Math and Alternate Representations

Since linguistics invokes mathematical formalism (i.e. phrase trees, feature bundles, rules or tableauz, etc), I am interested in some aspects of how math is taught.

One question that comes up a lot is why is it important for all students to learn algebra or trigonometry if only a small minority will ever use these tools in daily life. The standard answer is that algebra teaches you “mathematical thinking,” but I’m pretty sure most students (especially those who hate math) miss the point.  Actually, I would say that if you want to learn “deductive” skills, you’re better off taking formal logic or rhetoric.

However, there is one aspect of algebra that is important in real, but rarely pointed out and that’s its ability to provide multiple respresentations for “the same thing”. For instance the concept of “1” can be represented as “1”, 4/4 (four-fourths), x0, |i2| and my personal favorite – .999999… And believe me I haven’t even touched the tip of the iceburg. Although these formulations all represent the same quantity, they do not quite the same meaning.

You normally use “1” in real life, but if you’re working on a weird property issue where an piece of lanf is divided into quarters maybe the formulation “4/4” would have meaning. Or maybe you have a formula which you raise x to a certain power – whatever it is. It’s just that when it’s zero, the result is 1.

My point isn’t just that the “same” item can have multiple
representations but that the different representations can be selected
to help you focus in a different aspect. To borrow a concept from
Semantics class, the meaning of something is partly fixed by your
context – but you have to know EXACTLY what your context is.

The use of multiple representations does extend beyond algebra (and I don’t just mean linguistics either). For instance, there are lots of places around the world which have multiple place names, and sometimes you select one based on what era you are studying.

For instance modern historians may study be studying Turkey“, but historians from the 14th-early 20th century may be studying the heartland of the “Ottoman Empire” while those who specialize in the Bronze Age probably study “Anatolia” and Roman historians are probably studying “Asia Minor.” It’s roughly the same place, but the different names not only establish the time context, but can be used fudge minor details like changing political borders.

You don’t want to start calling modern Turkey “Anatolia”, but the use of the term “Anatolia” is useful for referencing the set of Bronze Age cultures in the region (none of which are now related to the modern Turkish culture in terms of language or religion)…so you don’t usually call ancient Anatolia “Ancient Turkey” either (unless you’re writing a tourist brochure). And no matter what – you never want to confuse Turkey with Turkestan (not cool).

This kind of mathematical thinking isn’t about accepting one “right answer,” but systematically determining what the possible answers are and when to deploy them while understanding that some answers are just plain wrong!

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