Sunday, June 29, 2008

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. 




Saturday, June 28, 2008

Tuesday, June 24, 2008

"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.


Saturday, June 21, 2008

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.

Saturday, June 14, 2008

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?

Wednesday, June 11, 2008

Who Determines "THE" Standard Algorithm for Subtraction?






One popular complaint amongst anti-reform pundits is that so-called reform/"fuzzy" math advocates and the programs they create and/or teach from "hate" standard arithmetic algorithms and fail to teach them. While I have not found this to be the case in actual classrooms with real teachers using EVERYDAY MATHEMATICS, INVESTIGATIONS IN NUMBER DATA & SPACE, or MATH TRAILBLAZERS were being used (the so-called "standard" algorithms are ALWAYS taught, and frequently given pride of place by teachers regardless of the program being taught), the claim begs the question of how and why a given algorithm became "standard" as well as how being "standard" automatically means "superior" or "the only one students should have the opportunity to learn or use." It strikes me that it is almost as if such people are stuck in some pre-technological age in which we trained low-level white collar office workers to be scribes, number-crunchers who summed and re-summed large columns of figures by hand, etc. The absurdity of seeing kids today as needing to prepare for THAT sort of world is evident to anyone who spends any time in a modern office, including that of a small business. Desktop and handheld calculators are commonplace. So are desktop and laptop computers. There is a need for people to understand basic mathematics, but not to be fast and expert number-crunchers in that 19th century sense.



Thus, it seems reasonable to ask what should be an obvious question: if the goal is to know what numbers to crunch and how (what operations need be used) to crunch them, and, most importantly, to correctly interpret and make decisions based upon the results of the right calculations, and further if it is glaringly obvious that the actual number-crunching itself is done faster and more accurately by machines than by the vast, vast majority of humans can reasonably expect to do, why would any intelligent person be obsessing in 2008 over the SPEED of an algorithm for paper and pencil arithmetic? For the big argument raised for always (and exclusively) teaching one standard algorithm for each arithmetic operation seems to be speed and efficiency.



I have argued repeatedly that the efficiency issue is only reasonable if one fairly assesses it. And to do that is to grant that a student who misunderstands and botches ANY algorithm is unlikely to be performing "efficiently" with it. Compared with a student who uses even a ludicrously slow algorithm (e.g., repeated addition in place of any other approach to multiplication) accurately, the student who can't accurately make use of the fastest possible algorithm is going to be taking a long time to arrive at the right answer, which will be reached, if at all, only after many missteps and revisits to the same problem. For that student, at least, the "algorithm of choice" is not efficient at all. So finding one that the student understands and can use properly would by necessity be preferable. But not, apparently, in the mind of ideologues. For them, there's one true way to do each sort of calculation and they are its prophets.



Of course, I'm not favoring teaching alternate algorithms because I dislike any particular standard one or feel the need to "prove" that, say, lattice multiplication is "better" than the currently favored algorithm. On the contrary, I'm all for teaching the standard algorithm. But not alone and not mechanically, and not at the expense of student understanding. Indeed, from my perspective, it's difficult to understand why it is necessary to mount a defense for alternative algorithms in general, though any particular one may be of questionable value and might need some justifying or explaining. If anything, it is those who hold that there is a single best algorithm that is the only one that deserves to be taught who need to make the case for such a narrow position. In my reading, I've yet to encounter a convincing argument, and indeed most people who hold that viewpoint seem to think it's glaringly obvious that their anointed algorithms are both necessary and sufficient.



