On Sep 29, 2012, at 12:52 AM, Louis Talman <firstname.lastname@example.org> wrote:
> The pluses are outweighed by a very big minus: Questions like this convince too many kids that the heart of mathematics consists of knowing the hidden trick---and that, therefore, no real thought is ever needed. That's absolutely the wrong message.
I did think through the problem. I noted the implicit relationship between a and b and realized that substituting that back into a^2 - b^2 would result in an expression of just a. After that (simple) realization the problem solved itself as fast as I could type that first line. You are not suggesting that my (semi hard) way is not "thinking", are you? Or are you saying that students would not recognize the "thinking" involved in solving such problems and mistake it for a "hidden trick"?
I think we have to be careful when using the phrase "hidden trick". I know what you mean, but my bar is a bit higher. To me the "trick" must be very unique to be called "hidden". In the case of Dave's problem, all the student needs to be aware of is that relationships do not have to be explicitly spelled out with the variables, they may be implicit in the constants as well.
Can you provide an example problem that is more thoughtful as you say?
> And the point isn't that one should "work the algebra FIRST". It's that algebra is for making life easier.
Well, this is what I have struggled with, the use case. I have not found in people lives in general a use case for algebra. I am not saying that you couldn't invent use cases (recipes, mortgages and such), just that the majority of people are not interested in them nor do they ever use them. There are paths that do use algebra, but if you interviewed and monitored 100 people's lives, chosen at random, I doubt you would find 5 that ever put an x or a y down on piece of paper, post school, in their entire life. Now spreadsheets on the other hand, they beat algebra at making people's lives easier 100 to 1.
> Kids should be able to do multi-digit addition where it's sometimes necessary to carry a one. Even in their heads, given that they can write down the digits of the answer as they produce them.