The Power of Visualization in Mathematics

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I think the best way to describe teaching math is by overcoming a monster. I believe that math is often an underdog like a lot of the stories we know.

Once upon a time, I was going to be a math teacher. Because I love mathematics, I thought all my students would love math like me. However, when I started working with students from all different grade levels, I saw that they feared this monster called math. If I wanted to be a successful math teacher and make my students good at math, I had to figure out why my students see mathematics as a monster. There should be a way to help them overcome their fear and become more confident when dealing with mathematics.

There are all kinds of theories out there as to why students don’t like math and don’t do well in math. If you bring up the topic of math teaching and conversation, sometimes these fears break out, and, in some places, they call it math wars. What I do know is that we need to change the culture and the attitude around mathematics. These are the kind of messages that are out there for our students to see. However, they’re not the kind of words we want to see. If we ask students if they like mathematics, we can see that the attitudes change for students when they go from grade 3 to grade 6 to our not applied classrooms. So what do people think of when they think about mathematics?

Back in the 19th century, Bertrand Russell talks about his experience in math classes and says,

“The beginnings of Algebra I found far more difficult, perhaps as a result of bad teaching, I was made to learn by heart: ‘The square of the sum of two numbers is equal to the sum of their squares increased by twice their product.’ I had not the vaguest idea what this meant, and when I could not remember the words, my tutor threw the book at my head, which did not stimulate my intellect in any way.” The Autobiography of Bertrand Russell, p. 31

Then they reminded me of this poem by Charles Carl Sandburg, who says,

“Arithmetic is where the answer is right, and everything is nice, and you can look out of the window and see the blue sky — or the answer is wrong, and you have to start over and try again and see how it comes out this time.”

“Not enough evidence, God, not enough evidence.” Attributed to Russell in the New York Times article So God’s Really in the Details?

So for most people, math is about right and wrong answers. It’s an exact science, and if you think back to your math classrooms, the math that you did, such as math homework, math questions, and math worksheets, was about getting that correct answer. There wasn’t a lot of ambiguity. If we ask your students what they think about math, they’ll tell us it’s like remembering the rules, memorizing formulas, and the steps that they have to remember to get to an answer. Then students are often seen in classrooms to keep solving questions. Sometimes we even add in the timer, and we say how many questions you can do before the timer runs out. We’re using technology in our classrooms more and more, and so now we see students in front of our classes doing questions that look incredibly tedious. Then we add in a timer, and we say how many questions you can do before the timer runs out or the stars disappear.

Math isn’t just about being correct for teachers. It’s actually about doing it quickly too. Then we wonder why our students dislike mathematics.
Somewhere along the way, teachers started to ask students to show their work or explain their answers. What that ended up being is students started giving us solutions complicated.

They’re just neat and tidy rows. You’re not allowing any independent thinking. It still really rules evident. So all we’re doing now is taking correct answers one step further and asking them to have one right way of doing their math.

I work as a consultant with teachers, and we often look at students’ work together. I find it interesting that the teachers always go to the bottom of the page and see if the students got the question right or wrong. Sometimes they give the wrong answer for lack of understanding. I was observing the class where a teacher was posing this question to the students:

2,6, …………..

It was just 2 and 6, and you want to continue the pattern. The teacher was looking for 2, 6, 10, 14, 18, which is correct because you’re adding 4 each time. A student comes along and gives them to 6, 11, 17, and I explained to the teacher that that’s still correct because you’re adding 4, then 5, and then 6. Also, you can have 2,6,2,6, ……. or 6,3,7,4,8……….

So, this is a common mistake when teachers teach. They expect their students to assume that there’s one correct way of completing the pattern. However, the only way you can do that is if you give them the rule to start. So if you give them two numbers like 2 and 6, various answers are possible and an opportunity for students to defend their answers. Regardless of whether the answer is right or wrong, it is worthwhile for teachers to investigate with the students whether their solutions make sense.

After you get the answers, where we can consolidate and help with the mathematical understanding — but speaking of questions, what about “when will I ever use this?

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That is a question that plagues math teachers and math classes. We could say if you know algebra, then you can figure out how old Mary is. If you see that she is two years older than twice old age seven years ago because I’m sure that’s going to come up one time or another. I don’t think that students want to know how to use their newfound knowledge. I feel more likely that they had a misunderstanding about the math that they were learning, and they’re just secretly hoping that it’s going to be of no use.

Math anxiety seems to occur in students when they’re about grade five or six early adolescent years. There’s a lot of reasons for that. It could be about poor math performance, bad classroom experience, or parental influences. But the question that I have is, do adults experience math anxiety? Let’s see!

You’re ordering a pizza, and you know that a pizza with four toppings costs $13, and a pizza with seven toppings cost 16.75. I want to know how much it’s going to cost for you to buy pizza; that’s one of those deluxe ones that are twelve toppings. Or maybe you need to figure out how many cans of paint you need to paint a room with two windows and two doors, and you know that the can of paint is going to cover 32 and a half square meters of wall. Or maybe you’re out there shopping on the Black Friday weekend and trying to figure out how much you’re going to save.

So if you’re uncomfortable with mathematics, maybe you’re starting to feel some of those symptoms of math anxiety. Maybe your heart starts beating more quickly, or you’re feeling queasy. Then you start looking around desperately for help and start looking for a calculator, an app for math, or a guy who has the math gene.

