Out on Good Behavior, and a Final Narrative

Editor’s Note:  At long last!! This is Chapter 20, the last chapter in this 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: “I thank all my faithful readers for staying with this til the end.  The book will be out in the fall. That said, there will be no book tour.”  The previous chapters can be found here: Chapter 1 , Chapter 2 , Chapter 3 Chapter 4 , Chapter 5, Chapter 6, Chapter 7Chapter 8Chapter 9Chapter 10,  Chapter 11Chapter 12, Chapter 13, Chapter 14, Chapter 15 and Chapter 16, Chapter 17, Chapter 18, and Chapter 19.

Chapter 20 Out on Good Behavior, and a Final Narrative

My meetings with Diane during my second year at Cypress had taken place once a week, initially at a coffee shop near the school. When that proved to be too noisy we moved to my classroom. The day finally came when all electronic checklists had been filled out and discussions ended.

The potential for ending our discussions reminded me of a conversation I had in the Math 7 class regarding how you cannot divide by zero, nor zero by zero. I asked about the latter. Someone said “It’s zero”. I said, “Yes, that would be an answer.”  Someone else said “two”, another said “seven” and others threw out numbers until I said “There are lots of answers which is why we call it indeterminate.” I overheard Jimmy whispering to a classmate: “We could have kept this going for a long time.”

This struck me as a fitting description of my talks—first with Ellen and then Diane—which, like the mathematical concept of zero divided by zero, seemed to be indeterminate. On the one hand they were meant to help me be a better teacher. On the other the discussions were often fueled by a chain of misconceptions and ideologies built on the magical thinking found in most ed schools. And they could go on for a long time.

But all that was ending at long last. The principal joined our final meeting which Diane started by saying “This has been quite a year for you. The seventh grade class really gave you some challenges.”

“Yes, they did,” I said.  “You can lead a horse to water, as they say. But I think some of them drank it.”

 “You weren’t just leading them—you dragged them to the water. Kicking and screaming,” she said. This was an exaggeration of course, but it was in my favor so I let it go.

“Any words of wisdom for us on the mentoring process?” she asked.

I’ve had the “any words of wisdom” question asked of me by HR people as part of exit interviews at other jobs I’ve had. It’s one of those questions where they want to hear good things, but are willing to take their lumps.

 “I know we didn’t always agree on things,” I said.

“Yes,” she said. “It’s been interesting  You’re certainly not what I expected when we first met.”

Which was probably true. I’m definitely not someone in their twenties right out of ed school.  And while I had successfully avoided getting into knock-down drag outs about things like “productive struggle” and “differentiated instruction”, I did feel bad about some of our disagreements and how I had expressed them.

“In any teaching situation there are going to be people we don’t agree with,” I said with a bit of hesitation. I wasn’t sure where this was going to end up—a not unfamiliar feeling for some of my math lessons.

“Maybe we don’t agree with the way they teach or their philosophies about education. But somehow we all have to get along—we have to make it work,” I went on. “And even though I disagreed with you on some things, there were things that I did agree with and which were helpful. So that’s what I’m taking away from this.”

I don’t know if she viewed this as an apology, but I intended it as one. I could tell she meant well for me, and she had a good heart.  She seemed pleased with what I said.

We then moved on to other business, signing papers, and getting instructions on how to retrieve my final teaching certificate from a certain website. And then a picture of me holding my certificate of completion.

And that was that. I was now out on good behavior as a fully credentialed teacher, free to continue putting into practice my ideas about teaching math.  Free, that is, to the extent possible with having to attend PD sessions that are about engagement but pretend to be about instruction. Or hearing teachers talk about particular students’ learning styles. Or having discussions about how to instill students with a growth mindset, or being asked how I differentiate instruction in my classes, or being exhorted to engage students in productive struggle and, of course, having to be intentional. And above all: getting along with others.

I’ve been out on good behavior for over a year now and am tremendously happy at St. Stevens. As of this writing, I just completed my second year there. In keeping with my “We all have to get along” apology to Diane, I keep my views to myself. 

There is the occasional PD that I have to attend, but nothing as bad as what I’ve had to endure in the past. I hear teachers talk about blended learning and intentionality and growth mindsets now and then, but we all get along. And based on my evaluation from the principal, they’re willing to look the other way.