What compounds my outrage at the narrower viewpoint is the fact that it is based for the most part on woeful historical ignorance. Elsewhere, I've addressed the question of the lattice multiplication method, which has come under attack from various anti-reform groups and individuals almost certainly because it has been re-introduced in some progressive elementary programs such as Everyday Math and Investigations in Number, Data, and Space. The arguments raised against it are very much in keeping with above-mentioned concerns with speed and efficiency. Ostensibly, the algorithm is unwieldy for larger, multi-digit calculations. The fact is that it is just as easy to use (easier for those who prefer it and get it), and while it's possible to use a vast amount of space to write out a problem, it's not required that one do so and the amount of paper used is a social, not a pedagogical issue. But please note that I said RE-introduced, and that was not a slip. The fact is that this algorithm was widely used for hundreds of years with no ill effects. Issues that strictly had to do with the ease of printing it in books with relatively primitive technology and problems of readibility when the printing quality was poor, NOT concerns with the actual carrying out of the algorithm, caused it to fall into disuse. Not a pedagogical issue at all, and with modern printing methods, completely irrelevant from any perspective. Yet the anti-reformers howl bloody murder when they see this method being taught. The only believable explanation for their outrage is politics. They simply find it politically unacceptable to teach ANY alternatives to their approved "standard" methods. And their ignorance of the historical basis for lattice multiplication as well as their refusal to acknowledge that it is thoroughly and logically grounded in exactly the same processes that inform the current standard approach suggests that bias and politics, not logic, is their motivation.



Subtraction algorithms

I raise all these questions because I've recently had my attention drawn to a "non-standard" algorithm (actually two such algorithms and some related variations) for subtraction. Tad Watanabe, a professor of mathematics education whom I've known since the early 1990s both through the internet and from meeting him at many conferences, posted the following on MathTalk@yahoogroups.com, a mathematics education discussion list I've moderated for the past seven years:




Someone told me (while back) that the subtraction
algorithm sometimes called "equal addition algorithm"
was the commonly used algorithm in the US until about
50 years ago. Does anyone know if that is indeed the
case, and if so, about when we shifted to the current
conventional algorithm?


I couldn't recall having heard of this method, and so I was eager to find out what Tad was talking about. Searching the web, I discovered an article that repaired my ignorance on the algorithm: "Subtraction in the United States: An Historical Perspective," by Susan Ross and Mary Pratt-Cotter. (This 2000 appearance in THE MATHEMATICS EDUCATOR was a reprint of the article that had originally appeared several years previously in the same journal. It draws upon a host of historical sources, the earliest of which is from 1819. And there are other articles available on-line, including Marilyn N. Suydam's "Recent Research on Mathematics Instruction" in ERIC/SMEAC Mathematics Education Digest No. 2; and Peter McCarthy's "Investigating Teaching and Learning of Subtractions That Involves Renaming Using Base Complement Additions."

The Ross article makes clear that as far back as 1819, American textbooks taught the equal additions algorithm. To wit,



1. Place the less number under the greater, with
units under units, tens under tens, etc.
2. Begin at the right hand and take the lower figure
from the one above it and - set the difference
down.
3. If the figure in the lower line be greater than the
one above it, take the lower figure from 10 and
add the difference to the upper figure which sum
set down.
4. When the lower figure is taken from 10 there
must be one added to the next lower figure.


In fact, according to a 1938 article by J. T. Johnson, "The relative merits of three methods of subtraction: An experimental comparison of the decomposition method of subtraction with the equal additions method and the Austrian method," equal additions as a way to do subtraction goes back at least to the 15th and 16th centuries. And while this approach, which was taught on a wide-scale basis in the United States prior to the late 1930s, works from right to left, as do all the standard arithmetic algorithms currently in use EXCEPT notably for long division (which may in part help account for student difficulties for this operation far more serious and frequent that are those associated with the other three basic operations, it can be done just as handily from left to right.

Consider the example of finding the difference between 6354 and 2978. Using the standard approach, we write:


6354

-2978

------



and work as follows: 1) 8 is greater than 4 so we "borrow 1" from the 5 and then subtract 8 from 14 and get 6. We cross out the 5 and write 4 to account for the borrowing; 2) 7 is greater than 4 so we borrow 1 from the 3 and then subtract 7 from 14 and get 7. Again, we scratch out the 3 and write 2 to account for the borrowing; 3) 9 is greater than 2, so we borrow 1 from the 6, subtract 9 from 12 and write 3. Again, we scratch out the 6 and write 5 because of the borrowing; 4) finally we subtract 2 from 5 and get 3, leaving us with the answer 3, 376.


(Although it may not be obvious, we could do that subtraction from left to right using an approach similar to what I will show below for the equal additions algorithm).






