In these posts I will share some of my reddit comments that I feel are worthy of saving.

You ask a good question, “How do you teach the act of proving without showing proofs?” In short, I don’t think you should not show them– but I don’t encourage memorization of them, and you should show more “proto-arguments” early in the year, becoming more and more formal as the year goes on.

I think my top level reply answers a little bit of my thoughts on the matter:https://www.reddit.com/r/matheducation/comments/3nhn28/motivating_geometry_students_to_do_proofs/cvon7ffbut I’ll say more here…

So first, I’m definitely NOT saying get rid of the theory. Proofs are the most important part of a typical geometry course. The course should be all about justification. But, I take issue with the development of proof in many textbooks– they often leap into it as either chapter 1 or 2 and say, “ok got it? good now the rest of the chapters will just ask for harder or longer proofs”

Consider this analogy. You’re taking a class in how to build a car. It would be silly for the class to have you first build a fully functional toy car, then a fully functional half size car, and finally a fully functional full size car. The size of the car is irrelevant to how difficult it is to create it. If we don’t know how to prove, then doing any proof is difficult.

So we should back up and think: what are the interior parts towards proof? i.e. what are the wheels, the engine, the gears?

I argue some of those internal parts are conjecture, precision of description, representing an idea in a format other than what it started as.

So consider something like the huge timewaster traditional vertical angles proof (see https://www.youtube.com/watch?t=201&v=wRBMmiNHQaE for how to turn a simple idea into 4:51 of boredom).

Instead of showing that proof or anything like it, ask students to draw two lines intersecting at any angle. Have them notice and wonder ( https://www.youtube.com/watch?v=a-Fth6sOaRA ) about what they see. You WILL have a kid claim that there are angles that are always equal– if you wait long enough. And then you can pounce, “hey, Bobby says that the angles he calls ‘across from each other’ are always equal, who agrees?” Use the vocab that the students came up with– or if its unclear, ask for clarification… ask other students if they can help clear up Bobby’s naming… reach a class consensus.

What you’re doing there is building the pre-requisite skills needed for proof. The students first need to be able to talk about their mathematical ideas with some level of precision. Many of them have never been asked to try (and then how are they going to prove anything?) As they get better at making mathematical statements and asking mathematical questions, they get better about linking answers together, and formalizing their informal reasons for belief in certain truths.

As you do these, you can also provide tools to students like examining how to draw conclusions from conditional statements (I wear my boots only if it is snowing. Its snowing… do I need my boots? (not necessarily) I only wear my boots if it is snowing. (oh, aren’t you cold then?) I wear my boots if it is snowing. (sounds reasonable)) and making a chain of logic using things like the law of syllogism or equivalent statements.

But these things should be developed over the course of the year. Students should always be justifying — but its is the formality and rigor that should increase. In the beginning they are making guesses, then conjectures, then argument ideas, then providing possible reasons, then linking reasons together, then referring to larger concepts and tie-ins… and then by the end of the year they can write a formal proof instead of a rough ‘because’.

I said this metaphor in the other post, but I’ll say it again here. Instead of doing ‘easy’/short proofs, then ‘harder’/longer proofs, building the structure of proof skills as if stacking blocks — think of proof skills going from unfocused to focused as if you are adjusting a lens throughout the year, bringing the structure of proof skills into sharper and sharper clarity.