Talking Openly about How to Do It Better

The Hardest Questions Aren't on the Test: Lessons from an Innovative Urban School Last week I somewhat impulsively picked up and read cover-to-cover the new book of an important mentor of mine.

The Hardest Questions Aren’t On the Test: Lessons from an Innovative Urban School, by Linda Nathan

Linda is the principal of the Boston Arts Academy, where I did my student teaching a decade ago, in what I believe was the school’s 3rd year of existence. The book is largely a collection of vignettes from the BAA’s 12-year history. The vignettes have a theme:

Education involves facing difficult dilemmas. The thing that needs to be done is to bring together the people involved, open up the lines of communication, and try to figure out jointly what to do.

Some of these dilemmas are pedagogical, some pragmatic, some political, and some interpersonal. Some are a combination. The community of people involved may be administration, teachers, students, parents, or a combination. But however configured, Nathan is saying this process is at the heart of education: put the hard choice to the community, and keep everyone engaged with each other as you undertake to work it out.

This book was a very refreshing read for me. We are deep in the days of Arne Duncan, Michelle Rhee, Race to the Top, the Common Core Standards, and the tendency among journalists¹ to regard the KIPP schools as the greatest thing that have ever happened anywhere in the universe because they have high test scores. Now I have some nice things to say about some of these things. The Common Core Standards in 6th to 8th grade math are an order of magnitude better (i.e. shorter and less concerned with trivia) than the New York State standards have been, and while I have no firsthand knowledge of KIPP schools, I’ve been curious about them in a good way since my student teaching year at BAA, when a fellow student teacher came back from a visit to a KIPP school very excited about SLANT. But what this list is meant to capture is that I can’t escape the feeling that the highest-profile conversations about education in this country, in their frenzy regarding accountability and competition, have totally lost sight of the following facts:

a) Students are people and they have cares and values.
b) Teachers are people and they have cares and values.
c) Everybody involved has cares and values.
d) Education takes place in a community. (Corollary: improving education involves improving community.)

Reading The Hardest Questions… felt like walking into a room full of people who had never lost touch with any of this. Nathan is talking about thinking through educational dilemmas with her staff and students and being guided by what all the people involved value. Stating and working for what matters to her, and asking her teachers and her students what matters to them. It’s absurd that this should feel like a refreshing notion, but to me right now, it does. The Race to the Top funding criteria include a lot about assessments and data that will be used to measure teacher and principal effectiveness, and no encouragement whatsoever for students, teachers, principals or even state superintendents to reflect on what they value.

Another refreshing aspect of The Hardest Questions… is that it doesn’t uniformly make Linda or the BAA look good. (Often – and from firsthand experience they are good – but not uniformly.) The book narrates some play-by-play encounters with some difficult conundrums that don’t have clear resolutions, so it airs some missteps. (Different readers will probably count different moves as missteps.) One of the most pernicious elements of the accountability-and-results orientation in the national conversation about education is that it gives everybody (states, schools, teachers and students) a great reason to hide every mistake. You can’t learn math while you’re trying to hide your mistakes and you can’t learn to teach that way and you can’t learn to run a school that way. You can’t learn that way, period.²

Some specific themes and highlights:

* Schools need to develop a “unifying framework” – what the school stands for educationally. This is not a mission statement that collects dust in an administrative folder but a vision articulated frequently to students of the most important themes in their education. The faculty needs to be involved in developing it. The administration needs to be willing to commit to it in a long-term way. The school community needs to periodically revisit whether and how the school is implementing this shared educational vision. At the BAA, the unifying framework the faculty eventually came to, after 2 years of discussion and debate, is a list of four “habits of the graduate” – refine, invent, connect, own. The idea is that these words are the faculty’s answer to the question, “what we are committed to cultivating in every student?” and that this goal defines the school. Nathan makes a point that she initially tried to have faculty sign on to other lists of words (that to an outsider now don’t look so different), but it turned out to be necessary for the faculty to go through the intense and time-expensive process of answering this question for themselves.

I am suspicious of statements that begin with “All schools should…” But this is one I truly stand behind: all schools should develop and use a unifying framework. The “new initiative every year” model doesn’t work. Teachers need to be involved in articulating the framework, and a school must be willing to commit to the implementation of the framework over the long haul. Finally, I would argue that schools without a unifying framework still have an unspoken one – a de facto assumption of what this school is about. If it were expressed in posters on the wall, these frameworks might be “We Are Failing: Who Should We Blame?” or “High Scores and College Admissions – Everything Else Be Damned!” To honestly answer the question “What does your school stand for?” takes a willingness to ask again and again how your practices are improving, what students know and can do, and how day-to-day realities in the classroom match the ideals you have articulated. pp. 30-1

* Developing a school’s commitment to social and moral values also takes a community-wide process, and this one has to go beyond the faculty to the students. And it needs to be continually recreated, because new kids come every year. Chapter 2 of Nathan’s book describes how the BAA faculty first articulated a group of “Shared Values” in response to a community crisis (a “white power” graffito in the bathroom), and then slowly learned more and more, over the course of a series of other community crises (involving theft, homophobia, alcohol…), about what it would take to make these shared values a part of student culture. Some highlights:

As Shared Values became a way to talk about what was important in our community, and even the way to address some of our rules, a few students suggested that we change our quarterly honor-roll assemblies to be called Honor Roll/Shared Values assemblies. They wanted the school to recognize students when they were “Caught in the Act of Shared Values,” a phrase they coined. Students or faculty could nominate students who had done something to exemplify a shared value. The action wouldn’t have to be a big deal, but it had to be something that everyone could applaud. We have, for instance, acknowledged students “caught in the act” of putting up posters that someone had ripped down, staying behind to help clean up a classroom, bringing in doughnuts for everyone in the class after a strenuous day of testing… pp. 38-9

In the spring of 2005, some BAA music students performed at a local music club… It was a wonderful concert; the house was packed… However, the next day the owner of the club called to report that alcohol had been stolen from her establishment.

Ms. Torres [the assistant principal] gathered all the musicians together, and initially had an awful time getting any of the students to say they had seen anything. Finally, one of the young musicians, Martin, a leader in the band, said to the whole group, “Hey, listen, someone saw something. It will be terrible for our school and our reputation if we don’t figure out who did it and make sure it doesn’t happen again….” Martin spoke fervently, but still nobody talked, not for another few days. During these days, the entire school was buzzing with talk about expulsions and rumors that the music department would never be able to perform outside of the school again. In the meantime, Ms. Torres and security personnel managed to uncover the truth: which students had actually stolen the alcohol, which had looked away but knew what was going on… They were all suspended and the ringleader… was expelled…

Even though this incident only directly involved one group of students, so many students were talking about it that Ms. Torres decided to hold another whole school assembly. She also decided to have students talk to students rather than… expect administrators to chastise everyone. Ms. Torres asked Martin if he would address the student body and explain why this was such a big deal… Ms. Torres explained, “I need you to talk about the larger issues, Martin…” He agreed.

At the assembly, Martin got out of his seat, twirling his drumsticks in one hand. “We all know this school is pretty amazing,” he began. “Sure, we’ve got beefs and there are things that we all think are stupid and try to change. Sometimes we do. I know all you freshmen want to have lunch off campus, for example. Well, maybe you can change that. But, you know, one thing that keeps us together is that we have these Shared Values. Sure, some of us might laugh when Ms. Torres gets on the intercom every morning and tells us to live one of the Shared Values, but it’s cool. We do believe in diversity with respect. Just look around at how many different kinds of people are in here. And passion with – ” And then he held his mic out to the audience like a DJ as they responded, “Balance!”

