Friday, November 28, 2014

Conversations on the Rifle Range 17: Boundaries of Behavior, Parallelograms, and the Art of Forgiveness

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 17:

There are a variety of methods one can use to discipline students: detentions, referrals, sending the student outside of class, contacting the parents. I was confused about most of them and resisted using them. Lunch-time detentions were especially tricky because of a dual lunch schedule at my school. Because of the limited space for lunch there were two lunch periods for the two grades. This meant that during the eighth grade lunch period, I was teaching my fourth period class (pre-algebra).

The first person I ever referred was Peter in my fifth period algebra 1 class. He showed disrespect in a number of ways. He would sometimes say in a sarcastic Eddie-Haskell-like tone: “I think you made a mistake—oh but I know you’re a great teacher,” which would elicit knowing giggles from others. One time when he was particularly disruptive, I sent him outside which in this school meant outdoors. The school was a collection of modules—all classrooms opened to the outdoors. Sandra, another disrupter, waved to him on his way out and called “We love you, Peter.” He has a fan club, I thought—just what I need.

Her seat was next to the wall on the other side of which Peter now stood. She pounded on the wall to get his attention. I heard the pounding, and saw Peter’s head appear in the window as he jumped up to see what was going on. Not knowing the details of the event, I assumed wrongly that Peter had been doing the pounding. I got him back inside and gave him a referral. As I filled out the form, Peter protested and Sandra quickly confessed. “It was me who was pounding on the wall,” she said. I knew Sandra was telling the truth but I decided I had no time for details; the die had been cast. I needed an example. Plus, if the class thought I was acting irrationally or in error, then it was a signal that they better be quiet and not risk my irrational actions.

My referral was thus performed as an irrevocable act. I called the office and said that I was referring Peter and he was on his way. I then turned to Peter and handed him the completed form. “They’re waiting for you.” Butter would not have melted in my mouth.

It worked. The class was like a tomb. I remember thinking “I didn’t mean for them to be this quiet.” But I went on as if nothing had happened and the class noise level eventually increased—but not to its usual level. At the end of class Sandra again confessed and told me she would give herself an after-school detention. “That’s fine,” I said and thought it a good opportunity to give her some of the tutoring that she desperately needed. She would keep putting it off and would not show up; I never pushed the issue.

The next day I dreaded my fifth period algebra class, knowing I would have to confront Peter. I didn’t know how that would go, or even if there would be one. Earlier that day, I sent out a boy named Chad in my third period pre-algebra class. He was noisy, and given to saying “Sweet Jesus!” when amused or surprised or frustrated, and would make what I thought to be frog-like croaking sounds. After hearing such a croaking sound I told him to go outside. He croaked “Why?” and I explained “Those croaking noises are precisely the reason I’m sending you outside.” He left and the class became tomb-like as had my algebra class the day before. After a few minutes, I went outside to talk with Chad.

“I want you to control your outbursts. I know you’re a nice boy and can do this work, but I can’t have the class disrupted like this.”

“Well I wish you’d tell Bryan to stop barkin’ like a dog!” he said. While I was unaware of Bryan’s dog imitations, this was the first time I heard Chad in conversation. Except for outbursts, he did not participate in class. I suddenly realized that Chad’s croaking was his natural voice—perhaps the first stage of a changing voice which starts around middle school for most boys. I felt terrible. I told him I’d keep an eye on Bryan.

What with behavior boundaries established (and clarified) we went back inside where I proceeded to demonstrate a more mathematical view of boundaries—namely that the area of a parallelogram is the same as that of a rectangle. I did it by means of magnetic shapes I had made that stuck to the whiteboard. The parallelogram had a right triangle indicated as below. “Now it so happens that this triangle…”

The class didn’t let me finish and many shouted: “You can move it to the other side!”

Which I did, thus transforming the parallelogram into a rectangle without altering its area. “Sweet Jesus!” Chad croaked. I took this as a Q.E.D. for my informal proof.

The day wore on until at last the fifth period that I had been dreading arrived. Peter came into the room and said "Hi, Mr. G!" I greeted him back with a wave and while I began taking attendance he asked "Do you hate me, Mr. G?"

"I don't hate you and I never have,” I said. “I felt you were showing me disrespect so I gave you a referral."

"I wasn't disrespectful yesterday," he said.

"You've been disrespectful for quite some time. I think you know that." He nodded and said nothing else. I realized then that no matter what disciplinary action is taken, it necessitates the equally important act of reconciliation. Our brief conversation was it, and we moved on.

