A common problem geometry teachers face is that people have overly narrow perceptions about what counts as math. This makes sense. 90% of what kids learn in math class is arithmetic or arithmetic’s older sister algebra. These associations often don’t fit with what goes on in geometry, to the detriment of geometry. For some kids, these perceptions of unmathiness undercut their motivation to try to understand the central skills of geometry. For others it can lead to a very, very slow start to the year, where they feel as if perpetually stuck in an introduction to a subject they never get to. (When are we going to do the real math?)

One approach to this problem is to be explicit and unwavering. This is math, and we will spend class working on drawing, defining, sketching, debating, justifying and explaining. This is what math is about, just as much as it’s about practicing skills or developing procedures. Don’t like it? Tough cookies. This is geometry, and you need to learn what geometry is all about.

I don’t like this approach. It doesn’t fit with what I know about learning and teaching. I don’t teach kids to solve equations by being explicit and unwavering: This is how you solve an equation. What is this guess and check stuff? Algebra isn’t arithmetic, time to get with the program kids! Instead, we take what kids know about arithmetic and we build on it. This is one way to solve an equation, but it’s inefficient. Here’s a new tool, it’s called algebra.

I’d argue that just as we ought to take a developmental approach to teaching any mathematical topic, we ought to take a developmental approach to developing students’ perceptions of math. All we can expect by being staunch and strident is for lots of kids to not actually understand what we’re talking about.

If we want to take a development approach to solving equations then we’re tasked with articulating students’ existing thinking, eventual thinking, and proposing thinking that could bridge those two poles. If there’s an analogy here, then those are the questions we need to answer about how geometry students are likely to perceive math.

What will my geometry students likely perceive as legitimately mathematical coming in to my class? What is my goal for their mathematical perceptions?

I don’t know. Evidence — student work or research — would be helpful here. And maybe I should find a way to smoke out their beliefs about math early on in the school year. In the meantime, here’s a guess as to what students’ think counts as mathematical:

• Numbers are involved
• After learning some math you can solve a new type of problem
• Math questions are either right or wrong
• Math is essentially communicated in symbols or technical language

Now, all of these beliefs are wrong in general for mathematics. That doesn’t mean these beliefs are entirely wrong. These beliefs are true for most of the math students are likely to have seen in their schooling. A shame, yes, but that it’s a shame has nothing to do with kids’ thinking, which is as legitimate as it is wrong.

Prior beliefs aren’t valuable so that we can lambaste kids for having them. Imagine starting a unit with a formative assessment that reveals that kids have all these wrong beliefs about exponents. Next class, everyone sits down and then I say, “Well so many interesting beliefs about exponents. But they’re all wrong! Here, let me show you.” I know that some teachers do this, but it’s a silly activity. You aren’t really helping kids develop those ideas and see their limitations, you’re telling them not to think, that their opinions aren’t worthwhile. You’re not giving them a chance to be smart. And yet this is conventional wisdom about how to deal with students’ beliefs about the nature of math and geometry.

What should we do once we know what students beliefs about exponents are? We should creative opportunities for kids to understand the limitations of their beliefs, give them time to formulate new theories, make explicit the truth about exponents once they have the experiences that can help them make sense of that. And shouldn’t we do that with students’ beliefs about math also?

I have a favorite style of activity. It’s an activity that kids can do with their current knowledge, but can also be done with more sophisticated knowledge, and it can be done better with the more sophisticated knowledge. In this case, that would mean that I’m looking for math that is both recognizably geometric but also makes sense as math in the limited theories that my students are likely to come into the year with.

What would that mean? My favorite geometry textbook spends its first major of unit focused on constructions. But the above puts some constraints on what sorts of constructions it would be a good idea to start with:

• Constructions that include numbers
• Constructions that can help students solve a new type of problem
• Constructions that are either right or wrong
• Constructions that involve symbolic language

Do such constructions exist? There are certainly constructions that include numbers — my favorite textbook includes a bunch of them, things like “make a triangle whose sides are exactly 3 in, 5 in and 7 in long.”  And there are certainly types of construction problems that are awfully similar: “construct a triangle with these sides”; “draw the angle bisector of this angle”; etc. And these are either right or wrong, in a sense. I could also introduce precise, symbolic and pictorial language to describe these constructions. This would feel like math.

But that’s not going to be enough. I also want to nudge my students beyond these conceptions. Math involves explanations and proofs. It often is done in informal, everyday language. It’s often more than just right or wrong. How do we get at this?

I think my plan is to build this on top of the other constructions. We’ll ask questions about the numerical constructions — why does this work? why doesn’t this work? why can these sides make a triangle? why can’t these? — and in dong so we’ll make conjectures, create informal proofs of theorems. It will be important for me to make explicit that we will spend much of the year focusing on these sorts of arguments.

Experience before formality. That’s a great principle, and one that we should use not just for mathematical content but also to help students develop their perceptions of what it means to be mathematical.

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1. trigotometry says:

For some constructions, I think there is an underlying set of concepts that we can take for granted if we approach the topic so close to the start of the year. For example, I think of copying an angle. What’re we’re really doing when we construct a copy of an angle is creating a congruent triangle using a side-side-side argument. Unfortunately, some students are just taught a process for this construction over understanding. If we are aiming for an informal approach to constructions, I think we need to start even more basic and build that desire of knowing why something is true as we add more complexity. Before copying an angle, we need to appeal to a student’s ability to copy segments. Why does this even more basic act work with a compass? I picture students responding to this question by arguing about measurement. The idea of mathematical truth (knowing why) appeals to the sense of one right or wrong answers and curiosity that students harbor, while opening the door to more interesting ideas such as proof and multiplicative reasoning. From this explanation, we introduce copying an angle and ask students why this construction works. The explanations don’t need to be formal or even use mathematical terminology, but this event provides a gateway for introducing some vocabulary to overcome the vagueness of explanations and a beginning for seeing connections between geometric structures like lines, angles, and polygons.

Perhaps an informal approach to constructions can build a need for vocabulary and proof in students? Perhaps the very choice of constructions can lead students to the goal teachers want for geometry students, which is proof without constructions that use axioms and relationships between angles, segments, and lines? Similar to the progression you noted for moving students from arithmetic to algebra for solving equations, maybe constructions can serve as the formative stepping stone towards formal geometric proof.

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