“Yeah, that’s right,” Martin continued. “And we believe in community with – ” And again the audience responded, “Responsibility.”
…
“So, like you’ve heard from Ms. Torres, they’re dealing with the students who did this, but I just think we all have got to think about what this means for our whole community and our reputation out there. We live by our reputation as artists, and if it gets tight out there for us, we won’t be performing…”

We didn’t want students to dismiss the incident as “just something that happened to the music majors.” Dumb, destructive behavior like this is common among adolescents… As sad as I was that BAA students had stolen alcohol, and as disappointed as I was that other students hadn’t turned them in, I was proud of our school’s overall response to the incident. Martin’s leadership meant so much to me. It established a norm that respected student leaders could support school values publicly… pp. 48-51

* Great teachers are empowered to be great by the community they’re a part of. The principal needs to work for the creation and maintenance of this community in order to empower teachers to be great. Building a great school involves “transforming a faculty into a professional learning community.”

Success truly begets success… This plays out in Ms. Chan’s [dance] class, but we see it even more clearly in Mr. Ali’s [humanities class], where students are not all here by choice. Mr. Ali can build on Aleysha’s engaged identity as an artist to encourage in her an engaged identity as a scholar. He has listened to her concerts over the years, and he knows she has a gift and love for music. It is his challenge to create the same set of expectations and joys in his own humanities classroom. p. 78

Teaching at BAA is decidedly not a solitary activity. While I have very little influence on what goes on moment-to-moment in Ms. Chan’s or Mr. Ali’s classroom, I can, and do, work on the schedule (the skeletal system of a school) to ensure that teachers help each other, that worries and questions are shared among team members and the entire faculty. Mr. Ali meets weekly with academic and arts colleagues to discuss students and to develop curriculum. At the end of the year, he will spend two days with his team reviewing and critiquing each other’s units and lessons, and creating notebooks on the year’s courses so that they continue to build a collective archive of work.
…
Mr. Ali and Ms. Chan are not “one-offs” or “the exceptions” at BAA. I tell their stories here as representative of the ways in which our teachers can be successful. As a leader, it is my job to build a school in which all teachers work in teams, and have the time built into their schedules to talk, to visit each other’s classrooms, and to create curricula as carefully and self-critically as artists create their pieces. pp. 80-1

* A school that wants to make progress on the achievement gap needs to have frank and potentially uncomfortable conversations with faculty and students about race.

There are a lot of really compelling passages to quote on this one but it’s already several hours past the time I told myself I would have finished this post. Read the book.

More info:

Here is a video of a half-hour talk that Linda and some BAA students gave. (At Google I guess??) I found it much harder than the book to follow thematically, but it’s cool because the students do a performance based on the unifying framework (refine, invent, connect, own) and talk about it afterward.

Here is a review of the book written by a former BAA student for feministing.com.

[1] I’m thinking of Malcolm Gladwell (in Outliers) and Daniel Coyle (in The Talent Code), for example.

[2] As an aside, one of the reasons I think The Wire is such a significant show is its persistent exploration across different urban institutions (school, law enforcement, city politics) of the way that numerical “accountability” incentivizes maintaining the status quo and hiding the dirt rather than digging into the problems and seeking real improvement.

Despairing vs. Working: Learning Classroom Management and Learning Math

[Virtual Conference on Soft Skills]

I. Prelude

One of the great challenges of teaching math is the fact that many students walk into class with trauma surrounding the subject. One way or another they have absorbed the idea that the difficulties they have had solving math problems say something important and damning about their intellect.

Trying to do math makes them feel stupid.

J, whom I taught as a junior in Algebra I, was a very developed writer and poet. He would talk about math as a mythical dragonlike beast waiting at the end of his quest to destroy him after he had surmounted every other obstacle. A, whom I ran into on the street two years after teaching her, told me that her life would be great if she could just understand math. O, a professional adult in the financial industry who took a workshop with me, looked like she wasn’t making progress by herself at one point during the workshop, so I asked another participant to join her. She ran out of the room. I found her in tears in the hall. She had fled rather than let someone else “find out how stupid she was.”

If they are going to learn anything, the this tragic association needs to be disrupted, and as quickly as possible. I know you have all already read Dan’s lyrical description of the problem and one part of how to take it on. For now, what I want to call attention to is the mechanism by which this association renders it impossible to learn.

The mechanism is this: when you feel stupid, you are not thinking about math. Like driving a car and playing basketball, it is not possible to think about math and feel stupid at the same time.

I am using “thinking about math” in a strong sense here. It is possible to execute an already-known algorithm like the multiplication algorithm while feeling like the biggest dumb*ss in the world, although it is harder than doing it when you’re feeling better about yourself. What it’s not possible to do is solve a problem new to you, think creatively or resourcefully, see a surprising connection or a pattern, notice your own curiosity, or any other type of thinking that would cause you to grow mathematically. What I am claiming, in short, is that the activity of feeling stupid excludes all activities that allow you to grow.

To make this concrete:

In the workshop for adults I mentioned above, I had posed the sums of consecutive integers problem in a fairly open-ended way. (What numbers can and can’t be represented as sums of at least 2 consecutive natural numbers? Why? What else do you notice?) Most of the participants in the workshop were having conversations with themselves and each other along the lines of:

“What’s going on here?”
“Can I get this number [as a sum of consecutive naturals]? How about this one?”
“Is there a pattern in the numbers I can/can’t get?”
“If you give me a number is there a system I can use to represent it [as a sum of consecutive naturals]?”
“What patterns are there in the representations I’ve found so far?”
Etc.

Here are the conversations O was having with herself before I asked someone else to join her and she ran out of the room:

“Everybody else is having all these insights. Why am I not?”
“What’s wrong with me that I’m not?”
“What will they think of me?”
Etc.

I didn’t realize this by looking at her, although perhaps I should have. I thought maybe she just wasn’t making progress for whatever reason. She is a pro at hiding it (along with all other people who have this type of conversation with themselves). Lots of practice. But the point I’m making here is this:

The conversation that O was having with herself was of a totally different character than the other participants. The thoughts she was having, and the work that she needed to be doing in order to grow mathematically, live on different planets. When students begin to have this conversation with themselves, they have gone to Mars as far as learning math is concerned.

I listened to O talk about how she was feeling, gave her a hug and told her something to the effect that it made me mad to think anyone had ever made her feel bad for taking her own time to explore something. I brought her back to the workshop and partnered her with another participant who hadn’t come up with a whole lot yet (and who was also very empathetic). She let O explain herself and vent a bit; I let this happen for a few minutes and then said I thought it was time to get back to the math.

Maybe you have seen this miracle yourself: when that traumatized person unloads their pain and finds it accepted and not judged, or just plain has the cycle of self-doubt/paralysis/self-doubt interrupted in any way at all, and then takes a fresh look at the problem… the natural dynamics of the process of problem solving take hold and they instantly become a frickin genius.¹ Not by everyone’s standard but by the only standard that ought to count: they start to see the problem from new angles. This amazes them. I’ve lost count of the number of times I’ve seen this happen and it’s breathtaking every time. They then often invalidate their accomplishment through an unfair comparison with others, but that first moment of seeing-the-problem-in-new-light is there, available, and needs to be highlighted. “When you said, ‘oh, I could simplify the other side first’ and that opened up a path to make progress… that’s what being a mathematician is. That’s the whole game right there. Looking at what’s there and playing with it and working with it till you get a new angle. There is one secret to ‘being good at math’: do that as much as possible.”

O, a reflective grownup, got the lesson in a powerful way. How much of her mathematical paralysis was really entrapment in the self-doubt cycle. How much she was capable of, that she didn’t even know about, whenever she could switch off that cycle and be present to the problem.

