LEARNING MATH BY THINKING - Hassler Whitney, Louis P. Benezet, and how many more wasted lives and decades will it take?

It was 1986, folks (or perhaps 1929), for those keeping score at home. Twenty-three (or eighty) years later and the same arguments are going on, the same mistakes are being made, as if nothing at all has been said like what Louis P. Benezet or Hassler Whitney offered. As if Constance Kamii's work has never been done or published.

My thanks to G. S. Chandy for pointing me to this article. It was published while I was in the process of taking undergraduate mathematics courses in NYC and slowly gravitating towards changing fields from literature to mathematics education. I'd never heard of NCTM or any of the other organizations involved in mathematics teaching and research, the Math Wars hadn't officially started yet, and had I read this piece at the time, I would have naively wondered how anyone could be on the other side from people like Benezet and Whitney. Having suffered through a K-12 mathematics education that was about as inspiring as a dead fish in the gutter, it is remarkable to me even today that I took it upon myself in my thirties to go back to school just to prove to myself that I could indeed learn more mathematics, from calculus to where I was led. Didn't plan to go into the field, and it was fortune, more than anything, the attention of one of my instructors, that led me to start teaching remedial mathematics and, eventually, to do graduate work in math education at the University of Michigan in the 1990s.

Yes, there has been some progress since then, but the entrenchment of traditionalists is fiercer than ever. The lies, distortions, selective quotations, meaningless and carefully culled data, shifting criteria for what "counts" when it comes to evaluating programs, teachers, schools, kids, materials, etc., and many other shady tactics continue unabated, fueled by a hatred for innovation and purveyed by politically-motivated, educationally conservative and reactionary pundits, think-tanks, and foundations, all fiercely determined to see to it that mathematics teaching and learning in this country remain in the hands of a smug, patronizing elite. As long as they are successful in reducing us to a standardized-test crazy culture, as Whitney so accurately describes below, the country as a whole and millions of children will suffer unnecessary torment and boredom when it comes to mathematics. And a populace that is mathematically ignorant is a populace that is far easier to lead by the nose.

Are Benezet's and Whitney's ideas really just those of a couple of isolated cranks, as the anti-progressives from groups like Mathematically Correct would have us believe? Consider for a moment the following anecdote about the great mathematician, Augustin Louis Cauchy:

A mathematical friend of Cauchy's father, Lagrange, recognized the young boy's precocious talent and commented to a contemporary, 'You see that little young man? Well! He will supplant all of us in so far as we are mathematicians.' But he had interesting advice for Cauchy's father. 'Do not let him touch a mathematical book till he is seventeen.' Instead, he suggested stimulating the boy's literary skills so that when eventually he returned to mathematics he would be able to write with his own mathematical voice and not one he had picked up from the books of the day.

It proved to be sound advice. Cauchy developed a new voice that was irrepressible once the floodgates protecting Cauchy from the outside world had been reopened." (Marcus du Sautoy's THE MUSIC OF THE PRIMES pp. 65-66)

No, the analogy to Benezet's experiment is not perfect by any means. But it does suggest that there has long been an awareness that some aspects of formal instruction as they become institutionalized can be stifling to creativity and originality.

Does this mean I am advocating for the destruction of public (or any) formal education and schooling? Not quite. What I am advocating for is a coming to sanity on the part of educators in this country when it comes to mathematics teaching (if nothing else). We are destroying our children, en masse, with the most stultifying approaches imaginable to learning and doing real mathematics, substituting instead a phony "school mathematics" that serves no one truly well, and from which only a small minority emerges able to actually do mathematics, in spite of, rather than because of, the way the subject is taught in most instances.

Few American K-12 teachers have the smallest idea what mathematics is, what it means to do mathematics, or what it means to be a professional mathematician. And what these teachers wind up doing, consciously or not, is to guarantee that very few students will ever find out.

Every time I bump into a piece like the one about Whitney and Benezet below, I am both amazed and sickened: amazed that I hadn't seen this before (though I've been aware of Benezet for over a decade now); sickened that the same lies rise like a foul smokescreen every time Benezet's name or any idea that sounds even vaguely like his, is presented. How much longer do our children have to be tormented by meaningless mathematics education? When will it be time for real mathematics to be taught and learned, in a manner suitable to how children are? If not now, when? If not us, who?

ABOUT EDUCATION; LEARNING MATH BY THINKING

By FRED M. HECHINGER

Published: June 10, 1986

SCHOOL reformers, business executives and politicians are demanding more mathematics for American children. Schools are responding, at least in terms of the hours given to math. Not all mathematicians are cheering. They worry that pressures for more hours of mathematics may hurt rather than help, unless mathematics is taught differently.

Dr. Hassler Whitney, a distinguished mathematician at the Institute for Advanced Study in Princeton, says that for several decades mathematics teaching has largely failed. He predicts that the current round of tougher standards and longer hours threatens to ''throw great numbers, already with great math anxiety, into severe crisis.''

Dr. Whitney has spent many years in classrooms, both teaching mathematics and observing how it is taught, and he calls for an end to what he considers wrongheaded ways.

Long before school, he says, very young children ''learn in manifold ways, at a rate that will never be equaled in later life, and with no formal teaching.'' For example, they learn to speak and communicate, and to deal with their environment. Yet the same children find much simpler things far more difficult as soon as they are formally taught in school.

Learning mathematics, Dr. Whitney says, should mean ''finding one's way through problems of new sorts, and taking responsiblity for the results.''

''This has been completely forgotten'' in most schools, he finds. ''The pressure is now to pass standardized tests. This means simply to remember the rules for a certain number of standard exercises at the moment of the test and thus 'show achievement.' This is the lowest form of learning, of no use in the outside world.''

Dr. Whitney, in a recent report in The Journal of Mathematical Behaviour, recalled an experiment begun in 1929 by L. P. Benezet, then superintendent of schools in Manchester, N.H. Mr. Benezet was distressed over eighth graders' poor command of English and their inability to communicate ideas.

''In the fall of 1929,'' he wrote in 1935, ''I made up my mind to try an experiment of abandoning all formal instruction in arithmetic below the seventh grade and concentrate instead on teaching the children to read, to reason and to recite'' by reporting on books they had read and on incidents they had seen. The children were no longer made to struggle with long-division. ''For some years,'' Mr. Benezet went on, ''I had noticed that the effect of the early introduction of arithmetic had been to dull and almost chloroform the child's reasoning faculties.''

Over the years numbers crept into the children's experience, Mr. Benezet said. They learned to deal with ''halves'' and ''doubles,'' with estimates of size, with a natural development of multiplication tables and slowly, with formal arithmetic.

Mr. Benezet concluded that children who had not been dragged into early but only dimly understood mathematics eventually outdistanced those who had. Literacy in English and a capacity to think independently and to speak and write clearly helped many to do well in mathematics, too.

Dr. Whitney points to that experiment as he looks at today's mathematics teaching. He cites the responses to a problem on a recent test given by the National Assessment of Educational Progress: John and Lewis are planning a rectangular garden 10 feet long and 6 feet wide, and they want to put a fence around it. Ignoring such real matters as the need for a gate, the question was simply how many feet of fencing was needed.

Of the 9-year-olds who took the test, 9 percent chose 32 feet; 59 percent, 16 feet; 14 percent, 60 feet, and the remaining 18 percent gave other answers. Of the 13-year-olds taking the test, 31 percent said 32 feet; 38 percent, 16 feet; and 21 per cent, 60 feet, with 10 percent giving other answers that apparently did not use any arithmetical formulas.

