I have taught arithmetic for 38 years and am confused using why the analogies I became taught in the course of my children had been misplaced. I consider that scholars often miss the point of an idea if it’s miles taught without an analogy to make clear.
Here’s an analogy I learned inside 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 factor is neglected. Consider the motion of setting to your hat and your coat as opposed to putting in your shoes and socks. If you placed on your hat after which your coat, or vice versa, the final results are unaffected by way of this choice.
However, in case your motion entails putting to your footwear and socks, the final results is surely one-of-a-kind. If you placed on your footwear then cowl them together with your socks, the result is quite different from when you put on your socks after which put on your shoes.
Commutativity is set to order and whether you may exchange it below an action without affecting the final results. So, when the movement does not order touchy, this indicates it is commutative. Since addition and multiplication revel in this freedom of order, they are analogous to your hat and coat, whereas subtraction and division order touchy as analogous on your shoes and socks.
The Green Bucket
Next is associative, which is about whether changing the placement of the parenthesis impacts the outcome. This is ready emphasis—who comes first? It is recorded as (A+B)+C=A+(B+C), however is excellently seen through an analogy.
Consider the following to make clear what it approach not to be associative:
(light inexperienced) bucket as opposed to light (green bucket)
The first says it’s far a bucket, mild green in color. However, the 2d says it is a green bucket that isn’t heavy. Changing the position of the parenthesis is ready emphasis and whether it affects the translation. Addition and multiplication experience this freedom, whereas subtraction and division do no longer.
Train Stops
Another instance I find that analogies can help clarify an idea relates to features. Sometimes the mathematical definition of what it means to be a feature isn’t always clear to college students.
In the dialogue of features, I first provide an explanation for that it’s a rule (connection) from one set to another. It says that each ‘x’ price goes to handiest one ‘y’ fee.
In the graph underneath, I talk over with the left column as people on a educate and the second column as teach stops. So, it qualifies as a characteristic if nobody claims they were given off at distinctive stops, which is not viable on the identical train. Notice that a characteristic allows two human beings to get off on the identical forestall.

Person “-2” claims they were given off at prevent 3 and prevent -2, which is not possible, and is consequently not a characteristic.
The following poem I wrote describes my very own philosophy on teaching math:
Formula regurgitation
Explains why this kingdom
Is mathematically poor
A toddler learns what they need
To temporarily prevail
Passing the check appears enough.
­­­­
Somehow arithmetic
Moves to the attics
Of many humans’s intellects
In boxes separated
By walls corrugated
Soon the dirt collects.

However, it’s miles my perception
This fulfillment is short
It feeds the mind for a quick time
To never take delivery of
Any empty concept
Is the motive for my rhyme.

When the commutative assets
Is visible as A+B=B+A expressly
It is uncertain that order is the problem
Like putting in your hat and coat
The order, please be aware
Is immaterial to you.

Addition enjoys this freedom
So too, in a multiplication state
But some moves have order problems
When considering subtraction
Or the division action
These placed on socks and footwear.

When taught to memorize
Math fails to mesmerize
Giving the solution for a day
If shown the connections
With its real instructions
One can resolve come what may additionally.

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