# Re-invitation

Dylan Kane’s recent post about prerequisite knowledge has me wanting to tell you a story from my very first year in my very first full-time classroom job, which I think I’ve never related on this blog before, although I’ve told it IRL many times.

It was the 2001-2002 school year. I taught four sections of Algebra I. I was creating my whole curriculum from scratch as the school year progressed, because the textbook I had wasn’t working in my classes, or really I guess I wasn’t figuring out how to make it work. Late late in the year, end of May/early June, I threw in a 2-week unit on the symmetry group of the equilateral triangle. I had myself only learned this content the prior year, in a graduate abstract algebra course that the liaison from the math department to the ed department had required of me in order to sign off on my teaching degree, since I hadn’t been a math major. (Aside: that course changed my life. I now have a PhD in algebra. But that’s another story for another time.)

Since it was an Algebra I class, the cool tie-in was that you can solve equations in the group, exactly in the way that you solve simple equations with numbers. So, I introduced them to the group, showed them how to construct its Cayley table, and had them solving equations in there. There was also a little art project with tracing paper where they drew something and then acted on it with the group, so that the union of the images under the action had the triangle’s symmetry. Overall, the students found the unit challenging, since the idea of composing transformations is a profound abstraction.

In subsequent years, I mapped out the whole course in more detail beforehand, and once I introduced that level of detail into my planning I never felt I could afford the time to do this barely-curricular-if-awesome unit. But something happened, when I did it that first and only time, that stuck with me ever since.

I had a student, let’s call her J, who was one of the worst-performing (qua academic performace) students I ever taught. Going into the unit on the symmetry group, she had never done any homework and practically never broke 20% on any assessment.

It looked from my angle like she was just choosing not to even try. She was my advisee in addition to my Algebra I student, so I did a lot of pleading with her, and bemoaning the situation to her parents, but nothing changed.

Until my little abstract algebra mini-unit! From the first (daily) homework assignment on the symmetry group, she did everything. Perfectly. There were two quizzes; she aced both of them. Across 4 sections of Algebra I, for that brief two-week period, she was one of the most successful students. Her art project was cool too. As I said, this was work that many students found quite challenging; she ate it up.

Then the unit ended and she went back to the type of performance that had characterized her work all year till then.

I lavished delight and appreciation on her for her work during that two weeks. I could never get a satisfying answer from her about why she couldn’t even try the rest of the time. But my best guess is this:

That unit, on some profound mathematics they don’t even usually tell you about unless you major in math in college, was the single solitary piece of curriculum in the entire school year that did not tap the students’ knowledge of arithmetic. Could it be that J was shut out of the curriculum by arithmetic? And when I presented her with an opportunity to stretch her mind around

• composition of transformations,
• formal properties of binary operations,
• and a deep analogy between transformations and numbers,

but not to

• do any $+,-,\times,\div$ of any numbers bigger than 3,

she jumped on it?

Is mathematics fundamentally sequential, or do we just choose to make it so? I wonder what a school math curriculum would look like if it were designed to minimize the impact of prerequisite knowledge, to help every concept feel accessible to every student. – Dylan Kane

Acknowledgement: I’ve framed this post under the title “Re-invitation”. I’m not 100% sure but I believe I got this word from the Illustrative Mathematics curriculum, which is deliberately structured to allow students to enter and participate in the math of each unit and each lesson without mastery over the “prerequisites”. For example, there is a “preassessment” before every unit, but even if you bomb the preassessment, you will still be able to participate in the unit’s first few lessons.

# A Thought on First Days

One of the ideas I’ve encountered in my wanderings that has ultimately been most useful to me in shaping my teaching is about the needs of students on day 1. It’s this:

Students come to the first day of class with a number of important questions. They almost never ask you these questions out loud, and they are often at most barely conscious of them. But how you respond to these questions will have a very significant impact on how the class goes.

The questions are things like:

Can you teach me effectively?

Will I feel safe and supported here?

Do you believe in me?

Different students have different questions, and it often happens that an effective way to respond to one student’s question is an ineffective answer to another’s. Nonetheless, it’s not hopeless to try to figure out something useful about what questions are dominant in a given class and how to respond to them effectively.

I forget where I first heard this idea. I remember thinking about it a lot during my 4th year in the classroom, in conversation with a particular colleague I’ll call Leslie.

In those long-ago days, I taught:

* Algebra I to 9th graders
* Algebra II to 11th and 12th graders
* AP Calculus AB to mostly 12th graders

I struggled a lot with classroom management with the 9th graders. I almost never had any management problems with the 11th or 12th graders. This was not about “strong” vs. “weak” students: on average, the Algebra II kids were the “weakest.” (The scare quotes are intended to communicate that I don’t buy these labels, but that’s a story for another time.) My in-the-trenches conclusion was that 9th graders are just hard.

Leslie was a history teacher. Like me she taught mostly 9th graders and 12th graders. I was extremely surprised when she told me that she got along great with the 9th graders and was in an epic struggle with the 12th graders.

She eventually resolved it, but I remember being extremely confused and curious when she first told me about the difficulty. Twelfth graders, acting like that? I don’t remember what I asked or what she said. But my takeaway was something like this:

“I like math; I know a lot of math; I work very hard to make lessons clear, creative and engaging. I’m curious about kids and excited about their thoughts, and I will spend a lot of extra time with you to try to understand your mind and help you understand the content. On the other hand, I do not like it when students don’t cooperate with my plans or engage with the lesson I worked so hard on, and I wish they would just cooperate and engage.”

