## Wednesday, February 8, 2012

### Vocabulary or Jargon?

While we're entering a new era or spelling in our culture, the debate reigns whether to join the revolution or fight for the original ways. It's a debate on which I can argue both sides and about one in which I have a hard time deciding my own position. While "kids these days" are writing things like "lol ur rong" does seem somewhat uneducated, it also gets the point across and isn't the point of language to communicate ideas?

While that example may fall more squarely into the English teachers' realm, I have an internal debate about it in my own classroom, too. How important is it really that I use the words "denominator" and "numerator," when it's so much quicker (and more easily understood) to say "bottom" and "top?" When a line with negative slope is changed to get more vertical, how wrong is it for me to say that the slope gets "more negative?"

I realize that there are some words that need to be defined and used correctly so as to avoid confusion. I just finished a section of Algebra 2 where we talked extensively about the difference between permutations and combinations. The words here are important to the understanding of the logic and the formulation of the mathematical expressions needed to solve these problems.

On the flip side, though, in precalculus we are discussing vectors. What is the real difference between calling two vectors orthogonal versus just the familiar "perpendicular" word they already know? Later we introduce the term "normal" for this same concept. Do we really need three words to express the same concept? (Or are they not the same concept and I'm thinking about it incorrectly?)

I realize that even the words "denominator" and "numerator" are important when we involve more complex expressions. But even they get confusing when we discuss things like "multiply both the numerator and denominator of the entire fraction by least common denominator of the terms in both the denominator and the numerator" to simplify an expression like $\inline \frac{\frac{1}{x}+\frac{2}{x+1}}{\frac{3}{x-1} + \frac{5}{x^2 - 1}}$

So, what say you? When teaching my students math, how important is it to indoctrinate them to the traditional language of mathematics and somewhat confusing vocabulary? Is it possible to communicate some concepts using more colloquial language?

1. I was just teaching limits today in calculus, and every time something approached negative infinity, I had to decide whether to say it was "getting bigger, but staying negative" or "getting smaller" (a decidedly counterintuitive phrase, even if it matches the ordering on the reals) or "becoming more negative" or "moving away from zero in the negative direction". Most of the time I ended up saying several. So I feel your pain.

My general rule is to use colloquial language when possible, but use the terms consistently. I also like to explain the origins of the more technical language, as well as how mathematical meanings relate to broader usage, but that's just my philologist side. :-)

2. I am the author of the fraction cat image. I made it in fun (I took the left-hand picture, saw the possibilities, manually rearranged the cats and took the other side) but I'm glad to see it used in a serious discussion on language.

Well, two languages really; the intersection of math and English.

I can see both sides. On one hand, math is a language, and to speak it properly means learning the formal terms for things. Imagine a complex 10 page proof for a theorem that has never been proven before. The language must be formally correct if anyone else is to read it and try to reason about its validity. English is mushy and many words are colored with connotation; mastering this is great for fiction but not so great for math. So, for any serious student, a firm grounding in formalism is a requirement.

However (and I don't know at what level you are teaching; your simplification example reads like pre-algebra or algebra so I'm guessing junior high or high school) many students will never take another math course after graduation. Many won't need more math in their life other than checkbook balancing, basic geometry like figuring out the square footage of a room or house, or computing a 20% tip. A lot of people, too, just don't have the head for formalism -- their minds just don't work this way. I have a good friend who is very smart and has the stereotypical creative mind. I spend many hours tutoring him on logarithms so he could make it through college algebra. Even though he's a smart guy, the formal definitions of exponentiation and logs just didn't click with him. For these people, having to memorize formal names and rules and reason in the strict way that math requires just doesn't work, and it probably is a reason why so many claim to hate math.

So, I would probably target problem solving and concepts, and mention the formalism but not use that as a main measure of success. As I remember (it's been a while since high school) it's pretty easy to identify the kids who are really into math, and giving a little incentive they'll find ways to challenge themselves. In my case, I had a great math teacher that encouraged me to become involved in Mu Alpha Theta and regional math contests.

3. Oh, per the other question, to me "perpendicular" specifically refers to the 90-degree relationship of two lines. I think of "orthogonal" as referring to objects of perhaps higher dimensions. "Normal" is a more generic concept, and again confusingly, is used as both an adjective and noun (short for "normal vector"). If I talk about a normal to a sphere at point P, what I really mean is a vector (probably unit length) that's orthogonal to the plane tangent to the sphere that intersects the sphere exactly at point P.

4. On re-reading I note you say Algebra 2. Clearly I need to learn to read English better :)