kitchen table math, the sequel: Definitions, Precision, Coherence

Sunday, October 23, 2011

Definitions, Precision, Coherence

What's missing from today's school math, and particularly, from middle school math?

Definitions, precision, and coherence.

Without a proper introduction to definitions, students don't get clarity in what a math statement is and what it is not. They don't get a sense of the abstractness of it. Without having actual definitions, they don't learn to work with definitions, so they can never learn how to use them to derive new things that are true of "all" of a given set. Without definitions, they cannot learn to REASON about mathematics because they have no basis for reasoning.

Without precision, students cannot make clear, unambiguous statements. They are not able to properly manipulate the symbols they are given, and they can't even say what their manipulations refer to and what they do not refer to. (As Wu says, In the same way that we do not ask “Is he six feet tall?” without saying who “he” is, we do not write down xyzrstuvw = a + 2b + 3cdefghijklmn
without first specifying what a, b, . . . , z stand for either (this is
an equality between what? Two random collections of symbols??
What does it mean??))

Without coherence, math is a series of unrelated facts designed merely to trick students. Without coherence, math is just a test of how big your working memory can be when you can never integrate any understanding together. There is no notion that the results you get follow from other results. Reasoning can't exist without coherence.

The reason SAT math feels so "tricky" to so many students is because they were never taught math in a coherent, reasoned way with definitions and precision. So the test seems to bejust a set of tricks designed to "catch" you, as opposed to a test of how what you know relates to other things you already know. That is why it seems to just test working memory. That is why it seems to have so much associative interference. The reason is because you were never taught that math was reasoned, with every piece of it following from the other pieces, a seamless whole that reinforced the same truths from a zillion different directions. And you were never taught what it meant.


MagisterGreen said...

One could make similar comments about English instruction today as well.

RMD said...

absolutely .. .well said!

and the lack of definitions continues

in advanced math, no one tells you what the fancy notation really means.. . so if you don't know how to read, for example, series notation, it just seems to be a series of symbols . . .

Anne Dwyer said...

And this is why calculus is so hard for students.

Because we are done with review and we expect students to have mastered everything that came before and...

because we start using definitions and expect students to be able to work with definitions.

When I took advanced calculus, one of the things that really helped me was counterexamples.

We would learn the definition and how to work with it. But we were also given counterexamples so we knew when the definition did not work.

I try to do that as much as possible with my math students, depending on their level.

Bonnie said...

I am old enough to have been educated when "New Math" was the dominating trend. This sounds rather like New Math, which was all about working from precise definitions. Parents hated it, which is why it went away. Actually, my father loved it, saying it was how math should be taught - but he was a physicist! Anyway, New Math didn't hurt me, and I went on to do well on the SAT. I especially appreciate the way that approach forced us to really understand the base-10 number system, which is important in computer science - and which I find many students do not understand.

Anonymous said...


I completely agree with you about symbols. One of the first tests that I give new a new student goes along the lines:

Solve, x^2-16=0 for x (or some other simple eq.)

I then ask them to solve the equivalent a^2-4=0 for a. I have occasionally come across students who said they were never taught how to solve for a. Another example would be graph b=2a^2-4 or something similar.

I recall in my private high school math books, there were always a few exercises in which the symbols were different. The math texts were written by two of the math teachers. They were traditional, using all the notations used in Thomas's Calculus (3rd or 4th edition). We were so used to problems beginning with "Given ..." that if a teacher gave a problem in which a symbol was not properly defined, we'd give him grief.

But to get back to the earlier discussion about "a", one of the important conceptual bridges to cross is the translation from equation to graph. What I mean here is that the values of a, b & c in a x^2 + b x + c influence the shape of the curve. A trivial example of the general case would be for y=f(x) where f(x)=ax^2+ bx+c, what would the form of the family of curves be when a=0. Obviously, y=bx+c, straight lines with slope = b and y intercept = c. Would this confuse students who had learned y=mx+b?

Are students asked to sketch the general shape of the curves for a<0 and a>0? Or, comparing two curves y=ax^2 and y=bx^2 for a,b>0, how do the relative values of a & b influence the shape of the curves?...and so on. It is through exercises like these that on would acquire and understanding of the meaning of the coefficients of the quadratic equation as they relate to the shape of the curve. Once the equation -> curve exercises were done, we were given exercises where a curve was drawn and we had to come up with the general properties of the coefficients a, b & c (e.g., a>0 or a<0). I can't recall any of the practical word problems for these equations, but I am sure that there were plenty.

Playing the coefficients had an even greater pedagogical value than just understanding the relationship of the coefficients to the shape of the curve: it laid the foundation for the time when the coefficients were replaced with functions, e.g., a(t)x^2+b(t)x+c(t)=0 where a, b & c are functions.

I guess that I am a bit old school when it comes to graphing functions. I prefer that students draw them out by hand with pencil and paper. The value here is that students don't just look at the curve on a computer/calculator screen, they get the physical sensation associated with the shape of the curve. One example from tutoring in calculus, I had the student draw curves with an inflection point and then point out location of the inflection point. I even had them draw curves with their eyes closed and then point to the inflection with their eyes closed. Having a physical interaction with the math, I feel is important for development of understanding beyond the algorithmic execution of problem solving.

Anonymous said...

pt 2

With this sort of experience, the problem that has evoked all of this discussion did not appear tricky at all. Though I must comment that the curve does not appear to be parabolic and the graph was of y vs x and y was never defined, but the conventional assumption would be y=f(x). When Catherine pointed out that the "a" in the problem was not the "a" from high school, I was confused because I could not remember a particular "a". It was only after I read that it was the coefficient of x^2 that I understood confusion. It got me to thinking...there are only two letters that have universal meaning, pi and e (both are numbers). One of the reasons you find Greek letters used so often is that after a while you run out of letters in the English alphabet. The only meaning that I could come up with for "a" was that it stood for "activity" in thermodynamics and the "scale factor" for the universe in cosmology. Each branch of science have their own conventions, and journals used to be very strict about using abbreviations: they had to be defined at the first use in a manuscript. My pet peeve is when an equation appears and all of the symbols are not defined.

This symbol confusion, though annoying, is nothing compared to the brain addling effects of using everyday words in a scientific context - "spontaneous" comes immedediately to mind.

rocky said...

Civilization advances by extending the number of important operations which we can perform without thinking about them.
Alfred North Whitehead

I want the kids to understand how things work. I don't like "magic math". So I would introduce multiplication by having them draw rectangles of dots nXm and count the dots. Then I tell them: You can do that each time, or you can memorize the answers. Here's the n times-table...

I would teach fraction division the same way, first dividing fractions by whole numbers, then multiplying fractions by fractions, then discovering the reciprocal multiplies any fraction to 1. Finally, when I show them "invert and multiply", they will know how it works. Do I have them do it the long way more than a few times? No. I build on the shortcut.

In other words: yes, it all hangs together as a "seamless whole", but we must build it carefully because we don't often go back and check the connnections. Sometimes the "jigs" are never used again after the structure is built.