Active Calculus

In my last post, I noted that I’ve been doing some interesting reading about the history of calculus instruction and some recent developments.  Earlier today, I gave a talk that summarized much of that reading at the Kansas City Math & Tech Expo, a wonderful conference that happens each fall in KC.  I thought it fitting to post some similar observations here, with a link to my talk slides at the end of this post.  Throughout what follows, the emphasis added in quotes is mine.

One of the quotes in a 1997 publication said the following:  “Courses and materials developed under the aegis of calculus reform emphasize direct experience with methods and processes of inquiry.”  This statement – juxtaposed with two other recent articles I’ve read – make me wonder how fully we have lived into that aspect of the goals and progress of the calculus reform movement.

In David Bressoud’s CSPCC study, one aspect is a survey of over 700 Calculus I instructors.  One summative conclusion is that “Most instructors see themselves as fairly traditional. They view lecture as the best way to teach students and believe that procedural fluency precedes conceptual understanding.”  This struck me as being incongruent with the statement from calculus reform regarding “direct experience with methods and processes of inquiry.”  Then, I read Jerome Epstein’s article in the Notices on the Calculus Concept Inventory (CCI), which observes that “Both the FCI in physics and the CCI in calculus show that traditional instruction has remarkably little effect on basic conceptual understanding, and this has been the greatest shock to faculty. Research dating back more than thirty years has shown that most students emerge from standard introductory courses without a solid grasp of the basic concepts.”

Read that again:  traditional instruction has remarkably little effect on basic conceptual understanding.  The physicists have known this since the late 1980s, but we mathematicians have been slow to acknowledge it.  Indeed, from the survey of calculus instructors, “Most instructors see themselves as fairly traditional. They view lecture as the best way …”

Everyone in our community should read Epstein’s article in full, but here are some important points from it:

– “Hake’s findings [for physics] are striking. They show that [the normalized gain] is independent of the level of the class at entrance … and largely independent of instructor and text. It is, however, strongly dependent on the teaching methodology used.”

– The teaching methodology that promotes the strongest normalized gains Epstein calls “interactive engagement”.  “Interactive Engagement (IE) methods are those designed at least in part to promote conceptual understanding through interactive engagement of students in heads-on (always) and hands-on (usually) activities which yield immediate feedback through discussion with peers and/or instructors.”

– Hake’s findings have been confirmed in independent settings for calculus, too.  This might be seen most prominently in a 1998 study of the University of Michigan’s program — known for its model of interactive engagement — where the overall normalized gains across 51 sections exceeded those at essentially every traditional program.  One of the most striking features of this study was that of the instructors involved in the study, 1/3 of them were new to teaching calculus.  Think about that:  novice instructors, carefully trained to used methods of interactive engagement, were able to lead their students to greater normalized gains that any instructor of any level of experience and expertise using a traditional lecture model.

Novice instructors.  Not lecturing.  Generating demonstrably more conceptual understanding.

So, as I said to my audience today at the Kansas City Math & Tech Expo, I say to all of us: “how ever much interactive engagement we already use, we should use more.”  Interactive engagement moves the needle on conceptual understanding.  And, for lots of reasons that are well established, I submit that conceptual understanding matters far more than procedural fluency.

If you are interested, see the slides from my talk, which include some additional quotes beyond what is in this post, as well as a full list of my bibliographic resources.