I could feel a chill that day amongst the people who were his friends. Perhaps they felt as I did 50 years ago when, one day during spring semester, I mouthed off to Mr. Dombey, my algebra teacher. He raised his voice and I felt he hated me just as Peter felt I hated him. I remember feeling betrayed, then confused. I don’t recall how it was reconciled; I just remember that the next day it no longer mattered. I knew where the boundary was, and things were both different and the same.

Thursday, November 27, 2014

Turkey Grammar Answer Key

1. Even more ridiculous is the idea of cooking it in a bag.
2. Overstuffing the turkey makes the stuffing come out dense and the turkey difficult to cook properly.

A happy thanksgiving to all!-- And may your turkeys be neither ridiculous nor improperly cooked.

Tuesday, November 25, 2014

Your syntactic toolkit, I: Two tools for Turkey Grammar

Last year I posted a Turkey Math problem; this year it’s time for Turkey Grammar.

Here are your two syntactic tools:

Tool #1: Inversion:

Some sentences contain phrases that can be moved to the front, inverting the subject and verg. For example:

People make pies out of pumpkins only here in America
Only here in America do people make pies out of pumpkins.

Tool # 2: “Tough movement”:

Sentences containing words like “easy” and “difficult” allow a variety of possible word orders. For example:
Cooking turkeys thoroughly without drying them out is notoriously difficult.
It is notoriously difficult to cook turkeys thoroughly without drying them out.
Turkeys are notoriously difficult to cook thoroughly without drying out.

Exercise 1: Invert the second sentence to make it link up better with the first one:
Some people think that soaking a turkey in brine overnight makes it tastier. The idea of cooking it in a bag is even more ridiculous.

Exercise 2: Sharpen this sentence via tough movement.
Overstuffing the turkey makes the stuffing come out dense and makes it quite difficult to properly cook the turkey.

Sunday, November 23, 2014

More Common Core-inspired issues: the communication skills of non-native English speakers... and of Common Core Authors

Two side-by-side articles in this past week’s Education Week show a disconnect between what the Common Core authors vs. actual classroom teachers think are the biggest challenges posed by the Common Core. First, there’s an interview with William G. McCallum , the lead author of the Common Core Math Standards. McCallum cites coverage of fewer topics as the biggest change brought by the Common Core, and fractions, ratios, and proportional relationships as the biggest challenges to teachers.

For teachers, on the other hand, what seems to be most novel and challenging is the Math Standards’ emphasis on conceptual understanding and verbal communication. This is particularly true in the case of teachers of language-impaired students and students whose native language isn’t English. The latter are the focus of the other article.

When he began working the Common Core State Standards into his instruction three years ago, New York City middle school mathematics teacher Silvestre Arcos noticed that his English-language-learner students were showing less progress on unit assessments than his other students.
"It wasn't necessarily because they didn't have the numeracy skills," recalled Mr. Arcos, who is now a math instructional coach and the 7th grade lead teacher at KIPP Washington Heights Middle School, a charter school in New York. Rather, they were struggling with the linguistic demands of his new curriculum, which was oriented heavily toward word problems and explication of solutions.
To address the issue, Mr. Arcos began incorporating strategies that are typically the province of language arts teachers into his math lessons. Especially when working with his English-learners, he provided detailed instruction in close reading, sentence annotation, and writing fluency.
Nor is Mr. Arcos alone:
Mr. Arcos' recognition that the new math standards may require greater attention to the needs of English-language learners is not uncommon among educators who work with such students. Particularly in the Standards for Mathematical Practice that preface and inform the grade-level objectives, the common core emphasizes the importance of explaining solutions and relationships, constructing arguments, and critiquing the reasoning of others. While such expectations are proving difficult for many students, educators say, they pose unique challenges for those not fully proficient in English.
When I was a 6th grader in a school outside Paris, immersed among native French speakers, math class offered refuge from the linguistic challenges of my other classes. From the beginning, with minimal knowledge of French, I was able to follow what was going on on the chalkboard. And could figure out what to do on homework and tests. Nor do I feel like my math experience would have been any richer had the word problems involved more elaborate French sentence structures and vocabulary, or had I been required to explain my reasoning in French. In fact, the best elementary school math class I had was that 6th grade math class, with its solidly conceptual and engaging French math curriculum. When I returned to the U.S. for 7th grade, I was, in fact, ahead in math relative to my peers. Had my French math class gotten bogged down with “detailed instruction in close reading, sentence annotation, and writing fluency,” I would surely have instead ended up significantly behind.

So is “detailed instruction in close reading, sentence annotation, and writing fluency” in math class really what’s best for our students—whether or not they are non-native English speakers?

