Hi, On Dec 19, 1:51 pm, michel paul <mpaul...@gmail.com> wrote: > Since most on this list probably work at the college level, as a high school > teacher I'd be interested in the math expectations you'd have for incoming > high school graduates today?
(1) Some kind of evidence that the student is accustomed to reading a textbook on simple mathematics, and has the ability to read and understand it. Honestly, if all you do is teach your students to read mathematics, *really* to read it, everything else becomes trivial. (2) The ability to perform simple algebraic operations without a calculator. This includes arithmetic with negative numbers and fractions. (3) The ability to use a scientific calculator. (Not a graphing calculator. Those things are bulky, overpriced, and obsolete.) (4) The understanding that the only math problems that can be done in a minute or less aren't worth the paper they're printed one (except in grade school textbooks). Corollary: "exciting" and "flashy" problems are often trivial, and therefore not especially worthwhile. (5) *Most important* The habit of wondering about something you don't understand, and asking questions with a sincere openness to learning it. Mathematics is a kind of philosophy, which as Aristotle rightly said, begins in wonder, yet the word "wonder" itself seems to have fallen out of common parlance. 100% of the freshmen I have taught in the last five years lack (1). At least 50% lack (2), which really slows the class down if I ask for the result of a computation. Too many lack (3). I'm pretty sure all of them lack (4); I remember my PhD adviser's remark that students' eyes would glaze over the moment any process required more than 3 or 4 steps. And too many students who think they're good at math lack (5). Students need more experience with challenging problems that require them to stop & think about the solution starting in middle school. Rote mechanism won't solve it; neither will technology. > 4. In a computational age, it is more important to grasp relations > between concepts than to memorize particular formulas. Better to learn how > to analyze a concept as a set of inter-related concepts. Example - the > quadratic formula. Yes to the first, but an emphatic no to the example. First, the quadratic formula isn't really that hard to memorize. It isn't so very complicated, and it's used so frequently on problems that are actually useful at the high school and college levels that memorization could come simply through practice. The cubic or quartic formulas, now: *their* complexity exceeds their utility in high school. (One of my undergraduate professors told me, twenty years ago, that when she was a child they studied the cubic & quartic formulas in high school.) > The traditional schoolish expression minimizes the > number of hand calculations necessary. The adjective "schoolish" sounds disdainful here. Maybe you didn't mean it that way, but "schoolish" expressions can be quite sophisticated. An undergraduate student of mine recently used the quadratic formula to devise a simple test of whether a biological problem was feasible through algebraic methods. It impressed the biologist we're working with. The coefficients were purely symbolic, so a graph (and, by extension, a graphing calculator) would not have helped. Without the quadratic formula, she would have had to complete the square. Joy! > However, a more conceptually > valuable expression might be to express it as h +/- r, where h =axis of > symmetry, and r = distance to the roots. I don't see how technology will help here; it isn't hard to draw this on the board, or on a sheet of paper. I don't think an interact would be helpful here, either, but I'm open to the possibility that it would be. Don't misunderstand; I think you have some good points; I just think the quadratic formula is a terrible example. > 6. There is always a tension between the use of calculators and 'showing > ones work'. Kids hate having to write it all out if the calculator has > already done it. All kinds of discussions go on about how 'much' work > needs > to be 'shown'. All of this becomes irrelevant if we instead focus on the > 'work' being a functional decomposition of a problem or a concept. If one > does ones 'work' correctly, the 'work' will then work for you! You can use > it! I don't think the discussion becomes irrelevant at all. It will always be relevant because different people are interested in different aspects of a problem. > 8. China is already uniting Computer Science and math classes at the high > school level. Doesn't surprise me, and it's a good thing. But, > "Computer science is the new mathematics." > > -- Dr. Christos Papadimitrious I disagree strongly. The statement implies that computer science supersedes or replaces mathematics, when it is merely another subset of mathematics. regards john perry -- You received this message because you are subscribed to the Google Groups "sage-edu" group. To post to this group, send email to sage-...@googlegroups.com. To unsubscribe from this group, send email to sage-edu+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/sage-edu?hl=en.
