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.

A Cathartic Discussion, Putting the Bell on the Cat, and Business as Usual

Editor’s note: This is the ninth 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 and Chapter 8

Ch 9.  A Cathartic Discussion, Putting the Bell on the Cat, and Business as Usual

My meetings with Diane at my previous school were not always confrontational. Sometimes we got into my interactions with other teachers which I found fairly enjoyable. While it’s not quite gossip, it does have cathartic benefits.

Shortly after James and I met with the moderator who wanted us to meet for six two-hour sessions and collaborate on how best to teach math (see Chapter 6) Diane asked “Are you getting along with James any better?”

“Well, he was definitely friendlier towards me after that meeting.”

 “So there’s been a breakthrough.” Diane said.

“I wouldn’t go that far,” I said. “He was friendly for a few days. He’s back in his non-talkative passive-aggressive mode.”

“I’ve met his wife,” Diane disclosed. “She works as a counselor at a high school near here. She’s very nice,” she said and sipped her coffee while looking at me out of the sides of her eyes. “Except when she isn’t,” she added.

“Sounds like a marriage made in heaven,” I said. She almost spit out her coffee.

“I find when I talk to teachers, that one of the biggest complaints about teaching is not always the teaching itself. It’s frequently about getting along with other teachers,” she said.

This made a lot of sense. One event in particular came to mind when she said that. During my first year at the school, the Superintendent was pushing for having seven periods rather than six. This meant that our classes would be forty-five minutes long instead of fifty-five. It would be even shorter on Wednesdays when we were dismissed early because of the weekly staff meeting.

The seven period day, in fact, was the topic of discussion at one such staff meeting, led by the Superintendent. Prior to the meeting, two of the teachers were in the room, and they agreed with me that shorter class times were not a good idea. “You’re right, Barry,” one of the teachers said. “It would really end up forcing us to cram a lot of things in.” She said she would speak out against it.

When our meeting began, the Superintendent talked up the benefits of the new schedule since it would allow students to now have two electives instead of just limiting them to one. And an elective could also be two periods long: sixth and seventh periods. With more electives, this could open up more teaching opportunities—an important consideration, given that the largest class was graduating and enrollment numbers were dwindling.

 “But I want to hear from you now,” he said. “What are your feelings about this schedule?”

The silence that followed reminded me of Aesop’s fable about which mouse was going to put the bell on the cat.

Given the discussion prior to the meeting, I felt I was on safe ground to start the discussion.

“I’m not for it,” I said. 

All eyes were suddenly on me.

“A shorter class period will make it difficult to teach. Right now Wednesdays are my worst day because class length is 45 minutes and I often don’t get done what needs to be done. With seven periods, every day will be like Wednesdays are now—and Wednesdays will be even shorter.”

“So I take it that you would be voting ‘no’ on this?”

I wasn’t aware that this was a vote, but now so informed I replied “That would be safe to say.”

Discussion continued. The Drama and PE teachers concurred with me, and then it was James’ turn.

“I think this would be a good step forward,” he had said. “I would like the opportunity to reinvent myself as a teacher…” and other words to that effect.

After he spoke, others now seemed to approve of the seven period day including the teacher who had previously agreed with me that it was a bad idea.

“I can’t blame her,” I told Diane. “She’s worried about her job and didn’t want to be against the Superintendent. Plus I think James being the union rep kind of makes him the thought leader.”

“Don’t get me started on teachers’ unions,” she said. “I’m starting to get a whole different take on this now.”

The union influence, such as it was, didn’t hold a candle to the history teacher’s final words on the subject.  He had taught at the school for 30 years and was well respected. Although he voted in favor of the seven period day he offered this reflection after the yeas were seen to outweigh the nays: “I think it’s a shame that we’ll be going into the next school year with some people not happy about this change and a cloud hanging over them. So I propose we think about this some more.”

Apparently the Superintendent did. At the next staff meeting he announced that the current six period schedule would remain, though he was disappointed in the reaction. I guess he wanted unanimity. Apparently James was disgruntled about it as well.

“I don’t know if he’s held it against me,” I told Diane.

“He might have,” she said. “It’s hard to say.”

It’s hard to say a lot of things that go on in any school. I recall another time—this one a party—James was talking with the third grade teacher who mentioned the downward trend of state test scores on math at the school.

James gave a reason for that. “It’s because we don’t have students collaborate with each other enough,” he told her. She nodded in agreement at what is accepted as educational wisdom. Perhaps James thought there wasn’t enough collaboration in my classroom. Who knows? Hard to say.

In any event, I decided to not bring up that particular conversation at this session with Diane. Despite her dislike of unions and distrust of people who represent them, she was likely to say “Well he does bring up a good point; what are your thoughts?”

