More on Devlin and the Nature of Multiplication

Mark Chu-Carroll

My previous post, "Devlin On Multiplication (or What is the meaning of "is"?)" engendered a great deal of discussion on several lists and blogs. However, primary credit for both serious and heated conversations about whether it's sacrilege to teach kids that "multiplication IS repeated addition," "repeated addition is a valid model for learning multiplication," "multiplication of integers can be effectively thought of as repeated addition," and many variations on this theme clearly goes to Professor Devlin himself. 

One particularly interesting conversation has arisen on a blog I enjoy reading on occasion, GOOD MATH, BAD MATH by Mark Chu-Carroll, a PhD in computer science. His July 25th, 2008 post, "Teaching Multiplication: Is it repeated addition?" has resulted in over 65  responses as of this writing and seems likely to continue to give rise to interesting comments about mathematics, its teaching, and much else. While I don't always agree with what Chu-Carroll has to say about mathematics education (not his main focus in any case), he's worth reading, and for the most part those who comment on his blog manage to refrain from the sort of sniping that dominates more politicized venues (e.g., math-teach@mathforum.org). 

Devlin On Multiplication (or What is the meaning of "is"?)

I generally like to be pretty sure I'm right about something before I blog about it here (or at least that I'm not absolutely and irredeemably wrong). Some topics, of course, are relatively safe in that it's pretty hard to determine a definitively correct answer (though perhaps that doesn't prevent some answers from being almost certainly wrong nonetheless): matters of just how to teach something in mathematics or other subjects rarely avail themselves of a single right answer that serves all students equally well for all times and places (regardless of what my antagonists at Mathematically Correct, NYC-HOLD, and similar nests of absolutism may believe or claim). On the other hand, it's easy to be shown to be entirely wrong about specific pieces of mathematics, especially if one is not a mathematician, something I neither am nor have ever claimed to be. Not that members of that priesthood never err or that we laypeople are invariably wrong when we disagree with one of the elect on matters mathematical, of course. But a betting person would do well to give the odds and go with the mathematician when such disputes arise about mathematical particulars. I've known of errors in mathematics problems published by the Educational Testing Service, too, but you'd clean up betting on them against the likelihood they've erred. 

Imagine then my trepidation at taking on someone like Keith Devlin, the popular mathematician who writes regularly for the general public and the profession about mathematics in books, in a monthly column for the Mathematics Association of America, and on National Public Radio. His June 2008 column ("It Ain't No Repeated Addition") came to my attention this morning because several people who blog about home school mathematics teaching (Maria Miller and Denise,  an Illinois-based homeschooling mom who blogs under her first name only at Let's Play Math) were puzzled by what Devlin had to say.

In particular, as his title suggests, Devlin argues quite vehemently that "(m)ultiplication simply is not repeated addition, and telling young pupils it is inevitably leads to problems when they subsequently learn that it is not." It's hard to be more unequivocal than that. Instead, he argues, we need to teach students from the beginning that addition and multiplication are two basic things we can do with numbers that share some similarities in places but are not the same thing (I insert the similarity issue that he mentions in a way that likely gives it more attention than he would like, so responsibility for where I plan to go with that point is mine alone).

By his own admission, Devlin's not a K-12 mathematics teacher and he is perhaps a tad disingenuous when he mentions this at the end of his article, stating, "I should end by noting that I have not tried to prescribe how teachers should teach arithmetic. I am not a trained K-12 teacher, nor do I have any first-hand experience to draw on." But considering the respect both K-12 teachers and the general public have for professional mathematicians, it's hard not to think that his intent and deep hope is that his position might carry enough weight to have a great deal of influence in how arithmetic is taught. This isn't, after all, a major league baseball player endorsing a presidential candidate, something that would carry zero weight with most of us; it's a world-famous mathematician with a good deal of media exposure (more than the vast majority of his colleagues) making a claim about what multiplication is and is not. That would likely influence more than a couple of elementary school teachers, school administrators, and other interested and concerned stake-holders. Or so I would expect.

So now I need to go out on a limb and suggest that while Devlin isn't wrong, he also isn't quite right, and the problem lies with the nature of school mathematics and its teaching, as well as issues of mathematical maturity. His argument reminds me of a similar one I have when I do professional development with K-5 teachers regarding things they tell students that simply aren't true. Well, that's actually not always the case, either. The problem is that so much depends on what set of numbers one is working with. Teachers state without hesitation to lower elementary students that "you can't subtract a larger number from a smaller number" and "you can't divide a bigger number into a smaller number" (alternately on this last one, we hear "six doesn't go into three, so. . . "). And if one is working in the natural or whole numbers, both statements are true. But in the integers, the first statement is false, and in the rational and real numbers, both statements are false. (And one could go on to look at other sets of numbers, but for K-5, this suffices to make the point). So are teachers wrong to make the statements they do?

If you say "No," unequivocally, you're suggesting that you likely agree with the hypothetical teachers Devlin cites who believe that kids simply aren't adequately sophisticated at a given age (or individual point of development not directly attributable to age alone) to deal with more advanced or sophisticated mathematical ideas. It's tough enough to understand subtraction and division of whole numbers without throwing negatives or rationals (let alone irrationals) into the mix. We must walk before we can run, after all.

If you say "Yes," unequivocally, you likely find the reasoning above objectionable, as does Devlin, and conclude that the solution is to eschew making any untrue mathematical statements to kids at any stage of the game (assuming one knows all the requisite mathematics, something perhaps much to be wished for, but not so easy to guarantee in practice, even if we raised not just the bar, but the actual minimal mathematical knowledge of anyone teaching K-5 mathematics in this country in public education). 

Naturally, I wouldn't be writing this if I thought Devlin's analysis was sufficient and proposed solution ("Don't tell kids that multiplication is repeated addition"), such that he's made one , adequate. One thing I find lacking in his piece is a solid example that would communicate well and clearly to K-5 mathematics teachers (based on the ones I've known and worked with) how multiplication differs in some deep way from addition. I does not suffice merely to assert that the two are, for the most part, not the same. Teachers at the K-5 level may be intimidated into accepting an order not to claim otherwise, at least when they're being watched, but if in their hearts they believe otherwise, that belief will surely inform their teaching and the idea is going to be communicated to their students. Similarly, research indicates that college mathematics students struggle to accept fully such concepts as the equality of 0.9999... and 1 even when they can be taught to repeat that this is factually true. This gap between surface knowledge and deep understanding and concomitant belief should not be dismissed.  