Equal additions





The equal additions method works as follows for the same problem above: 1) "8 from 14 is 6; 2) 8 from 15 is 7; 3) 10 from 13 is 3; 4) 3 from 6 is 3"


That is to say, each time the digit in the subtrahend is greater than the digit in the same place in the minuend, 10 (or 100 or 1000, etc.) is added to the digit in the minuend and also to the digit in the next largest place in the subtrahend. However, for example, in step #1 above, adding 10 to the units digit in the minuend appears to be "compensated" for by adding only 1 to the 7 in the subtrahend. In reality, they both represent additions of 10 because of place value. The algorithm really does involve equal additions at each step as necessary. And of course, because adding equal quantities to the minuend and subtrahend does not change their difference, the resulting computation is correct.


Left to right subtraction?


Could the calculation be performed correctly operating from left to right? Consider the following approach: 1) beginning on the left, take 2 from 6 and write down 4; 2) since 9 is greater than 3, add "10" (really 1000) to the 3 and take 9 from 13 getting 4 which we write down. Take "1" (really 1000) from the previous 4; 3) since 7 is greater than 5, add "10" (really 100) to the 5 and subtract 7 from 15 and write down 8. Take "1" (really 100) from the previous 4; 4) since 8 is greater than 6, add 10 to the 4 and take 8 from 14 and write down 6. Take "1" (really 10) from the previous 8. The answer is 3,376.


Is this "better" than the standard ("compensation") algorithm? Is it worse? I only mention it because research has suggested that many young students, left completely to their own devices, are likely to develop similar left-to-right strategies, correct or flawed. It seems highly likely that this is a natural outgrowth of the fact that we read English from left to right, and we teach students to read numbers the same way. It seems almost bizarre, once one thinks about it, that the standard algorithms for addition, subtraction, and multiplication, as well as some alternative methods, insist upon working from right to left. I suspect, too, that this both causes some students a lot of difficulty (there is plenty of evidence of kids who have later problems TRYING to do arithmetic from left to right and getting into difficulty)

I will not discuss or describe in detail the Austrian algorithm other than to say that it doesn't feel "right" to me. That's not saying it's "wrong," but rather that I can't see it as one I would use. And here is one major difference between me and the reform-haters: that doesn't mean I wouldn't revisit it, wouldn't show it to teachers, and perhaps if I saw a particular student or class for whom it might prove helpful, I'd teach it. My "taste" isn't the issue, but rather keeping a large number of options available for my practice and for my students. I suppose that's just not very "efficient" of me.

Finally, it bears noting that there is reference in the above-mentioned articles to research on the use of these algorithms, and at least some reason to think that equal additions needs to be looked at again very seriously by mathematics teacher educators and K-5 teachers. If you read the historical treatment of subtraction algorithms in the US, you'll likely note how much chance and arbitrariness there can be in how one particular algorithm comes into fashion while others fall into disuse. I see no firm evidence for the "superiority" of the current most commonly-taught algorithm, and there is clearly a history of it's causing difficulties for particular students. Would the universe collapse if we were to teach both? Even more, would it collapse if we didn't rush to teach it right away, but rather, as has been proposed by more than a few researchers and theorists on early mathematics education, let students play and invent their own algorithms first, before trying to steer them toward one or another of our own? Sadly, the anti-reformers amongst us, the activist educational conservatives who are constantly trying to narrow rather than open up K-12 education, believe that there's always a best way to do everything. And not coincidentally, that way always turns out to be the one they learned as a child. That, more than anything, is why I think it reasonable to call the not-so-traditional math that they push on everyone "nostalgia math." It's not that what they learned is better. It's just what they learned back in simpler times when life was easy and there were no Math Wars and no one like me to suggest that their emperor is stark naked.


Friday, June 6, 2008

Quirky Investigations: More Nonsense From an Old Source (Part 2)

In the first part of this look at the questionable notions of William G. Quirk about the TERC elementary math curriculum, INVESTIGATIONS IN NUMBER, DATA, AND SPACE, I analyzed the interesting deceptions in the title of his recent propaganda effort, "2008 TERC Math vs. 2008 National Math Panel Recommendations." Now the time has come to look at some of the specific charges against INVESTIGATIONS leveled by Mr. Quirk and expose them for what they are.