Our conversation turned to the usual business of filling out her online checklist. There were plenty of avenues for her to pursue what I thought of dubious practices, so no need to give her ideas.

The General This ‘n That Shop, Negative Numbers, and Faith

Editor’s note: This is the eighth piece 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 and Chapter 6 , Chapter 7

Chapter 8: The General This ‘n That Shop, Negative Numbers, and Faith

I went to school at University of Michigan in the late 60’s/early 70’s, long before the proliferation of 24 hour 7/11’s, Starbucks, and other brand name franchises. A few blocks away from me was a small mom and pop grocery store that was fairly new called “The General This n’ That Shop”. During my senior year I noticed that the items in the store were dwindling. On a particularly cold day in December, I stopped by to find that most of the shelves were bare, and on one, a single loaf of bread remained, which I bought.

 “Are you going out of business?” I asked the cashier.

Her response: “I hope not.”

This reminds me of that bread aisle, except they are in better shape.
Photo Credit: Rusty Clark (CC-By-2.0)

It is this type of optimism that I try to instill in my students. And among my seventh grade students the first such test has been negative numbers.

I do not like to prolong the topic. I once observed a teacher taking three weeks to teach it. After two weeks of adding and subtracting negative numbers and the students had it down fairly well, the teacher introduced a new explanation using colored circles. While the approach can be effective, its use at this point only caused confusion. One girl asked “Why are we doing this?”

The teacher answered “I know you know how to add and subtract with negative numbers. Now I want you to understand why it works.”

The girl’s response: “I don’t want to understand!”

This incident was not unique. I’ve found that a lot of the confusion with the addition and subtraction of negative integers is that students are given more techniques and pictorials than are really needed. They are left with the impression that it is a complex process and that there are many different ways to do it. This is ironic considering that subtraction is an extension of addition. In mathematical terms, a – b = a + (-b) where a and b can be positive or negative.

I keep it relatively straightforward with the first day spent using number lines with arrows to compute addition of negative numbers. The next day, the addition is done without pictures.

I then introduce subtraction. I tried this my first year teaching at my previous school with an accelerated seventh grade class.  I asked them to compute 6 + (-4) which they knew how to do from the previous lesson. Two was their answer. I then asked them to compute 6 – 4. They saw the connection almost immediately, leading to the general rule of “adding the opposite”.

While it worked well with my accelerated class, I thought maybe I wouldn’t have the same success the next year with the seventh grade class which had large deficits in their math knowledge. But the technique worked just fine, with Kyle shouting out: “Adding a negative number is the same thing as subtraction”. This became a quote that I posted on the quote wall.

The only rub in all this is the subtraction of a negative number.  I used to introduce this by first asking if anyone could solve 10 – (-5), and then linking the question to football for those who liked or played It.: “After a loss of 5 yards, how many yards do you need to get a first down?” As I mentioned in an earlier chapter, Jimmy had answered “Can’t you just punt it?” I have since changed my tactics. I specify that the ball has to be run and then use any number of non-football examples such as: “It was -10 degrees yesterday and 20 degrees today. By how much did it increase?”

My goal in teaching adding and subtracting of negative numbers is to achieve a level of automaticity, so that students can ultimately solve a problem like 3 – 7 without using pictures or writing it as 3 + (-7). At the same time, I try to get them to develop a number sense as to whether their answer is going to be negative or positive. I give them models to use such as: “If I gain 3 yards and lose 7 am I ahead or behind and by how much?  If I earn $3 but owe $7, am I ahead or “in the hole” and by how much?  They do get it, though they need reminders through the year.

And as far as multiplying negative numbers I provide an illustration of why things work as they do. I use an example of making a video of someone riding a bike backwards, and running the video backwards.

“What if they were skateboarding?” Jimmy asked.

“Whatever you want,” I said.

If the backwards rate is represented as -3 mph and we run the video backwards at 2 times the normal speed—represented as -2—the person appears to be riding a bike or skateboarding forward at 6 mph. A similar model can be used to show why a negative number times a positive is a negative number. (For the more curious students, and certainly in accelerated classes and in eighth grade algebra, I show the proof using the distributive rule.)

While my backwards video example generally does the job, it didn’t with Jimmy even when I used skateboarding rather than a bicycle rider.  He tended to be quite literal. “That’s in a video; does it work in real life?” he asked thus opening up the question of whether mathematics can apply to images.  He finally accepted the example used in JUMP Math where someone on a mountain is descending at a rate of 30 ft per minute or -30 ft/min.  The example asks how one would represent where the mountain climber had been relative to his present position 3 minutes earlier, or -3 min. Jimmy agreed that the person would be higher 3 minutes previous and further accepted that the situation is represented by -3 x -30, or +90.