''Why did not all the children get the correct answer?'' Dr. Whitney asks. ''If they were involved in it as a real problem they could have drawn a picture or made it real in some way, and looked to find the answer.'' Instead, he said, they did it ''the school way,'' guessing at what kind of ''operation'' to use - multiplying or adding the numbers.

Numbers, Dr. Whitney says, become a tool when you use them for a purpose. In a class of 6-year-olds, he recalls, the teacher explained how to find the sum of 3 plus 5 by drawing ducks on the board, not noticing a boy in the back of the room saying to another, ''Yesterday I gave you 10 cards; now you gave me 7, so you still owe me 3.''

In the traditional school climate, Dr. Whitney writes, children's natural thinking ''becomes gradually replaced by attempts at rote learning, with disaster as a result.'' In high school, students increasingly say, ''Just tell me which formula to use,'' a way of saying, ''Don't ask me to think.''

Because teachers must ''cover the material,'' Dr. Whitney adds, there is less time to think. When students are called on, they must answer instantly. Wrong answers are not discussed.

''Students and teachers are all victims'' as national commissions clamor for more mathematics without realizing, Dr. Whitney warns, that they may create less knowledge and more anxiety. He says it is crucial to stop just learning the rules.

Dr. Whitney's views are controversial, as were Mr. Benezet's in 1935. Some mathematics teachers and other experts may denounce them as soft on mathematics, but others may welcome relief from demands that turn youngsters off mathematics. Of course, some teachers, ignoring the demands of the moment, actually do teach in the Benezet-Whitney fashion.

However controversial his views, Dr. Whitney deserves a hearing. Present attitudes, he writes, ''lead to the lowest of goals, passing standardized tests,'' instead of encouraging the kind of thinking ''essential for true progress in science, techology and elsewhere.''

The mathematics teaching Dr. Whitney talks about makes children want to know the answers in situations that are real to them. It makes mathematics come alive for them as they do their own thinking and take control over their work, not for tests but for themselves.

Stoller’s Rules: Thoughts on Psychoanalysis and Education Research

In his 1985 book, OBSERVING THE EROTIC IMAGINATION, the late psychoanalyst, Robert J. Stoller, offers a number of rules for psychoanalytic research that could well be of usefulness in considering what exists and what is possible in conducting (and consuming) education research (as well as the research in other social sciences. I first give Stoller’s list of rules, without putting them in the precise context in which they were originally presented, for fear of prejudicing some readers about taking these ideas seriously. Note that words in brackets are changes I have inserted to make the rules more obviously relevant in the context of educational research. Original language being replaced is specifically psychoanalytic.

Rule 1: anyone can assert anything.
Rule 2: no one can show anyone is wrong, since no one can check anyone’s observations (including his or her own).
Rule 3: ignorance can be wisdom (“The way toward better understanding, then begins with our understanding how little we understand.”)
Rule 4: use [description of motives] warily
Rule 5: ease up, forswear rhetoric, love clarity, relax.
Rule 6: describe people as we see, hear, or otherwise sense them, carefully and in detail. Do not use [educational jargon] in the midst of . . . descriptive sentences.
Rule 7: When it comes to [using educational jargon], less is more
Rule 8: stop picking on students or teachers
Rule 9: let us then, regarding [education], start afresh.
Final note: this is not really a report about [psychoanalysis], but, rather, one that uses psychoanalysis] as an example of the failure of [educational research], so far, as science.

So, what is this? More bashing of education research but from an unexpected source? Hardly. In fact, it is a call for honesty on the part of educators (at various levels), administrators (from buildings, to districts, to states, to Washington, DC), researchers, policy-makers, politicians, journalists, talking heads, business leaders, think-tank pundits, foundation funders, ideologues, and various other stake-holders when it comes to what education research is, can be, or should be.

Like Stoller in his remarks on psychoanalytic research literature and its jargon, I offer this commentary out of concern that both practitioners and critics are destroying education research by trying to make of it something it is not: a scientific literature akin to that of physics. I will resist at this juncture the temptation to go further by looking at the work of Paul Feyerabend and other post-modern philosophers of science who are critics of the notion that even the hardest science is not devoid of subjectivity, judgment, and, for want of a better term, humanity. It suffices to state that it is an error to try to make a "pure" science out of an area of inquiry that cannot afford to lose contact with its essential humanness and humanity.

Thus, allowing those who think that "the answer" to problems in education and education research are the same, and that this alleged single answer is foundational, I suggest that we turn to the instructional precedent of the history of the philosophy of mathematics itself, where logicism (as exemplified by the efforts of Russell and Whitehead in PRINCIPIA MATHEMATICA) failed to establish purely logical grounds for all of mathematics, due to underlying problems with completeness and consistency revealed by the work of Kurt Godel. The failure of the Russell and Whitehead project did not lead to the death of mathematics, of course, but rather to the demise of foundationalist attempts at creating some "ultimate" underpinnings for mathematics. Among the implications of Godel's Incompleteness Theorem are the realization that there will always be unprovable but true theorems as well as false hypotheses that forever resist refutation. Is this a tragedy, or in fact, as I believe, a signal to mathematicians that new ideas and inventions in their discipline will never be exhausted?

Returning to education research, if mathematics, the "queen of sciences" cannot be ultimately grounded, why should it be necessary to ground education or its research literature strictly in quantifiable, statistical/mathematical terms? The fact is that no such grounding in some sort of absolute and objective reality is possible. Neither is such a chimerical pursuit necessarily desirable if it could be attained.

One of the peculiar things I noticed as a new (but hardly young) graduate student in mathematics education at the University of Michigan in July, 1992, was the on-going controversy and conflict in my newly-chosen field between those who advocated for purely quantitative research and those who supported primarily qualitative methods. Given that my adviser and the principal investigator on the project that funded my graduate work at the time was engaged in fundamentally qualitative research, it might seem predictable that I would be unduly prejudiced against quantitative methods. It bears noting, however, that I had already taken two graduate-level courses in statistics and quantitative methods and experimental design while doing graduate work at the University of Florida in psychological foundations of education. I was not ignorant of or intimidated by statistics. I did, however, find myself somewhat skeptical of the notion that educational issues would readily be settled through the kinds of experiments that could be well-analyzed by the quantitative methods I studied (somewhat ironically, the other students in the courses I took were all doctoral candidates in clinical psychology, many of whom planned to be psychotherapists, not research psychologists. And the text we used, the classic STATISTIC FOR EXPERIMENTERS by Box, Hunter, & Hunter, seemed to draw all its examples from industry: nothing could have been more quantitative and, seemingly, objective. I could make sense of it, but I didn't see it applying readily to educational research and still do not.

Neither, I suspect, would Robert J. Stoller. He makes clear in book after book that he feels obligated to remove a false sense of objective truth (conveyed in no small part through the use of psychoanalytic jargon, which he both decries and eschews) from his psychoanalytic studies of particular patients (several of his books deal primarily or exclusively with but a single case history) and issues of gender identity, sexual attitudes, practices, and feelings, etc. In the latter part of his career, he worked directly with an ethnographer/anthropologist, Gilbert Herdt, and even traveled to meet with and study the Sambia tribe in New Guinea that Herdt had been investigating (they co-authored a book on this collaboration). Stoller began doing a form of what he termed clinical ethnography that extended to studying several marginalized segments of American society.

