mathematics ⊃ (arithmetic ∪ analysis)

An early January CNBC article entitled "Best Jobs in America 2011" discussed jobs from the perspective of its overall environment: stress, work environment, salary, and physical demands. The article rates mathematician as the #2 position. Being a mathematician, this is of course great to hear. The problem with the article? Look closely at the picture that accompanies the description.

The above image belongs to CNBC.
The person has factored the polynomial incorrectly.

While I'm all for positive press for mathematics and find the error in the image amusing, what bothers me is how mathematics is portrayed.

For most non-mathematicians, the end of their mathematical journey at best ends with a first semester calculus course in college (or perhaps a first course in statistics). For the most part, their journey exposes them to arithmetic, rudimentary geometry, trigonometry, and, if they make it, calculus, the tip of the analysis iceberg.

I don't think there's an argument against the need to have core arithmetic and problem solving skills as they are needed in some of the simplest aspects of life. While most will find utility in geometry the same isn't necessarily true for trigonometry. For those heading to college, trigonometry is required by calculus which seems to be the staple for anyone needing to demonstrate math skills beyond those offered in high school. Of course, there are those that think statistics would serve the purpose better than calculus (see below) - and I agree.

The problem is that students are not exposed to other areas of mathematics: number theory, abstract algebra, linear algebra, set theory, topology, real analysis (beyond calculus), complex analysis, and so on. These areas go beyond equations and computations. You definitely need arithmetic skills to engage the problems contained in the other areas but students could easily be exposed to other branches of mathematics earlier.

I went to high school in NY in the mid/late 80's and completed the Regents track: Math I, II, and III as they were called then. Though I could be wrong, I have distinct memories discussing groups, rings, and fields and even matrices somewhere along the way. Those discussions piqued my curiosity and fueled my desire to pursue of mathematics after high school. I don't know that these topics are part of the curriculum anymore. A quick search of NYSED's APDA's collection of past exams led me to a couple questions regarding an operational inverse in a Math A exam so the topics aren't perhaps completely lost. From discussions with my students, it doesn't seem like the topics are are part of PA's high school curriculum and after skimming the Pennsylvania Department of Education's Standards Aligned System that seems to be the case. (The site is pretty dense and not trivial to navigate so I may have missed it.)

When my students comment that they "don't like math" what they really mean is they "don't like the math they have been exposed to so far" or they "don't like they way math has been introduced". By the time these students arrive at the university they are soured and convincing them there is an interesting world beyond what they know is difficult.

This isn't to say that everyone should be interested in mathematics. It's more that they haven't had the opportunity to really understand it. I should also mention that there are teachers who, despite what the curriculum may or may not include, find ways to show their students how vast the horizon is. Further, many students arrive thirsting for more as well as students who appreciate what mathematics involves though are not interested in pursuing it beyond the tools they need to support their other interests.

To be fair, this discussion could be applied to any subject. Despite the efforts of my history teachers, I never particularly liked history. I think that has to do with the fact that I didn't really understand the true purpose of studying history and, further, never really understood what historians do. (An aside: I attended a talk by a historian colleague of mine years ago. In his talk I managed to finally understand what it is that historians do. Of course, days later, I couldn't remember what it was. In speaking some time later to a mathematician colleague about it he was in the same position: he suddenly understood and promptly forgot.)

By the way, the #1 job according to the CNBC article? Software engineer. So, if you're a mathematician-turned-software engineer...

-- Hal

Here's a quick discussion arguing for the replacement of calculus with statistics in college education by Arthur Benjamin at a 2009 TED conference:

1 comment:

  1. This has implications for the second thread of mathematical enrichment -that of the teaching approach adopted. how to master math