While Jimmy understood it, a girl said she did not understand either example.

“For now, just work with the rule,” I said.  “You’ll get it the more you work with these kind of problems.”  The girl did understand the examples a few days later. “Sometimes you just have to have faith in the math,” I told her. She evinced no expression so I said nothing more. Which is a good thing because I doubt she really cared about how we’re all sometimes like the cashier at the General This n’ That Shop.

A Gnarly Problem, Critical Thinking, and Authentic Struggle

Editor’s note: This is the seventh piece 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 and Chapter 6

Ch 7: A Gnarly Problem, Critical Thinking, and Authentic Struggle

My meetings with my parole office/mentor Diane occurred once a week in the early morning at a local coffee house a few blocks from my previous school. At one particular meeting she showed me her notes from an observation she had made of a lesson I gave my seventh grade class. Her notes were typical “hunting for problems” comments, such as my not noticing a particular boy who was unfocused, or another student who was talking, and so on.

“Any comments?” she asked.

I resisted the urge to say that it sounded like she was hunting for problems to enter on her online checklist. I talked instead about the lesson itself. It had been about taking a situation like “A bowling alley charges $5 for shoes and $3 per game bowled” and writing an algebraic expression for the cost of x games. (5 + 3x). Knowing that Diane wanted to see me extend JUMP’s scaffolded approach to more “gnarly” problems, I told her about a problem I gave the seventh graders on one of the Warm-Up questions the day after the lesson she observed.

They were to write an expression representing the cost for n hours, if a babysitter charges a flat fee of $10 and $15 per hour, but with the first hour free. I was met with the usual questions of “How do you do this?”

In my highly scientific approach I helped the first person who asked “How do you do this?” which happened to be Kyle.  He was a talkative boy who was quite good at problems when he put his mind to it.

“How many hours does the babysitter charge for 6 hours of work if the first hour is free?” I asked.

“Five,” he answered.

“Right: 6 -1 = 5. So for five hours of work what does he charge?”

“Five minus one,” he said.  I gave him a few more numbers and then asked “How much for n hours work?”

“Oh! n-1,” he said. He was then able to see it was 10 + 15(n-1), though he and others needed help with the parentheses. 

“Yes, that’s a good problem,” Diane said.  “But it wasn’t really critical thinking.”

“Why not?” I asked.

“You led them there.”

I said nothing, hoping for an awkward silence and got my wish.

“There’s nothing wrong with what you did,” she said. “But true critical thinking would involve them struggling to come up with a solution.”

I recognized this immediately as the “struggle is good” philosophy which holds that if students aren’t struggling they aren’t learning. There are nuances to this philosophy including “productive struggle”, “desirable difficulties” and “students should be able to use prior knowledge in new situations without scaffolding because otherwise it is inauthentic.” I’ve read variations of this thinking in books that I’ve thrown across the room.

“Let me give you a problem that I want you to solve,” I said. “Two cars head towards each other on the same highway. One car starts from the north heading south, at 80 mph. The other car starts from the south heading north at 70 mph.  They meet somewhere on the highway. How far apart are they one hour before they meet?”

She took a gulp of coffee and tried to smile.

“You do not need to know the distance they are apart to solve it”, I said.

She looked perplexed and gave me a look I see on my students’ faces when they ask “How do you do this?”

“Tell me this,” I said.  “How far does a car going 80 mph travel in one hour?”

“Eighty miles,” she said.

I drew a line on a napkin and marked a point near the middle with an X.  “Where was the 80 mph car 1 hour before he got here?”

“Well, that would be 80 miles north of that point.”  

” What about the 70 mph car?”  

“Uh, 70 miles south of the point?”

“Good. Can you put that together somehow?”

She suddenly saw it.  “Oh, I see! They’re 150 miles apart one hour before they meet.”

“Good work.  Now let me ask you something.  I gave you some hints.  Would you say that you used those hints in thinking about the problem and coming up with a solution?”

She smiled knowingly. “Ah, I see. Critical thinking.”

“So would you say that what you did qualifies as critical thinking?”  She agreed.

“Then why would you say that what I did with the baby sitting question did not qualify as critical thinking.”

“I’ll have to think about what I mean by critical thinking,” she said. “I think applying an algorithm repeatedly does not entail critical thinking.”

“Even if it leads to a conclusion? And in essence that was what I had you do when you think about it. And you put it together like my students did. Why would you not call that critical thinking? In your mind is there no difference between thinking and critical thinking?”

“I guess I might have to look up the definition of critical thinking.”

“I’ll send you a definition tonight by email,” I said.  “My concern is this. If your goal is to look for examples of critical thinking in my classes using the definition you’ve presented, you will probably never see critical thinking in my classes. I use worked examples and scaffolding and problems that ramp up. That’s how I teach. You’ll see this more in my algebra class, and I hope you observe one of those.”