I’m also pleased to say that Lucy, my algebra student, took algebra again in high school and got A’s all the way through. I recall during the final exam in my algebra class, she asked me for help on a point slope problem. It asked for the equation of a line passing through a point, and perpendicular to a specific line.

“I don’t know how to do this,” she said.  I allowed them to have a cheat sheet and I pointed to the point slope formula that was on her cheat sheet. It was clear that my lesson deriving that formula didn’t stick with her.

A few minutes later I came back to see how she was doing. She was crying. 

“Oh, you’re upset,” I said.  “What’s the matter?” 

She pointed to her answer to the problem.

I looked at her work. “It’s correct! You got it right.”  

“But it doesn’t make sense,” she said.  

I’m fairly sure she thought the problem was asking for an ordered pair of numbers. Getting an equation for an answer – well, it didn’t fit her narrative, so to speak.

And as long as we’re on the topic of “narrative”, and also in the spirit of getting along with others, I offer my readers a choice of narratives that this episode represents, with varying nuance:

 1) Understanding always trumps procedures.

 2) It’s all part of formative assessment.

 3) At the novice level students focus on the procedure. Sometimes the understanding will come later. And for some, never.

 4) Teach understanding as best as you can but don’t obsess over it.

There are probably other narratives, but I’m somewhat new at this and therefore take a rather narrow and un-nuanced view of the world. So I will leave it to my faithful readers to add their own.  Just don’t tell me about them. I’m happy in my ignorance and from what I hear, doing just fine with what I know.

An Evaluation, the Red Book, and Checking for Understanding

Editor’s Note:  Chapter 20, the last chapter, will be out this Friday! This is Chapter 19 in a soon-to-be-ended 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. With the series almost over, the paparazzi are following me, so I’m probably looking forward to its end more than you are.”  The previous chapters can be found here: Chapter 1 , Chapter 2 , Chapter 3 Chapter 4 , Chapter 5, Chapter 6, Chapter 7Chapter 8Chapter 9Chapter 10,  Chapter 11Chapter 12, Chapter 13, Chapter 14, Chapter 15 and Chapter 16, Chapter 17, and Chapter 18.

Chapter 19: An Evaluation, the Red Book, and Checking for Understanding

Marianne, the principal at St. Stevens, would occasionally do informal observations of teachers without notice. I had such an observation about the second week of the school year during my Math 7 class. As she got seated at the desk she was greeted by John, one of my students hiding underneath. 

 “Sometimes they like to hide from me and surprise me,” I said.

This sufficed as an explanation and gave her a window into how I run my classes. It was a good lesson and on the Data Walk-Through form she was very positive about what she saw. She didn’t mention John’s hiding underneath the desk, but later in the day a teacher told me he heard about it, so apparently word gets around quickly.

A more formal evaluation occurred later in the school year. Prior to the event, I had to fill out a form outlining my plan, stating what standards would be the focus of the lesson, and how I would “differentiate the lesson to meet the needs of all learners”. I had said that I would give the stronger students more complex problems to do. I wasn’t sure whether that would occur in class or part of their homework, but I felt my answer was good enough.

The observation occurred in my algebra class, on graphing quadratic functions. The students in that class were a noisy bunch and quite spontaneous. Earlier that year during a sudden downpour the class cheered and before I could stop them, ran outside to get soaked in the rain, including Lucy my recalcitrant student.

There was nothing to worry about for my evaluation; the students were well-behaved. I had students come up and do graphing at the board as part of the lesson, I asked questions and called on those I knew would have answers, and then called on weaker students who, I was glad to see, were able to answer as well.

After going through the lesson, I then assigned students to small groups.  I did this to act in the manner that I assumed was expected of teachers aligned with the educational party line. I paired strong students with weaker ones, and gave each group an equation to graph. I circulated around to answer questions and inspect what was done.

About a week later, I met with Marianne in her office, to go over her observations of that particular lesson. She handed me her written comments while she talked to me about her observation of my class.

“I really thought that was a good lesson,” she said. “They followed the explanations, they were engaged, it was well-scaffolded, and it’s clear that they really like you.” 

