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.

Saturday, November 24, 2007

The Book Gods?

Something that always blows me away is how teachers will follow books blindly in the face of what should be big warnings from what they know about their own students. All too few teachers are immune to book worship, having been led to believe by their own experience as kids and education students: the math textbook (and its magical authors) knows more than any regular old K-6 teacher.

Two cases in point: I was coaching upper elementary teachers in math at a low-performing K-6 school in a district near Detroit a few years ago. They were using the Everyday Math program for the first time. I was asked to guest teach some lessons on fractions in a couple of the 4th and 5th grade classrooms. I noticed that in one lesson, involving pattern blocks, there were three problems for classroom discussion. The first one was clearly needed to establish the relationships amongst the smaller shapes (triangles, parallelograms, trapezoids, that could be fit together to make a hexagon (if you have the standard pattern blocks, the relationships were that two triangles formed a parallelogram; three triangles made a trapezoid; six triangles made a hexagon, which in the first problem was the "whole" or "1." Also, you could make the trapezoid from a triangle and a parallelogram; and you could make the hexagon from two trapezoids or from three parallelograms, etc.

In the first problem, therefore, students establish that the triangle is 1/6 of the unit hexagon (which is an outline drawing in the book that you cover with these various combinations); the parallelogram is 1/3 of the unit hexagon; and the trapezoid is 1/2 of the unit hexagon.

While some students had difficulty with this, most did pretty well as I had figured. But when I saw the second problem, I sensed danger (interestingly, the third problem was much easier, and relatively few students struggled with it). #2 was an outline drawing of TWO of the hexagons with an adjacent edge, but the line where the edges met was erased, so you had a double hexagon as the new unit. The idea was that students would cover this new figure and see it as a new "unit"; it would take, for instance, 12 triangles to cover it, and so the triangle would now represent 1/12 of this unit, and so on. Each figure would represent a fraction half its previous size, since the new unit was double the old unit.

I would like to say that I brilliantly smelled a rat, but I didn't. Or that is to say, I smelled it, but I didn't trust my reactions sufficiently. Then again, I can plead lamely that I was not experienced with this book (though the mathematics wasn't a problem for me) or teaching kids this age. I was there because I had some previous experience as a coach, and because I knew the math well, and because I am very quick to adapt to new ideas and approaches).

In any event, I went ahead with the first group and had them do the problems in the order given by the book. As I had sensed but failed to act upon, the students struggled mightily with the second problem. They couldn't wrap their minds around the shift in the unit. The picture, I suspect, was perceived by most of them not as a new "unit" but simply as "two"; "obviously" it WAS equal to two of the OLD units connected to one another. They likely were visually filling in the removed boundary line where the adjacent edges met. It was very difficult to get a lot of them to make the leap to seeing this as a new "one." Even when they did the individual tasks ("cover this new shape with the triangles. How many triangles does it take?") and got the correct numbers, they were not going to budge from the notion that if a little green triangle was 1/6th in the first problem, then it was still 1/6th in the second problem. When I asked about why it now took 12 triangles, not 6, to cover the figure, they just said in essence, "Well, sure: there are two hexagons there."

I hope I have made this clear enough without the exact problem drawings that would likely made it more transparent.

So, when I guest-taught the same lesson in another class, I changed the order of the problems. I got much better results, overall. And I warned them that the problem just described, now their last problem, was challenging and might prove upsetting until they played with it a while. Interestingly, some of the metaphors I tried that bombed the previous time worked well here (e.g., a quarter is a half of a half-dollar, but it's only 1/4th of a whole dollar, etc.). Was it really that simple? Just change the order, build their confidence, warn them of quicksand, and things would go better? I'm not sure.

But the teacher in the second class was truly SHOCKED that I changed the order, and after class, when we debriefed, it took a lot to convince her that I had made an informed choice based on the previous class and my own previously ignored intuition that there was too great a leap in that second problem to go to it directly from the first one and without offering some warning bells. Her feeling was that the textbooks authors knew more than she did and must have a good reason for the order of the problems.

I assured her that SHE was the expert on her students in her class, and no author would dare usurp that position. I was making an educated guess, and had the experience of the previous class to back me up. (I know one of the main authors of EM personally, I told the teacher I would be e-mailing him with a summary of the experiences I had that day. I told her that I had no doubt at all he would understand my decision. May he might suggest a change in the next edition as a result. This was all amazing to her. And she was not a rookie teacher.

In another example from the same school and book, I chose to omit entirely a couple of problems that introduced mixed numbers into a situation where they were NOT the main point of the lesson. I anticipated (and this time trusted my instinct) that throwing mixed numbers into the fray was an error that would distract students from the real mathematical residue I wanted them to take away. Again, the classroom teacher (not the same one), was surprised by my choice and not confident that she was allowed to make a decision to remove problems, temporarily or permanently (I made clear in class to the students that we would come back to them another day).

I think this is tragic. And I think it extends even to home schooling parents, even though they know their own child(ren). They can make pedagogical choices based on that knowledge, as well as other factors. There's no textbook that can anticipate the needs of each kid, and a good teacher must intervene to make the best choices s/he can given the limitations of either whole-class or individual instruction. Clearly, teaching one child or only a few is a huge advantage and allows great flexibility. Not being under the thumb of a district or even the state or NCLB makes things even better. But the god of the book is intimidating. What if you're wrong?

But I must add this caveat: if math isn't "your thing," you need to think carefully about your choices and you are going to make some definite mistakes. Few of them will be fatal (I'm talking about the sorts of pedagogical choices already described, not mathematical errors, which are a separate issue entirely). There is what is called "pedagogical content knowledge": an understanding of both the subject and effective ways to teach it. If you're weak in content, it's hard to be strong in this area, but if you are strong in content, you may not be strong in it anyway. (And college math departments prove this daily throughout the land). It takes a lot of thinking, before, during, and after teaching lessons to be effective. It takes a willingness to really anticipate how a reasonable-seeming problem could be a mine-field for your student(s), to think about where your student(s) may go astray in the task and/or topic at hand, based on their strengths and weaknesses AND your knowledge of the mathematics (obvious example: kids learning fractions will be prone to do addition by adding the numerators and adding the denominators. It's a good idea to be prepared with an example like 2/3 + 1/5 so that you can ask the student(s) if it makes sense that the answer would be 3/8, since 2/3 is already greater than 1/2, but 3/8 is less than 1/2).

So I don't advocate just tossing things out because they look unappealing or you don't like the topic or something like that. Or because you read somewhere that a particular method isn't good. You want to think it through and in terms of the student(s). Then, you do what you can, being prepared to change course if necessary. No shame in that. Indeed, it's a wise teacher who can admit error, doubt, and change course. Too bad some people in Washington, DC seem to lack that wisdom.

Friday, November 23, 2007

Mastery of What?

