Tuesday, December 20, 2016

Order of operations (a mini-rant)

Order of operations is a major pet peeve of mine for two reasons:
(1) Some people love to use it in "gotcha" challenges to make other people feel bad about their math abilities. Here are some examples: facebook meme, a similar one, and this:

(2) For some students, it stands as a clear example of the idea that math is a set of arbitrary rules they have to accept and/or memorize (It isn't!)

I think there is a very different way of approaching this issue which is much more mathematically rich and fun.

1a. Multiple answers
The first and most obvious is to treat these memes as games and see how many answers you can justify by making the order of calculation explicit (use parentheses). Implicitly, this is the idea behind games like 24 or the traditional New Year's challenge (use the digits of the new year to make all values from 1 to 100).

2. Talk about history
Some launching questions: Where did the order of operations come from? Is it universally agreed?
This article from Tara Haelle does a good job of talking about this perspective and gives some further references.

3. Ask students for their own thinking
What do they think the order of operations should be? Why?
A tantalizing question: should there be an agreed (implicit) order of operations at all?

4. Talk about redundancy and error flagging (and error correcting) codes
Jordan Ellenberg's How Not to Be Wrong has a great discussion about redundancy in language and related code concepts. One key point is that we always face a trade-off between brevity and transmission errors. In other words, we can write short messages where every character carries critical, independent information, or we can use a system in which our messages are longer, but carry duplication and internal references that make our meaning more robustly clear. (compare the previous two sentences!)

Relying on a convention, like the order of operations, means that we can use fewer symbols to convey a mathematical expression. The great danger comes if the author and the reader don't share the same conventions! A less obvious danger is if a symbol gets garbled in the transmission, it may be hard to identify the error or even see that there was an error.