She offered me more praise and it was obvious from this and previous conversations with her that she thought well of what I was doing. But evaluations being what they are she then brought up her concerns.

“I notice that the textbook wasn’t used in this lesson.  How are you using it? Do you use it for homework?”

I wasn’t sure whether she was asking about my use of Dolciani’s textbook or the official textbook. So I proceeded cautiously.  “The lesson was actually taken from the blue textbook,” I said. (This is the official one). “I incorporate problems from the book into my lesson and yes, the homework is generally taken from the book.”

I could see that she was concerned over whether we were adhering to the Common Core standards as did public schools. I understood this—it would be bad for business if students graduating from St. Stevens would be at a disadvantage in public high schools. In her mind, sticking to the official textbook meant compliance with Common Core.

And while I had mentioned in my initial interview my extensive use of the Dolciani book in teaching algebra, I discerned that such information had not stuck with her. So I offered further clarification.

“I primarily use the book by Dolciani.”

“That’s the red book I’ve seen students with?”

“Yes. I dislike the blue book; I think Dolciani is much better.”

“But you do use the blue book?”

“Yes; to cover what isn’t addressed in Dolciani—like the lesson you saw, and pretty soon exponential functions.”

“So you’re saying you use the red book as a supplement?” she asked. 

I said “Yes” even though I believe we both knew that “supplant” would have been more accurate. But people hear what they want or need to hear.

Our conference ended positively and afterward I read her written notes. She questioned how I checked for understanding, noting that I paired weaker students with stronger in my small groups. How I would assess whether the weaker could have completed the lesson without the stronger?

Excellent question; I had to agree. In acting the way I thought I was expected to act I hadn’t considered that perhaps Marianne disliked small groups as much as I do. In fact, I hate small groups. I took her question as a sign that perhaps Marianne hated them as much as I do. As far as how one checks for understanding, there are many ways. I’ve made a note that next time I’m asked, I will rely upon the ed school catechism that formative assessment is a process not an event.

She also noted that I did not do what I said I would in the pre-observation form; namely give challenging problems to the stronger students. Her recommendation: “More intentionality to provide challenge to those students that need it.” To be honest I forgot that I said I would do that.

She reiterated her concern about the textbook and recommended “Intentional correlation between lesson and text to ensure Common Core standards need for Algebra 1 mastery are adequately covered.” I’m always a bit confused about the word “intentional” as used in education. But I think I assured her that my use of text (both “red” and “blue”) was not by accident.

Common Core Advocates’ Latest Strategy to Deceive Parents

Supporters who want to keep Common Core’s failed standards in place have come up with a new twist for deceiving unhappy parents.  First, they point explicitly to Common Core as a failed strategy to increase the academic achievement of low achievers in order to alert parents to what has happened

They do what seems at first confusing because it is widely known that most parents and teachers (if they felt free to speak their minds) detest Common Core’s standards, tests, and aligned textbooks or readings.  All Common Core’s failings and limitations are real. While the many articles on the decline in student achievement in a Common Core-aligned educational environment tell the truth, there is malice in the schadenfreude expressed about the many disadvantaged kids who have been deprived of the educational equity that Common Core was initially touted as creating. 

The strong possibility of public deception is suggested by two phenomena.  First, there has been no media clamor in reports of Common Core’s failures for stronger standards and curriculum materials.  Second, Common Core’s major supporters—the bureaucracy at U.S. Department of Education (USED), most if not all state departments of education (e.g., DESE), and the Gates Foundation—have kept their financial and political commitment to the failed strategy. Not one major foundation, or USED, or NAEP educator has advocated that Common Core’s materials and approach be replaced with more effective ones despite their failure.  The problems, they and many others claim, lie with the mandated tests—and lack of federal money.  Explicitly, they don’t like “standardized” tests but think “performance-based assessments” would do the trick, even though they are costly, time-consuming, and unreliable. 

Strange.  State commissioners and departments of education have long known that in order to get rid of Common Core-based tests and get really different tests they must get rid of Common Core’s standards. They also know that Common Core-aligned standards and textbooks are in each state’s 4-year state education plan—all approved by the USED bureaucracy in 2016 or 2017 and that these Common Core-aligned standards MUST be used until 2020.  That’s why the strategy of public deception is taking place this year.  