“9th graders are developmentally different from adults. Though they are anxious to be seen as grown-up, they still find it difficult to self-regulate their emotions. In this context, a family of questions they have for their teacher on day 1 is, ‘how will you help me stay focused when I find this difficult? how will you help me self-regulate? will you keep us all safe from undue disruption stemming from ourselves’ and each others’ difficult feelings?’

“I have up to now been bad at responding effectively to this suite of questions. I have resented and wished-would-go-away the part of my job that is about helping students stay in control of themselves. I am sure the 9th graders sense the implied power vacuum. They probably find it terrifying. They want to know class will be happy and productive, and they find out the answer is, ‘only if I, and all of my peers, simultaneously, spontaneously stay focused and positive for the whole period.’ Yeah right.

“Meanwhile, Leslie understands and enjoys this part of her job. Her 9th graders relax quickly as they learn what she is willing to do, happily, to make sure they as a community stay their best, most productive selves.

“On the other hand, 12th graders are much closer to being adults. They self-regulate much more easily. They don’t need you to prove to them what you can do to help them with that. On the other hand, they are anxious to know that you are not on a power trip and that their time won’t be wasted.

“In this context, the deal I was subconsciously offering — I know this stuff really well and I’ll work really hard to help you learn it; I won’t condescend to you about how to act, but I need you to cooperate and engage without much structural help from me — actually probably sounded like a great deal to 12th graders. They were ready to do the self-regulating without me, and I probably implicitly answer the questions ‘do you know your sh*t?’ and ‘can you help me learn it?’ very quickly in the affirmative. That explains why their affect was always like, ‘ok, cool, let’s go.’

“On the other hand, from what Leslie is telling me, she did not successfully reassure her 12th graders that she knows her sh*t early on. She does in fact know her sh*t, but somehow they didn’t get that sense at the beginning, and eventually went into open rebellion. Probably sexism was involved; who knows what the whole story is. But, for whatever reason, that question did not get successfully answered, and it led to a big problem.”

I have no idea if any of that is the truth. But it seemed to explain the puzzle to me, to fit my experience and my colleague’s story, and has shaped my thinking about what needs to happen on day 1 ever since.

All of this was at the front of my mind not too long ago when I started a new class in a new context. It was an advanced college level math course, and I had been told that the students had taken a full sequence of prerequisite courses but that their grounding in that content was uneven. Having been told this, it was hard to plan anything and feel confident it would be appropriate. Would it be too easy and they’d feel condescended to? Too hard and they’d be lost? I was really stuck on this.

I reached out to the students for info: “what do you know about X subject?” My first inquiry went unanswered for weeks. I followed up. One of them said, “To answer your question I’d need to look at my previous syllabi.” I asked an administrator for help with this and they turned up several syllabi. A few more days went by with no word, so I followed up again. A second student wrote back: “Every professor has a different idea about these courses. Maybe if you tell us what you want us to know, we can tell you if we know it.” I replied, listing specific topics. Nothing, for a few more days. With the class beginning the next day, I wrote one last time: “Now’s your last chance to tell me something about what you know before we get going. Can you reply to that list I sent before?”

Another student wrote back to the effect of, “Look, we have taken numerous classes before. Nothing on this list is foreign to us. Our mastery over specifics will vary from topic to topic.”

This email told me so much. I mean, it told me almost nothing in terms of their actual background — how that mastery varies from topic to topic was exactly what I had been asking. And yet, it told me just what I needed to know to make the planning decisions that had been tripping me up.

These folks need to know I stand ready to challenge them!

Underneath that, I supposed they might be anxious to know I planned to take their minds seriously. And my attempts to get some orientation for myself could have exacerbated that anxiety! My “are you familiar with X?” questions had all been about content they were supposed to have seen before! If indeed they were concerned I might not think they were up to a challenge, perhaps these questions had fed that concern. (This would at least be a plausible explanation for their slow and unforthcoming responses.)

So, I felt I knew what question I had to answer on day 1. I put together a lecture full of rich, hard content, outlining a grand sweep for the whole semester. I erred on the side of more and grander content. During class itself, I erred on the side of telling them more stuff, rather than probing what they were making of it. I wanted the experience to say, “I know you are not here to play, and neither am I. We are going to go as far as you’re ready to. Maybe farther.”

At the end of class, I mentioned to the student who’d sent the email that I’d enjoyed its tone of “c’mon now, bring it!” He smiled, like, “yeah, you know it.”

The course is behind us now. In fact, that first day was the fastest, most content-packed day of the class. It is not generally my style to construct class in a way that pushes forward without much information about what sense the students are making of the ideas. Once the students became willing to show me what they actually knew and didn’t know, it was possible to properly tailor the course, and we were able to drill down on key points and really get into what they were thinking. To be clear, it didn’t get any easier — I would say it actually got harder and harder over the course of the semester. By the end we were line-by-line in the thick of intricate, pages-long proofs. But we never again zoomed forward at the breakneck pace of that first day.

That said, with hopefully due respect to the fact that I haven’t had this conversation with the students directly, I do believe it was the right choice for day 1. A different first class might have been a little closer to what the rest of the semester would look like minute by minute, but it wouldn’t have spoken to the question I believed then and still believe that my students really needed answered.

By the same token, for different students, it could have been exactly the wrong choice. If my students’ incoming burning question had been, “are you willing to meet me where I am?,” then that first lesson could have come across like, “no, not even a little bit,” and we might have had a real long semester. And I honestly did not know which question my students had! This is why I’m grateful to the one who emailed me to say, “Look, we’ve done a lot.” That told me what I needed to know.