More Calculus, Less Understanding?

Sooner is not necessarily better! If you’re a long-time reader of this blog, you may remember my posts about hyper-acceleration. (I have combined those into one article on my Web site.)

Today, a guest post on acceleration by Robin Pemantle, a mathematician at Penn, who addresses this topic from his point of view as a university professor.

–Henri

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In January, 2016, I was on a panel at the 2016 joint AMS/MAA meetings, on the topic of calculus for college freshmen.  A part of the discussion centered on the issue of “acceleration”, by which was meant the practice of modifying the K-12 math curriculum by moving through the material at such a rate that an extra year or two is added, corresponding to a semester or two of calculus from the college freshman curriculum.

One should distinguish the terms “acceleration” and “enrichment” as follows.  There is a more or less standard K-13 curriculum, where year 13 is AB calculus (fall) and BC calculus (spring).  Typically each semester of college mathematics corresponds to a year in the high school curriculum. Acceleration means covering the K-12 curriculum at a sufficient rate that one fits in 1/2 or all of year 13 by the end of year 12.  Enrichment concerns the addition of curriculum throughout the K-12 curriculum that is outside the usual boundaries, though often relevant to the college math major curriculum.

Various members of the AMS/MAA panel described the harmful effects of acceleration on their incoming freshman class.  The panelists were mathematics professors at a number of different kinds of universities: Ivy League (Penn), a liberal arts college (Macalester), a smaller private university (Pacific U.) and large state universities (Illinois and Texas). The common perception is that all over the US, not only in STEM fields but also in Business, Medicine, Economics, Law, etc., students are arriving at college with more semesters of calculus on their transcripts and a less solid grasp of everything they have learned along the way.

At Penn, our recently revamped placement exam for incoming freshman added some data to our understanding of this phenomemon.  The new placement exam contains 24 questions sampling the K-13 curriculum in various places: algebra and pre-algebra, analytic geometry, pre-calculus, AB calculus and BC calculus.  The bulk of the questions are from pre-calculus and AB calculus, as the chief question of interest is whether students should begin in BC calculus (the modal choice, due to acceleration), AB calculus (the vast majority of the remainder) or something else (either advanced or remedial). According to the traditional, linear curricular model, a student should get nearly all the questions right up to some point, then get nearly all the rest wrong or unanswered.  In fact what we see is quite different.  Most students answer a portion of problems correctly throughout the diagnostic, but this portion varies greatly from student to student.  It seems that the weaker students have learned 20%-30% of everything and the stronger ones 70% to 80% of everything, including not just the calculus courses but pre-calculus, algebra, and analytic geometry.

The consensus about this problem across diverse post-secondary institutions was striking, as was the match between what we suspected from anecdotal evidence and what we found by testing. I think one can consider the above description to be generally accepted among the group of post-secondary educators represented by joint AMS/MAA task forces.

It seems clear that porous learning, as described above, is exacerbated by acceleration.  This is particularly true when the milestones achieved by acceleration are adopted as the single measure of mathematical accomplishment: what score did a student achieve on which AP exam?  It is not easy to fit 13 or 14 years of instruction into 12.  What gets sacrificed is everything not measured by the AP exam. Specific deficiencies we see most commonly are in graphing (most students are utterly lost without a graphing calculator), number sense (poor arithmetic, poor estimation skills), and ability with word problems and applications.  Analytic geometry and facility with logs and exponents are also surprisingly weak, surprising because they are part of the testable and tested curriculum, apparently learned in a way that can be quickly forgotten. Interviews with students show that, across the board, they learned to compute without understanding.  This strategy is in fact optimal for the short term goals of both students and teachers.  More can be learned in a short time horizon if specific procedures are learned and drilled while time is shaved off the part of the lesson spent on solidifying understanding.  Such teaching practices prevail throughout the K-12 curriculum, becoming noticeably worse in grades 11-12 when the time horizon beyond which the students need to remember what they have learned is the shortest (the goal of college admission eclipsing all longer term goals).

The emerging consensus among college math instructors is that the best prepared students, mathematically, are those that know a reasonable amount of math and know it well.  What is reasonable depends on the program of study.  For most students going into non-science fields, even somewhat quantitative fields such as economics, social science, business or life sciences, we would rather see solid understanding and no calculus than the porous learning we are seeing now.  In hard sciences and engineering, at least at a school such as Penn, students should know AB calculus.

The issue is somewhat muddied because many students do not have the option of choosing deep learning over acceleration.  Therefore, acceleration is taken as a proxy for how mathematically apt a student is and for whether the student has sufficiently “challenged him/herself”, a favorite buzzword in college admissions.  At Penn, the majority of students in any field have AB calculus on their high school transcripts, and the majority of engineering students have had BC calculus. This greatly confounds our efforts at teaching freshmen.  It is a delicate matter to design courses for students who think they are beyond freshman math but actually need to learn it over again in greater depth.

–Robin Pemantle