Create+Share Math Interactives

This thread comes from 2 thoughts and if / when you have time I would really appreciate any thoughts on this. Even quick reactions are good.
(a) I read this headline - "Teaching with Skepticism and more ..."    and
(b) I didn't easily see that the diagonals split the rectangle into 4 equal areas and was willing to say so in my video.

I enjoy being skeptical and wondering why things work and whether I can prove them and if not - where they go wrong. I don't mind being confused and trying to figure out what confuses me. How do we bring this into the classroom - into the teacher's and student's psyche? The skepticism and the wanting to wonder, to explore, to prove, to make mistakes and by continuing to test, find and correct our mistakes is important.

"Rich problems" don't do it for me and I don't think they do it for students. Often they take up huge amounts of time trying to understand the text of the problem and not the math.

In contrast, small problems like we are doing here and screencasting and/or constructing with GeoGebra work for me. With screencasting, we talk/show our thinking process. And when I think aloud, I become more conscious of what I am saying/doing and skepticism has a chance to enter my work, i.e. "Why am I sure this is right?", "What facts/formulas am I using?" .
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First let me say that the headline was actually for "media messaging" not mathematics, but it seems to me that we teach math so "boldly".

For example, we state: the ratio of circumference of a circle to its diameter is pi. Like it is the most natural thing in the world. I think it is EXTREMELY odd. and I will bet that the Greeks (or whoever discovered pi) thought God must be having some fun. Firstly, it is odd that it is always the same number, and secondly, it is odd that pi is not a particularly nice number. But we talk about pi like it is a natural number (pun intended).

Also, it took me a while to see (b). I am not giving it away though - it was a wonderful ah-ha moment. I walked around with my folded rectangle in my pocket.   I am relatively certain that in almost any hs geometry class, the teacher would think me slow for not knowing this. (I found that if I teach something over and over the same way - I begin to think everything is obvious.) My point is that I allowed myself to be skeptical, to be confused and then - using my mathematics and logical thinking skills - I worked it out.

Esther and I have been talking (see her page) about the skills that make you successful in life and the fact that these skills don't seem to be encouraged in today's classroom. Jumping in head first, being willing to make mistakes, testing our results and if they don't work, figure out why and fix them.

Re: Square from Diagonal Problem
I asked Steve if we could give a diagonal to a student and ask them to create a rectangle and he said "no" - the non-uniqueness would confuse them.

1. Is confusion bad if we teach them how to logically confront it? (Of course, we must start this process when they are very young.)

And, Cathy was doing her "trying to think like my students". Just looking at the "image", I could NOT figure out why her construction wasn't a square. I finally realized that because the construction was done in GeoGebra, I could follow it (view->construction protocol) - like a screencast. Three points here.

2. I could "see" it wasn't right, but I didn't know why - However, I WANTED to spend the time to find the problem (do our students??)

3. By following the construction protocol, I could say to myself  "Yes, we see the diagonals crossing perpendicularly at the midpoint. That's good. Ahh, C isn't right" and to my student  "Something about C doesn't look quite right. How did you find it?"

4. I see GREAT value in giving this construction as part of the problem and asking "Why isn't this a square?" Why? Because they may go right for the easy construction. Part of learning is recognizing something is wrong and finding out why.

As teachers, we want to help our students learn. To do this we must allow them to make mistakes, but teach them to test their results. With screencasts and GeoGebra, finding where they are not testing their thinking is easier both for us and for them.

I tend to go on and on....