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concur with the idea that Toom's earlier article, "A Russian Teacher In America," had much more of value than does the unpublished "Wars In American Mathematical Education."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.