Euclidean geometry is for many students the first time they get a taste of what math is really about. The problems don’t all fit the same pattern; it’s natural and expected that students will come up with their own arguments to prove something, rather than following a set of rules. Ideally, geometry class also involves experimentation and conjecture.
I don’t think Park students are encountering these things for the first time in geometry. Our students are used to investigating and asking their own questions. And they are used to making careful arguments to support their claims. Still, for Park students as much as students anywhere, geometry tends to be the first time that they are asked to write formal proofs. Anyone who has taught geometry knows that writing proofs can feel to students like wearing a straightjacket. For the first time, arguments that are correct but either vague or not axiomatic are inadmissible:
- Opposite sides of a parallelogram have to be congruent because lines with the same slope stay the same distance apart.
- Opposite sides of a parallelogram have to be congruent because there is no way to extend one of those sides without changing the angle of the side coming to meet it.
- The base angles of an isosceles triangle have to be congruent because the triangle is symmetric.
I’ve stopped telling students that these arguments are not convincing. Anyone who understands the terms they’re using would be convinced. And I’ve even stopped telling students that they are incorrect. They’re not incorrect; they’re just not arguments from first principles. They appeal to intuition and common sense, as most arguments we’d make in daily life do.
Acknowledging those things, we still need to make rigorous arguments that appeal to specific principles we’ve studied in class, such as theorems about parallel lines, and theorems about congruent triangles. For this reason, I stick to the “statement/reason” model of proofs taught in most geometry classes. I find that if students don’t write proofs this way it is too easy for them to fall into arguments that are merely intuitive. It’s also easy for them to fool themselves into thinking that they have enough evidence to conclude that triangles are congruent when, say, they’ve really only found two pairs of congruent sides.
I don’t, however, insist on the degree of rigor that most geometry books do. Students in my classes do not write proofs that contain the sequence, “If angles form a linear pair, then they are supplementary. If angles are supplementary, then their measures add up to 180 degrees.” They can go right from linear pair to adding up to 180 degrees. I don’t think that this level of following tiny steps in a chain serves the purpose of helping students to build new theorems out of the knowledge they already have.
Generally, my rule is that if students are using congruent triangles to prove something, they need to
- Name the three pairs of sides/angles that they need to justify the congruence, providing a reason for each.
- Name the pair of congruent triangles and say which theorem (SSS, ASA, etc) they are using.
- Only after they’ve done all that, name the pair of sides or angles that they can now say are congruent. To justify this, they will sometimes use the infamous “CPCTC,” or, since many students have trouble remembering what the acronym stands for, just say that they are using triangle congruence.
I think that, among Park faculty, I am one of the teachers who insists the most on some kind of standard template for proofs, even though I allow much more leeway in what can be used for a reason than most textbooks do. I’d be interested in what other teachers ask of their students when writing geometry proofs.