First, Mr. Quirk asserts:

A major objective of elementary math education is to provide the foundations for algebra, the gateway to higher math education. Although we call it "elementary math," K-5 math content is quite sophisticated and not easy to master. But constructivist math educators believe that concrete methods, pictorial methods, and learning by playing games are the keys to a stress-free approach.


It can't be easy to squeeze so much inaccuracy into so small a space, but Mr. Quirk is adept at doing so. While it is arguable as to whether the correct focus for elementary math is or should be "the foundations for algebra," it's even more questionable as to what is meant by algebra. If Mr. Quirk's other writing is indicative (and I'm sure it is), algebra to him is just as mechanistic as is his take on all other school mathematics: a series of rules and definitions to be memorized to "mastery," such as: theirs not to reason why, just invert and multiply, etc.

On the other hand, we see in the writing and talks of Liping Ma the notion that for Chinese students, the goal of arithmetic is to have a profound understanding of arithmetic. Algebraic facility flows from them rather naturally, according to her, because students have a great familiarity with how numbers are "made": the idea of "composing" and "decomposing" numbers (here, I think is meant more than just factoring, but also what I've seen elsewhere called "number bonds" - the different ways to make a given number by adding two numbers (of course, we're mostly talking about positive integers here for young kids). In some curricula to which Mr. Quirk vehemently objects (as do some more progressive writers, such as Van de Walle), this is explored through fact families or similar approaches. I like the idea of exploring in a progressively focused manner, letting kids start to realize that a given number can be the sum of lots of pairs of whole numbers (and when negative numbers are introduced, this becomes an infinite number of pairs of integers).

I'm unconvinced that what is being done with INVESTIGATIONS (and other progressive programs) is drastically different from one or more of the above ideas. Van de Walle's notion of developmental arithmetic is a sound one that seems better reflected in INVESTIGATIONS and EVERYDAY MATHEMATICS than in the more rote-based approaches that Quirk and other anti-progressives advocate.

Math Wars and Literacy Wars: similar rhetoric, similar tactics

I suspect it's no coincidence that such folks are often found to have been in the forefront of scurrilous attacks on whole language. The tactics there have been very much like those in the Math Wars: claim that the reformers are "destroying" traditional educational methods which are alleged to have been effective in the past (the key word is, of course, alleged; there's no sound data that supports the idea that a higher percentage of kids were effectively taught math or literacy "back in the day" (and exactly when that day was depends directly on the age of the critic. Having been educated back in the '50s & '60s, I'm a bit less sanguine about any so-called golden age of phonics and times tables. I know too many people who didn't learn to read or write adequately (if at all) who went through the same sort of schooling I did. I taught far too many kids at the U of Florida c. 1975 who had high school diplomas from districts that weren't exactly on the cutting edge of reform literacy education. They couldn't write a meaningful sentence about their day at the zoo, let alone an actual college paper. A current article in the ATLANTIC MONTHLY, "In the Basement of the Ivory Tower" by 'Professor X,' discusses an adjunct English professor's concerns about the disservice he believes is being perpetrated on many adult and non-traditional students at various community colleges and four-year institutions, by allowing them to enroll in courses they are unqualified for and cannot reasonably hope to pass given the skills they enter them with. Having taught TRADITIONAL college students at the "flagship" public university of Florida in the '70's, I can attest to the fact that there's nothing special at work in what Professor X observes, criticizes, and bemoans. The only difference is that he's seeing it today with students he feels shouldn't be going to college. I saw it 30 years ago not only with undergraduate students in the most selective public university in one of our most populous states, but with fellow graduate students of English who couldn't write a passable undergraduate piece of literary analysis.