She said she definitely would. I thanked her for having the discussion with me. “I felt it was important that we understand the language we’re speaking and what I’m about.”

“Yes,” she said. The conversation then shifted to lighter topics. I felt a bit bad for putting her on the spot with my math problem. But then again, her struggle with  critical thinking was productive if not authentic.

The Prospect of a Horrible PD, a Horrible Meeting, and an Unlikely Collaboration

Editor’s note: This is the sixth piece 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 and Chapter 5

Ch 6  The Prospect of a Horrible PD, a Horrible Meeting, and an Unlikely Collaboration

Many schools require teachers to attend some kind of professional development and St. Stevens was no exception. Fortunately it was rather benign even though it was the whole day and was about technology in the classroom.  But other than that it was fine.

In my previous school, the principal (and Superintendent) “asked” me and James, the other math teacher to attend six all day professional development (PD) sessions over the course of the school year. The PD, held by the County Office of Education, was to be a forum for “collaboration” among math teachers in the county.

While I don’t mind collaborating with teachers, I don’t like the collaboration to be prescribed. And certainly not for six times. I was also leery of James who hardly gave me the time of day and was passive aggressively hostile. As I told Diane during one of our more productive mentor sessions, “I dislike the idea of going to the sessions with him more than I dislike the idea of the PD itself.” She advised me that most complaints from teachers were about problems with other teachers. “And passive aggressive types are the worst,” she added. It was a valuable piece of advice.

As it turned out, the PD was cancelled. James and I were the only two people in the county who had signed up. Our delight was rather short-lived, however. The moderator met with our principal and suggested having a series of two hour meetings with us at school during the early part of the day when we weren’t teaching. Neither James nor I were too thrilled at the idea of collaborating with each other.

We all met one time. The moderator, a middle aged woman who talked in the cheery tones of a facilitator began describing how she loved math while in school but was just “following the rules and getting an answer”.  Later when she taught math, she found she couldn’t explain to students the underlying concepts.  Which led her to say, that the Common Core standards were all about “understanding”, and teachers had better teach for understanding because as she explained, “California’s Common Core-aligned tests are not about ‘answer getting’ anymore!” Students had to explain their answers and the tests evaluate whether students are able to solve problems in more than one way.

She went on almost breathlessly: “Students can get full credit on problems where they have to provide explanations—even if they get the numerical answer wrong.” 

James and I said nothing.

 “Provided the reasoning and process are correct, of course,” she added. “Explaining answers is tough for students and for this reason there is a need for discourse in the classroom and ‘rich tasks’ ”.

My years in education school had taught me the skill of keeping my mouth shut appropriately but at this point I couldn’t contain myself and asked “Could you define what a ‘rich task’ is?”

Her answer was extraordinary in its eloquence at saying absolutely nothing: “It’s a problem that has multiple entry points and has various levels of cognitive demands.  Every student can be successful on at least part of it.”

I had had some experience with rich problems so I knew exactly the type of problem to which she was referring; problems like “A rectangle with a perimeter of 20 inches has what dimensions?” or something similar.

At this point James could take it no longer. He said that meeting for two hours for five sessions was superfluous if it was just the two of us. “I teach three different math classes plus doing the I.T. for the school and don’t have time to delve into alternative approaches other than to follow the script and curriculum as laid out in the book.”

The two of us must have seemed like a rich problem. “Books are just tools,” she proclaimed. “They may be strong in one area but weak in another. Traditional textbooks tend to be lacking in opportunities for conceptual understanding and are old school in their approach.”

She sensed that both of us were more than willing to let her dig her own grave here.  “Though there’s nothing wrong with old school,” she quickly added.

As tempting as it was. I saw no need to tell her that I used a 1962 textbook by Dolciani for my algebra class.

She asked if we relied on our textbook for a “script”, meaning scope and sequence “Do you read just one textbook?” she asked me.

“I read lots of textbooks,” I said.  She looked surprised.

“He’s also written books,” James said. I was surprised that he knew about them, but I had slipped “Math Education in the US” surreptitiously in the bookcase in the teacher’s lounge so maybe he had read it.

“How nice!” she said and feigned an interest by asking what they were about. I gave a “rich” answer. “Math education,” I said.

“Wonderful!” she said.

I then tried to summarize our thoughts. “Neither of us teaches in a vacuum,” I said. “I read lots of textbooks and talk to lots of teachers.” Since James had played up my authorship, I decided to return the favor.

“And James has a lot more experience than I do so he isn’t exactly ignorant about how to teach math. I really don’t think that this two-hour collaboration is going to add much more.”

