Thursday, March 25, 2010

You Just Can't Make This Stuff Up (or can you, if you're James Milgram?)

If someone sent this to me without the accompanying link, I'd never believe it was real. Sadly, it's very real.

R. James Milgram, a renowned mathematician who has long been in the forefront of opposition to progressive mathematics education reform, was interviewed for something called the Baltimore Curriculum Project in 2006 and the entire interview appears here.

However, the following excerpt is perhaps the single most bizarre-sounding thing I've ever encountered in the history of the Math Wars (and that's saying something).

Where Dr. Milgram got these notions from I have no clue. Who the mathematics educators are that he claims hold the views he mentions below would be amazing to discover, if such people actually exist. My guess, however, is that if they do, they're almost certainly not in the mainstream of the mathematics education research community today.

Perhaps he found someone from a more conservative era of NCTM and the research community who actually believes what Milgram states. If so, that is hardly grounds to make the assertions about a large number of mathematics educators.

Dr. Milgram cites NO specifics. He seems to be in fact making this up as he goes. And on my view, it has absolutely nothing to do with the real world. I've been involved in mathematics education officially since I started graduate work at University of Michigan in 1992. I've attended many conferences of mathematics education researchers and teachers. I participate in many on-line discussion about mathematics education and have corresponded individually and in small private lists with dozens of university mathematics educators and researchers, both in the US and abroad.

I have NEVER heard or read anyone else say anything even remotely like what Milgram asserts below. And I believe he has no facts to support his claims. Even the interviewer appears completely caught off guard by the assertion that there's a conflict between mathematics educators and other professors of education, and that the former feel constrained by the latter and disempowered by them.

Further, his off-the-cuff claim that there is "a lot more [agreement] than you would expect" amongst mathematics educators with the anti-reform traditionalists Milgram represents appears to be another howler he invents for the occasion.

If this is actually what Milgram believes, and the general community of mathematics educators and other school of education researchers and professors has gotten wind of it yet has remained to a person silent about it, I'd be amazed. I think this interview has sat unread by anyone in a position to expose it for what it is. Or else people are being far too polite to reply.

Not being quite so polite, I challenge Dr. Milgram to back up his claims about what professors of education see as the purpose of education, what mathematics educators generally believe about their colleagues in schools of education, and what mathematics educators believe regarding the views that Milgram espouses about mathematics teaching and learning. I think he'd be very hard-pressed indeed to find more than the tiniest trickle of support, if he can find any current working mathematics educator or other ed school professor who backs any of his points.

Here is the excerpt to which I am referring:

Q: Why do the philosophies of mathematicians and educators seem to vary so widely?

A: The people holding the power in the education community today hold the belief that the major function of the public schools is to keep children out of the workforce.

The recollection is the horrors of child labor from the 19th century. The objective was to keep them out of the workforce as children, but that was it. They also believe that kids should have a good time in school because implicit in their belief is the conviction that kids will not have a good time as adults.

Q: That's shocking to me. I've read about the Math Wars, but I've never hear that viewpoint expressed.

A: The debate in the Math Wars was between math educators and mathematicians. Somehow the people in the education schools proper stayed out of it. But when you come right down to it, you have to deal with the people in the education schools.

Ultimately and what really was remarkable to me when I got to know a number of these math educators is they were consistently telling me of their feelings of powerlessness. We were assuming they were the ones that are responsible. They don't necessarily agree with us 100 percent, but they agree with us a lot more than you would expect.

Sure, Dr. Milgram. And maybe I'm a Chinese jet pilot.

Sunday, March 14, 2010

Where Are The Explanations? Davydov, Vygotsky, Measurement, and Scientific Knowledge

Someone asked today on why it is that students fail at math. His answer?

The short answer: They don't get it.

The long answer...

What is it that they don't get? They don't get the chain of reasoning involved in mathematics.

This struck me as a rather useless tautology, a non-explanation purporting to reveal all. I could not see what he subsequently offered as any sort of realistic answer. I thought instead about Davydov's curriculum.

I first became aware of the mathematical curriculum of V. V. Davydov and his colleagues early in 2009 when several people independently wrote to ask me my opinion of his "measurement-based" approach to elementary mathematics. I had no idea who Davydov was or what his approach entailed.

