School systems are killing the art of mathematics

Have you ever claimed that you are “not a math person” or heard someone say it about themselves? This phrase is utter nonsense in my opinion and here’s why. Stating that you are “not a math person” is just about the same as saying that you are not a person at all. Math is an inherent part of our imagination and our nature. I trace a large part of this “not a math person” mentality back to the devastating structures and habits of math classrooms. In this article, I will often reference one of my favorite books, A Mathematician’s Lament by Paul Lockhart. I challenge you to open your mind towards the meaning of mathematics and encourage you to explore the universe of it yourself.

I’ve often heard mathematicians describe math as poetry. It’s true! The feeling of flow and struggle, the drawing and exploration, the painfully rewarding feeling of finally making sense of it. This is all part of the beauty of the subject. Math is playful. It is the art of discovery.

“If you deny students the opportunity to engage in this activity — to pose their own problems, to make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs — you deny them mathematics itself.” — Paul Lockhart

I feel strongly that the reason so many students hate math originates from school systems that force students to prioritize memorization and exams over learning. I went through these systems my whole life and it wasn’t until college that I was given space to explore the subject.

I also recognized the problem when I noticed the difference between how children view math versus adults. When I began teaching young kids math, it was incredibly refreshing to watch them ask questions out of curiosity and see their eyes light up with discovery, as opposed to seeing high school students staring at textbooks, drained of energy. This is not necessarily their fault. I think it is the education systems that children grow up within that drain them of their curiosity. So often, students are bombarded with lists of formulas to memorize with no context or space for exploration. 😵‍💫 Unfortunately, schools are obsessed with drilling these formulas, while ignoring the origins of them. We need to replace textbooks with real mathematical literature like Change is the Only Constant by Ben Orlin and Shape by Jordan Ellenberg. And we need to change the mindsets within math classrooms to encourage visualization and discovery.

Now, I’m not saying that every classroom is this way, but the majority are. If you’re lucky, you may have had a teacher that gave you space to be curious. Usually, this is because they have taken the time to be curious themselves. When teachers have not genuinely explored mathematics, they are simply regurgitating what has been passed along on a syllabus. Imagine taking an art class from someone who had never freely painted. Or a jiu jitsu class from someone who had never sparred. It’s no wonder the majority of people despise the subject when they have never been exposed to it in the first place.

Mathematicians create theorems and axioms not for memorization, but as a vocabulary for the language of the subject. You need to learn how to read and write words before creating poetry. ✨

One of the main issues in math classrooms is teachers give students the question and then spoilers! They give away all the answers. Now you may challenge this statement, claiming that questions are given without answers. Teachers do give students space to “solve” them.

Let’s look at an example of some of these so-called questions that are given in school.

Here we have a formula and then practice problems. What about questions like why are the left and right terms related? What does an integral represent on a graph? Why do we have both u and v variables here and have dv on the left, but du on the right? What does an integral and derivative even mean on a graph? What does this look like visually? The visual aspect of math is core to its beauty. Drawing things out allows you to notice connections and make new ones.

We do not learn well by being told. We learn by doing, struggling, solving. This process is skipped completely when students are told to “apply” formulas and “evaluate”.

Here is a real explanation of integration by parts from the above worksheet when we are given space to explore.

The questions asked in most math classes are not mathematical questions. Ask instead— Is infinity a number? Why can’t we divide by zero? About how many steps would it take for a knight to return to its starting space on a chess board? Now that's more like it. Mathematics has evolved because of the conversation and struggle of understanding and attempting to answer these kinds of questions.

Let’s play some more with our imaginations.

Think of a rectangle with a triangle inside it. The base of the triangle sits along the full base of the rectangle. The top point wanders anywhere on the top line. How much of the rectangle’s space does the triangle consume?

In this picture, it doesn’t matter that real physical objects are made of atoms pulsating, ultimately changing precise sizes. The edges in our shapes are perfectly straight because I want them to be.

“You have endless choices; there is no reality to get in your way” — Paul Lockhart.

Once you make those choices though, the shapes react however they like. The triangle takes up a certain amount of space within the rectangle. I don’t know exactly how much that is, but I want to find out.

After a bit of play and struggle and wandering, I see something beautiful! I can draw a straight line from the top of the triangle down to the bottom.

When I draw this line, I see that two smaller rectangles have formed, with diagonals splitting each in half!

One half of each rectangle is filled by the triangle and one half is not. So, the triangle must take up half of the rectangle.

Since we know that a rectangle’s area is the length times its width, we can state that the area of a triangle is half the rectangle surrounding it, or 1/2 times the base (length) times the height (width).

Area of a triangle = 1/2 bh

This is one of the first formulas we are “taught”. However, so many of us skip the process of understanding.

You might ask, when will I ever use this in life? Fair question. It’s not necessarily the specific proofs you learn that will apply to your life (although the topics of geometry, probability, and calculus are building blocks to our world and are used to create many powerful models). It is the act of exploration, struggle, and discovery that will truly benefit you.

Engaging with math is a creative task that requires us to use the imagination. Unfortunately, our culture likes to shove creativity to the side even though imagining is one of the freest, wildest capabilities of the human mind. Math is a playful, powerful universe and way of thinking. The world becomes a more creative place when you allow yourself to explore it!

Keep learning! ✌💛

• 3Blue1Brown (my favorite youtube channel ❤)
• PBS Infinite Series — YouTube
• Mathematicians Lament — How schools cheat us of our most fascinating and imaginative art form by Paul Lockhart
• Shape — The hidden geometry of information, biology, strategy, democracy, and everything else by Jordan Ellenberg
• Change is the only constant - The wisdom of calculus by Ben Orlin