Throughout this work, he makes crystal clear in the books he authored that in the sort of science he engaged in, he, the analyst, interviewer, ethnographer, IS the instrument of (not under) investigation, and hence cannot be kept out of the awareness of readers. The idea is not to guarantee a sort of false reliability by, in effect, telling readers: "Look, I'm drawing your attention to the fact that I'm the lens through which all of this is being filtered (it's important to note that Stoller made a major point of reviewing analytic notes and manuscripts with patients and those he interviewed for other studies before publishing and always got their explicit approval), and so you can trust me because I'm telling you this." Rather, he repeatedly states that the best he can do is to periodically remind readers of his own choices of what to report and how, and to do his best to get his viewpoint, biases, and methods "out there on the table" for readers to consider and examine. This practice doesn't make him right, or reliable, or objective, or anything of the kind. But it does make him much more honest than many other writers. And it does go a long distance towards demystifying and "descientizing" what he is up to.

On my view, this approach of Stoller's is precisely what educational researchers should be doing. While there are places where purely quantitative research may be possible, I believe those places are far fewer than many would have us believe, especially those during the last eight years who have pushed for so-called "data-based research" as the alleged gold standard. My sense of this move is that it has been a smokescreen for marginalizing research that looks closely at individual and small cases, the very sort of work that brings to life what happens in real classrooms with real teachers and kids. It may be possible that a blend of qualitative and quantitative research methods will emerge during this century that will allow research teams to include close studies of individuals with "bigger picture" statistical studies, the former fleshing out and giving real life to the latter. My fear, however, is that the biases both outside and inside the research community will continue to make such work difficult. The misguided desire to appear scientific will allow pressures from politically-motivated forces to keep qualitative researchers on the defensive, when in fact it may well be that it is purely quantitative research that we need to be most suspicious of. The idea that "data speak," that there is some objectivity about deciding what data to collect, how to collect it, what experiments to conduct, what statistical methods to employ, and how to interpret the results of statistical analysis, are all highly doubtful and dangerous. This sort of think is similar to that of people who believe that documentary films are somehow objective, when in fact every single aspect of them entail subjective choices by the filmmaker(s) that shape what the viewer gets to see and attempts to craft a particular sort of response. The fact remains that such films can be, if anything, less true, less honest than "fictional" movies.

If it were possible to do so, I would strive to convince all educational researchers to cast off the guise of objective science and to turn to methods similar to those employed by Stoller. I would urge them to put themselves and their prejudices and viewpoints more explicitly into their research reports, while preserving the sound tradition within social science research of examining alternative interpretations and hypotheses, something Stoller does with great frequency. And I would counsel that they abandon as much as possible jargon-filled writing that alienates teachers in the field, parents, and other stakeholders from being able to make sense or use of much educational research, jargon that primarily is aimed, I believe, in making the research appear to be weightier and more objectively scientific than in fact it is or could possibly be.

Like Stoller, I suspect that I am here trying to swim upstream. But if his work is as valuable to psychoanalysis as I believe it to be, and if he, a physician who could readily have hidden behind both psychoanalytic and medical jargon, refused to do so in the belief that his work would be far more meaningful and beneficial if written in more accessible language, then how much easier should it be for educational researchers to abandon much of the pseudo-scientific trappings of their work? And how much more effective would that work be, in the long run, if through honesty and simplicity they stopped trying to pretend to be physicists and stood up to those who demand that educational research produce theorems and laws to which the enormous complexity that comprises both teaching and learning can be supposed to have been reduced.

Making Pedagogical Choices in Algebra Class

From 2000 to 2003, I taught the same intermediate algebra course semester after semester to (mostly) high school sophomores whom I was trying to prepare to take and pass with at least a C the same course given by a community college mathematics department for dual-enrollment credit (this was at what is called a "middle college," located in Ann Arbor and serving a diverse population of students drawn from around eight counties in southeast Michigan).

I taught from a variety of materials during the nine semesters in which I taught this course, from very traditional to more contemporary and progressive textbooks, all with accompanying use of graphing calculators to varying degrees. Following the order of topics in the books always resulted in presenting quadratic equations, their graphs, and the relationships between their transformations and parameters before exploring the same issues with absolute value equations and their graphs in the Cartesian plane. And student understanding and mastery as evidenced by performance on assessments was often poor on the first topic and abysmal on the second. When we had to look at quadratic and absolute value inequalities and their respective graphs and transformations, things deteriorated further for many. (Of course, many kids "got it" all the time, based on their test results and occasional class participation, and overall my students did well both in my class and when they moved into the college course, but I'm speaking here of the ones who did not).

One semester, for reasons I don't recall, I reversed the order. Remarkably, or so it seemed to me at the time, many of the students I had at the time who had seemed indifferent and/or lost when we worked on linear equations gave evidence both in classroom discussions and on subsequent assessments of "getting" how the graphs of absolute value equations graphed and moved around as they played with the parameters (and vice versa). Later, when we looked at the same issues for quadratic equations, their understanding seemed to carry over. The overall success of the two units went up dramatically compared with past semesters. Had I inadvertently stumbled upon something of value, or was the result an utter fluke that could rarely, if ever, be replicated by other instructors or me?

Before considering that question, let me share my speculations on why things may have gone as they did. It struck me that for students who had some minimal understanding of the behavior of linear equations and their graphs, it might have been easier to move to a look at absolute value equations and their graphs because those graphs are comprised of two linear "legs" that meet at a vertex. A look at the interplay between the graphs and the algebraic expressions that produced them was easier to gain if one graphed by hand, because all that was needed once students understood the basic shape of these graphs was to find the vertex and one point on each leg. Naturally, with the use of graphing calculators (or computer software) it would be even easier to play with and think about the graphs, but for students who did not have access to these tools or who were expected to work with out them at the beginning of (or even entirely throughout) each unit, graphing a symmetric pair of line segments that meet at a common vertex is relatively easy by hand. It is also very easy to find the coordinate pairs needed to produce the graphs.

By contrast, calculating the y-values for quadratic equations can be more challenging for many students. And anticipating how the graphs will look seems to be complicated by how certain parts of the graph (e.g., for non-zero x-values between -1 and 1) will turn out because squaring numbers in that interval results in smaller absolute value outputs for simple quadratic expressions, a somewhat counter-intuitive concept for many students.

My sense was afterwards that students were able to deal with these absolute value equations and their graphs more easily when they saw them immediately after looking at linear expressions and graphs, and before they had been (possibly) confused by the quadratic ones. They were then more able to look at the transformations and subsequently apply what they learned to the quadratic situation.

Of course, it would be wildly irresponsible to claim that my experience with these students would obtain consistently or even in a majority of cases with other students and/or other instructors. While my analysis may be plausible, to know whether it's correct would require significant further research. To expect teachers, textbook authors, policy makers, and other stakeholder to consider seriously that this may be a more effective order to teach the topics in question, there would need to be reliable data that support the above. Indeed, for some, only controlled a double-blind experiment would suffice.

Unfortunately, it is not possible to conduct such an experiment. Teachers know the order in which they teach topics. Students know the order in which they are being taught them. If different classes in the same school are taught in different ways, it is extremely difficult to keep the differences between the approaches walled off from one another. Students have friends in other classes. These are only a few of the obstacles. Indeed, it would not be easy to conduct the simpler, non-blind experiment with controls. Because as a rule, parents are not happy about letting their children be subject to "educational experiments." And when it comes to investigating mathematics education, they are perhaps least inclined to do so, given the hostile propaganda against meaningful reform that has been spread by the American media, fed to them by conservative think-tanks, foundations, pundits, and propaganda groups like Mathematically Correct and NYC-HOLD. Testing even as simple a question as in which order is it more effective to teach two basic and related topics in elementary algebra would likely face real opposition, should it come to the attention of such groups. The seeds of suspicion have been sewn and in many places already have taken root.