After reading some papers that these folks sent me, I knew that Davydov followed and grounded his work in some key ideas of Vygotsky, and that at least one attempt had been made in the US to develop and implement a measurement-based elementary curriculum: MEASURE UP! at the University of Hawaii's Curriculum Research and Development Group. One of those developers was someone I knew, Barbara Dougherty, now at Iowa State University.

Further, I learned that Jean Schmittau at SUNY@Binghamton had written several papers about the Davydov curriculum and had, in fact, translated Davydov's elementary books (which comprise an early-grade three-year program) into English in the late 1990s and used these translations to work with elementary students in a local school district.

Perhaps coincidentally, at the time I became aware of and began looking into Davydov's work, I became interested and embroiled in the subject of my previous post (and many others): the controversy Keith Devlin generated beginning with his June 2008 column on multiplication, "It Ain't No Repeated Addition." By the time I jumped in, first to oppose Devlin's views and later to agree with them, the arguments about whether there was any reason NOT to teach young students that multiplication is repeated addition (MIRA) and whether in fact it was ever true to claim that MIRA is correct was raging hot and heavy in the blogosphere. That didn't stop me from putting both feet in my mouth (I took umbrage at Devlin, a mathematician I'd once met briefly and whom I respected as a solid thinker, trying to tell elementary teachers how to teach basic mathematics to kids.) And then I actually started to think, both about the nature of multiplicative reasoning and problems I saw in typical elementary school mathematics education (and our way of "schooling" so many students into intellectual passivity) that Davydov's approach seemed well-positioned to address.

As I began to post about both Davydov and the multiplication/MIRA issues both on my blog and on a couple of on-line discussion lists, I noticed how quickly the educational conservatives lined up against Devlin, against Davydov, against anything that wasn't precisely how they were taught or taught others themselves in the cases where they instructed on elementary mathematics (usually in remedial situations).

I wondered why they would not find Vygotsky's theories and Davydov's implementation of them in early mathematics curricula appealing or at least interesting. After all, that viewpoint is all about the idea that kids, left to their own devices or to typical educational approaches to teaching them mathematics, will have little reason to grapple with abstraction. Schmittau and Morris make this clear when they write regarding Vygotsky and Luria that they

"found in their studies of the development of primates, 'primitive' peoples, and children, that cognitive development occurs when one is confronted with a problem for which previous methods of solution are inadequate [Comment: not unlike how much mathematics has developed, by the way]. Hence, Davydov's curriculum is a series of very deliberately sequenced problems that require children to go beyond prior methods, or challenge them to look at prior methods in altogether new ways, in order to attain a complete theoretical understanding of concepts. More importantly, their consistent engagement with this process develops the ability to analyze problem situations at a theoretical rather than an empirical level, and thus to form THEORETICAL rather than EMPIRICAL generalizations, which is the distinguishing feature of Davydov's work." ("The Development of Algebra in the Elementary Mathematics Curriculum of Davydov," THE MATHEMATICS EDUCATOR, 2004, p. 62)

One might expect that this approach would have great appeal to anyone concerned with how little is asked of US students for the most part in elementary mathematics teaching and curricula. And given the fact that for many disadvantaged students, reading is a huge impediment to using many of the often-maligned progressive reform elementary books such as EVERYDAY MATHEMATICS and INVESTIGATIONS IN NUMBER, DATA & SPACE, the following should be enormously heuristic:

"The curriculum itself. . . consists of nothing but a carefully developed sequence of problems, which children are expected to solve. The problems are not broken down into steps for the children, they are not given hints, and there is no didactic presentation of the material. There is nothing to read but one problem after another. The third grade curriculum, for example, consists of more than 900 problems. Teachers, in turn, present the children with these problems, and they do not affirm the correctness of solutions; rather, the children must come to these conclusions FROM THE MATHEMATICS ITSELF [emphasis added]. The children learn to argue their points of view without, however, becoming argumentative." (loc. cit.)

I find it interesting, however, that in speaking with several US mathematicians and mathematics educators who either have developed or are in the process of developing curricular materials (or both) based on Davydov's ideas and work state that they don't feel that Davydov's original materials would fly in the US (in translation, of course).

