Masquerading as Algebra

Jerry Dancis

Washington Post    Free For All , the Saturday's Post's large and glorified letter-to-the-editor page (the page preceding Saturday's editorial page)   September 7, 2002

Charles R.L. Power wrote, "I don't know who could solve such an equation without algebra" [Free for All, Aug. 24]. The question was: "Eva bought 5 pairs of identical socks and a $6. 50 hairbrush. The total cost for the items was less than $29. Which of these inequalities best describes the cost (c) of each pair of socks?"

I would solve this as an arithmetic problem: Suppose that the total cost is exactly $29. Then the five pairs of socks cost $29 - $6.50 = $22.50. So one pair of socks costs $22.50 / 5 = $4.50. Because the total cost is less than $29, the cost of a pair of socks is less than $4.50. This translates into Answer A: c < $4. 50.

I am an associate professor of mathematics at the University of Maryland, College Park. I was the expert who told your reporter that this is an arithmetic problem ["Algebra = X in One School, Y in Another," front page, Aug. 19]. I was a bit hasty; I overlooked the last part of translating "less than $4.50" into "c < $4.50." This raises it up to a sixth-grade-level, background-for-algebra problem.

After some modification, 15 of the 49 problems on the sample Maryland High School Assessment on Functions, Algebra, Data Analysis and Probability test should be included in fifth-grade instruction; the modifications would be ones of form, not of mathematical substance.

Some problems are clearly arithmetic. For example, Problem 24 states:

"A local baseball league separates its season into two parts. The win/loss records for each team are shown in the two matrices [tables]below.

Which team had the most wins and which team had the most losses during the entire season?"

One out of three students did not obtain the right answer when this problem was field tested. This is absurd. Of course, Problem 24 is not listed as an arithmetic problem. According to the Maryland Department of Education Web site, in solving this problem the student "will demonstrate the ability to investigate, interpret, and communicate solutions to mathematical and real-world problems using patterns, functions, and algebra."

Charles Power, by setting up the inequality and then solving it, does present a nice algebraic solution to the socks-and-hairbrush problem. But setting up an equation or inequality and then solving it is useful on only five of the 49 problems on the Maryland algebra test. When problems can be solved by both arithmetic and algebraic methods, the arithmetic method is often simpler, and it often provides more conceptual understanding of the problem than an algebraic solution.

A student could score 100 percent on Maryland's algebra test and on the math parts of Maryland's eighth-grade school performance assessment test and still not have the math background to take a real algebra I course. The needed background is fluency in arithmetic, including fractions.