Some math experts believe that math anxiety exists because when you begin to feel that you are not good at math, you just avoided doing any more math for the rest of your life. You don’t want to take any more math-related courses in high school or post-secondary studies. But we know that primary students are confident and are highly capable of doing mathematics. They don’t show any signs of anxiety. So what I think happens is when students are in junior and intermediate grades, they start to feel the fears of doing math. They’re so worried about getting those right answers. Those worries in their head divert their brainpower and working memory away from the math itself that the lack of self-esteem and the lack of confidence doesn’t allow their brain to function correctly.

We know that the working memory is housing the prefrontal cortex of the brain. The left side is of our brain responsible for the verbal tasks to write the spatial? When students are in pressure field situations, what happens is their verbal brainpower goes into overdrive. It becomes more difficult for them to access those oral resources, which is sometimes why we see students struggle with math word problems.

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We also know that children that have math anxiety show increased activity in the fear center. Still, it’s accompanied by a decrease in the numerical information processing regions of the brain. So a math-anxious student actually will have difficulty reasoning through a math problem. What do teachers do when students exhibit these signs of math anxiety or struggling with math?

Well, we want to help them, and so we’re going to intensify what they don’t know, and you’re going to make them do it over and over again. Are we going to break it down at these little small steps and hope it helps them? Or better yet, math is just dull. We’re going to make it more exciting, and we’re going to put it into these real-life contexts and make it relevant for them no matter how contrived that is. But remember, a math problem is still a math problem, even if it’s about baseball.

When I started teaching, a friend of mine gave me a book, and it was called Ish. In this book, there’s a boy named Ramon, and Ramon loved to draw, but he had a brother that was mean and told that his pictures weren’t very accurate. He became discouraged, and he stopped drawing. Then his sister came along, and she put the pictures up on the wall and said, “I liked them because that vase is a vas-ish.” So she has a new way of looking at his drawings. He restores his confidence, and he starts to draw again.

So I began to wonder if maybe we should be teaching more “mathish”. So I’m planning to share some strategies that allow teachers to meet most of their students’ needs in their classrooms at the same time. Mathematics is about giving students a good question and an enjoyable task. That task has to be inclusive, which means that different classroom students can use different strategies and other ways to approach it. It also means that it’s inclusive and it’s accessible to all students. That means that students at various stages of development will benefit and grow from doing that task. What that means is that students in the classroom will be part of the learning conversation. It also means that they will be valued and essential members of that learning community, which means they will gain confidence during perseverance with their math tasks. We’re going to see improvement in our students.

So let’s take a look at this question: 1/2 + 4/5. That is a question that might have been given to students that are doing a class on fractions. But what if I’d say, “Choose any two fractions you want with different denominators and that their sum is going to be just a little bit more than one. Then tell me what that sum is and explain to me how you did it.”

As you can see, there are no specific numbers here. Students can choose whatever fractions are comfortable with to do this problem. Or instead of saying, “Use this formula and tell me the area of a circle.” I could say, “One of the measurements of a circle is six units. Tell me something about at least one other measurement of the circle.” Or, what if I give them a graph and say, “Tell me a story about this graph, imagine the possibilities.” Or I say, “The answer is five, then what could the question have been?” Well, some students might tell me it’s fingers on their one hand. Some of them will be thinking about operations and say, “3+2” or “5–0.” They may probably think and say the atomic number of boron.

The point here is that a number five or any number can be represented in many different ways. But the way you describe that number tells you something about the number like 5 follows 1 2 3 4 when you’re counting.
In algebra, we always ask students to solve for “x,” but what if I say to give them four equations and to ask them which two of them are most alike and why. Maybe they’re just going to choose equations, or they will make a table of value, or they’ll look at the graph.

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The big idea here is that when you compare mathematical relationships, they will behave in similar ways. We know that visual representations are vital in teaching and learning. Mathematics and children from a very young age make sense of the world by looking at picture books. Why can’t we show a picture to students and help them make sense of the math they’re learning?

Well, math is no longer black and white anymore. It’s not about the kids in the class who get it and those who don’t. More students are likely to try these questions and not shut down. We’re now turning those passive struggling learners into active learners in their classrooms.

More like other subjects, we’re allowing students to express their point of view, have their interpretations, and be able to share their solutions. These approaches enable teachers to set up what we call a math talk learning community, which we know is vital for students to understand math concepts.

These examples also allow teachers to create an environment where students are building each other’s ideas. It’s no longer just about recalling facts, but they’re responding to open and rich questions. We have teachers out there who are working with these strategies. The students in their classroom tell us that they understand math better. They’re more confident. It makes more sense that math is better when you can visualize it than trying to picture numbers in your head. That math is more straightforward and more fun.

We also know that students from these classrooms are showing increases in student achievement. So we want students to like mathematics and also be more confident and persevere with problem-solving. We can’t be satisfied with graduating a generation of high school students that are quantitatively literate. We want our students to be outstanding in mathematics. We want them to be our future innovators in science, technology, engineering, and mathematics.

So what good is it for people to walk around with vague memories of geometric figures and formulas in their head and clear memories of how they hated it or how they were afraid of it? Would it be better if people remembered their math classes as inspiring, where they could be creative and critical thinkers? There are likely several different factors, a combination of factors that determines successful students in math today, including their parents, teachers, peers, and their attitudes towards learning. But what we can do is we can make a great start by changing the culture around mathematics. So that being bad at math is socially unacceptable. We also want to provide our students with opportunities to answer questions that pique their curiosities accessible to all of them and allow them to think about mathematics and talk about mathematics. If you want more innovation in our classrooms and fewer math anxieties in our students, we need to move beyond the idea that we only focus on optimizing what we know is the right answer.

Instead, we want to encourage discovery and creativity in places where we’re not so sure about the outcome but be willing to learn with our students along the way.

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