Barry Garelick - Chapter 4

Mr. Garelick informs me that what with teaching duties coming up for the new year, he has to leave his writing aside.  He left us with this as a memory of where he was as he now approaches his new assignment at a Catholic school, teaching middle school math.

He thanks you all for reading.

4.  Haunted by Waters, Procedures vs Understanding, and the Company of Eighth Graders

During my second year at the school, my classes were later in the day. This meant that when Indian Summer was active, my classroom was a hothouse. In addition to a fan at the front of the room, I thought to bring in several cartons of bottled water.  I started the ritual of passing out the waters during my eighth grade algebra class which was held after lunch during fifth period when temperatures were nearing their peak.

This became a daily event, with students asking “Are we going to get waters today?”

I would distribute the waters and got into the habit of saying “I am haunted by waters” and then inform them that this was the last line of “A River Runs Through It”, a book by Norman Maclean which was made into a movie.  Only one student had heard of the movie.

The passing of the waters became a ritual that extended beyond the heat of Indian Summer on into winter, and eventually included my fourth period class of seventh graders.  My utterance of “I am haunted by waters” was confined only to my algebra class, but I continued saying it throughout the year.  The phrase was always greeted by “River Runs Through It” as if we were practicing some procedure in math.

The students in my algebra class had been in my seventh grade accelerated math class the previous year. As a result, I knew them quite well—or thought I did. I’m new at teaching so I’m still getting  used to adjusting to the changes between seventh and eighth graders that occur during the summer. Eighth graders are bigger and seem more knowing about life, as if the summer let them in on some eternal secret.

This is not to say that they were completely mature and serious. Connor, a very bright student who had a penchant for singing in class, making puns and playing with words, is a prime example.  I chose to call my classroom “The Batcave” my second year, in honor of Adam West who played Batman in the 60’s TV series, and who had passed away the previous June.  I had various bat paraphernalia in the room: Batman coffee cups filled with pencils, a serving tray with the Bat insignia on it, and eventually I acquired a trick-or-treat container in the shape of Batman’s head.  As the class was settling in on the first day, I heard Connor singing the theme from the Batman TV show.

“Connor, is that the Batman theme you’re singing?” I asked.

“Yes,” he said.

“Get up here now,” I said in a stern voice that no one took seriously.

He came up to the front of the room and I handed him a Kit Kat bar.  “Yes, this is the Batcave and it is entirely fitting and appropriate that the Batman theme be sung here.” 

This set the tone for the school year. Or one of the tones, anyway. 

I went over the usual classroom rules and handed out the textbooks.  One was the official one—a big oversized purple covered book—and the unofficial 1962 algebra textbooks by Dolciani.  

“You won’t be using the purple book much, so you may as well keep those at home until I ask you to bring it in.  You’ll be using these red books that I managed to get for a penny a piece on the internet.”

The students looked at the books as if they were a strange but beguiling insect. 

“Why would you want us to use a book written in the 60’s?” someone asked.

“Math books written in the 60’s actually had math in them,” I said.  Hearing no reactions to this statement, I began the first day’s introductory lesson.

“If I asked whether an odd number plus an even number results in an odd or even number, what would you say?” I asked.

There was a moment of silence and then voices shouted out “Odd!”

“How did you arrive at that?”

 “I just added some odd and even numbers and saw that they were odd,” someone said.

“OK.  But does that prove that that particular pattern holds it for all possible combinations of odd and even?”

Vijay, whose observations of my teaching patterns were quite acute said “You always get happy when you talk about proving something in mathematics, and so you’re probably leading up to talking about proofs. So just adding numbers is not what you would consider a proof based on what makes you happy.”

I had to agree.  “In a few weeks you will be able to prove that using some of the tools we’ll develop in algebra.”

Connor appeared to be in thought. “I can see why it’s true,” he said. 

I was somewhat prepared for that eventuality. “Tell us.”

“If you add an even number to an odd one, you’ll be counting by two’s from that number, and no matter how many times you do that you’ll always land on an odd number. If you start with an even number, the only way to get another even is to skip numbers by two—but with an odd number, you won’t be able to land on an even number.”

“Good argument,” I said. “But I could ask how you know you’ll always land on an odd in the examples you gave. It’s intuitive but there are ways to say it mathematically that leave no doubt.”

“And we will learn that?” Vijay asked. 

I hold to the belief that with the development of more mathematical tools, procedures in many cases lead to understanding in math. At least I hope it does. What do I mean by that?  The ritual of my saying “I am haunted by waters” and the class’ response of “A River Runs Through It” was a procedure that I hoped would lead to someone interested enough to read the book, of which I kept a copy in my room.

No one borrowed it, which leads to what I mean by “I hope it leads to understanding”: not all the concepts behind the procedures in math are going to be understood.

In “A River Runs Through It” Maclean says: “You can love completely without complete understanding.” And though Maclean wasn’t talking about mathematics, he just as well could have been.

Some students noticed that there were three books that bore my name on the front cover and they were impressed that I was an author (even though self-published).  I always welcomed them to read them, but like “A River Runs Through It” they went unread.

I felt I was in good company.