But even more importantly, what Devlin proposes presumes that there is one correct model or metaphor for understanding multiplication. This assumption runs counter to vast teacher experience in K-5 mathematics and beyond. The multiplication-is-repeated-addition metaphor fits a number of physical models of multiplication grounded in even more fundamental and deep metaphors of arithmetic discussed extensively by Lakoff and Nunez in WHERE MATHEMATICS COMES FROM. Further, it is clear from the perspective of how arithmetic developed historically that the use of multiplication from a practical perspective can readily be called "fast addition," just as exponentiation can be called "fast multiplication." The associated algorithms that emerged for doing whole number and integer multiplication compress some of the individual steps through the clever use of place value in ways that are sometimes confusing to many students, and it is often helpful to show students how this viewpoint is valid by "unpacking" the algorithms and recreating the "repeated addition" that could be done to obtain the same results. Of course, at the same time, we want students to understand why using multiplication algorithms to find the product of even moderately large numbers is preferable. 

The modern view of multiplication Devlin advocates as "one of the things we can do with numbers" that is different from addition is not necessary for students who are learning arithmetic with any set of numbers up to and including the integers. Therefore, it is questionable that students would be advantaged by being given that more sophisticated idea immediately. Further, Devlin's claim that they are actually disadvantaged by being told something else is something he would need to provide proof or at least strong research evidence for, something that is absent in his article. I'm reminded in part of those who argue that failure to stress or thoroughly teach the long division algorithm by, say, fourth grade, will make students unable to learn and do synthetic division in algebra class. This argument sounds plausible to some people (generally those who know little or nothing about teaching K-12 mathematics), but I have yet to see anyone provide even a shred of evidence to support the claim.

The "obvious" solution to this seeming quandary (lie to students about what multiplication "is," or run the risk of over complicating things for them unnecessarily and losing one of the more effective and understandable metaphors/models for it) is, on my view, the same one I have always suggested would solve the above-mentioned issue of lying unnecessarily to students about the nature of subtraction and division: make clear to students that how we define specific arithmetic and other mathematical operations depends upon what set of numbers we're speaking of and that while they may currently be dealing with numbers that do not allow us to answer questions like 7 - 9 = ? or 3 divided by 5 equals ?, the fact is that there ARE numbers that they will learn about in a few years (if they don't know about them already!) that will allow us to answer such questions. It's not that multiplication "is" repeated addition, but rather that there are contexts in which that is one reasonable way to think about it. However, like all the arithmetic operations, it also "is" many other things, each of which will prove valuable to consider in turn.

In higher grades, teachers constantly tell students that "We can't take the square root of a negative number," but the fact is that this is true only if we are restricted to the set of real numbers or one of its subsets. Using Devlin's argument, that practice should be outlawed because we might be making it "too confusing" for these students when they encounter complex numbers or any other new set of numbers (e.g., quaternions). We shouldn't tell them that multiplication is commutative, because they will by high school encounter matrix methods for solving systems of linear equations and be taught that matrix multiplication with elements drawn from the real numbers is NOT commutative. Surely THAT should be a mind-blowing experience that will destroy students' abilities to progress in mathematics! 

Of course, nothing of the sort happens. And to the extent that it might with some pupils, the same approach I've mentioned previously would smooth the way: simply be honest with students about what restrictions apply to broad generalizations we ask them to accept (or even prove!) in class. This practice would actually raise the level of sophistication students bring to the mathematics they are doing, making them more careful about simply accepting any given statement as true for "all numbers" and any given operation. Instead, we will have impressed upon them the necessity of considering what set of numbers is under consideration and whether statements made about a particular operation are restricted to a given set and situation. Multiplication is NOT always commutative. It also isn't always repeated or fast addition. But it surely is commutative in many circumstances and it is important for students to be taught this and to see that it makes sense. Similarly, it is valuable for students to explore the notion of multiplication as repeated or fast addition, as long as they are given the tools to understand that this might not always make sense as they consider other, more abstract sorts of numbers or applications of the multiplication concept. Students so taught might well develop the valuable mathematical habit of mind of questioning how broadly a particular mathematical idea applies. And that is one of the things upon which they will be able to build a deeper and more mature knowledge of higher mathematics. 

Manipulating Cato: More on Games, Gods & Grades

Victor Hobson/ Cato June/ Fred Goodman

I recently posted on some ideas of Fred Goodman, Layman E. Allen, Sheldon Wolin, and the relationships among puzzles, games, democratic values and mathematics education. I've received some very interesting and encouraging replies, both on and off the comments section of this blog in response to that entry, but none more engaging and useful than the one Fred Goodman sent recounting an anecdote he'd told me once or twice before, the exact details of which had escaped me (though I knew that Vic Hobson was the central figure). With Fred's kind permission, I include it here in its full context:

At a very theoretical level, John Searle builds directly on games as the ideal illustration of the "construction of social reality" in his book by that name.

I distinguish between Layman's emphasis on "literal" games and my emphasis on "metaphorical" (although I strongly believe in and build both "literal" and "metaphorical" as well as "simulation" games). I felt the need for the term "metaphorical" after so many people complimented me on my knowledge of things I KNEW I DIDN'T KNOW when I thought I was building "simulation games." I had not built the relationships they said they saw into my "simulation model." In the early days I drew on Max Black's "interaction theory" of metaphor (as contrasted to what he called the "substitution" theory) in his MODELS AND METAPHORS. Now I find myself referring to "cognitive metaphors" defined by Lakhoff and Nunez as "grounded, inference preserving, cross-domain mappings" (quoted from their WHERE MATHEMATICS COMES FROM).