I strongly suspect that the concern is not so much for the ability of kids to succeed with Davydov's materials (after all, if they work for kids in Russia, and they are still successful there (at least up to about 10 years ago; I have no more updated information) as evidenced by the 1999 testing at the end of third grade of 2300 Rusian students who used the El'Konin-Davydov curriculum. According to Catherine Sophian's latest book, Vorontsov concluded based on the results of the testing that

"'pupils learning in [this] educational system completely fulfill the requirements of the existing state standard.' The percentages of students who succeeded on 'tasks corresponding to the "standard" level of elementary school' ranged from 86 to 96% with the exception of a task that involved dividing multi-digit numbers, on which 76% of the pupils were successful. In a comment reminiscent of the debate surrounding reform mathematics in the United States, Vorontsov goes on to state, 'The results obtained dispel the myth, current in pedagogical circles, of the poor results of mastery of subject matter among children in [these] programs.'" (Sophian, Catherine: THE ORIGINS OF MATHEMATICAL KNOWLEDGE IN CHILDREN, 2007, p. 168).

In fact, I suspect the problem, such as it is, has little to do with kids, but rather with the low ability and willingness of so many US elementary teachers (and beyond) to teach mathematics without step-by-step guidebooks and their relatively weak knowledge of the relevant mathematics and content-based pedagogy. Considering that any non-traditional mathematics program introduced in the US that does not have the imprimatur of such educationally conservative groups as Mathematically Correct, NYC-HOLD, and a host of think tanks that condemn each and every "progressive math" program being used in this country, but praise uncritically particular programs from high-scoring Asian countries (particularly Singapore) for reasons that may not be completely non-political or grounded in reality for American kids and teachers, are subject to criticism for being TOO verbal, TOO hard, TOO sparse AND TOO watered-down (I am not kidding; the willingness of these self-proclaimed curriculum experts to offer utterly contradictory criticism in order to destroy programs they don't like is remarkable) it takes a daring teacher, school or district indeed to risk using something that is so starkly non-verbal as are the original Davydov materials.

Nonetheless, I think that teachers in inner-city, poverty-stricken, and other challenging environments would, if properly helped to understand how to teach and implement a Davydov-like program, be interested in mathematics books that don't have the serious disadvantage of being inaccessible to non-native and native English speakers alike who struggle with reading English.

Obviously, to fully evaluate the potential of a program like that described by Schmittau and Morris, it would be necessary both to examine the problems and see the instructional guides that come with them, and/or to view classrooms in which children are taught by competent teachers who have been trained effectively in its use. Thus, it is impossible to draw definitive conclusions about exactly what students are doing and how they are led to do it. While the reported results are naturally intriguing, it remains an open question, to my mind at least, as to whether American teachers could be brought to use willingly (and ultimately successfully for their students) a translation, with or without adjustments, of the original Davydov problems and guides. Absent DIRECT access to the problems or teacher materials, and knowing that replication of Schmittau's and her colleagues' research has not been attempted elsewhere. However, neither is it fair to draw any definitive conclusions from that. One would also want to look at the research emerging from various Americanized programs that purport to take an approach similar to Davydov's (though clearly they do not follow the entire Vygotskian theoretical approach as Davydov and his colleagues implemented it).

All the above said, I find it curious that some critics feel well-positioned to draw any conclusions either about Davydov's work or what is actually behind much of American student failure in mathematics. No expansion upon the tautology that students fail because they "don't get it" is likely to be adequate to the task unless one has looked adequately at approaches that purport to be able to take students to much higher degrees of success in mathematics, on average, by instead of giving students pre-algebraic experiences that are numerical, giving them pre-numerical experiences that are algebraic. I'd say that 2300 Russian students succeeding with such an approach is worthy of serious consideration as at least one piece of the puzzle as to how to remedy student failure. Offering up tautologies and then speculating about the causes based on one's own limited imagination? Probably not so much.

Saturday, March 13, 2010

Terezinha Nunes and Peter Bryant Dole Out The Multiplicative Harshness

From CHILDREN DOING MATHEMATICS by Terezinha Nunes and Peter Bryant:

"[I]t would be wrong to treat multiplication as just another, rather complicated, form of addition or division as just another form of subtraction.

The reason for this is that there is much more to understanding multiplication and division than computing sums. The child must learn about and understand an entirely new set of number meanings and a new set of invariants, all of which are related to multiplication and division but not to addition and subtraction. [p. 144]

One is tempted to stop there. The above comes from the opening of the authors' chapter on multiplication and division, in a section called "Multiplication, Division, and New Number Meanings. For the debate that has been raging all over the internet since Keith Devlin published his first column about multiplication in 2008 called "It Ain't No Repeated Addition," it seems that for many people, the ideas raised in the above quotation are absurd. After all, one can compute products by doing repeated addition and quotients by doing repeated subtraction. If you can get the right answer to a computation by two different operations (which begs the question as to whether repeated addition is actually a well-defined operation), aren't those operations the same? Isn't multiplication, when all is said and done, repeated addition? Certainly this is true for the whole numbers, right? And the analogy used there can readily be made to fit integers, rational numbers, maybe even the real numbers in their entirety. It's merely common sense, and this is, after all, the way CHILDREN think, isn't it?

