*Barry Garelick, who wrote various letters under the name Huck Finn, 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 19:*

In all my classes, I required my students to answer warm-up questions at the beginning of class. I used two types of questions:. One was a review-type question to apply what they recently learned. The other required them to some apply their prior knowledge –or what was familiar—in a new or unfamiliar situation. Some may view this as an inquiry-based approach, or an application of the “struggle is good” philosophy that adherents of Common Core seem to say is necessary to develop perseverance in problem solving, as well as the all-important and frequently undefined “grit”. I view a short amount of struggle as appropriate provided that explanation is provided shortly after. That way, even if students do not succeed in solving a problem, most are receptive to explanations that they might otherwise tune out.

In my Algebra 1 class, one of my problems was: “Simplify (√5)

^{2}” , which had stumped my 2-year Algebra 1 class at the high school earlier that school year. As was usual, students asked me how to solve it. I advised them to solve (√4)

^{2}to see if that helped. After some thought they realized the answer to the original problem is 5.

"Yes", I said "It's like the question 'Who's buried in Grant's Tomb?' " (I thought I’d try that one out again.) At least this time someone got the joke. Luanne, my star student in my 5th period class shouted "OH! It's Grant!" The class started buzzing about the joke although one boy said he thought it was Fred. In the interest of time, I didn’t pursue his reasoning.

I wanted them to be familiar with the “Grant’s Tomb” example of radicals because the lesson that day was on fractional exponents. We had been covering the rules of exponents, and negative and zero exponents. I wrote 5

^{1/2}on the board and asked if anyone knew what that was. They didn’t, though that didn’t stop them from trying to guess. “Stop shouting!” I said. “If you have something to say, raise your hand!” They did, and I got an array of guesses: “Two and a half”, “Ten”, “Grant!”

In proceeding with the lesson, I referred to a poster I had on the wall from Alice in Wonderland in which Alice is talking with the Cheshire Cat. The dialogue is as follows:

“Would you tell me, please, which way I ought to go from here?”

“That depends a good deal on where you want to get to,” said the Cat.

“I don’t much care where—” said Alice. “

Then it doesn’t matter which way you go,” said the Cat.

“—so long as I get somewhere,” Alice added as an explanation.

“In math, sometimes we’re like Alice in the poster on the wall,” I said. “We don’t know where we’re going or where it will lead. So let’s do that with 5“Oh, you’re sure to do that,” said the Cat, “if you only walk long enough.”

^{1/2}and see where it leads. If we have (5

^{3})

^{2}, then what do we do with the exponents?”

The usual shouting occurred, with a variety of answers and I heard “multiply”. “Right, you multiply them,” I said. “So even though we don’t know what 5

^{1/2}is yet, let’s see what happens if we raise it to the second power, like this.” I wrote 〖〖(5〗

^{1/2})]

^{2}on the board. “Now if we follow the rule for raising a power to a power, we get 2 times 1/2 which is 1, so we have 5

^{1}, otherwise known as 5. Now, switching from Alice in Wonderland to Grant’s Tomb, does this suggest something to you?” More shouting. “Raise your hands, don’t shout!” I shouted. Luanne came through. “You square the square root of five,” she said.

I went on about unit fractional exponents, and one boy picked up on something right away. "So if we have 5

^{1/4}, if you raise it to the fourth power, will it be 5?"

"Yes!" I said., I felt victorious and invulnerable. In the next few days, we covered fractional exponents in the form a

^{m/n}). A few more topics came after that, and then there was the chapter test.

Despite their understanding of exponents, both my Algebra 1 class’s performance on the chapter test was not as good as I thought it would be. When I passed back the test, students asked what the class average was, as they always do. It was 79. “Oh, that isn’t good,” said Luanne (who got a score in the high 90’s). “Yeah, we usually get like an 89 class average,” another student said.

The assistant principal was sitting in the back on one of his occasional visits to my class. “It’s bad, isn’t it?” Luanne asked the assistant principal.

“That’s not a bad average—anything above 70 is a fair showing,” he said. But the students they were disappointed, as was I. A few days later, I received an email from the mother of a boy named Brian:

"Hi! Just wanted to let you know I am concerned about my son. He was receiving an A from Mrs. Halloran but now has a C and received a D on the last test. He is continuing with the same amount of effort but is telling me he thinks he understands the material but this is obviously not the case. I am concerned that there is a disconnect at the teaching/comprehension level. Can you please give me some feedback? Thanks!"

As disarming as the “Hi” was at the beginning, I knew she thought it was my fault—the giveaway was that she had copied the principal.

I wrote back that the material this semester is more complex than in the first semester, so that may be contributing to the problem. I also offered to help him after school if he wanted to come. And I copied the principal.

Brian tended to be quite noisy and disrespectful. He was one of a number of students who tried to get a jump on the homework by doing it while I was still giving the lesson. I mentioned none of this to his mother. I never heard back from her. I asked Brian if he wanted help after school. He said he had sports and couldn’t do it.

I didn’t quite know what to do next regarding my teaching approach or whether I needed to do anything. After asking myself the eternal question—“What the hell do I do now?”—I knew there had to be some things to try, taking the Cheshire Cat’s advice that even if I didn’t know where I was going, I was bound to get

*somewhere*. What those things were, I hadn't the slightest idea.

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