I realized this opened me up to her protesting that perhaps I could benefit from his experience so I needed to head that off. “Besides, I’m getting mixed messages,” I said. “On the one hand I’m told by the administration that I’m doing great, and I hear from parents that I’m doing great. But then I’m told that I must attend this PD. Is there something about my teaching that’s lacking?  What is this about?”

She assured us that there’s nothing lacking in our teaching and that she’s sure we are both fantastic teachers. “What is it then?  Is this about test scores? They think this will raise test scores?”

She had no answer for this except something that I can’t remember. She saw the handwriting on the wall and said “No use beating a dead horse” and said she would talk to the administration about it.  And that was the end of our PD.

I decided the next time I met with Diane, I would tell her about the success of James’ and my “collaboration.”

The Rituals of School, an Unusual Communion, and the Vast Wasteland of Math 8

Editor’s note: This is the fifth piece 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 AtlanticNonpartisan Education ReviewEducation 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 3Chapter 4.

Ch 5. The Rituals of School, an Unusual Communion, and the Vast Wasteland of Math 8

Each day at St. Stevens starts out with the entire school of 200 students, plus teachers, gathered around the flagpole to say one or two prayers, and the Pledge of Allegiance. I enjoy the ritual, particularly seeing everyone—first graders through eighth—cross themselves in unison. Before I started at St. Stevens, a friend asked me if knew how to cross myself.  “Yes,” I said. “But at this point it’s procedural; I think the understanding will come later.”

My days are built on a set of procedures resulting in an ever-changing understanding of where I am. After “flag”, came a quick walk to my classroom, walking fast to stay ahead of the rapidly dispersing horde of students. I call my classroom The Batcave, partly because my classroom is out of the way and cavelike. I have come to love the room and wouldn’t change it for the world.

It appears to have been a storage closet in a former life and is right next door to the gym. It has a door to the gym which sometimes gets bumped by stray basketballs and other objects. Then there is the music that is played during exercises or dance which usually elicits conversation among my students about the songs being played. I squeezed eight desks in the room to accommodate the students in Math 7 and 8.

The Math 8 class was segmented from the rest of the eighth graders who were in the eighth grade algebra class. During the first week, a rather stubborn and outspoken student, Lou, stated what the rest of the class was feeling. “It’s obvious we’re not too good at math which is why they put us in this class.”

I had heard something similar at my previous school from my seventh graders. In neither case did I respond by talking about “growth mindset”.  It was the first time I had taught Math 8 and I was rapidly discovering that the course was a vast wasteland of disparate topics that did little or nothing to prepare them for algebra in the ninth grade.

After my Math 7 and 8 classes, came my algebra class—held in Katherine’s classroom since there were sixteen students. Like most of the algebra classes I’ve taught, this one was full of energetic and motivated students. They were also quite noisy and extremely competitive. Unlike the Math 8 class, they had both confidence and curiosity. The difference between the two classes was never more obvious than when I showed a magic trick to both classes.

It was on a day in which school was dismissed at noon (another cherished tradition in which one’s best plans and schedules written over the summer start to resemble a game of Battleship—a day you thought you’d have for a complicated lesson turns out to be a short one). I had performed this trick many times over the years, starting when I was a sub.

Given the following five cards, someone picks a number from 1 through 31, writes the number on the board and erases it after everyone has seen it so everyone but me knows the number.  

I then ask what cards the person sees their number on.  I immediately tell them the number. The trick is based on the binary number system. One student in the algebra class who is interested in computers knew how it was done so he kept quiet.

I embellished the presentation for the algebra class by having them help me construct a table of binary numbers from 1 through 31, and then transfer the information onto the five mini whiteboards to make the five magic cards.

“You can fill this table out by looking at the patterns,” I said, realizing that any onlooker who happened to poke their head in to my class would think “Oh, good, Mr. Garelick is teaching them that math is about patterns!”—a characterization that I dislike for reasons I won’t get into here.

I started filling out the first four rows; once I got to the fifth row, they started to see the pattern.

“Oh, it just keeps repeating itself: 01, 10, 11, and 100,” a boy said.

“What do the numbers mean?” someone else asked.

“They correspond to the numbers on the left.”

“But why?”

I tried to explain, showing that you’re adding powers of 2 just as in base 10 the number 11 is (1 x 10) + (1 x 1). Some students understood, but most didn’t.

I then said “Just keep filling out the table. It doesn’t matter right now whether you understand what the binary numbers mean.”  If the same onlooker who liked my comment about patterns was looking in again, the reaction would probably be “Oh wait; he’s having them ‘do’ math without ‘knowing’ math”. Or some equivalent bromide.