Let's look further at what Nunes and Bryant have to say before considering the above question again. Summarizing the number meanings and situations for additive reasoning they write:

Additive reasoning is about situations in which objects (or sets of objects) are put together or separated. All the number meanings in additive situations are directly related to set size and to the actions of joining or separating objects and sets. Number as a measure of sets involves putting objects into a set where the starting-point is zero; number as a measure of transformations relates to the set that is joined to/separated from another set; number as a measure of a static relation (in comparison problems) relates to the set that would have to be joined to/separated from another in order to make two sets equal in number. [p.144]

Anything too radical there for the folks who seems to cling so passionately to the Multiplication Is Repeated Addition (MIRA) point of view? I would think not. For all that Nunes and Barnes are doing is reflecting the essential nature of addition itself. And there is no real debate about that raging underneath the battle over multiplication. The trouble comes when Devlin and others suggest that multiplication not only need not be defined as "repeated addition" (even though it often is in many places, including reputable ones), but also that it simply should not be so defined. It is a separate operation that stands on its own, with no need to be defined as something else.

To be sure, as previously stated and as Devlin and others on the other side of the aisle in this particular debate have stated many times, we can readily arrive at the result of multiplying two whole numbers, say, 3 x 2, by adding 2 + 2 + 2. No one will argue that the resulting product and sum are not the same whole number, 6. But is that coincidence (in the literal sense of two things "falling together") sufficient to determine that we are looking at the same operations (indeed, there is good reason to question whether repeated is a well-defined binary mathematical operation (Devlin says it is not)?

Well, if not, why not? What IS different about multiplication and multiplicative reasoning? Nunes and Barnes state:

Situations which give rise to multiplicative reasoning are different because they do not involve the actions of joining and separating. We will distinguish three main kinds of multiplicative situations: (1) one-to-many correspondence situations; (2) situations which involve relationships between variables; and (3) situations which involve sharing, division, and splitting. [pp.144-145]

Those of us familiar with some of the arguments of those in the MIRA camp know that one typical response from such people is to suggest that their antagonists are presenting "distinctions without difference." Is that in fact the case with the analysis of Nunes and Bryant? Let's examine each of the situations they mention above in this light.

One-to-many correspondence

Most of us are familiar from ordinary experience with the notion of one-to-many correspondence. Some obvious examples are: a car has four wheels (1-to-4); a hand has five digits (1-to-5); a triangle has three sides (1-to-3); a piano has eighty-eight keys (1-to-88), and so on. As Nunes and Bryant point out:

There are some continuities between these multiplicative situations and additive situations. The most salient is that some of the number meanings here are also connected to sets.

In the examples above, "four wheels," "five digits," etc., do refer to set size. But Nunes and Barnes claim that there are four greatly significant differences:

First, the multiplicative situations involve a constant relation of one-to-many correspondence between two sets. This constant one-to-many correspondence is the invariant in the situation, a type of invariant which is not present in additive reasoning. The one-to-many correspondence is the basis for a new mathematical concept, the concept of ratio. In order to keep, for example, the correspondence '1 car to 4 wheels' constant, each time we add one car to the set of cars we must add 4 wheels to the set of wheels - that is, we add different numbers of objects to each set. This contrasts with the additive situation where, in order to keep the difference between two sets constant, we add the same number of objects to each set. [p.145]

In raising the issue of ratios (and thereby proportional reasoning) quickly, Nunes and Bryant cut to one of the core issues that mathematics educators must take seriously: if multiplication is fundamentally not about joining/separating sets, but about things like ratios, is it reasonable to suspect that one major cause for the well-known difficulties students have with rational numbers and proportional reasoning comes from the propensity of so many teachers to introduce multiplication as if it were simply repeated addition, a viewpoint that is, of course, naively reinforced by those teachers and other adults who were taught to think about multiplication the same way?