I like to start with a few rules that players can't change and then encourage players to construct their own additional rules. That was the approach I used as early as 1968 in "They Shoot Marbles, Don't They?" when requested to design a game that taught about "police-community" relations for Detroit area kids. Then, honoring the notion of "metaphor," I always "debriefed" by talking about how what happened WAS NOT like police-community relations as well as how what happened WAS like police-community relations. Of course the real value of the exercise was that the participants could not agree as to what had happened IN the exercise despite the fact that they all had just experienced "it." What this really adds up to is analogous to a "chemistry set" ... I see a metaphorical game as a "kit" for performing experiments in rule-governed behavior.

As for my experience with Vic Hobson, I built directly on the four big, basic math metaphors in Lakhoff and Nunez's "Where Mathematics Comes From." I found that he responded to fairly complex NIM games as long as I used poker chips or any other PHYSICAL representations of ideas ... and as long as it was a COMPETITIVE exercise. One day I said "I guess these poker chips are what math educators call MANIPULATIVES." He said, sort of under his breath but politely, "I don't care what they're called as long as I can MANIPULATE Cato" ... Cato June being his roommate at the time. I REALLY sat up and took notice of that ... recognizing perhaps that this was the key to why Layman's games worked so well. When you manipulate and are manipulated by others, it makes the "social reality" that you are creating something that is felt in a different way than when you do problems (puzzles) in school. I recall so distinctly playfully criticizing Vic for "gloating" after a tremendous tackle that I saw him make on TV. I said "you know you shouldn't do that, but it FEELS SO GOOD you can't really help yourself can you?" He said, "Right." A week later he beat me at a 3-D color coded game of NIM played with poker chips and the look on his face led me to say, "You're GLOATING again, Vic. Cut that out. But it does feel so good to beat the professor, doesn't it?" A long dragged out "RRRIIIGHT" was his response.

While it may seem of only passing interest, one reason I find the above story so compelling is that Fred did a lot of work with University of Michigan athletes in his educational gaming classes, and has a host of stories from his experiences with them that indicate how much university professors tend to miss about the intellectual abilities of elite athletes. I have no idea what sort of student Vic Hobson was in mathematics class or any other academic area, but I am certain from conversations with Fred and my own experience with athletes at U of M and the University of Florida as a graduate teaching fellow in English in the 1970s that there are a wealth of readily-overlooked insights and abilities they bring to college classrooms. In that regard, I suspect that many universities fail to exploit (in the good sense of that word, "to make full use of and derive benefit from") the non-physical resources elite athletes often possess. The intellectual snobbery of academia is not infinite or all-pervasive, but it does tend to place blinders over the eyes of many.

From the perspective of K-12 mathematics education, I find the Hobson anecdote heuristic in a number of ways. First, there is a lot of conflict between more traditionalist/socially conservative educators and progressive educators over the appropriateness and usefulness of "competition" in the classroom. In general, I tend to side with Alfie Kohn and others in questioning the use of grades, praise, and other forms of extrinsic rewards in education. I personally blanch when my administrator sets up competitions amongst the staff in ways that strike me as potentially much more likely to promote divisiveness, resentment, and discontent than the goals I'm sure are intended.  I don't care for the idea of making education into a contest.

And yet, some people I respect in various educational venues see competition in a different light. Are we all talking about the same thing? Probably yes and no. There is the sort of economic Darwinism idea of survival of the fittest in the economic market place that I do not inherently trust. Markets are notoriously prone to manipulation by insider players who have access to both information and other tools that give them unfair advantage. The best product doesn't always win. Witness the fate of the Dvorak keyboard, the beta video format, the Tucker automobile, etc. Furthermore, if we're talking about education, do we want to view it in a primarily economic model as do so many anti-reformers and educational conservatives? I think not. As previously stated a few posts ago, I don't accept the idea that "schools should be judged by their contribution to the economic health of society." I don't think kids are commodities and I am very disturbed by some of the ways business models are routinely applied to public education as if our children were precisely products to be sold in some cut-throat marketplace. While no doubt that viewpoint is held without revulsion by some Americans, it's not one educators should unquestioningly accept, let alone swallow whole with some sense of guilt that they should have been asking business for advice in how to educate students from time immemorial. 

But there is another side to competition, and it is likely that side that Fred is tapping into in his story about Vic Hobson and his ideas about educational gaming. Putting aside the related issues of cooperative vs. competitive games, there is in many of us, perhaps nearly all of us, whether inherent or culturally derived, a desire to excel. And often that means putting ourselves, our ideas, our abilities, etc., in "conflict" and competition with others and theirs. Certainly there is no dearth of zero-sum games that allow us to (relatively) safely play out scenarios that pit us against the physical, intellectual, and strategic talents of others. And in Vic Hobson's remark there is both a very adult and very child-like sounding desire to win, regardless of what others might see as going on in a given situation. But perhaps it isn't even quite about "winning." After all, the word he chooses is "manipulate" rather than "beat" or "defeat." Perhaps there is a sort of competition here that doesn't depend on a final score so much as a feeling of mastery (in various senses). As Fred comments, it has something to do with how one feels about what one is doing, perhaps meaning that it puts a different sort of emotional tone on one's investment in an activity, including an educational one. 

How does that vague sense of things play into my usual concerns about mathematics education and my recent theme about democratic issues in pedagogy? I'm not entirely sure at this juncture, but I feel that it is significant. I know that there is a deep issue for the (mostly) at-risk students I teach that hinges on owning one's own education and taking responsibility for the choices one makes about spending time in and out of school. How much time per day is spent engaging in behavior that is utterly divorced from the ostensible goal of getting a high school degree or learning about mathematics or whatever the student is supposed to be pursuing, short- or long-term? How much energy is invested in just about anything but the subject matter? At times, it appears like students engage in arguments and even fights not because there are real conflicts but because to do so is to be able to avoid the schoolwork that sits there waiting to be done or, as the case too often is, avoided entirely.

I don't want to suggest that the schoolwork is all it should or could be. Much of it is deathly dull as presented and could easily be made far more relevant and engaging. But still, most of them choose to be in school when many of them could do otherwise yet also choose not to make productive use of the time given what their alleged reasons are for being there. 

