Editor’s note: This is the eleventh chapter in a series called “Out on Good Behavior: Teaching Math While Looking Over Your Shoulder” by Barry Garelick, a second-career math teacher in California. He has written articles on math education that have appeared in The Atlantic, Education Next, Education News and AMS Notices. He is also the author of three books on math education. Says Mr. Garelick: “At its completion, this series will be published in book form by John Catt Educational, Ltd.” The previous chapters can be found here:
Chapter 11: More on Making Sense and a Fickle Bookseller
What making sense in math means varies for different people. For Lucy, a lot of the time it was often the monotony of the process. Same for most of the seventh graders I’ve taught, though there are other “nuances” depending on the person and at what level of silliness or seriousness they are operating on any particular day.
My Math 7 class at St. Stevens was a mix of different abilities and personalities. John was an aspiring athlete who had difficulty with math facts and remembering procedures. He worked earnestly and trusted me, but felt that ultimately math wasn’t something he would need. His vision of the future was that he would be a superstar in the sports world and have enough money to hire people to do various chores—math being one of the things.
While Lucy from my algebra class might utter “That doesn’t make sense”, John was more likely to say “That’s a lot of work” when faced with tedious procedures like adding or subtracting large mixed numbers.
He once asked in all seriousness why I assigned so many problems. I asked if there was a particular play in baseball that he had to practice a lot. There was—it was a tricky play that first basemen had to perform automatically and perfectly. “It’s the same thing in math,” I said. “We have to practice certain procedures so we can use them automatically to solve problems.”
Two second pause; then: “But Mr. G., I like baseball.”
My reply was performed automatically and perfectly. “You don’t have to like math; you just have to know how to do it.”
Donna, another student in that same class had a different idea of sense which vacillated between childish whimsy and pubescent whimsy.
Example of childish whimsy: After I explained that letters representing numbers were numbers going by different names, she proclaimed that the number 10 should be called “Jerry”.
Example of pubescent whimsy: I had passed out a worksheet that had on it a problem asking for the area of the shaded portion of the figure below:
Upon seeing the figure, Donna shouted “What the?!” and covered her mouth to stifle a giggle. When I came over to see what was the matter she turned the paper over so the figure would be out of sight. She did not disclose the source of her outburst to anyone in her class, but started to work on the problems.
Looking at the picture a few minutes later, I could see that one could interpret it to be any of two portions of human anatomy, one of which lacked nipples.
A completely different facet of the word “sense” came from my student Jimmy at my previous school. In an earlier chapter I described his penchant for asking questions during a lesson on multiplication of negative numbers. Before I could teach multiplication of negative numbers, however, JUMP Math required covering how to evaluate expressions such as 3-(2-x).
Knowing how to multiply by negative numbers would make this a lot easier. But JUMP decided on a micro-scaffolded approach which in retrospect I would not choose to do again. JUMP’s JUMP Math’s approach was to first look at something like 10-(5-2).
“We know we can do this easily by just doing the subtraction in the parenthesis first,” I said. “So we get 10 – 3 or 7. But suppose I wanted to do it by distribution.”
“Why would we want to do that when we can just subtract what’s in the parentheses?” Jimmy asked.
“Because pretty soon we’re going to evaluate expressions like 3 – (2-x) where we don’t know the value inside the parentheses.”
This quieted him for the moment so I went on. I decided to make up a story to go along with the problem. “Say you visit a book seller and he says to Jimmy, ‘I’m going to give you a special deal. I’m going to reduce the price of this $10 book by $5.’ ”
“Yeah, that would be a good deal,” Jimmy said.
“Yes, it is but then at the last minute he says ‘I changed my mind. I’m only going to take off $3.’ ”
“Wait a minute, he said he was going to take off $5,” Jimmy said.
“Right. So you’re going to pay more aren’t you? Originally you would have paid 10 -5 which is $5. But he reduced the discount by $2. So how much more are you going to pay now?”
Jimmy thought a minute. “Two dollars more.”
“Right,” I said. “If I wrote it now as 10 – (5 – 2), we can see that you end up paying two more dollars than what you would have paid had he not changed his mind. And what you end up paying can be written as 10 – 5 + 2.”
The whole idea being that we evaluated the expression using an intuitive approach, thereby sidestepping multiplication of each number by – 1. As I say, I wasn’t fond of the approach. Jimmy was strangely silent.
“Now, let’s suppose at the last minute the bookseller says, ‘Wait, I changed my mind; I’m going to take off $7.’ Now you’re paying less than you would have if he only took off $5. How much less?”
“Two dollars,” he said with a sigh.
I then summarized it as a rule: the signs of the numbers inside become the opposite. The homework problems were to evaluate various expressions in this manner, including those with variables, like 10 – (5-x).
“It just doesn’t make sense,” Jimmy said.
“What doesn’t?” I asked.
“I don’t understand why he would give less of a discount than he said he would. The guy said he would take $5 off, and then he only takes $3 off. Why would he do this? What sense does that make?”
“The bookseller is a bit strange, I admit,” I said. “But on the other hand, he also took off $2 more than he said he would. So he’s not all bad.”
“I don’t trust him,” he said. “I wouldn’t come back to his store.”
With Jimmy it was hard to tell whether his questioning was serious or a means of wasting time. Either possibility made sense.