Opposition aside, a sole practitioner would find it daunting to try to conduct research of this kind. Finding support for it within a public school setting would be far from trivial. Why, after all, should other teachers, let alone administrators, take the question seriously? And if they did, why should they take the risk of investigating it given some of the things I've raised above? Where would the funding come from, especially in these difficult economic times? Why not leave well enough alone?

Perhaps the most viable option for a single classroom teacher looking to investigate the sort of question I've raised here would be to connect with a university-based researcher who has or could obtain funding. With the funding and (relative) influence and authority of a professor, it might be possible to convince a district to allow such research, though many of the above-mentioned concerns and limitations would still obtain.

I'm not suggesting that research is impossible or that teachers shouldn't be reflecting on practice and using it to inform future teaching. But I do despair to some extent that in the recent and current educational climate, rhetoric about "data-based research" is probably the biggest obstacle to actually conducting meaningful research there is.

Mind The Gap? The Sky Still Ain't Falling

In the Feb/Mar 2009 MAA FOCUS, current President David Bressoud's column is entitled, "Mind the Gap." In response, Dom Rosa, former associate professor of mathematics at Teikyo Post University in Waterbury, Connecticut and full-time crank, posted to math-teach@mathforum.org:

According to the article: "For four-year undergraduate programs, calculus and advanced mathematics enrollments dropped from 10.5% of all students in 1985 to 6.36% in 2005. This occurred while high school students were taking ever more mathematics at ever higher levels."

This should not surprise anyone who is minimally aware of the current state of pseudo-education in the U.S. I keep meeting more and more students who do nothing more than scribble on photocopied handouts that are distributed in so-called Geometry, Algebra II, and Precalculus courses. They never open their bloated doorstops. Recently I met a student who was told, "You don't need your [junk] book; just leave it at home."

The article also points out: "... more students arrive at college having earned credit for Calculus I, but they have not produced larger enrollments for Calculus II. Over these same 20 years, Fall term enrollments in Calculus II dropped from 115,000 to 104,000. Across the board, students are arriving at college and failing to take the next course in their mathematical progression."

My question is this: How is it possible that Bressoud who is the current MAA president, his predecessors, and other hierarchs appear to be so clueless about the mathematical pseudo-education of American students? It seems to me that this is where the real "Gap" exists.

This post drew a couple of rather skeptical replies, the first from Metropolitan State College of Denver mathematician Lou Talman, who asked:

Dom, have you considered the possibility that "the current MAA
president, his predecessors, and other hierarchs appear to be so
clueless about the mathematical pseudo-education of American
students" because *you* perceive a problem that isn't there?

And Lou's response engendered the following from anti-progressive Internet ghost, Haim Pipik (aka, Edmund David):

The premise of the argument is,

"According to the article: 'For four-year undergraduate
programs, calculus and advanced mathematics enrollments
dropped from 10.5% of all students in 1985 to 6.36% in
2005. This occurred while high school students were
taking ever more mathematics at ever higher levels.'"

Do you accept the above as an accurate representation of the facts? If this is a fact, do you accept that declining enrollments in the calculus is a problem?

Dom thinks this declining enrollment is a problem, and he asserts that the source of the problem is bad teaching in K-12. If you agree it is a problem, what do you think are some of the major contributing causes?

It's simple enough to dismiss Dom Rosa as a conspiracy theorist: he has a track record of posting and writing letters to various media outlets about his three favorite themes: the pseudo-education of America's youth in mathematics; "doorstop" textbooks; and various "rackets" in education, from standardized testing to calculators to any notion, book, method, or tool conceived after his beloved golden era of learning math in Massachusetts, a time when, if he is to be believed, teachers, books, and students were all simply wonderful, and everyone graduated with a genuinely deep, rich knowledge of mathematics. The question of how we still managed to produce yet another generation of Americans who fear and loathe mathematics during that period never seems to cross Dom's mind, or, if it does, he ignores it.

Nonetheless, the far less "cranky," but far more dangerous neo-conservative cum Libertarian, Mr. Pipik, uses Dom's latest bit of crankery to once again raise the specter of evil on the part of America's public schools. For those not playing along at home, Pipik's recurrent themes are that: we spend too much money on public education, much of it wasted, mis-spent, pilfered, given inordinately to lazy teachers and administrators who fail to show accountability and don't produce high test scores, etc.; the private sector would do a much better job of educating kids; we're wasting time and money trying to teach mathematics to most kids, who really can't learn it, don't want it, and don't need it; and anyone who disagrees with him is stupid and a communist who loves Stalin, Mao, and wants to eat the rich (well, maybe I'm exaggerating with that very last part).

Thus, while I don't think Dom or Haim has much of a case, I offer the following serious reply:

My Reponse

Having just read Bressoud's piece, I'm wondering if either Dom or Haim bothered to do so. Had they actually read it, they'd see that Bressoud gives us a good deal to chew over, all of which needs to be looked at carefully, not skimmed to find the one or two tidbits that can be spun just the way the reader has already concluded the "truth" requires.

It's necessary to look at Bressoud's analysis paragraph by paragraph to get a realistic picture of what real questions it raises:

DB: “Mind the Gap” is an appropriate metaphor for one of the greatest challenges facing undergraduate mathematics education today. There is a significant gap between students’ experience of mathematics in high school and the expectations they face on entering college, and there are troubling signs that this gap may be widening. There are serious problems in K–12 mathematics education, but college faculty also need to look to their own house and think about the first-year experience of their own students.

MPG: It's a safe bet that Mr. Rosa is NOT looking to his own house. He never does. And Mr. Pipik? We don't even know what his house might be, but he rarely criticizes departments of mathematics (too many Mathematically Correct and HOLD allies there to risk doing so). No, his focus, like Dom's is always on the other guys: K-12 public school teachers, not because he gives a rat's patootie about how much math gets taught or learned by most kids, but because of his political agenda: destroying public education and turning things over to the private sector. But since he has nothing to contribute to any conversation about how to actually improve teaching at any level in any subject (unless being snide and smug count as tips for bettering things), he can be safely ignored for the duration of this analysis.

DB: In my article “Is the Sky Still Falling?” (2009), I observed that four-year college mathematics enrollments at the level of calculus and above declined from 1985
to 1995 and have since recovered to slightly below the 1990 numbers.

MPG: Well, let's be sure to gloss over that last statement. I hate to be the first person to point this out, but there are cycles in most areas of study. And the reasons vary enormously from cycle to cycle. But if there was a drop from 1985 to 1995 and then a recovery to just below 1990 levels, is the sky actually falling or merely moving up and down rather lazily? And one might assume that four year colleges, not community colleges, represent the majority of stronger, better prepared students.

DB: Two-year colleges saw calculus enrollments rise in the early ‘90s, then fall to well below the 1990 number, while the number of their students requiring remedial mathematics exactly doubled. In percentages, the picture is dismal. For four-year undergraduate programs, calculus and advanced mathematics enrollments dropped from 10.05% of all students in 1985 to 6.36% in 2005.

This has been debated and discussed widely. Bressoud doesn't offer any real analysis here. No hypothesis, just numbers and the word "dismal," which of course implies a great deal but tells us nothing. If enrollments in community colleges are growing rapidly, it's quite conceivable that what this means regarding the PERCENTAGES of students who enroll in calculus and beyond SHOULD be dropping. Indeed, it would almost have to drop, because we're getting vastly more students in such colleges who are coming from the lower echelons of high school graduates. It does not follow from his statements that things are any worse in terms of the qualifications of comparable percentile ranks of high school graduates today and at various points in the past. Until someone actually comes up with that sort of comparison, with concrete examples to illustrate the nature of any apparent decline or growth, we really don't know squat based on the above other than that more kids are enrolling in two year colleges but a lower percentage of them go into higher math. And when one considers who goes to these schools and why, this seems utterly NON-dismal.