The key word there is presence. If you are present to a mathematical question, and the reality it is asking about; in other words if the question and its reality are available to you, vivid for you, there before you to touch and probe; then doing math is the most natural thing in the world, and growth is inevitable.

But you can’t be present to the math when you are busy thinking/worrying/stressing that you suck. This takes your attention away from the actual problem and the process of looking for a solution stays shrouded in mystery.

II. An Analogy

All of this is set-up for what I really wanted to talk about.

In my six years as a full-time public school classroom teacher, I spent a lot of time and emotional energy thinking about and struggling with classroom management. I was, of course, not alone here. It’s a major issue for beginning teachers.² Everybody knows this.

I learned a fair amount about classroom management in that time, but there’s something important that I don’t think I ever understood, till this year when I worked as a teacher trainer. I feel like I could have accelerated my learning curve immensely and spared myself and my students a lot of pain if I’d understood it earlier. Consider this true statement:

Struggling with classroom management made me feel like sh*t as a person.

My intention is for this sentence to have landed with some echoes in the background, but just in case:

… Trying to do math makes them feel stupid.

Like math itself for so many of our students, classroom management struggles have left many teachers traumatized. And with reason. Math’s power to hurt is based on the perverse culturally taught belief that accomplishment in math is a manifestation of some important inborn intellectual attribute and struggle to understand is evidence you don’t have it. The power of struggles with classroom management to make you feel bad are likewise amplified by the current cultural milieu, in which the idea that teachers need to be more minutely and exhaustively judged is the coin of the political realm. But the fact is that the experience of being treated rudely by a room full of children pretty much bites, whatever the cultural context.

Reasonableness aside, though, just as math trauma paralyzes the growth of the math learner, feeling bad about yourself because your kids aren’t listening to you is an activity essentially different from, and incompatible with, in fact on a different planet than, growing as a classroom manager.

Let me make this point more concrete. This year, as a supervisor for an MAT program and as the math coach at a high school, I had the privilege of witnessing a lot of different people teach and thinking with them about how to improve their teaching. Frequently this role called upon me to help them think about their classroom management. I found myself, to my surprise, with lots of advice. What was happening was that it was much easier to perceive the dynamics of the classroom as a third-party observer who knows what they look and feel like but is not presently involved. If you’ve got at least a few years experience but have never stepped into the classroom of a fellow teacher with the intent to give management advice, do it – you’ll be surprised how useful you are. It’s the essential awesomeness of what not actually being caught up in it lets you see.

What really threw me, in a good way, is that the suggestions I was making were things that by and large

a) I was sure I would have benefited from during my own full-time classroom practice; and yet

b) most of them were in areas I had never thought about. They were like a whole new angle on the classroom. More specifically, they were smaller and more concrete than most of what I had thought about in all those years of stressing about management.

When things went badly in my classroom, and I thought about what to do about it, my questions were most often like:

“How do I convey strength?”
“What’s the appropriate response to insubordination?”

On bad days,

“What’s wrong with me that they don’t listen to me? (and is it possible to fix? probably not…)”

These are big, abstract questions. What I’ve come to understand this year is that this abstract level is not where the answers live. They live in the minute-to-minute, real-time interactions that constitute a class period. They are solid, tangible, low to the ground. A discipline problem would develop, and then boil over, so that I found myself furious with a student or multiple students, and feeling like a failure. I would then ask myself these big abstract questions. In so doing, I would totally divert my attention from the tiny incremental steps by which the problem had built itself, and from the tiny, concrete things I could have done to head it off before the axe fell. I would also make myself feel horrible for no reason. I felt weak, as though the difficulty I was having had been caused by my fundamental inadequacy as a human. In reality, it was caused by a chain of extremely small and concrete failures of technique. These techniques can be taken on and learned one at a time. They are all individually too small to be worth feeling bad about.

To get specific. Here are some of the suggestions I found myself giving to teachers repeatedly this year. They may be individually useful to you if you are struggling with management and recognize your classroom in the situations they are designed to address. But the big thing I am trying to communicate is that these suggestions do not relate to anything it makes any sense for a teacher to feel bad about. They’re just bits of technique. If your class is messing up because you’re not doing one of them, all this means about you is that you haven’t learned this bit of technique yet.

* In the 1-3 minutes following a transition in which you issue an instruction to the whole class, do not converse with any individual kids. Keep your attention on the whole class. Make it your only job to see that your instruction gets implemented.

(I gave this advice, for example, when I saw teachers give an instruction and then immediately begin to help or reprimand an individual kid, while the rest of the class implemented the instruction inconsistently or not at all.)

* If you have assigned classwork and are trying to help the whole class through it one desk or table at a time, stop the work and call the class back together. The work wasn’t ripe for doing yet it turns out.

* Do not communicate disappointment when a student fails to do something you didn’t communicate a clear expectation about. Communicate your vision of how the class should behave before they have an opportunity to fulfill or disappoint that vision.

(This piece of advice was usually coupled with specifics.)

* Do not make capricious decisions about your students’ attention. For example, if you set them to work 3 minutes ago and someone asks you a question that you think deserves the class’s attention, don’t take lightly the decision to interrupt the work to share the question. If you want to be able to direct students’ attention you need to be willing not to ask too much of it.

(This is a piece of advice I could really have used myself.)

Again, the point is not about these specific suggestions, which I gave to particular teachers facing particular challenges that may or may not be yours. The point is that each suggestion connects to a bit of learnable classroom technique that can be taken on one at a time; that there’s nothing here to feel bad about, since each bit of technique is nothing more than that; and lastly that the big heavy questions of self-worth that plague so many teachers struggling with management are really distractions from these techniques. They pull your attention up and out, to the broad and abstract, and carry you away from what is actually happening in your room between you and your students.

Now I want to be clear: it’s not that the individual techniques are easy, and it’s not that you can just learn them by deciding to. Sometimes, the techniques involved get deep into your being. One of the deepest: communicate the intention that your instructions be followed. This bit of technique is totally natural to some teachers before they walk into the classroom. Others (I’ve been one) need to learn and sometimes relearn it, and learning it may not be as simple and external as the other techniques I’ve listed.

The point is that in spite of this, it’s still just a technique. You just learn how to look, sound and feel like you mean it when you tell your students to do something. This skill can be broken down into smaller components that also can be worked on individually: relaxation and confidence in the tone of voice; relaxed posture; steadiness in the body; a steady gaze. Follow-through: the maintenance of all this personal force in the second and the minute following your instruction. Doug Lemov’s “stand still when you’re giving directions” is the same thing. You can get better at each of these components. Because they have to do with deep habits of your body and social M.O., they may be hard to work on. It may help to videotape yourself or work with a coach, mentor or colleague. But the point is just this: there is nothing mysterious in improving these skills. They are nothing more than techniques. Underdevelopment of any one of them, or many of them, is simply something too small and concrete to feel bad about. That heavy burden of self-doubt is ironic because it’s simultaneously an awful experience and an obvious gambit by the lazy-bum part of our brains to distract us from the real job of learning these techniques. (Isn’t being a lazy bum supposed to be kind of pleasant?)

So: the kids are battered by self-doubt because they think struggle with math impugns their intellectual worth. This cycle distracts them from the math. Free them from this cycle and they grow. The teachers are battered by self-doubt because we think struggle with classroom management impugns our worth as people/professionals. This cycle distracts us from the real job of getting better at the techniques that comprise classroom management. Free us from this cycle and we grow.

I hope if you’ve been there that this post can be part of helping you stay free.