It is hard to believe that many young children who run into difficulty grasping multiplication will NOT be told by most adults, "Look, honey: it's just repeated addition. See it now?" Amazingly, I have read some MIRA supporters claim that many kids don't know this "fact" (which of course these same teachers presume is correct), and go on at length about how they must remediate such students by showing them this "obviously true" model. One of them has gone so far as to argue that Devlin and his supporters wouldn't be making claims against MIRA if only he and they had studied with her as a teacher. I'm not confident that this was intended as humor.

Getting back to Nunes and Bryant: they state that this new kind of reasoning

leads us to the second difference: the actions carried out to maintain a ratio invariant are not joining/separating but replicating (to use Kieren's expression) and its inverse. Replicating is not like joining, where any amount can be added to one set. Replicating involves adding to each set the corresponding unit for the set so that the invariant one-to-many correspondence is maintained. For example, in the relation 'one car has four wheels', the unit to be considered in the set of cars is one, whereas the unit in the set of wheels is a composite unit of four wheels. The inverse of replicating is removing corresponding units from each set. If we remove one car we must remove four wheels, in order to maintain the 1 : 4 ratio between cars and wheels. [p. 145]

Here we see one of the most problematic points in trying to convince MIRA supporters that multiplication is not repeated addition. They do not see that in a question like, "If each car has four wheels, how many wheels are there on six cars?" that there are two sets being operated upon differently from the way sets are in additive situations. I suspect strongly that this has something with the difficulty they have in seeing (or simply the refusal to grant) that there are implies or explicitly stated units associated with each set in the problem as stated and they are not the same for each set, whereas in addition, we would be talking about joining or separating sets of the same kind of object (with the same unit, say, 'tires') in each set. With the one-to-many concept grounding this view of multiplication, we have a set of cars with a unit "one car" and a set of wheels with a composite unit, "four wheels per car." Thus, if there are six cars, this problem can be thought of as six cars * four wheels/car. Multiplying the numbers yields
24 cars * wheels/car which equals 24 wheels. In my view, a final non-composite unit, in this case, wheels, can emerge by itself only from multiplication or its inverse. Were this truly repeated addition, we would start with a set containing zero wheels and add four wheels at a time. Joining sets of wheels can only result in wheels. Cars have nothing to do with the case. We could, in fact, add 1 wheel, then 3 wheels and then 2 wheels and then 5 wheels and then 4 wheels and then 6 wheels and then 0 wheels and then 3 wheels and get the same set of 24 wheels. But it would not have the same meaning as 6 cars times 4 wheels/car = 24 wheels. This is the point the authors make when they state that in joining "any amount can be added to one set." And I am convinced that MIRA supporters simply do not see this crucial difference. (There is something here worth looking into regarding how the units behave in rational number arithmetic that I shall delay until another post.)

Turning to the third crucial difference between addition and the one-to-many correspondence situation, Nunes and Bryant state:

[A] ratio remains constant when replication is carried out even if the number of cars and the number of wheels change. In a set where there are 3 cars and 12 wheels, the ratio is still 1 : 4. This is the case because the ratio does not represent the number of objects in either set but is an expression of the relation between the two sets. [pp. 145-156]

This distinction, too, seems utterly absent when reading what MIRA defenders speak about. They harp repeatedly on the fact that the calculations are the same, and thus the operations must be the same or at least that multiplication is reasonably viewed as definable in terms of addition (or the "functions" must be the same, terminology that on my view does absolutely nothing to clarify the underlying meanings or fundamental mathematical ideas under consideration, though perhaps it does help obfuscate those ideas for adults, including some teachers, who aren't sure what a function is or whether calling something a function will help focus, rather than obscure, the issues.)

The last distinction Nunes and Bryant raise for the one-to-many correspondence situation is that:

a new number meaning can be identified in the number of times that a replication is carried out. For example, if we start with the simple situation where we have 1 car and 4 wheels and replicate this starting situation six times, "'6' refers to the number of replications - called the scalar factor. A scalar factor is neither about cars nor about wheels; it does not refer to the number of objects in the sets but to the number of replications relating the two set sizes of the same type. 'Six' expresses the relation between 1 and 6 cars and between 4 and 24 wheels. For the ratio to remain constant, the same scalar factor must be applied to each set.

It is worth pointing out that ratios do not need to involve a unit: for example a recipe may involve a 2 : 3 ration between the number of eggs and cups of flour. When you increase the number of cups, you also need to increase the number of eggs so that the ratio remains constant.