DB: This happened while high school students were taking ever more mathematics at ever higher levels. In 1982, only 44.5% of high school graduates had completed mathematics at the level of Algebra II or higher. By 2004, this had risen to 76.7%. In 1982, 10.7% had completed precalculus. By 2004, it was 33.0%, over a million high school graduates arriving in college ready — at least in theory — to begin or continue the study of calculus. Yet over the years 1985–2005, Fall term enrollments in Calculus I dropped from 264,000 to 252,000.

MPG: Hmm. Why do I not find myself panicking over a drop in freshman calculus enrollment of 12,000 students? Aside from many issues that even Bressoud touches on below, anyone paying attention may have noticed that we're not exactly dying for professional mathematicians in this country. Any opening to teach mathematics at the post-secondary level has dozens of qualified applicants. In some parts of the country, it's no easy task to find a public school that is hiring in grade 6 - 12 mathematics classrooms. Engineers are getting laid off in many disciplines. No general cry for more physicists has gone out. Straight A's in four semesters of calculus is no guarantee of admission to medical school, even less so into veterinary school (though exactly why those courses are prerequisites for either program remains a bit of a mystery). That 12,000 student drop, which remains unanalyzed as to possible causes, doesn't seem linked to the current economic crisis: indeed, one might reasonably assume that all those brilliant economists and MBAs took lots and lots of calculus.

DB: Admittedly, many more students today arrive at college already having earned credit for Calculus I, but they have not produced larger enrollments for Calculus II. Over these same 20 years, Fall term enrollments in Calculus II dropped from 115,000 to 104,000. Across the board, students are arriving in college and
failing to take what should be a next course in their mathematical progression.

MPG: It would be important to know if Bressoud's figures account for two things: first, how many of those students who come to college with a semester's credit already in hand wisely choose to wait until second semester to take Calculus II with friends and to give themselves an easier first semseter in college, something I have frequently recommended to high school seniors. There isn't a ticking clock, and there are lots of other courses in which a bright student might wish to enroll regardless of whether s/he is ready for second semester calculus in the fall of freshman year; and second, what percentage of those students NEVER take another mathematics class at the level of calculus or above? Without providing information on either of these concerns, Dr. Bressoud really has no grounds for concern on the above numbers, yet he seems to imply we should be terribly worried.

DB: The college community is not blameless. Too many good students are turned off by their initial college experience in mathematics. Too often, first-year courses are large and impersonal, instructors — especially adjunct faculty and graduate teaching assistants — are under-prepared, and little thought has gone into implementing appropriate pedagogies. Moreover, a common complaint that I hear from high school teachers is that colleges focus exclusively on what students do not know, with the result that many students find themselves assigned to classes they find stultifying.

MPG: I'm not willing to put too much blame on adjuncts or graduate students. I have had some very good instruction from some of them. At the University of Michigan, one of the finest mathematics professors I studied with was a post-doctoral student. When I decided to take a pair of refresher summer classes in calculus, both instructors were graduate students and were very clear, concerned and competent instructors. On the other hand, one of the most horrid mathematics classes I had was taught by a full professor. He lacked just about every requisite skill imaginable for good teaching other than subject matter competence.

That said, Bressoud would have it right and do students a huge service if he didn't try to foist the responsibility for bad pedagogy on lower-rank teachers and admitted that there is a vast amount of bad instruction in mathematics departments regardless of the rank of the teacher. Of course, there are also outstanding mathematics professors of higher rank (and I don't merely mean great mathematicians who can communicate with three or four truly gifted students, but the Polyas, the Edward Burgers, and others who are deeply committed to teaching mathematics to as many students as they can possibly reach.

DB: This last is a tricky issue. The answer cannot be that colleges lower their expectations of what it means to know algebra or calculus. It does mean that colleges need to rethink how to get students from where they are as they enter college to where they need to be. It does mean offering more routes into good mathematics and restructuring existing courses so that they acknowledge and build upon what students do know while remaining mindful of and addressing the gaps in this knowledge. Especially when a student needs to relearn a topic that appears familiar, we must ensure that the course is structured so that it provides fresh challenges that entice students to keep moving forward.

MPG: No argument with the above, especially if Dr. Bressoud can make a commitment for the professional mathematics community in post-secondary institutions to communicate the above to parents, employers, K-12 educators, administrators, guidance counselors, college admissions officers, politicians, and the media. Further, to try to minimize the counterproductive activities of the small, vocal minority of activists within or affiliated with the professional mathematical community who are set on preserving the status quo at all costs and who actively oppose any sort of reforms along the lines he mentions. It is high time that both major organizations of professional mathematicians took a principled stand against groups such as Mathematically Correct and NYC-HOLD.

Additionally, if Dr. Bressoud is serious about creating alternative routes and attractive options for students without sacrificing mathematical standards, it is high time that the professional mathematics community support teaching discrete mathematics, the underlying mathematics of computer science, and powerful connections to technology and the Internet as important paths for youngsters to explore and study before they reach college. The notion that calculus is the exclusive meaningful goal for K-12 mathematics education is an outmoded idea that will be modified in the minds of the public and teaching community only when it comes with the support of professional mathematicians.

DB: We have learned a lot about teaching undergraduates in the past 20 years. There are proven programs for bridging the gap. The Emerging Scholars Program is one. Stretching Calculus I over two terms with precalculus topics treated on a just-in-time basis is another. But there are no magic bullets. Each college and university must examine what others have done and adapt to its own situation those programs that are most appropriate.

MPG: I can agree with the statement about the lack of panaceas. But if the focus remains solely on getting students into calculus, all the other fine things Dr. Bressoud mentions will simply break against the walls of tradition, in all likelihood. As long as mathematics is viewed so narrowly through the lens of analysis, the traditionalists will feel they still have a mandate to demand control over the K-12 curriculum. The preceding sixteen years of the Math Wars show clearly that the hard-core educational conservatives will never concede an inch towards meaningful reform of mathematics teaching at ANY level, K-12 or post-secondary, as long as they can promulgate the idea that they represent the entirety of professional mathematicians or nearly so.

It is incumbent upon anyone in Dr. Bressoud's position, if s/he wishes to make positive changes feasible, to be much more careful in the examination, analysis and interpretation of statistics such as those cited in "Mind the Gap." While few people doubt or deny that we can continue to improve the quality and effectiveness of mathematics education at all levels in this country, the sky really is not falling. American kids are neither more stupid nor more ignorant today than in the past. Nor are they being "pseudo-educated." They could use, however, better guidance from the professional mathematics and mathematics education communities about what mathematics really is and the many ways in which they could positively engage with it. As always, the question is whether those who understand the need for change have the courage to stand up for it, even against respected colleagues desperate to stand in its way.

Testing and America's Unspoken Abuse of Children

In a recent post to the Gerald Bracey's EDDRA (aka, Education Disinformation Detection and Reporting Agency) discussion list, Mark Shapiro, who blogs as The Irascible Professor, wrote regarding FairTest, a group that monitors and fights against the abuse of standardized and other tests:

I'm no great fan of PISA. But really "Fair Test" is basically "No Test".

While I don't think that high stakes tests alone should be used to determine a student's fate, I do think that standardized tests have a useful place in the assessment of educational progress both at the individual and global level.

Just as there are no perfect teachers, no perfect teaching techniques, there likely will never be perfect tests. But that doesn't mean the tests are worthless. Imperfect knowledge is better than complete ignorance.