A lot of conversation on one of the lists I read about spiraling vs. mastery in mathematics curricula. Nothing new about that. Just another issue that strikes me as adding more confusion than clarity by creating false dichotomies instead of seeing that most of these things go hand in hand. Frankly, I'm hard-pressed to see how it would be possible to teach or study something as enormous as merely the tiny slice of mathematics we want all students to learn and be able to use in K-12 education (which brings us up to pretty much nothing invented in the field as recently as the 17th century and still excludes enormous amounts of what was already known when Newton and Leibniz were inventing the differential and integral calculus), without doing a reasonable amount of "spiraling" (which is to say that we must revisit already-explored ideas again when students have more sets of numbers to look at, say, or have developed sufficient mathematical maturity to delve deeper into things that many of us mistakenly think of as simple, elementary, "easy," basic, etc. At the same time, it's hard to move forward (at least in the linear way we teach the subject in the US), if you haven't attained some facility with the procedures (if not the actual ideas behind them) that are often referred to as "the basics" (that's the stuff educational conservatives keep telling us we need desperately to "get back to," as if we've somehow been skipping that and teaching partial differential equations, differential geometry, and category theory to elementary students while they weren't watching us crazy progressives carefully enough, and now we must return to sensible arithmetic, a taste of geometry, some faux algebra, etc.) However, interesting questions arise about whether it is absolutely mandatory to shove every child through the same narrow funnel, and whether much wouldn't be gained by giving kids (and teachers) a lot more options about what route(s) they want to explore on their ways up and around in the tree of mathematics. (see Dan Kennedy's provocative "Climbing Around In The Tree of Mathematics"

In any event, here's some of what I've been thinking about regarding the whole "mastery" thing that so many people worry about (or say that they do):

People talk a great deal about "mastery" in K-12 math, as if mathematics was somehow like typing. You practice and attain mastery.

But that part of mathematics, while important up to a point, isn't really what mathematics is except for kids (and then, only a small piece of it).

Consider the following quotation:

"There is something odd about the way we teach mathematics. We teach it as if assuming our students will themselves never have occasion to make new mathematics. We do not teach language that way. . . the nature of mathematics instruction is such that when a teacher assigns a theorem to prove, the student ordinarily assumes that the theorem is true and that a proof can be found. This constitutes a kind of satire on the nature of mathematical thinking and the way new mathematics is made. The central activity in the making of new mathematics lies in making and testing conjectures." (Judah I. Schwartz and Michal Yerushalmy, quoted in "Geometer's Sketchpad in the Classroom" by Tim Garry, in GEOMETRY TURNED ON, p. 55).

You could readily change a few words above and talk about problem-solving as well. We pose problems to students the solutions for which are well-known. Students cry "Foul!" when confronted by: 1) problems they haven't specifically been trained to do. That is, they think it's dirty pool to be asked to solve problems that aren't identical to others the teacher/book has explicitly worked through with them, or nearly so. These, of course, aren't problems, but rather they are exercises, just like typing drills. I show you how to use the keys with your right index finger, then drill you on that, and so forth; b) problems for which the solution pushes them beyond the immediate topic, perhaps calling on general strategies they've learned, things they've worked on, but also a bit more, stretching their minds, asking them to reach a bit, speculate, imagine; and c) any sort of problem that has no known solution (or maybe just no solution with the methods or numbers they're familiar with) just so they learn that math never ends, mathematics is always being extended and invented, and part of what drives that is unsolved problems (along with new problems, new math, and so on). It could be something as simple as asking a student who hasn't learned about negative numbers what 6 - 8 equals, or someone who hasn't learned about complex numbers to consider the equations x ^2 + 1 = 0. Or it could be having students explore accessible but unsolved problems like Fermat's Last Theorem (when it was still unsolved) or the Goldbach Conjecture. Or something like the three utilities problem or Konigsberg Bridge problem which led Euler (see photo above) to invent new mathematics (graph theory). Fun stuff, really, but many kids think all problems in math class should be trivial exercises, not real problems for them to speculate about and experiment with.

I think much of the error we make in putting so much emphasis on mastery lies in cheating students of knowing what it means to think mathematically, even though they are quite capable of doing so. There is a body of work out there that suggests kids can do much more mathematical thinking than we give them credit for. But for most, by the time they get to do some, they hate mathematics (even though they actually don't know what it is, really).

Just a little late-night food for thought.

Thursday, November 15, 2007

Language, division, and calculators.

Since several related issues are floating around on various math education lists I'm reading these days, I thought the following problem and what I observed recently with some African-American students would be worth sharing.

This problem appeared on an actual ACT exam; it should be noted that it was only #5 of 60 questions in a section of a test which allows 60 minutes total time for those problems:

The oxygen saturation level of a river is found by dividing the amount of dissolved oxygen the river water currently has per liter by the dissolved oxygen capacity per liter of the water and then converting to a percent. If the river currently 7.3 milligrams of dissolved oxygen per liter of water and the dissolved oxygen capacity is 9.8 milligrams per liter, what is the oxygen saturation level, to the nearest percent?

A) 34% B) 70% C) 73% D) 74% E) 98%

First, I want to observe that there are quite a lot of words in the above problem, most of which make it difficult to read and which don't flow terribly smoothly. The actual mathematics is hardly at the high end of difficulty for high school students, if they get to it, or at least we would hope that to be the case. But the language seems quite "high end" for where this problem appears in the section, and that would potentially be an obstacle for students who are not good readers, not native English speakers, or who may be thrown by a science situation with which they are unfamiliar. Is it a good idea to embed this particular mathematical task in language like this? What, exactly, is being tested? Here in Michigan, where the ACT is now the official "exit" exam for high school students, these concerns are not trivial.

That said, let's look first at how I went over this problem with some students, once they had it set up as 7.3/9.8 or agreed that such was a reasonable way to begin. Of course, the ACT allows calculators (but unlike the SAT, does not permit the TI-89, which has a computer algebra system included). But I always start with the assumption that it might be quicker and safer to do mental math, and that knowing how to do the problems several ways is worthwhile. (That's also a personal holdover from the period prior to 1995, when calculators first were allowed on the SAT. I learned how to excel on these timed tests by studying them starting in 1979. Mental math, estimation, and knowing that fractions are often the easiest form in which to do quick arithmetic has stayed with me and I still encourage students to think that way on timed tests. I rarely use calculators for doing them, though it's not a bad idea to have one, if you know how to use it intelligently.)

First, I pointed out that we could multiply by 10/10, getting the equivalent expression, 73/98. Then I asked about 34%, hoping that students would see that it was far too low, 73 being clearly less than 50% of 98. Most students agreed. I suggested further that 98% was unlikely, as 73 is not "almost all of" 98. Again, this line of reasoning was amenable to most of the students. I next asked about 73%. Some students saw that for that to be correct, the denominator would have to be 100. That left 70 or 74, and most (though not all, of course) students at this point saw that if the denominator was less than 100, then 73 would comprise more than 73%, so that 74% was more reasonable than 70% would be.

The advantage of this sort of estimating and "number sense" approach avoids some of the errors that I found occurred with alarming frequency among those who attacked the problem with more traditional methods or with calculators. One student in particular ran into interesting mistakes, not unique to him. Without my help, he correctly set up 73/98. He articulated that he should divide, but then proceeded to start dividing 98 by 73. Since this yields approximately 1.34, it's not a coincidence that one of the choices is 34%: students simply assume that they should "drop" the "1" and round. Rather than considering that they've converted a fraction considerably less than one half into a percentage that initially (before the groundless dropping of the 1) represented more than 100%. Would they be likely to do this with the mental math approach?

But this student took it further. He grabbed his low-end calculator (non-graphing), pressed some buttons, and told me that he had been right to begin with, because he still got "34%" as the answer. Of course, he'd failed to realize or recall that when entering 73/98 into a calculator, his "wrong" impulse in the hand calculation was now the correct order (and such would be the case as far as the numbers are concerned were he using a HP or other Reverse Polish Notation device).