III. Related Reading

* I started to put together the thoughts in this post in some comments I wrote in response to a beautiful post from Jesse Johnson.

* Jesse and Sam Shah, who have been at PCMI the last 3 weeks, have both been writing about teacher moves, and distinguishing teaching from teachers. Meaning, learning to focus on the actions that are being taken in the classroom, rather than on judging a person. This distinction seems to have been introduced at PCMI in the context of looking at video of other teachers, but both Jesse and Sam recognize you can use it on yourself as well. This is closely related to what I’ve talked about here: the realization that just like a kid learning math, getting present to the real, actual, concrete process of teaching both empowers and is empowered by letting go of judging yourself.

* Here’s a 3-year-old post from Dan Meyer drawing an analogy between the process of subdividing our job into small, concrete bits that can be worked on one at a time, and integration (as in, $\int$ ). Closely related and very cool.

* When I looked up that New York Times Magazine article about Doug Lemov to link to it, I realized that some of the same issues are being dealt with there. Maybe this is part of why (except for acting like Lemov is the first person to wonder how good teachers do their job) that article was so refreshing to me.

IV. In Other News…

The New York Math Circle has their Summer Workshop coming up in a week! It’s about the Pythagorean Theorem and based on talking to the organizer, Japheth Wood, I think it’s going to be both mathematically and pedagogically interesting. (Y’all know this theorem is the greatest single fact in K-8 education. If I may.) The program is housed at Bard College and doesn’t cost very much for a week-long residential thing. ($375.) Clearly the place to be.

Notes:

[1] Assuming that the problem is at an appropriate level of challenge. Another way to put this is, assuming that the reality the problem is asking about is available to the student. (This could be a physical reality or a purely mathematical one.)

[2] Totally unnecessary citation: “A significant body of research also attests to the fact that classroom organization and behavior management competencies significantly influence the persistence of new teachers in teaching careers.” Effective Classroom Management: Teacher Preparation and Professional Development, p. 1 (issue paper of the National Comprehensive Center for Teacher Quality, 2007), citing Ingersoll & Smith (2003), The Wrong Solution to the Teacher Shortage. Educational Leadership, 60(8), 30-33.

Pattern Breaking II

This is a followup to Pattern Breaking.

Since I got interested in apparent patterns that break later on, I’ve discovered an article on this very subject by Richard K. Guy, published in the American Mathematical Monthly over 20 years ago! Also, Justin Lanier has directed me to a discussion on this topic at MathOverflow with many more examples. Lastly, I’ve got a solution to the computational half of the problem that many of the examples found in these sources are not conceptually or computationally accessible at the secondary level. Namely, I’ve come across a couple of computational resources that might help to bring some of these examples (especially the ones involving primes) within striking distance of a lesson you might create at the middle or high school level.

The article is called “The Strong Law of Small Numbers” and is found in Vol. 95, No. 8 (Oct., 1988) of the AMM. Here it is as a pdf. It is somewhat silly in tone, but it is actually more academically serious than anything else I’ve pointed you to thus far. Here is how it starts:

This article is in two parts, the first of which is a do-it-yourself operation, in which I’ll show you 35 examples of patterns that seem to appear when we look at several small values of n… The question will be, in each case: do you think that the pattern persists for all n, or do you believe that it is a figment of the smallness of the values of n that are worked out in the examples?

Caution: examples of both kids appear; they are not all figments!

In the second part I’ll give you the answers, insofar as I know them, together with references. (p. 697)

The compendium of patterns is excellent. Credit where credit is due: some of my favorites so far, e.g.

The points on a circle pattern;
The fact that 31, 331, 3331, …, 3333331 are all prime;
The fact that 3!-2!+1!, 4!-3!+2!-1!, …, 8!-7!+…+2!-1! are all prime

were in this article before they were anywhere else I’ve found them. In addition, Guy has interesting commentary on what’s going on – for example, he offers some insight regarding why the points-on-a-circle pattern matches the powers of 2 in the first 5 cases and then breaks.

Now, I became interested in these pseudo-patterns because I think they’re pedagogically important. Guy has a different angle: he is actually trying to say something about math. In his view, pseudo-patterns like this are quite common, because of his eponymous “Strong Law of Small Numbers”:

There aren’t enough small numbers to meet the many demands made of them (p. 698)

It’s worth it for you to download the pdf. The article is a good and enlightening read all the way through. I’ll share one more pattern (or almost-pattern? which is it??) from it but really I recommend downloading the whole thing.

Example 24. Consider the sequence
$x_0=1$ , $x_{n+1}=(1+x_0^2+x_1^2+\cdots+x_n^2)/(n+1)$

$n$ 0 1 2 3 4 5 6 …

$x_n$ 1 2 3 5 10 28 154 …

Is $x_n$ always an integer? (p. 704)

Okay. Secondly. Justin Lanier pointed me to a source of a good many more examples of patterns breaking: a discussion of this very topic at MathOverflow. (This is a link to the Wikipedia article to tell you what MathOverflow is. There is a link to the actual site at the beginning of this post.) Here it is:

MathOverflow: The phenomena of eventual counterexamples

Justin also directed me to a particular link within this discussion:

“Law of Small Numbers”… more examples!

So, here are two more sources for patterns that break. The repository is growing…

Now a drawback of some of Guy’s and many of MathOverflow’s examples are that they are not really accessible at the middle or high school level. Now in some cases this is because of the concepts involved. But in other cases, it’s just a matter of the computations being out of range. For example, Guy mentions a famous example several of us also noted when I first raised the issue: Fermat’s conjecture that $2^{2^n}+1$ is always prime. This is true for n=0, 2, 3, 4, and wrong for n=5, but $2^{2^5}+1$ is over 4 billion and its smallest prime factor is 641. Not something the kiddies can find out, even with a calculator.

Enter PARI/GP. This is an open-source computer algebra system that does number-theoretic calculations at ludicrous speed. For example, just now I had it execute the following self-explanatory code:

{t=0; for(n=1,20000000, if(isprime(n),t=t+1) ); print(t)}

Running on my new MacBook, it produced the output (1,270,607) in about 24 seconds.

In other words, it individually assessed the primality of twenty million numbers, most of them fairly large, in under thirty seconds.

In other words, if you want to do a lesson based on Fermat’s conjecture and want to make it computationally possible, PARI/GP has you covered.

Now it is freely downloadable at the link above. It is not particularly user-friendly, however. Downloading and installing it forced me for the first time to interact with my Mac’s UNIX interface. The installation instructions at the PARI/GP website are not designed to be comprehensible by someone who doesn’t know anything about UNIX. For some reason, no one I asked for help seemed to be 100% able to comprehend the existence of people like me who want to install a piece of software like PARI/GP but don’t know anything about UNIX. I had a lot of trouble getting the answers I needed. The whole experience was pretty frustrating. If you want to download and run this program on a Mac but don’t know how to install something in UNIX, feel free to email me.

Now in order to take PARI/GP into the classroom, if you have a smartboard or projector or something and can get it on one computer, you can do experiments with the whole class. If you want to have the kiddies interact with it, you need it on more than one computer. Also, it doesn’t produce files so to have kids save their work you have to show them how to have it import text from a text editor. All this has to happen in UNIX. This could be a pain.

Enter SAGE. SAGE is another open-source computer algebra system with the following key advantages:
a) You can run the whole thing online!
b) You can save your work online!
c) You can run PARI/GP from inside SAGE!
PARI does not run as fast if you’re using it inside SAGE online as it does if you download and compile it on your computer. Also, SAGE is not the most dependable piece of internet software there is. (It struggles in particular when accessed through Safari. I recommend Firefox.) But, if you want to bring this computational power to the classroom, without requiring any administrators to invest in any piece of software, this is the way to go. All you need is computers with internet access.