My response

I think that in fact there are times where ignorance is actually better than partial knowledge, especially if one uses the partial knowledge as if it were complete knowledge with the power for prediction we normally reserve for physical laws. People who recognize the extreme limitations of their partial knowledge and proceed with utmost caution seem to be all too few when it comes to educational and psychological testing and measurement, unfortunately for all of us (except maybe those who profit from acting as if limited knowledge is omniscience).

So-called "intelligence" testing has a particularly long, ugly, racist, xenophobic, and highly politicized history, perhaps more so in the US than anywhere else on the planet (England may be the runner-up, but I'm not certain of that). Much of that history spilled over quite predictably into education, with destructive results for millions of Americans (not to mention those would-be Americans left stranded to die in Nazi camps in Europe because they could not emigrate to the US under restrictive immigration laws passed by Congress in 1923 due in no small part to Carl Brigham's testimony that was grounded in the meaningless results of Yerke's data from the Army Intelligence Tests given during WWI recruitment:

The decline of American intelligence will be more rapid than the decline of the intelligence of European national groups, owing to the presence of the negro. These are the plain, if somewhat ugly, facts that our study shows. The deterioration of American intelligence is not inevitable, however, if public action can be aroused to prevent it. There is no reason why legal steps should not be taken which would insure a continuously upward evolution.

The steps that should be taken to preserve or increase our present intellectual capacity must of course be dictated by science and not by political expediency. Immigration should not only be restrictive but highly selective. And the revision of the immigration and naturalization laws will only afford a slight relief from our present difficulty. The really important steps are those looking toward the prevention of the continued propagation of defective strains in the present population. (Brigham 1923) (The Mismeasure of Man, Stephen Jay Gould, p. 260)

The horror and wrongness of what he'd done led Brigham to later publicly recant:

“This review has summarized some of the more recent test findings which show that comparative studies of various national and racial groups may not be made with existing tests, and which show, in particular, that one of the most pretentious of these comparative racial studies—the writer’s own—was without foundation.” [Brigham, C. C. 1930. Intelligence test of immigrant groups. Psychological review 37:158-165. (p.165)]

though this didn't help the dead people he helped doom.)

The damage done to kids through the abuse of tests of all sorts (psychological, academic, and so forth) is never going to be fully documented. It goes on every day. It is not for the most part a matter of willful abuse, of course: few educators have adequate training in assessment. Far fewer still have any real understanding of statistics or psychometrics. Given some comments on this and other lists I read, I'd say that some opinions here appear to reflect similar ignorance.

The conclusion isn't to do away with tests. Assessment is necessary. It's the nature of assessment, the quality of the tests, the way in which test results are used (or misused) and the purposes behind testing that must be vigilantly interrogated at all times. I see nothing to lead me to the conclusion that FairTest is against testing. What it is against, as we all should be, is testing abuse. The idea that any data is better than no data is ridiculous if the damage done by collecting the data and then misusing it can be reasonable shown to far outstrip the good done with it.

If someone knows and openly admits that the meaning and usefulness of data s/he collects is highly speculative at best, and vigilantly seeks to protect those being assessed and others from unfair punishment and needless suffering (assuming anyone really has that much power), then it may be ethical to collect that data in hopes that it will lead to better, less speculative understanding and assessment in the future. But it is far more common that what in fact occurs is that test results become the evils unleashed from Pandora's Box, and no one has the power to return them to that box once it is opened. Kids are wrongly made to feel stupid and to suffer, pushed around in the school system in idiotic ways, all because of bad testing practices. Teachers, schools, administrators, districts, states (and now even nations) are wrongly viewed and mistreated due to unethical and/or ignorant abuse of basic principles of psychometrics. Parents and non-parents alike suffer as their property values are hurt by test scores, regardless of whether those scores mean a great deal or nothing at all.

And what makes this all so heinous is that there are ostensibly intelligent people, some with backgrounds in mathematics and/or science (but apparently not in statistics or psychometrics) who become apologists or even cheerleaders for bad testing practices and attack mercilessly anyone who dares even question what's going on. In my experience, such people are invariably politically and/or economically motivated, and the most polite word I can use for this practice is "unethical," though some may prefer one of the following: immoral, amoral, hypocritical.

Once again, there is nothing wrong with the idea of assessment, but there is much wrong with how it's done in this country and a great deal of evil perpetrated as a result of testing abuse.

For further reading on this vital subject, I recommend the following:

Gould, Stephen Jay: The Mismeasure of Man

Hoffman, Banesh: The Tyranny of Testing

Mansell, Warwick: Education By Numbers: The Tyranny of Testing

Nichols, Sharon L. and Berliner, David C.: Collateral Damage: How High-Stakes Testing Corrupts America's Schools

Owen, David: None of the Above: Behind The Myth Of Scholastic Aptitude

Constructivism? We Don't Need No Steenking Constructivism! (or do we?)

In a recent post on math-teach@mathforum.org in response to my comments about pedagogical content knowledge, Jonathan Groves wrote, in part:

Perhaps it is tricky to define constructivism precisely and to know exactly what it is. I am not even sure if I know exactly what it is though I do have at least an idea of what it is. I have learned some about constructivism in some education classes I took at Austin Peay State University when I originally was trying to become a high school math teacher (which I later abandoned so that I can graduate in four years and teach college instead).

But the constructivism I learned I remember vaguely, and I learned some more about it when I was completing my online faculty training for Kaplan University, where I will begin teaching math online on March 25. The faculty training consisted not only on
information on how to use their online system for their online classes and about Kaplan University itself and some of their expectations for faculty but also about some of the theories of online learning and how online learning differs from traditional higher education learning. Constructivism is used a lot in their classes since direct instruction can't be used well in online classes. Even with live seminars, instructors still can't use direct instruction as their primary teaching method since the seminar is only one hour once a week.

The constructivism I learned from the training is very general; it says nothing about constructivism in mathematics or in any particular subject for that matter. That is perhaps not a good idea, but the faculty training course is designed for all the new faculty, regardless of what subjects they are teaching.

It is extremely important to know how constructivism is relevant to mathematics learning and teaching not only for those discussing it but also for those teaching mathematics. This knowledge can help us fellow mathematics teachers discuss constructivism intelligently and without confusion to others and to teach mathematics more effectively. We first need to learn exactly what constructivism is and
how it is relevent to learning and teaching mathematics.

[F]lexibility and content knowledge of mathematics itself is extremely important for teachers. I myself would hate to lead a discussion and it sidetracks into another related interesting topic but then find myself not knowing how to handle this situation. Class would then become a disorganized mess, and no students would benefit from it in the least bit. Lesson plans don't have to be followed rigidly and shouldn't be; the teacher should allow for digressions if they are relevant. But if the teacher cannot handle these digressions, chaos will reign, and chaos never works in any classroom.

I would also hate it if this new related interesting topic is something I know little about. I doubt that will happen with the classes I will soon teach in the next few weeks. One such course is Basic Algebra (Developmental Mathematics) for the Spring 2 quarter at Florida Tech (starts March 9) and Math 103 College Mathematics for Kaplan University (starts March 25). Kaplan uses 10-week terms, and they have new terms starting about every 3 weeks or so. Thus, one of your colleagues' classes may start
at the beginning of one term and yours start at the beginning of another and yet your colleague hasn't finished teaching his class yet. If I am wrong and something does arise I know little about, but since the courses are online, getting around this problem will be easier since I will usually have time to think about such posts
before I reply to them.

It seems obvious that subject knowledge is required for the mathematics teacher, yet the lack of mathematics knowledge potential teachers have when they graduate and then get their teaching licenses later makes one wonder if it is as obvious as it seems. At least that is a major problem in states like Tennessee and Kentucky that are known
for providing education that is not so good.