Once all 31 rows were filled, I had five students transfer the numbers to five mini whiteboards. I then proceeded to do my magic trick. The first time I got the number the entire class shouted. 

“Do it again!” I did, and got it right again. Each time I revealed their number they were now screaming. “He’s a wizard!” a boy named Sam shouted out.

I finally revealed the trick: “I look at the first number on each card you told me contained your number and then added them up.”  I showed them that for the number seven, the first numbers on those cards are 1, 2 and 4, which sum to seven.

There was a collective “Oh, that’s how!” Their excitement was in stark contrast to the Math 8 class who, although curious and amused, took it in stride as just one more thing that didn’t concern them. I had sets of magic cards that I printed up and asked if anyone wanted them. No one in my Math 8 class had wanted them, but the algebra class immediately surrounded me, some with hands cupped as if receiving communion.

During the prayer before dismissing for lunch (“Bless us, Oh Lord, and these thy gifts…”) I realized I had to do something to fill in the vast wasteland of the Math 8 course. I wondered if perhaps I could sneak in more algebra. And for an extra challenge, doing so without the ritual of “growth mindset”.

Who Knows? Who Decides? Who Decides Who Decides?

I was recently reading a section of the book “The Age of Surveillance Capitalism: The Fight for a Human Future at the New Frontier of Power” by Shoshana Zuboff.  This is an eye opener and an excellent book about surveillance capitalism.

Many of the things being addressed in this book are things I see related to or can relate to what I observe happening in our country today, especially in education.  Chapter Six, Hijacked: The Division of Learning in Society asks and addresses three questions:  Who Knows? Who Decides? Who Decides Who Decides?  While the chapter addresses these questions at more length, below are two quotes from the book.  The first briefly explains the questions and the second provides an extremely brief answer to each question.

“The first question is “Who knows?” This is a question about the distribution of knowledge and whether one is included or excluded from the opportunity to learn. The second question is “Who decides?” This is a question about authority: which people, institutions, or processes determine who is included in learning, what they are able to learn, and how they are able to act on their knowledge. What is the legitimate basis of that authority? The third question is “Who decides who decides?” This is a question about power. What is the source of power that undergirds the authority to share or withhold knowledge?”

“As things currently stand, it is the surveillance capitalist corporations that know. It is the market form that decides. It is the competitive struggle among surveillance capitalists that decides who decides.”

To me, this is scary to think that we are not only heading in this direction but that we are well on our way.  I would say we are already there except things related to technology are always evolving.

Think about this in terms of our education system during three periods of time:  1) prior to the recent education reform era, 2) during the recent education reform era, and 3) the surveillance capitalism present and future.  Shifts have taken place in the transition from one time period to the next.  The answers to the questions Who Knows? Who Decides? Who Decides Who Decides? with regard to our education system in this country have shifted from parents/local community to state/federal government to surveillance capitalist corporations.  These shifts, like their corresponding time periods, as simply stated here does not capture or adequately convey the complexity.  The shift to surveillance capitalist corporations driving our education system is well underway, or has already taken place and continues to evolve.

Below is a table that in a simple but incomplete way shows Who Knows? Who Decides? Who Decides Who Decides? for each of the three time periods.

QuestionPrior to Ed Reform EraRecent Education Reform EraSurveillance Capitalism Future
Who Knows?Educators and subject matter experts
parents/local community
state/federal governmentsurveillance
Who Decides?Educators and subject matter experts
parents/local community
state/federal government
Who Decides Who Decides?parents/local communitystate/federal governmentcompetitive struggle among surveillance capitalists

Different people may place different entities in the various boxes in the table.  There are definitely more players involved than just those mentioned.  It is possible that others dubbed as “experts” may be included with state/federal government.  Such “experts” may be more driven by an agenda or ideology than by any real expertise based on evidence, factual data, or true knowledge.

Foundations and the influential wealthy seem to go hand in hand with the surveillance capitalist corporations in deciding who decides.

These shifts have taken place gradually over time.  Has it happened so gradually that most parents and local communities have yet to realize their rights/responsibilities have been usurped?  Are parents and local communities okay with this?  Have they willingly turned those rights/responsibilities over to the surveillance capitalist corporations?  If not, what can be done to restore those rights/responsibilities back to parents and local communities?

What is Surveillance Capitalism?


The Parole Officer’s Check List, the Dialectic of Competition, and Gnarly Problems

Editor’s note: This is the fourth piece 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 AtlanticNonpartisan Education Review, Education 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.

4. The Parole Officer’s Check List, the Dielectic of Competition, and Gnarly Problems

Upon starting at St. Stevens, I had completed my two years compulsory Teacher Induction Program (TIP) under two different parole officers, otherwise known as mentors. The TIP process consisted of discussions, observations, and the mentors/parole officers having to fill out an online checklist of items which would serve as the final authentication of having gone through the induction.

