## I have a little note that I’ve left for myself that I keep in front of me at work

Reminders for the classes where I don’t have a curriculum I’m marching through

Retrieval practice from yesterday
Something too hard
Something too easy
Something just right

Such a little thing, but it makes a huge difference for me.

## Responding to Criticism from @blaw0013

I wasn’t sure whether to respond to this or not. I want to be the sort of person that gives people stuff to think about, and (just like in the classroom) there’s a point where you have to step back and give people a chance to speak.

But: “deny joy of and access to maths for many”? It’s an interesting criticism, one that I have a lot of thoughts about.

I don’t see micro-skills as denying joy and access to students. And I think it’s partly about seeing joy in maths as something that happens in the abstract versus something that happens in the context of school.

If you think “abstractly” about what joy in math involves, your mind would probably start thinking about the sort of math that is joyous and exciting, the very coolest stuff that math has to offer. You would think of noticing surprising patterns, of unusual theorems, the endorphin release of cracking a puzzle.

Francis Su is the current leading expositor of this side of math, the beautiful, joyous, elegant side:

Pursuing mathematics in this way cultivates the virtues of transcendence and joy.  By joy, I refer to the wonder or awe or delight in the beauty of the created order.  By transcendence, I mean the ability to embrace mystery of it all.  There’s a transcendent joy in experiencing the beauty of mathematics.

If you think abstractly, and ignore the context that students of math actually encounter math in, then you’d look at something like “micro-skills” as just the opposite of all this. And yet I think if you look at the reality of students’ lives (instead of a radical proposal for what students’ lives should be) then I think you can see where joy comes into the picture.

Yesterday I gave students a no-grades quiz in algebra. A student who, I had been told at the start of the year, frequently struggles in math, has been having a lot of success lately. She knew exactly how to handle both of the systems of equations that were on this short quiz, but she got stumped at one of the resulting equations:

$-1.7x = 4.3x + 3.6$

I didn’t know what to say when she got stuck, exactly, but I was fairly confident that this was an example of a micro-skill that she was missing.

She and I agreed that she’d like me to write a little example on the side of her page, so I wrote this:

$-2x = 5x + 7$

[I drew some arrows going down from each side labeled “+2x.”]

$0 = 7x + 7$

My student read the example and then exclaimed (in a way I can only describe as “joyous”), Oh wait, you can make 0 there?!

You can! It’s very cool, and to the mind of a child learning algebra it’s surprising, elegant, beautiful, joyous. This is what I’m talking about — not treating the moments when kids get stuck as “forgettings” or “bugs” in some universal algorithm, and instead thinking of them as opportunities for students to prove mini-theorems, try mini-strategies, learn mini-skills.

And to treat these as moments lacking joy is also to ignore the major impediment to joy in a classroom: feelings of incompetence, worries about status, anxieties about math.

I’m no psychic and my students’ story isn’t mine to tell, but she showed all appearances of being happy and relieved when she understood how to go about solving this problem. How could she experience this, given that she was dealing with the drudgery of a micro-skill? Well, part of it is that (it’s easy to forget) things that are drudgery to teachers are often rich, problematic (in a good way) terrain for students.

But part of it is that these are children in school, surrounded by other children in school. Joy can’t be separated from that social context. Students can’t experience joy if they don’t feel competent, and conversely there is joy in competence. I see this every day.

If, like me, you care both about helping kids experience joy in math and joy from competence in math (hard to separate) then you need to find opportunities in your teaching to do both. The above is how I’m currently thinking, and I’d be interested to read Brian’s take on all this — maybe he and I can find a way to write up a case that illustrates the different choices we’d make in a situation like this. I love the idea of collaborations to resolve differences.

## Here’s what I think I do differently

I just read a really interesting post called ‘Applying Variation Theory.’ It’s by a teacher from the UK who I don’t yet know, Naveen Rizvi. The core mathematics is familiar territory to teachers of algebra. To factor an expression like $x^2 + 10x + 16$, you can ask yourself “what pair of numbers sum to 10 and multiply to 16?”

Answer: 8 and 2.

Therefore: $x^2 + 10x + 16 = (x + 8)(x + 2)$

I actually first learned about this from Dan Meyer, and for my first few years teaching quadratic factoring I used his “Diamond Problems” as a factoring lead-in:

In her post, Naveen talks about intentionally designing problems of this sort to draw out this underlying structure — that the factors of “c” have to sum to “b.” Factoring then becomes a quest to search the factors of “c” for things that sum to “b” (or vice versa).

