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.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.

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.

## Sunday, June 29, 2008

### Manipulating Cato: More on Games, Gods & Grades

## Saturday, June 28, 2008

### In A Spacious Place: Games and Grades

## 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 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.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)

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.

## Saturday, June 14, 2008

### An Idea Whose Time Has Come?

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.”

## 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:

## Friday, June 6, 2008

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

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.

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.'

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).

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.