What I find provocative is that students can wind up going wrong on the same problem in so many ways and for reasons that aren't strictly mathematical. Not only is the language in which the calculation is imbedded more dense than one might expect, but the language of division "73 divided by 98" is in the opposite order from the order in which students are taught to divide, leading many to then try to divide 98 by 73. And then, having made that error, they find their mistake reinforced by the fact that the calculator wants them to enter things exactly in the order they appear in print (at least in THIS example). Our way of expressing division in language is at least partly to blame for a lot of the confusion in even setting up a correct calculation, let alone carrying it out successfully.

There are other difficulties with division surrounding the conceptual basis for the algorithm, but here we see how students can be led astray regardless of whether they know how to do long division by hand on an exercise that simply offered 98, the side-ways "L" we put to its immediate right, and 34 placed above it. This student did the wrong problem correctly by hand. He then made precisely the error the test-makers set him up to make by selecting 34%. I would have been interested to see how he would have done with the calculator alone, or with the calculator first, at any rate, as far as the order of the numbers was concerned.

To conclude, I would hesitate to say that this student didn't know how to do long division. He could successfully carry out the algorithm. Where his problems lay had to do with issues that may have been more linguistic than anything else. How he said the problem to himself or heard me expressing it was not reliably leading him to set up the right calculation. And the answers were, of course, tempting him to make some unsound and unjustified moves to retrofit his results to the answer choice that seemed to fit. This occured both when he did the problem by hand AND with a calculator. The mistake had nothing to do with his relying on the calculator, in fact, since it was his choice initially to do the problem without it. He did the division he set up properly. Yet his answer was dead wrong, regardless of the technology or its absence. I don't claim this proves anything, but it is thought-provoking. And this incident does help emphasize the potential power of estimation and mental math.

Wednesday, November 7, 2007


I am writing to support David Wasserman's decision to refuse to administer a test in which he did not believe and to decry the way in which he was subsequently dealt with by his superiors. I am a mathematics teacher educator, teacher, and expert on standardized test preparation with more than 30 years' experience working with students on various instruments (e.g., SAT, GRE, ACT, LSAT, and GMAT) as well as with grading state tests from Michigan, New York, and Connecticut. With that experience and expertise in mind, I am deeply troubled by the manner in which this nation has been pushed further and further towards accepting an ill-founded religious belief in the power of (for the most part) multiple-choice, multiple-guess tests to measure not only student achievement, a concept which is at best open to question, but teacher, administrator, school, district, and state competency (not to mention national status when viewing similar international tests such as the TIMSS), in total violation of one of the basic principles of psychometrics: never use a test to measure something it has not been specifically designed and normed to measure. This country has long been enamored with numbers and rankings, going back to the early decades of the 20th century, when we shamefully abused IQ scores to restrict immigration in ways that can only be viewed as unscientific and utterly racist. I urge everyone to read Stephen Jay Gould's definitive work on the abuse of "intelligence" testing, THE MISMEASURE OF MAN, for a shocking and sobering account of how standardized tests have been misused and abused in the United States, generally out of racist and chauvinistic ignorance and bias.

It takes a brave person to risk his job and his livelihood, to put himself and his family in jeopardy, in the face of blind obedience on the part of so many of his fellow teachers and education professionals to what is nothing more than an outlandish political ploy to destroy public education, undermine teacher authority and autonomy, punish students, parents, teachers, administrators, schools, and districts MOST in need of support, and to shamelessly promote vouchers and privatization to help those most advantaged and least in need already. Sadly, there is not a single member of the US Congress (and, I suspect, of any state legislature) who has a balanced view of educational politics, who actually has K-12 teaching experience, who has a background in either education or psychometrics, and who understands that measuring something is not the way to improve it. Regardless of political party or identity as liberal, conservative, reactionary, moderate, or libertarian, our politicians have no little interest in reading educational research, theory, or case studies beyond what they need to figure out which way the political winds are blowing and what policies will sound good to the voters, regardless of what professional educators believe or know from hard experience. No Child Left Behind has devolved, predictably, into No Child Left Untested, Untraumatized, Unabused by high-stakes, high-pressure exams that for the most part are not grounded in state or district curricular frameworks, do not reflect best practices, and are not scientifically sound when misapplied for a host of purposes for which they were never intended by the test authors. I have personally witnessed teachers being instructed by administrators in my home state of Michigan to lie to elementary school students about the implications for individual students of their scores on the state tests, to wit that they should tell these children that their scores will "go on their permanent record and follow them throughout their lives." This is a bald-faced lie, as everyone reading this knows full-well, but 7 year olds are not so savvy and often neither are their parents. These administrators were not evil, but merely people succumbing to immoral and unethical bullying pressure originating with a president and federal government bureaucracy that has been acting cynically in the name of helping the very children they are helping to destroy.

Few individuals whose careers and families depend upon their acting like good little Germans and pleading ignorance of wrong-doing have had the courage to do what David Wasserman did. And how was he rewarded? He was immediately made to feel the power of the institution and individuals that wield power over him and his students. Instead of applauding his integrity, they endeavored to crush him or force him to capitulate. And as a parent and husband, he did so, though I am sure it was very painful for him to have to choose between his family and his students, between his principles and his fear.

It is shameful that the administrators in the Madison Metropolitan School District did no more (or less) than those fine citizens of Munich, Berlin, Frankfort, and so many other cities in Germany and other European countries brought under Nazi rule: they used their power to snuff out the first sign of real bravery and dissent, the first act that really was designed to see that no child is forced to take meaningless tests that measure little, if anything, that most educators value. A multiple-choice test can easily reward luck over knowledge, mechanical regurgitation of mere facts over thought, imagination, and creativity, and, of course, conformity over individuality. I would be very much surprised if there is a state or national high-stakes test being used in conjunction with NCLB that is not primarily or entirely of this type. David Wasserman had the guts to stand up for his students and for meaningful assessment over shallow, cheaply processed "data"-gathering and number worship. His colleagues, principal, and superintendent should have applauded him. I suspect many of his students were grateful for even a moment's thought for their plight. Instead, we saw no acts of courage from those with a little more power than a mere classroom teacher. It was business as usual, full speed ahead, and testing uber alles. How utterly sad, and how utterly tragic for real kids and real learning.

Tuesday, November 6, 2007

The Games Ideologues Play

In a 2006 article for the Hoover Institution, Barry Garelick wrote:

"It was another body blow to education. In December 2004, media outlets across the country were abuzz with the news of the just-released results of the latest Trends in International Mathematics and Science Study (TIMSS) tests. Once again despite highly publicized efforts to reform American math education(some might say BECAUSE of the reform efforts) over the past two decades, the United States did little better than average." ["Miracle Math" by Barry Garelick, EDUCATION NEXT, Fall 2006 (vol. 6, no. 4)].

This example of Mr. Garelick's purple prose, replete with "body blows," and the laughable image of media outlets buzzing over ANY education-related item, is a masterful piece of conservative propagandizing in which he clearly sets the groundwork for a classic Hoover Institution tactic: having it both ways.

You see, in 2006, Mr. Garelick was all abuzz, if no one else was, about the shortcomings of U.S. mathematics achievement as measured by TIMSS, with the blame neatly dropped on US reform efforts (and you know who THAT means) of "the past two decades." Hmm, now. He's writing in 2006, so these efforts have been going on since 1986. . .

