## Saturday, December 16, 2006

Over at edspresso there is a fascinating series of articles written by an ed student using the pseudonym John Dewey. In one of the installments, the student quotes from the textbook the class is using:

A case in point is the textbook we are reading in my math teaching methods class. The textbook is "Teaching Mathematics in Secondary and Middle School" by James S. Cangelosi. Excerpt from Chapter 4:

"Because mathematics is widely misunderstood to be a linear sequence of skills to be mastered one at a time in a fixed order, some people think teaching mathematics is a matter of following a prescribed curriculum guide or mathematics textbook. ...
This statement makes me shudder. Math is brutally cumulative and a linear sequence of skills is often necessary. For example, how can you reduce fractions without having learned division? Leave it to ed schools to mistrain teachers in such an egregious fashion. One can only hope math continues to be "widely misunderstood."

## Monday, December 11, 2006

### Long division

If you listen in on the math wars, you'll notice that long division is one of the more divisive issues in this war. It's hard to comprehend why fuzzy math zealots are so opposed to teaching long division and making such a fuss. For example, NCTM in its so-called standards calls for "decreased attention" to long division. I've taught long division and the kids caught on quickly.

In a paper called The Role of Long Division in the K-12 Curriculum that should be must-reading for educationists, David Klein and R. James Milgram show that long division is a crucial skill necessary to understand more advanced math concepts:

Abstract
We discuss the role of long division in the K - 12 mathematics curriculum. We begin by reviewing the reasons that most math educators today depreciate the topic and other topics in the curriculum that derive from it, such as polynomial long division or polynomial factorization. Later we show that this view is simply wrong mathematically. The role of long division is not just to divide one rational number by another, but the algorithm itself contains the initial exposure of topics which become crucial in the core
applications of mathematics in our society today. Following the introduction, we discuss methods for teaching long division in such a way that the underlying concepts can be understood by students. We then provide more details about the ways in which these concepts develop in later mathematics course, and why they are so important.
After reading this, the mysteries of real numbers and the conversion of fractions to decimals will become much clearer. Isn't conceptual understanding what the fuzzies purport to be after? So why are they opposing a tool that leads to a conceptual understanding of major math topics?

UPDATE: Reader Katie has left an incredible link to an actual conversation between a phone customer and a number of customer reps (supervisors) that sounds more like an Abbott and Costello routine. It's about telling the difference between dollars and cents expressed in decimal form. The reps never get it and instead rely on a calculator without realizing what they are doing. This should give pause to NCTM and fellow fuzzies and their enthusiasm for calculators.

What is absolutely hilarious is that at the end the rep (supervisor) declares that the difference between the customer's correct math and the rep's fuzzy math is a matter of opinion.

More here.