Beware of Procedural Knowledge

I'm embarrassed to say that I just learned the State of Virginia is in the process of revising its math standards. I will provide updates as I learn more, but apparently they've already taken comments online from parents, teachers and others. There may be other opportunities to comment however.

I learned about it when I approached my school board representative about the possibility of using Singapore Math in Fairfax County where I live. She seems interested in it and passed on my note to the Assistant Superintendent's office. In the reply back which covered several different items, the following sentence leaped out at me:


"We have not looked at the Singapore curriculum in a couple of years; the last time we did it focused on procedural knowledge and was not as aligned to the POS as other texts."

Uh, excuse me? Could you explain why procedural knowledge is a bad thing?

On the other hand, please don't.

Real world vs mathematics for math's sake

Arguments about how math should be taught frequently include the issue of "real world" math problems. I.e., students need relevance, otherwise they'll tune out. This attitude excludes a whole host of problems that one might find in a geometry book, let's say. So a problem that is real world and makes use of the Pythagorean Theorem is OK, but a proof of the Pythagorean Theorem is not. Well, no, people might object to my extension. So let's use another example. "For any point within an equilateral triangle, the sum of the perpendicular distances of the point to each respective side of the triangle is constant, and equal to the altitude of the equilateral triangle."

This wonderful theorem probably wouldn't qualify as "real world" and even though understanding the theorem and how it is proven opens up higher order thinking skills and mathematical reasoning, a math reformist would not teach it in its pure form. They would have to find some application, however contrived, to provide relevance.

Math for math's sake is out. All math must be relevant. In fact, when the Fairfax County Public School Board (in Virginia) was adopting math textbooks back in 2001, they argued about the criterion for "real world" applications. During the ensuing debates, two school board members found the following criterion too narrow: "Materials and concepts are related to real world situations". They argued for, and lost, the following (excerpted from the minutes of a School Board meeting found here):

"Mrs. Brickner moved, and Mrs. Thompson seconded, to amend the main motion to add the words, “mathematics and” to the eighth bullet under criteria #2, so that it would read, “Materials and concepts are related to mathematics and real-world situations.”

"Mrs. Brickner said that a mathematics textbook could not relate everything to a real-world situation; that there should be a balance between the presentation of math concepts and their relationship to the real world to help students understand the need for those concepts; that applications of mathematics should come from both within mathematics and from problems arising from daily life; that a strict application of the criteria, as originally written, would cause an evaluator to find a textbook less effective than they otherwise might on the basis that the text did not wholly focus on the real world; and that the objective was clearly to teach math concepts and skills.

"The motion to amend the main motion to add the words, “mathematics and” to the eighth bullet under Criterion #2, so that it would read, “Materials and concepts are related to mathematics and real-world situations” failed 4-7, with Mr. Braunlich, Mrs. Brickner, Mr. Reese, and Mrs. Thompson voting “aye”; with Mrs. Belter, Mrs. Castro, Mr. Frye, Mr. Gibson, Mrs. Heastie, Mrs. Kory, and Mrs. Strauss voting “nay”; and with Mrs. Wilson absent. "

So, the upshot is that math textbooks must base all examples and applications on real world situations; mathematics for mathematics sake does not count. That would make calculus textbooks rather challenging to write!