So is the sky really falling more today than it was in the '70s? If not (and I think it is not), when was the golden age of teaching literacy? The students I taught were not much younger than I was and finished high school during the "back to basics" era. My less competent colleagues in graduate school were my contemporaries. How did they manage to get into decent (if not great) colleges and graduate programs and still be embarrassingly unfit in their chosen field? Whole Language cannot be blamed for these failures or for the students in remedial English in the 1960s. We had phonics in my school district. We also had the famous "Dick and Jane" readers that were about as engaging as a poke in the eye with a sharp stick. Whole Language advocates called for the use of more relevant, more challenging literature as part of the basis for literacy education. They didn't call for the death of phonics instruction. Every elementary school teacher I've spoken to who supports Whole Language tells me that s/he teaches phonics as part of balanced instruction and always has. Where are the alleged extremists who aren't teaching phonics that we hear so much about?

Similarly, I have a hard time finding elementary math classrooms that don't teach math facts, that don't test kids under time pressure on addition, subtraction, multiplication, and division of single digit numbers starting by third grade if not sooner. Every district in Michigan with which I've worked in any capacity (and that's quite a few as either teacher, coach, university supervisor, university instructor, or consultant) does the same. While there is some variation in the approach individual teachers use, everyone stresses knowing basic math facts.

On my view, the problem cuts across curricula, both reform, traditional, and what-have-you. Unless there's a strong district commitment to the sort of developmental approach that Van de Walle and like-minded mathematics educators advocate and to employing and supporting elementary teachers and other staff who have all three components of effective teaching in place, there's going to be less than optimal mathematics teaching and learning in K-5. Once students are allowed to emerge from elementary school with significant holes in their mastery of basic arithmetic (I'll leave alone for now other things that students should be given an opportunity to learn and - dare I say it? - play with in mathematics while in K-5), algebra is going to prove far, far more difficult than should be the case and that is, reportedly precisely what many mainland Chinese students are able to avoid suffering through because of their facility with and intimate understanding of how arithmetic works.



Quirky Lies and Misdemeanors

Mr. Quirk asserts correctly that elementary math content is quite sophisticated. He also claims that it's not easy to master. There I think he errs in make an all-too-typical over-generalization. Is it difficult to master in all its particulars? What, exactly, comprises mastery in his view? And do all students suffer equally in trying to master it? One doesn't get the sense from Liping Ma that Chinese students are suffering as they learn. Constance Kamii, one of the mathematics education researchers much-despised by the anti-reform crowd, certainly makes it sound like students she studies enjoy playing with mathematical activities and ideas. Well, maybe that's the rub: the folks who seem to think there's more to math then computation are often the ones who think mathematics isn't torture and need not be taught as if it were. And then they are accused of a rafter of sins: dumbing down the curriculum, promoting "self-esteem" over real learning, making math into fun and games (instead of torment, I suppose), and so on. The more traditional view is what I refer to as the fraternity hazing perspective: math is hell and damn it, my (kids/students/successors) will suffer just as much as I did (if not more). No Pain, No Gain.


However, all that is the least of my concerns with Mr. Quirk. It's the next claim that wins the prize for setting off major alarms: "constructivist math educators believe that concrete methods, pictorial methods, and learning by playing games are the keys to a stress-free approach. " If you thought my analysis in the previous paragraph was speculative, Mr. Quirk here has removed all doubt. It's not, of course, that games are designed to remove stress: they're designed to enhance understanding AND in the case of two of the curricula he despises (EVERYDAY MATH and INVESTIGATIONS), to do the work of some of that much vaunted drill and rote learning. Of course, one would have to actually read the curricular materials on the games with an open mind to see that. It's so much easier to simply used the word "games" and link it with some sort of Dionysian hedonism.

On the other hand, why are folks like Mr. Quirk so enamored by stress, especially in the case of K-5 students? Captain Quirk is, like so many of his fellow traditionalists, a great lover of high stakes tests. And there is little that has been used to create MORE stress for young children and their teachers and parents than these tests since the passage of No Child Left Behind. I'm sure I'm not the first person to note that putting such emphasis on tests for young kids is a wonderfully effective way to destroy the pedagogical relationship between students, teachers, and administrators. One merely need spend a little time in most schools, especially inner-city and rural poor schools, in the month before and the month during a high-stakes test to see schooling at its worst, and kids at their most tense and miserable. And not just the kids. For a progressive, humanistic educator like me, nothing could be more demoralizing, upsetting, and depressing than to hear administrators tell principals and teachers that it's okay to lie to THIRD GRADERS about the impact of the local state assessment on the PERMANENT RECORDS of those kids. I am not making this up: I witnessed it in 2005 in an inner-city elementary school northwest of Detroit.



