The Logic of Change: Do the same, expect different results

You have likely heard the quick definition of insanity:  Doing the same thing, and expecting different results.  Presenters often apply this statement to teaching, frequently stated as “Why should we teach it the same way as they had before, when that obviously did not work?”.  The interesting thought in this logic is the ‘same’ descriptor can be applied to many aspects of the current environment; in spite of this, most discussions focus on the pedagogy and on the teacher behavior in particular.

What about the content?  Perhaps we can improve our results if we first improve the content.  The appropriateness of the current content is questionable, and some have argued that the current content is damaging.  You might take a look at the New Life course outcomes (MLCS Course Goals & Outcomes Oct2012  and Algebraic Literacy Course Goals & Outcomes Oct2012).

However, perhaps we are tragically over simplifying the conversation.  What do we mean by content?  “Algebra” does not always refer to the same content, nor do we use it to refer to the same assessment standards.  We also ignore, I believe, the issue of student perceptions of content.  If you want to trigger a uniform reaction to content, put a simple problem that involves fractions and variables in front of students; in developmental courses, most students will perceive this type of problem as a threat and as something they can not understand.    We could improve our courses tremendously if we would invest time in improving the accuracy of student perceptions of content.  Yes, this takes time, and we would have to give up something … look at it this way:  Most students do not achieve deep understanding of most topics anyway; perhaps the net result would be better if we went a lot slower, with fewer ‘topics’.

You might try this experiment:  After you have covered a topic in a class like you have usually done, where the class went as well as you normally see, ask your students to write their answer to this question: “What are we learning about?”  [I ask individual students this question, and suspect that your students will struggle to provide a good answer just as mine do.]

I see still another over simplification in this conversation: is our content described by objects and procedures, or is our content better described by concepts and relationships?  We do not share perceptions of content, which makes it harder on students.  My hope is that we can, through many professional discussions over an extended period, involving all parts of the country, develop shared language to communicate our perceptions of content.  Of course, I would like us to emphasize reasoning and mathematical problem solving (beyond ‘real world’ problems).  In any case, our students would benefit from our accurate use of a shared language for content.

In many cases, speakers who use quotes like ‘do the same, expect different results’ are using this as a rhetorical device in their efforts to convince us to adopt their solution.  Our profession needs us to take a deeper look at the situations and problems.  If simple statements could solve the problems, they would have been solved long ago.  Making progress at scale (in location and in time) depends on broadly shared conceptualizations and collaboration on solutions.  We each are part of the solutions.

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1 Comment

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  By schremmer, December 10, 2012 @ 11:45 am

  I certainly agree with the fact that content matters.

  Underlying your argument, though, seems to be that the mathematical contents are cast in stone. This is not so. Any mathematical subject can be mathematically treated in more than one way—not to dwell on those ways that are essentially incorrect such as looking at -x as shorthand for (-1)•x rather than as OPPOSITE of x. (The second one involves only the definition of integers while the while the FIRST one involves multiplication and is therefore a theorem.)

  For example, boolean algebras can developed in terms of ordering (binary relation) as well as in terms of meet and join (binary operations).

  For another example, differential calculus need not be developed in terms of limits—whether explicitly or implicitly—but can also be developed algebraically, that is in terms of polynomial approximations.

  In consequence of which, which mathematical path we choose matters.

  Regards
  –schremmer

  P.S. Beyond contents, though, there is also the issue of the mathematical language.

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