That leads to an extremely important question: What content knowledge should math teachers have? Of course this depends on the grade level or level of college they are teaching.

If new topics are related to the curriculum but not part of the curriculum itself and there is little time to devote to them, then having students think about it as homework or to lead a short class discussion or as extra credit research assignment are wonderful options to pursue. The last one is good since it encourages students to become curious about mathematics and to learn to study it on their own, which is a great benefit for them. And it teaches them that mathematics studied do not have to be topics restricted to what is learned in the classroom; in other words, it encourages them to learn whatever mathematics interests them, even if it is not discussed in the classroom.

Being able to make these decisions effectively is vital for preventing classrooms from turning into chaos and into doing what is best for the students.

[W]hat is best depends on who the students are, what the required curriculum is for the class, what math classes the students will or are likely to take in the future, and too many other factors to list here. Depending on the experience of other colleagues, whether in your school or not, when facing a tough decision in
teaching is vital. Kaplan University encourages us instructors to ask our department chair and our mentor and other colleagues when we face difficult decisions and other problems in teaching. And reading widely about teaching and discussing with others may
help prepare us for these difficulties later and to make good decisions. That is why several colleges I have seen take professional development seriously. Kaplan University does.

We definitely need better teacher education and professional development programs in this country. This will help not only in mathematics teaching but in the teaching of any subject. Like mathematics, most other subjects aren't being taught very effectively
these days. For example, schools didn't teach me much world geography or world history (our high school world history class was really European history; our teacher skipped all the chapters on history in the Americas, in Africa, in Asia, etc. except when the history pertained directly to European history. Even then, that was really
history on European colonization). And our U.S. government and economics classes weren't very good either. I wouldn't be shocked if that is the case throughout the U.S. since somehow I get the impression that many young American adults these days don't know much about how economics (except maybe business and economics majors!)
and the government work. I could give more examples, but I think these suffice in illustrating my point.

Where we start these improvements, I am not sure. One idea is to make sure our mathematics teachers have enough subject knowledge and knowledge of teaching to get started. And we should make sure mathematics teachers at all levels appreciate and enjoy mathematics. I have heard this problem is most severe at the elementary school
level, and I am not surprised since these teachers are forced to teach math,
regardless if they want to. Higher up, they can choose whatever subject they want. And I suppose elementary school math is harder to get excited about than more advanced math. And a lot of these teachers who don't like math may too have had teachers in elementary school who didn't like math who turned off their interest permanently, which in turn hurt their ability to learn the mathematics they ought to know to teach elementary school math effectively. If students learn to dislike math early, say in elementary school, they may learn to dislike it the rest of their lives. Exactly how to fix these problems, I don't know, and I strongly wish I knew! Does anyone have any ideas here?

My reply

I think the relevant ideas from constructivism are not complex. From my point of view, constructivism simply asserts that people construct their own individual understanding of anything they learn (in or out of formal instructional settings) through an INDIVIDUAL interaction between past knowledge/experience and new information.

Why this is important for teachers, most particularly for teachers of subjects in which there is a propensity for many instructors to think that "it is obvious" that something is true, or that it suffices for them to state a logical sequence of definitions and reference to previously-taught axioms and theorems in order for students to learn a new theorem, is that very little is obvious to many students, and it's lousy pedagogical practice to assume otherwise. Further whipping through the carefully constructed steps of a proof, while it may be beautiful to those who understand and are capable of appreciating it aesthetically, is lost on most students seeing the ideas for the first time and just learning to grapple with proof-theoretic mathematics. This approach not only leaves most students in the intellectual dust, but gives them the VERY false impression that if they were mathematically competent, all would be clear, just as it is to the instructor and must have been to the mathematician who first constructed the proof.

But of course, that is precisely NOT the case. In all likelihood, the professor did NOT grasp the workings of the proof (if it was of any depth and complexity) at first blush. And it is certain that the mathematician who constructed the proof didn't have it spring from her/his head like Athena from that of Zeus. There were generally a series of fits and starts, a good deal of creativity, intelligent/inspired guess work, going up blind alleys, retracting steps, and much frustration requiring sustained effort on the part of the mathematician or mathematicians who ultimately succeeded in constructing the proof in question. (Yes, there are "simple" proofs at various points in the development of any mathematical subject, but it's not long before those are left behind, and those who are going to follow the development of the topic will need to be able to juggle a host of definitions, axioms, previously-proved theorems, and, of course, abstract concepts, in order to make sense of new material and proofs). We do an enormous disservice to many students by hiding ALL the rough edges as if they never existed and as if we "got it" first time and every time.

At a lower level (namely K-14, where there is a lot less formal or even informal proving and a lot more teaching of definitions and procedures, much of the same still holds except that it's conceivable that the teacher has at best a tenuous hold on what's being taught, making many lower-grade or poorly-qualified higher-grade teachers loath to go off the reservation of any lesson plan. Hence, they are wont to fail to offer alternative explanations for concepts and procedures, often holding to the dreaded "my way or the highway" philosophy, sometimes from fear, sometimes from laziness, sometimes from just being a rigid character, but in all cases failing to serve well many students.

Teachers who have recognized the usefulness of the constructivist approach to understanding how we learn are less likely to come at teaching as a "one-size fits all" process. Constructivism doesn't tell anyone how to teach, but it surely points towards the notion that we can't pour our own understanding of anything into anyone else's head. That's simply not how folks learn. There is NO unmediated learning. There is NO way to get through the filtering and processing EACH individual will do every time we teach and s/he tries to understand and learn. Any teacher who gives a "dynamite" lecture and leaves the classroom with a confident belief that the students must have learned (if they were prepared and paying attention) is likely deluding him/herself. It's not that no one learned: everyone who WAS paying attention learned. But just what each one learned is unknowable, and to have ANY idea at all requires doing more than simply lecturing, no matter how well.

That is not to say that all lectures are equally good (or bad) or that lecturing doesn't have its place. But it does suggest that there needs to be a lot more. And given how much lecturing and its variants (so-called direct instruction, teacher-centered instruction, and similar approaches in which teachers can say with straight faces, "Boy, I taught good, but they learned lousy") dominate math teaching in this country, we might expect that mathematics teachers at all levels would welcome ideas about how to be more effective by varying instructional strategies and styles. If a group of mathematics teachers and teacher educators called for a shift in emphasis towards more student-centered instruction and less of the sage-on-the-stage approach, folks could look at what has been going on (and not going on) for decades honestly, recognize their past successes and failures, take some ownership and responsibility for both, and respond fairly to the 'new' ideas (of course, many of them actually aren't new, but rather are a return to some things that fell out of fashion for various reasons, not all of them because the methods didn't work).

Thus, it would perhaps shock a neutral observer to note how some mathematicians and K-12 math teachers responded, and the vehement attacks that have been launched on various notions that were introduced in the 1989 NCTM standards volumes (though not necessarily solely there, as there were individual practitioners using a host of more student-centered practices in their classrooms long before the first of those volumes was published). Twenty years later, the same distortions, lies, misinformation, epithets, etc., about this call for a shift in emphasis continue to appear on the internet, in the media, and in public and private discourse. Reform is portrayed as utterly wrong-headed, extremist, throwing babies out with bath water, ad nauseum, with little sign that the original opponents and those allies they gained will ever lighten up at all.

We're a long way from seeing broad-based and meaningful reform practice in math classrooms in this country, but it is clear, too, that reform ideas aren't going to go away, despite the best (or worst) efforts of its hard-line foes. And so it becomes necessary to go over a lot of the same ground periodically to try to sweep away the enormous amount of dreck about reform ideas that spews from a few persistent sources.

