Friday, December 21, 2007

"If math were a color. . ." and other crimes against humanity.

If you're a fan of the Math Wars, you've heard about the question in the first edition of the EVERDAY MATH K-5 curriculum, "If math were a color, it would probably be ______ because ________." This particular question has probably engendered more ridicule and garnered more notoriety than anything associated with the NCTM Standards of 1989 on, and the progressive reform curricula that emerged from NSF grants in the early 1990s. After Googling on the question, I've concluded that no one on the planet has the slightest idea what could possibly be the point of asking this question. It has resulted in so much negative feedback that it has been dropped from revisions and subsequent editions of EVERYDAY MATHMATICS. Indeed, as recently as last week, Andy Isaacs, an author of the program who works at the University of Chicago Mathematics Project (UCSMP) posted on math-teach at the Math Forum:

As for the "If math were a color, ..." question, it appeared in our
first edition in a Time to Reflect section at the end of the first
unit in fifth grade, a unit that focused on prime and composite
numbers, factoring, figurate numbers, and so on. The first edition of
fifth grade was published in 1996. The "If math were a color, ..."
question was deleted in the second edition, which was published in
2001, and the entire Time to Reflect feature was deleted from the
current edition, the third, so this question has not appeared for
some years in EM.
I suppose that since Andy, whom I've met and consider a friend, is washing his hands of the question and it's been dropped entirely from the series, I should let sleeping inquiries lie, even if the folks at Mathematically Correct, NYC-HOLD, and countless "parents-with-pitchforks" groups and web-sites (as well as right-wing pundits like Michelle Malkin continue to make "big" political points about how programs like EM are sending us and our children down the fast track to enslavement by those ever-inscrutable Asians (that Ms. Malkin is herself of Filipino extraction should in no way be taken as a mitigating circumstance in her willingness to help scare the James Jesus Angleton

out of hard-working Americans with the reliable threat of hordes of Asians (Chinese, Japanese, Singaporeans, and Indians, in particular) overtaking us in the constant battle to be #1 in anything and everything. Never mind that it's actually those sneaky Finns who top the latest list of countries that outperform the US in mathematics on an international test. No one is yet recommending that we adopt the Finnish national math curriculum and load up on glug and besides, the Finns themselves are not quite so sanguine about the situation. However, racist that I am, I just can't get sufficiently frightened by images of invading Finns. Asians, of course, are notorious for "teeming," and there's no question that they're going to overwhelm American any second, no doubt due to that famous "math gene" they're all born with.*

But let's get back to the point (sometimes my own digressions frighten even me). What about that "If math were a color" question? Am I seriously planning to defend it as not only "harmless" but actually sensible? Those of you who know me are no doubt sure that's exactly what I claim.

My argument is brief, but I believe it's quite reasonable:

It is increasingly common practice for some high school teachers and some professors of mathematics and mathematics education, especially those working with future or would-be future mathematics teachers, to ask their students for a brief "mathematical autobiography" at the beginning of the term. It's a way to get to know more about students and to find out their attitudes towards and experiences with mathematics (does that seem like a trivial thing to anyone? While I wouldn't be the least bit surprised if it did given some of the folks who teach mathematics, it's nonetheless the case that some educators think it's quite valuable to know how education students think AND feel about the subject(s) they propose to teach, and how students think and feel about a subject they are expected to learn).

Now, if you're teaching K-5, especially if you're teaching K-2, how would you propose to get students to relate their beliefs about and attitudes towards mathematics? Direct questioning, like direct instruction, is not always effective, which is why many child psychologists work with their clients through playing games and other indirect methods for getting insight into experiences and feelings about them. The question"If math were a color. . .?" is a perfectly imaginative and reasonable way to get children to explore and reveal some of their feelings and beliefs about mathematics. The important thing is not, of course, to elicit a particular response or "right answer" (and isn't that a radical concept?), but to help students to find a non-threatening, non-intellectualized way to talk about math. Heaven forfend that child psychology should enter into any elementary school teacher's classroom or pedagogical repertoire, of course. Sensible, rational, intellectual questions only, most particularly in math.

*For the irony-impaired, that's satire.

Sunday, December 9, 2007

Book: OUT OF THE LABYRINTH: Setting Mathematics Free

Ellen & Robert Kaplan and The Math Circle

If you only read one book about mathematics teaching and learning this coming season, let me suggest that it should be OUT OF THE LABYRINTH: Setting Mathematics Free by Robert and Ellen Kaplan, founders of The Math Circle in Cambridge, MA.

