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.