As readers familiar with the Math Wars know, one of the most active groups who attack reform and progressive ideas in mathematics education (and other educational areas) calls itself HOLD, which ironically stands for "Honest, Open, Logical Debate." That is a grand slam of linguistic abuse, given that there is nothing honest or open about this group of angry professionals and parents, they do NOT welcome debate, and while there is sometimes logic in their arguments, just as there is a kind of logic in MOST opinions, that doesn't make what they assert true. If you start with false assumptions, you can draw some lovely logical conclusions that are utterly false, even if they are logically valid.

While HOLD began in California as part of the assault on whole language and reform math, it leapt across the country to form a second (and as far as I can tell, the only active branch), NYC-HOLD. What has made this offshoot effective is that many of its members are professional mathematicians affiliated with NYU's prestigious Courant Institute (though not all the members are so affiliated, not all from NYU are mathematicians, and the ostensible head of the group, Elizabeth Carson, is a self-described "actress." And yes, she certainly CAN act, as her testimony before various public hearings and at school board meetings, etc. attests.

Like any number of other leaders and members of "parents with pitchforks" groups dedicated to overthrowing any and all instances of what they perceive of as progressive education (if you, like I, grew up thinking that "progressive" is a positive word in this regard, rest assured they would vehemently disagree), including "whole language" literacy instruction and reform ideas in mathematics education that they dismiss as "fuzzy" (one of the less offensive epithets they and their forerunners at Mathematically Correct have used to attack ideas they dislike), Ms. Carson can whip up all the drama needed to gain attention of the media and scare parents and other educational stakeholders about the evils they associate with progressive reform.

While it would be impossible to debunk the entirety of the disinformation that has appeared over more than a decade on the NYC-HOLD web site, I have chosen one of their web pages as an appropriate place to focus my first substantive blog entry following yesterday's announcement and introduction of the Rational Mathematics Education blog. I'm going to look at their "Ten Myths About Math Education And Why You Shouldn't Believe Them" by Karen Budd, Elizabeth Carson, Barry Garelick, David Klein, R. James Milgram, Ralph A. Raimi, Martha Schwartz, Sandra Stotsky, Vern Williams, and W. Stephen Wilson. This appeared on May 4, 2005 at

While you may go to their page for the original, I will include here only the ten "myths," the "reality" the NYC-HOLD folks offer up, and my own responses, taking them in order.

Under the misleading heading of "NCTM (Fuzzy) Myth we get:

Myth #1

"Only what students discover for themselves is truly learned."

And the alleged "Reality" :

"Students learn in a variety of ways. Basing most learning on student discovery is time-consuming, does not insure that students end up learning the right concepts, and can delay or prevent progression to the next level. Successful programs use discovery for only a few very carefully selected topics, never all topics."

My take: This is one of the innumerable assaults on constructivism in mathematics education. Nearly every such attack distorts or completely misrepresents both what constructivism is (a theory of human learning) and what constructivists believe are the theory's implications for mathematics teaching and learning.

One classic distortion is to write as if "constructivism" and "discovery learning" were the same thing, and that there is only one flavor of "discovery learning" (and it's horribly ineffective, time-wasting, etc. Opponents of progressive education are very fond hollering about precious instructional time, and this leads some of them to simultaneously lobby for longer school days and a longer school year, as well as a reduction or elimination of "extra" things like recess, music, art, etc. The result for kids and teachers has been quite horrific, and one need only spend a day in most elementary schools where recess has been cut back to 15 minutes once a day for fewer than five days a week (if not entirely done away with) to see the deleterious effects of such a short-sighted policy. Not only do kids absolutely need time to move their bodies and play, but teachers, too, need time to rest their brains a little, move, and do a little reflecting on what's happened thus far and how that might lead them to modify their plans for the rest of the day. While I will likely be critical at times in this blog of the lack of professional reflection on the part of most teachers of mathematics in K12 that I've observed, the blame should not fall solely on the shoulders of the teachers, especially not those who genuinely WANT to reflect and revise practice even in the course of a single day, but who don't have a moment to breathe or even go to the bathroom, let alone reflect on lessons and craft changes based on the day's experience (for the teacher and the students). Making teachers "teach" more and kids "learn" more is, of course, part of the big, fraudulent push for more "accountability" in public education: someone things teachers are sitting on their thumbs getting "big bucks" for goofing off. Do some teachers in fact goof off? Of course. There's nothing unique to education about lazy, cynical employees. But for the vast majority of teachers who care about their students and are trying to do a nearly impossible job at pay levels that generally do not reflect either the amount of education they've had to qualify them for their work or the number of hours they work on their craft and attend to their duties outside of their classroom and building.