Some states may seem to be changing their K-12 math and ELA standards right now.  But what the USED bureaucracy approved for the Every Student Succeeds Act or ESSA (the Obama/Duncan administration’s title for the revision of the Elementary and Secondary Education Act, passed in December 2015 and sponsored by Senators Lamar Alexander and Patti Murray) in a state plan written in early 2016 and approved soon afterwards is in control.  Every state department of education knows that, even if the public still doesn’t know who wrote ESSA and paid for it.    

But ways to strengthen the K-12 curriculum are available.  Most of the old pre-Common Core standards (pre-2009) are still available even if archived away (e.g. Massachusetts’ original pre-2009 standards). Or states could do what many countries like Finland have always done at the high school level: (1) create syllabi (i.e., course outlines showing content and readings to be taught) for all the courses that students are required to take in high school for their particular curriculum program (in Finland, there may be over seven to choose from) and (2) also require all students who want to go on to a 4-year college to take a “matriculation” test. 

If states like Massachusetts do this, the governor or secretary of education has to ensure that the committees creating syllabi for high school courses consist of experts in both pedagogy and content, such as classroom teachers in grades 11/12 and college profs who teach math and science to freshmen in engineering schools.  If high school syllabi (or standards) are created by school administrators or teachers for learning disabled or low-achieving students below grade 11, they are useless and invalid.  They will be Common Core standards warmed over

In the meantime, the Heritage Foundation seems to think that removing cabinet-level status from the USED will cure the ailments to public education inflicted on public education by the Gates Foundation. Making the USED a lower-level education agency, it claims, will enable parents to regain control of the local K-12 curriculum.  How this miracle would remove the damage the Common Core project has imposed on public education is anyone’s guess. But if both “conservatives” and “liberals” support the idea, like the “lockdown,” it will happen and in another decade we will all wonder why this country chose to shoot itself in the foot.

Cross posted from New Boston Post

An Unexpected Narrative, a Limbic Dialogue and a Note from Ellen

Editor’s Note:  This is Chapter 18 in a soon-to-be-ended 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. I appreciate your devotion to the series and hope you will not regret the time spent doing so.”  The previous chapters can be found here: Chapter 1 , Chapter 2 , Chapter 3 Chapter 4 , Chapter 5, Chapter 6, Chapter 7Chapter 8Chapter 9Chapter 10,  Chapter 11Chapter 12, Chapter 13, Chapter 14, Chapter 15, Chapter 16 and Chapter 17.

Ch 18  An Unexpected Narrative, a Limbic Dialogue and a Note from Ellen

Having worked for forty years in both the private and public sectors prior to my retirement, I can say that the axiom I formulated in Chapter 14 applies to both the teaching and non-teaching worlds. Restated: You never know for sure what’s really going on, but general suspicions suffice to fit the narrative at hand until nuances prove otherwise.

I had kept my notice of termination at the Cypress School under wraps, having told only Diane, and one teacher who I trusted. I didn’t want the students to know. I heard stories from other teachers about receiving lay-off notices every year and getting hired back. Even the Superintendent had said that it could be rescinded.

This bolstered a vague belief that the same would happen to me. I was therefore surprised when on a Monday in May, the teacher who I had told said “I’m sorry to hear you were let go.”

Clinging to the narrative I had been given to understand, I said “That can be rescinded.”

She looked at me, and speaking rapidly, informed me of a new narrative. “Sandra has been told she will be replacing you.”

This development had occurred the previous Friday; the whole school knew but me.  Sandra who taught third grade was told that she would be teaching my math classes the next year. In fact, she had taught middle school math with James a few years ago. They would be hiring a third grade teacher to replace her.

My first thought before going into limbic mode was that at least I wasn’t being replaced by Sandra’s student teacher who had observed my algebra class the previous year. “Was anyone planning on telling me?”

“The principal is in her office,” she said in very kind tones.

I took the hint and stormed into the principal’s office. By way of greeting I said “Well, I just heard that despite declining student enrollment, there’s still going to be two math teachers and I’m not one of them. What the hell does this mean?”