Could a more game-based curriculum have an impact on this issue (as opposed to the one we are currently using, which is entirely grounded in puzzles)? Is the sort of "social investment" that appealed to Vic Hobson a key missing component for some/many/most of my charges?

Unfortunately, as things are currently structured at my school, there's been no real opportunity to find out. However, things are open to change starting with the new school year (which begins in two weeks). This is something I need to explore with my administrator and colleagues. Perhaps the set of Layman Allen games I have in my desk drawer will come in handy after all. 

More on this and a return to my friend Becca up in Saginaw and her alternative education students shortly. 

In A Spacious Place: Games and Grades

In A Spacious Place: Games and Grades

If you found the recent post about Fred Goodman and Layman Allen of interest, you might enjoy the above.

"Games, Gods and Grades": Fred Goodman Runs the Voodoo Down

Pictured above are University of Michigan Professor of Education Emeritus, Frederick Goodman, and UM Professor of Law, Layman E. Allen. I was lucky enough to have the former as my informal mentor during my graduate studies in mathematics education at the UM School of Education The latter I have known of by reputation since I was in my early teens and bumped into his game of symbolic logic, WFF 'n' Proof, and then via his professional and personal friendship with Fred Goodman. It was Fred who was kind enough to recently introduce me to the work of the brilliant and eloquent Sheldon Wolin, whose essay on matters pertinent to the Math Wars inspired the previous entry to this blog.

In a private response to that blog entry, Fred mentioned some ideas about educational games and democratic values that he thought were relevant to what I had posted. I share what he sent me below with his kind permission:

Games, Gods and Grades (Fred Goodman, 1/27/07)

School grades may be misleading because the problems students learn to solve in school may not be the kind of problems they face after they graduate. Solving a puzzle brings closure to a problematic situation. The creator of a puzzle must not pose a problem that does not have a solution. Success at puzzle solving can be measured by comparing the speed, completeness and elegance of different solvers’ performance and by assessing the relative difficulty of the puzzle. Closure in a game is defined by the game rules not by a problem being solved the way the creator specified. The creator of a game constructs a situation in which players are both the posers and solvers of one another’s problems. Success at games is measured in a startlingly surprising variety of ways, not just in terms of whether a player’s team wins or loses. These characterizations lead me to the following points.

First, an analogy: Games are to puzzles as mysteries are to secrets.

Second, a claim: The more you know about a mystery, the more mysterious it becomes. The more you know about a secret, the less secret it becomes.

Third, a comparison: A puzzle creator is “God-like” in that the creator constructs both the problem and the correct solution to it. A game creator is “God-like” in that the creator constructs the rules that enable participants to make choices that affect each other, provide a criterion by which to compare the participants’ overall success, and specify when the activity ends.

Fourth, an observation: Schools tend to pose problems to students in the form of puzzles far more than in the form of games. This can result in students being taught to think that there is an answer to every question, a solution to every problem. There is an endless array of secrets that others know and you don’t. When students leave school they frequently find that problems in the “real world” tend not to have “once and for all” solutions. Many problems seem to have no solution at all. People create problems themselves and solve problems created by others. They begin to think in terms of strategies for coping with their problems, strategies that serve their ends but can be expected to conflict with other people’s goals. Therefore a puzzle-based education might not prepare people for life after school as well as a game-based education might.

These four points call into question the importance that our society assigns to school grades. In many contemporary upwardly mobile families getting good grades is right up there with “Godliness.” (In some families good grades are probably ranked even higher than “Godliness.”) Grading is intended not only to give feedback to students in a manner that might help them learn better in the future, grading is intended to sort people out in terms of their future value to others. If pernicious grade inflation is to be avoided, some students must learn to adjust to the fact that they just aren’t as good at solving certain kinds of problems as others are. Further, they learn that some kinds of problems are more important than others. But what if the problems that are the basis for such conclusions aren’t the kind of problems that people need to solve when they get out of school?

The answer to that question might well have economic implications but there could be even more serious consequences. As the world moves closer and closer to a world where Gods collide and their followers depend with greater and greater certainty on the correctness of their God’s solution, we need to look more closely at the relations that might exist between games, Gods and grades. If learning is conceived primarily as a matter of finding the one correct answer according to the teacher who already knows the answer, and students’ sense of worth is tied to their ability to discover, understand and accept that correct answer, we may be encouraging, even in our secular schools, a tendency towards sectarian thinking.

There are practical alternatives to the puzzle approach, alternatives that encourage people to reflect upon, cope with, and even enjoy mysteries. That games are analogous to mysteries does seem to be the case insofar as progress towards higher and higher levels of game playing proves to bring greater and more confusing challenges. “Solutions” that worked at one level are exposed quickly as solutions that were only relevant to the prior situation. This follows whether the game is bridge, chess, football … or to move closer to the topic at hand … Equations: The Game of Creative Mathematics. Equations, created by Layman Allen, has been played by generations of students nationally for forty-some years. The game speaks profoundly to the question of what it means to be right, focusing attention on imaginative and efficient use of resources. Students are continuously shifted to learning environments that maximize the challenges to each one and are provided with opportunities to make tangible, positive contributions to their team. Their performance is recorded and shared in a constructive, motivating form of grading.

Similarly Allen’s Queries ‘n Theories: the Game of Science and Language offers students the opportunity to practice performing the act of asking good questions, guided by the construction and testing of theories, in a way that illustrates the very essence of the scientific method. Further, it does so in a way that teaches the relationship between “facts” and “theories” in a manner that is worthy of the attention of anyone concerned with how those two words are used and abused in contemporary discussions of science, religion and policy. (See wffnproof.com for more on both games.)

The example of Equations and Queries ‘n Theories is offered to demonstrate that the points being argued are not solely theoretical. There is a great deal of experience with the use of soundly constructed educational games that manage competition constructively. The example, however, might also serve as “the exception that proves the rule.” That is, even the best of educational games tend to be marginalized and channeled in the direction of extra-curricular activities.

