Re: Rule to Apply for Rates and Ratios

[John Clement <clement@HAL-PC.ORG>]

> Quoting Bob LaMontagne <rlamont@POSTOFFICE.PROVIDENCE.EDU>:
>
> > No wonder kids aren't
> > learning anything in school - teachers can't formulate
> > proper questions!
>
> Maybe that's part of it.........
>
> But one could at least as plausibly argue the opposite point,
> that kids are not sufficiently trained to deal with ill-posed
> questions.
>
> Real life is *not* a multiple-choice question where one of the
> answers is clearly right and the others are clearly wrong.
>
> In real life, almost all questions are ill-posed in one way
> or another, perhaps underspecified, perhaps overspecified,
> perhaps self-contradictory, et cetera.

This is certainly true, and attempting to answer rich context questions
helps students become expert problem solvers.

However, the basic issue is that students do not understand proportional
reasoning.  The school system is not doing what is necessary to help them
understand proportional reasoning.  Indeed the teacher's question shows that
the teacher assumes you must always calculate one of the particular ratios
to solve the problem.  While one often does calculate one of the ratios,
this may not be necessary for problem solution.  Good teaching would equip
the students to understand and pick one of several strategies.

So yes, John is right to point to rich context problems as a good problem
type, but the students must have the necessary understanding of the various
forms of reasoning identified by Piaget as markers for formal operational
reasoning ability.  I have yet to see a PER curriculum which has adequate
emphasis on proportional reasoning as an outcome.

John M. Clement
Houston, TX