On Mon, Oct 1, 2012 at 9:40 AM, Wayne Bishop <email@example.com> wrote: > > At 08:34 AM 9/29/2012, Robert Hansen wrote: > >> 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. >
I think it's a "trick" if you're taken by surprise by the answer, which likely means something about the context you're in.
What's missing from any single problem is more context. If we've been hashing through (a + b)(a - b) as a pattern, a common topic in algebra books, and if we're familiar with this style of problem, to reward understanding of common patterns, then yes, we stand a better chance of not hating the problem and not hating the test makers (the ones who make us take them).
What's disappointing is when there's no context, or not much. There's "school" but it's not a safe place, and the kids are distracted, aren't being left to learn, have to get out there and make a way for themselves, mom and dad already crushed by the system or whatever. No time to noodle these symbols around if they're not gonna promise you a meal plan, or shelter. No time to jump through all these hoops, learn all these tricks, with gratification "to be delayed" for any number of years. That all assumes "a protected childhood", a neo-Victorian invention to some degree.
That level of economic insecurity (no "safety net") sounds "3rd world" (a misnomer for "non-aligned") but is the reality in much of North America for many school aged. "Math" (so called, the pre-STEM version) is for the privileged with the leisure time to do uncompensated school work without experiencing overwhelming opportunity costs.
That opportunity for safe study is not provided to everyone equally, lets be clear. Not all environments are created equal, when it comes to safe niches for "curling up with a good book, and a cup of Folgers [tm]". And lets not immediately jump to "race" as a reflex, thinking inequality is so simple minded. If you want to be fair, consider yourself a walking victim of the poor society you live in. You're only half as smart as you could have been, only half as informed. Much was denied you, no matter who you are, because you've lived in a dark age. Hard to admit that about oneself. We all like to think we turned out OK.