Schools pose problems in the form of puzzles, almost to the complete exclusion of problems posed in the form of games. That observation deserves serious attention because how a problem is structured makes all the difference in the world.

It is to be hoped that those of you who read my previous post see the connections to The Math Wars and questions of what comprises worthwhile, meaningful, and ultimately democratic kinds of activities in math classrooms that are likely to support independent-thinking students who do not quietly and passively go along with authority simply because they are unable, unwilling, or flatly terrified to question it. The mentality that has been used to teach mathematics to the masses in this country (and in many others) has for far too long been grounded in authoritarianism. It cannot be a coincidence that progressive-minded reformers continue to call for approaches to classroom teaching that are more student-centered and which stress communication of mathematical ideas, offering sound reasoning for mathematical answers and procedures, while anti-reformers decry this as "time-wasting," "fuzzy," and somehow too "touchy-feely" to matter. Oddly, many of these same skeptics claim to be very much about choice in mathematics education. However, it turns out that "choice" for them means finding ways to undermine the use of the sorts of curricular materials and teaching methods that are grounded in exploration, investigation, problem-solving, and justifying one's thinking and reasoning. Again, this sad fact cannot be a coincidence. One merely needs to explore the websites of Mathematically Correct, NYC-HOLD, and many local "parents-with-pitchforks" groups to see how much energy and rhetoric is put into trying to ban the use of specific programs, methods, and tools in mathematics classrooms. Choice? Apparently that only applies in situations where something is going on that these folks don't care for. If a conservatively-approved program is offered and nothing else, choice doesn't matter and democracy is for "the other guy."

Once again, it seems difficult to escape the underlying totalitarian and, as Fred Goodman terms it, sectarian nature of the discourse and community created by the vast majority of traditional mathematics teaching. We hear teachers say all the time, "This isn't a democracy; I make the rules here." As an experienced classroom teacher, I understand what motivates such statements when they pertain to classroom management (which is not to suggest that such an approach is the only or best one possible. It is, however, seemingly the one that reflects the role teachers are expected to play when they are evaluated by their administrators and colleagues, as well as by parents and even by kids). However, my concern here is for the way that subjects are taught and what the political lessons are that aren't explicitly stated or acknowledged. And those lessons are fundamentally anti- and undemocratic. The focus upon single right answers that are arrived at by (generally) one approved method speaks volumes towards the underlying values of the teacher, the school, the district, right on up through the state and federal governments. The job of students becomes not learning and thinking, but anticipating what teachers expect exactly as they expect it: no less, and generally no more. And therein lie a host of tragedies, even were there not the anti-democratic issues to consider.

But once we focus on the behavioral lessons being taught, the intellectual hamstringing that discourages independent thinking and teaches and reinforces passivity and fear in students, we begin to see how mathematics class is a particularly good place to help create citizens unsuited to thrive in a truly democratic state, but perfect for life in the sort of system Wolin calls Economic Polity, a state of passive consumers for whom democracy is a shibboleth but not a living thing. How ironic is it that many of the same anti-reformers who insist that their views are about promoting "freedom" for minorities and impoverished people in fact undermine freedom both by doing a poor job of teaching math and by promoting an attitude often reflected in phrases like "ours not to reason why, just invert and multiply"? In one of the subjects that most heavily depends upon reasoning (having and giving reasons for solutions and methods and interpretations of results), the anti-reformers have managed to turn education into as dull, mindless, and dependent an activity as can be imagined, made worse by the enormous feelings of fear and loathing so many of our citizens are taught to associate with it. How better to guarantee the status quo, preservation of the corporate state, and the continued disempowerment of the least privileged Americans?

That Goodman and Allen are onto something powerful with games is undeniable. The sorts and nature of the games that Goodman and Allen develop and promote are precisely the kind that enhance democratic values directly and indirectly through both the content of the games and, more subtly but perhaps even more importantly, through the ways in which social interactions about rules become an inherent part of the game-playing process. It is truly tragic that the vast majority of our schools and teachers do, as Goodman suggests, marginalize games and advantage puzzles (with all the inherent control, rewards, penalties, "grades," and consequent sorting they entail).

It is often said that teachers test what they value, which for me has always meant both that teachers choose to place on tests what they (and/or the system, and by implication the state) value, but also that students quickly learn that the only thing that matters is what's on the test and finding ANY means to pass it. The notion that the learning process or even the course content is what matters, as well as what students choose to do with what is learned might be the sole or primary point, as opposed to the grade or the degree being "earned," strikes the contemporary student as a definition of insanity. Beating the system is viewed as perfectly normal and reasonable. Actual independent thought and effort beyond or outside of what is clearly defined by the test is nearly unthinkable. A perfect fit for future manipulators in the market place, and a third-rate "education" into no where and general political passivity for the enormous majority who really need "the knowing of things" to have any chance to contribute and thrive to the community and themselves without resorting to unethical and/or illegal but self-serving, "system-beating" enterprises. Wolin's sorting process is clearly one of the major effects of such miseducation. Cui bono? Certainly not very many of us, but for those who ARE benefited, the profits are astronomical. Too bad the numbers involved will be lost on most of the people, who'll no doubt be intensely engaged with AMERICAN IDOL, SURVIVOR, and other bread and circuses.

Wolin, Democracy and The Math Wars

The new vision of education is the acquisition of the specific job skills needed in a high-tech society. There are some striking consequences of this definition of education or, rather, the redefinition of it.

One is that the principal purpose of education is no longer conceived primarily in terms of the development of the person. In the past, the person was understood in complex terms of diverse potentialities. The academic subjects to be studied represented not only different methods of understanding but elements of a different sensibility. Becoming a person meant embarking on a quest for the harmonizing of diverse sensibilities.

The rejection of that conception of person can be measured by the disappearance of the older rhetoric about “personal discovery,” “the exploration of diverse possibilities,” or “initiation into a rich cultural heritage.” In its place is an anti-sixties rhetoric which is really an attack on education as the representation of human diversity. Or it is the rhetoric of “core courses” which work to dismiss the very subjects they profess to be defending. The new education is severely functional, proto-professional, and priority-conscious in an economic sense. It is also notable for the conspicuous place given to achieving social discipline through education.

