To the Editor:

In their article "Make Math a Gateway, Not a Gatekeeper" (*The Chronicle,* April 23), Anthony S. Bryk and Uri Treisman describe a problem and how the Carnegie Foundation for the Advancement of Teaching is going to solve it. The problem they describe is that some students are unable to meet the algebra requirements for their degrees—specifically they speak of students who need statistics for their degrees but cannot meet their algebra requirements. The authors claim that they are going to develop a new curriculum that will prepare these students for statistical work but will somehow avoid much of the algebra. I am going to state a set of observations on algebra and statistics and then point out where I see difficulties.

1. Generally people do not retain algebra for more than two years unless they have had calculus. That is where they really learn algebra. Students who have not had algebra for two years and need to use it generally need to retake their last algebra course. Hence many students who are quite capable give up on math because they were encouraged or allowed to take a course they were not ready for. Sometimes the capable student gives up simply because of a bad teacher.

2. Unless students go into areas that require calculus, they are unlikely to use any algebra beyond the most basic.

3. Students who do not need mathematics for their majors can be accommodated with a "math for humanities" course. The ubiquitous course in "finite mathematics" offered at the precalculus level can serve this function.

4. The basic statistics course, whether taught in business or psychology, is meant to familiarize the student with statistical concepts. The algebra required is minimal, and even though the student performs various statistical tests, the course does not give the student proficiency in performing statistical procedures.

5. An enormous body of statistical tests can be performed without use of calculus. However, in practice this statistical work requires the mathematical maturity associated with learning calculus. Statistical theory, even at the undergraduate level, is calculus-intensive.

If the students Mr. Bryk and Mr. Treisman are talking about need only to understand basic statistical concepts, that can be accomplished with a course in basic algebra, a well-designed statistics course, and good teaching. However, if these students need genuine statistical skills, their inability to pass algebra courses implies that they have chosen to pursue the wrong fields—and no curriculum design will change that fact.

James M. Cargal

Professor of Mathematics

Troy University

Montgomery, Ala.

## Comments

1. lexisaro - June 04, 2010 at 02:44 am

I am not sure of the above argument. I somehow missed calculus along the way (long story), but seemed to do just fine in all of my PhD level statistics courses that I took at a very top school in my field. Courses that included anova, manova, regression, structural equation modelling, factor analysis, social network analysis, cluster analysis, multidimensional scaling and I am probably missing some. Moreover I have been publishing decades using these techniques (and learned enough to be able to teach myself as needs have arose over the years). I sort of picked up the calculus part during the classes as it appeared, but it hardly seemed like I was at a disadvantage.

2. petelclark - June 04, 2010 at 08:12 am

Reply to lexisaro: a sufficiently intelligent and driven student can overcome lacunae -- and even gaping holes -- in their background. Based on your brief description of yourself, you sound like you are way towards the high end of the spectrum: presumably most of the students Prof. Cargal is talking about are not going on to a PhD in any field, let alone at a top school.

I am a math professor at a very good university in a state which is ranked in the high 40's in terms of K-12 education (i.e., almost the worst). I have seen many students who have clearly not learned enough pre-calculus level mathematics to succeed in freshman calculus. It has come to the point of me starting to be frustrated at being asked to teach a course which seems beyond the reach of our average enrollee.

However, there are always exceptions. Once I was teaching a calculus class in a building far away from the rest of campus, so I would often have brief conversations with students while walking back to my office. One day a student caught me on the way out and mentioned that he was concerned that some trigonometry ("sin x") had appeared on the first exam; he wondered whether these concepts would continue to appear. I allowed that they probably would, although they were not the primary emphasis of the course. Would that be a problem? Yes, he said, because he had never had trigonometry. Did he mean to say that he had never learned it or never been taught it? Never been taught, he replied. Fighting against my natural but unproductive reactions of shock and horror, I said: "Well, it's not so bad, really. Consider a point moving along the unit circle...and the cosine and sine are just the x- and y-coordinates of the point." Amazingly, his eyes lit up: I could see that he had, in the course of a two minute stroll, picked up what the sine and cosine were. And when these things came up on future exam problems...he got the problems correct! On one homework assignment or quiz, something slightly different like the inverse tangent came up, so he asked me briefly about that. Problem solved.

In order to properly appreciate how unusual this is, you need to have had the experience of trying to explain trigonometry to someone in less than, say, three months. Most people just can't pick it up that quickly. Some can! But my horror that a student who was in the college preparatory track at his high school and was never even taught trigonometry is undiminished. This was a terrible disservice to most of his classmates.