Her main points is a good one, which is that if you keep everything else the same, and then vary just one thing, that thing will draw a lot of attention. Here is how she uses these ideas to design a practice set of quadratics, one that isn’t so unlike Dan’s:

My experience of teaching this topic is that, even knowing the relationship between c and b, these problems can be very difficult for students.

Why? What makes these factoring problems hard? Part of the reason has to do with fact automaticity, to be sure. While there are many topics in algebra that a student can handle without automaticity, this is definitely one of the times when things get much hairier for kids if they don’t know e.g. quick ways to find all the ways of multiplying to 60.

But take a closer look at some of these problems, and you can see that there is more going on than just knowing the relationship between b and c and knowing your facts. Consider one of these problems from Naveen’s page, which I just chose randomly:

$a^2 + 29a + 28$

This is a problem that I can imagine my students having some trouble with, at first. Not because the facts are difficult or because they don’t know the relationship…it’s just that the solution (a + 1)(a + 28) might not occur to kids. One thing I’ve noticed is that a lot of kids don’t think of 1 x N for a while when they’re searching for ways to make N. This makes sense — it’s so computationally straightforward, they don’t spend a lot of time thinking about multiplying by 1. It can slip under the radar.

Now, might a kid working on Naveen’s problem set become familiar with this tiny nugget or structure, that if the “b” term is one off from the “c” term, you should try sticking “1” into one of your binomials?

A student might make this generalization from the examples in the practice set, but this would essentially amount to learning by discovery. Which absolutely happens sometimes, but especially since this activity isn’t structured around giving kids instances that would cue-up that generalization…it probably won’t happen for most kids.

Now, the question is whether this sort of “micro-strategy” is a good use of classroom time. Maybe it’s too narrow a class of problems to be worth making it an instructional focus, I don’t know. Maybe you just go for the main strategy, and hope that kids are able to apply what they know to this little side-case.

But then again, maybe you give a quiz and kids end up mostly baffled by this problem. Teaching is full of surprises — this could happen.

That’s when I say, OK, let’s design a quick activity that would focus entirely on this micro-skill. Maybe a mini-worked example, or maybe a string of mental math factoring problems ala Naveen or Dan’s that puts that entire “variation theory” focus on this one, specific corner of the mathematical landscape — just the one that the students in your class need.

$a^2 + 8a + 7$

$x^2 + 14x + 13$

$j^2 + 176j + 175$

$q^2 - 14q + 13$

$s^2 - 11s - 12$

And then I call this “feedback” and don’t spend a lot of time writing up stuff in the margins of their quizzes.

There’s something nice about these little micro-skills. For one, it’s an alternate way of thinking about what’s leftover after you’ve taught the “main” skill. (Meaning, it’s not just that kids are forgetting what you’ve taught and need to be reminded — it’s that there are little corners of the mathematical world that haven’t yet been uncovered for kids.)

One thing I’ve been struggling with has been trying to figure out what exactly my pedagogy involves that’s distinctive. It’s not about the activities I tend to choose or design — since I’m pretty boring in that regard. I’m not an amazing motivator of people. Kids like me, I think, but not in the “oh my god he was my best teacher” way. More like “he’s nice.” (“Being nice” is an important part of my pedagogy.)

Though I still don’t have a snappy way to put it, I think that this is part of my story:

• I’m really curious about how kids think
• So I try to use that to come up with a more systematic understanding of how they think about different types of problems, especially when it’s something that people typically think of as constituting a single “type” of problems (e.g. factoring quadratics, solving equations, adding fractions)
• While teaching a topic I try to figure out which types of problems the kids understand how to handle, and which they don’t yet
• Then I focus in on a micro-skill for handling one of those little types of problem and I teach it with a short little activity followed-up by practice, in place of feedback

I don’t think any of this is exciting or inspiring, and I don’t really think I can make it so. There really is something here, though.

I think the exciting action comes in the second of those bullet points, in describing the mathematical landscape in a way that’s pedagogically useful. One of my favorite things in math education is Carpenter et al’s Cognitively Guided Instruction. I’ve moved away from some of the pedagogy it has grown into, but the problem breakdown is my paragon of pedagogically useful knowledge. It’s what I always come back to.

