Learning mathematics in school and doing mathematics in general are not the same thing.
This might seem obvious, but I worry a lot about students that don’t have a chance to realize this before they are turned off from mathematics forever. The reason is that the message which is sent to students throughout their years in elementary and secondary school is that mathematics is all about numbers, but this is false.
Sure, a lot of mathematics involves numbers, but it’s a mistake to make the leap that mathematics is all about numbers1. Mathematics is a way of thinking. There’s so much more than equations and formulas within mathematics.
Students get a quick glimpse of this when they are young. During their first years being exposed to mathematics, they learn about shapes and patterns. There’s no algebra involved. Their sense of numbers is only beginning to get sharpened, so the curriculum is focused on other areas of mathematics. Notice how they are still learning without needing to transform everything into equations.
Fast-forward a few years, and the focus has shifted. Now, students are getting used to doing arithmetic, and applying this knowledge to algebra. From here on out, most of the emphasis is on getting students to be proficient with equations. Students learn about probability and statistics, graphing equations, solving quadratics, doing word problems, solving trigonometric relations, exploring vectors, and thinking about geometry. However, in all of these subjects, the emphasis is almost always on using algebra to solve problems. Even in the case of geometry, the use of symmetry is often substituted for brute-force equations. And what do students learn equations are for? Plugging in numbers to get an answer out. The theme of a problem becomes finding the right equation that will spit out the desired answer.
I’m not saying that we should ban equations from ever being used in class. They are a great tool to get a handle on the essence of a problem. But this total emphasis on algebra shoves aside other areas of mathematics. Areas like graph theory and it’s parent field, discrete mathematics, are quite accessible to students at the secondary level, and do away with a lot of algebra2. The point isn’t to do away with algebra, but to look at some subjects which don’t emphasize its use.
What if the student didn’t have to touch an equation, unless absolutely necessary?
I think this change would help some students see that mathematics isn’t just about “finding the number” to solve a problem. Rather, it’s a method of thinking about how we can boil down the essence of a question or a problem into something we can manipulate. This is not limited to the areas I highlighted above. Heck, I can even see a project where students learn how to use equations in order to create mathematical art. Even this is different, because it removes the emphasis on equations (though they are still there!) and more toward creation.
Students tend to have strong opinions on mathematics, which is informed by their experience in secondary school. I’m hoping that we can do more to remove the idea of mathematics being all about numbers and equations. It’s so much more, and I think diversifying our offering to students is key to changing that mindset that some students come away with. For some, this is the mindset they will carry through for the rest of their lives. Even if they don’t study mathematics anymore, I want them to leave with a fair view of the subject, not one informed only by solving endless equations.
In fact, I would argue that students don’t even get to encounter the mathematics of numbers, which is the study of number theory. This isn’t part of the curriculum (at least where I’m from). ↩
I also realize that we do use a lot of numbers in discrete mathematics. But it’s of a slightly different type. Instead of formulas, we often compute quantities like permutations and combinations which can be visualized and don’t require as many equations. ↩