>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.
My concern with Dave's problem (and his expressly as well) is that someone can know Algebra 2 well and still miss the trick unless a hint is given. Our statewide Entry-Level Mathematics has shown that simplifying a single "rigged" to be simple but compound rational function (numerator and denominator each consisting of sums/differences of rational functions) correlate so well with future success in anything needing algebra that it could be a one item test. Success with such a problem, however, does not mean that student would spot this problem as trivial.
>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.
That has never bothered me. Out of that hundred, how many will need to know or ever use who Carl Sandburg was? And a ton of other such stuff. What really does bother me - and leads to the huge inequity at later stages - is being genuinely algebra ready at algebra time. At that point, becoming algebra competent or not (especially at the Algebra 2 level) is probably irrelevant for those whose ambitions lie other directions (if known; if not known ?) Hence my huge interest in quality K-7 mathematics with K-5 (both math and reading) being the most important and 6-7 having lots of ratio/percent/etc. word problems that have to actually be read and understood for the mathematical content and how to interpret it back in terms of the original setting.