I sometimes wish there was more to what I do, and I also wish I didn’t have such a hard time figuring out how to describe what it is that I’m into. I’ll be doing a thing with teachers this spring that will give me another shot at refining my little message. I sometimes get jealous when I see all the cool things that people make and share online. I’ve never been cool, and none of what I’ve written here is cool either, but it’s what I’ve got.

(Last year’s version of this post: here.)

## Some of the things I stand for in education, these days

When we critique an idea, we should critique the best version of it.

When we critique a pedagogical idea, we should critique the idea itself, not its misinterpretations. (Unless we’re saying the idea is easy to misinterpret.)

Most of the time when a pedagogical idea is critiqued, it’s critiqued for bad choices the teacher might make that have nothing to do with the idea itself. When we imagine an idea we don’t like, we imagine a classroom that we don’t like. That’s not fair, though.

Every pedagogical idea gets misinterpreted.

Nobody knows how to make ideas about good teaching scale, so we might as well talk about what good teaching actually looks like, without worrying about how the truth will or will not be misinterpreted at large.

You and I might teach in very similar ways but have wildly different ways to describe how we teach. The fiercest debates in education are also the vaguest. When you get down to classroom details, or even the tiniest bit of additional specificity, a lot of disagreement vanishes. It’s not that these debates don’t matter, it’s that they are highly theoretical.

OK maybe these debate don’t really matter.

Teachers have access to classroom details and specificity. One way that teachers can contribute to the knowledge base of education is to resist the urge to move to generalization, to spend some more time in the greater specificity that classroom life encourages. This is what teachers can uniquely contribute.

Non-teachers will often tend to think about scale. Teachers usually don’t, and this is something else that we can contribute.

Some people will tell you that being a jerk is important, as long as you’re being a jerk to the right people. Those people are jerks. Stupid jerks.

I think the core of what I do is trying to have a detailed understanding of how my students think about the stuff I’m teaching, and also of how they could think about the stuff I’m teaching. And then I’m trying to create as many opportunities for them to think about that stuff in the new way.

I care about my students’ feelings, a lot.

The way I see it, teaching is best understood through its dilemmas and tensions. This is not a new idea, but it’s one that hasn’t been sufficiently explored. And I’m very suspicious of people who claim to have resolved one of these tensions or dilemmas.

We shouldn’t ad hoc create rules for political discourse that only apply to people who we disagree with.

Debate is essentially performative. I mean, it doesn’t need to be, but it usually is.

Trying to understand someone and how they think is an almost complete improvement over debate in every way.

Lots of research is interesting. It’s especially fun to try to figure out how two different perspectives on teaching can fit together.

Teaching is a job, but it can be a great job if you like ideas and little humans.

Writing and reading should be fun and interesting. When writing instead aims to be useful, it’s usually not useful either. And 99% of writing about teaching is supposed to be useful.

At a certain point you have to decide if you’re trying, primarily, to change the world or to understand and describe it. It’s great that some people are trying to change the world, but I think inevitably those people end up having to be less-than-honest at times, for the sake of their projects or reputation. Personally, I prefer to understand and describe it when I write or think. (I’m not opposed to helping people, though!)

Blogging is dead, but it’s all we got for now, so let’s keep at it.

I like to think of my teaching life as a bubble, and I like to think of all other aspects of my life as little bubbles too. And a question I ask myself frequently is, can I grow these bubbles? Wouldn’t it be something if all the other bubbles could float and sort of merge into each other, turn into just one big bubble that encompasses everything? I feel like that would be nice, someday.

## What I wish the NCTM journals were

It’s hard for me to separate what I like doing as a writer from what I enjoy reading. Back when I was critiquing the NCTM journals it felt like I was being sort of vague about what I wanted…well, I had to be vague, because what I want to read are roughly the kinds of things that I write. (Emphasis on kinds; I don’t want to read me, I want to read you.)

But just to put the pieces together, here is what I would love to read in an NCTM (or any other) math education journal:

Toss in a few features — crosswords, math problems, a curriculum resource, etc. — and this is the sort of thing that I, Michael Pershan, would be interested in reading. I have no idea who else would read or pay money for this sort of thing, so please NCTM don’t rashly decide to change the journals to more closely resemble my blogging output. But this is close to what I imagine would be my ideal publication, i.e. the sort of pretentious garbage that’s my calling card.

