Neat Knowledge? Messy Landscape?

We all spend quite a bit of time talking with students, and we also look at massive amounts of student work.  Sometimes, we get in to “homework system mode” where we only provide feedback on the answer.  The answer, by itself, is very weak as a communicator of the knowledge a student possesses. 

I have been thinking about how messy student knowledge is (about mathematics); perhaps this is true for all domains.  In my classes, I try to create an atmosphere where people feel safe asking questions and participating; of course, sometimes the questions asked indicate significant misunderstandings … which can temporarily cause some damage to the overall learning in the class.  In this post, I want to focus more on the implications of errors in student work.

Okay, in our intermediate algebra class we just had a test on ‘quadratics’.  The material is a mixture of procedural and conceptual, with a few ‘applications’ included.  One of the applications involves compound interest (annual) for 2 years — so it creates a quadratic equation like this:

 Most students managed to write this (based on the verbal description and the provided formula).  The most common error?  Subtracting 4000 from each side, a disturbing error.  Since this problem was on a test, I did not have an oppportunity to discuss the error with the students to identify the source.  My primary suspect:  An early rule (early in education) to the effect that “large and small numbers mean divide, similar size numbers mean subtract”.  Every  one of these students can solve linear equations requiring division (like -4x=30), and most solved a quadratic involving a common factor (like 2x²=100).  What triggers  the “we must subtract” response?

In another class (the quantitative reasoning course), we have been doing geometry this week.  As for other topics, the formulas are provided — we are much more interested in the reasoning involved.  One of the problems dealt with finding the perimeter of this shape:

Two consistent problems came up.  First, students thought that ALL perimeter problems dealt with “P = 2L + 2W” … causing them to count the widths of the rectangle (which are interior, therefore not part of ‘perimeter’).  Second, even when convinced that some perimeter problems dealt with a different relationship, they did not see  why we should omit the perimeter (they still wanted to include the interior dimension).  Since I was able to discuss these issues, I have some idea of what is behind them. 

My point here is not to complain about what students do or don’t do, nor to suggest that they have had bad teaching  [I mean, beyond having ME as a teacher :)].  Rather, perhaps we need to think more about the root cause for many student difficulties: 

The basic problem is an oversimplification of a connected set of ideas down to a single ‘if-then’ clause.

Students are sometimes desperate to learn math, and we want to help them.  Too often, this results in only dealing with one thing (or one thing at a time) — which simplifies the learning, but which avoids building connections (contrast, similar).    The geometry instance of this is easily described:  by only dealing with one shape for ‘perimeter’ we encourage students to connect just one formula with that work — and to disconnect this from the concept involved (‘distance around’).   Our procedural work in algebra is even more prone to the oversimplification, because the correct learning is abstract about a combination of ‘properties’, ‘undoing’, and ‘maintaining value OR balance’. 

I’ve got to anticipate what some readers are thinking (due to that last sentence) … are manipulatives the solution for this problem in algebra?  Not really.  The research I have read on ‘kinetic learning’ suggests that this might provide some of the scaffolding to abstract ideas — but is counter-productive if the manipulative remains a focus.  This is a very difficult task, to shift from manipulative to concept; the stages vary with each learner.  If you can do manipulatives on an individual basis with continual feedback (conversation) with a highly skilled faculty, then I believe this can support better learning.   The skills involved are complex, and most faculty will not have them — including me; this is combining the skills of a cognitive researcher, qualitative researcher, and math faculty.  The problem with most manipulatives in learning is that the entire process is over-simplified to be practical in a group setting; a group of 3 students is too large, and possible 2 students is too large.

What is the answer?  We need to clearly articulate the few critically important learning outcomes in a math class (perhaps a dozen for a course), and provide diverse learning experiences around those outcomes.  “Simple” is not the solution; simple is part of the problem.  The solution is a planned sequence of activities to deeply explore the ‘good stuff’ of mathematics.

 
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