The number meanings in one-to-many correspondence situations are schematically represented in figure 7.1 [Note: the figure consists of two drawings, each with a truck on the left and four wheels on the right. The first drawing has the words "1 truck, 4 wheels: each replication keeps the same ratio." The second has the words "2nd replication of 1 truck, 4 wheels"] In short, one-to-many correspondence situations involve the development of two new number meanings: ratio, which is expressed by a pair of numbers that remains invariant in a situation even if the set size varies, and the scalar factor, that refers to the number of replications applied to both sets maintaining the ratio constant. It should be clear that neither of these meanings relates to set size: the ratio and the scalar factor remain constant even when the set sizes vary. [p. 146]

I will remind the reader that the above four differences are only raised about the first of THREE situations Nunes and Bryant examine. Still to come are those that involve relationships between variables - co-variation, and those that involve sharing and successive splits. We will examine these in subsequent posts. However, I believe that there is enough in just this first situation to seriously damage the idea that multiplication is just repeated addition, even looking strictly at whole numbers. The supporters of MIRA as a reasonable way to introduce children to multiplication and to help children who are struggling with multiplication and multiplicative reasoning have a lot of 'splainin' to do. Or, more likely, explaining away.

Wednesday, March 10, 2010

An Open Letter to Wayne Bishop (and the MC/HOLD posse)

Responding to some positive remarks about progressive mathematics education, Wayne Bishop (seen above), a founding member of the anti-progressive reform group Mathematically Correct wrote:

Such speculation sounds beautiful, of course, but I have yet to meet
any mathematician who was taught in a full-blown "discovery"

This prompted me to write the following open letter:

Dear Wayne and Posse:

Reading your comment about what sort of mathematicians you've never met, I must point out that I've yet to meet one who was taught in a post-racial American classroom, either. That's because neither environment exists, or perhaps like post-racial American classrooms, a fully-realized (is it just a coincidence that you used 'full-blown,' a term I only hear used in connection with cases of AIDS?) K-12 discovery environment exists in some tiny little isolated pockets of the country, so tiny and so few that it's merely a drop in the ocean of indifferent, mostly traditional teaching, materials, and curriculum. Of course, there are districts that use some progressive reform curricula in K-5 or K-8, a very few who use them in K-12. But then, books aren't "discovery learning" or student-centered teaching. They're books. Last I checked, teachers and pedagogy are a major component of what goes on in mathematics classrooms. Classroom culture. School culture. Loads of other factors. Which textbook was purchased may not even reflect which resources are used. I've been in many classrooms where teachers have two or more sets of books and pull from all of them, none of them, or some other variant. At any rate, finding those discovery classrooms that might or might not produce mathematicians, doctors, lawyers, or members of Mathematically Correct is a challenge. Finding ones that fit MY view of high-quality, inquiry-based, student-centered discovery learning with good mathematical content and good problem tasks is not a trivial matter.

This is especially so if we're discussing, ahem, full-blown discovery from K to 12 and then through college and graduate school. And mathematicians would necessarily have to have completed graduate school, right? So you're complaining about not meeting someone who under current conditions cannot likely exist. Let's not miss the fact that the curricula and methods you decry weren't even in the wind for the most part until the early to mid 1990s. Doing the math. . . gee, whiz: it would be pretty surprising to find a professional mathematician who was in a discovery learning oriented mathematics classroom from K - 20 (Kindergarten through Ph.D) Because the environment required doesn't exist in all likelihood and certainly not in one district in enough classrooms to cover K-12 over the requisite years, let alone a university where one would find it from freshman year through the end of doctoral classes.

So you've given us a tautology. No doubt you've not found a lot of dead people who are alive, either. Great job, Brownie!

Then again, have you met many mathematicians at all lately, Wayne? Particularly recently-minted ones? I didn't think so.

You went on:

Many of us correctly believe that we should have been
taught more, and more quickly, but the idea of not learning as much
as possible (ostensibly, from knowledgeable teachers and/or
well-written mathematics books) before embarking on discovering new
and exciting mathematics is purely the stuff of ed school insight,

You made that up. Where is it written in "ed school insight" that we should not want students to learn as much as possible, and from excellent teachers, texts (and other sources you seem to always forget about).

But then we have your usual intentional distortion of what "discovery" entails.

