Am I the only person who feels like this list has turned into NCTM-TOOM lately? It's really heart-warming to read the self-congratulatory exchanges of "ideas" among Andrei, Kent, Hsu, Jack Roach, and others, but frankly, I think the lot of them are utterly full of crap (an opinion they and others no doubt hold about me).
The latest garbage about problems by type is (once again) the result of Andrei's extremely questionable reading of the Standards, supported by the equally questionable reading of Hsu. But the claim by Jack Roach in his last post that the Standards would discourage the use of an old brain teaser because "it's a coin problem" is just about the last straw.
Let me see if I can shed a different light on this discussion:
Look through any number of mathematics texts from the the past 30 or so years, including books designed to help students review mathematics for exams like the NY State Regents' Test, the SAT, and the ACT. How many of them DON'T have sections dealing with a host of word problems broken down by type: problems involving mixtures (coins, candy, etc.), travel with and against some force (current, wind, etc.); work problems; age problems; percent mixtures (acid solutions, and other dilution problems), as well as a host of others?
Then ask yourself the following questions:
1) Are these problems connected to students' real world experience?
2) Are the solution strategies being taught to help students increase their ability to think and reason mathematically, or are they being taught (and, more importantly, experienced by students) primarily as algorithms to be memorized, applied on tests, and then forgotten?
3) Are students being led to make connections between the mathematics underlying these problems and the rest of what they study in mathematics, or are these problems presented so as to make it highly likely that students will view each problem-type in isolation both from all the others and from the rest of the corpus of their mathematical knowledge?
An honest answer to the first question would be, I think, that some are, some aren't; that some students will connect more easily than others to the situations described, but that virtually no one has ever had a real world need to algebraically figure out how many nickels and dimes s/he has given the total value of the coins and the total number of coins. That doesn't make coin problems inherently "bad" or "wrong"; it simply suggest that there may be problems with similar mathematical issues that come closer to something an actual student might care about or have experienced.
In general, we need not throw out any or all of these problems, but rather we need to consider the MANNER in which they are presented. If we answer questions 2 & 3 above honestly, it should be "intuitively obvious" that there is vast room for improvement in the breadth and depth of the problems we choose for students to tackle.
I realize that there are folks on this list for whom the sentence "We need to do less of __________" immediately becomes translated into "We should NEVER do ________"; and that the suggestion that "We need to do more _________" magically becomes "We should ONLY do ________." I would suggest that such folks need remedial reading instruction.
Perhaps this sounds petty and intolerant. And perhaps it is. But observing Andre, et. al. pat one another on the back as they staunchly defend a bankrupt approach to mathematics teaching primarily because they have little clue or desire to learn what it would mean to teach mathematics any other way than what they've experienced and mimicked has become truly tiresome. And yet, I think I understand what may inform much of what these folks believe. I got out of English teaching, despite winning various accolades, because fundamentally I couldn't understand what it would mean for someone to have virtually no facility for reading and/or writing his/her native language. I was one hell of a literature student; success came to me with relative ease. Why wasn't what I saw so readily "intuitively obvious" to my students?
As a mathematics teacher, it's far easier for me to empathize with student difficulties and to be willing to find more effective, flexible approaches to various topics. I even think I can LEARN from my students' insights, questions, and mistakes. Unlike Andrei, who frequently expresses disdain for the shortcomings of American mathematics students, I am able to put myself in their shoes rather easily. Perhaps Andrei should spend a few semesters teaching English; he might gain a modicum of humility when he returns to the mathematics classroom. Of course, as Daniel Benzali said this week on MURDER ONE, "I'd like to believe that the meek will inherit the earth."
|--------------------------------------------------------------------------- |Michael Paul Goldenberg |University of Michigan 310 E. Cross St. |School of Education 4002 Ypsilanti, MI 48198 |Ann Arbor, MI 48109-1259 (313) 482-9585 |(313) 747-2244 | |"Truth is a mobile army of metaphors." |Friedrich Nietzsche |---------------------------------------------------------------------------