If you are having any inklings you’d like to try these resources out in the classroom, I recommend you go play with SAGE right now. Go to the website, click “try SAGE online,” sign up for a new account, and then click “new worksheet.” To access PARI/GP, choose “gp” from the 4th drop-down menu below the worksheet title. (The one whose default is set to “sage.” Btw, PARI is only one of the many amazing computational software packages you can access from inside SAGE. SAGE is kind of its own one, for example. In fact, it can do a lot of what PARI can do. But PARI is faster with the large-scale computations.) You can find users-guide typed materials (none particularly user-friendly, but oh well…) in a link inside the SAGE worksheets. For PARI you can find them on the PARI website. It helps to be familiar with the basic structure of computer programming languages.

Anyway, I think there are a lot of awesome classroom activities waiting to be made out of SAGE and PARI and a few of the pseudopatterns found in the Guy article and at MathOverflow. Here’s one thought, based on a pseudopattern mentioned in Guy (although actually I learned about this example in a workshop led by John Cullinan, which is also where I learned about PARI). Question: how would you turn it into a classroom-able activity at whatever level you teach?

Are there more primes of the form 4k+1 or 4k+3?

(What does this question even mean?)

Just by counting the small cases:
For primes less than 10, one (5) is 4k+1, while two (3 and 7) are 4k+3
For primes less than 30, four (5, 13, 17, 29) are 4k+1, while five (3, 7, 11, 19, 23) are 4k+3
For primes less than 100, I resorted to PARI. I opened a SAGE worksheet, set it to PARI mode (i.e. picked “gp” from that drop-down menu), and entered:

Ones(n)=\ i=0;\ forstep(k=5,n,4,\ if(isprime(k),i=i+1)\ );\ return(i)

(Some basic syntax info:
*In PARI/GP, everything is really meant to happen on 1 line. Every time you press return, the program executes whatever you entered. If you’re running GP inside SAGE, you use backslashes to split something up over several lines that is really supposed to be 1 line.
*Commands always enclose their arguments in parentheses.
*Use semicolons to separate commands from each other.
*forstep(k=this,that,stepsize,dosuchandsuch) has k count up from this to that by whatever the stepsize and executes suchandsuch for each k.)

Then I entered:

Ones(100)

PARI says there are eleven primes less than 100 of the form 4k+1.

Threes(n)=\ i=0;\ forstep(k=3,n,4,\ if(isprime(k),i=i+1)\ );\ return(i) Threes(100)

PARI says there are thirteen primes less than 100 of the form 4k+3.

So it looks like the 4k+3’s are consistently outnumbering the 4k+1’s. Will this stay true?

How could we find out?

If you want to go experiment on your own, do it.

If you want some guidance, try this code:

PrimeRace(n)=\ i=0;\ j=0;\ forstep(k=3,n,2,\ if(isprime(k),\ if(Mod(k,4)==1,i=i+1,j=j+1);\ if(j-i<0,print(k))\ )\ );\ return(j-i)

This creates a routine that runs through all the odd numbers from 3 to n and asks if they’re prime. If they are, it checks to see if they’re of the form 4k+1 or 4k+3, and keeps track of the totals. If at any point the 4k+1’s are leading, it prints out whatever prime numbers this happened at. (The if command can be either if(condition,dothisthing), or if(condition,dothisthing,andifnotthendothisthing). I’m using it both ways here.) At the end, it prints out the total amount by which the number of 4k+3’s exceeded the number of 4k+1’s.

Okay, now you can play around with it.

…

More guidance? Alright, but ***SPOILER ALERT***

Have it evaluate:

PrimeRace(10)

Yup, in the first 10 numbers, the 4k+3’s are up 1 as we previously counted.

PrimeRace(100)

Up 2 now, matching what the One(100) and Three(100) functions told us before.

PrimeRace(1000)

Up 7 now.

PrimeRace(10000)

Up 10 now. Aren’t you in suspense?

PrimeRace(30000)

Up 22 now, but what is that other number it printed out? 26861?

PrimeRace(26861)

Yes, for one moment, at 26,861, the 4k+1’s take the lead! But 26,863 is prime as well, evening it back out, and then the 4k+3’s retake and stay in the lead all the way to thirty thousand. How crazy is that?

PrimeRace(700000)

I’m not gonna say anything. Just run it.

The History of Algebra, part II: Unsophisticated vs. Sophisticated Tools

Math ed bloggers love Star Wars. This post is extremely long, and involves a fair amount of math, so in the hopes of keeping you reading, I promise a Star Wars reference toward the end. Also, you can still get the point if you skip the math, though that would be sad.

The historical research project I gave myself this spring in order to prep my group theory class (which is over now – why am I still at it?) has had me working slowly through two more watershed documents in the history of math:

Disquisitiones Arithmeticae
by Carl Friedrich Gauss
(in particular, “Section VII: Equations Defining Sections of a Circle”)

and

Mémoire sur les conditions de résolubilité des équations par radicaux
by Evariste Galois

I’m not done with either, but already I’ve been struck with something I wanted to share. Mainly it’s just some cool math, but there’s a pedagogically relevant idea in here too –

Take-home lesson: The first time a problem is solved the solution uses only simple, pre-existing ideas. The arguments and solution methods are ugly and specific. Only later do new, more difficult ideas get applied, which allow the arguments and solution methods to become elegant and general.

The ugliness and specificity of the arguments and solution methods, and the desire to clean them up and generalize them, are thus a natural motivation for the new ideas.

This is just one historical object lesson in why “build the machinery, then apply it” is a pedagogically unnatural order. Professors delight in using the heavy artillery of modern math to give three-sentence proofs of theorems once considered difficult. (I’ve recently taken courses in algebra, topology, and complex analysis, with three different professors, and deep into each course, the professor gleefully showcased the power of the tools we’d developed by tossing off a quick proof of the fundamental theorem of algebra.) Now, this is a very fun thing to do. But if the goal is to make math accessible, then this is not the natural order.

The natural order is to try to answer a question first. Maybe we answer it, maybe we don’t. But the desire for and the development of the new machinery come most naturally from direct, hands-on experience with the limitations of the old machinery. And that means using it to try to answer questions.

I’m not saying anything new here. But I just want to show you a really striking example from Gauss. (Didn’t you always want to see some original Gauss? No? Okay, well…)

* * * * *

I am reading a 1966 translation of the Disquisitiones by Arthur A. Clarke which I have from the library. An original Latin copy is online here. I don’t read Latin but maybe you do.

I’m focusing on the last section in the book, but at one point Gauss makes use of a result he proved much earlier:

Article 42. If the coefficients $A, B, C, \dots, N; a, b, c, \dots, n$ of two functions of the form

$P=x^m+Ax^{m-1}+Bx^{m-2}+Cx^{m-3}+\dots+N$

$Q=x^{\mu}+ax^{\mu-1}+bx^{\mu-2}+cx^{\mu-3}+\dots+n$

are all rational and not all integers, and if the product of $P$ and $Q$ is

$x^{m+\mu}+\mathfrak{A}x^{m+\mu-1}+\mathfrak{B}x^{m+\mu-2}+etc.+\mathfrak{Z}$

then not all the coefficients $\mathfrak{A}, \mathfrak{B}, \dots, \mathfrak{Z}$ can be integers.

Note that even the statement of Gauss’ proposition here would be cleaned up by modern language. Gauss doesn’t even have the word “polynomial.” The word “monic” (i.e., leading coefficient 1) would also have been handy. In modern language he could have said, “The product of two rational monic polynomials is not an integer polynomial if any of their coefficients are not integers.”

