Lumpers and splitters

Considering a “stupiphany”

The secondary school math curriculum used to be extremely predictable in the middle decades of the twentieth century. High school freshmen took elementary algebra, sophomores enrolled in geometry, juniors refreshed and extended their first-year curriculum with intermediate algebra, and college-prep seniors took trigonometry. That's just the way it was in many places across the United States (and certainly when I was in high school).

Eventually, though, many changes occurred. While that old-fashioned core curriculum survives in many ways, it certainly drifted. Algebra trickled down into middle school and high school seniors started taking introductory calculus (or something called “analysis”). Perversely, however, the old high school courses also migrated into the college curriculum, where we originally called them “remedial” and then relabeled them with the less pejorative “developmental” tag. I think that such remedial courses used to be the province of what we call “continuation” high schools, but today most developmental math is taught in community colleges. My college, for example, teaches more developmental math than anything else. We even teach basic arithmetic to those who failed to learn it in elementary school.

And we teach developmental math over and over and over again to students who fail it the first, second, or even third time. Success rates hover between fifty and sixty percent for most of the classes, indicating the degree of recycling that goes on. It's maddening to both students and instructors.

Most of my colleagues in the math department know the secrets to success in a math class. In fact, they're hardly secrets because we share them constantly with our students: attend class regularly, pay attention, study the material, do the homework, and ask for help when you're stuck. While luck plays a role (catastrophic illness, financial distress, and family emergencies can derail anyone), most student failure is based on the neglect of those fundamental guidelines.

Of course, we don't just enunciate the principles of successful math learning and then sit back and wait for our students to succeed. We try to meet them halfway (or more than halfway). We offer tutoring centers, accommodations for learning disabilities, on-line support, and different course formats. These days the traditional classroom-based lecture class is often supplemented with on-line instruction or hybrid classes that combine in-class and on-line elements. Students may be able to enroll in self-paced computer-based math labs, too.

And then there are the “splits,” which try to slow down the pace of the curriculum by slicing the courses in half. Students having trouble with our one-semester elementary algebra might be permitted to take half of the course during fall semester and the second half during spring. You can spot these courses in college catalogs where they bear labels like “Algebra 1A” and “Algebra 1B.” Many schools have even done this with arithmetic. You can struggle during the fall to learn your times table and save fractions till spring.

I wish I were kidding, but I exaggerate only slightly.

Guess what? The students who enroll in the splits aren't particularly more successful than those enrolled in regular lecture classes, on-line classes, or math labs. Are we rescuing a few additional students with each new approach, or would they do as well (or as badly) if we just ushered them all into a classroom and made them sit in rows?

I believe we do achieve some marginal additional success with the multiple formats because students do learn in different ways and one-size-fits-all is almost never true. Still, I wish the benefits were more than marginal.

This reflection on student success and failure in developmental math was stimulated by a recent post by a pseudonymous community college dean. “Dean Dad” traveled to California last month to attend the San Diego meeting of the League for Innovation in the Community College. He was particularly struck by the remarks of a Bay Area faculty member:

Prof. Myra Snell, from Los Medanos, coined a wonderful word: “stupiphany.” She defined it as that sudden realization that you were an idiot for not knowing something before. The major “stupiphany” she offered was the realization that the primary driver of student attrition in math sequences isn’t any one class; it’s the length of the sequence. Each additional class provides a new exit point; if you want to reduce the number who leave, you need to reduce the number of exit points. If you assume three levels of remediation (fairly standard) and one college-level math class, and you assume a seventy percent pass rate at each level (which would be superhuman for the first level of developmental, but never mind that), then about 24 percent will eventually make it through the first college-level class. Reduce the sequence by one course, and 34 percent will. Accordingly, she’s working on “just in time” remediation in the context of a college-level course. There is definitely something to this.

Um. Under the given assumptions, I can't fault the math (0.70⁴ = 0.2401 and 0.70³ = 0.343), but it is just a tiny bit simplistic. If we squeeze all the remediation into one course, then we'll be rewarded with a 49% overall success rate at the end of the college-level course. Yay!

Except that it certainly wouldn't work.

This is a classic optimization problem—the kind that you see in calculus. Two countervailing factors have to be balanced in order to achieve the best possible outcome. For example, if you want to enclose the maximum possible rectangular area with a given length of fence, you have to balance the contributions of length and width, because one can be increased only at the expense of the other—yet both contribute equally to area. (Thus the ideal figure turns out to be a square. Big surprise!)

In the case of developmental math classes, the splits offer more failure opportunities. On the other hand, they reduce the curriculum to bite-size chunks that more students might be able to master. The more you cram into a course, the more likely the students are to be overwhelmed. The trade-offs are rather obvious.

(Frankly, I prefer that split classes be taught at the same pace as regular classes, because stretching them out to semester length attenuates the reinforcement that most students need. At the halfway point the successful student moves on to the second-half split while the unsuccessful student repeats the first-half class without having to wait till the next term.)

I don't think that Prof. Snell's “stupiphany” is quite as significant as suggested by Dean Dad, although I presume her presentation would be more nuanced at greater length than it is in a one-paragraph summary. (She did, apparently, couch her presentation in terms of timely intervention.) The tension between length of sequence and course content will continue. The experiments will certainly continue. In fact, I can even tell you the direction in which they will go. The splitters having had their day, the lumpers anticipated Snell's observation and are putting accelerated curriculum into place. Courses are being designed and curriculum is being implemented. Hang on to your hat as developmental math tries to speed up.

That's probably a future post.