## My Answers to “Six Questions Math Educators Need to Answer Honestly”

[Questions can be found here.]

1. What is mathematics?

Don’t know. Pass.

2. Where should we learn mathematics?

Don’t know what “should” means here. Pass.

3. Should mathematics be a compulsory subject?

Question: What would the immediate effect of changing of making math non-compulsory tomorrow would be?

We’ve had decades (centuries?) to learn a bit about how to teach what we teach. I’m not saying it’s easy to teach systems of inequalities well; I’m saying that if I’m going to teach systems of inequalities there are dozens of resources I can turn to, curricula that have thought carefully about how to put the pieces of the course together to reach many students.

Have you ever taught a new course or an elective? It’s hard, isn’t it, not knowing how to go about the work. A world where we radically change the curriculum or where we eliminate mandatory classes is that world. It’s not like number theory and graph theory is easier to teach than polynomial division. Yeah, number theory is really cool, but it’s hard to turn cool things into learning. That’s the whole art of teaching.

So go ahead, make math non-compulsory. Honestly, I don’t care, what we ask kids to study in school is mostly arbitrary anyway. In the 1920s it was unclear whether math would be a compulsory subject in high schools — NCTM was created, in part, to protect math compulsory math offerings. If they’d lost that fight, who knows what hoops we’d be asking students to jump through before graduating. Mandatory home ec? Mandatory workshop? What reason do we have to think that these mandatory offerings would be better taught?

And if we make most of the curriculum non-mandatory? That’s like college, and if you don’t like how k-12 is taught then, wow, get excited for higher ed.

The fact is that there are tremendous pedagogical benefits from having a lot of people teaching and learning the same thing. School requirements are arbitrary, but there are benefits to keeping things the way they are.

4. How should we learn mathematics?

5. Who should learn mathematics?

Don’t know what “should” means here, pass.

6. What is the purpose of mathematics education?

Tempted to pass, but I think there’s something interesting to think about.

The purpose of math education depends on the purpose of math, and the purpose of math has varied across history, across cultures, across math departments, varied so much in the time of Fermat that there wasn’t even a single thing called “mathematics.” From The Mathematical Career of Pierre de Fermat:

With due allowance for exception, one may sort those practitioners into six broad categories: the classical geometers, the cossist algebraists, the applied mathematicians, the mystics, the artists and artisans, and the analysts. Although only one or two of these categories constitutes what one might call a “school” of mathematics, and although the work of many individuals falls into several categories, nonetheless each category distinguishes itself from the others by characteristic attitudes toward the nature and purpose of mathematics, its problems and methods of solution. Each category has a distinctive style, and the different styles often conflict in essential ways.

When people talk about the “values of math” or the “purpose of math education,” know that the question is ill-formed. Math education emerges from the competing needs of students, parents, communities, our government. It’s inherently contradictory, and given that the conflicts are genuine it probably should be sort of a mess. Teaching is not hampered by these conflicts, teaching is the job of making sense of these conflicting needs. That’s the job.

Math in 2018 is used for joy, pleasure, getting rich, pursuing justice, attacking foreign governments, passing tests, teaching, and dozens of other purposes too. When educators say “we have to get back to the values of math” that’s a lateral move; that’s no clearer. Math has as many purposes as math education.

“Math education is a very young child of mathematics,” Singh writes, a completely incorrect statement. One thing I learned from Learning Modern Algebra is that pedagogy is a major source of mathematical innovation. Part of why you’d need a way to find Pythagorean triples is to help you come up with nice problems for your students.

Singh calls math education “a spoiled and rotten child” of mathematics, as opposed to mathematics, whose purposes are clear and pure:

Mathematics is not up for debate. It is what it is, and it has been tattooed in so many civilizations and cultures for many millennia. Its purpose and mandates, might run parallel to math education at times, but in reality, they operate on a higher plane of truth, justice, beauty, play and love

Historically speaking, this is nonsense. Math changes, math has different purposes. Math is contradictory, math education is contradictory. Math is beautiful and ugly, so is math education. In math education we’re not the spoiled child of math — we’re the grown sibling, sometimes in touch, sometimes not.

So I’m entirely against Singh’s idea that we can look to the values of math and measure math education against them. In sum: Stop pooping on math education and educators. The end.