But this is not the most dramatic difference between Gauss’ statement (and proof – just give me a sec) and the “modern version.” On page 400 of Michael Artin’s Algebra textbook (which I can’t stop talking about only because it is where I learned like everything I know), we find:

(3.3) Theorem. Gauss’s Lemma: A product of primitive polynomials in $\mathbb{Z}[x]$ is primitive.

The sense in which this lemma is Gauss’s is precisely the sense in which it is really talking about the contents of Article 42 from Disquisitiones which I quoted above.

Huh?

First of all, what’s $\mathbb{Z}[x]$ ? Secondly, what’s a primitive polynomial? Third and most important, what does this have to do with the above? Clearly they both have something to do with multiplying polynomials, but…

Okay. $\mathbb{Z}[x]$ is just the name for the set of polynomials with integer coefficients. (Apologies to those of you who know this already.) So a polynomial in $\mathbb{Z}[x]$ is really just a polynomial with integer coefficients. This notation was developed long after Gauss.

More substantively, a “primitive polynomial” is an integer polynomial whose coefficients have gcd equal to 1. I.e. a polynomial from which you can’t factor out a nontrivial integer factor. E.g. $4x^2+4x+1$ is primitive, but $4x^2+4x+2$ is not because you can take out a 2. This idea is from after Gauss as well.

So, “Gauss’s Lemma” is saying that if you multiply two polynomials each of whose coefficients do not have a common factor, you will not get a common factor among all the coefficients in the product.

What does this have to do with the result Gauss actually stated?

That’s an exercise for you, if you feel like it. (Me too actually. I feel confident that the result Artin states has Gauss’s actual result as a consequence; less sure of the converse. What do you think?) (Hint, if you want: take Gauss’s monic, rational polynomials and clear fractions by multiplying each by the lcm of the denominators of its coefficients. In this way replace his original polynomials with integer polynomials. Will they be primitive?)

Meanwhile, what I really wanted to show you are the two proofs. Original proof: ugly, long, specific, but containing only elementary ideas. Modern proof: cute, elegant, general, but involving more advanced ideas.

Here is a very close paraphrase of Gauss’ original proof of his original claim. Remember, $P$ and $Q$ are monic polynomials with rational coefficients, not all of which are integers, and the goal is to prove that $PQ$ ‘s coefficients are not all integers.

Demonstration. Put all the coefficients of $P$ and $Q$ in lowest terms. At least one coefficient is a noninteger; say without loss of generality that it is in $P$ . (If not, just switch the roles of $P$ and $Q$ .) This coefficient is a fraction with a denominator divisible by some prime, say $p$ . Find the term in $P$ among all the terms in $P$ whose coefficient’s denominator is divisible by the highest power of $p$ . If there is more than one such term, pick the one with the highest degree. Call it $Gx^g$ , and let the highest power of $p$ that divides the denominator of $G$ be $p^t$ . ( $t \geq 1$ since $p$ was chosen to divide the denominator of some coefficient in $P$ at least once.). The key fact about the choice of $Gx^g$ is, in Gauss’s words, that its “denominator involves higher powers of $p$ than the denominators of all fractional coefficients that precede it, and no lower powers than the denominators of all succeeding fractional coefficients.”

Gauss now divides $Q$ by $p$ to guarantee that at least one term in it (at the very least, the leading term) has a fractional coefficient with a denominator divisible by $p$ , so that he can play the same game and choose the term $\Gamma x^{\gamma}$ of $Q/p$ with $\Gamma$ having a denominator divisible by $p$ more times than any preceding fractional coefficient and at least as many times as each subsequent coefficient. Let the highest power of $p$ dividing the denominator of $\Gamma$ be $p^{\tau}$ . (Having divided the whole of $Q$ by $p$ guarantees that $\tau \geq 1$ , just like $t$ .)

I’ll quote Gauss word-for-word for the next step:

“Let those terms in $P$ which precede $Gx^g$ be $'Gx^{g+1}$ , $''Gx^{g+2}$ , etc. and those which follow be $G'x^{g-1}$ , $G''x^{g-2}$ , etc.; in like manner the terms which precede $\Gamma x^{\gamma}$ will be $'\Gamma x^{\gamma+1}$ , $''\Gamma x^{\gamma+2}$ , etc. and the terms which follow will be $\Gamma'x^{\gamma-1}$ , $\Gamma''x^{\gamma-2}$ , etc. It is clear that in the product of $P$ and $Q/p$ the coefficient of the term $x^{g+\gamma}$ will

$= G\Gamma + 'G\Gamma' + ''G\Gamma'' + etc.$

$+ '\Gamma G' + ''\Gamma G'' + etc.$

“The term $G\Gamma$ will be a fraction, and if it is expressed in lowest terms, it will involve $t+\tau$ powers of $p$ in the denominator. If any of the other terms is a fraction, lower powers of p will appear in the denominators because each of them will be the product of two factors, one of them involving no more than $t$ powers of $p$ , the other involving fewer than $\tau$ such powers; or one of them involving no more than $\tau$ powers of $p$ , the other involving fewer than $t$ such powers. Thus $G\Gamma$ will be of the form $e/(fp^{t+\tau})$ , the others of the form $e'/(f'p^{t+\tau-\delta})$ where $\delta$ is positive and $e, f, f'$ are free of the factor $p$ , and the sum will

$=\frac{ef'+e'fp^{\delta}}{ff'p^{t+\tau}}$

The numerator is not divisible by $p$ and so there is no reduction that can produce powers of $p$ lower than $t+\tau$ .”

(This is on pp. 25-6 of the Clarke translation.)

This argument guarantees that the coefficient of $x^{g+\gamma}$ in $PQ/p$ , expressed in lowest terms, has a denominator divisible by $p^{t+\tau}$ . Thus the coefficient of the same term in $PQ$ has a denominator divisible by $p^{t+\tau-1}$ . Since $t$ and $\tau$ are each at least 1, this means the denominator of this term is divisible by $p$ at least once, and so a fraction. Q.E.D.

Like I said – nasty, right? But the concepts involved are just fractions and divisibility. Compare a modern proof of “Gauss’ Lemma” (the statement I quoted above from Artin – a product of primitive integer polynomials is primitive).

Proof. Let the polynomials be $P$ and $Q$ . Pick any prime number $p$ , and reduce everything mod $p$ . $P$ and $Q$ are primitive so they each have at least one coefficient not divisible by $p$ . Thus $P \not\equiv 0 \mod{p}$ and $Q \not\equiv 0 \mod{p}$ . By looking at the leading terms of $P$ and $Q$ mod $p$ we see that the product $PQ$ must be nonzero mod $p$ as well. This implies that $PQ$ contains at least one coefficient not divisible by $p$ . Since this argument works for any prime $p$ , it follows that there is no prime dividing every coefficient in $PQ$ , which means that it is primitive. Q.E.D.¹

Clean and quick. If you’re familiar with the concepts involved, it’s way easier to follow than Gauss’s original. But, you have to first digest a) the idea of reducing everything mod $p$ ; b) the fact that this operation is compatible with all the normal polynomial operations; and c) the crucial fact that because $p$ is prime, the product of two coefficients that are not $\equiv 0 \mod{p}$ will also be nonzero mod $p$ .

Now Gauss actually had access to all of these ideas. In fact it was in the Disquisitiones Arithmeticae itself that the world was introduced to the notation “ $a \equiv b \mod{p}$ .” So in a way it’s even more striking that he didn’t think to use them here when they would have cleaned up so much.

What bugged me out and made me excited to share this with you was the realization that these two proofs are essentially the same proof.