Not surprisingly, professors of mathematics education, as opposed to professors of mathematics, applaud this continued dilution of math with English:
In addition, the common core's emphasis on verbal expression and reasoning in math are widely seen as beneficial to English-learners. "The more language you use in the math classes, the more [ELL] students are going to learn, both in math and language," said Judit N. Moschkovich, a professor of mathematics education at the University of California, Santa Cruz.
Especially because it becomes one more excuse to have students work in groups:
At the same time, a Teacher Notes panel provides specific activities teachers can use to help English-learners engage with the language of the lesson. One such exercise says: "Have students work with partners to discuss the graphic organizer and fill in the sentence frames [provided]. Then have them use the word bank [provided] to fill in the summary frame."
As for Common Core Math Standards lead author William McCallum, he seems blind to the problems posed by the Standards’ perceived emphasis on verbal expression. When asked if there’s anything he might change about the Common Core, all he mentions are the geometry progression in the elementary school Standards and the level of focus in the high school Standards:
“I think the geometry progression could be evened out a bit in elementary school. I think in high school there could be more focus. High school was difficult because everybody has their pet topic, and it was difficult to resist those pressures.”
As for challenges of particular Common Core-inspired problems or of conceptual understanding, McCallum blames these on mis-implementations or misinterpretations:
“What's interesting to me is that both the supporters and the critics of the common core, I think, are overemphasizing conceptual understanding—and understandably because everybody's always demanded procedural fluency, and the conceptual understanding is what's new. But that doesn't make the other requirement go away.”
Well, what’s interesting to me is that the lead writer of the Standards (a) thinks that conceptual understanding is something new in math education (b) has written something that he acknowledges is being mis-implemented and misinterpreted, and (c) has failed to do anything to stop this.

Thursday, November 20, 2014

Math problems of the week: Common Core-inspired geometry problem

The Common Core Standard in question:


Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

The source:

The problem:

The justification, solution and scoring:

Student reactions to this kind of problem:

An alternative, more tradition proof:

Extra Credit:

Which proof do you like better?

Use transformations to prove that the two proofs are (or aren't) similar.

Tuesday, November 18, 2014

Conversations on the Rifle Range16: Parallelograms, the Mercy of the Court, and Kit Kat Bars

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 16:

A “prep period” –a period in which teachers have no class—is one of education’s most sacred and cherished traditions. Mine was first period and involved making copies and putting finishing touches on lesson plans, as well as pacing nervously in anticipation of the day ahead. The stage fright dissipated when class began but would return if I didn’t pace things right and had slack time left over at the end of class. The resulting restlessness of students spelled disaster and invited clock-watching, students getting out of seats, general disruption, and lining up at the door (though I put an end to that practice quickly).

My fourth period pre-algebra class was the one class I dreaded most, though in the end it turned out to be my favorite. It was the most crowded, and also had six students who were “English Learners”. A Spanish-speaking aide was in class to help them. Mrs. Halloran had explained to me that because they were “low” in ability, they weren’t part of the regular class, and were relegated to the back of the class where they worked on an online course on computer tablets. They were given a special “pass/no grade” credit for the class. With the aide working with six students, the conversations at the back of the room served as a stimulus for other students, many of whom had self-control problems to begin with, to start talking. When the din got to a certain volume I had to raise my voice to quiet them. If that failed, Allysandra, a rebellious Mexican girl would yell “SHUT UP” at the top of her voice. This would generally do the trick.

Then there was Trevor, a disruptive boy, not well-liked by the other teachers. He got into fights and was even suspended for a week for one of them. He had a gift for arguing convincingly even when wrong. I first noticed this when he put some make-up work in the wrong place. When I told him where the correct bin was, he said “Chill.” My reaction was so swift it caught both me and the class by surprise: “Don’t you EVER tell me to chill!” I said. The class became a tomb.

“No, I didn’t mean it like that,” he said. “I meant ‘chill’ in the same way you say ‘cool’ or ‘OK’. That’s how I meant it.” He was convincing so I backed off and the class din resumed after about a minute. When I related this tale the next day to a teacher, she told me “Oh, that’s Trevor! He always has some excuse about how he didn’t mean this or that. He’s good at that.”

I mentioned to her that he was on the school’s mock trial team. This team competed with other schools in a mock trial, judged by a real court judge. “Given his gift for arguing and being on the mock trial team, I would guess he’ll end up being a lawyer,” I said.

“I hate to think who his clients will be,” she said.

