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New Math Vs. Reform Math

Education column for the April/May 2007 issue of the Newsletter of the Association for Women in Mathematics:

Back in my days at the wrong end of the red pencil, I observed that when I was told to compare and contrast two things I often wound up with a clearer picture of both of them individually than I had previously had of either. With this in mind, I recently decided to assign myself the task of comparing and contrasting New Math and Reform (or NCTM Standards-based) Math, along with the controversies and polemics engendered by each of them. I wouldn’t say that all is now clear to me, but the process certainly generated some new perceptions.

Before I launch into the comparison, I had better explain my own place in each of the events, since that determines the perspective from which I view them. The explanation involves a little history, a considerable amount of time having passed since the New Math was new. The point of origin of New Math was very straightforward: in 1957 the Russians put a satellite into orbit before we did. Sputnik, it was called, and its impact was dramatic. Spotlights were instantly focused on our scientific community, and it was noted that not enough mathematicians were coming out the upper end of the educational pipeline. The mathematical community snapped to attention and observed that the K-12 curriculum was A) mathematically incoherent and B) boring. A group of extremely dedicated and hard-working mathematicians set about to repair this lamentable state of affairs. The School Mathematics Study Group (SMSG), funded by the National Science Foundation and directed by Ed Begle, worked from 1958 to 1977, and produced a mathematically elegant series of texts.

All of this I watched happening from a very close vantage point on the sidelines. My father, E.J. McShane, was on the National Science Board from 1956 to 1968, and was president of the AMS in 1959-60. His support of SMSG was whole-hearted and energetic. I was correspondingly quite excited and completely convinced by the whole project.

Returning to historical mode: New Math ran into serious difficulties, which I will discuss later, and the result was a massive swing back to basics. There the pendulum sat for a while, until the country’s mathematical inadequacy once more came into the public consciousness. Reports like A Nation at Risk sounded a clarion call to Do Something. This time the mathematics education community – specifically, the National Council of Teachers of Mathematics -- took up the challenge. They assembled a huge multitude of people to whom the mathematical education of the country’s children was relevant, from teachers and school administrators to business leaders. They even reached across the chasm produced by the New Math difficulties to invite in some mathematicians. Several years of hard work later, the NCTM Standards appeared, advocating a very different set of emphases from the accustomed ones and a correspondingly different approach to teaching. With massive effort, a large part of the mathematical community set about changing the whole system over. This change is the locus of the infamous Math Wars.

Completing the perspective issue: During and after the New Math times I continued along the course I had set out for myself and completed a PhD in mathematics. Shortly thereafter the family passion for teaching, which was second only to the family passion for mathematics, came into ascendancy in my life. Eventually this led to an interest in building bridges between mathematics and mathematics education, which in turn led to my having the opportunity to work closely with colleagues in the College of Education. With some concentration I can simultaneously see things from the mathematician’s point of view and the mathematics educator’s. And when the two diverge I shuttle madly back and forth to try to find the elements that could be used to promote communication.

What then are the similarities between New and Reform Math? By far the most visible is that both proposed sweeping changes and each was produced with massive effort by a community determined to meet a clearly perceived need. Another similarity is that each ran into difficulties of a scope that was far beyond the expectations of its creators. In the case of the New Math it took me years to develop any kind of clear picture, because I was too closely enclosed in the community that produced it. My father was badly hurt by its rejection, as were the others who worked so hard to produce it. To the end of his days my father felt that he had received an unfair slap in the face from the educational community. Coming into the educational community I was presented with an image of mathematicians as arrogant clods who came trampling in where they really didn’t have any business to be. Each perception had a grain of truth. The mathematicians’ arrogance had much more to do with their love of mathematics than personal pride – they were not able to imagine that their beautiful field, if presented with a carefully thought out axiomatic structure, could fail to be clear and inspiring in the eyes of the learner. They also, I suspect, lacked a real perception that there was a difference between teaching, say, an inner city class with children from four different cultures and teaching the classes in which they saw their children and their friends’ children learning. That wasn’t because they were mathematicians – very few people had a handle on that problem at that point in history – but it was certainly damaging. On the other side of the chasm, people in the educational system were part of a culture we have built ourselves that regards mathematics as obscure and frightening and best left in the hands of the severely gifted. For the average teacher, the important thing was to protect children from it, which is a good way to guarantee they will fear it (generally expressed as hating it) as well. I should add that there were in fact many teachers who took to New Math like ducks to water and still light up at the mention of it. Unfortunately they were emphatically a minority.

The contrasts between the two movements are in general easier to spot. Underlying them are three major changes. One is that at the time of the New Math research into learning was an area in which there were a few outstanding people whose work was regarded as very interesting, but relevant to theory rather than practice. By the time of Reform Math it had become a respectable research field. Like any academic field, it has its share of silly articles, and unfortunately anybody can read them, unlike silly mathematical articles, which unduly damages some folks’ reactions. What’s important is that good, solid work has been going on since mid-century, and work on the NCTM Standards was based on it.

