Language and Concepts

A language generally has rules; then we use that language to describe concepts.

Let’s again at ELA.  We use the English language to describe the concept of the hero’s journey or the idea of revenge or violence.

So let’s look at math.  Where does the language start?  Where do concepts start?  Well, since it’s my blog, here’s what I think.

Languages have rules.  “I before E” and all that jazz.  So what are the rules in the mathematical language?  Well, firstly, there are those obvious ones, like the axioms.  There are rules surrounding the number 0.  Rules of when to switch a negative sign.  Perhaps rules that are basic, that are those building blocks to mathematics?

However, perhaps it’s more the rules that dictate how we describe mathematics, how we write down and communicate what math is.  How to write those symbolic symbols, how to write something down so that it means one thing instead of another.  Being from an engineering background and simply taking applied math (which is a far different world than theoretical math, let me tell you), we were not necessarily concerned about how things were written down or the theories.  That isn’t entirely correct, my professors were strict with how we communicated certain theories and ideas but not to the degree of my ‘actual’ math professor.  When we wrote down theories and ideas, there was a specific way we had to write it down.  There was a specific method and language that, if we got wrong, suddenly what we were trying to say wasn’t what we said at all.

It reminds me of grade 9 English, actually.  We had to memorize Romeo’s soliloquy from ‘Romeo and Juliet”.  Not only did we have to memorize the words and the lines, we had to memorize the punctuation.  Our teacher said that the punctuation was just as important, that if something was off, then what you were trying to say and what Shakespeare was trying to say was different.  Interesting...

Anyways.  Perhaps the rules of the mathematical languages are those that allow us to communicate, that dictate how math may be written and passed on.

The concepts on the other hand are theories and ideas that describe certain properties.  They often help describe our natural world.  For example, Pythagoras’ theorem was not actually his theory.  The relationship between the three sides of a right angle triangle was well know before Pythagoras.

These concepts often do not stand alone; in fact, the do not.  They often use or are used by other ideas and concepts.  Pythagoras’ theorem is the basis for much of trigonometry.

Mark Prensky noted how much is changing that we no longer need to focus on things on certain subjects like (he noted) Euclidean geometry.  That they are like Latin and Greek, we no longer need them.

First of all, why on earth did Mr. Prensky choose Euclidean geometry?  Pythagoras’s theorem is an important branch of Euclidean geometry and, if I still understand it, a basic understand of angles and how they relate are still keys in this world; hence why they are taught in the general grade 9 math course.

But I digress.  What Mr. Prensky does not realize is that he is abolishing a concept.  We do not speak Greek or Latin anymore but we still discuss many of the concepts that they talked about.  In fact, part of Greek philosophy was logic, which is part of the “future math” that Mr. Prensky mentioned.

We do pick and choose what concepts in math is necessary for what student.  I do agree that we really do need to put a lot more focus on statistics and teaching students about bias and how to read sets of data (a skill that I believe applied mathematics students gain but is not necessarily touched upon in pre-calculus).  More emphasis should also be put on showing students how they are thinking and problem solving (just a note: not until I was practically full immersed into the world of computer scientist that I began to see myself as a problem solver, so perhaps that is something to also look into).  There are certainly concepts that are not deemed necessary for high school students (such as advanced graph theory, which no one will really need...well, except that it’s really fun so I wish I could teach graph theory).

All right, talking it out makes me kind of understand what Mr. Prensky was getting at.  I can see we have to re-look at what we’re teaching.  Manitoba’s math curriculum actually went through some significant changes, so I would believe that future needs of students were kept in mind (I’ll have to look this up).  But what do we lose when we forget certain concepts?

I don’t know whether I can answer this right now.  I will direct you to my first entry.  Try reading the story, I think it has some meaning here.