Meaningful Mathematics: That is Worth …

A few years back, my dean informed me that returning adult students wanted to know how their learning would be applied to their lives as opposed to understanding theory.  I was quite surprised by this statement, given all I’ve learned over the years; I had reached the conclusion that the more ‘seasoned’ students wanted to understand the why as well as the how and when.  This cognitive dissonance resulted in a non-discussion as the Dean would not believe my statements.  Of course, these generalizations (hers and mine) are seldom true over a broad range of situations.

This idea of ‘application’ and meaning continues to be a hot-button for me.  Somehow, academia has accepted the strange notion that learning needs to be justified by seeing how knowledge is applied to individual lives.  In the guided pathways movement, mathematics is specifically designated for ‘alignment’ with the student’s program of study.  I’ve written a bit about that; see GPS Part III: Guided Pathways to Success … Informed Choices and Equity and other posts.

There is a need for balance here, as in most things.  Traditionally, college mathematics courses were theory-driven gauntlets designed to ensure that only the fit students reached the point of seeing how mathematics is applied to significant problems and processes.  No ‘meaning’ is permitted until the student has survived entry into calculus with some sanity, and then meaning is only explored within a limited range of classic problems (‘maximize the area of …’).  The absence of meaningful uses of mathematics is only part of the problem with traditional courses.

At the other extreme are some modern courses in quantitative reasoning or statistics.  A note came out this week from Carnegie Math Pathways (the folks doing Statway™ and Quantway™) about how great it was that students could see how to apply the mathematics in their lives (“for the first time …”).  Some of my colleagues emphasize finance work in our QR course for similar reasons.    Yes, that adds some ‘fun’ to the course, and helps with motivation for some students.

A few years back I did an invited talk at a state meeting dealing with general education mathematics.  The talk was apparently well received for the wrong reasons — members of the audience thought I was advocating for a focus on applications and context that students could understand.  I left that meeting dismayed.

Why the dismay?

• Are we only able to motivate student engagement and learning in mathematics if we can convince them with immediate applications?
• Is the value of learning mathematics constrained by specific utilitarian advantages of a constrained set of content?
• Are we so unskilled in teaching mathematics that we see a need to focus on context instead of understanding mathematics?

Some readers might see these statements as disparaging inquiry based learning (and ‘problem based learning’).  My concerns with those pedagogical approaches centers on the balance issue.  As a matter of learning and cognition, context is the classic double-edged sword — yes, context can provide an initial anchor for learning and supports motivation.  However, context also tends to constrain the learning making it difficult for students to transfer their knowledge.

At the heart of my concern is this:

If we focus on utility of mathematics, how are we to inspire the next generation of mathematicians?  Is that inspiration going to be limited to applied mathematics?

For years, I have been saying that every math course should engage students with “useless and beautiful mathematics”.  We should show students how we became inspired to be mathematicians; for most of us, this inspiration combines theory and application — not ‘or’.

Let’s keep mathematics in every math class.

2 Comments

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  By Susan Jones, November 25, 2019 @ 7:12 pm

  Do people do puzzles only if they are “relevant”? No. People like being able to figure things out. Oh, and practice working with the language of math before actually buildnig stuff is just normal learning progression. Skipping that part is one of the more pernicious filters — only people who have other resources to get the missing learning steps succeed.

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  By Jack Rotman, November 26, 2019 @ 2:53 pm

  You raise a good issue … and a complicated one … about when to have learners ‘build stuff’ in mathematics. I think “balance” is a good guideline here, since beginning too early can cause the type of filtering you mention and delaying too long can limit opportunities that can make a critical difference for some students.

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