Quick: ignoring the question of what percentage of U.S. public school classrooms are now using or have exclusively used for the past eleven years since his piece appeared what he and the rest of us would agree are "reform" methods AND materials, what are the numbers for 1986-1996? Pinning the blame on all that alleged failure for twenty years on programs that didn't even exist in 1986 is pretty neat. But it gets better.

Because Barry Garelick argues in Sunday's in a piece entitled, "It Works for Me: An Exploration of 'Traditional Math' Part 1" that actually we HAVEN'T been failing. And why not? Well, he cites The Way We Were?: The Myths and Realities of America's Student Achievement, a book by one of the best and most liberal-minded education reporters in the country, Richard Rothstein, to argue what many folks, most notably Gerald Bracey, have been pointing out for years: the sky isn't falling, or at least not in general, when it comes to US education.

But Mr. Garelick, of course, wants it both ways.

Instead of an honest assessment of the successes AND shortcomings of traditional instruction in mathematics in this country (let alone something similar for progressive mathematics programs and methods), Mr. Garelick is offering a completely skewed version of reality in which all failures are BECAUSE of reform efforts, whether they even were in existence, let alone implemented, while all successes are due to the good old traditional instructional methods of his youth. Or maybe due to Singapore Math. I'm not completely clear on his notions of causality.

In either event, this is remarkable sophistry on his part. Because there's only one undeniable thing in this debate, and that is that there were no NCTM-style reforms going on to any discernible extent prior to 1986 (and pretty much prior to the mid-1990s, if in fact they are truly the dominant texts and methods of today and the past dozen years or so, a very doubtful proposition in itself), but the instructional methods through the mid-1980s were indeed Mr. Garelick's beloved "traditional" teaching.

So we're asked to believe: a) all was well, or nearly so, up until 19___ (I fear to complete the date, because whatever is put there will be so ludicrous that I may be struck dead for writing it even as a hypothetical or belief of someone else); b) we went down the slippery slope in 1986 or so; c) but, see, the sky wasn't really falling after all; d) except that it was, in 2006; e) but it wasn't, in 2007; f) and anyway, if it wasn't traditional methods that had us all rosy-cheeked and healthy in mathematical competence up until that 19___ year, but Singapore Math is a MIRACLE that will bring us all back to health again; g) and all this regardless of whether Singapore Math is actually a diverse set of materials and methods, that there is no clear picture of what methods it allows or proscribes, and despite the fact that the US has so little in common with Singapore (4 million people, ethnically homogeneous by comparison, a highly rigid, restrictive form of government, and a level of affluence that any US politician would KILL to be able to brag about as TYPICAL of his constituents), that it's just a tad difficult to believe that some books alone are going to save us. And this is NOT coming from a negative view of Singapore Math. Far from it. It's coming from a sense that Mr. Garelick is not an educator, but rather an ideologue, that he is like other Hoover Institution folks in his dedication to conservative views of education and education policy, that he is playing a dishonest rhetorical game, and finally that not even Singaporeans are so naive (or dishonest) as to believe that they have all, or even most, of the answers to teaching and learning mathematics.

But don't expect moderation from Mr. Garelick. Or modest claims. Or honesty. Just expect propaganda, trying to have it both ways, trashing NCTM, NSF, and any and all reform ideas and methods (even when they are strikingly similar to what people do in Singapore). Don't expect him to mention that it's difficult to find an Asian mathematics educator who comes to the US who isn't interested in learning from us and isn't critical of the shortcomings in their own countries methods, materials, and accomplishments.

No, from the Mathematically Correct/HOLD and right-wing think-tank groups that Mr. Garelick represents, there are only Singapore Math "miracles," wonderful, golden days of traditional instruction that worked for pretty much everyone, and horrible failures and stupid ideas, books, and practices from progressive educators, the NSF, and NCTM. It may be fun and easy to live in a fantasy world in which things are either black or white, wonderful or horrid, perfect or completely flawed, but unfortunately, US educators have to work in the real world, with real kids, in real communities, with real poverty, real malnutrition, real parental ignorance, real abuse of kids, real drug and substance abuse, real crime, real ethnic and religious conflict and hatred, and a host of other challenges. These are not excuses: they are simply part of reality. We have that reality and the many professionals working with it and doing research to try to improve the quality, depth and effectiveness of mathematics teaching and learning, and then we have Mr. Garelick and the Mathematically Correct/HOLD/Hoover folks with their simplistic viewpoints, propaganda, rejection of research, phony talk of miracle cures, and even in the case of one outspoken former member of NYC-HOLD, a person so cowardly he refuses to use his real name on the internet, the assertion that there are no open questions about mathematics pedagogy.

Yes, the cartoon at the top of this post pretty much says it all.

Thursday, October 18, 2007

Andre Has Risen From The Toom, and More From W. W. Sawyer

The following was posted to the list earlier today in response to the posting of a piece by Andre Toom and some comments by Michael Sakowski (quoted at the end). It was Dave L. Renfro who posted earlier that he preferred the first piece by Professor Toom, as do I:

I would concur with the idea that Toom's earlier article, "A Russian Teacher In America," had much more of value than does the unpublished "Wars In American Mathematical Education." Skimming the latter (I'm reading on a full stomach and I don't like to waste food), I was reminded of all the aspects of his prose and his "thought" on mathematics education that I find shallow and offensive.

Aside from his blind bias against Americans and his utter lack of understanding of or appreciation for our culture (imagine what a Russian would say about an American writing something similarly chauvinistic about another nation), Toom never seems to consider alternate explanations for anecdotal evidence that he uses to "prove" his assumptions. For example, the fact that students don't answer a question put by an over-bearing, hostile instructor who is prone to sarcasm and put-downs, as Dr. Toom appears to be from his writing about American students, is hardly proof that they don't know the answer to the questions (as in his query about conservation of energy) or that they've actually never heard of it, as he assumes. It may simply be an indication of fear of being asked to explain the concept by a teacher who is likely to cut their legs out from under them if their answer fails to satisfy. Indeed, given his attitude (not to mention his anti-American biases and male chauvinism, which he repeatedly expressed on the math-teach list in the 1990s through the most annoyingly patronizing responses to female posters), it's remarkable that any student would dare venture a public reply to a question about basic arithmetic in one of Toom's classes. I doubt it ever crossed his mind that his style might be just a teensy bit intimidating to women or to students in general. So much easier to blame the victims and to use the "evidence" to support his presuppositions.

Furthermore, it takes enormous arrogance, something of which Professor Toom appears to have an endless supply, to analyze a document such as PSSM or any other of the NCTM publications of which he is so disparagingly critical, when one has at best a tenuous command of the language in which it is written. I have corresponded at length with a Russian computer scientist/mathematician in France, Alexander Zvonkine, regarding the translation of a book of his, SMALL CHILDREN AND MATHEMATICS, into English. He knows not only that his translators thus far, not native speakers, have been inadequate, but to defer to the expertise of people who know the language and subject well enough and write with sufficient skill to guide him in developing an English edition of the text. His modesty is admirable. Unfortunately, it's not something shared by Toom.

I could readily dissect the "Wars" article in great detail but doubt it would have much impact on those already predisposed to take any attack on NCTM as correct, be it "mathematically" or otherwise. I just think it's too bad that Toom, whose earlier article was good enough that I used it in my weekly seminar for student teachers in secondary math at the University of Michigan in the early to mid 1990s, even before he became the darling of the anti-reform crowd, is so pig-headed and uncharitable when it comes to reform ideas in mathematics teaching. He had some useful insights about problems in American education, but he allowed his anti-American biases to get the better of him, and that is so overwhelmingly evident in this more recent foray into the Math Wars as to make the piece mostly incoherent and useless. Had he tempered his bile with the kind of rational thinking he no doubt employs when he does mathematics, he might have made another moderately useful contribution.

