Archive for June, 2007

Posted today at Grey Matters. I’m not in it because, um, I didn’t write anything interesting for the last couple weeks.

Tudors and Teachers

(Originally published inThe Ten Page News #24 (December 1998).


“Reeling and Writhing, of course, to begin with,”the Mock Turtle relplied; “and then the different branches of Arithmetic—Ambition, Distraction, Uglification, and Derision.”
—Alice’s Adventures in Wonderland

My wife and I, prompted by a display at the public library, have recently finished viewing the BBC video series Elizabeth R and The Six Wives of Henry VIII. I wasn’t very familiar with 16th Century England, so, naturally, I got out a few reference books to help me keep track of what was going on. Readers more familiar with the period will kindly bear with me: a brief summary follows.Recall that the Protestant Reformation began in 1517 (the year of Martin Luther’s famous 95 theses). This was the end of centuries of Papal supremacy in Western Europe (or, as it was then known, Western “Christendom”). King Henry of England, obsessed with his lack of a legitimate male heir, divorced Catherine of Aragon in defiance of the Pope and married his mistress Anne Boleyn in 1533. For good measure, he got Parliament to declare him supreme head of the Church of England. Catholics denying him this authority were executed for treason. These included the prominent scholars (later saints) John Fisher and Thomas More. Henry went right on killing his enemies (wives, bishops, and so on) throughout the remaining fifteen years of his reign, meanwhile suppressing monasteries (and nationalizing their treasuries).

Henry died in 1547 and was succeeded by his son Edward; Edward died in 1553 and was succeeded by his half-sister Mary, a devout Roman Catholic. Mary reigned for five years and executed hundreds of Protestants in the religious persecutions that earned her the nickname “Bloody Mary” (her victims include Archbishop Thomas Cranmer, the scholar largely responsible for the English Book of Common Prayer). Mary died and was succeeded in 1558 by her Protestant half-sister Elizabeth, whose reign was long and glorious. Elizabeth put down the odd rebellion now and then, and executed her Catholic cousin Mary Queen of Scots, but was generally (and justly) renowned for her moderation in the sphere of religion.

Over four hundred years have passed since then; the violence between Protestant and Catholic factions of the Christian Church continues (though not on so spectacular a scale). This is a great mystery; anyway, I sure don’t understand it. This much is clear: capable leaders on both sides were sacrificed along with large numbers of their followers. Officially, the issues involved concerned Christian doctrine—things like “what does the Last Supper mean?”. The real question in my interpretation usually comes down to“who has authority over whom?”.

Now let us turn our attention to “Calculus Reform”. Here again, though the surface issues concern doctrine (How useful are graphing calculators? How much writing should students do?), the underlying “real question” in my reading comes down to “who has authority?”. To my knowlege, no one has yet been beheaded or burned at the stake to settle these questions. Careers have been made and ruined in fairly large numbers, however, so our interest is not entirely, so to speak, academic. [Readers alert to contemporary academic jargon will very likely find these remarks reminicent of the deconstructivist school of “literary criticism”—one major theme of that school being the interpretation of “texts” (i.e., everything) in the light of power relations. These critics apparently take their cue from Karl Marx’s dictum: “All history is the story of class struggle”, and neglect his remark: “The philosophers have tried to understand the world. The point is to change it.”.]

My own particular ejection from the profession of college mathematics, for example, was precipitated by my public reactions to a consultant hired by the administration of my ex-college. Now, if the President of the College wants to pay some guy a hundred bucks an hour to make it rain, that’s her business. As Scott Adams says: “Consultants have credibility because they are not dumb enough to be regular employees at your company.” [The Dilbert Principle, 1996, HarperCollins New York] But the consultant knew nothing at all about my ex-job and I told him so.

Here is a story from Calculus: The Dynamics of Change (1996, the Mathematical Association of America; 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 engineeering departments, the Deputy Director of the Computing Center and members of his staff, the Associate Vice Chancellor of reasearch 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.

My ex-college suffers from many of the same problems brought out in this passage: a percieved need for mock participation by a bloated bureaucracy in complete and willful ignorance of the actual needs of the student body, for example.

I seem to be getting carried away; sorry; I do tend to rant. Let me return to my theme.

To the extent that great historical events can be explained, there is a general agreement that one of the “reasons” the Protestant Reformation occurred when it did was the emergence of a powerful new information technology: movable type and cheap printed books. Our current reform is likewise associated to some extent with a new technology: cheap computers. Incorporating the new technology into existing institutions calls for great and rapid change; certainly an opportunity for the ambitious and possibly also for the devout. Not without risks, of course.