“Yes, sometimes people are shuffled to make things come out right.”

“Meaning what, exactly?” I asked. 

The principal was a very nice woman who genuinely liked me but was not one to go against the Superintendent.  “Why don’t we talk to the Superintendent about this?” she suggested.  

I was in full limbic mode when we entered his office. I broke the ice.

“I have been more than professional about all of this and I am not happy that I am finding out from another teacher about Sandra taking over my job,” I said. “Why wasn’t I informed about this decision?”

“Do you think I have to share information about staff decisions?”

“Given that this affects me, I think that would be safe to say,” I said.

“Look, as I told you before, this had nothing to do with performance. I can’t go against the law though and I had to let someone go. Since you were the least senior, it was you.”

This reference to the law was interesting. I was thinking:  he lets me go, moves the more expensive Sandra into my spot, gives her other assignments so she retains full time status (I was part-time), and then hires a teacher to replace her. Seems like there are just as many teachers as before and more money expended. I felt like saying “You do realize I teach math, don’t you?” 

“It’s like in baseball,” he said.  “They let players go with trades all the time. It’s the same here.” I said nothing. “Look, you know I think highly of you. You taught a tough class of seventh graders who love you. I was even going to make you permanent.”

This was true; he had told me that. But something happened to make him change his mind. I knew I’d never know what it was.  As he went on talking, I realized it was over for me and nothing was going to change.

My limbic mode having subsided I became professional once more, like a baseball player being traded. I told him I appreciated the opportunity to vent, because it would have festered had we not had the discussion. I didn’t apologize for my initial outburst

Later as I was sitting in my classroom during lunch there was a knock on the door. It was Sandra. I had the door closed, since this was one day I didn’t want the usual group of my algebra students congregating, as much as I enjoyed them.

“I just want you to know that I had no idea you hadn’t been told what happened. I feel terrible about it,” she said.  “And believe me, I really don’t want to teach math. I want to stay teaching third grade. I was told that this is what I’m going to do.”

“So you and James will be the math teachers?”

“Yes.  We taught together before. I’ll be teaching algebra. We’ll both be teaching sixth grade because it’s a big class.”  I could tell she felt bad about the situation. She said she was going to tell the principal she didn’t want to teach math and to be kept in her position teaching third grade. I knew that nothing would change and it didn’t. 

I kept my silence about events; my students didn’t know what happened. Over the next week, with the exception of James, teachers expressed their regrets. I notified Diane and Ellen. Diane said she would be happy to be a reference. And Ellen wrote me the following note:

“I am so sorry that you have had to endure this undeserved trauma.  I have worked with new teachers for 20 years and have witnessed this process over and over again. Most of my teachers have gone on and taught at other schools and never looked back.”

I would shortly be hired by St. Stevens, but in the meantime I found it oddly comforting to know that despite conflicting narratives about education, some things never change.  And with that, I began the narrative of preparing my final exams.

“Math Talk”, Stalin’s Hemorrhoids, and Murder of Crows

Editor’s Note:  This is Chapter 17 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. Thanks to my faithful readers for hanging on. There are only three more chapters to go until the end.” The previous chapters can be found here: Chapter 1 , Chapter 2 , Chapter 3 Chapter 4 , Chapter 5, Chapter 6, Chapter 7Chapter 8Chapter 9Chapter 10,  Chapter 11Chapter 12, Chapter 13, Chapter 14, Chapter 15 and Chapter 16.

Chapter 17: “Math Talk”, Stalin’s Hemorrhoids, and Murder of Crows

The energy that accompanies the start of the school year begins to dwindle noticeably around Thanksgiving, continuing through the approach of Christmas. It starts up again for a short time in January. Around February or March, when the rains bring tree frogs, students (and teachers) start to sense that spring break is near with summer vacation soon following.

Seventh graders are growing and starting to look like eighth graders. And eighth graders are now looking ahead and becoming nostalgic for what will soon be a big part of their past. It is a nostalgia in advance—a holding on to the familiar at the same time as saying goodbye.

