Context in mathematics

Ever since reading about ethnomathematics I’ve been wondering how to identify what is western about western mathematics.  Moreover, if we view culture more abstractly as groups of individuals connected by traditions, beliefs, practices etc then 19th century woodworkers could have a ‘mathematics’ just as well as the Polynesian tribe of the Malekula.  This idea of each culture (or community of practice, to use another common term) having it’s own mathematics is a fairly powerful one.  Do physicists use math sloppily or do mathematicians use math too carefully?  Viewing mathematics as a cultural practice, they each have their own mathematical practice.  It’s not that one is better than the other, they’re different.

Isn’t there a right way and a wrong way?  Are facts facts and that’s that?  I admit to being in a crisis of sorts over such questions.  Even within the culture of western mathematical practice there are many different logical foundations. Which should you use?  The consequences of your choice are severe in terms of what you can prove.  Why do we make the choices that we do (in western foundations)?

Viewing mathematics as an inherently human activity but also a cultural practice can be an asset for a teacher like myself trying to understand my audience.  They are not in my culture, by that I mean the culture of a western mathematician.  In fact, I’m only peripheral to that culture as I don’t (and haven’t) conducted an original research (though I have engaged in a caricature of it when playing with mathematical ideas).  The way in which I think about mathematical ideas, mathematical truth, context, explanation and evidence are all tied to my cultural practice of mathematics.

What are these cultural practices?  There’s so much to write about (and many have written far better about it and I’ll discuss their contributions eventually).  For this post, just to get it off my chest, I want to address context.  Western math tends to strip context from the mathematics.  This can be strength certainly.  We can weave disparate ideas together in a uniform tapestry of knowledge that can be applied to seemingly anything.  This is done despite the varied origins of those ideas.

While powerful as a tool for abstraction I think we forget how important context is in our everyday lives and how important it might be to the mathematical practice of our students.  For example, I could show my students a system of three linear differential equations.  We could analyze the coefficient matrix and find that there are two complex eigenvalues and a real eigenvalue of zero.  I could talk about what those mean (spirals etc) and write down the general solution and then, given an initial condition, a particular solution.  Or I could say, suppose there has been a wildfire and there are now ten acres of bare ground.  In this region, the first plant to grow in such conditions is a certain grass.  The rate of reclamation is 30% per year.  The grass dies off at 5% per year and yields to a second species, shrubs, at 20% per year.  The shrubs then die off at 15% per year.  What happens?  What’s the region look like in 10 years.

As far as the math content goes there’s nothing new here.  But we know some people will be more interested in this problem now because it relates to something they care about.  We could restate this problem in several other ways as well.  Each context would not change any of the essential mathematical content but it would change the mathematical context.  I think our students need and want that context.  As an anecdote last semester while covering logarithms I became frustrated by the usual precalculus content.  Logarithms are presented as a weird afterthought, a consequence of the exponential function passing the vertical line test or as some need to solve seemingly arbitrary exponential equations.  I took a small detour and talked about log tables and how those worked.  I showed my students a picture of a room full of computers pre WW2.  That was a room full of people of course.  I could hear the surprise.  I mentioned that such engineering marvels as the Hoover Dam were built with those types of  computers, people, using mechanical adding machines and tables of logarithms.  You could hear the difference in the room.  A small number of people came down afterwards and we talked about the pre-analog/digital computer age.  It was only a few, but the context woke them up.  They were interested in the fact that logs, log tables and incredible structures had been built with them. Of course that’s just the tip of the iceberg, historically, with regards to logarithms.

Context is important.  This morning as I came into school it occurred to me that this might be like the analysis of a joke.  I’m sure someone has categorized joke structures and form.  I imagine there’s a rich theory of humor which describes why certain ideas work and others don’t.  As powerful as that method might be, you would still need to add context to that joke form or joke structure to make it funny.  Western mathematics removes context and places value on abstraction.  With that value it has made important discoveries and contributions.  However, when we share that math back to our students, just like when we tell a joke, we need add that context back in for it to make sense, for it to mean something.


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One Response to Context in mathematics

  1. To provide a provocative analogy I might compare the removal of context in mathematics to the reduction of love making to masturbation. It can be done and it’s convenient. It reduces potentially distracting information and confusing terms. However it can be rather artificial and limiting. Not a bad exercise, a simplification to a more complicated phenomenon. However, most of us are willing to engage in the deep, rich and often confusing context of love making. The acquisition of information about the other person’s states and reactions to stimuli. The unknown parameters, bifurcations and equilibria that exist. The lack of confidence, at least at first, in long range forecasting. Context can make things cumbersome, difficult, confusing, and frustrating. However, in this case, it might be worth the trouble. In mathematics education I think it’s also worth the trouble.

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