A Classroom Observation, an Evaluation and Sense-Making Again

Editor’s note: This is the twelfth 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 AtlanticEducation NextEducation 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 1 , Chapter 2 , Chapter 3 , Chapter 4 , Chapter 5  Chapter 6  Chapter 7Chapter 8Chapter 9, Chapter 10, and Chapter 11

This will be the last chapter for a while since school is starting up next week. Enjoy the intermission; the story will resume and conclude at a later date.

Ch 12  A Classroom Observation, an Evaluation and Sense-Making Again

The nice thing about being observed by someone in administration is that the students are well-behaved for fear of being punished. Students are receptive, answer questions, and cooperate. Such was the case at my previous school during my second year there. The Superintendent informed me that he would be observing my seventh grade class on a particular day. The lesson that day was on multiplying decimals. The lesson was straightforward. I didn’t have them work in groups or do posters at the back of the room as I had done for Diane. I thought I’d see if that affected my evaluation—I live an exciting life.

A week later I met with the Superintendent in his office to go over the results.  He handed me his filled out form, had me read it, and asked if I had any questions. The evaluation was all very positive, going beyond what was observed in the classroom and ending with the following: “Mr. Garelick has done a good job of being a professional who takes care of business without stirring up unnecessary controversy or conflict.”

“So, do you want to come back next year?” he asked.

“I think so,” I said.

“Well I hope you will; you’ve been doing great. The students like you; I hear good reports from the parents, you teach well. And I can get things going to make you permanent,” he said.

“I thought I couldn’t be permanent unless I was full-time,” I said. I taught two classes and a remediation session.

“No, you don’t have to be full-time. We can make you permanent.”

“Sounds good,” I said.  “Would I be teaching the same classes next year that I’m teaching now?”

He pointed to a schedule he had on the wall, showing that my classes would be the same.

“Sounds good,” I said again.

“Let’s do it.”

The next day I ran into James in the copy room and, feeling in a somewhat confident mood, said “I had my evaluation yesterday and found out we’ll both be teaching the same classes we’re teaching this year. So since you’ll be teaching Math 8 again, I’ll be writing up my observations of my seventh graders so you know how best to work with them.”

“I haven’t heard anything about that,” he said.  Talking with James at times was like walking in a mine field; you never knew what was going to tick him off.

“Well, he showed me the schedule.”

“He hasn’t talked to me about it,” he said.

“OK, fine,” I said and acted like I suddenly remembered something that I had to do, and left.

Three weeks later, the Superintendent called me in to his office.

“I hate to tell you this,” he started, “but I need to because it will come up at the next school board meeting, and I didn’t want you to hear it from someone else.”

Call me a pessimist but I knew that this couldn’t possibly be good news.

“We’re going to have to let you go, and I want you to know it has nothing to do with performance. You’re doing great as I told you, but the way schools work, last one hired is the first one to go in situations like this.”

I asked the most logical question I could think of: “What is the situation?”

“I’m not at liberty to say. But under contract rules I have to give you this notice prior to mid-March.  It could be rescinded if things change, but I have to issue this notice now.”

“I don’t know what to say,” I said.

“I’m sorry,” the Superintendent said and that ended the meeting. He did not look happy.

A week or so later, after the School Board met, the Superintendent gave me a copy of the Board’s vote to eliminate a position; 4 to 1 in favor of elimination.  The document included the rationale: because of declining enrollment, the number of students would fall below a certain level. By law the number of permanent positions had to be reduced. And—as the Superintendent had told me—last one hired is first one fired.

 He had me sign it and said I could request a hearing if I wanted to appeal it, but he advised against it. “The chances of convincing the board to reverse the decision is pretty small,” he said.

At my weekly meeting with Diane, I told her the news. “I’m sorry to hear that,” she said. “But when I was a principal, I had to sign a lot of those notices, and they were rescinded in the fall. So there may be a chance.”

“It doesn’t look like it, what with declining enrollments,” I said.

“Well, you never know. But it sure doesn’t help to have that happen. Have you told anyone?”

“No,” I said. “I don’t want to make an issue of it, and I certainly don’t want the students to know.”

“Good decision,” she said.  “Good decision.”

“I have a question, though. I don’t know too much about teachers unions and all that—.”

“Consider yourself lucky,” she said.

“—but shouldn’t the union rep have been there at the meeting when the Superintendent told me the news?”

“He wasn’t?? You didn’t tell me that.”

“No,” I said.

“That doesn’t make sense.”

“You’re starting to sound like my students,” I said.

“Has James talked to you about it at all?”

“He’s continuing to not give me the time of day.”

“Are you planning on talking to him?”

“No,” I said. “It doesn’t sound like there’s anything he can do. And even if he could, something tells me he wouldn’t.”

“Hard to say,” she said. “Well, maybe not that hard. I hope it gets rescinded.”

I told none of the teachers nor the students. My classes went as normal. Normal for me being struck by a kind of stage fright before the start of each class. I’m like an actor before the curtain goes up, suddenly struck with panic that he has forgotten all his lines and even what the play is about. But once the students come into the room, the curtain is up and I am too busy tending to my immediate task than to worry about school politics and its various nuances.

More on Making Sense and a Fickle Bookseller

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 AtlanticEducation NextEducation 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 1 , Chapter 2 , Chapter 3 , Chapter 4 , Chapter 5  Chapter 6  Chapter 7, Chapter 8, Chapter 9, and Chapter 10

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.