I put that thought aside and tried to reach out to him. He would sometimes read a book quietly if he either 1) finished his homework early or 2) was avoiding doing the homework. Rather than try to find out which it was, I would ask him about the book he was reading and he would tell me. I sensed a more cooperative side to him, but wasn’t sure whether it was because I was showing an interest in him, or because of the impending championship mock trial coming up that was helping him focus his energies. Since I had members of the mock trial team in all of my three pre-algebra classes, I decided to tap into these students’ argumentative gifts during the unit on geometry.

I was finishing up a chapter on parallelograms. I decided to put a figure up on the screen in each of my pre-algebra classes and asked if anyone could tell me if two particular line segments in the figure were congruent and why. Some parallel lines were marked as such. The two line segments of interest were marked as being perpendicular to one of the two parallel lines and were, in fact, opposite sides of a parallelogram and therefore congruent.

I offered a Kit Kat bar to any student who could answer the question. Students immediately rose to the challenge. In all classes, some student would inevitably say “Can’t you just measure the two segments?” to which I would reply “Inadmissible evidence! The court will not allow rulers or any type of measurement devices in this trial. Demonstrations must be made using definitions and theorems only.”

As expected, the members of the mock trial team rose to the challenge. Some tried to get around my restriction of no measurement devices by saying “They look equal” but I easily put that to rest. “Not adequate. Visual comparisons are not allowed.”

In fourth period, upon hearing the “Inadmissible evidence” warning, Trevor rose to the challenge. He stood up and said “I got this! I got this!” and then proceeded to make spirited, breathless demonstrations that didn’t quite make the case.

I gave him some hints. “Do you think these two line segments are parallel?” I asked.

“Yes, definitely,” he said.


“Because they’re both at right angles, at right angles!” Trevor said as if pleading to a jury that his client did not deserve the death sentence. “What about the right angles?” I asked.

“It proves it,” he said.

“Proves what?” I asked.

“Proves that the lines are congruent.”


“Because they’re right angles, they’re right angles!”

I paused as if giving the matter great thought and the class quieted.

“Do you mean to say if two lines are perpendicular to the same line they are parallel?”


“So why are they congruent?”

Someone shouted “Because it’s a parallelogram!”

In the best spirit of courtroom drama, Trevor protested: “Unfair! I was going to say that!”

“There will be silence in the courtroom,” I ordered to no avail. “Counsel will be seated, please,” I said and continued: “The court will show mercy and recognize that counsel’s observations and arguments have merit and has provided indications that he knows that opposite sides of a parallelogram are…what?”

“Congruent!” Trevor shouted.

“One Kit Kat bar is awarded.” The class applauded, though the person who identified the figure as a parallelogram wanted one also.

“Come up and claim your Kit Kats,” I said and presented the awards. To Trevor’s credit and my satisfaction, after he took his Kit Kat, he shook my hand.

Sunday, November 16, 2014

Autism Diaries: reversing heart rates

J spent at least the least the first 15 years of his life relentlessly raising the heart rates of everyone around him. In years 1-5, he'd constantly throw things around and break them; turn appliances off (the lights in the evening; the refrigerator and/or freezer) or on (the heat in summer; burners); run into other people's yards or ahead of us into the street (face turned towards us, grinning); push and grab people and poke them in the eye; and vanish in pursuit of ceilings fans. In years 5-10 he'd disrupt his classes and alienate his teachers; charge through crowded hallways and thoroughfares, force random people to sign two; bother the heck out of his siblings; and vanish in pursuit of ceilings fans. In years 10-15 he continued to disrupt his classes and alienate his teachers and charge through crowds, as well as engaging in increasingly sophisticated mischief and vanishing in pursuit of ceiling fans.

Fast-forward a few years. The mother of a dear friend who works with J has been in the ICU all week. And so she, too, has been in the ICU all week, at her mother's side. But she misses J tremendously--as she often does when separated from him for more than a few days. She misses, in particular, the fresh air and levity he provides when times are tough: his innocent questions ("How is your mom's heart?") and hopes ("When she gets better, do you think we can go to the restaurant with fans?"). So, two days ago, she asked if I could drop him off near the hospital so that she and her mother (another fan of J's) could "get some J time."

Afterwards she wrote me a text message commemorating what has to be one of the biggest milestones we've seen in his 18 years:

I am literally in tears over how sweet J is. Thanks for sharing him with us. He was a calming presence and brought my mom's heart rate down to the lowest it's been since she got here.
Who could ever have predicted that, 18 years on, J could not only provide comfort when times are tough, but enter an ICU and bring someone's heart rate down?