the livingston review

Constructivism, High-Tech, and Multi-Culti
Draft of a talk at the AMS Special Session on Mathematics Education and Mistaken Philosophies of Mathematics, January 1999.

Introduction
I cheerfully admit that I’ve bitten off much more than I can chew: my title was published before I’d written the paper. I’ll barely scratch the surface of any of the three subjects in my subtitle here.

Constructivism.
EXHIBIT A: Quotes taken from “The Harmful Effects of Algorithms in Grades 1—4”, by Constance Kamii & Ann Dominick in The Teaching and Learning of Algorithms in School Mathematics (NCTM Yearbook, 1998).

“The teaching of algorithms is based on the erroneous assumption that mathematics is a cultural heritage that must be transmitted to the next generation.” (p.132)”Some leaders in mathematics education also began to say that we must stop teaching algorithms because they make no sense to most children and discourage logical thinking.” (p.141)

I only wish these quotations could be allowed to speak for themselves. But apparently some people don’t find them as outrageous as I do; these ideas are taken seriously by the NCTM, after all.(The National Council of Teachers of Mathematics needs no introduction to the audience of my talk—suffice it to say here that they’re the professional organization most aligned with Math Ed “reform”).

Kamii & Dominick base their pedagogical strategy on “Piaget‘s Constructivism”, which contrasts [by their account] “logico-mathematical” knowledge with “social (conventional) knowledge”. The sum of two numbers, for example, is culture-independent, whereas the algorithm for calculating the sum is not. Of course this much is true. It does not follow that there is no benefit in choosing one particular algorithm and studying it in detail.

Kamii and Dominick say that

“When we listen to children using the algorithm to do

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for example, we hear them say, “Nine and four is thirteen. Put down the three; carry the one. One and eight is nine, plus three is twelve . . .” The algorithm is convenient for adults, who already know that the “one”, the “eight”, and the “three” stand for 10, 80, and 30.” (p. 135)

But the adults know it because they learned it in elementary school! Because somebody transmitted it “from generation to generation”!

Of course, there is a sense in which everyone will agree that students must “construct” their understanding of a mathematical theory. Sometimes memorizing formulas is enough to get students through exams; we’re all agreed that these students haven’t learned the mathematics. Let’s call this uncontroversial idea “weak” constructivism. “Strong Constructivism” (as I here propose to call it) would have us believe that we should not urge our own models on the students. But isn’t it clearly unreasonable to expect our students to recapitulate the entire evolution of techniques discovered by geniuses over the course of thousands of years? No matter how much guidance on the side we provide?

Now, admittedly, the position that algorithms are evil is at least consistent no matter how wrong headed, and my title has advertised contradictions of education reform. So consider this:

It’s completely absurd that anybody should try to persuade us to believe in the constructivist model. After all, if the model is valid, then it must be necessary for each of us to construct our own understanding of teaching methods. Or does constructivism only apply to what somebody else is trying to do?

(I’m not the first to have noticed this irony. Similar remarks can be found at Applications and Misapplications of Cognitive Psychology to Mathematics Education 1, by John R. Anderson, Lynne M. Reder, and Herbert A. Simon of Carnegie Mellon University.)

High Tech
Maybe the most glaring “contradiction”in the computerization of mathematics education is the claim that it represents a reform at all. In fact, it’s very much a done deal. For example, high-tech teaching experience is frequently mentioned in job descriptions.

Here’s some data from the classified ads seeking faculty in the latest Chronicle of Higher Education (as I prepare these notes: 11/13/98).

Of the ten ads in Mathematics, five list high-tech in their desiderata (and one is actually a Math Ed ad). The wording varies in strength from “experience . . . integrating technology into the curriculum is a plus” to “ability to integrate computers into teaching is required”. (There are also four classified ads in Mathematics/Computer Science and four in Mathematics Education.) I used to read the ads every week and can tell you for sure that this is about typical. One effect of all this is that programmers and administrators have more and more to say about the curriculum — and mathematicians and teachers have correspondingly less and less to say.

What are the effects of including computers on the math curriculum itself? Well, I don’t at all doubt that Mathematica, for example, can do lots of useful things. Neither do I doubt that it’s well worth teaching students how to use. But it’s just not Calculus. A “calculus” is by definition a collection of calculational tools. Of course, for historical reasons, capital-C “Calculus”, refers to what is more formally called “Differential and Integral Calculus” — a set of techniques that Mathematica actually replaces rather than supplements. The user, for better or worse, is allowed to avoid having to do calculations by partial fractions, integration by parts, L’Hôpital’s rule, or what have you. Maybe the famous Calculus & Mathematica should have been called “Calculus XOR Mathematica”!

In “The Case Against Computers in K–13 Math Education (Kindergarten through Calculus)” (The Mathematical Intelligencer Volume 18, No. 1, Winter, 1996), Neal Koblitz says of computerization that “The downside can be divided into several broad areas:

1. drain on resources (money, time, energy)
2. bad pedagogy
3. anti-intellectual appeal
4. corruption of educators[.]”

Under 1., Koblitz points out that “Ironically, it is sometimes the colleges with the highest proportion of working-class students that become most enamored of expensive new gadgetry for teaching mathematics” and goes on to conclude that “All of the fuss about computers serves to divert attention away from the central human needs of the school system — better conditions for teachers and better teacher training.”. As for 4., he says “About 2 years ago, the NSF asked me to help evaluate calculus reform proposals. But when they learned that I am skeptical about computers and graphics calculators in calculus, they changed their minds and decided not to send me any proposals to review. They did not want any input from a nonbeliever.”. Which, as we shall see, brings us in a roundabout way to . . .

