The (trigonometric) vicious circle
Just in time for the first days of class in several parts of the world, I decided to write a small play to honor the event. Enjoy!
Act IThe scene is a modern Japanese bar. A group of people is enjoying some drinks at a table. LoliAlgebra is identifiable by his cute anime t-shirt.
人間1 : [To LoliAlgebra] What did you study in college?
LoliAlgebra : [Stylishly holding a glass of Yamasaki on the rocks] Oh, I studied mathematics.
人間1 : So cool! I always hated math though... [Everyone laughs except LoliAlgebra, who pretends to be drinking]
人間2 : I was so bad at it! I can barely remember how to solve a formula! [More laughter, LoliAlgebra smiles politely]
人間3 : Me too, I gave up during middle-school! You must be so good at calculations, LoliAlgebra! Guess we'll leave the check to you! [LoliAlgebra downs his entire drink in one gulp]
LoliAlgebra :
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I'm not a playwright so you'll have to excuse my first attempt to write one, but I can tell you that I've lost count of how many times something like the above has happened to me (minus the outburst). There are few things that are as poorly understood as mathematics; yet, it's also one of the few things most people think they understand what it's about. They "know" that it's about memorizing formulas, that mathematicians are human calculators, that it's rigid set of rules, etc...
It doesn't take much reflection to realize this misunderstanding starts in school and the question of how to solve it is one of the biggest ones in pedagogy. I wouldn't be surprised if you've read about terms like "math anxiety" for example. Originally, I had intended this post to be about my general thoughts on math education and how it is failing a significant amount of people. It quickly became clear that it would become a monster read, so instead I'll talk mostly about the things that I find most troublesome with middle/high school education (low-level undergrad and "math for the arts" courses in university may well deserve a future post).
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I'll start by acknowledging that education in general has been going into a better direction for some years now. The whole inquiry-based learning is certainly much better than the rote memorization and emphasis in calculations of the past, and most education systems seem to recognize that. This is, however, very much hindered by the one thing found everywhere in the world: the curriculum. Each country's curriculum may be different, but they are all made by people who seem to have no idea what math is actually about; they only care about the most efficient way to fulfill whatever is needed to send people into university.
The true essence of mathematics is that, at its core, it is art. I don't only mean that there are pretty things like fractals or golden ratios in leaves; it really is an art requiring as much creativity in its application as poetry or painting, the only difference being it's not recognized as such. Math is just as majestic, mind-blowing, and inspiring as any other art; perhaps even more since it's not constrained by the physical universe, it lives in the realm of imagination!
What does this have to do with the curriculum you ask? Well, do you know of any art that is taught in an even remotely similar way to how math is? Can you imagine a music class where you don't play music?
Since the people in charge don't understand this, of course their curriculum will be completely fucked up, no matter how many times it's revised or tweaked. Worse than that, it almost seems purposely designed to make people think it is a boring and stale subject, quite a feat in my opinion. Discovery, curiosity, struggle, the feeling of success... things that anyone would agree are vital for art, are all killed even before they get a chance to be born. I for one am required to make unit plans for the whole year with contents, skills, resources, assessments, etc... I'm going to use; as if there was a set order to it all. Talk about drowning math.
As long as there are things that students need to know, it is very hard to be authentic in trying to foster true inquiry. Sure, you can make up all types of games or opening problems to try to awaken students' curiosity; but in the end no matter how much the teacher wants it, s/he needs to guide everything back to the "point" of the lesson. Can you really have any of the feelings I described above when you're just repeating what the example in the textbook/board does with only superficial differences?
Imagine a math class that was completely unchained and free. Teachers could give a small idea but it would be up to the students to make it grow. A class where whatever it is that piques the students' interest is something that would be pursued, even if that means that they won't learn the sigma notation or the formula for interior angles of a regular polygon or how to integrate $\tan(x)$...
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| Graphic depiction of the effects of the curriculum on young minds. |
Curriculum aside, there's another issue that masquerades as something benign: the attempt to make math fun and to try to show its usefulness.
The other math teachers at my school, and plenty at seminars I've been to, spend a good amount of time and energy trying to find ways to make math interesting to kids; to find games, real world situations, and applications. All tailor-made to make some completely random and inane math topic seem relevant. They end up creating or finding in some textbook gems such as:
"Boat B is located Y miles away to the north of boat A and going east. Boat A goes west for 10 minutes and turns 30° north...."
"4 years ago Abel had twice the age of Britney; in 6 years Britney's age will be three fourths of Abel's. What is Abel's current age?"
I ask you, dear reader, can we really blame students for thinking math is dull after asking them this nonsense?!
Back when I was their age I couldn't care less about the distance to the boat, or the angle of the shadow, or a stupid kid's age. I couldn't care less now! I liked math because of the endless possibilities, the beauty of the explanations! Even genuinely interesting problems in physics/biology/whatever won't get much excitement because they're just brought up as an excuse to apply the "lesson"; not born from the students' curiosity and often lacking important historical context.
So, what's the point with all this? One answer is that the whole cross-curricular study seems to be pretty popular with educators lately. I'm not opposed to this, and it certainly can work as long as it's not shoved in just because "it happens fits" in the corresponding lesson. The other, and the one that irks me the most, is that it is done in the hope of avoiding the dreaded question: Am I ever going to use this?
