Benny, Research, and The Lesson

The most recent MathAMATYC Educator (Vol 6, Number 3; May 2015) has a fascinating article “Benny Goes to College: Is the “Math Emporium” Reinventing Individually Prescribed Instruction? ” by Webel et al.  This article describes research in a emporium model using a popular text via a popular online system.  A group of students who passed the course and the final exam were interviewed; some standard word problems were presented, along with some less standard problems.

At the heart of the emporium’s approach to teaching and  learning we see the same philosophy that undergirded Benny’s IPI curriculum: the common sense idea that mathematics learning is best accomplished by practicing a skill until it is mastered.

I would phrase the last part differently, though you probably know what the authors mean … this is more of ‘the common mythology that mathematics …’ (common sense implies a reasonableness that seems lacking, given students attitudes about mathematics).

The phrase “Benny’s IPI” is a reference to a prior study by Erlwanger (1973) wherein the author looked at an individualized prescribed instruction (IPI) system; Benny was a similarly successful student who left the course with some very bothersome ideas about the types of topics that were ‘covered’ in the course.  In both studies, the primary method involved 3rd party interviews of students.

The current study had this as a primary conclusion:

We see students who successfully navigate an individualized program of instruction but who also exhibit critical misconceptions about the structure and nature of the content they supposedly had learned.

Although I am not a fan of emporium-related models, I am worried about the impact of this study.  These worries center on what the lesson is … what do we take away?  What does it mean?  The research does not compare methodologies, so there is no basis for saying that group-based or instructor-directed learning is better.  The authors make some good points about considering the goals of a course beyond skills or abilities.  However, I suspect that the typical response to this article will be one of two types:

• Emporium models, and perhaps online homework systems, are clearly inferior; the research says so.
• Emporium models, and online homework systems, just need some adjustment.

Neither of these are reasonable conclusions.

I spend quite a bit of time in my classes in short interviews with students.  Most of my teaching is done within the framework of a face-to-face class combining direct instruction with group work, with homework (online or not) done outside of class time.  Typically, I talk with each student between 5 and 15 times per semester; I get to know their thinking fairly well.  Based on my years of doing this, with a variety of homework systems (including print textbooks), I would offer the following observations:

1. Misconceptions and partial understandings are quite common, even in the presence of good ‘performance’.
2. Student understanding tends to be underestimated in an interview with an ‘expert’, at least for some students.

I have seen proposed mathematics that is equally wrong as that cited in the current study (or even worse); granted, these usually do not appear when talking to a student earning an A (as happened in the study) … though I am reluctant to generalize this to either my teaching or the homework system used.  Point 1 is basically saying that the easy assessments often miss the important ideas; a correct answer means little … even correct ‘work’ may not mean much.

Point 2 is a much more subjective conclusion.  However, I routinely see students show better understanding working alone than I hear when I talk with them; part of this would be the novice level understanding of mathematics, making it difficult to articulate what one knows … another part is a complex of expectations — social status — and instructor expectations by students.

Many of us are experiencing pressure to use “best practices”, to “follow the research”.  The problem is that good research supports a better understanding, but almost all research is used to advocate for particular ‘solutions’.  This is an old problem … it was here with “IPI”, is here now with “emporium”, and is likely to be with us for the next ‘solution’.

The “Lesson” is not “use emporium”, nor is it “do not use emporium”.  The lesson is more important than that, and involves each of us getting a more sophisticated (and more complicated) understanding of what it means to learn mathematics.  Most teachers seek this goal; the problems arise when policy makers and authorities see “research” and conclude that they’ve found the solution.  We need to be the voice for our profession, to state clearly why it is important to learn mathematics … to articulate what that means … to develop courses which help students achieve that goal … and use assessments that measure the entire spectrum of mathematical practice.

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2 Comments

• 

  By schremmer, May 29, 2015 @ 5:18 pm

  Re: the common sense idea that mathematics learning is best accomplished by practicing a skill until it is mastered.

  In other words, for the author, mathematics is nothing but a bunch of skills and logic does not enter it in the least. Sad. Coming from the AMATYC, very sad.

  –schremmer

• 

  By Sue Jones, June 24, 2015 @ 4:53 pm

  What s/he said. I, too, get sad when the focus is on the package instruction comes in, instead of the actual instruction. I get frustrated when we’re urged to teach understanding — but solutions are what’s used to measure that understanding.
  For most of the non-academic worlds and a whole lot of the academic world, math *is* nothing but a bunch of skills, and logic doesn’t enter it in the least.

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