Not Making Sense, and a Conversation I Never Had

Editor’s note: This is the tenth 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 AtlanticEducation NextEducation 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 1 , Chapter 2 , Chapter 3 , Chapter 4 , Chapter 5  Chapter 6  Chapter 7, Chapter 8 and Chapter 9

Ch 10: Not Making Sense, and a Conversation I Never Had

“Math doesn’t make sense.”  This was the chief complaint that Lucy evinced when seeking help with algebra. She was a bright girl in my eighth grade algebra class at St. Stevens.

Lucy’s statement will no doubt serve as evidence for those who view me as an unbending traditionalist hell-bent on teaching procedures at the expense of “understanding”. While I do provide the underlying concepts to procedures, there are students, like Lucy for whom math had always come easy and the connection between procedure and concept was obvious. With algebra the level of abstraction ramps up and things were no longer as obvious. Lucy thought that if math didn’t come easy then either something was wrong with her, or math made no sense.

The range of abilities in the St. Stevens algebra class was much wider than my previous classes, and likely more typical of most schools. There were about five students at St. Stevens who were at the very top of the class. At the lower end there were about four or five. Lucy was starting to fall into that lower group. She made a good effort in my algebra class in the beginning but increasingly got caught in waves of confusion starting with multiplying and dividing powers.

She had begun to make a good comeback with factoring of trinomials such as x²+5x+6 into two binomials: (x+2)(x+3). She even volunteered to do a problem at the board. But the next day we had more complex trinomials like 6x²-5x-6. Students were having a hard time with these and Lucy was back to sitting with arms folded, answering questions I asked of her with a shrug and a response of “I don’t know” laden with teenaged insouciance.

I had taught this particular type of trinomial by using a trial and error method in which you try various factors like 2x and 3x, and 6x and x to get it to work. (If you’re curious, the factorization of 6x²-5x-6 is (2x-3)(3x+2).

There is another method, sometimes called the “diamond method”, which involves some steps that I won’t go into here, but results in the trinomial being expressed as 6x²-9x+4x-6. This can then be expressed as 3x(2x-3) + 2(2x-3). Since (2x-3) is a common factor, this now can be expressed as (2x-3)(3x+2). I’ve tried to teach this method in the past with mixed success; many find it difficult. Given the problems I was having with Lucy and others, I decided to stick with the trial and error method.

I allotted time in every class for students to start on their homework to allow me to offer help and guidance. She accepted my help grudgingly. After working through a problem I asked “Does it make sense now?”

She gave her usual response. “Sort of.” I took this as “no”.

Katherine would sometimes use that period to catch up on paperwork, and in so doing would observe what was going on in class. She never offered any criticism or comments on anything that happened unless I asked. (And when you get down to it, that’s how I like to be mentored.) When I saw Katherine later that day I told her “I’m at my wits end with Lucy.”

“I know,” she said. “One look at her body language tells you she’s given up.”

“I’ve tried everything,” I said. “I’ve communicated with her mother, let her know she can get help, but she doesn’t even try. I feel like saying ‘I’m bending over backwards for you; the least you can do is show some respect and make an effort.’”

“You should tell her that,” she said. “Just talk with her and tell her what you told me, and what your expectations are. She’ll be real honest with you, but you need to reach an understanding.”

I lost sleep that night, rehearsing how that conversation would go. I decided I would pull her aside when the rest of the class was doing their warm-up questions, have the talk. But when I arrived in the classroom, I was greeted by a very cheerful Lucy who offered to help me pass out the day’s warm-up questions to the class. She then excitedly told me “I found a way to do the factoring.”

She showed me. It was the diamond method I had decided not to teach because I thought it would be too confusing for her.

“Where did you learn this?” I asked.

“I looked it up on the internet. It’s really easy.”

“Fantastic,” I said. “Do you want to show the class how it’s done?”

She didn’t want to, so I demonstrated the method. There were the sounds of people getting it as I put some problems on the board for them to work. I left the class elated that Lucy had taken the initiative and was getting it.

I ran into Katherine after class was over and excitedly told her about Lucy’s miraculous turn-around. As it turned out, after Katherine had talked with me the previous day, she decided to talk with Lucy at the end of the day.

“That explains her change in attitude,” I said.

“I should have told you,” she said. “I’m sorry. But she was in the classroom getting something so I just talked to her.”

“What did you tell her?”

“I told her that her body language is telling us she’s given up.”

“Anything else?”

“I said ‘Mr. Garelick thinks you don’t like him.’”

I wished she hadn’t said that. “What did she say to that?” I asked.

“She said ‘Oh no, that’s not true.’  She felt bad.”

That evening, my wife, who was brought up Catholic, said this was part of Catholic guilt. I have chosen to remain agnostic on such matters.

In the end, the top students were able to work the diamond method, while the other students relied on the trial and error method. Lucy would forget the procedure she had found on the internet and even simple trinomials would elude her despite the fact that factoring trinomials doesn’t go away in subsequent lessons.

There is an advantage to continued practice should anyone have their doubts. It leads to proficiency and eventually can connect with the understanding and “sense” that Lucy felt was lacking.

She would continue to be a challenge. And I would learn to take my victories if and when they occurred.