What?

I’m not gonna spell it out, because what’s the fun in that? But here’s a hint: that term $Gx^g$ that Gauss singled out in his polynomial $P$ ? Think about what would happen to that term (in comparison with all the terms before it) if you a) multiplied the whole polynomial by the lcm of the denominators to clear out all the fractions and yield a primitive integer polynomial, and then b) reduced everything mod p.

(If you are into this sort of thing, I found it to be an awesome exercise, that gave me a much deeper understanding of both proofs, to flesh out the equivalence, so I recommend that.)

* * * * *

What’s the pedagogical big picture here?

I see this as a case study in the value of approaching a problem with unsophisticated tools before learning sophisticated tools for it. To begin with, this historical anecdote seems to indicate that this is the natural flow. I mean, everybody always says Gauss was the greatest mathematician of all time, and even he didn’t think to use reduction mod $p$ on this problem, even though he was developing this tool on the surrounding pages of the very the same book.

In more detail, why is this more pedagogically natural than “build the (sophisticated) machine, then apply it”?

First of all, the machine is inscrutable before it is applied. Think about being handed all the tiny parts of a sophisticated robot, along with assembly instructions, but given no sense of how the whole thing is supposed to function once it’s put together. And then trying to follow the instructions. This is what it’s like to learn sophisticated math ideas machinery-first, application-later. I felt this way this spring in learning the idea of Brouwer degree in my topology class. Now that everything is put together, I have a strong desire to go back to the beginning and do the whole thing again knowing what the end goal is. The ideas felt so airy and insubstantial the first time through. I never felt grounded.

Secondly, the quick solution that is powered by the sophisticated tools loses something if it’s not coupled with some experience working on the same problem with less sophisticated tools. The aesthetic delight that professors take in the short and elegant solution of the erstwhile-difficult problem comes from an intimacy with this difficulty that the student skips if she just learns the power tools and then zaps it. Likewise, if the goal is to gain insight into the problem, the short, turbo-powered solution often feels very illuminating to someone (say, the professor) who knows the long, ugly solution, but like pure magic, and therefore not illuminating at all, to someone (say, a student) who doesn’t know any other way. There is something tenuous and spindly about knowing a high-powered solution only.

Here I can cite my own experience with Gauss’s Lemma, the subject of this post. I remember reading the proof in Artin a year ago and being satisfied at the time, but I also remember being unable to recall this proof (even though it’s so simple! maybe because it’s so simple!) several months later. You read it, it works, it’s like poof! done! It’s almost like a sharp thin needle that passes right through your brain without leaving any effect. (Eeew… sorry that was gross.) The process of working through Gauss’ original proof, and then working through how the proofs are so closely related, has made my understanding of Artin’s proof far deeper and my appreciation of its elegance far stronger. Before, all I saw was a cute short argument that made something true. I now see in it the mess that it is elegantly cleaning up.

I’ve had a different form of the same experience as I fight my way through Galois’ paper. (I am working through the translation found in Appendix I of Harold Edwards’ 1984 book Galois Theory. This is a great way to do it because if at any point you are totally lost about what Galois means, you can usually dig through the book and find out what Edwards thinks he means.) I previously learned a modern treatment of Galois theory (essentially the one found in Nathan Jacobson’s Basic Algebra I – what a ridiculous title from the point of view of a high school teacher!). When I learned it, I “followed” everything but I knew my understanding was not where I wanted it to be. Here the words “spindly” and “tenuous” come to mind again. The arguments were built one on top of another till I was looking at a tall machine with a lot of firepower at the very top but supported by a series of moving parts I didn’t have a lot of faith in.

An easy mark for Ewoks, and I knew it.

This version of Galois theory was all based on concepts like fields, automorphisms, vector spaces, separable and normal extensions, of which Galois himself had access to none. The process of fighting through Galois’ original development of his theory and trying to understand how it is related to what I learned before has been slowly filling out and reinforcing the lower regions of this structure for me. Coupling the sophisticated with the less sophisticated approach has given the entire edifice some solidity.

Thirdly, and this is what I feel like I hear folks (Shawn Cornally, Dan Meyer, Alison Blank, etc.) talk about a lot, but it bears repeating, is this:

If you attack a problem with the tools you have, and either you can’t solve it, or you can solve it but your solution is messy and ugly, like Gauss’s solution above (if I may), then you have a reason to want better tools. Furthermore, the way in which your tools are failing you, or in which they are being inefficient, may be a hint to you for how the better tools need to look.

Just as an example, think about how awesome reduction mod $p$ is going to seem if you are already fighting (as Gauss did) with a whole bunch of adding stuff up some of which is divisible by $p$ and some of which is not. What if you could treat everything divisible by $p$ as zero and then summarily forget about it? How convenient would that be?

I want to bring this back to the K-12 level so let me give one other illustration. A major goal of 7th-9th grade math education in NY (and elsewhere) is getting kids to be able to solve all manner of single-variable linear equations. The basic tool here is “doing the same thing to both sides.” (As in, dividing both sides of the equation by 2, or subtracting 2x from both sides…) For the kids this is a powerful and sophisticated tool, one that takes a lot of work to fully understand, because it involves the extremely deep idea that you can change an equation without changing the information it is giving you.

There is no reason to bring out this tool in order to have the kiddies solve $x+7=10$ . It’s even unnatural for solving $4x-21=55$ . Both of these problems are susceptible to much less abstract methods, such as “working backwards.” The “both sides” tool is not naturally motivated until the variable appears on both sides of the equation. I used to let students solve $4x-21=55$ whatever way made sense to them, but then try to impose on them the understanding that what they had “really done” was to add 21 to both sides and then divide both sides by 4, so that later when I gave them equations with variables on both sides, they’d be ready. This was weak because I was working against the natural pedagogical flow. They didn’t need the new tool yet because I hadn’t given them problems that brought them to the limitations of the old tool. Instead, I just tried to force them to reimagine what they’d already been doing in a way that felt unnatural to them. Please, if a student answers your question and can give you any mathematically sound reason, no matter how elementary, accept it! If you would like them to do something fancier, try to give them a problem that forces them to.

Basically this whole post adds up to an excuse to show you some cool historical math and a plea for due respect to be given to unsophisticated solutions. There is no rush to the big fancy general tools (except the rush imposed by our various dark overlords). They are learned better, and appreciated better, if students, teachers, mathematicians first get to try out the tools we already have on the problems the fancy tools will eventually help us answer. It worked for Gauss.

[1] This is the substance of the proof given in Artin but I actually edited it a bit to make it (hopefully) more accessible. Artin talks about the ring homomorphism $\mathbb{Z}[x] \longrightarrow \mathbb{F}_p[x]$ and the images of $P$ and $Q$ (he calls them $f$ and $g$ ) under this homomorphism.

ADDENDUM 8/10/11:

I recently bumped into a beautiful quote from Hermann Weyl that I had read before (in Bob and Ellen Kaplan’s Out of the Labyrinth, p. 157) and forgotten. It is entirely germane.

Beautiful general concepts do not drop out of the sky. To begin with, there are definite, concrete problems, with all their undivided complexity, and these must be conquered by individuals relying on brute force. Only then come the axiomatizers and systematizers and conclude that instead of straining to break in the door and bloodying one’s hands one should have first constructed a magic key of such and such a shape and then the door would have opened quietly, as if by itself. But they can construct the key only because the successful breakthrough enables them to study the lock front and back, from the outside and from the inside. Before we can generalize, formalize and axiomatize there must be mathematical substance.

Nonlinear

From Alison Blank:

Math Is Not Linear

How awesome is that?

