COMMENTARY: What was wrong with old math?

Re: “Let teachers teach” (@Issue, Nov. 29)

I read with interest the interview with Jack Curtis, a middle school principal, relative to the testing in New Jersey schools. I am continually perplexed by methods used to teach in comparison to the insights I was taught in the 1960s, particularly related to mathematics.

Here’s an example: 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.

If you put on your hat then put on your coat or put on your coat and then put on your hat, the outcome is unaffected by this choice. However, if your action involves putting on your shoes and socks, then the outcome is definitely 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, then your shoes.

So 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. Order-sensitive says “not 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; they are analogous to your shoes and socks.

So put to work, this says that if the problem is 683 x 14 vs. 14 x 683:

If you have 14 x 683, you can reverse it and calculate 683 x 14 instead.

Next is associative, which is the grouping issue for sure, but the change in the placement of the parenthesis is about emphasis. 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 to be not associative: (light green) bucket vs. light (green bucket). The first says it is a bucket that is a light shade of the color green but the second says it is a green bucket that is not heavy to carry.

Changing the placement of the parenthesis is about emphasis and whether it effects the interpretation.(This is, the outcome.)

So putting these to work shows that you can group efficiently when adding.

For example, with 48 + 29 + 52 + 11, see that 8 + 2 and 9 + 1 are 10, so the first column adds to 20 so carry the “2” says you carry two 10s. Then in the second column, 5 + 4 + 1 is 10 as well, so the sum of the second columns (tens) is 14.

I was taught these insights in fourth grade, in 1961, and therefore always understood that under addition and multiplication I could change the order without affecting the outcome. These concepts were readdressed in algebra I, and I already knew where it applied and why.

I have successfully taught mathematics for 36 years and continue to use these insights that enlighten students and I am surprised by the fact that most students have never heard these interpretations. My question continues to be: Why did the methods used in the ’60s disappear?

Thanks for an informative article. I can only hope that these insights are once again taught in every mathematics class.

Jo Johansen has taught math at Rutgers University in Camden since 1989. She previously taught at Camden High School and Camden County College. She lives in Voorhees.