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Archive for January, 2009

Language and Counting

There are two topics that I teach which allow me to discuss counting.  When we cover the order of operations (or what I like to call the hierarchy of computation, since exponentiation is not an operation), I begin by saying that the most basic form of mathemtics is counting.  By repeated counting, we arrive at a consideration of addition.  By repeated addition (despite some vigorous dissent) we arrive at multiplication and so on.

In teaching about the set of complex numbers, I begin by saying that the simplest form of mathematics is counting, so the most basic set of numbers is the set of counting numbers.  But a problem arises if we write an equation using the counting numbers – we can write an equation whose solution is not a counting number.

x+5=1

This then requires an extension of the Counting Numbers to the Integers.

When I discuss counting, I mention that almost all cultures count.  From there, different cultures develop different mathematics depending on their different needs.  Ethnomathematics is a very interesting field.  Researchers in ethnomathematics study not only how different cultures use math, but also how different professions use math.  Professor Tod Shockey at University of Maine is a specialist in ethnomathematics and wrote his dissertation on the use of math in the medical sciences.

I mentioned that ALMOST all cultures count.  There are some cultures that do not have words for numbers above 3 or 4.  Why not?  Probably because they don’t need them.

One culture which seems not to count at all is the Piraha people of the Amazon basin.  Here is a link to a pretty good article on this.  Two researchers have argued about whether or not the Piraha count and whether or not they have words for colors.  One researcher who thought that the Piraha had words for numbers was contradicted by another who said that they just have words for smaller and bigger and that these were being mistaken for words for one and two.

What makes this particularly interesting is that this disagreement between the linguists also arose over whether or not the Piraha have words for colors.  In describing a color, they will say that it is the same color as something in their everyday life.  This seems to get at a fundamental property of the Piraha language – it does not have abstractions.  This is an insight into the way that this culture views the world, just as any language is a snapshot of the collective mind of the culture that uses it.

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Change

At the end of last term, I was working on two ideas that are seemingly not related – population growth and the mathematics of music.  The population equation was for the MTH 111 College Algebra class and the mathematics of music was for a presentation I plan to give sometime this spring.

What caught my attention about these two subjects was that the equations which describe the most important behaviors in each case – the population equation and logistic equation in the case of population and the wave equation in the case of music – start out as differential equations.  That is, they are concerned with change.  Calculus allows us to take an equation describing change and transform it into an equation that tells us about the behavior of the system itself.

Why do the equations for these two situations (and many others) begin as differential equations?

Because – change is generally what scientists and researchers can MEASURE and thus have information about.  Once the change is measured and described, then the other pieces of the puzzle can be put together.

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