“Why do we need this?”

My students in precalculus class have asked a few different times over the last couple of weeks, “When will I need this?”. I encourage them to ask this question of their teachers, because the teacher should be able to answer it. But when it comes up in the middle of a lesson I don’t really have time to answer it fully, so here are my thoughts, in a thought out format.

  • The blunt answer is that this is an elective and no one is making you take this. And if someone is making you take it (your parents, for example) then I can’t help you. I wouldn’t be satisfied with this answer if I was you, but it is true.
  • We don’t know and you don’t know what you’ll end up doing, so it’s better to err on the side of a broad knowledge base in high school.
  • These are things you need to understand if you’re going to take more math. Why might you take more math?
    • You might want to study math
    • You might want to go into a math related field
    • You might want to go into a science related field where calculus, at a minimum, will likely be required
    • You might want to go into pre-med or pre-vet (or pre-law?), where you’ll likely have to pass a calculus class
    • That’s what this class is designed for. That’s it’s stated purpose. It’s to get you ready for calculus, regardless of the reason you might end up taking calculus.
  • I admit, math is also a gatekeeper. It’s a way to check your level of competence. I’m not making a statement about whether that is right or wrong, but it is true. See a great exploration of this idea in Bad Drawings here.
  • But more broadly, math is giving you a language to describe and understand that world. Sinusoidal behavior (which is what we’re studying now) is everywhere. Anything that works on circular motion or has periodic behavior can be described by sine or cosine waves.
  • It also helps you exercise your rational thinking and pattern finding muscles. We don’t always do a great job of helping you work the pattern finding muscles, but the reasoning muscles are exercised every day. When you solve a problem you are reasoning. You put together a chain of logical statements, either explicitly or implicitly, to arrive at a conclusion. And those statements aren’t the same for each person, which is a great thing. There’s often more than one logical way to arrive at the same conclusion. But this mindset, of taking on assumptions and arriving at logical conclusions based on those assumptions is a vital component of thinking critically. It’s how we persuade other people. It’s how we make progress. It’s how we call BS on people when they make claims or statements that don’t make sense.
  • Mathematics, and the different parts of mathematics, can be viewed as games of sorts. You are given some rules (axioms, theorems, etc.) and then you see what is true given those axioms. You see how far you can push the rules and how far you can get in the game. But they’re often better than an arbitrary game. Many times they have implications in reality (although this isn’t a necessary condition for interesting mathematics and is not necessarily where mathematicians spend all their time). And sometimes those implications aren’t seen at the time, but turn out to be useful later on in science. A mathematical truth is a truth about reality itself.

The fact that we attach grades to mathematics does minimize some of this. It’s not a fun game to play if there are real consequences (in terms of grades) for not understanding the rules quickly enough. And I get that. But I reject the notion that we should throw the whole project out the window because what you’re learning on a Tuesday in precalculus might not help you navigate your grocery shopping trip.