I had the distinct privilege several years ago of attending a session Bob and Ellen led at Northwestern University's Math Club. We were taken through a short series of problems that led to the main question of the day: is it possible to cover a particular rectangle with non-congruent squares? The way things were led was truly masterful, perhaps the best teaching I've ever seen or experienced. The preliminary questions helped scaffold the main one, but once we entered into trying to solve the main problem, for which most in attendance seemed to believe the answer was "No," very few comments or questions were offered to help us. And yet those "hints" that were forthcoming seemed to be just the right ones needed to steer us out of ruts or stimulate our best thinking to move forward, and within about an hour, the students (who included undergraduate math majors, graduate students in mathematics, and some faculty members from Northwestern's mathematics department, none of whom seemed to be familiar with this problem or area of math) had successfully found a correct solution. The entire process was beautiful to see.

Afterwards, I introduced myself to the Kaplans and commented on how impressed I was with the way in which the lesson had been taught with such minimal but incisive questions and comments from them. Bob stated that they did the same problem recently with a group of 5th graders in Cambridge. Before I could express any skepticism or incredulity, he added, "Of course, it took about twelve weeks!" (The problem was presented as part of an on-going Math Circles course, and a great deal of time was spent building up the necessary tools so that the students could tackle the bigger question. I had no doubt, however, that the basic approach to helping these students was the same as what I experienced in Chicago.

When I asked Bob about who else taught in the Math Circle classes, he said that it was no problem finding people who knew the requisite mathematics. The difficulty was in finding such people who could also keep their mouths shut. And indeed, that is not only a problem in Cambridge, but nationally, including in my own teaching. I struggle mightily to resist the temptation to bail students out prematurely, to succumb to the pressure, both internal and external, to show what I know and to relieve the students of responsibility for thinking and learning mathematics the only way that really makes a difference, in my view: by doing it themselves. The cliche "Mathematics is not a spectator sport" is clever, but it is also very true. It's difficult to really "own" a piece of mathematics without struggling to wrap one's brain around it. This is not to say that we can gain nothing at all from a lecture, of course. But when the money is on the line and there's no one there to help us solve a challenging problem (where "challenging" is, of course, a very personal and relative term), will we do best by recalling a step we saw someone else use, seemingly pulled effortlessly from thin air, or by calling on methods we've used ourselves and built up sufficient experience with to sense that one might be of particular value in the current case?

The story the Kaplans have to tell in their new book is far more powerful than any argument I could offer here as to why it is so crucial for students to be led to believe correctly that they CAN do it themselves and, perhaps more importantly, that they SHOULD do it themselves, collaboratively, as a community of learners, without the usual competition to be the first or the best, but out of a collective passion to know.

I will say in all honesty that what I saw in Chicago had me nearly weeping with pleasure and frustration. Pleasure at the wonderful way in which the Kaplans taught, but frustration at the rarity of the experience and my knowledge that far too few Americans ever get to see and learn mathematics (or much of anything) in this powerful way. Reading their new book, I continue to experience these strong emotions. If you care about mathematics and really empowering people to DO it, go to the Math Circles web site: check into their books, look at some of the notes you can download from recent courses, look at the photos, and generally get a taste of what they do. You'll be very glad you did.

Wednesday, December 5, 2007

Finding The LCD, or Why Does My Math Teacher Insist On Making Me Hate Math?

Recently, there's been an on-going conversation/argument on, unusual in its frequently bordering on civility and even some degree of agreement among antagonists, regarding the role of teaching/learning/using the Lowest Common Denominator (LCD) or Least Common Multiple (LCM), which are effectively the same thing, when working with, say, addition and subtraction of fractions. It should be noted that similar issues arise later in working with rational algebraic expressions, particularly when there are polynomials in one or more of the numerators or denominators that can be factored in some of the standard ways that students have worked with previously. But my concern is with something else, specifically the arbitrary way in which many K-5 teachers insist that to add/subtract fractions one MUST find the LCD. Failure to do so often results in all sorts of negative consequences, from stern glances to loss of credit on classwork and tests. On my view, this is a classic example of abuse of teacher authority, as well as an indication in many instances of mathematical ignorance on the part of some teachers.

It is false to state that in order to add or subtract fractions (or rational algebraic expressions, for that matter) that one must first find the LCD. What is needed is only to find A common denominator and create equivalent fractions for the numbers or expressions one wishes to add or subtract that have the same denominator. The rest is fairly trivial.

Since one can ALWAYS find a common denominator by simply multiplying together the denominators of any two (or more) fractions with unlike denominators, the issue becomes a matter of taste, to some extent, but also convenience, speed, and perhaps elegance or aesthetic qualities as well. But it suffices to find ANY common denominator if one so chooses. If the goal is to find the sum or difference, we just need equivalent fractions with the same denominator.