Let's keep this simple for the slower readers in the audience: when is 15 - 9 = ? a problem, and when is it an exercise?

Not to keep you in suspense, it's a problem when you haven't been explicitly shown how to do it or make sense of it. It's an exercise when it's something you already know and are asked to merely repeat what you know to demonstrate that you know it.

If I give that question to the vast majority of first grade students, for them it's a problem. If I give it to the vast majority of high school students, it's MOSTLY an exercise, though for some it's STILL a problem, sad to say. Left to figure out what this could mean, students will figure out one or more ways to have it make sense to them. Given the chance to share ideas under the guidance of a wise and knowledgeable teacher, they will decide what makes mathematical sense and choose the method(s) that work well for them. And given the chance to think, they'll likely keep right on thinking. For 13 years of K-12 and well beyond that. Of course, your own genius children and yourself aside, you don't trust most kids. You really think most kids need Saxon Math or something equally dull. And that most teachers (whom you trust even less than you do children) can't possibly learn to do anything better for their students than take them through a million years of a billion exercises culled from the teacher-proof materials of the late Saint Saxon of the Increments. So utterly pessimistic. So utterly mind-killing. But I suspect that for the majority of kids, that's just what you would love to see.

What IS discovery learning? Is it requiring that students "re-invent" all of the K-12 mathematics curriculum as they go? Of course not. No one has ever suggested anything of the kind except for you and your buddies when you try to scare the pants of the ignorant and gullible. And you do SUCH a good job of it. Kind of like the mathematics educational equivalent of WMD and yellow-cake uranium from Niger, etc. Tell your ugly Big Lies often enough, of course, and no one who didn't already know what you're up to might have a hard time distinguishing reality from your fantasy spook stories.

Is discovery learning creating "new and original" mathematics in K-12? Yes and no. It's new and original to each student as she constructs her own understanding of mathematical ideas. (And once in a great while, K-12 kids actually DO come up with something original. But that's not really the point and you know that fact fully well (or at least that we over here in the real world know it), despite your willingness to feign otherwise). But kids are generally not going to go to many places that they haven't been put in a position to go. If the questions asked and the manner in which they are asked and the classroom and school cultures in which they are asked are suitable, the sky may well be the limit as to what is possible or even likely that kids will do in mathematics or any subject. And when the opposite is the case, then kids will almost never go anywhere worthwhile or meaningful when it comes to thinking about mathematics as much more than a set of rules, facts, and procedures to be memorized at least long enough to pass today's test.

It's interesting that someone as utterly lacking in intellectual joy or curiosity as you managed to become a professional mathematician, though not a particularly prolific one from what I can gather of your output. Seems like you settled into a very mundane position at a less-than-demanding school and decided that to puff up your own importance you'd declare yourself an expert on K-12 mathematics education. And you did a nice job of blustering your way into some level of national prominence (or notoriety, depending upon one's perspective). And so you got to show up at some school board meetings and some state or local hearings, maybe a national one here and there, and declare that the sky is falling because people want to provide kids with a richer, more exciting environment in which to learn mathematics than you ever had. You must be VERY proud of yourself, indeed. Seriously.

But boy, does that progressive education stuff threaten and disturb your little world. And so you latched onto the post-Reagan rhetorical ploy of how to undermine progressive thinking and work: preemptive strikes! It's brilliant. If you're politically, socially, personally, educationally,or philosophically regressive, accuse the other guys of being what you clearly are before they get a chance to point out the obvious about you.

MC, HOLD and similar groups are just a small part of the national manifestation of this sort of tactic. You get to call black people, native Americans, and anyone else you choose "racists" if they advocate an approach to math education that goes against your grain. What could be sweeter? You get to call yourself a reformer, when your idea of reform is "Back to the one-room Iowa school house of my youth" or just back to SOMETHING, even a something that for the vast majority of us never existed and never will. Or back to Saxon Math. Gevalt, it's enough to evoke tears from a gargoyle.

It's hard not to laugh when you cite the seminal group who created Mathematically Correct. Even though you managed to attract a couple of self-proclaimed "socialists," to the last member MC comprises people with conservative souls when it comes to education. You tried to pass yourself off for years on the math-teach list-serve as a "life-long liberal Democrat." That may be the single most absurd and transparent lie ever told.

