I got a promo email from Virtual Nerds yesterday. They got a *little* publicity in theNew York Times a while back; they do what Khan does but more formally, and at cost. The videos are better, IMO — just because it’s done more professionally, and I suspect also with a bit more regard to the math behind things.
However, when I’d looked back when NYT covered it, it was… procedure, procedure, procedure. A brief email exchange revealed that they were interested in being more conceptual…and I said I was very interested in participating in developing more concrete, visual materials, but … dead end. I replied to this promo asking if they were still procedural. I got back an enthusiastic response saying my students could be helped so much! And could I look at two videos in particular…
I replied that well, these videos did focus on concepts as well as procedure — it was about the nature of additive inverses — but… yes, it was 100% abstract. So, I explained that my studentsneeded visual and concrete connections to the math symbols and procedures.
I got back another enthusiastic request to look at a video which had symbols “at the end,” but relied “less” on them… and it was an entirely abstract, symbolic, logical explanation of why anything to the zero power is one.
I resisted the urge to ask if they had a dictionary handy.
Now, I’m thinking that if our folks out designing material have that level of what “concrete” means, no wonder we’re having issues with the curricula. I’m almost tempted to see how often this happens… ’cause it’s certainly not the first time I’ve gotten the answer to a question about building concepts and confidence *before* hammering on a procedure… with “here is the correct procedure. It should solve your problem!”
I also wondered if it really was the problem.
Well, maybe students just don’t really need to understand math at all. Maybe really the point of math is to teach them to flex their brain muscles for Memorizing Procedures and Working with Abstractions. If so, then how about finding procedures and abstractions that are more based on their areas of interest?
I happen to know that an awful lot of teachers (and I was one of ’em) honestly, sincerely believe that students have a basic foundation and make connections I’m teaching… but this morning I was sitting in on a math class and the teacher put up a triangle and asked what the area for it would be… base of 3, height of 4.
It’s 12, don’t you know?
And despite the drawings and Socratic questioning, the primary answers from the class were that of course, the area of the rectangle (drawn with the squares so all could see, albeit not really well drawn, which makes me think that that might be important — *I* just need it to be close enough to generalize to the abstraction, but the learner needs it to be TRUE – perhaps becasue most if the visuals don’t really make sense to them anyway, but (I know I do this) the student just looks for the formula)
… do I digress?
… the area of the rectangle was 12… and so was the area of the triangle that was taking up half the space.
Area is, don’t we know, base times height?
Yes, our students generalize.
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Rachel
September 23, 2011
I agree that students need to see concrete examples. When I was teaching angles a few weeks back, I taught my students to use the protractor, and–even though it’s moderately complicated–they liked doing it. It really helped some of them to see and measure the complementary and supplementary angles, and they moved on to the abstraction fairly well from there.
I have one gentleman who insisted on measuring all the angles in the problems instead of doing the math. Even though he was able to do the math, he preferred the more concrete measuring. I eventually had to warn him that not all the pictures in the books are drawn exactly to scale.
I would love to see a computer program that allowed students to change the angle with the mouse and see the number of degrees change–or type in a number of degrees and see the angle, and then maybe move another angle next to it and compare the measurements of the two old angles and the new one…I wish I knew some programming. I think interactivity is key.
xiousgeonz
September 23, 2011
Geometer’s Sketchpad can almost certainly do that.
I remember figuring out that usually, the “not necessarily drawn to scale” thing was a lie and that the angles that looked equal usually were 😉
This same teacher, last year, after the quiz on perimeter and volume — which they did well on — had given them a cereal box to measure the volume of. They were **clueless** even when walked through “measure the length, width and height” processes. They had no idea that they were applying what they had “learned.”
Rachel
September 23, 2011
I just looked into Geometer’s Sketchpad. It seems pretty interesting. Have you used it before?