The other changes are of an even larger scope. We may cringe to admit it, but racial and gender inequities were so entrenched at Sputnik time that those working with the image of filling up the scientific pipeline unquestionably had a vision of lots of middle class white males emerging. By the late 80’s that situation had at least been faced, and the task the NCTM set itself was not simply to improve the mathematics of the cream of the educational crop, but to see to it that all children in every walk of life have the opportunity to learn what they really need to know in mathematics.

That last sentence subtends the third major change. A century ago the job market held lots of possibilities for anyone who could competently carry out all the basic operations with whole numbers and positive fractions. Even negative numbers were a frill. Today those skills remain a part of basic literacy, but the fact is that any can be done by a dime store calculator. Correspondingly, just being able to do them is way short of enough. People need to be able to fit them into a whole framework -- to see them as tools and be able to use those tools freely and comfortably. In terms of job skills, the business community has made it clear that what it needs is people who can solve problems, and who can communicate and cooperate with others. In terms of functioning in society, people need to be able to analyze a situation, reason correctly and recognize false reasoning. All of these things need to be taught to all children. That’s a very different mandate from the one to which the SMSG was responding.

As the situation is more complex, so too are the challenges Reform Math has faced. For the New Math there was a single basic sticking point: the mathematicians who produced it were unaware either that the mathematics that was so crystal clear to them was obscure and frightening to the majority of teachers and administrators, or of how damaging that obscurity and fear were. Standards-based teaching, on the other hand, involves a shift in the viewpoint of what constitutes good teaching, along with many changes in mathematical emphasis. It involves seeing to it that students are intellectually engaged and producing their own ideas, while also making sure that those ideas are tied together in a way that forms a foundation for further learning, and that that learning converges to the essential elements of whatever level of mathematics they are learning. It involves really listening to students in a way that the classical mini-lecture-plus-worksheet tactics never did. It is hard! Fortunately when it goes as it should it is also extremely rewarding.

A change of that scope can’t and shouldn’t happen too swiftly. There was a field-testing phase, initially involving materials brought over from the Netherlands, where many elements of this kind of teaching have been in use for years. After that came a time of creating and testing and re-creating materials of our own, with the support of the NSF. Then finally began the expansion phase, with a lot of assessment accompanying it. Assessment can’t happen very swiftly either, since the point is what happens to children who are consistently taught in this way. A cheer went up across the country when in 2001 results of a multi-year study in Pittsburgh were released and the theoretical benefits began to have support from data: for children in the schools that made thorough use of Standards-based curricula the computational skills did not diminish and the problem-solving and reasoning abilities shot up – all as measured on standardized tests. Other places have since supplied similar data.

Meanwhile, however, an opposition developed. It was swift, effective and extremely politically savvy. After a brief period of stunned disarray, the supporters of Reform sprang to its defense, and the all-too-aptly named Math Wars were launched. I watched them for a number of years with slightly smug sympathy, because I knew my state was too reasonable to be susceptible to such tactics. I was wrong, of course, and am now somewhat battle-scarred and as of this writing still fighting like mad – but that’s another story.

All of this brings me to one final contrast and similarity. The contrast is in the nature of the opposition. In the case of the New Math what happened was that the educational system said, “You’ve thrust at us something that we just don’t want to deal with” and was duly supported in rejecting it. Opposition to Reform Math was, as I understand it, launched as a power play by someone completely outside of mathematics or education. It was a professional lobbyist who came up with the “Fuzzy Math” smear that has served them so well. They’re operating on different hypotheses from mine, so their outrageous actions must be in some way reasonable within their framework. I can even understand the collection of mathematicians who signed an open letter to the Secretary of Education decrying the NSF curricula, many of them without having looked at a single one of them. A colleague had asked them to, and we are all pretty collegial. What I can’t understand is that the attack is still being carried forward by a small cadre of active, well-established mathematicians. Pointing to specific (sometimes genuine) flaws, they advocate eliminating everything and returning to the good old days – the ones that brought us a country where it is far more acceptable to hate math than to enjoy it, and where an electorate quietly accepts whatever data and “reasoning” the media present. There are a few tadpoles and algae in the bathwater, but have they no concern for the baby?

The final similarity, on the other hand, points up a glaring omission on the part of those of us who are trying to make the change happen. One of the fatal weaknesses for the New Math was that parents couldn’t understand their children’s homework and couldn’t help them. When the education establishment told them it was actually nonsense, they were easily persuaded. It was their support of the opposition that helped the tide to turn against the program. Now we are again sending home homework that parents don’t understand, and to make matters worse, instead of looking to the parents like something alarmingly abstruse, it looks like a race around invisible obstacles towards an unfamiliar goal. Small wonder that when they are told that “mathematicians think this is nonsense” they find the statement easy to accept. We urgently need their support, and to gain that support we need their understanding, and it is up to us to produce that understanding. How to achieve that? That’s not just another column, it’s another whole book!

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