It is as though social planners, both public and private, had suddenly realized that education forms a system in which persons of an impressionable age are “stuff” that can be molded to the desired social form because for several years they are under the supervision of public and private authorities. (Parenthetically, it is in this light that private religious schools have found great favor in the eyes of public and private policy makers: these are perceived as superior means for imposing social discipline, although that discipline is usually described as moral or religious education and as anti-drugs and -sexual permissiveness. Such schools represent the privatization of public virtue.) Third, and closely related, the new conception is tacitly a way of legitimating a policy of social triage. High-tech societies are showing themselves to be economic systems in which a substantial part of the population is superfluous, and so is the skill-potential of perhaps a majority of the working population. Such economies tend to dislocate workers, replace them by automation, or relegate them to inferior, less demanding, and less remunerative types of work. If such a population is not to be a menace, its plight must be perceived as its just desserts, that is, the failure must be theirs, not the system’s. More prisons, not social and educational programs, must be seen as the rational response. The schools should operate, therefore, to sort out people, to impose strict but impersonal standards so that responsibility for one’s fate is clear and unavoidable.

Sheldon S. Wolin: THE PRESENCE OF THE PAST: Essays on the State and the Constitution, “Elitism and the Rage Against Postmodernity” (pp. 60-61)

The above may be and probably should be some of the most important words you read this year about education, mathematics or otherwise. Written by Sheldon Wolin, emeritus professor of political philosophy at Princeton University, they are part of a brilliant essay on, in part, Alan Bloom's THE CLOSING OF THE AMERICAN MIND, as well as the vehement reaction from various quarters against postmodernism and contemporary university education. Moreover, they show that this reaction is above all else part of the opening salvo of neo-conservative Straussism, or at least aspects of Strauss' work as interpreted by what Wolin refers to as that philosopher's epigones that informed or provided justification for, however doubtful it might be that Leo Strauss himself would have agreed with it, the neoconservative movement that dominated the first George W. Bush administration and shaped much of the second until the ship clearly began to sink in 2006.

In reading the quotation from Professor Wolin, I was struck by the eerie similarity between what he describes there and the mind-set of so many anti-reformers in mathematics education that I and many others have struggled against during the past 15 years or more. His essay continues a couple of paragraphs later with this insightful commentary on the Bell Commision's oft-cited A Nation At Risk report:

Although the Bell Report never suggests how it had come about that the nation was at risk, its remedy was remarkable for its pared-down vision of education, its emphasis upon the disciplinary role of schools, and its martial rhetoric. It warned of 'a rising tide of mediocrity' in educational performance, and it likened that prospect to 'an act of war.' It compared current educational practices to 'an act of unthinking, unilateral educational disarmament.' While one might dismiss this as mere hype, the uncomfortable fact is that this rhetoric was chosen in order to establish the context in which the problem of education was to be resolved. Military language is inherently uncongenial to thinking about individual growth but not about adapting individuals to organizational functions. Its barracks language of pseudo-democracy is also a way of brushing off problems of minorities and of the poor. Indeed, the coercive language of war, crisis, and mobilization is so antithetical to what education has traditionally symbolized that it should alert us to the radical recontextualization being proposed for education.

One need not be a scholar of the Math Wars to recognize that it is no coincidence that the military language Wolin mentions informs this particular phenomenon. Words like "entrenched," "battle," "fight," "shots fired," and many other martial terms are common to the passionate debates about how to better or "best" teach mathematics to American students. Undeniably, the rhetoric is often inflammatory and combative. But what really resonates here is the mentality Professor Wolin describes, and the philosophy and politics that inform so much of the commentary in the Math Wars, formal and informal. Articles appear almost daily that reflect an enthusiastic embrace or tacit acceptance of the shift in focus from education as something to develop diverse and individual potentialities to one of creating drones for the workforce. And anti-reform pundits and commentators, as well as some journalists who no doubt see themselves as either neutral or even progressive, buy into the notion that "schools should be judged by their contribution to the economic health of society." Of course, this assumption is hardly one Wolin would accept as a sound basis for effective education, and neither would I. But it fits well the mentality that informs the writing of most educational conservatives and, I believe, goes some distance towards accounting for their opposition to many pending or already-implemented changes in math teaching and curricula.

Progressive Math Education and Democracy

To begin with, NCTM-style reformers and many progressive educators who work independently of its various reform volumes, have long believed that a major detriment to effective teaching and learning of mathematics is the idea that the authority for mathematical truth lies primarily or exclusively outside the student and even outside the classroom teacher (at the K-12 level). Generally, where the teacher isn't seen as the authority, "the textbook" is. Of course, a book can't be an authority. The book is merely a symbol of the authority of one or more authors who are presumed to be authorities and whose work has been sanctioned by even more expert authorities in the general communities of mathematics and mathematics teaching. But a distant author, let alone an even more distant reviewer or endorser, can't know the needs of individuals teachers or students. The author can, at best, offer a finite number of topics, along with treatments of them, along with examples, applications, exercises, etc. in one specific and fixed order. If the author is John Saxon, there is explicitly no room for a teacher or student to skip or change the order of anything. Saxon books are intended be taken as holy texts. No wonder that many educational conservatives adore Saxon Math and promote it above everything else as the absolutely best way to teach the subject. And better yet, in their minds, it has been touted by its creator as "teacher-proof." Surely this claim represents an Eden for anti-reform advocates: no room for variation, individuality, independence, or freedom. Just Prussian military precision and strictures, in keeping with the anti-progressive educational restructuring of the early 20th century in this country.

But even where John Saxon doesn't rule, teachers and students alike often defer to the textbook as the authority. Teachers often do not dare to stray or modify math lessons from district mandated texts, even when they are not proscribed from doing so by anyone. This is typical in the lower grade bands. At the high school level, it is more likely that the teacher will believe and promote the belief among students that s/he is the authority. It goes without saying that this is the rule at the university level, where instructors are, for the most part, professional mathematicians with PhDs who for the most part are (and desire to be) viewed as gods.

