Don’t Tell, Active Learning, and Other Mythologies about Learning Mathematics

For one of the projects I’m involved with, I was providing feedback on a section providing concepts and suggestions for the use of active learning in college mathematics classrooms.   One of the goals of this project is to connect practice (teaching) with research on learning … a very worthy goal.

This particular section included “quotes from Vygotsky” (Lem [or Lev] Vygotsky); see https://en.wikipedia.org/wiki/Lev_Vygotsky for some info on his life work.  I put the reference in quotes (‘quotes from …’) because none of the quotes were from Vygotsky himself.  Vygotsky wrote in Russian, of course, and few of us can read the Russian involved; most “quotes” credited to Vygotsky are actually from Cole’s book “Mind in Society” (https://books.google.pn/books/about/Mind_in_Society.html?id=RxjjUefze_oC).  That book was “edited” by scholars who had a particular educational philosophy in mind, and used Vygotsky as a source (both translated and paraphrased).

I talked about that history because Vygotsky was an influential early researcher … in human development.  As far as I know, the overwhelming portion of his research dealt with fairly young children (2 to 6 years).  That original research has since been cited in support of a constructivist philosophy of education, which places individual discovery at the center of learning.

Most of the research in learning mathematics is based on macro-treatment packages.  The research does not show whether this particular feature of learning results in better learning … the research looks at treatments that combine several (or dozens) of treatment variables.  Some “educologists” use this macro-treatment research to support very particular aspects of those treatments (like inquiry based learning [IBL]).

The “don’t tell” phrase in the title of this post comes from the original NCTM standards, which told us not to tell (ironic?) based on some macro-treatment research.  I’ve never seen any research at the micro-level showing that “telling” is a bad thing to do.  Some of us, however, have concluded that the best way to teach any college math course (developmental or college level) is with discovery learning in context with an avoidance of ‘telling’.

I want to highlight some micro-level results from research, but first an observation … in addition to the problems listed above about macro-treatment research, the Vygotsky research dealt with children learning about material for which they had little prior learning.  In our math classes, the majority of students have had some prior exposure to the concepts up to pre-calculus; when these students are placed in to an IBL situation, the first thing that will happen is that the process will activate their prior knowledge (both good and bad).  This existence of prior knowledge complicates our design of the learning process.

So, here are some observations I offer based on decades of reading research as close to the micro-treatment level as possible.

• Lecturing (un-interrupted talking by the instructor) can be effective as a component of learning.
• Small group processes can be effective as a component of learning.
• The effectiveness of either of those treatments depends upon the expertise and understanding of learning on the part of the teacher.
• Teachers need to deliberately seek to develop expertise and understanding about learning the mathematics in their courses.
• Students assume that their prior knowledge is sound and applies to everything.
• The  amount (frequency) of formative assessment should be directly proportional to the amount of inaccurate prior knowledge in the students.
• Feedback on student learning should not be instantaneous but timely, and qualitative feedback is just as important as information on accuracy.
• The primary determinant of learning is student effort in dealing with the material at the understanding levels (as opposed to memorizing).
• Repetition practice (blocked) is okay, though mixed practice (unblocked) is more effective.
• Classrooms are complicated social structures, and the teacher does not have influence over significant portions of those structures.

Those are the “Rotman Ten”, presented without their references to research.  Many of them are based on a sabbatical I took a few years ago, and much of this is based on extraction from multiple sources.  A few (like blocked and unblocked practice) have an extremely sound historical basis in micro-treatment research.  None of them suggest that the adoption of a particular teaching method will result in general improvements.

Hopefully, you see some wisdom in that “Ten List”, and perhaps some food for thought.

 
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1 Comment

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  By schremmer, December 3, 2016 @ 8:26 pm

  Re: This existence of prior knowledge complicates our design of the learning process., Students assume that their prior knowledge is sound and applies to everything.
  See Adult Resistance to Learning

  Re. The primary determinant of learning is student effort in dealing with the material at the understanding levels (as opposed to memorizing). As Hung-Hsi Wu in the response of the Notices of the American Mathematical Society to Why Do Americans Stink at Math?, Elizabeth Green’s New York Times article, “If Americans do “stink” at math, clearly it is because they find the math in school to be unlearnable. […] For the past four decades or so the mathematics contained in standard textbooks has played havoc with the teaching and learning of school mathematics.”

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