Wednesday, September 3, 2014

Conversations on the Rifle Range 8. Repeating Decimals in Post-Cold War America

Barry Garelick, who wrote various letters under the name Huck Finn and which were published here is at work writing what will become "Conversations on the Rifle Range". This will be a documentation of his experiences teaching math as a long-term substitute. OILF proudly presents episode number eight:

I saw Gray the other day at the coffee house I frequent in the small town where I live. He was with someone who at my age looked to be an attractive young woman. At the same time, she looked a bit old for him—old enough to be his mother, I thought. I realized that in fact this probably was his mother who was taking him to school. I’ve made that same mistake whenever I teach in a high school; young teachers look like students sometimes.

Gray had grown quite a bit. He was well mannered and was turning into a nice young man. I asked him how he did in Algebra 1 last year. He said he passed—barely—but he got through it.

“Who do you have for geometry?”

“Mr. Lake,” he said.

“Oh he’s really good,” I said.

“Yeah, he’s awesome,” Gray said and we went our separate ways.

I was glad Gray was in Lake’s class; he would probably do well with him. Lake was a young teacher who looked to be 18 years old at times. His classroom was next door to mine and I would often confer with him at the end of the day to get advice on how to handle my sixth period class. In fact, it was at Lake’s urging that I decided to go ahead and do one of the “formative assessment” projects that Sally wanted us all to do. I would have gladly skipped the project, except that in the six week period of my assignment, I had 13 sessions, and was given strict instructions by the teacher for whom I was subbing to not go beyond the chapter she wanted covered. No matter how I structured the lessons, there was one period left over to be filled with something—and that something, I decided, had to be the dreaded group project. For my 4th and 6th periods, this would have to be the project on repeating decimals. Those who are curious may view it here.

It essentially covers the conversion of fractions to decimals, and vice versa. I had learned the “vice versa” (decimal-to-fraction) procedure in algebra 2. This was in October of 1964, an era when such things were taught later rather than earlier. At that time, the 60’s New Math that had its genesis in Sputnik was still alive and well, and the space race was well on its way. Lyndon B. Johnson was running against Barry Goldwater for president. Nikita Khruschev had been removed from power in the USSR and replaced with Leonid Brezhnev. The general fear in the nation (as I perceived it) was that if Goldwater got in, we would be in a nuclear war. The purpose of math in general and algebra in particular (also as I perceived it) was rarely questioned.

Now, in the classes I taught, particularly sixth period, aside from about five or six students, most appeared to be out in never-never land, thinking about being the hero of their rather sad universe. Nevertheless, I tried to prep them for the activity. A few days prior, I asked my students to divide 1 by 3. There were maybe two students who knew how to divide well enough—which necessitated knowing multiplication facts—to do this. I then tried to show how to convert 0.333… to a fraction. I don’t think anyone followed it; at least not very well. The procedure for 0.333… is as follows:

Set x = 0.333….

Then 10x = 3.333…

Calculate 10x- x = 9x = 3.333… - 0.333… = 3.0

Solving for x, x = 3/9 = 1/3.

Other repeating decimals take a bit more ingenuity, like 1.0303… which requires multiplying by 100 so that you end up with 100x = 103.03… and x = 1.0303… The end result is 99x = 102 and x = 34/33. The students in both fourth and sixth period could do the last step; i.e., divide both sides of the equation. Other than that, however, they generally did not follow the general procedure.

When the day came for the activity, I split the classes into groups of about four students. Each group was given three sets of cards. One set of cards consisted of various fractions; the second various repeating decimals, and the third various equations such as 9x = 3, or 99x = 102. And in all the groups there were some blanks. The object was to match up the cards and paste them on a piece of poster paper. Thus, the fraction 34/33 is matched up with 1.0303… and with the equation 99x = 102. In some cases, two out of three pieces of information was given, and the third piece (fraction, decimal or equation) had to be supplied.

In light of all this I bought some calculators so each group would have one (the school didn’t have a supply I could use). I bought rolls of tape to attach the cards to the paper since I suspected that if I used glue sticks the sixth period students would throw them at each other. I gave instructions, plus worked through an example.

I was delighted to see that both my fourth and sixth periods got into it. The game aspect of the activity intrigued them and even Patrick made an effort to match up the cards. “How do I figure out what 1/6 is as a decimal?” I showed him again how to figure it out on the calculator. “And how do I figure out which equation goes with it?” he asked. We looked through the equations. I had him solve for x and then convert the fraction obtained to a decimal. For 1/6, the equation was 9x = 1.5.

If a repeating decimal had no matching equation—requiring the student to derive it—I tried to walk him through it. But, like the previous day, Patrick didn’t follow, nor did most other students. I told them to leave those blank. For students like Elisa who showed a keen enough interest, they followed enough that they could do one—with help. No serious disruptions occurred in sixth period. Some students made little loops out of the tape and threw them at each other, which told me my decision to not use glue sticks was a good one. I saw Mr. Lake before I left that day. I told him it went well. “Glad to hear it,” he said.

“Maybe six or seven students actually learned something,” I said.

He looked confused. I felt he needed to hear something else. “In retrospect, I think I should have spent the time teaching division and converting fractions to decimals,” I said. I was going to add that it was coincidental that the Berlin wall came down the same year that the National Council of Teachers of Mathematics (NCTM) came out with their math standards, but I’m not even sure what that means.

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