Some people seem to believe, however, that there is a huge need to put students through the grind of doing lots of examples in which they factor a bunch of denominators and possibly numerators (not just with numbers, but in rational algebraic expressions). They argue that this is essential stuff for algebra and above. But of course that is true only to the extent that textbook authors join in the conspiracy to present cooked problems that always submit to just the techniques of factoring that we make the heart and soul of Algebra 1 courses. It's hard to deny that if you rig the game you can guarantee the outcome. If in the real world, we knew that most problems people have to solve were going to have convenient factorizations in them, then by all means it would be worthwhile to obsess on this sort of thing. But that's not really the case, no matter what math textbook authors might mislead us into believing. And so all this focus on finding the simplifications, while mathematically nice, is not necessarily what people will want or need to do outside of math class: they're going to be more interested in getting the numbers crunched efficiently and accurately and then doing what actually matters: making sense of the results. Elegance and "oh, isn't that slick" mathematical aesthetics are indulgences for those who have the time and inclination. Nothing wrong with that at all, but do we really think misleading students that such will be the situation most of the time when they need to deal with fractions or rational algebraic expressions? Do we really want them to think that most people don't grab the calculator, possible with a CAS system, or sit down at their computer? Please. Pull the other leg, it's shorter.

Yet I read people arguing with very straight faces that we have to teach this and teach it thoroughly and that it is essential for algebraic readiness (so to not make finding LCDs a MAJOR emphasis is to deprive students of their futures. Seriously. You can't make this stuff up).

When I and others retort that the main goal should be to understand what fractions are and how their basic arithmetic operations work, and that methods and techniques, while worth exploring, are not in and of themselves valuable without understanding, and that further, it's possible to learn to do an algorithm by following steps but really not having much or even any understanding of the symbolic manipulations one is doing, I'm told that being able to repeat steps IS an indication of understanding, though maybe not of a deep and thorough kind. I find this truly bizarre.

I think my preference would be to say that when students make choices about how and what to do and can intelligently explain their reasons for the choices they've made, then they show understanding. Following cookbook steps to solve cooked-up, convenient problems may be little or nothing more than donkey-like behavior, what Douglas Hofstadter has described as "sphexishness."

I'd prefer not to focus specifically on procedures/algorithms, but not to exclude them from consideration, either. That's because as students move through elementary mathematics, it is easy to forget that there are deep levels of understanding relative to their degree of mathematical maturity and experience that can easily be forgotten, overlooked, or consciously ignored by overly-focusing on procedures. Not only does making algorithms for calculation THE primary focus (not that it isn't a part of what should be attended to) of early mathematics instruction do a disservice to the students, it readily can hide and distort important information about who actually has a very good mathematical head on his/her shoulders AND which students may have significant difficulties despite seeming success at following steps they don't really understand at all.

That has long been my gripe with how many people talk about mathematics at the K-8 level or so: as if computational proficiency suffices to be mathematically on track. Lack of such proficiency MAY indicate serious lack of understanding, or it may only indicate a student who has not yet developed specific knowledge, yet still has clear ideas about the concepts involved (e.g., a student may very clearly understand what addition means and can recognize when to add in a given situation, but may not have mastered all the facts yet, or may not have them at finger-tip recall. My son took his time in that regard, but without flash cards or empty drills, managed to get them all eventually. Some of his teachers doubted his mathematical abilities because he didn't seem overly concerned about such immediate recall: he had various compensatory strategies (none of which involved calculators or counting on his fingers, etc.) that worked, generally within a few seconds, and I thought they were fine. He didn't obsess about passing those medieval 100 problems in 60 seconds tests that have become widespread in this part of Michigan, regardless of whether the elementary math curriculum was EM, INVESTIGATIONS, or something more traditional. Such tests drove his very intelligent half-sister to tears. He just told me that he took his time, got most of them right, and would know them all, quickly, sooner or later. And he was, of course, completely right. I was very gratified at his recent parent-teacher-student conference to have the principal make a point of being there to tell us what a great kid he is, how friendly and open he is with both adults and kids, and what a great student he is. I reminded him and his mother that in second grade, we were told by his teacher, the principal, and several other experts that he likely wouldn't be ready socially or intellectually for middle school (he's in 7th grade and got straight A's first quarter). Without wanting to draw any definitive conclusions from n = 1, I find it telling that it's easier for some teachers to focus on arbitrary and shallow tasks like 100 facts in 60 seconds than to recognize that a student actually THINKS mathematically and likes doing so.

Further, I wouldn't conclude that his situation is proof that we should stop breathing down the throats of kids who are working at a slightly different pace at fact-mastery than might be mandated by some arbitrary scope and sequence guide, or local, state, or national curriculum or norm. We're never going to establish such proofs to the satisfaction of the skeptics, and we're unlikely to ever reach the point where we generally tolerate AND support such differences, or recognize when a student is doing just fine (as my son truly was) or really is in trouble even if s/he can calculate or do recall on demand. As long as we put the focus on an extremely narrow band of skills and call that mathematical proficiency for elementary school, we're going to screw up a lot of kids. We've been doing a lovely job of that in schools when it comes to mathematics and other subjects for more than a century, by and large. It's one reason I fully empathize with those who choose to home school, even if I think that, too, has drawbacks, and even if I don't agree with all the reasons that some people believe they should home school their kids. It's a free country, though, and as Kirby Urner says, there's room for intellectual competition in how we educate and what we educate about.