I have taught mathematics for 38 years and am puzzled by why the analogies I was taught throughout my youth have been lost. I believe that students often miss the point of a concept if it is taught without an analogy to clarify.

Here’s an analogy I learned in the 60s which seems to have disappeared.

### Shoes and Socks, Coats and Hats

The concept of **Commutative** is recorded as A+B=B+A, but it is enlightening to see the concept analogously or the point is missed. Consider the action of putting on your hat and your coat versus putting on your shoes and socks. If you put on your hat and then your coat, or vice versa, the outcome is unaffected by this choice.

However, if your action involves putting on your shoes and socks, the
outcome is undoubtedly different. If you put on your shoes then cover them with
your socks, the result is quite different from when you put on your socks and *then*
put on your shoes.

**Commutativity** is about order and whether you can change it under an action without affecting the outcome. So, when the action is not order sensitive, this means it is commutative. Since addition and multiplication enjoy this freedom of order, they are analogous to your hat and coat, whereas subtraction and division are order sensitive as analogous to your shoes and socks.

### The Green Bucket

Next is **associative,** which is about whether changing the placement of the parenthesis affects the outcome. This is about emphasis—who comes first? It is recorded as (A+B)+C=A+(B+C), but is best seen through an analogy.

Consider the following to clarify what it means *not* to be associative:

(**light green**) bucket *versus *light (**green bucket**)

The first says it is a bucket, light green in color, but the second says it is a green bucket that is not heavy. Changing the placement of the parenthesis is about emphasis and whether it effects the interpretation. Addition and multiplication enjoy this freedom whereas subtraction and division do not.

### Train Stops

Another instance I find that analogies can help clarify a concept pertains to functions. Sometimes the mathematical definition of what it means to be a function is not clear to students.

In the discussion of functions, I first explain that it’s a rule (connection) from one set to another. It says that each ‘x’ value goes to only one ‘y’ value.

In the graph below, I refer to the left column as people on a train and the second column as train stops. So, it qualifies as a function if no one claims they got off at two different stops, which is not possible on the same train. Notice that a function allows two people to get off at the same stop.

- -5
**→**5 -5 goes to 5 **-2 → 3 -2 goes to 3 and also to -2**- 2
**→**-6 2 goes to -6 - 3
**→**-4 3 goes to -4 - 5
**→**6 5 goes to 6

Person “-2” claims they got off at stop 3 and stop -2, which is not possible, and is therefore not a function.

The following poem I wrote describes my own philosophy on teaching math:

Formula regurgitation

Explains why this nation

Is mathematically deficient

A child learns what they need

To temporarily succeed

Passing the test seems sufficient.

Somehow mathematics

Moves to the attics

Of many people’s intellects

In boxes separated

By walls corrugated

Soon the dust collects.

However, it is my belief

This achievement is brief

It feeds the mind for a short time

To never accept

Any empty concept

Is the purpose for my rhyme.

When the commutative property

Is seen as A+B=B+A expressly

It is unclear that order is the issue

Like putting on your hat and coat

The order, please note

Is immaterial to you.

Addition enjoys this freedom

So too, in a multiplication kingdom

But some actions have order issues

When considering subtraction

Or the division action

These put on socks and shoes.

When taught to memorize

Math fails to mesmerize

Giving the answer for a day

If shown the connections

With its true directions

One can solve come what may.

*Josephine Johansen has taught for 38 years and has been teaching at Rutgers University for 30 years. Johansen has overseen the developmental courses for the math department. She was also involved in the certification to teach processes for math majors.*