Many adults don’t remember their time spent with math, specifically geometry, too fondly. Thinking back to our middle and high school courses in geometry, we may remember the unpleasantness we felt, but perhaps not much of the actual geometry we were supposed to be learning! There are many reasons why we might have such bad memories of geometry. Those annoying two-column proofs? The fact that it may have seemed so completely disconnected from anything we experienced in real life? Well, this is the first of a series of occasional blogs where we address some of your potentially grim attitudes towards geometry, and try to convince you to think otherwise. However, we’re not going to slap you on the hand and say “hey, why did you forget all your geometry!” To the contrary, we’re going to argue that geometry was good for you, even if you can’t remember any of it!!

One thing that puts many of us off from geometry is the flood of vocabulary we were inundated with in those courses. In this week’s problem, you may have had those old feelings of discomfort come flooding back, as we encountered many vocabulary words, some old, and maybe some new (like *annulus*; if it’s new to you it’s an interesting one to explore!). And although all that vocab can be really tedious, it’s really just the opposite: it’s fabulous! Because if you *didn’t* have all those words flying around in your head, it would be so much *more* tedious!

For example, take a look at the following figure:

It’s a triangle, of course. Whenever you see a shape like that, you refer to it as a triangle, nice and simple. On the other hand, you could also refer to it as “the union of three segments whose endpoints form a set of three non-collinear points.” Would you like me to say that phrase every time I see one of those things, or should I just call it a triangle? OF COURSE I should use the vocabulary here. And there’s the fabulousness of vocabulary: it allows us to use a single word, instead of a long phrase, to describe something. As long as you put the effort in to understand the meaning of that long phrase, you can happily replace it with that single word instead, and get on with your business. Imagine if there were no geometry vocabulary… every textbook would quadruple in size! How terrible would that be?

So, perhaps you hated all that vocab, and now maybe you don’t even remember most of it. I guess it just didn’t matter, right? Wrong! We believe that mathematical thinking benefits us all, even if we end up not actually remembering the math we were thinking about! Generally speaking, vocabulary empowers us to think. How can we say what we mean if we don’t have the right words? Of course, it’s much the same in math. But what if you’ve forgotten all that math vocab? Was it all a waste? We say it isn’t. Because math vocabulary is special, and requires us to really *think*, and think hard, to master. Sure, you can memorize the subtle differences between the words “envy” and “jealousy”, and that’s that. But mastering math vocabulary takes a lot more effort than just memorizing; it takes *understanding.* What’s the definition of a parallelogram?

Of a rectangle? Of a rhombus? Okay, great, you’ve got it. But wait, now there are *theorems* about those things, and to understand those theorems, you *really* need to understand the vocabulary. And many times, the definitions *of* the words sound so much like the theorems *about* those words! Have you ever confused the *definition* of a parallelogram (a quadrilateral with opposite sides parallel) with a *theorem* about that word (a parallelogram has opposite sides and angles equal)? It takes serious thought to keep all these things straight, and to understand how the definitions and theorems interact with each other.

That’s where the lasting effects of geometry come in. You may have forgotten the definitions, but really that’s okay, because mastering these concepts way back in high school helped to train your mind on how to think. This is one of the most important reasons to study geometry: to learn how to think logically. And understanding and applying geometry vocabulary goes beyond memorization; it demands that we think things through very carefully. That mode of thinking, if we’re serious about it, lives with us long after the math words are forgotten. Geometry vocabulary helped us to continue to build the foundations we need to think critically, clearly, and deeply. So, you may not remember your geometry vocab, or even *want* to remember it, but those words had a place in your mental development, and helped nudge you towards the thinking adult that you are today. Thanks parallelograms!

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