That said, I'm not going to fight about the issues raised below. They each have some use, but they aren't sufficient. And the "traditional word problem" argument was one he used to push incoherently here, rambling on about "problems by type," never seeming to grasp that the objection wasn't to any given problem but to the idea of teaching mechanical approaches to problem solving (see any of the algebra books by Dressler and Rich for a sense of what it means to teach a method for students that, if mastered, may well allow them to solve any problem that fits a type they recognize, regardless of whether they have the smallest clue as to why the method works or what the answer actually means, but woe to such students who encounter a curve ball problem that requires any original thought on their part because something doesn't quite fit into those neat rectangular diagrams they've been taught will work every time. It's a bit analogous to the idiocy of teaching "FOIL" to algebra students instead of addressing polynomial multiplication as an extension of the distributive property. Great if you will spend the rest of your mathematical life multiplying binomials and nothing but binomials. Not so good when a trinomial rears its ugly head.

Let me close with an extended quotation from another mathematician, W. W. Sawyer, whom I believe had many more useful and accurate insights into the issues with which we're still grappling in mathematics teaching and learning (I'd say "education," which should suffice, but I know how that makes some people's eyes go all glassy and prevents them from being able to see their noses in front of their faces):

"We do wish, in planning a syllabus, to take account of all the mathematics that is known: we want our pupils to be able to cope with the mathematical aspects of a scientific and technological age; we do not want to waste their time and effort on work that could be more efficiently done by a machine [apparently Dr. Toom and some other readers of this list feel otherwise]; we want them to have the best teaching possible. Satisfactory mathematical education can only be achieved by a proper balance between these considerations, and this is by no means easy to achieve in a world that is rapidly changing and in which there is no one competent to speak on all the departments of knowledge involved [except maybe for a couple of readers of this list, it seems]. A mathematician has to work very hard to learn even five per cent of the mathematics in existence today; he can hardly be expected to be well informed on the various sciences, on industry, AND ON TEACHING IN SCHOOLS. [emphasis added] Other specialists are in a like plight. Teachers are confronted with the difficult task of drawing on the specialized knowledge of a variety of experts, and of wielding their divergent ideas into a coherent whole.

"This task sounds, and indeed is, extremely complex. But great harm is done by any approach which ignores this complexity. In some countries, at an early stage of the education debate, mathematicians have been asked what they thought important, and it seems to have been assumed that their answers would automatically provide material relevant to the problems of industry and attractive to teach to young children. But the evidence for this mystical harmony is hard to find. Indeed, there is considerable evidence in the opposite direction. For specialists differ not only in what they know; they differ in their philosophies of life and in what they regard as important. To ignore this is to run the kind of risk you would take if you bought a car on the advice of a friend, and only afterwards discovered that, while you judged a car by the power its engine and its mechanical performance, he judged it by its colour and artistic appearance." W. W. Sawyer, A PATH TO MODERN MATHEMATICS, pp. 10-11.


All comments in brackets are, of course, my own. To connect Sawyer's last paragraph to the questions on the table, asking the mathematical community as a whole for advice on K-12 mathematics teaching and curricula and taking it at face value as the best possible advice available would be like taking a poll of presidential preferences via telephone in Chicago in 1948. You just might get a very skewed set of data that led to very and embarrassingly wrong conclusions.

On Oct 18, 2007, at 12:34 PM, Michael Sakowski wrote:

Thanks for posting! It was an interesting read.

Dr. Toom stresses 3 components:

1. Mastery of algorithms
2. Logical proofs
3. Traditional word problems, even if not real world

I will keep this in mind as I develop my curriculum in my sabbatical studies. I think he (Toom) is right on. Implementing will be tough. One person on this forum stated I am "going against the flow". This critic is probably right. But I think with the right motivation and using more of an entertaining setting (groups in competition), I think I can pull this off. My materials (in progress) are located at
They are a work in progress.

Monday, October 15, 2007

My First Piece for DA/THE PULSE

At the invitation of Gary Stager, I have written a piece for the on-line magazine THE DISTRICT ADMINISTRATOR/THE PULSE about the Math Wars.. Specifically, I've offered a perspective on K-12 mathematics viewed through the lens of W.W. Sawyer's introduction to A CONCRETE APPROACH TO ABSTRACT ALGEBRA.

My first contribution is called, "A Mathematician Weighs In On Math Course Construction, or Who Is W. W. Sawyer and Why Is He Saying These “Mathematically Incorrect” Things?"

Saturday, October 6, 2007

Who Invented "Lattice Multiplication"?

I can't seem to get the issue of lattice multiplication off my mind. The negative reactions to this perfectly sound algorithm are not grounded in any reasonable arguments, as far as I can see. But I can't help but suspect that many people, both those who are vehemently opposed to teaching alternative approaches such as lattice multiplication, as well as those who are open to or neutral on this and similar ideas, believe that somehow some hippie math ed reformers at the University of Chicago Mathematics Project (UCSMP) dreamed it up one night and stuck it into EVERYDAY MATHEMATICS (unless other hippie math ed reformers at TERC beat them to it with their INVESTIGATIONS IN NUMBER, DATA & SPACE curriculum). Alternate theories, some involving Satan, Osama Bin Laden, or Josef Stalin have not been verified as of this writing.

However, looking at Frank Swetz's CAPITALISM & ARITHMETIC, I came upon the following earlier today. I hope this doesn't cause riots among some of the anti-reformers:

A ready variant of this method is the following gelosia or graticola technique, so named for its similarity to the contemporary lattice grillwork used over the windows of the high-born Italian ladies to protect them from public view. Following Byzantine custom, the wives and daughters of Venetian nobles were usually kept sequestered. Spying unobserved from such vantage point, these ladies often saw scenes that disturbed them or made them "jealous."

The computational technique employed is basically the same as that used in per quadrilatero; however, the cells are partitioned by diagonal lines so that when the product of a row entry and a column entry result in two digits the units digit is written below the diagonal line and the ten's digit above. In this manner, carrying becomes an integral part of the algorithm. When all partial products are obtained, the entries within diagonal columns are added as before and the resulting total product written along a side and base of the array.

This method is quite old. It probably originated in India, was known to be popular in Arab and Persian works, and was finally accepted into European arithmetics in the fourteenth century. Due to its organizational efficiency and the ease it provides in multipling any two multidigit numbers, it was quite popular as a computational scheme; however, it was difficult to print and read and thus fell out of favor. It is from the gelosia grid and principle that the computational device known as Napier's Bones (1617) evolved.

Horrible to discover that this "fuzzy" approach has been around for many, many centuries, was popular and fell into disfavor only because of the limitations of 15th century printing methods (Swetz's book is focused on arithmetic in 15th century Europe). But of course, if it has organizational efficiency and ease of use, it just might be an okay thing to teach to kids as ONE way to do multiplication.