In only very rare cases does one see mathematics teaching that promotes the idea that the authority for mathematical truth must in no small part rest within the students themselves, both individually and collectively. This is not to say that if a student or class believes that something patently false is in fact true that the false belief becomes "true" through force of will, or a cockeyed notion that mathematical truth depends upon a majority vote. However, if students do not themselves have to struggle with mathematical truth and sense-making as an issue, it is unlikely that they will ever engage in the questions that real mathematicians grapple with on a daily basis. Moreover, they will not even be able to successfully work through the problems of mathematics they are asked to deal with "at grade level." Instead, they will and nearly-always do become passive recipients of received mathematical truth, always grounded in and authenticated by external authority. And they will predictably resent being asked to determine truth for themselves.

Ironically, this implies that the very complaints one hears so frequently from anti-reformers about how "fuzzy" reform mathematics education makes students unwilling to engage in proof, the real culprit may well be the sort of elementary mathematics teaching promoted by those same anti-reformers. The lack of opportunity for students to struggle with their own ideas about mathematics should be anathema to real mathematicians, who would be expected to recognize the necessity for this process based on personal experience. Why, then, are some mathematicians, both prominent or relatively obscure, so vehemently opposed to progressive ideas about how to promote mathematics learning for young children?

I think Professor Wolin has identified correctly the anti-democratic sentiment that fuels regressive ideas about education, particularly in the public sphere, where the masses of our children are likely to have the opportunities to receive whatever version of education we support as a culture and society. The kinds of activities and discourse that progressive mathematics educators promote generally call for learning communities that are highly democratic in nature. On multiple levels, a democratic approach to mathematical discourse communities is prone to grow children who are a serious threat to the anti-democratic forces currently at work in our country (and elsewhere in some nations that ostensibly are democratic). It makes a great deal of sense that those who fear that the poor will become better equipped to struggle against the gross imbalances that are increasing daily between haves and have-nots (particularly in the United States), should the latter become more literate and numerate, would oppose precisely the kinds of educational practices that inherently promote independent thought, meta-cognition, the challenging of assumptions, the questioning of authority, and self-reliance for determining truth.

Once again, I need to stress that I'm not arguing for mathematical or any other sort of anarchy. The idea is not for students to come to believe that anything goes in mathematics, but rather that they are obligated to improve their abilities to judge mathematical validity themselves, (in opposition to passive acceptance of whatever explanation the teacher or book might offer, particularly when it is quite possible that either source might be in error). So-called traditional education as currently construed is very much about social control, with truth imparted from above and passively accepted. The last thing one would expect those who support that vision of education to accept would be active, student-centered learning that stresses free arguments in class in the context of honest, open, logical debate (hmm, a clever acronym might be hiding there).

Finally, I think Wolin is particularly on point when he talks of the sorting function implicit in the current vision of education, and the need to create a system in which the haves can readily rationalize that the have-nots were given a "fair shake" to succeed and didn't make the most of it, hence that they deserve their sad lot and those who "have" need not feel either guilty or responsible for inequities past, present, or future.

For all the rhetoric to the contrary, the anti-reformers at Mathematically Correct and NYC-HOLD have long ago given their game away with their hatred of student-centered teaching, their obsession with direct instruction, their aversion to anything that smacks of either discovery learning of content or "self-discovery" in the broader sense. Wolin's vision of developing diverse potentialities is at the opposite pole from the regimented classrooms being promoted by the anti-reform side of the Math Wars. And so they undermine the sorts of real mathematical learning that they claim to be fighting for, all because their real agenda is something that can't survive the sort of independent thinking that such learning necessitates.

An Idea Whose Time Has Come?

On p. 7 of the current issue of the MAA FOCUS one may read the following, which I presume is presented as biting satire of an event the author would apparently find heinous should it become reality:

Outsourcing Mathematics: Is a News Story Like This Possible?
A nightmare from Michael Henle, Oberlin College.

Dystopia Times

Mathematics Department Shuts Down

Monday, May 3, 2010. Nemesis College announced today the dissolution of its mathematics department. No details were given, only the statement that the future mathematical needs of its students would be met outside the traditional Department of Mathematics setting. We wondered what this meant. Could it really be true that students at Nemesis would no longer be subjected to the universally unpopular subject of mathematics? To find out, we interviewed Professor Earnest, the former chair of the mathematics department. He met us in his old office, surrounded by half-packed boxes of books. We asked first if this action on the part of the College administration had come as a surprise to him or to other members of the mathematics department. “Not at all,” Professor Earnest said. “This has been in the works for some time. For example, we haven’t taught statistics for at least a year. It’s outsourced to economics, psychology and other client departments. They prefer it like that. The last statistician left the Department of Mathematics several years ago.” We were curious about the calculus, that most dreaded of mathematics courses. How would Nemesis students be taught calculus? “Not a problem,” Professor Earnest told us. “Most of our students get their calculus in high school now. The few that don’t will be given workshops. They’ll be shown how to do calculus using a computer algebra system. The Engineering Department will handle this. That’s what they want. Likewise, students who need remedial work in algebra and trigonometry will be trained on software.” What about current members of the Department of Mathematics? Where would they end up? “Well, a few will retire,” Professor Earnest said, “but most of us will be right here. Let’s see. A few colleagues are joining the Computer Science Department and some others will be in Engineering. They’ll teach the workshops I mentioned. Then a few more will work in Information Technology. They will update software, trouble-shoot email problems, replace spent print cartridges, and the like. Oh, and a few lucky chaps are joining Environmental Studies. They’ll teach modeling software. Maybe even a course or two.” All this seemed very well planned to us. Our last question concerned Professor Earnest himself. Where would he be? “I’m fortunate,” he said. “I’ll be in the Physics Department. I get to teach transform theory and advanced analytical methods.” He paused. “There’s only one problem.” For the first time in the interview he looked a little sad. “What was the problem?” we asked. Professor Earnest sighed. “No proofs,” he said. “I have strict instructions. There must be no proofs in my classes.”