The necessary content knowledge for teachers is a majr focus of the research and writing coming out of the University of Michigan's mathematics teaching program, particularly that of Hyman Bass and Deborah Ball and their colleagues. Indeed, it is the major research question they've been looking at for about a decade.

Of course, part of the issue here goes beyond simple concerns about curriculum and towards questions about "the system." Recent calls for a national curriculum, national standards that apply to every state, etc., are simply an extension of the idea that there's some blueprint we could devise that, if followed to the letter or nearly so, would produce a mathematically competent country. I think this idea is in fundamental conflict with constructivist theories of learning, and these calls are an attempt, well-meaning or not, to make it increasingly impossible for individual classroom teachers to make the sorts of pedagogical judgments which in fact should be a major part of what they do. I have argued here and elsewhere that no one is better-positioned than the individual classroom teacher to make such choices about his/her students. Rather than managing curriculum from the top down in some blindly monolithic fashion, the people in charge should be putting energy into ensuring as much as possible that teachers are well-equipped to make those decisions as they arise.

The phony "accountability "movement would appear to be all about finding out which practices work and which don't and rewarding the successful and "dealing with" those who are not, but the real agenda is to reduce teaching to formulas: a wacky stop-watch brandishing "management science" approach out of the worst nightmares of those of us who realize that there's no such formula. A genuine accountability effort would entail creating structures in which teachers are helped to make better choices while not hamstrung so that they can't dare do anything that isn't "by the book." A real approach to accountability would entail ensuring that teachers give "accounts" of some of the important choices they make, both to encourage reflective practice on the part of each individual teacher and to collect a body of knowledge that can be shared with other teachers. Of course, when education is viewed as a business and creativity, originality, and any sort of "risk-taking" behavior is likely to be punished, this sort of model can't happen (and it's hard not to see the irony in these management notions coming from various executives whose companies may have been burning through money like a '65 Buick badly in need of a ring job burns through oil, while the genius managers are collecting millions of bonuses and perks. Apparently accountability is only for others).

What pedagogical content knowledge is, exactly, would take books to contain. What it is in small part is some of the choices I pointed to previously when I looked at roads not taken by Bill and by each teacher every time s/he chooses to either stay on the prescribed path of the lesson (and of course in constructing the lesson in the first place) or not. It's unlikely that I or anyone could cover every possible choice (even just the reasonable ones and some examples of less reasonable ones) that could arise. However, there are books starting to appear that, if their titles are to be believed, focus on mathematical pedagogical content knowledge. I don't have any of them and can't comment on how effectively they accomplish any of this. If and when I get my hands on some, I will try to report on what I see.

I don't think there are any simple or exact fixes to the current state of mathematics teaching and learning, naturally, but being open to real reflection on the issues, one's practice, and much else makes for a good start. The closed-mindedness exhibited all-too-often in response to constructivism and its relation to mathematics education helps no one, not even the entrenched.

Pedagogical Content Knowledge: The Missing Link In Math Education

In a recent post to the math-teach@mathforum.org list-serve, Bill Marsh wrote, in part:

I continue to think that guided discovery can be a powerful teaching tool, of definitions, as well as of proofs, but I should emphasize that in GUIDED discovery there will be a teacher in the room who knows the theorems that are coming and some of the pitfalls on the way to getting them.

If you don't know how to use them, power tools can dangerous. If you don't like them or have to use them, you won't use them very much. Unless you use them a lot, you are unlikely to learn how to use them well. If you don't know how to use them well, you may underestimate what can be done with them.

Suppose students arrive in a seventh grade math class to see 1+2+3 = 6 on the board. After a moment or two, the teacher might mention that numbers like six are called perfect and ask if anyone can say what's going on. I'd expect that pretty quickly someone would suggest adding up all the divisors, which can then be tweaked into a good definition. The class could look for another perfect number less than a hundred. If this were done near Valentine's Day, amicable numbers might be mentioned.

At a higher level, I might, on the first day of a real analysis course I was teaching, say that we were going to be looking at things like the epsilon-delta definitions and proofs they saw briefly in calculus. I might ask them to consider, for a real number a, the sequence the intervals defined by |x-a| < dk="" education="" 01330="" pdf="">http://www2.mat.dtu.dk/education/01330/Chapter1Math3.pdf

I don't claim that guided discovery is the best way to teach. I will claim that it is a good way, and that it is an especially good way in K-12. But only for those who like it enough to be willing to try to do it well.

I think Bill's post points to something far more important than "constructivism" (a topic that generates lots of heat and virtually no light here, in my experience). Who is and who isn't a constructivist, and what comprises applications of constructivist learning theory to mathematics lessons is rather useless to talk about amongst folks who can't even approach the slightest agreement about what "constructivism" actually is or what its theorists have to say that is relevant to mathematics teaching and learning.

What Bill's post touches upon, however, is the issue of pedagogical content knowledge. A teacher who proposes to teach a lesson on perfect numbers and offers the example Bill originally posted may well, as Dave Renfro subsequently pointed out, find herself in the middle of a conversation about triangular numbers (though it's quite possible that no one in the classroom, the teacher included, may know that term). Or perhaps (though less likely) the reverse situation takes place, with a lesson on triangular numbers potentially branching off towards a conversation about perfect numbers. One important consideration is, as always, what do teachers do when the unexpected or unplanned for arises? And clearly, one consideration in that regard is whether the teacher has the requisite content knowledge (awareness of the mathematics being pointed to, the mathematics that leads into what's arising, and some of the mathematics that is pointed to by what's under consideration AND being raised by the "surprise" issue). But more than that, the teacher must be able to make quick but reasonable choices as to how to respond to what arises.

Bill's proposed response seems to me to be one exemplary option. And in order to produce it, a teacher would need to be prepared to deal with the math that comes up, and also have a strong sense of whether it is a better idea to continue with the plan or to go down the new road. That means pedagogical content knowledge and knowing the class and what is most likely to serve them well, collectively and individually. If the original plan is chosen, how is the new idea addressed (never? in the next class? by the teacher raising it? by the teacher asking the class to think about it as a homework assignment? by asking the student who raised it to report on it or lead a class investigation/discussion, etc., as an extra credit assignment? some other option?) If the new opportunity is pursued, what happens to the previous plan?

My view here is, of course, that content knowledge alone and general pedagogical knowledge alone will not suffice. While strength in both these areas is necessary and contributes to making good decisions in this and many other situations in math class, they are not sufficient. There is a third kind of knowledge pointed to in what I raise above that is peculiar to this domain of teaching (and so saying does not preclude the idea that similar but different specific pedagogical content knowledge is needed in other subject areas) without which teachers are less likely to make choices that adequately serve the vast majority of students.

None of the above is intended to imply even remotely that there is a single "correct" decision to be made in the situation Bill describes. I'm perfectly comfortable that he made a good one, but there were other possibilities that, depending upon information that only the teacher has access to at the point the decision is made, might have been as good. There's no way to know with certainty what the results of these other choices would have been, but experienced, reflective teachers who have the requisite knowledge and use it actively are the ones most likely to pick from a "menu" of better and more productive options. Naturally, there are worse, less effective choices, based on previous experience, observation of and/or conversations with colleagues, consultation with the literature, interaction with coaches, master teachers, etc. It is naive to think that without reflective practice and competence in the three domains of teaching knowledge mentioned here that teachers will "naturally" make better choices when these situations arise, based simply on content knowledge alone. It is for this reason, among others, that effective teacher education and professional development is vital if we are to improve mathematics teaching and learning in this country.