getting started with math enrichment for all

Now that you understand the why behind enrichment for all, it’s time to get down to the nitty gritty of how to provide it. In their paper, Schnell and Prediger propose seven design principles for fostering all students’ mathematics potential :

• Provide enrichment in whole-class settings
• Enrich and deepen topics related to the on-grade curriculum
• Utilize low-floor/high-ceiling problems to allow for differentiation and challenge
• Engage students with rich mathematical problems
• Use open-ended problems to give students experiences of autonomy and competence
• Value cognitively demanding processes over “perfect products”
• Support positive engagement through discussion of students’ ideas and thinking

Below are three specific ways to put some of these principles in action. The good news is you may already be doing some of these!

1. Support positive engagement through discussion of student’s ideas and thinking

Many college- and career-ready standards include practice standards requiring students to articulate their thinking and critique others’ reasoning. Thus, many teachers already include mathematical conversations as a regular part of their classroom. The importance of this cannot be understated. In his research on high-growth strategies , Chase Nordengren states that “Student conversation is the most concrete representation of students’ higher order thinking. By introducing student discourse early and often, high-growth teachers create opportunities for all students to engage in higher-order thinking around grade-level topics.”

But not all student conversations are alike. To be effective, the discourse should be focused on high-level questions and big topics rather than solely on procedural questions. Research has shown that high-level questions that prompt students to reflect on and consolidate their learning improve student performance. While students may start by explaining their approach to a problem, they can be prompted with high-level questions to see connections to previous problems and big mathematical ideas. Delving into big mathematical ideas supports the design principle of deepening and enriching on-grade topics.

When planning questions for a unit or lesson, think about the mathematical ideas that underlie the topic of the lesson and what you might ask to help students make these connections. For example, when discussing a problem about dividing fractions, students can be directed to discuss larger topics like the meaning of division, the relationship between division and multiplication, as well as fundamental fraction concepts. NWEA’s free Formative Conversation Starters provide a great model for using a single problem as a jumping-off point for a deeper discussion of big math ideas. For tips on how to implement these conversations, check out my colleague Kailey Rhodes’s post “Formative conversations and the pursuit of equity in math instruction: 4 light bulb moments.” Or read my colleague Ted Coe’s post on student discourse to learn more about the connection between high-level questions and strategies for high growth for all .

Open-ended, general questions like “How does that work?” “Is that always true?” and “What do you notice/wonder about…” also help promote deeper thinking over quick responses.

2. Engage students with rich mathematical problems

Problem-solving is a standard part of all math classes. Traditionally, this takes the form of routine word problems where students apply the skill learned in the previous lesson to a real-world context . Such problems rarely demand deep thinking of students.

In his TED Talk , math teacher Dan Meyer talks about how he revises the problems in his textbook to support “patient problem-solving,” where students must ask questions, rely on their intuition, build the problem themselves, and actively and iteratively make decisions. His Three-Act Tasks present students with limited information, usually in multimedia form, and a question to answer. Through discussion, students ask questions, generate ideas, determine what information they need, gather that information and then work on answering the original question, self-monitoring, and changing course as needed. They are actively engaging in mathematical thinking in a way that they don’t when solving a rote problem with a straightforward solution path.

Fermi problems are another example of problems that support creative thinking. Named after physicist Enrico Fermi, these are open-ended problems that push solvers to determine a solution path, make and test assumptions, and sometimes make reasonable estimates to solve. They also support a collaborative group approach and mathematical modeling. An example of a Fermi problem is, “How much water does your household use in a week? Can you answer this without looking at a water bill?” As with Three-Act Tasks, students must determine what information they need and determine a solution path, which they must monitor and change as needed.

3. Utilize low-floor, high-ceiling problems to allow for differentiation and challenge

Low-floor, high-ceiling tasks are designed to allow all students to participate and contribute to the conversation while giving those with deeper understanding something chewier to wrestle with. Or as described on NRICH’s site , “everyone can get started and everyone can get stuck.” Problems are structured so that there is more than one way to solve them, and there is room for students to explore different approaches and wrestle with bigger mathematical concepts.

