Towards a Balanced Approach

When I hear somebody suggest that we take a ‘balanced approach’, my first thought is that the speaker either does not have confidence in their judgments about what is important … or does have confidence but does not want to offend the audience.  The phrase ‘balanced approach’ is often used in reference to a reform model balanced with traditional ideas.

I suggest that we think about the phrase in a new way.  Let’s begin with the assumption that the traditional curriculum has limited value and that the reform curricula have limited value.  What would we build from a blank slate?  How would we use scientific evidence in the process?

A balanced approach looks at implementing two basic properties of human learning:

1. Understanding (connected information) results in more transfer of learning and facilitates long-term retention.
2. Repetition (deliberate practice) results in efficient recall and abilities to apply information.

Some reform curricula emphasize (1) almost to the exclusion of (2).  I have taught courses like this, and talked with my students; few of them have a good report about the experience.  We all have students who approach a math course in that fashion — the students who usually are in class, and do very little ‘homework’ because they understand what they are doing (occasionally they are correct).

As mathematicians, we are drawn to ideas with power — ideas that can represent relationships among quantities, communicate the information, and help reach conclusions about some future state of those quantities.  [We are also drawn to special cases, as well as mathematics that is aesthetically pleasing.]  Our students need the ‘basic ideas with power’ so they can handle the quantitative demands of academic and social situations.  I think we can have fairly strong consensus on the mathematics that most students need.

The balance we need is about pedagogy.  Having a better ‘table of contents’ will not help if students do not learn any mathematics.

I see this issue of balance as our basic problem over the next 5 years.  We know that our courses are going to change in basic ways.  We understand what mathematics is important for all students.  Our issue is to address both the understanding and repetition in the learning process.

Currently, an ‘understanding’ method is based on students dealing with a situation and using guided questions so that they discover the basic idea.  In some cases, this works surprisingly well.  However, discovering an idea has little connection with understanding mathematics.  Here’s an example:  By looking at a set of ordered pairs (bivariate data for a situation), students are led to the idea of slope so that they can predict another value.  This forms the beginning of understanding, not the end: understanding takes extended work with diverse views of the same idea.  Students often over- or under-generalize.  In the case of slope, students think this applies to any set of values … or that it does not matter which ones ‘go on top’ … or that slope is like an ordered pair.  Understanding is a natural human process, but does not happen spontaneously with correctness.

As for ‘repetition’, we seldom get this right.  Textbooks often confuse ‘any sequence of problems’ with ‘repetition for learning’.  Much is known about properties of practice that result in different degrees of learning — a sequence can highlight the most important idea (or hide it), a sequence can reinforce good understanding (or prevent it), and a sequence can reinforce accurate recall (or prevent it).  We somehow make the mistake that good authors can design good assignments; these are vastly different sets of expertise. We also make the mistake that computer systems provide appropriate repetition.

We can (and need to) focus primarily on the big ideas in mathematics; our courses need to match the amount of material with reasonable expectations for students learning with understanding and repetition.  With a balanced approach, we can help students succeed.  With a balanced approach, we can show policy makers that we have the professional skills to solve the problems that they are concerned with.

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2 Comments

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  By Sue Jones, July 15, 2013 @ 11:14 am

  Truth!
  Love this summary…
  I’m not sure that just *saying* these things will convey. Heavens — the table of contents doesn’t do any good if the students don’t learn it ?!? Understanding doesn’t happen with correctness? Computer systems don’t automatically provide appropriate repetition? I’m thinking about the percentage of materials and practices you’ve just stamped as “incorrect” …. but correctly so, adn that’s *before* tossing in the reality that a different sequence to reinforce accurate recall will work for different students…

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  By Sue Jones, July 15, 2013 @ 11:24 am

  (There are so many details in here that are like creeping charlie, infesting our attempts at teaching. *We* are drawn to special cases. When we bring them in before “regular cases” are understood, then *everything* is “special,” as in unrelated and confusing…)

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