Failing algebra, in calculus

See here for the idea behind writing kata.

Once upon a time, I worked at a university. I worked a hodgepodge of jobs, including tutoring and adjunct teaching. Students came to me for help with material at many levels, from algebra to calculus to mathematics education, and over time I observed certain patterns.

In particular, students don't fail calculus; they fail algebra, in calculus class.^[1]

As an example: the limit definition of derivatives (the value of the slope of a function as ∆x approaches zero) is fundamentally an algebraic definition. Students who weren't completely comfortable with their algebra struggled with even this basic definition, because their algebra wasn't up to snuff. Sure, there was a conceptual gap to be covered. But in the absence the kind of algebraic fluency that would have allowed those students to adeptly work through exercises, they never stood a chance of understanding the core idea of the derivative.

WIthin the system I was a part of, students generally found a way to repeatedly pass to the next level of education, at least until some failure caught up to them, like antelopes gradually getting picked off by lions.

Now, mind you, I don't think understanding was beyond these students. A fair number of the students who underperformed were, in my mind, capable of not only passing my class, but excelling in it. However, when challenged, their most common course of mental action was to retreat to memorized formulas; they viewed novel problems not as an opportunity to stretch out their cognition, but as a demand to pick the "right" formula from a mental menu, and, failing that, to complain about how unfair it was. In my time in front of the class, I never did find a reliable way to unwind this level of learned helplessness.

As an example: when composing a test, I picked a problem from the homework, and changed some coefficients so that it wasn't exactly the same as a problem they had already done. (IIRC, it was a fence optimization problem, "what is the largest area you can enclose with a fixed amount of fencing?") There was an additional trick: instead of being of the form x + y = z, where z was known, it was of the form x + y = z, where y was known.

Out of 24 students, two got anything resembling a correct answer.

You might, like many of my students, ask "when will they ever have to use this?" Abstract symbolic reasoning shows up time and time again in course material and in job duties... later in my career, I saw psychology grad students who couldn't figure out how to use SPSS;^[2] students reading research papers who couldn't understand statistical significance; pre-meds who would could not reason about exponential decay in drug metabolism.

So, here, I propose an alternative grading scale for calculus-bound students:

• A+: should probably be a TA in the next class
• A: ready for the next class
• Below A-: might be able to pull off a cargo-culted passing grade in the next class, but they have not gained understanding of the material
• Below B-: failure

This may sound harsh, but consider: if math is actually required for understanding the material in their major, then anything below B- suggest that there is nothing resembling "understanding" going on in the students' heads.^[3]