Quirk Conflates Apples and Oranges


Let's return to the substance, such as it is, of Quirk's complaints about INVESTIGATIONS. First, he offers what he calls the NMP's "2008 View of the Foundations of Algebra." However, what he quotes in fact is a definition of school algebra:


The report first defines 'school algebra' as the 'term chosen to encompass the full body of algebraic material that the Panel expects to be covered through high school, regardless of its organization into courses and levels.



Quirk then goes on to complain:

NMP carefully defined 'school algebra.' TERC counters with 'algebra is a multifaceted area of mathematics content.'
Yes? Is this supposed to be some sort of ruinous critique? Aside from the ploy of inventing a "TERC" that is trying to "counter" statements that were made by a panel of no clear authority to dictate national policy or content AFTER the publication of what TERC published. Nice try, Mr. Quirk, but this is deeply flawed.

More significantly, it is completely absurd to compare a definition of "school algebra" with a definition of "algebra." I don't wish to argue whether the NMP "carefully defined" school algebra, nor whether that definition is at all satisfactory. But it is clearly a definition of "school algebra." That is a far cry from what algebra actually is, as Mr. Quirk knows full well, being a Ph.D in mathematics. Is he really so naive as to believe that "algebra that 'should' be taught in schools" is the same as what algebra IS? Or is he simply being conveniently hazy on what's being discussed in the two quotations?

I believe it's glaringly obvious that the latter is the case. This is simply the classic rhetorical methods of the anti-reform crowd. They don't play fair, they don't worry about truth, they simply do whatever they think will win. And to "win" is to fool the public and anyone not paying close attention to the specifics of the debate. Shoddy, but so often effective.

One might ask Mr. Quirk whether he contends that algebra is NOT a multifaceted area of mathematics content. If so, to what, exactly, would he restrict it? And would his restrictions be acceptable to working mathematicians? If not, then really, of what use is it to compare apples to oranges in this context other than as a cheap method for attacking a program he hates on general principles (and I use that last word quite loosely).

What is "too much" emphasis on patterns, and who says?

Another "devastating" complaint from Mr. Quirk is that INVESTIGATIONS over-emphasizes patterns. His authority is none other that another Mathematically Correct hack, David Klein. According to Quirk, Klein states that "the attention given to patterns is excessive, sometimes destructive, and far out of balance with the actual importance of patterns in K-12 mathematics." One might well ask what comprises "excessive" emphasis on patterns, how exactly focusing on patterns could be "destructive" and in what ways, and who - other than himself and his like-minded reform opponents - legitimated Klein's claim to be an expert on mathematics teaching and learning. Similar questions apply to Quirk, of course. There is a fascinating and utterly false assumption on the part of far too many people that expertise in mathematics (to the extent that having a Ph.D in the subject makes one such an expert) automatically qualifies someone as an expert on the learning and teaching of the subject, and at any and all grade levels.


