What does bad notation buy you?

As I was teaching my geometry class on Friday, I went into the need for good notation and labeling as ways to understand and begin to problem solve. Labeling items helps to identify different characteristics and enables our brains to start to begin absorbing pertinent facts. Hopefully, the brain makes connections to similar situations or relates one idea to another to begin the problem solving process.

Notation is an often overlooked weapon in the problem solving arsenal. We teach lots of notation to be memorized but too often overlook how important good notation is to solving problems and making math useful.

I have a couple of examples that I can think of where notation actually made mathematics more useful and expanded its role in the world. The first being the use of Hindu-Arabic numerals and the use of 0 in writing numbers.

The Romans had an expansive empire that lasted for centuries but did little in the advancement of mathematics. Yes, they were wonderful engineers and ruthless conquerors but they did little new in the way of mathematics. One reason that has been hypothesized is that Roman numerals hinder mathematics.

Let's try this. Don't convert these values, try to use them as a Roman would. What is the answer to the following addition problem?

XCIV + LVI = ?

It's not the easiest problem to work out, is it?

Now, what about this problem?

94 + 66 = ?

There is quite a difference in working through these problems. One yields and answer of CLX, the other answer of 160. Oh, wait, they are the same value but the connection between the problem and result are not as clear when adding with Roman numerals. 

The use and spread of Hindu-Arabic numerals led to a rapid expansion of mathematics in the western world.

Flash ahead to the 1600's and early 1700's in Europe. The calculus wars raged during this period and while most of Europe adopted Leibniz's notation, England stuck with Newton's. The weaker notation that Newton developed slowed mathematical progress in England while mathematics across Europe flourished, expanding mathematics influence across many fields.

The next time you work with notation or have students work and understand notation, think about whether the notation is flexible and useful or cumbersome.

Mathematics should help students to easily model situations and to make their lives easier not more difficult. Good notation can help in this effort.