My first mentor (who I will call Ellen) told me that she would be making suggestions and giving me ideas, but I was under no obligation to follow any of them. This was good because she had no shortage of dubious ideas and quickly learned that I was going to do things my way. At one point early in our meetings, having determined that I didn’t assign group work and rarely had activities, Ellen asked “What are you going to do about Common Core, which requires activities and group work in teaching math?”

“The Common Core standards do not prescribe such pedagogy,” I said and pointed out that the website for Common Core states clearly that the standards do not mandate pedagogical approaches. I expected an argument, but instead she quickly moved on to other business. The mentors are used to teachers fresh out of ed school who are in their twenties and believe whatever they’ve been told about how to teach and what Common Core requires. In any event that was one response she did not expect that remained unrecorded on the online checklist.

My second mentor, who I will call Diane, was assigned during the second year of my induction. Like Ellen, Diane also had teaching experience—second grade mostly—and was now in charge of the mentor/parole program for the county in which I teach. Like Ellen, she would make suggestions that I could either follow or ignore. She would occasionally evince educational group-think that passed as sound advice. 

Our first meeting took place in my classroom. “Tell me about your classes,” she said.

I had two that year: a seventh grade math class (non-accelerated), and eighth grade algebra—the latter made up of students I had taught the previous year.

“My seventh grade math class had a rough year last year so they’re coming in with an ‘I can’t do math’ attitude right at the start,” I said.

“That’s never a good thing,” she said.

 “And on top of that, they have significant deficits. Like not knowing their multiplication facts.”

Her eyes widened. “Really? How can that be?”

Actually, it can be and is in many schools across the US. I wondered how on earth she could not know this, being in education as long as she had.

“So how are you addressing that?” she asked.

“I’ll let you in on a secret,” I said.  She looked intrigued.

“I’ve been giving them timed multiplication quizzes every day to start off the class. My principal told me that timed quizzes stress students, but these kids love the competition, plus I show them how their scores are increasing.”

“Of course!” she said. “Kids love to compete.” I was heartened at this for two reasons: she wasn’t against memorizing multiplication facts, and she appeared to be going against the educationist dialectic of “competition is bad”. But then she added, “Of course, it isn’t good to do that in the first and second grades because it can stress kids out, but it’s perfect for seventh graders.”  A few minutes later when I told her I posted the top three scores on quizzes or tests, the dialectic clicked in.

“Are you sure that’s a good thing to do? Some of the students who didn’t make the top scores might feel left out,” she said. 

“They ask me who got the top scores, and they don’t seem upset when I post them, so I’m assuming it’s OK,” I said. She had no answer to that and looked around the room.  “I like the way you’ve set up your classroom.”

Diane liked various quotes I had tacked up on my walls; random things uttered by my students that I felt worthy of posting.  Like “I never get used to math; it’s always changing.” and “Variables don’t make sense and make sense at the same time”.

I kept the ones from my previous year’s classes on the wall, as well as those from my current students. “Seeing quotes from previous classes gives students a sense of legacy and tradition,” I said. I remember being intrigued when my seventh grade English teacher would show us examples of work done by her previous classes, and we would see the names of students from years past—some were brothers and sisters of my classmates.

I pointed out one quote from a student named Jimmy in my current seventh grade class. It had emerged from a dialogue I had with him on subtracting negative numbers:

Me: You lose 5 yards on a play.  You have to make a first down.  How many yards do you have to run?

Jimmy:  Couldn’t you just punt it?

Jimmy had had a particularly rough time in math the previous year and had very little confidence. The first time we had a quiz I had made sure that the students would do well, giving them lots of preparation. Jimmy did do well: 97%. When I was handing back the quizzes he kept saying “I know I failed it.” When he saw his quiz he was silent and then asked if anyone in my class last year had failed.

“No,” I said.

“Do you think it’s possible that I’ll pass this class?” he asked.

“Yes, it’s entirely possible,” I said.

I told Diane about this. “Fantastic that he got 97%. How did that happen?”

I explained how I was using JUMP this year and how it breaks things down into manageable chunks of information that students could master. Given where they were coming from I felt that building up their confidence was very much needed.

“Are you planning to go beyond just mastery and give them some gnarly problems?”

I was tempted to ask her to define “gnarly problems”, though I’m fairly sure it had something to do with “If they can’t apply prior knowledge to new problems they haven’t seen before, there is no understanding”. But my answer to her was “Of course”. JUMP does in fact provide “extension” problems in the teacher’s manual. I made a note to self which when roughly translated was something like: “Come up with something.” Given the deck I had been handed with this class, I had other things on my mind besides giving the class gnarly problems, however it was being defined.