I read only a couple of days ago on an anti-reform math list that "[s]ince your child is only in first grade one thing you really need to be aware of going forward is that it is unlikely you child will be taught the standard techniques (now elevated to "algorithms" ) for doing basic arithmetic operations in EM." This was quite a surprise to me, since Everyday Math most definitely presents the "standard techniques or algorithms" for addition, subtraction, and multiplication. Long division, a point of much contention in the Math Wars, is not presented in the EM curriculum, though during my work in Pontiac, MI, the teachers at all five of the elementary schools where I coached introduced it anyway. I suspect they are not isolated cases. But aside from division, there is no doubt that the claim by this parent about EM is completely false. Sounds great, though, when you want to scare the bejeezus out of folks, especially those inclined to believe anything they read or hear that puts unfamiliar or "new" mathematics teaching tools in a negative light.

In any event, it appears that we can't lynch the authors at UCSMP or TERC for lattice multiplication. I have no doubt, however, that some charge or other can be made to stick and that sooner or later those who dare to try to help kids think about and do mathematics more deeply and effectively will be punished for their temerity.

Wednesday, September 26, 2007

Looking Further at Multiplication

For some reason, the lattice method of multiplication really annoys some people. I wish I could understand what appears to be an irrational rejection of a perfectly sensible approach that is just as mathematically sound as the "traditional" way most Americans were supposed to learn to do multi-digit multiplications.

I recently had another fruitless "dialogue" (read: smash-your-head-against-a-brick-wall, you'll-get-further argument) with an entrenched foe of anything and everything viewed as tainted by the evil worm of fuzzy, reform thinking. I will try to edit it into something that some readers may find useful. What started this was the posting by another notorious reform opponent of a link to a YouTube video that shows a way to do multiplication by drawing a series of crossing lines. Of course, the underlying mathematics is the same as what makes the lattice method (and pretty much all methods I've seen taught) work. But the video appears to be presented more as a "neat trick" than as something to be taken seriously, and I know of no one who is teaching it to kids. It wouldn't be disastrous if someone were, but it seems a bit too much like a bit of showmanship than something anybody wants kids to learn.

The person who posted this to math-teach sarcastically called it "the lattice method made easy" or something like that. I commented that it WASN'T the lattice method, but that this video hardly constituted a reasoned critique of lattice multiplication or any other alternative model or algorithm. And once again into the breach stepped an old antagonist to insist that the Vedic multiplication really WAS the lattice method. What follows is comprised of some of the things I wrote in response to this anti-reform critic.

Why Not Teach Multiple Methods?

It's rather hard to get away from the fact that many students like and prefer the lattice method to the "traditional" one. So foes need to either "get over it," as the saying goes, or deal with the real issue: this and any method that makes mathematical sense should be taught "transparently": that is, the sense as well as the procedure needs to be investigated by teachers and students. Otherwise, it's no more or LESS of a trick than is ANY algorithm. If you deny that, you'd best have some logical argument to support why it's different to do what you learned in school, that the "traditional" method is "real" math, while the lattice method is not, etc. I can't help but insist that such a viewpoint has no merit, as I've yet to see any logical argument to persuade anyone who isn't simply entrenched in the Mathematically Correct/HOLD dogma

Who cares whether something is "standard" as long as it works and makes sense to the person(s) using it?

Of course, we don't expect teachers to teach EVERY single algorithm they've ever heard of or seen. If a KID were to present the algorithm from the video, it would be nice for the teacher to be able to recognize the underlying structure. But why identify it specifically with the lattice method? What you really seem not to get (though maybe you do and just can't acknowledge for political or religious reasons) is that ALL the algorithms one sees in EVERYDAY MATH and other books, reform or not, are based on the same idea: find a fast way to do repeated addition based on the fact that we have a place-value system.

Four Models Worth Exploring

Area Model

1) Note: the example illustrated above uses smaller numbers than the example I have explicated in the text, but the basic procedure and interpretation are the same.

Take a piece of decimal graph paper and find the product of 32 and 27 on it in the manner I will describe: write along the top of the page "32" and draw a line along the top boundary line of the graph (not of the paper itself) that runs across three blocks of ten and then two single squares in the next adjacent block of ten. Then write 27 down the left margin and draw a line down the left boundary line of the graph that runs down two blocks of then and then seven unit squares in the next adjacent block of ten.

Next, extend a line from to the left at the end of the line you just drew equal to the length of the line you drew at the top. Extend a line from the right end of the new line up to meet the end of the first line drawn.

The rest, which is easier to show than write, involves picking four colors and with markers, crayons, or colored pencils, color the 10 x 10 squares one color, the 10 x 7 rectangles a second color, the 2 x 10 rectangles a third color, and the 2 x 7 rectangle a fourth color. Counting up, you have six 10 x 10 squares + three 10 x 7 rectangles + two 2 x 10 rectangles + fourteen unit squares.

This gives us 6 x 100 + 3 x 70 + 2 x 20 + 14 = 600 + 210 + 40 + 14 = 864

Expanded notation

2) Compare this with the "expanded notation" model:

32 x 27 = 30 x 20 + 2 x 20 + 30 x 7 + 2 x 7 = 600 + 40 + 210 + 14 = 864.

The Lattice Method

3) Note: the example illustrated at the beginning of this entry uses different numbers. I am staying with the same two numbers throughout my explanations, but again, the directions and analysis remains the same.

Compare this with the "lattice method." You draw a 2 x 2 box and draw a diagonal in each box from the lower left to upper right corners.

You write 3 and 2 over the top boxes, respectively.

You write 2 and 7 down the right side next to the top right and bottom right boxes, respectively.

You multiply 2 x 7 and write 1 and 4 in the upper and lower compartments of the bottom right-hand box, respectively. You multiply 2 x 2 and write 0 and 4 in the upper and lower compartments of the top right-hand box, respectively. You multiply 3 x 2 and write 0 and 6 in the upper and lower compartments of the top left-hand box, respectively. And then you multiply 3 x 7 and write 2 and 1 in the upper and lower compartments of the lower left-hand box, respectively.

Now starting in the lower right hand corner, you do diagonal addition from "top to bottom," moving from right to left and writing one digit under each box, carrying any extra digit, if it occurs, to the next diagonal. You get in this example, from right to left, 4, 6, 8, and reading this from left to right as we normally do, we have the correct answer, 864.

Because of the numbers chosen for this example, there are no carries. I've found that they are the only stumbling block likely to emerge that isn't related to errors in the individual multiplications or final additions due to carelessness or lack of knowledge of one-digit math facts. After practice, most kids handle this carry issue with ease. The model extends to more digits and to decimal numbers as well.

Is this algorithm "better" than others? Not for me, personally, but then, I didn't learn it until a few years ago. Lots of students like it. It seems like a "neat" (in more senses than one) way to do the partial products.

What's clear is that it's really NOT significantly more time consuming, no matter what the nay-sayers insist upon arguing, and certainly not to the extent that they would have us believe. It's a compact algorithm, like all of them. The only one discussed so far that I do not present to teachers or kids as a way I actually would like to see them do their work for more than the purposes of THINKING about (and in this case VISUALIZING) the partial product <=> area similarities is, of course, the first one, because we really don't need to do a drawn out set of rectangles and coloring (which clearly DOES take a relatively long time by comparison to all the others) to get the answers. We only want to see that there is some underlying relationship between area and multiplication. For very visual kids, however, sometimes the coloring, which I present fourth, not first, really helps tie everything together, and for some it's a key breakthrough model, so it's definitely worth spending some time on it, at most part of a period, and then re-examining ALL the models one uses to see how they relate to one another.

Keep in mind, too, that we could use other tools here, including blocks, tiles, or other hands-on models. But I think we have enough for now to make the important points.