Those of us who have suffered through pedestrian mathematics teaching and worse on college campuses (which isn't to say that there aren't inspired mathematics professors who understand and care deeply about high-quality teaching), might wish to suggest that there are a lot more folks who know mathematics at a higher level who haven't a clue about how to teach it to anyone except perhaps another mathematician. The true horror, however, is when those who are at best indifferent classroom teachers for college mathematics presume that because K-12 mathematics is so "simple," their ability to breeze through such trivia makes them highly-qualified to speak about what those of us who actually teach and work in K-12 mathematics classrooms should be doing. Not surprisingly, many of such people strongly believe in one-size-fits-all instruction (what could be simpler) regardless of reality. They justify this, in many cases, with phony arguments about racism, equity, and the like. For the most part, they don't even care about these issues, but recognize the necessity of paying lip-service to them. Klein is a relative exception in the Mathematically Correct/NYC-HOLD camp in that he is officially a socialist. This seems to lead him (and his more conservative associates) to believe that these anti-reform organizations represent a "diverse" cross-section of the political arena. However, without wishing to go into an in-depth analysis of the modern American Left, I will simply assert that there are ways in which some strains of that Left are as regressive as many aspects of more conservative or reactionary political thinking, just as there are ostensible conservatives who can come up fairly progressive on some issues. Further, being left-wing in politics appears not to be a guarantee of being in favor of progressive education (and as Alfie Kohn has recently pointed out, progressive education in theory isn't always terribly progressive in any given instantiation of it, sad to say): not only is it hard to find truly progressive schools, it's well-nigh impossible to find politicians who favor progressive education or have a clue about any meaningful aspects of classroom life. Suffice it to say that David Klein is no progressive educator and that his views of K-12 mathematics teaching and learning could hardly be more conservative or wrong.


Returning to the question of patterns, we see well-respected mathematicians like Lynn Arthur Steen arguing that mathematics very much is about pattern.


The rapid growth of computing and applications has helped cross-fertilize the mathematical sciences, yielding an unprecedented abundance of new methods, theories, and models. Examples from statistical science, core mathematics, and applied mathematics illustrate these changes, which have both broadened and enriched the relation between mathematics and science. No longer just the study of number and space, mathematical science has become the science of patterns, with theory built on relations among patterns and on applications derived from the fit between pattern and observation.


I'm sure David Klein and William Quirk know better. Certainly better than mathematician Keith Devlin, author of MATHEMATICS: THE SCIENCE OF PATTERNS. Yes, it's just wrong of the authors of K-5 mathematics books to stress pattern. Where DO they get such wild ideas? Well, apparently from renowned mathematicians.


The Myth of "Standard" Arithmetic Algorithms


Much of the Quirk critique of programs like INVESTIGATIONS hinges upon a single crucial untruth: that there are somewhere enshrined a list of "standard" arithmetic algorithms from which one must never depart in K-5 education and beyond. This stance allows Quirk and like-minded people to bash reform-oriented programs without having to ever make a coherent argument as to why a particular algorithm is "bad," while another is inherently "good." It suffices from this perspective to argue from tradition regardless of the fact that so-called traditions may be relatively recent or limited to particular countries and cultures. France and countries like Haiti that were once French colonies, write long division in a way that would seem "upside-down" to those of us who learned the "standard" algorithm in the United States, yet the workings of the algorithms are identical. Would Quirk & Co. contend that the French are all confused, all "hostile," to use his term, to "standard language, standard formulas, and standard arithmetic"?

In fact, "standard language" is itself another red herring. Mathematically-knowledgeable people are well aware that it is hardly unusual to find multiple terms for the same mathematical concept, as well as multiple notations. This is true in calculus, in abstract algebra, and in other areas of mathematics. Not unlike the squabbles about Macs vs. PCs, the dispute between supporters of one set of symbols or terms may be more about which one the individual learned first than about truly substantive issues. Regardless, however, it is difficult to make the case that mathematics as it has evolved depends on one standard set of terms or symbols, however awkward that reality may be for folks like Quirk. He might wish that were the case, but mathematics appears to have thrived despite that fond hope of his.

It's unlikely that Quirk would have the temerity to suggest that Leibniz didn't know what he was talking about because he used a different notation for calculus than did Newton, or vice versa. Yet he has no compunction about trying to dismiss INVESTIGATIONS for similar "sins." He also fails to make any reasonable distinction among terms as to what might be vital to know and what might not, especially given that INVESTIGATIONS is aimed at K-5 students. Should they have to know "subtrahend," "minuend," and "difference," or does the latter suffice? Should their teachers know all three? Is it fatal for kids and/or teachers to use the term "borrowing" or "carrying"? Just how anal does one need to be about such matters?