Thanks Sue VanHattum for (indirectly) calling my attention to it.

Reflections, Part II: Try It vs. Hear How You Did It

This is a followup to my last post, processing a provocative conversation I had with my awesome friend and colleague Justin Lanier.

I want to try it vs. I want to hear how you did it

In my previous post I was saying I’d kind of like to take an Algebra I class and alternate units between my profound homespun curriculum design (if I may) and carefully group-digesting the textbook, and then at some point start letting the class choose which approach to follow. This gets to something that Justin brought up, which broadens this question beyond textbooks:

You’re presented with a mathematical problem. Unless you’re on the research frontier (and often even then), there are always two things you can do:
1) Try to solve the problem
2) Find out how somebody else solved the problem
Both things are, to paraphrase Justin, essential parts of mathematical experience. Both are options that the two of us sometimes go for.¹

When, if ever, do we give our students this choice?

What percentage of the time do we make it for them?

Justin was struggling with this question (what percentage of the time am I making this choice for my students and consequently what percentage of the time am I letting them make it?). In particular, with the sense that the former percentage is quite high, and not feeling entirely right about that. Having now had a month to mull on it,
a) I dig that Justin raised it as a question to be concerned with because now we get to choose deliberately what we want the answer to be and then try to implement that; and
b) depending on the class, it’s fine to make the choice for your students a very high percentage of the time.

More detail about (b) – this is related to another thing on my mind a lot lately, partly due to the conversation with Justin and partly due to its obviousness as a thing to think about:

In what (pedagogically relevant) ways am I similar to my students as a math learner and in what ways am I different?

I have a partial answer to this question, in the form of a principle:

The most important difference among math learners is not who is visual and who is auditory, or who is linguistically-oriented and who is nonverbal, or who is algebraic and who is geometric, or anything like that. These are interesting, useful differences, but they’re not the most important.

It is certainly not who is “fast” and “slow” or any such meaningless and pernicious bullsh*t. (More to come on this topic.)

The most important difference among math learners is, when faced with a problem to solve, an idea to grasp, a theorem to prove, who believes that they can they can solve/grasp/prove it and who doesn’t.

Category I: People who believe they can solve/grasp/prove it.
Category II: People who do not believe they can.

I believe that the best thing math teachers can do for the world is take people in Category II and move them to Category I.

To repeat my question: in what (pedagogically relevant) ways am I similar to my students, and in what ways different?

My partial answer: One difference is that I am in Category I and most of my students are in Category II. I think this is probably the main difference.²

Now, when faced with a math problem and thus the choice above between option (1) (try to solve it) and option (2) (find out how somebody else did), people in Category II always choose option (2). (Why? Clear: because they don’t really believe (1) is an option.) Meanwhile, in order to learn math, you need to be taking option (1) a big percentage of the time. This is true of everybody but especially true of the folks in Category II. How could you ever come to believe you can solve it except by solving it, a lot of times? And how could you do this if you don’t first try to solve it?

Now as I write this I can’t escape the feeling that I’m writing stuff down that is obvious to the point of being tautological. But I think it adds up to something useful in considering Justin’s question. (How often do we choose for our students between (1) and (2) and how often do we let our students make this choice?)

What it adds up to is this: if you teach a lot of Category II folks, you need to do a lot of choosing (1) for them, since they need it desperately and won’t be choosing it on their own. In fact, you need to be all up in their sh*t, not just at the moment they are faced with the choice between (1) and (2) but
*When they have a sound thought you need to point it out to them
*When they have a wrong thought that leads to a productive conversation you have to call their attention toward the productive conversation it led to
*When they see something from a new angle that illuminates it in a new way, you have to make sure the reflexive feeling of exhilaration doesn’t go unnoticed
*When they jointly solve a problem you need to make sure they noticed their own contribution to it
*Etc.
In other words you need to be in control of quite a few elements of their math learning process. Their choice between (1) and (2) is only one of these elements.

So under this circumstance (namely a room full of Category II folks), it’s really actually quite appropriate to be choosing for them – in particular, choosing option (1) for them.

What you don’t want to be doing is choosing option (2) for them when they were ready to choose option (1). I think one of the important sensitivities to develop as a math teacher, along with the ability to recognize at a glance the “I don’t understand but I don’t want to ask a question” face and the difference between somebody thinking about it and somebody being totally stuck, is a sharp ear for the quiet, tentative, tinkly sound of a Category II student who is about to choose option (1). If you hear it, do not intervene.

This all said, one of the things I took from the conversation with Justin was a reminder that although Category I is always a better place to be than Category II, option (1) is not essentially better than option (2). They are both legitimate choices. Category II people need a lot of option (1) and won’t choose it for themselves, so you gotta take control there, but in general there is a time and place for each kind of mathematical experience. Each one’s got something going for it the other one doesn’t:

By far, all the most exhilarating mathematical experiences of my life have followed the selection of option (1). These were the times there was adrenaline involved, so that my hand was shaking as I put pen³ to paper. Nothing in mathematical life can top the peak experience of realizing something.

By the same token, option (1) immediately preceded all my most profound experiences of mathematical power. (Empowerment, if you like.) I pretty much walk through life feeling like I can accomplish anything with enough patience, and solving mathematical problems myself has been maybe the biggest thing that’s made me feel that way.

However, by definition, I had chosen option (2) prior to every time I’ve had the privilege of seeing something beautiful that I would never have thought of myself.

Relatedly, most of those experiences of empowerment I mentioned above involved some option (2) that came before the option (1) and empowered my actually solving something. In an abstract algebra class last year, my professor said, “now I’m going to solve a cubic… with my bare hands.” I didn’t get that sense of power till I did it myself later that summer, but I probably wouldn’t have done it myself if I hadn’t watched him do it.

So Justin’s train of thought feels like it’s raised my consciousness a bit. This is something to be deliberate about. For students to appreciate both options (1) and (2) and have the choice as much of the time as possible feels like a beautiful ultimate goal. But the main thing is just to keep in mind that not only [option (1) vs. option (2)], but also [our choice vs. students’ choice], are pedagogically significant arenas, and we should stay aware of this as we navigate them.

One other thing I recollect from the conversation with Justin that I want to record before wrapping this up:

I really love the act of explaining things I think are awesome. But this is not a reason to do it in a teaching situation. The idea is to be pedagogically deliberate about making the choice between options (1) and (2) for your students. If I decide you’re going to listen to me share a solution, I should be doing it because I think it’s what’s best for you, not because I love doing it. I will repeat myself:

The impulse to share something awesome needs to be entirely repressed while teaching. Sharing something awesome should come from a judgement of the pedagogical need, not a desire to share.

(We are allowed to love it once we’ve decided it’s pedagogically called for; but the desire to do it can’t be driving this call.)

Notes:

[1] We didn’t talk about it at the time, but there is a third very important option, which is to work on the problem with somebody else. This is sort of “between” the other two and also sort of qualitatively different. But the two choices above work to define the extremes of a spectrum, so I’ll stick with them for the purpose of this discussion.

[2] It is not the subject of this post but I have to acknowledge what I think is the second most important difference among math learners: when faced with a problem to solve, an idea to grasp, a theorem to prove, who sees value in solving/grasping/proving it and who doesn’t.

[3] Yes, pen. I do all my math in pen. I am not going to tell you to stop telling your kids they have to use a pencil, but honestly I don’t like this pencil-only doctrine. “Put that pen away right this minute! If you don’t use pencil, you won’t be able to hide your mistakes and pretend they never happened!!” This is the message we’re going for?

(If you want to tell me you have a particular kid who really should be using pencil, I believe you. It’s the general principle I’m objecting to.)