I have taught arithmetic for 38 years and am confused about why the analogies I learned in my children’s course had been misplaced. Scholars often miss the point of an idea if it’s miles trained without an analogy to make it clear. Here’s an analogy I learned in the ’60s, which seems to have disappeared.
Shoes and Socks, Coats and Hats
Commutative is recorded as A+B=B+A, but it is enlightening to see the idea analogously, or the factor is neglected. Consider setting your hat and your coat instead of putting in your shoes and socks if you place on your hat, after which your jacket, or vice versa, the final results are unaffected by this choice. However, if your motion entails putting on your footwear and socks, the final result is surely one-of-a-kind. If you place them on your footwear and then cowl them with your socks, the result is quite different from when you put on your socks, after which you put on your shoes.
Commutativity is set to order, and you may exchange it below an action without affecting the final results. So, when the movement does not call 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 equal to 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 the ready emphasis—who comes first? It is recorded as (A+B)+C=A+(B+C). However, it is excellently seen through an analogy.
Consider the following to make clear what it approaches 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 the 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 explain that it’s a rule (connection) from one set to another. It says that each ‘x’s price goes to the handiest one ‘y’ fee.
I talk over with the left column as people on a educate and the second column as teaching stops in the graph underneath. 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 feature allows two humans to escape an identical forestall. Person “-2” claims they were given off at preventing three 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:
However, it’s beyond 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 Are visible as A+B=B+A expressly, It is uncertain that order is the problem, Like putting in your hat and coats; the order, please be aware, Is immaterial to you. Addition enjoys this freedom to,o, in a multiplication state, But some moves have order problems When considering subtraction Or the division action. These are 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.