This is a “puzzler” to prove your point: http://www.computermagic.gr/Fun/StrangeMaths/FromWhereComesThisHole/DefaultUK.aspx

I would also note that this is a powerful entry into the need for formalism around labeling diagrams and requisite symbols. I often stress that the formalism saves a poor artist like myself, as a quickly sketched square is not really a square until I label all angles as right and all sides congruent. Internalizing this helps my students avoid the intuitive potholes of “well it looks like a square” or “that angle looks right” or “those triangles look congruent”

I love how GeoGebra clarifies this particular issue. I can fully acknowledge that a shape looks “close enough” to a square to be a passable representation of one. However, what I require of students is that the shape stays square when I drag points and lines around, and otherwise change the construction within its degrees of freedom. If there is too much freedom, it won’t be a square anymore! Obviously so!

Computer construction provides a good language to discuss the issue.

I’m more referring to those examples when the concepts of proof are first introduced- when a student can intuitively apply the transitive property of congruence, say, and needs to understand the importance of slowing down and proving every step along the way. It’s a challenge to convey the utility of proofs when the answer is legitimately “obvious”.

But I completely get your point.

And to your last point, we sure do end up proving stuff we don’t know to be true, but it comes later after skills have further developed.

The problem I see with getting students to prove only things that are obvious is that this doesn’t challenge their misconceptions. If all you do in Geo is prove things that you already know are true, then you won’t see the point of doing it.

You put it perfectly: “The answer is in using the perfect set of specific examples- ones that strike that delicate balance between not obvious and possible for the novice geometer.”

There are some good examples in Lockhart’s book. There are other collections, such as The Art of Problem Solving accumulates in various forms. However, this brings us back to the issue of quality and time.

It is significantly harder, and takes significantly longer, to find these GOOD (let alone “perfect”) examples than random problems easy enough for novices.

Who is going to spend ten times more time?

Who is going to pay for it?

Why not prove things that aren’t obvious?”

As a Geometry teacher currently working through the foundations of logic and elementary proofs right now, I am often vexed by this question- I spend much time cautioning my students that “Although we can all tell this is obviously true, we need to slow our minds down and explain every step”. Indeed, we do end up proving much that isn’t obvious, but we end up needing more sophisticated parts put together in subtler and more sophisticated ways to get there. How do you prove things that aren’t obvious at the start, when the need is greatest for developing the motivation for learning proofs. I suspect the answer is in using the perfect set of specific examples- ones that strike that delicate balance between not obvious and possible for the novice geometer- but which ones work for this?