The Fuzziest of Disciplines
by Alexander Nazaryan and Alexander Wilson
In the early 1990s, a group of public school districts clustered in the American West — California, Arizona, Texas, Iowa — selected a set of Math curricula, for children from kindergarten through their last year in high school, that promised new approaches to learning and a more enticing mathematical climate. It was hoped these changes would increase students’ interest in and capability with Math and ultimately help close the lauded mathematical gap between lagging American students and those in the rest of the industrialized world.
These school districts received about $50 million in Federal aid from the National Science Foundation to implement these curricula, collectively (and colloquially) known as `The New New Math.’
Five years on, these programs are a universal educational disaster. In one California school district, the average standardized test score moved from the 86th percentile to the 55th in only the first two years after the New New Math curricula was implemented. The test scores quickly rose back to the 77th percentile after the school’s old, traditional Math curricula were reinstated.
This is consistent with other results, and though the National Science Foundation has announced it will withdraw funding from any school district which abandons the New New Math curricula, many of those school districts who initially adopted these curricula are reverting to more traditional (proofs and rote) methods of learning mathematics.
The academic tail, however, may well still be wagging the educational dog. Whatever the level of its popular rejection, the New New Math is still extensively taught in American graduate schools. Though New New Math was only developed in 1989, it is estimated that fully half of American Math teachers are trained in Fuzzy Math, the central component of the New New Math.
The impetus for the New New Math curricula is political and, as its name suggests, it is a political antecedent to the `New Math’ movement of the 1960s.
In the 1960s, Math teachers, overwhelmed by notions of cultural subjectivism and inclined, as was the contemporary zeitgeist, to question rigid theories of learning as potentially autocratic, decided that the Arabic number system’s (the number system we use) reliance on the number `10′ as a base for arithmetic, geometric, and algebraic learning was arbitrary and probably Euro-centric. Instead, the New Math devoted extensive time to learning alternate number bases, such as 7. (In a base seven system, the number `6′ would still be written `6′, but `12′ would be written as `15′ and `16′ would be written as ’22’).
The idea was not to develop a better basis for understanding higher math (since base 7 is pretty useless in, for example, calculus) but to convince students of the fallacy of the objective reality of numbers. `12′, in the New Math, is not necessarily ’12,’ since in a base 7 system it would be ’15.’ Numbers, for New Mathematicians, are nothing other than `numerals,’ signifiers. Though the concept of ’15 elephants’ is real, it would be insufficient to abstract that to ’15,’ since `15′ is unnecessarily filial to the base ten number system. Instead, to answer the question, “How many elephants are there?” a student would have to answer something along the lines of “The amount of elephants that, in a base ten number system, we would usually express using the numeral `15′.” The New Math also focused extensively on things like set theory and `congruence arithmetic’ rather than multiplication tables and long division.
This system, the `New Math’ of the 1960s, was, predictably, a resounding failure which did little more than confuse students. Test scores plummeted, students turned away from Math, and before long the New Math was dropped.
Thirty years later, enter the New New Math.
The New New Math, originally conceived as a curricular notion in 1989 by a report of the National Council of Teachers of Mathematics, takes the same impulse, the rejection of the objective abstract reality of numbers, and combines it with a series of voguish concepts in Mathematical education — `fuzzy math’, `ethnomathematics’, and `constructivism’.
Since numbers are not really `real,’ in the conception of New New Mathematicians, focusing the study of mathematics on finding a `correct’ answer is pretty trivial. The emphasis thus belongs to the process of learning math, recognizing patterns in the numeric symbols – not numbers – and on making “shrewd guesses” from visual models. There is a lot of emphasis, in the New New Math, on using two and three-dimensional models, which can show the reality (`three blocks’) from which numbers stem, with little emphasis on numeric manipulation itself. (Blocks and tiles are in, pens and pencils are out).
This branch of the New New Math (which holds that the teachers should discourage abstract numerical manipulation in search of an absolute answer and encourage shrewd, intuitive guesses, regardless of whether they are right or not, based on recognized visual patterns in objects like tiles or blocks) is called `Fuzzy Math.’
Martin Gardner, writing in The New York Review of Books, gives a good example of what Fuzzy Math means: “Teachers traditionally introduced the Pythagorean theorem by drawing a right triangle on the blackboard, adding squares on its sides, and then explaining, perhaps even proving, that the area of the largest square exactly equals the combined areas of the two smaller squares. According to Fuzzy Math, this is a terrible way to teach the theorem… [The students] cut from graph paper squares with sides ranging from two to fifteen units. Then they play the following `game.’ Using the edges of the squares, they form triangles of various shapes. The `winner’ is the first to discover that if the area of one square exactly equals the combined area of the other two squares, the triangle must have a right angle with the largest square on the hypotenuse. For example, a triangle of sides 3,4,5.”
Students who never discover the theorem are said to have `lost’ the game. Thus, with no help from teacher, the children are supposed to discover that with right triangles a(squared) + b(squared) = c (squared).” (This is `constructivism,’ perhaps best defined as the method of teaching New New Math, which stresses that students should figure out answers for themselves – elsewhere called `experiential learning’ – instead of being fed formulas and theorems by the teacher).
`Ethnomathematics,’ which stems from a whole range of multicultural impulses, is a secondary component of the New New Math curricula, though it is more sociological than mathematical in structure. It has two effective curricular influences.
The first is to encourage the study of the way primitive tribes counted and added. There is a popular text, released only in 1997 but already in full use, called Africa Counts: Pattern and Number in African Culture by a woman named Claudia Zaslavsky which gives extensive examples of different methods of dealing with Math and numbers. (As Carl Sagan and others have pointed out, the arguments for the mathematical and scientific success of ancient tribes are, in large part, fabrications).
The fact, of course, is that Math, like Physics or Chemistry, is a progressive effort. We don’t study Aristotle’s physics because we have Einstein’s. The effect of ethnomathematics is to reduce science from a way of understanding the world to a way of understanding culture.
The second effect of ethnomathematics is to suffuse the texts of the discipline with multicultural “examples” that have at best a dubious relation to mathematics. One popular textbook quotes a Maya Angelou poem in whole (there is an accompanying photograph of Ms. Angelou with President Clinton) and claims the parallelism in the poem is weighty evidence, contributing to an enhanced understanding of parallel lines and geometric structures.
The same text asks, “Is the time it takes to read an Alice Walker novel always a function of the number of pages?” John Leo, the US News & World Report columnist, examined several New New Math texts and found numerous “photos of President Clinton, and Mali wood carvings, lectures on what environmental sinners we all are and photos of students with names such as Taktuk and Esteban who … offer their thoughts on life.”
Since fuzzy math and constructivism stress the formation of subjective and personal mathematical algorithms and ethnomathematics stresses sociology and Maya Angelou, the bulk of responsibility for ensuring that students do well on traditional measures of mathematical acuity, like standardized tests, falls squarely on the shoulders of calculators. Calculators are a necessary part of the New New Math curricula, and are hence now being introduced as early as the first grade. The upshot is that students no longer need to know multiplication tables or how to do long division — they can simply press a few buttons.
Some states and standardized testing agencies have caught on to this effort to cheat the evaluation system, and so (California is notable here) have banned the use of calculators on tests.
Parents and educational policy-makers alike have been consistent in their opposition to the New New Math, and the associated dependence on the calculator.
Nevertheless, the development and persistence of contemporary trends in American schools of education seems to indicate that the New New Math may be here to stay.
It is ultimately teachers that determine the way students are taught, and the number of teachers who are trained in and teaching fuzzy math, constructivism, ethnomathematics, and the rest of it, is rising.