Is this a dystopian dream of mathematical hell? Or is it in fact much to be wished by sensible students and educators that teaching mathematics be taken from the hands of mathematicians for whom teaching is too great a chore? 

Let's consider Professor Henle's piece in more detail. He states, "the future mathematical needs of its students would be met outside the traditional Department of Mathematics setting." Why would this be a problem for students? As long as their real mathematical needs are truly met, what student would care about which department was meeting the need? Do students care or think about how academic colleges are structured? At the graduate level, the story is possibly different, of course, but even there, if a student could get the coursework, mentoring and degree s/he desired, would the name of the department supplying any/all of the above matter? 

When I came to the University of Michigan to do graduate work in mathematics education in 1992, one of the first classes I took was a 400 level undergraduate course in mathematical probability, taught by a full professor who came highly recommended to me by his office mate, the fellow in the mathematics department who taught all the mathematics for teachers content courses. I'm sure, in retrospect, that when I was told that Professor S. was outstanding, what was being communicated had absolutely nothing to do with his ability to teach. Rather, I was being told what in fact did not need to be said: that a person with a full professorship at a world-class university mathematics department REALLY knew the mathematics needed to be an expert about one or more tiny twigs on some tiny branch of some larger branch on one of the major branches of the tree of mathematics. And that he had climbed all the requisite branches to get there as an undergraduate and graduate student of mathematics, had done the expected original research needed to be officially accepted into the tribe of professional mathematicians, and had finally dotted all the i's and crossed all the t's needed to reach the plum position he currently held. 

What I wasn't being told anything about whatsoever was this professor's ability to help non-experts learn the smallest thing about mathematics. And in fact, he would rate as the single worst teacher, from a technical perspective, I've ever had for mathematics. That he spoke with a heavy accent I can't blame him for, though I did not get the impression that he had made much effort to work on his pronunciation. But he spoke consistently in a very soft voice, frequently spoke while facing away from the students, and often let his voice trail off at the end of sentences in an essentially inaudible manner, usually coupled with a little grin that, after a few weeks, became particularly annoying given that it almost seemed like he knew that he wasn't communicating very well. I made it a big point to get to class early so I could always have a front-row seat, but nothing helped. On the one occasion I went to him for help at his office, I left more confused than when I entered. He assigned homework problems from the text that he'd used for more than five years for which the book gave the wrong answer but didn't bother to mention this to us. I spent a tortured couple of days trying to figure out why I was consistently coming up with a different answer to a (for me) difficult problem in combinatorics than the one the book provided only to discover that my answer was in fact correct. Since the numbers involved were sufficiently large that it would have been impracticable to do the actual counting as a check, I wound up wasting a lot of time that could have been better spent doing problems for my other classes. Of course, Professor S. did not apologize for his thoughtlessness. I'm sure that for him it was no big deal at all. And his final outrage was to offer an optional review session before the final during which something a student asked led him to lecture briefly on something not covered during the course. He then put a problem on the final that was drawn directly from that topic. Not that my having been at the session prepared me for this problem, but imagine the shock it gave students who chose to miss the "optional" session. All in all, a horror show. 

I felt even worse when I was told by a senior mathematics major I'd become friendly with in another class that the same course was offered every semester in the statistics department, that the book they used was vastly more comprehensible and student-friendly, and that the professors who taught the course were generally considered to be much better teachers than those who did so in the mathematics department. It was too bad we hadn't met before I was already past the point of no return in Professor S.'s class.

Of course, I don't really wish to suggest that all mathematics professors, full or otherwise, are so unskilled in the classroom. Some are gifted and dedicated teachers. But the notion that having research mathematicians moved to various departments would be a bad thing for students as far as the teaching and learning of undergraduate courses is concerned simply is laughable. 

Dr. Henle later quips, through the mouth of the imaginary Professor Earnest, "Most of our students get their calculus in high school now. The few that don’t will be given workshops. They’ll be shown how to do calculus using a computer algebra system." Hmm. Doesn't THAT sound like a slasher movie? Not that I am a great believer in the necessity or desirability of teaching calculus in high school, of course. In fact, I see very little advantage to students in trying to complete a lot of calculus before students enter college. Or at least having it be the sole or primary option to seniors who are ahead of the curve in terms of state or district mathematics requirements. A course in discrete math, especially one that involved proofs, would be both more practical and potentially more challenging. It would likely provide a better sense of what mathematics majors are expected to do. And it would undoubtedly connect better with many common real-world applications that high school students are interested in. But then, I'm heretical in so many regards. 

And then there's the gratuitous and predictable pot-shot at computer algebra systems. This despite the fact that research mathematicians use a variety of high-end software that includes, of course, computer algebra and the ability to crunch a lot of difficult integrals and other calculus computations. I'm sure that those researchers COULD do the calculations by hand, though perhaps not as quickly or as accurately in all cases, but then they consciously choose not to, and to take advantage of the power of software like Maple and Mathematica despite the attitude of nay-sayers and, well, Luddites. I don't know if our satirically-minded Oberlin professor is in that camp, but I assume my sides are supposed to be split by laughing at the notion of college kids doing a lot of calculus number crunching with the aid of computational tools. Perish the thought. 

I'm afraid I just don't find the scenario Professor Henle offers us particularly scary. When it comes to teaching mathematics, at least at the college level, I suspect quite a number of high-level mathematics professors would be well-employed replacing printer cartridges as he suggests. Or, perhaps, just doing what they do well and actually wish to do passionately: creating new mathematics. Solving challenging problems of both theoretical and applied math. And communicating to the small number of folks with whom they are capable of so doing. Those professors who actually have both a gift and a desire for teaching should of course be allowed and encouraged to do so. I have been privileged to learn from a few such people and to know others professionally. I doubt very much that any of them would be troubled by Henle's satire. I can't imagine that many students would be. And I know for certain that I'm not. I want teaching to be done by those who not only know their subject but who have the ability to teach it well to real students, and not just one or two who happen to think and learn exactly the same way they do. No watering down of the material, but definitely shoring up the quality of the communication and pedagogy. Imagine that! And who would really care under the auspices of which departments such teaching came?