It's hard not to wonder if Quirk and his friends understand the difference so perfectly highlighted by Richard Feynman between knowing the name of something and knowing something. If I have students who understand how to do division, what it is, and what the results of doing division mean, I'm not likely to lose sleep over whether they can correctly tell me what the dividend is. Nor do I care whether they use the terms "partitive" and "quotitive," as long as they understand that sharing and measuring are two important ways to think about division. (Of course, being terms from mathematics education, rather than from typical mathematics coursework, these words are likely of no value to Mr. Quirk whatsoever). What wasn't taught to him in his own K-5 classroom is "non-standard," of course.

When it comes to actual algorithms, Quirk, like many other reform critics, resorts to an appeal to "efficiency and speed." However, given the reality of modern computational devices, it's hard to imagine that either of these matters very much to the vast, vast majority of people. What does matter and should matter is understanding of both mathematical procedures and what results from following them, as well as which to use for a given situation. There is a kind of narrow thinking seen far too frequently among some K-12 teachers of mathematics, as well as some professional mathematicians, who most certainly should know better, that reduces school mathematics to "right procedures." It is hard to justify, however, a "My way or the highway!" approach to teaching mathematics to school children. This is not, as reform critics are sometimes quick to falsely suggest, an invitation to "anything goes." Obviously, some procedures are not justifiable. Some are inefficient to the point of being almost useless. However, there is a vast middle ground, and it is there that programs like INVESTIGATIONS invite students to explore, but which Quirk and others refuse to grant ANY legitmacy whatsoever. I have yet to see a single argument from him that holds water against letting students invent their own algorithms (many of which, if not all, are likely to be well-known, possibly still in use in some cultures, and in any event methods that the students UNDERSTAND and can explain). The red-herrings of "speed and efficiency" don't hold up when weighed against utter confusion and error on the part of students. If a student consistently errs with a "standard" procedure, how can it sanely be argued that this procedure is faster or more efficient? Clearly, for the given student, it's quite the opposite. And this point is hardly restricted to elementary arithmetic, of course.

To my thinking, some teachers refuse to consider alternative algorithms for one simple reason: their own grasp of mathematics is so tenuous that they are afraid to think outside their comfort zone. I doubt that is the issue for Quirk, however. He is chosing consciously to close off alternatives because he thinks he has a rhetorical club with which he can beat the authors of reform programs, regardless of how valid and sensible their methods are. In the mind of many US parents who fear and loathe math, they may not know how to DO arithmetic, but they know it when they see it, no matter how poorly they grasp its workings. And so if it was bad enough for them, that, and ONLY that, is bad enough for their kids. The last thing they want to see, sadly, is an approach that the kids might just be able to understand but which confuses the parents even further if and when they look at it. Little is more humliating than not being able to help one's kids with homework, but if the homework doesn't even look vaguely familiar, well, that's sufficient grounds to become roused rabble. And Quirk is just the sort of fellow to provide a battle-cry.

The Bottom Line

I don't argue that INVESTIGATIONS, EVERYDAY MATH, or any other elementary, middle school, secondary, or post-secondary textbook or series is a panacea. But I do not see firm or even plausible evidence from Quirk or his cronies that these books are not offering sound mathematical content, thinking, or problems on the whole. They are not flawless, of course, but neither are those alternatives Quirk and company recommend. And they offer many valuable things one would never find in, say, a textbook from Saxon Math, not the least of which would be challenges to develop mathematical habits of mind, problem solving skills, and an ability to think outside the box. While all of these are prerequisites for real mathematicians, Quirk and his allies continue to decry programs that promote such thinking, all the while falsely contending that their only concern is to see all kids learn authentic mathematics. Were that really the case, they would at the least speak honestly about the positive things these programs offer, not the least of which is the ability to reach many students who are NOT reached or engaged by more "traditional" texts and pedagogy. What lies at the heart of the Quirk/MC/HOLD cabal is a pot pouree of mendacity and misdirection. I have on more than one occasion in this blog and elsewhere suggested some of the motivation for such consistent dishonesty. But regardless the truth of my suspicions in that regard, what matters is that fair-minded people look past the shrill dismissals Quirk and others offer of a host of progressive reform texts, methods, and approaches. Mathematics education is too important to allow a few ideological liars, few of whom spend any time working with real kids in K-5 or even K-12 classrooms, to call the shots for everyone.