In my Math 8 class at St. Stevens, the holding on to the familiar manifested itself in even more conversations than normal. The literature on math education does not talk much about eighth graders’ conversations. I recently saw an article claiming that “research shows” that students who talk about their math thinking are motivated to learn. In addition, this “math talk” is viewed as a form of formative assessment giving teachers a peek into student thinking and where they need help.  

I believe that motivation comes from proper instruction which allows students to carry out the tasks and achieve success.  “Math talk” is an effective tool only if the instruction they received allows them to make use of it. Otherwise, it is like children dressing up in their parents’ clothes to play “grownups”.

As far as a peek into student thinking, sometimes the conversations pertained to the math problems they were working on—and sometimes not. But as long as they were working on math, I didn’t mind. Shortly after the tree frog incident when Jared tried unsuccessfully to find the loud croaker, I was putting my plan to teach them more algebra into high gear. I had introduced some simple factoring exercises which  Jared found these fun and even said “Invigorating!” as he did them.

We were making fairly good progress with factoring, but when we got to algebraic fractions they got a bit bogged down. I had to continually remind them to factor in order to simplify.

“I don’t like factoring,” Jared said.

“Why? You told me they were invigorating a few weeks ago.”

“That was before they got complicated,” he said.

His friend Kevin chimed in. “Factoring messes things up,” he said.

“Look at it this way,” I said. “When you take algebra next year in high school, you will have seen all this already. You’ll be wondering why you thought this was difficult.”

Mary and Valerie had their own private conversations which would often merge with the others’. One particular conversation and its tributaries comes to mind. Valerie, avoiding saying the word “hell” said “H, E, double hockey sticks.”

Lou reacted to this. “There’s nothing wrong with saying the word ‘hell’. It’s a place,” he said.  Discussion followed about when “hell” was permissible to say and when it was not.

“I don’t see the big deal,” Lou said.

“You would if you were Catholic,” Valerie said.

“OK, I’m not Catholic, but I believe in Jesus. I just think Catholics are too strict about some things.”

Kevin chimed in “Well, this is a Catholic school so there are certain things you have to go along with.”

“Hell shouldn’t be one of them,” Lou said though it was unclear whether he meant the concept of hell itself or about saying the word.

Kevin then asked me how to find the lowest common denominator of two algebraic fractions.  As I was showing him, Mary, who clearly did not want to do any more work asked, “Lou, if I died would you cry at my funeral?” 

“Well, I would be sad,” he said. “But I don’t cry easily.” 

“What would it take to get you to cry?” she asked.

He appeared to be in thought. “I don’t know. When my grandmother died I didn’t cry, but when my dog died, I did. I don’t understand why.”

I had finished helping Kevin with his problem, and thought I might help Lou with his. “You don’t always cry when someone dies,” I said. “When my mother died last summer I was sad but I didn’t cry.” 

“Sorry about your mom,” Valerie said.

The room grew suddenly quiet; students are listening when you least expect it.

“But then I had a dream about her one night,” I said. “And when I was telling someone about the dream, I started crying.”

 “You probably cried because you knew you were saying goodbye,” she said.

Which, unbelievably was what the dream was about. I had to go somewhere but couldn’t take my mother with me so I had to say goodbye. There was no need to mention that so I didn’t and conversations returned to less somber topics—in particular, the history paper Lou was writing about Stalin. “Stalin died from hemorrhoids,” he said.

“How can you die from hemorrhoids?” I asked.

“From complications.”

“I assume that will not be covered in your history paper,” I said, but his answer was drowned out by a noisy chorus of crows in the tree outside.

“Can I chase the crows away?” Jared asked.

“Stay seated,” I said and shut the door to the classroom.

“The door won’t block out the sound,” Jared said.

I shut the door and the crowing stopped. I said “Do you hear them now?”

Never at a loss for a rejoinder, Jared said “The motion of the door scared them away.”

“Do you have proof of that?” I asked.

He didn’t answer.

“I’ll take that as a no,” I said.

“Can we play hangman?” Jared asked.

“If you’re all finished with your math problems.”

“There’s only two minutes left of class,” he said.

I gave the go-ahead, and Lou and Jared jumped up to put their hangman challenge on the board. I realized at that moment that I had grown quite fond of this class, and that our little rituals were starting to feel like we were saying goodbye.