That said, no one suggests seriously what is asserted to be the position of NCTM. That is, no one asserts that unless a student discovers something for her/himself, s/he doesn't learn it or doesn't "truly" learn it. The problem here to begin with is two-fold: we have an NYC-HOLD straw man version of what NCTM (and by implication other reform-inclined mathematics educators) supposedly believes, and no one at NCTM or anywhere else believes any such thing. But on top of that, we have this already-mentioned conflation of constructivism (which DOES speak to how we all learn) and discovery learning, which is one option among many.

The effect of this quite deliberate conflating is to lead unsophisticated readers to believe that when they hear someone advocate for a constructivist view of knowledge and learning, they are hearing a call for the exclusive use of discovery learning. However, constructivism as a theory of learning does not call for or lead logically to any particular teaching approach. What it asserts is that learning takes place in a cycle: no one starts as a tabula rasa, every person has an individual perceptual make-up that varies, and upon birth, every experience is processed in an ever-adapting, ever-changing, ever-modifying and being-modified framework. What we encounter is perceived and processed by us and at the same time changes us in ways that impact how we perceive and process each subsequent experience. This cycle of mental and physical experiences interacting with our already-existing but in flux constructs and mental and physical processes is what we call "learning." No two people perceive the same basic events identically; each brings unique constructs and perceptual frameworks to the experience and stands in at least a slightly different perspective to the "same" experience.

If the constructivist view of learning is correct, and I have yet to read one that makes more sense or undoes the fundamental tenets of the theory, then what, if anything, are the implications for teaching and learning mathematics? Does this viewpoint in fact lead to the conclusion being claimed by the NYC-HOLD authors to represent NCTM, et al?

Not in the least. There is nothing in constructivism that would say that we only learn what we discover, if by "discover" one means, as I think the HOLD folks intend us to think, "invent from scratch with no direction from teachers or peers (or texts or other sources." But that is simply nonsense. Constructivism doesn't assert, for example, that we don't learn when we read or hear a lecture or do a math worksheet or engage in other sorts of learning activities that the HOLD authors would find acceptable. What theory of learning could offer up something so obviously and empirically wrong and hope to win adherents among educated people?

["Students learn in a variety of ways. Basing most learning on student discovery is time-consuming, does not insure that students end up learning the right concepts, and can delay or prevent progression to the next level. Successful programs use discovery for only a few very carefully selected topics, never all topics."]

Thus, when we are offered the "reality" by HOLD, we first get a tautology that is supported by every competent educator on the planet: of course students do not all learn the same way. Indeed, this is an implication of constructivism. It has been more conservative and reactionary educational writers who have called for "back to basics" and "direct instruction" in mathematics, and for phonics-only literacy education. Yet here we have HOLD writers playing at being reasonable by offering a truism that more fittingly is associated with progressive educational theory and practice (this tactic, by the way, is rampant in the Math Wars and other battlefields in education).

Next, we get the "time-consuming" complaint about discovery learning, as if it were time-efficient to simply spoon-feed information to students who don't get it, don't engage with it, and who must be taught, re-taught, and often remediated for months, years, or even decades on the mathematics they never truly think about effectively when given "efficient" lectures.

