Sign Up For Our Newsletter

(This is the fourth in a series of articles by an anonymous ed school student working towards certification as a math teacher.  Click for parts 1, 2 and 3.  -ed.)

It has been a while since I have written; I thank my fans and paparazzi for not giving up on me.  I felt I had said enough about my previous class and wanted to wait until the start of the real thing: the Math Teaching Methods class, the first session of which I just attended.

A new class is always reason for anxiety, particularly after a semester with the agreeable professor.  You are faced with someone new who has different rules and expectations—and may not be as agreeable.  I am in a class in which the teacher is, shall we say, an adherent of the National Council of Teachers of Mathematics (NCTM) and its standards.  In fact, the NCTM standards and our understanding of same make up a portion of the syllabus.  Our first assignment is a comparison of those standards with the math standards for the state in which we reside for a particular “content standard”, grade level, and “process standard.  The content standard describes what students are supposed to learn.  The process standard describes how they are supposed to learn it.  I got assigned Geometry/11th grade/representation.  “What is ‘representation’?” I hear you asking.  Expressing things in different ways, I think.  You can use a graph to express a function, or a table of values, or a formula, for example. Which one is best to analyze the problem at hand, I think is what they’re getting at but they go on and on in the standards, bringing in all sorts of ways to show things which might be good things to mention as an aside, but to devote so much class time to it supplants the basics that they are supposed to be learning.  (And which educationists think is mundane, and mind numbing.)  

I was suffering from stomach flu when I was reading NCTM’s standards.  My wife asked why I was pushing myself like that.  My response was along the lines that as long as I was vomiting anyway, may as well take advantage of it.

In any class, there is a phenomenon of competition and sussing one another out, and trying to please the teacher.  And since we all have math backgrounds and can no longer pretend that we’re different or better than classmates who do not have math backgrounds, the competition is a bit more intense.  Although this is quite normal and, some might say, health, it can also be an insidious part of a learning experience, particularly where ideologies and the NCTM standards are concerned.  Case in point:  In our first class we were broken up into groups, and asked to look at a particular content standard for our state.  Our group got Algebra 1.  We were given 15 minutes to look them over and characterize them.  “What do you mean by characterize them?” one person asked.  The answer to that question was so vague I don’t remember it—something along the lines of “describe their relevance with respect to process and content” but even that’s too specific to do the answer justice.

What grabbed my attention was the standard that required that students be able to solve quadratic equations in one variable with a graphing calculator as the primary tool.  My feelings about graphing calculators aside, I noted to the others in my group that it said nothing about students learning the quadratic formula, much less its derivation.  A woman in my group, in apparent defense of the standard, told me her daughter didn’t have to learn the quadratic formula in Algebra 1.  I pointed to that standard and said “You’re looking at the reason why.”

When our turn came to report our findings to the class, I said the Algebra 1 standards were vague and allowed teachers to not teach the quadratic formula.  Some others in the room agreed.  The teacher—Mr. NCTM—in a thinly veiled, poker-faced support of anything resembling NCTM standards, responded that the standards were in fact, not “prescriptive”.  This generated some discussion about giving teachers flexibility and I found myself in a debate with a bright young man who although agreeing that the quadratic formula should be taught was also caught in an unconscious effort to please the teacher.  He found himself arguing that the standards were what must be taught “at a minimum”, that the non-prescriptive nature of the standard gave teachers flexibility to go beyond the minimum.  It apparently didn’t bother him that the minimum was inadequate and that some teachers—and textbooks—would not go beyond it.   He argued with gusto, however, enjoying the limelight, not knowing prior to tonight’s class that state standards existed, that NCTM existed, and that NCTM standards even existed.  All he knew was some standards are better than none and that he was pleasing Mr. NCTM, and wanted to be right about something of which he knew very little.  It did not bother him that the class was concerned with teaching to the NCTM standards and their look-alikes.

Later that week I worked on my assignment—comparing my state’s standard and NCTM for geometry.  NCTM’s standard, in part, says: “Students should see the power of deductive proof in establishing the validity of general results from given conductions.” My state’s standards said in entirety (emphasis added): “A gradual development of formal proof is encouraged.  Inductive and intuitive approaches to proof as well as deductive axiomatic methods should be used.”  After reading those is when the stomach flu hit, I think.

In prescriptive faithfulness to mathematics, I remain sincerely yours,
John Dewey