Myths that Many Mathematicians Must Still Believe because they Still Spread Them

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.

Contemplating the Problem of “New Math”

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:

  1. In our species deep past, only simple mathematical processing was generally required to survive.
  2. Therefore, through biological evolution, our brains are naturally wired in a way that can calculate these types of traditional problems.
  3. 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.
  4. 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.
  5. 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.
  6. All of this makes bringing these concepts back again a challenge.

Chain of Needs for Student Success

As I have been sharing, the debate about school effectiveness and student success so often centers around standardized tests and the teachers who teach, yet we ignore the box that the teacher is placed in, and the future life of the student after leaving the school system.

To have true student success, deductive logic tells us, that there must be at least four critical components:

  1. Content is Relevant to Students’ Ultimate Needs and Goals
  2. Content is Taught at an Appropriate Time
  3. Teachers are Knowledgeable about the Content
  4. Teachers have Activities that are Efficient at Helping Students Learn

In our public system, the first 2 links of the chain is being defined by the standards, while the last links are clearly in the privy of the teacher.  Although, I will argue that often the best teachers are the bold ones who also bring in new content that is important to their students, beyond what standards currently define.  In either case, each of these 4 components must be at a sufficient level so that students truly succeed.  If any one of these links is not sufficiently well done, then there will not be success for the student, unless the student finds the components outside of traditional schooling.

We spend the majority of our resources on improving our teachers, but  the amount spent on improving the first two links of the chain within the realm of standards is minute in comparison.  There are of course major reform movements that have occurred, but I think evidence would bare out that most of these have been more about political popularity than research of need, and the resources used to create them and improve them were still small in comparison to the resources used to implement them.

So I am trying to start this research, and find others who recognize the need of improving our standards if we are to truly have student success, such that no link will be broken.

Those who can, Do. Those who can’t, Teach. Those who really suck, Rule the world.

I’ve always wondered about the adage “Those who can, do. Those who can’t, teach.” While any generality will be incorrect for certain people. (For example, I personally believe I’m one of those who can, given my background in private industry) But what about the majority of teachers? Is there any evidence to truly suggest that this concept that teachers are not as good as those who go into other industries or majors?

Unfortunately, I think I found some. I am going to enter a masters program this coming year, and so I have decided to take the GRE test (even though the masters programs I’m looking in to don’t require it, specifically I think it could boost my chances with Drexel. I’ll talk more soon about my speculation about why they don’t require it.)

The GRE is used for many schools to determine admission into various masters programs. And while there can be arguments made about how well (or how poorly) it measures ones potential of success, it still is a benchmark that is used, and I’m sure has some merit.

What I found interesting, is in the math (quantitative analysis) part of the test, about 2/3 of the general group that takes the test score better than Educational majors.

But, when it comes to managers, it is mostly worse. For those majoring in Management for private industry about 60% of others did better, for those majoring in School Administration, about 69% of others did better, and for public administration it sunk to having about 71% of others doing better… So maybe the Peter Principle has some merit also!!!

Although, to be fair teachers and administrators do fair better on the English (Verbal Reasoning) portion of the test. On this part only 55% of everyone else did better than Teachers. Although in this case about 60% did better than private industry managers, about 62% did better than school administrators, and only about 55% of others did better than our public administrators… So, I guess in this case, our public servants, like our former President Clinton, are cunning linguists. 🙂

Oh well. I hope that I score well on the test to show that not all teachers “can’t”, otherwise my next post will need to be refuting this post, and telling you why the GRE is not a good gauge of why someone can or can’t… 🙂

BTW, I’m interested in anyone’s feedback about other objective methods that either show our teachers and public servants are more or less capable.

P.S. – To be fair to the MBAs of the world, most of them don’t take the GRE, they take the GMAT, and you really can’t compare them together, so private industry managers may not be as bad as I list here.