Next, we're told that discovery methods (note that we're far from talking about constructivism) do not 'insure" [sic] students end up learning the "right" concepts. This notion plays on fears that, left to their own devices, students will waste time, go irredeemably off-track, become stuck, lost, frustrated, etc. But "discovery" learning comes in a variety of styles and flavors. One of the most common, especially in the textbooks that emerged from various NSF-funded projects, as well as some of their predecessors and successors, is more accurately described as guided discovery. Lessons are crafted to lead students along a known series of questions that lead inevitably to certain conclusions. Students aren't "inventing" new mathematics, but simply being allowed to find something out that is already known. The argument here is that it would be faster to just tell them, and that's true, but if they don't get it when told, it's necessary to spend more and more time reviewing and re-teaching, as mentioned above. And there simply is no historical evidence to support the idea that the vast majority of Americans have reached a high degree of proficiency in mathematics and been able to go on to mathematics-intensive careers through this traditional direct approach. It seems reasonable to invest class time to let students make connections for themselves before a competent teacher leads classroom discussion in such a way that all students are convinced of the truth of what has been "discovered" by themselves and some or all of their peers.

The red herring of "efficiency" is then ramped up with the notion that students won't be able to progress to "the next level" if they haven't "discovered the right concepts." While this might make sense if progressive mathematics education actually left students completely to their own devices, if the textbooks and teachers' guides didn't try to lead students to important mathematical ideas, and if the teacher was completely incompetent regarding both what mathematical ideas are at the heart of the lessons s/he's teaching and the implications of the ideas the students generate themselves. Again, this is simply another HOLD straw man, easy to knock down, but not in fact what is being promoted.

Finally, we're told that successful programs use discovery on a limited basis. This may well be true, but if there is a program that uses discovery learning of ANY type that is approved and recommended by HOLD and its allies in the Math Wars, it's news to me, and likely news to anyone who has been following the debate over the past 15 or more years. The programs for which they advocate tend to eschew any sort of exploration, discovery (guided or otherwise), investigation of student thinking, discussion of the strengths and weaknesses of alternative approaches, etc.

Again, this has nothing to do with what most reform advocates believe or what constructivist learning theory states or implies. And none of this goes towards the initial claim that reformers believe that we only learn what we discover.

There is no doubt that students learn from EVERY lesson. The relevant question is, however, WHAT do they learn? They may be in the classroom while the teacher offers an explanation of fraction addition, multiplication of integers, the Pythagorean Theorem, or a host of other procedures and/or concepts. However, what the students may learn is that mathematics is hard, boring, irrelevant, useless, confusing, etc. Or they may learn that if they can parrot what teacher does, they will get a good grade on the next quiz or test, regardless of whether they have a clue WHAT is being done or why one might do it, let alone why it actually MAKES SENSE. Far too often, what is learned is that mathematics does NOT make sense (sometimes not even to the teacher).

Can "discovery" learning also result in student confusion, error, and other undesirable effects? Of course. No method or teacher or book or lesson is foolproof (regardless of what the late John Saxon or some people who advocate for Singapore Math or worksheets from Kumon may believe). But teachers who are well grounded in constructivism are far more likely to teach lessons that repeatedly encourage and develop student thinking and communication in class. As a good teacher helps her/his students grow from a room of isolated individuals into a learning community, it becomes imperative that students and teacher learn to listen to and respond to one another's "constructions" of mathematics, and this community learns to refine and improve their understanding, both collectively and individually. Instead of letting students sink or swim based on what they can make of lectures and attempts at solving problems at their desks and trying poorly-understood homework exercises away from school, teachers are helping the class reach consensus about what methods and explanations are most sensible and effective. The class is unlikely to agree on one single metaphor or approach, but having more than one way to do or think about mathematics is a strength, not a weakness. Whatever 'extra" time is used to arrive at a more collective understanding that emerges from student thinking (guided by a good lesson and the teacher) is easily justified by the quality and depth of the learning.

This, then, is a more accurate representation of what NCTM and other progressive reformers mean by constructivism and "discovery" learning. I close by pointing out that there is also room on occasion for both looser discovery in which students are encouraged to take more risks and follow paths that may not be so closely planned by authors and teachers, as well as for every traditional or direct approach. It is only when such approaches dominate that we see historically the utter stagnation of mathematics for the majority of American students. This has been the case for decade upon decade, and it is disingenuous in the extreme for groups like NYC-HOLD to suggest that a majority of our citizens were at some golden point in the past deeply mathematically literate by means of lecture and drill-and-practice instruction. It simply isn't so, and never was.

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