If I’m just bitter over a personal failure and complaining of sour grapes, well and good. I rather hope so. I will miss teaching—the only job I ever loved. I am indeed somewhat bitter. But I would act the same again; I will not study, nor will I teach, the Law of Authority: “Because I say so, that’s why.” I suspect that my case is not so exceptional and fear for my ex-profession.

When the smoke of the English Reformation cleared (briefly … but that’s another story), there was still enough talent left in the Church to produce the great religious treasure of the English Language: the “King James” Bible (James, the son of Mary Queen of Scots, succeeded Elizabeth in 1603).

Speaking of literary treasures; speaking of reform and my ex-career; speaking of “another story”: King Lear had three daughters . . .

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

                    89                    

                   +34                    

                   ----

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.

Summer Quarter Begins

I’ve been “bumped” by a full-timer out of one of the (two) classes I was planning to teach this summer. Enrollments are always down in Summer Quarter and it looks like this summer’s worse even than usual. Learning center hours are of course also made scarce in their turn … it looks like I’ll have to search  for honest work. Argh.

Today’s Links
\bullet The Philosophy of Mathematics Education Journal.
\bullet Donald in Math Magic Land on YouTube: here and here
(spotted at What the heck [05.22.07]).
\bullet Also, leingang on calculus books (same source, 05.21.07).

More Linkage

Turning our attention to beginning algebra courses. First of all, the text I’ve been using most recently is called Intermediate Algebra; “intermediate between pre-algebra and actual (university-credit earning) algebra” is the most charitable spin I can put on that. Anyhow, here again, set theory is used so clumsily that it’s hard not to attribute malice to somebody along the line (Hanlon’s Razor notwithstanding).

Consider, then, an abomination like
\{ x | x is a natural number less than 3\}.

If we’re going to go around calling a perfectly inoffensive set like \{1, 2\} out of its name in order to make some point about our notations, we’d be much better off to actually pretend we believed these very notations were actually good for something and write instead
\{x | x \in N, x \, \langle\, 3\}.

Or how about
\{x| x is a real number and x is not a rational number \}? Doesn’t it just make you want to, I don’t know, hurl the chalk at something? Actually, I have to admit that I’ve copied this display on several blackboards in my time … but only to illustrate a point (namely, that it was created by enemies of mathematics and that one of course really means
\{x| x\in R, x\not\in Q\}).

The symbols in question (N, R, Q, \in, \not\in) appear in this very section and are used in the exercises.  Though not, of course, in the exercises about “set-builder” notation — no, these have all been carefully contrived to reinforce the reader’s impression that our goal in presenting this material is to make easy things hard by way of the whole ignore-the-point word & symbol mishmosh I’ve just been complaining of.

But then, that brings us to the saddest part of the whole sorry business. We don’t actually need (still less want) these symbols — or the set-builder notation itself! — for whatever follows in the whole rest of the book! And pretty much every 9th-grade-algebra-for-college-students text that’s come out in several (admittedly very short) generations does things in exactly the same way!

I have what feels like a pretty coherent theory of how things got this way (though essentially no idea as to “what, then, are we to do?”).  But I promised myself I’d keep it brief, so I’ll just wave my hands in the direction of The Muddle Machine.

Random Linkage

Politics

There’s evidently been a tremendous amount of controversy over math-ed issues in Ridgewood, NJ lately. Kitchen Table Math has covered the story pretty extensively. Anti-counter-reformer Michael Paul Goldenberg takes the opposite postion here and here.

I haven’t followed the story up until now … in fact, I’ve posted the links here largely so I won’t lose track of ’em … but this appears to be as good a place as any to introduce the “Math Wars” issues that I’m planning to have much more to say about in this space.

Interview With a Blogger

The English translation of this interview with Mikael Johansson is well worth a look. Am I blathering?

Carnival!

The Tenth Carnival of Mathematics appeared today at Dave Marain’s MathNotations. And I’m not only telling you this because there’s a link there to my “Textbooks and Notations” rant of a few days ago. No, I’m telling you because the Carnival is always worth a look. Here are summaries of the first nine Carnivals (by Steve of Mathematics Weblog).
But the really big news is that I’ll be hosting the 14^{th} Carnival right here on August 10 (d.v.). If you can’t beat ’em join ’em.