Such a common question it is, but many teachers insult their students' intelligence with their answers, common ones being: "Look at this ridiculous real life example!" or "you never know, you may, engineers do!". They won't. I have yet to meet a single person that has ever used trigonometry/bearings/quadratic factorization/etc... outside of school, and most of my friends are in STEM. Does that mean it's useless? Not necessarily, exercises can have a place, but only after they have a context. The context being a problem that arises from a student's curiosity. I would liken it to sports: no soccer player does in a game exactly what he does during training, but it all helps in some way. That being said, it doesn't help with how tedious it is and at best it's putting the cart before the horse.
The crux of the question comes from the very beginning. We don't need to make math interesting, it is interesting! Curiosity is at the core of the human heart and math is its purest representation! You don't ask an art to show its "usefulness"; the fact that it is useful is an extra! Is literature taught so people can fill tax forms!?
It's heartbreaking to think of all the potentially good mathematicians that never spread their wings because how of boring it all is; to think about the disappointment people suffer when they discover that they're actually not that good at real math, even after the test scores told them they were. And just as ironic how many of those "good" students like "math" because it doesn't require creating anything...
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| I hope you discover this before getting to |
Given all this, if we were to somehow fix the above points, what would a math teacher need to do? What exactly is a good math teacher anyways?
As I stated before, school boards, education experts, textbook writers, etc... don't get mathematics, or maybe they do but don't care much (those revised edition textbooks and seminars sure bring in the money!). That being said, the people who actually apply them may be more vital. Unfortunately, and most worryingly, even most math teachers don't really understand math! In the current educational system, math starts showing its true face in grad school or in undergrad if you're lucky and pick the right courses; but a good amount of math teachers studied how to become teachers, not math itself. Would you take a painting class given by an art teacher who has never painted freely and knows nothing about art movements after the 17th century? Well then, why is it acceptable to have math teachers who have never produced a work of mathematics? Or who have no idea about its history and developments beyond the material they teach?
Now, I'm not saying that every math teacher needs to be a mathematician, or that they should have a paper published in some prestigious journal. But is it too much to ask that they have experienced the feeling of being stuck with a problem for weeks and solving it? Or that they know some things about set theory and the foundations of mathematics? Or that they actually enjoy it and not just as a teaching endeavor?! If for teachers math is nothing more than a group of arbitrary unmovable rules and symbols that must be followed, it will feel the same for their students!
Don't misunderstand me, I'm not blaming the teachers at all. Most genuinely want to help their students and if they were not restrained by the curriculum they would be able to do a much better job. They just happen to be a product of the same system; having never known what math is truly about they don't know better.
In fact, I think there's an extra issue with teachers, and that's teaching schools/degrees. I'm not going to pretend to know how effective they are for other subjects, but I can tell you that it's impossible to teach someone how to teach mathematics. One of my friends at work is a middle school math teacher. Like many, he went to an education school and is up to date with the current pedagogical trends. Sure enough he makes pretty nice lesson plans, introduces topics to students with games, and knows middle/high school math (whatever that means) like the palm of his hand... he's a great teacher by most standards.
And yet, when anything pulls the rug under his feet, he shakes quite a bit, if not collapsing right away. He prepares for his classes so much that they become too artificial. He knows the direction his class will take everyday and the answers to his "challenge" questions before even asking them. Because of this, the rare times he does run into slightly higher leveled math questions or a more profound look at familiar material, he gets quite a bit perplexed. A couple of times he has come to me while he's in class, just to ask about something that he doesn't know how to answer right away. He seems afraid to show any weakness in front of his students, perhaps not wanting to damage the "teacher" image.
This is where I also drastically disagree. For me, to be a teacher means to have an honest intellectual relationship with your students. It's about ideas and the history behind them and letting the students make their own discoveries. It's about refining their arguments and letting them develop a mathematical taste. I don't make lesson plans because
I happen to be pretty good at both real math and the curriculum approved one, but sometimes even I find a problem I don't immediately see how to solve during a lesson. What then? I do math with my students! I let them explore, conjecture, think of a ways to approach it, to try them and fail or see them as being too convoluted. Just as important, I do all that at the same time because that is math! They get much more out of me filling half a board with equations just to say "hmm, this doesn't seem to be working" and starting again; than they would from me solving the problem beforehand, polishing the solution, and regurgitating it in front of them. We get just as excited whenever any of us solves it!
So what do I suggest? Forget the lesson plans, the games, the posters, the falseness! Don't be afraid of history and of going on tangents! Talk to the students, listen to them; they are humans after all! And most importantly, actually do math!
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| Hopefully this isn't something you need to worry about. |
As you can tell, I'm not particularly happy with the current state of affairs in secondary level math education. I realize the biggest problems are structural and changing them in any meaningful way would be a monumental task. I also realize that most of my suggestions may be a bit too radical to be realistically applied. That being said, I hope you have gained a bit of a different perspective, compared to those of more traditional math teacher blogs, on these issues.
I may seem like a bit of a rebel, but like most I get to keep my job if my students have good scores and get into elite universities; so I'm bound by many of the same chains. However, I struggle and bend those chains as much as I can, and I hope you do too in your own way.
In the end I may not be in a position to help my students open their eyes completely to the true beauty of math... but at least I hope to inspire them enough so that they want to open them themselves; there is indeed light at the end of this dark tunnel.
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