I have taught arithmetic for 38 years and am confused using why the analogies I became taught in my children’s course had been misplaced. I consider that scholars often miss the point of an idea if it’s miles taught without an analogy to make it 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, if your motion entails putting on your footwear and socks, the final result is surely one-of-a-kind. If you placed them on your footwear then cowl them together 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 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 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 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 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 characteristic allows two human beings to get off on an identical forestall. Person “-2” claims they were given off at preventing 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:
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 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 So too, 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.