Nearly every leader in our nation is saying that we need to have students get more STEM education (Science, Technology, Engineering, and Math), so that our country will not fall behind technologically and economically from the rest of the world. But, what they don’t say (possibly, because they don’t know), is that the type of math that is needed for Information and Communication Technologies (ICT) and Computer Science (CS) is not the math that is normally taught in high school.
I have not wanted to post too much about the Common Core until I had some time to learn more about it, think about it, etc. So I’ve been sharing tidbits so far. But I think it is time to share some of my critiques about the math standards, which I hope may filter into the next set of standards.
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.
I am firm believer that in education, you must first start with where you want to get to, and then build a path to that goal. As I have lamented, there is a huge need for better and broader research to assist in knowing where we should go. (Although only part of this can be solved with research, as there are also inherently values and normative statements in this mix, which research cannot determine.)
But, with the research we do have, and my personal knowledge, I can see that in our technological environment, there is a need for different foundational skills and knowledge than the traditional. In fact many of these were part of “New Math”. But “New Math” was such a failure, that if I even mention it to someone, they nearly look at me in horror!
Here is what I see as the root cause of this and results:
- In our species deep past, only simple mathematical processing was generally required to survive.
- Therefore, through biological evolution, our brains are naturally wired in a way that can calculate these types of traditional problems.
- Our technological evolution has only recently needed different knowledge and skills as are contained in “New math”, and further only recently could we as a species even recognize that understanding the basis of ourselves (quantum physics, etc.) requires a different type of logical processing than the basic arithmetic of the past.
- Therefore, most people still have a past mental model of mathematical concepts, and democratically they can not see in sufficient mass the need for the different models.
- Because New Math was pushed onto the populous without them being able to understand it, and pushed onto children at a development cycle that was likely completely the wrong time for them to learn such concepts (or at least was presented in a way that did not match with their developmental stage), New Math was a failure.
- All of this makes bringing these concepts back again a challenge.