Understanding, and Outliers in a Sea of Outliers

Editor’s note: This is the third piece 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.” Chapters 1 and 2 can be found here and here.

3. Understanding, and Outliers in a Sea of Outliers

During my second week of school at St. Stevens the principal, Marianne, called me in to her office to tell me some good news.  “I just want to let you know that we heard from Mary’s mother and that Mary said she is really happy in your class; she says that “Mr. Garelick really wants us to understand.”

I was glad to hear that Mary’s mother was pleased, and while I haven’t taught for very long, I knew enough not to believe I was any kind of miracle worker—particularly during the first few weeks at school when everything is new and has the cast of a halo over it. At my previous school during back-to-school night one year before, a parent of one of my students in my seventh grade class said something similar. “My son said that this is the first time in any math class that he actually understood the math.”

In both cases, it didn’t hurt that the word “understand” was used in conjunction with my teaching, although the word has different meaning for me than what others in education think it means. I want students to be able to do the math. That’s pretty much what students mean when they say they understand. It isn’t something I obsess over.

Mary was one of two girls in my eighth grade math class (Math 8), who had to come in twice a week for intervention help for half an hour before classes began. The other student was Valerie who had been classified as special needs since the lower grades. They were both very animated girls; Mary was very outgoing and friendly with me. Valerie was more guarded. In her world of Smart Phone, songs, reality TV, I felt she viewed me as frightfully out of touch with what was really important. Math was certainly not on her list.

My Math 8 class was similar in some ways to my last year’s seventh grade class. My prior school, like St. Stevens, had two seventh grade math classes—one accelerated, the other not. I taught the non-accelerated group who considered themselves “the dumb class”. Their doubts were compounded by their last year’s math teacher who was not popular with parents or students, and was finally let go by the school.  My Math 8 class similarly knew they didn’t make the cut for the eighth grade algebra class (which I was teaching). Their previous math teacher was similarly unpopular—and also let go by the school.

In a school as small as St. Stevens, there weren’t enough students to form a remedial class by itself. As a result, in the midst of a class in which the students already doubted their abilities, Mary and Valerie felt they were outliers. I worked with them as best as I could. I called on them infrequently in the main class, and focused on them during my intervention time.

At first, I tried to get them up to speed with what the rest of the class was doing. During one of my sessions with them, I went over one-step equations. I asked them to solve the equation 6x = 12.   I had reached the point where neither one was trying to subtract the 6 from 6x  to isolate x. But while Marie understood that 6x meant 6 multiplied by x , Valerie could not see that; nor could she see that solving it meant undoing the multiplication by division.

“How do we solve it?” I asked.

 “You put the 6 underneath both sides,” Valerie said.

That is:

Putting one number underneath another meant divide to Valerie which is as procedural-minded as you can get. If ever pressed to justify my acceptance of her level of understanding to well-meaning doing-math-is-not-knowing-math types I could say that her method at least incorporated the concept that a fraction means division. No one ever asked, but just to make sure, I said “And what do we call the operation when we put another number ‘underneath’ another?”

She thought a moment.

Mary whispered in Valerie’s ear: “Division”.

“Oh; it’s division,” Valerie said.

Over the next few weeks, I worked with the two girls privately while trying to keep them on track in the Math 8 class. I realized that their deficits were so significant that to hold them to the standards of Math 8 would result in failure. Katherine, the assistant principal agreed with me, and said to focus instead on filling in the gaps and to base their grade on their mastery of those.

I leveled with them one morning when they came in for their intervention.

“The Math 8 class must be extremely painful for you,” I said.

Valerie for the first time let down her guard.  “I just don’t understand what’s going on.”

I found her statement true in a number of ways.  One was just the fact that she admitted it. But more, it brought home the issue of understanding. Of course she didn’t understand—she barely had the procedural and factual tools that would allow her even the lowest level of understanding of what we were doing.

While there are those who would say “Of course they didn’t understand; traditional math has beat it out of them,” such thinking is so misguided that, in the words of someone whose name I can’t remember: “it isn’t even wrong”. In the case of Valerie and Mary, they needed more than the two half-hour interventions every week. They needed someone who specialized in working with what is known as dyscalculia. But qualifying as a special needs student doesn’t guarantee the student will get the kind of help to deal with a disability.

They had finished what I gave them to do early that day, and Mary being in a celebratory mood said the two were going to make a drawing for me. They giggled while they drew a rather strange looking bird laying eggs and eating blueberries among other odd things going on and presented it to me.  I put it on the wall where it remained for the entire school year. Sometimes other students would ask what the drawing was, and they would explain it, excitedly.  The excitement was partly due to them having drawn it, and partly due to my keeping it on the wall for all to see.