## “Nobody is every going to invent an ethics class that makes people behave ethically.”

“Nobody is ever going to invent an ethics class that makes people behave ethically after they step out of the classroom. Classes are for riders, and riders are just going to use their knowledge to serve their elephants more effectively. If you want to make people behave more ethically, there are two ways you can go. You can change the elephant, which takes a long time and is hard to do. Or…you can change the path that the elephant and rider find themselves traveling on. You can make minor and inexpensive tweaks to the environment, which can produce big increases in ethical behavior. You can hire Glaucon as a consultant and ask him how to design institutions in which real human beings, always concerned about their reputations, will behave more ethically.”

From Haidt’s The Righteous MindThe elephant/rider metaphor is sort of clumsy, but his point is that rationality usually chases our moral intuitions; it’s rare for reason to override those intuitions.

Could an ethics class ever change the moral intuitions? Could a math class? What does ever change those moral intuitions?

Haidt’s answer seems to be that most changes to our intuitions come from our interactions with other people who we admire and want to admire us. I don’t know what that could mean for the possibility of moral education.

## Gutierrez on the tensions of teaching

Although teachers must recognize they are teaching more than just mathematics, they also have to reconcile that fact with the idea that, ultimately,  they are responsible for helping students learn mathematics. Teachers who are committed to equity cannot concern themselves with their students’ self-esteem and negotiated identities to the exclusion of the mathematics that the students will be held responsible for in later years. Yet preparation for the next level of mathematics must also not be the overriding feature of a teacher’s practice. In answer to which of the two foci are important (teaching students or teaching mathematics), I would answer “neither and both.” It is in embracing the tension…”

Rochelle Gutierrez, Embracing the Inherent Tensions in Teaching Mathematics from an Equity Stance

What does it mean to live in tension? I suppose it’s an emotional feeling, of not knowing at any given moment which way you should lean. I like the idea that teaching involves tensions that can’t be resolved, though I admit I’m still fuzzy on what it means to teach in a way that reflects such a tension.

## In response to Heidi’s post

https://tooteeter.blogspot.com/2017/12/8-12-by-12-inch-paper-cardinal-vs.html

How did you get into my 3rd Grade classroom?? I was dealing with all of this just a few weeks ago.

My situation was slightly different, and because it informs my thoughts about your questions, I”ll share it here.

I decided to move to measurement out of frustration with my students’ ability to subtract. In particular, I was asking them to find the distance between two numbers on the number line, and it felt like we weren’t going anywhere.

Based on previous years memories of students who start measuring at the 1′ mark, I began class with a picture of a ruler and a pencil on the board. I placed the pencil so one end was at the 1′ mark, the other at the 6′ mark. How long is this pencil, I asked the class? The first view was, as you might expect, 6′, and I don’t remember if another kid or I was the one to disagree, but I drew the class’ attention to the alternate view, which was based on two things:

a) Counting the spaces, not the lines
b) Our ability to slide the pencil to anywhere that we want on the ruler (i.e. it wouldn’t make sense for the length to change)

Then I gave out rulers and asked them to measure that piece of paper, that same Investigations activity.

OK, and one last thing: when kids end up with different measurements, one thing I’ve learned to do is to turn it into a statistics problem. At the end of their measurements I tallied their measurements on the board in a list. I point out that variation in measurement is totally normal — there’s no way to make a perfect measurement, there is always error. So, I asked, looking at our measurements, what do we figure the TRUE measurements of the paper to be?

What does all this mean? My first guess is that a lot of number line strategies don’t make a ton of sense to kids yet, but that measuring helps. When we want kids to realize that 20 + 30 = 15 + 25 because you can slide both down the number line? That seems like a strategy best built on something like experiences with a ruler.

(It also reminds me of this research paper that we talked about, where teachers who focused more on measurement seemed to help struggling students more than those teacher who dug in on arithmetic.)

Second, an alternative to asking kids to remeasure until they get the correct measurement might be to record their various measurements and then to ask the statistical question about them: what do we think the true measurement is, given the spread of measurements we’ve been making? can we remeasure to get closer to that true measurement? Because, unlike addition or subtraction, there isn’t a way to measure the true length — every measurement is an approximation.

All this said, I’m not really sure why it makes sense to kids to start at 1 instead of 0. I like your idea that moving between tiles and lengths might get at the true relationship, but I find myself still puzzled about how kids see the ruler.