The "Standard" Algorithm

4) Compare this with the "standard algorithm"

32 x 27 = "2 x 7 = 14. Write down the 4, 'carry' the 1'; 3 x 7 = 21. Add that 1 you 'carried' and write down 22. Now move down to the next line underneath and write a 0 under the 4 (alternatively, "mentally shift one place to the left before writing down the next digit); 2 x 2 = 4. Write down the 4 (under the middle digit above). 3 x 2 = 6. Write down the 6 to the left of the 4. Now draw a line underneath what you just wrote and do the resulting column addition. 4 + 0 = 4; 2 + 4 = 6; 2 + 6 = 8. Read your answer from left to right: 864.

(Yes, I purposely mechanized the description of the last algorithm, but not unfairly so: that's what kids ARE taught to do, after all, and it's not all that hard to see some of the places that they can and often do go wrong. No algorithm is fool-proof, and any algorithm is subject to errors in the sub-calculations as well as in where one writes the partial results (here, the partial products). What's GLARINGLY missing from this particular algorithm is any conscious consideration of place-value. Except for that shifting, which is generally taught as a mindless step to be religiously observed, there's no acknowledgment that with the exception of the first multiplication, everything you say to yourself is a lie. You never multiply 3 x 7, but actually 30 times 7, and so forth. The compression process gains speed but loses information.)

Now, if every kid who was carrying out this or any other algorithm had a reasonably good grasp of what s/he was being asked to do, none of that "lying" would matter. But for far too many kids, that standard algorithm makes as much sense as voodoo (perhaps less). Is that the fault of the algorithm itself? Not really. It's a fact, however, that our standard algorithms for both multiplication and division are all about speed, and so we sacrifice some information (what the REAL partial products are in the case of multiplication; what we're subtracting in pieces from the dividend at each step of the division algorithm) without, we hope, loss of accuracy, compressing the repeated additions or repeated subtractions, respectively, because place value lets us do this.

But you REALLY have to think like a kid: not a kid who either doesn't give a rat's behind about comprehension, but is aces at following instructions and knows the basic number facts well enough to do these two algorithms with accuracy and minimal screw-ups, nor like a kid who REALLY gets what's going on, practices, and then can do the steps automatically, but rather a kid who has holes in his knowledge of the facts and/or is not adept at following a set of steps that make little or no sense to him. This kid is going to make repeated mistakes, either in the sub-calculations or in where he writes things, or in the additions or subtractions, or, likely, all over the place. And this kid will be so bloody confused about where he's going wrong that he likely will sink into deeper confusion, quite possibly convinced that either multiplication or long division (or both) is way beyond him, or that these are some pretty messed-up algorithms.

There really are lots of places to mess up this algorithm (and the division algorithm). So rather than scream that these are perfectly good algorithms that EVERY kids MUST learn and that all other models and/or algorithms are stupid, inefficient, impractical, dumbed-down, not "real" math, etc., why not accept the fact they are all grounded in perfectly solid mathematics, that some models will connect first for some kids, while others will connect first for other kids, and eventually a competent teacher will do her best to see to it that all kids have the chance to make the underlying connections among these models and algorithms. Then, it's a tad easier, in all likelihood, to make a truly convincing case for the standard algorithm as the fastest (because you don't have to write down as much), but it will STILL be up to the student to choose.

If your goal is understanding and competence, I see no other choice. If it's a fanatical and irrational opposition to alternatives, well, be my guest, but you'll never get me or thousands of math teachers who work with real kids to accept such bizarrely rigid thinking.

The actual lattice method makes sense if you take the time to think about it. I find it objectionable, however, that it, like the "standard" algorithm and most of the rest of grade school math, is generally taught with no eye towards why and how it works, only as a black box to be followed mindlessly.

My antagonist wrote: "Of course the actual lattice method makes sense if you think about it. No one questions its validity, rather the questioning is whether it has any real instructive value, or lasting value."

How are the terms "real instructive value" or "lasting value" meaningful? They're undefined and unexplicated terms here that are, I suspect, offered for rhetorical rather than logical impact.

What I find objectionable to the anti-reform approach to "analysis" is that it seems to care only about any straying from the "traditional" black boxes.

I'll simply state that any black box is undeniably a black box. If you look inside the box and show or learn how it works, then for you it's no longer a black box. I've repeatedly said that I'm incensed by any teacher who teaches ANY method as a "black box." The fact that this one gets taught that way infuriates me. As for efficiency and volume of paper, I think those are clearly red herrings, especially the paper issue, though the time one isn't much more relevant in practice. A kid who knows this method can whip through it with facility. Ditto all the above models EXCEPT for the area model. There, both time AND paper are definitely issues. But it's a model to be explored only for understanding, not as a long-term strategy.

I was asked, "In the classrooms you prefer, does a lattice method get taught in addition to a standard algorithm, or instead of the standard algorithm?"

I have NEVER seen anyone teach the lattice method by itself. Never. What I prefer here is irrelevant to what is actually done, but to be perfectly clear: I want at least the four methods I outlined above taught, for reasons I hope I've made clear. For my money, the partial-products method is really at the heart of ALL other methods, so I would like to see it taught as the first "efficient, compact" algorithm. However, I believe that students should first do some one digit times one digit problems as repeated addition and then at least one two digit times one digit problems (e.g., 13 x 7 and 7 x 13) in "both directions so that the idea that repeated addition is far too time-consuming and space-consuming to be either efficient or (as the numbers get bigger) terribly accurate. How easy is it to write too many or too few 7's or to miscalculate when doing just the example given?

The idea is to have students appreciate what is gained by compression, but also to see what could be lost in term of information and what we "say" to ourselves as we do these other algorithms.

My opponent wrote:

For example, from the California Grade 4 Standards:
" 3.2 Demonstrate an understanding of, and the ability to use, standard algorithms for multiplying a multi digit number by a two-digit number and for dividing a multi digit number by a one-digit number; use relationships between them to simplify computations and to check results."

How is "demonstrate an understanding of" operationalized in any assessment you or the MC/HOLD crowd would accept? I know that "the ability to use" is operationalized on EVERY test I've ever seen.

He next wrote:

"There is still a use for teachers, Mike. And does your beloved lattice method makes it easier to detect 'understanding'?"

There's nothing more beloved about the lattice method, for me, than any of the others. They each have strengths and weaknesses, advantages and disadvantages. Any can be taught mindlessly. None SHOULD be taught that way.

"Demonstrating an understanding of ..." is not the same as 'mindlessly apply a method learned by pure rote drill and kill".

I agree

Then we're getting somewhere.

No, we're not. Because you still won't explain (and deliberately omitted my questions about) how you would assess the former in a way that you and I and Wayne and most reasonable, knowledgeable, intelligent people would accept as meaningful.

And you're still trying to denigrate all methods but one. I have no intention of doing anything of the kind, as I've explicitly outlined here and previously. I'm not trying to "sell" anything. I'm trying to get just one anti-reform person to finally admit that all these methods are grounded in real, valid mathematics, and that all can serve useful PEDAGOGICAL purposes. I don't really care which method kids choose to use, though I, too, would likely try to sell them on #2 and #4 over #1 (for sure) and #3 (though I have not objection to anyone who uses it if s/he uses it "well."

If your retort to all this is going to be more empty claims about time, paper, etc., leave me out of it. If you've got something non-rhetorical to offer, indicating that you've decided to drop your usual stance and engage in meaningful conversation, and if you want to answer that assessment issue and to operationalize the terms you used in your previous post on this topic, I'm all ears.