Multi-Culti
Here’s a quote from Fashionable Nonsense, by Alan Sokal and Jean Bricmont (1998, Picador USA New York):

“[Suzanne K. Demarin] is a prominent American feminist pedagogue of mathematics, whose goal — which we share wholeheartedly — is to attract more young women to scientific careers. She quotes [a text by Luce Irigary] and continues by saying:

In the context provided by Irigary we can see an opposition between the linear time of mathematics problems of related rates, distance formulas, and linear acceleration versus the dominant experiential cyclical time of the menstrual body. Is it obvious to the female mind-body that intervals have endpoints, that parabolas neatly divide the plane, and, indeed, that the linear mathematics of schooling describes the world of experience in intuitively obvious ways?

This theory is startling, to say the least: Does the author really
believe that menstruation makes it more difficult for young women to understand elementary notions of geometry? This view is uncannily reminiscent of the Victorian gentlemen who held that women, with their delicate reproductive organs, are unsuited to rational thought and to science. With friends like these, the feminist cause hardly needs enemies.”
—Sokal & Bricmont, p. 121.

Staggering amounts of research are devoted every year to differences in “learning styles” between the sexes or among groups from different ethnic backgrounds. The often-stated goal of much of this research is to allow greater fairness. Of course I share this goal. But it seems clear (to me) that in many cases, the effect has been exactly the opposite: to perpetuate stereotypes. I won’t consider this topic any further here; anyway there’s no apparent contradiction in pursuing such research. Suffice it to remark that there is room for reasonable people to differ on the question of its value.

The real contradiction of multi-culturism, in my view, is its co-optation by the mono-culture of managerial group-think. For me, this is the heart and soul of the “math wars”: what is mathematics education for? The NCTM, NSF, and suchlike enormous bureaucracies take it for granted that our mission is to prepare a “workforce”. Our job as teachers in this view is to train our students in such skills as they will require to fit into corporate culture. In short, we are to be managers.

An NSF (National Science Foundation) representative at a conference I attended (the Ohio Section of the MAA [Mathematical Association of America], Fall 1998) gave a presentation in which she listed several criteria used by that agency to rate grant proposals. Included in her list were consultation with outside agencies, other departments, and so on; these are apparently considered good for their own sake. But in some cases, the opposite is true: too many cooks spoil the broth. Consider this story from Calculus: The Dynamics of Change (MAA, 1996; pages 47 and 49):

. . . an existing calculus reform project was determined to be a suitable program for adaptation at the University of Mississippi. The University invited one of its developers to visit the campus for two days to explore more extensively the feasibility of the University’s being a test site for the project. During that time, the developer had separate meetings with the Mathematics faculty, the chairs of the other science and engineering departments, the Deputy Director of the Computing Center and members of his staff, the Associate Vice Chancellor of research and a member of the Office of Development,the Vice Chancellor for Academic Affairs, the Dean of the College of Liberal Arts, and the Director of the University’s Writing Program. In addition, he made a presentation to the university community on his project.After his visit, the Department’s faculty agreed that the goals and the materials provided in this project are excellent and match well with the goals and needs of the Department.

. . . As the Department continued teaching sections of the reformed course, it became apparent that students, and to a degree faculty, were having difficulty with the expectation that the text should actually be read. This led the Department to adopt a different reform text while continuing with laboratory assignments and group projects. The Department has developed its own laboratory manual and uses group projects and microcomputer laboratory assignments, but continues to struggle with the selection of a text.

As I remarked in The Ten Page News (#24), this passage highlights “a perceived need for mock participation
by a bloated bureaucracy in complete and wilful ignorance of the actual needs of the student body”. Notice in particular that graduate students and adjunct faculty are apparently not considered worthy of consultation — the very people who actually know what’s going on in the classes being “reformed”. Why has the NSF committed hundreds of millions of dollars to take power away from teachers?

Even the opponents of such “fuzzy math” as is found in controversial texts like Addison-Wesley’s Focus on Algebra: An Integrated Approach — a model of multi-cultural inclusiveness but very weak on actual mathematics — have generally assumed that their failure to inculcate skill in mathematics has been due to a misguided faith in false educational philosophies. (Richard Askey’s review of this textbook is available on the web [at the Mathematically Correct counter-reform site]).

But what if the suppression of mathematical thinking is actually deliberate? Mathematics is characterised by its intolerance of nonsense; it’s not surprising that such an attitude should meet with hostility from those in the business of pushing people around.

I’m afraid I’m starting to sound sort of paranoid. Anyway, for better or worse, there’s no time to follow up on this line of discussion; I can only hope that those who are unpersuaded will follow some of the links. Joe Esposito has documented a
“Tangled Web” of

interlocking relationships between National Center on Education and the Economy (NCEE), which produced America’s Choice: High Skills or Low Wages, the Department of Labor’s Secretary’s Commission on Achieving Necessary Skills (SCANS), the National Skill Standards Board (NSSB), and Achieve . . .
the deception and arrogance of the promoters of “School-to-Work”. . .

and their connection to TQM (Total Quality Management); another rich source [was an] “Education Deform” page by Arthur Hu.