Fake Math Models
Robert Kaplinsky wrote a note recently discussing fake math models and unnecessary context. This prompted an activity with the kids.
This issue seems to have come up a lot recently, so I've noticed a pattern: I really hate bad contexts.
Robert wrote: "it looks like the context is completely unnecessary to do all of the problems."
I would go farther: this context is harmful. The context creates a conflict between the specific new material (rational vs irrational numbers) and other important concepts (measurement and measurement error). Subtly, we are discouraging students from
(a) forming connections across topics. For my taste, surprising connections has to be one of the most beautiful and delightful aspects of math.
(b) using all of their ideas and creativity to understand a challenge.
(c) putting new mathematical ideas into a broader mathematical context (maybe I'm just repeating point a?)
I admit that the example only touches on these points lightly, but I suspect the accumulated weight over the course of a school math education is substantial.
If I were full-time in a classroom with a textbook, I'd be tempted to use it as follows:
1. create censored versions of all problems and examples (as you did)
2. work through the questions with the kids
3. Ask them what context they think the publishers originally included and why
4. show the published version
5. discuss (does the published version relate to the math, does it help them understand, does it add confusion, does it conflict with something they know, etc)