I wish I could report that the reply I received was useful. Unfortunately, as the reader likely has gleaned from just the dialogue in this entry, my opponent remains just that: a dedicated foe who has no interest in conceding anything positive about any "non-standard" approach to thinking about, teaching, or doing multiplication (or long division, or anything else in mathematics). Naturally, since I have a long history with him, I was not surprised. I no longer write on public lists in hopes of convincing those who are incensed about every and any reform idea or method with which they themselves were not taught, be it in mathematics, science, literacy, or any other educational area. I write to help clarify my own ideas and for those readers who are at least moderate or neutral or simply looking for varied viewpoints. When I mentioned that I would not continue with this fellow on these issues, but that I expected that what I presented about these models was likely useful to some readers, he retorted, "I doubt it." And of course, he really does.

Sunday, September 23, 2007

Reasons to be Cheerful, Part -3

Reasons to be cheerful (part three)

Summer, Buddy Holly, the working folly
Good golly Miss Molly and boats
Hammersmith Palais, the Bolshoi Ballet
Jump back in the alley and nanny goats

18-wheeler Scammels, Domenecker camels
All other mammals plus equal votes
Seeing Piccadilly, Fanny Smith and Willy
Being rather silly, and porridge oats

A bit of grin and bear it, a bit of come and share it
You're welcome, we can spare it - yellow socks
Too short to be haughty, too nutty to be naughty
Going on 40 - no electric shocks

The juice of the carrot, the smile of the parrot
A little drop of claret - anything that rocks
Elvis and Scotty, days when I ain't spotty,
Sitting on the potty - curing smallpox

Reasons to be cheerful part 3

Health service glasses
Gigolos and brasses
round or skinny bottoms

Take your mum to paris
lighting up the chalice
wee willy harris

Bantu Stephen Biko, listening to Rico
Harpo, Groucho, Chico

Cheddar cheese and pickle, the Vincent motorsickle
Slap and tickle
Woody Allen, Dali, Dimitri and Pasquale
balabalabala and Volare

Something nice to study, phoning up a buddy
Being in my nuddy
Saying hokey-dokey, singalonga Smokey
Coming out of chokey

John Coltrane's soprano, Adi Celentano
Bonar Colleano

Reasons to be cheerful part 3

If the above rings no bells, you're probably too young or too old to recall the sprightly Ian Drury and the Blockheads. You owe it to yourself to download the above song immediately and listen to it several times before proceeding. This way, there be monsters.

Bad Day At School

There's nothing quite like having a classroom full (and I do mean FULL: as in 36 of 42 allegedly enrolled students in a room with desks and books for 30) of "precalculus" students with too much heat in the room and too much ire in their souls as you try to take them through a review of something that on the one hand you know they SHOULD know, and on the other you suspect with good reason is going to be news to most of them, despite having allegedly passed two years of algebra and one of geometry. The review topic in question: the point-slope form of a linear equation. The first task, having ascertained that the majority of students at least believed they knew what "slope" meant and could find it given the coordinates of two points, was to find the equation of the line that passed through them. Little did I realize just what sort of tiger trap I was about to fall into.

Picking two points from one of the textbook problems, I began guiding the students through the process of using the point-slope form, with my usual plan of showing that regardless of the point one chose to use in the equation, it would result in the same graph, the same line, and, upon applying a little algebraic manipulation into the slope-intercept form, stark evidence that the equations arrived at with either point were equivalent, as of course should be the case if they have the same slope and pass through the same two points.

Unfortunately, I never got that far. So much time had been spent dealing with logistical issues regarding the shortage of seats and books, as well as dealing with various disciplinary issues, trying to take attendance on-line (PowerSchool, where is thy sting?) given that I was using my laptop, not the promised but as yet invisible Pentium-based desktop Mac that would have made the process much less time-consuming (for reasons not worth going into here), I knew that if things continued to go less than smoothly, it would be difficult to finish what was supposed to be the only major mathematical point I thought I might be able to look at with them before class ended. However, as I started to proceed with the equation with one of the two points, I was rudely and persistently told by a vociferous subset of the class who had been making things difficult (when not making them impossible) for much of the two weeks we'd been meeting that I was clearly wrong.

At first, I thought they meant that I'd made some sort of calculation error, hardly an impossibility under the best of circumstances, and these were anything but the best. I began to recheck my work, but the sound and fury from their part of the classroom made rational thought or even simple calculation a doubtful process at best. Finally, I realized what was amiss: they had taken PART of my lesson as gospel, but refused to attend to the crucial caveat. Since I labeled the points as "point 1" and "point 2" respectively, and since the coordinates were labeled with the appropriate subscripts, clearly I was in error when I chose "point 2" as the one to plug into the equation. After all, hadn't I written the point-slope form with subscripts x1 and y1?

How could I possibly be stupid enough to now be telling them that the point that had 2s in the subscripts were kosher (okay, no one said "kosher" and in fact, none of what was said approached in any way the slightest degree of civility, but this is a family blog, after all)? Obviously, I'd made a huge mistake and anything I did from this juncture on was going to be horridly wrong.

Now, this was hardly the first time I'd taught this topic, either to high school or community college students. Indeed, having taught a lot of algebra courses over the years, I was used to there being skepticism that it didn't matter which point was used and that it would be possible to show that whichever was selected would be fine, and that the resulting equations, though initially appearing to represent different lines, would ultimately turn out to be the same, without any doubt at all.

However, nothing in my experience with non-alternative education students had prepared me for a group of students who were supposedly ready for precalculus and who not only were so confused by a notation issue, but more importantly were NOT going to be patient or trusting enough to see if just possibly their teacher had a clue about what was going on and could, in short order, demonstrate that fact and part the clouds.

Be Careful What You Wish For

I hoped, of course, for a teachable moment. I figured that either I'd get to finish the problem with input from the students, and, having tried both points, they'd start to see (or in some cases REMEMBER from previous experience) that all was indeed well, or that if I were extraordinarily fortunate, the light would go on for most of them before I even got to that point, and, mirabile dictu! they would look at me with new-found respect, paving the way for a successful and productive year.

Instead, things bogged down as no one in the class would agree to see what happened if they finished the problem with those coordinates with the OTHER subscripts and I finished it with the point I'd selected: the hue and cry became such that more time was lost trying to maintain some order, and when the bell rang, the problem remained unfinished, several students loudly agreeing with my chief antagonist in the class when he proclaimed that I "didn't know what I was doing."

While of course there have been moments in my teaching career when I really DIDN'T know what I was doing, either mathematically, pedagogically, or a combination thereof, this was most decidedly not one of the times I was unclear about the mathematics. I was very confused, however, about what kinds of experiences would lead a class of seniors and juniors to be so invested in proving that the teacher couldn't possibly right, and not by making a convincing mathematical argument, but merely by making it effectively impossible for the teacher to show that he might actually be (dare I say it?) right.

What the fudge?

I'll stop at this point, leaving readers to consider their own experiences in similar circumstances, if any, giving everyone ample opportunity to contemplate what might have saved the day in this or similar situations. I'm sure that my "solution," such as it was, will not be terribly satisfying, so there's no hurry on my part to offer it. By all means, I'm sure others would have done much better in my shoes than did I, and I'm interested to learn about the alternatives that I might have employed but did not.

If only I'd had a Vincent Black Lightning 1952 and a red-headed girl waiting outside, instead of a Japanese sedan and a horrid, traffic-snarled 80 minute slog home.