Educator and researcher Marian Small has created a type of low-floor, high-ceiling task that she calls open questions. These types of questions allow for natural differentiation and, like the Formative Conversation Starters, also use a single question to explore big ideas in math through rich classroom discussions. Here’s an example of an open middle question: ____ is 4/5 of ____. This presentation allows all students entry to the problem. Every student can respond in some way, and the variety of responses can be leveraged to raise the level of the discussion for the entire class. The blanks can be filled with various types of numbers: whole numbers (4 is 4/5 of 5 or 72 is 4/5 of 90), fractions (12/25 is 4/5 of 6/10), or decimals (0.8 is 4/5 of 1). Students who show deeper understanding can easily be asked to stretch their thinking.

Small recommends circulating around the room to monitor student work. If you feel a student isn’t stretching themselves as much as they could, tell them that many people have similar answers and request that they find an answer with, say, greater numbers or a different type of number. All of this can lead to conversations about different types of numbers as well as the meaning of multiplication and scaling. Open questions like these help students understand that not all of math is about following a single path to an answer but, instead, can be about flexible thinking and creativity.

A note about technology

Technology has become an integral part of education. As with all educational tools, its uses should be carefully considered . While online programs can support math enrichment for all, care must be taken to ensure that they do not either become digital versions of the puzzles I gave to my fast finishers or claim to accelerate learning without building solid conceptual understanding. Online communication and collaboration tools can be a great way for students to model, map, and share their mathematical thinking with others. Check out “75 digital tools and apps teachers can use to support formative assessment in the classroom” to find some that might work for your class.

Whenever students are learning online, have them explicitly connect that work to whole-class content and teach them to think metacognitively about the material they are interacting with. Both strategies support higher-order thinking and retention of knowledge.

Changing minds

Shifting to an enrichment-for-all approach can unlock the mathematical potential of all students. To get started, think about how you can build on what you already do in your classroom to provide open-ended, rich explorations of math for all students. Feel like you need some more support? We’ve got you covered with the high-quality resources listed below.

• 3 Act Task File Cabinet . Educator Graham Fletcher has a host of resources on his site, including links to Three-Act Tasks for a wide array of grades organized by big ideas and standards.
• Dan Meyer’s Three-Act Math Tasks . This Google Sheet contains links to Three-Act Tasks, primarily for middle and high school, filterable by CCSS standard and mathematical practice.
• Formative Conversation Starters . This is our free resource for grades 2–8. Each grade-level document provides an overview of how to implement the conversations, problems to get the conversations started, questions, and possible student responses.
• Geogebra Open Middle Exercises . This site provides online, interactive problems similar to Marian Small’s open problems, organized by domain and grade band.
• NCTM Asking Questions and Promoting Discourse . This site provides a list of strategies and resources for supporting rich questions and conversations in your classroom. Links are provided to K–12 Notice and Wonder lesson plans designed to broaden student thinking and elicit conversations and creative thinking. A PowerPoint also provides tips for promoting discourse, including an overview of the five practices for orchestrating productive mathematical discussions .
• NCTM’s Game of the Year . Each year, NCTM creates a long-term, open-ended math challenge based around the numbers in the year. For 2024, the challenge is to use the digits in the year 2024, plus operational and grouping symbols, to write expressions representing the counting numbers 1 through 100.
• NCTM Notice and Wonder . This site contains a host of resources designed to support mathematical curiosity and engagement. Note that some resources require NCTM membership.
• NRICH . This site, developed by the mathematics faculty of the University of Cambridge, offers free curriculum-linked resources and challenging math problems plus low-floor, high-ceiling problems designed to engage students ages 3–18. There are teacher, student, and parent sections of the site, all of which contain problems and activities organized by grade ranges and content.
• One, Two…Infinity . Marian Small’s website lists her various publications related to enriching math activities and open questions. The presentations section provides copies of PowerPoints organized by grade band or mathematical topics, and there is also an open problem of the week. These presentations are loaded with free examples of open questions. You can also watch a free webinar where she discusses open problems on NCTM’s website .
• Openmiddle.com . This site contains a large number of printable K–12 open problems searchable by grade, domain, and Common Core standard. Student sheets can be printed in English, Spanish, and French.

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