While no one would suggest that a few of the MC/HOLD cabal are highly-regarded mathematicians, you don't quite get to declare yourself (or David Klein or Jerry Rosen) to be in the elite just by rubbing elbows or what-have-you with Jim Milgram. And having Jim Milgram in your fold doesn't make any of the rest of you (or him, for that matter) knowledgeable about ANYTHING that goes into effective K-12 mathematics TEACHING (I have to suspend my disbelief about college and graduate school teaching).

This really does come back to Lou Talman's recent question to Robert Hansen about how arrogant Lou would be were he to declare himself an expert in engineering because he's a knowledgeable professional mathematician. Robert didn't get it, or played at not getting it, but no one else can miss the point. It's arrogant to step way outside one's area of expertise (alleged or real) and then bash the knowledge and professionalism of the actual experts in that field, merely because there is some area of overlap (yes, the word 'mathematics' does overlap). But knowing math well and teaching math well or understanding what it takes to do so are not the same thing. The second two clearly require the first but don't necessarily follow from having it. And that's where you and your MC/HOLD buddies just go utterly off the rails and never come close to jumping back on again in the two decades or so that you've been trying to call yourselves everything you're not.

Deborah Ball and Magdalene Lampert, to name two non-mathematicians you no doubt would deride (behind their backs if not to their faces, or at least I recommend you not try the latter tack) as 'educationists' whose schools should be blown up (you're lucky: they work at the same one here in Ann Arbor), know more in their little fingers about teaching K-12 mathematics than you'll ever know if you live to be 1,000 years old. You are simply not ever going to figure out the things they figured out without all your advanced knowledge of abstract algebra (even when you try to pull the wool over the eyes of some readers here by throwing out a bunch of jargon to hide the fact that multiplication isn't repeated addition and never is going to be repeated addition. If it were, you'd long ago have addressed my inquiry about why those real mathematicians amongst whom you fancy yourself to belong all seem to think we need two fundamental operations for rings and fields and all the structures in between. You know it's not because they think that the latter is just some version of the former. But you can't bring yourself to say it because you'd be agreeing that there's something wrong about the traditional American curriculum. Horrors!!!)

Were you lucky enough to see Ball or Lampert teach kids, I think your head would explode. Well, not really, because your ability to shut out what you don't want to see, to come up with a thousand reasons why what you see can't be what's really going on is truly remarkable. It no doubt keeps you sane in the face of tons of facts that would produce an overload of cognitive dissonance in most people.

And if all else fails, you'll bring up test scores. Or religion. Or one of your dozens of other dishonest ploys.

I wonder if it ever occurred to you that we can tell more about what goes on in a classroom by actually observing what goes on in a classroom than by all the multiple guess tests in the universe?

Probably not.

But please, Wayne: no more red herrings and lies about what discovery learning is or why YOU'VE never met a mathematician who was trained in such an educational environment in K-12. Instead, talk about the millions who never had a chance in hell of becoming mathematically educated even minimally because they were never shown actual mathematics or any way to think mathematically. And hang your head in shame for continuing to try to prevent that from happening merely because it threatens you in a host of ways.

Well, not to be unfair, let me give you most of the last words:

not professional mathematicians let alone (and statistically
speaking, more important) those who need a strong mathematics
background to pursue their areas of interest. For example, none of
the seminal group who created Mathematically Correct word is
mathematics per se (although some of us became involved very early).
Two were PhD's from Stanford, one of statistics and another in
genetics (later recruited as full professor with tenure to Brown),
another was a PhD in geophysics from USC, another was an independent
contractor electrical engineer, etc., united serendipitously one
evening with a single common thread; all were teaching their children
(and sometimes small groups of their children's friends as well)
mathematics to compensate for their school's use of one of the better
of the math reform curricula, CPM under the misnomer College
Preparatory Mathematics about which I have had some experience:

Groovy. Just remember: your Ph.D wasn't from Stanford or anywhere close to it in quality. Nor do you teach at Brown, USC or anywhere near that caliber of institution. You don't get credit because some people who do happen to agree with you to sit in the same room sometimes and share your narrow and elitist views. But if you bet me one Jim Milgram, I'll see you with a Hyman Bass, and raise you a Deborah Ball and a Magdalene Lampert. You've got nothing on your side to match the likes of them, or the many outstanding K-12 mathematics teachers who get what all this is really about. You know: kids learning and doing mathematics and thinking mathematically. Not being little Saxon robots. Or robots of any kind.

Independent, democratic, student-generated thinkers and inquirers: they're not just in English class any more.