Saturday, August 16, 2014

Conversations on the Rifle Range 6: Grant’s Tomb and the Benefits of Boredom

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 six:

The block schedule alternated even and odd period classes every other day. My classes were all even period which meant I taught every other day. Between my fourth and sixth period classes came the lunch break during which, in nervous anticipation of my sixth period class, I would walk from one end of the high school complex to the other.

On the day I was teaching about powers and roots I thought about the upcoming lesson during my walk. An ongoing difficulty in teaching this topic is explaining what happens when you square a square root, cube a cube root and so on. How do I get students to make the connection that (Ö2)² equals 2? I recalled how I tried to get my daughter to understand this when she was taking algebra. She stared blankly at the problem. “When you square something what are you doing?” “Multiplying a number by itself,” she said.

“Right, so we can write it like this, right?” I wrote down Ö2 ·Ö2.

She still stared blankly.

“Come on, it’s like asking who’s buried in Grant’s Tomb,” I said.

Her response: “Who’s Grant?”

Despite this setback, I still decided to go into teaching after I retired.

What I learned from teaching my daughter years ago is that students need to know the connection between a square root and the square of a number. But as I also learned from teaching my fourth period class earlier in the day, the definition of “square root” should not be the one in the Holt Algebra textbook: "A number multiplied by itself to form a product is the square root of that product." The difficulty is not only the convoluted wording, but the book introduces and defines perfect squares after the definition of square roots. Definitions should be introduced in an order that makes it possible for students to use “prior knowledge”. Not that they hadn’t learned about these things before, but still. Matt, the ocarina player, relied on a purely procedural approach: “Oh, when you square a square root, the radical sign disappears.”

While I have no problem with procedural understanding, I thought I could do better with sixth period. I would lead with what perfect squares are, and then give my own definition of square roots: "A square root of a given number is another number whose square is the given number." When I returned from my lunchtime walk, I started writing down my order of attack.

As was her custom, Elisa came into the classroom ten minutes before it started and asked for a piece of computer paper. I gave her a piece which she took to her seat and quietly drew one of her many pictures of dogs and wolves.

I jotted down a few more notes until the class started to file in. Patrick sauntered up to me and complained about the homework assignment sheet that I handed out at the beginning of the semester. “My mother says it’s hard to figure out,” he told me. “Do you know what assignment is due today?” I asked. He pointed to it. “Good; not so hard to figure out. Now show me your homework so I can check it in.”

“I didn’t do it,” he said and snickered.

Patrick sat next to Elisa and liked spending his time making snide remarks about my attempts to teach the class. Although Elisa giggled at his remarks she told me once after class that Patrick was a wise ass. She was probably one out of five or six students who paid attention in class. When I did my teaching, I taught to them.

The class was boisterous as usual but after homework check-in and other rituals, they were sufficiently less noisy so that I could begin the lesson. I had defined perfect squares, and square roots, gave some examples and then asked: “Who can tell me what this is equal to?” I wrote (Ö2)².

A few guesses, all of them wrong.

“Let me try this,” I said and wrote  (Ö16)². “What is the square root of 16?” I asked.

Someone said “Four”, and I wrote (4)² and asked Patrick what that equals.

“Sixteen”, he said and whispered something to Elisa who giggled. “Correct,” I said and wrote (Ö16)² = (4)² = 16 on the board. I did this a few more times with other numbers: (Ö9)²,(Ö4)², (Ö36)², (Ö25)², hoping that they would see the emerging pattern. When I started hearing “Ohhh, I get it” I put up the original problem again:. (Ö2)². 

Elisa gave the answer: “Two,” she said.

“Absolutely right,” I said. I went into cubes, cube roots, and higher powers but the amount of paper wads being thrown (an ongoing problem in that class) was increasing. Roots and powers were no match for their unrest so I set them to work on their homework. I allowed them to get in groups of their choosing. Some worked on their homework, many did not. Elisa was one of the students who worked on her homework; she sat by herself. She summoned me over to her desk. She was stuck on finding the cube root of eight.

“You know how exponents work, right?” I asked. She said “I think so.”

“OK, if I write 33, what is that?”

“It’s three times three times three.”

“Good. Which is what?”

She thought a minute. “Twenty seven!”

“Right! So now, to find the cube root of 8 we work backwards: we want to know what number cubed equals 8.”

“What do you mean ‘work backwards’ ?” she asked. I explained how we're reversing what we do with exponents and suddenly the light went on. I asked her to try some numbers to see if we could find the cube root of eight. "Obviously, it's not 1, so try the next one up." She multiplied two by itself three times and got eight. We tried another: The fourth root of eighty-one. This took a little trial and error but she got it. “Oh. Three! Yeah, I get it now,” she said.

After a moment she asked: "Did mathematicians invent this stuff because they're really bored?"

I burst out laughing, but gave her a serious answer. I said these things were invented to solve certain types of problems and sometimes out of curiosity.

“I think they were just bored," she said.

“And I think you probably like math more than you realize,” I said, knowing I had nothing whatsoever to base this on other than a wishful hunch.

“I think you’re crazy,” she said, and I moved on to others on the rifle range.


R. Craigen said...

Concerning squaring and "rooting" as inverse functions, it is critical that this sort of thing be familiar to students in some form before they begin calculus because much in that course hinges upon elaborations upon that idea, or complex instances of it.

One needn't have a sophisticated understanding of inverse functions, but an operational comfort with them. Like that of the student Matt here, who thinks about the radical sign disappearing when you square.

That's about the level I need calculus students to understand things when they enter my class. What is a log function? I want them to understand that ln (e^x) = x and e^(ln x) = x when defined. From here we can work another useful version: y = ln x is equivalent to x = e^y, and to get comfortable switching between the two for convenience. For all of this it is best that they arrive in my class with a notion of inverse functions already intact. Unfortunately, few do, so I have to go to first principles with them -- and it's just too much to take in all at once. Things should be done in order, and students need time to master basics before moving up a rung on the ladder of abstraction.

Anonymous said...

"A number multiplied by itself to form a product is the square root of that product" [my emphasis]

I point out here that, in addition to the shortcomings articulately pointed out by the author, there is a serious mathematical deficiency in this so-called "definition". Namely, it is not well-defined. At least not well-defined for any student who knows of negative numbers. Mathematicians are precise with language, and the use of the word "the" here implies uniqueness in the definition. But by this definition, 4 would have two distinct square roots, namely 2 and -2. This is why we mean two different things when we write (or say) "a square root" (the definition given by the textbook) vs. "the square root" (some specified choice among all square roots, usually taken by convention to be non-negative and called the "principal square root"). This might seem very pedantic, but it's not. It can cause immense confusion when dealing with the quadratic formula or complex numbers later, and the central issue (finding a suitable 1-1 restriction of a non- 1-1 function) is important in many areas of higher math. Shame on you, textbook authors!