I was listening to the radio program The Best Of Our Knowledge this morning, and I heard Professor Colin C. Adams interviewed about the state of mathematical education in the United States. And while I have a deep respect for the work that Dr. Adams is doing in improving the general reputation of mathematics, I also believe that several of the answers that he gave on the radio show were myths or at least partial truths that are still promulgated as being gospel.
First, there is the assumption that is made that mathematics is a serial progression. While it is true that much of mathematics is cumulative, in both the sense that most topics require understanding of earlier topics, and that proof is always built upon more basic logical statements. But, it is easy to confuse this meaning of cumulative with assuming that math is linear. And math is anything but linear in how math has developed.
For instance, while Williams College seems to still follow the idea that Calculus is the entrance of advanced math, in reality understanding calculus is not necessary to understand many of the contemporary areas of mathematics that are critical for higher paid jobs. This is because discrete mathematics, including understanding number theory, such as base systems to understand binary, and prime numbers to understand contemporary encryption, is the basis of most technology that has fueled our economy over the past 20+ years. And while a deeper understanding of statistics does require calculus, a basic understanding of statistics and probabilities do not.
There is also the continuing ambiguity of what “advanced” means when it comes to mathematics. There are actually 2 ways of seeing math going from basic to advanced. There is the progression of how we cognitively understand mathematics, from basic to advanced, where we understand counting first, addition next, and so on. And there is also the logical progression of mathematics, where addition is not the basis of math, but instead set theory is the basis of mathematics. This distinction became very clear when the New Math was taught in the 1960’s. Schools attempted to teach what was mathematically basic but discovered it was a failure because this was not at all cognitively basic. To add to this, we again need to remember that both forms of progressions, the cognitive and proof progressions both can go different directions, as was discussed about how discreet math and calculus are fairly distinct areas where math branches.
Dr. Adams also talked about how math has “1 right answer”. This again is a half-truth. It is completely true that for a given set of postulates and definitions, given a mathematical problem, there is only one answer. But, in the real world there are often more than one mathematical model that can describe something, so often there are slightly different answers that approximate what reality is. Also, in pure mathematics, different postulates and different definitions produce radically different results. The parallel postulate is a clear example of this, which by following a different postulate, Einstein came up with many of his theories and mathematical models. Or in a more contemporary context, in Boolean algebra that is used in computer science, 1+1=1 is a true statement.
Further, even when there is only “1 right answer”, there are many ways of getting to this right answer. This is something that many math teachers still don’t seem to understand, and consequently many students who do math in a different way, but still get to the right answer, are often put down, or made to believe that they are wrong.
So in summary, while I am glad that more people are talking about math and the importance of math, we must stop spreading myths, even if they are being spread tacitly.