I flipped around this Illustrative Math lesson — it went pretty well!

Every year, this lesson has given my students trouble:

Here were the problems I had with the lesson:

• My students didn’t have strategies for making sense of why the faster bug would go on the bottom. Kids would quickly misidentify which line was which bug, and we’d have to back up and talk about that for a while before doing anything else.
• So they didn’t really get a chance to engage with the math. And I wasn’t sure what the math exactly was, beyond this little tricky graph that puts time on the y-axis.
• Also the formatting was tricky, because the ladybug/ant race happens on one page and the graph on the other. You don’t necessarily need cognitive science to tell you that swapping between two very separated images makes learning hard, but it does.

Each of the Illustrative Mathematics lessons has a Summary at the end of the lesson. It’s good, but meant as a reference — it’s not really designed for classroom use.

So here was my redesign idea:

• Turn the summary into sample student work i.e. a worked example
• Pair that with analysis prompts and a follow-up question i.e. an example-problem pair
• Redesign the actual materials so that the graph and the race are next to each other

Here’s what I did:

We used the original “warm up” from the materials. Then shifted into the example-based materials I created. Then the redesigned activity from the lesson itself.

It went well! Here’s how I know that it went well: most kids got to the extension questions, and the students were able to focus their on those.

That seems to me the basic tradeoff. If you leave ideas a bit more implicit, then kids will spend more time uncovering them. That can be good, mathematical thinking, of course. The other choice is to make things more explicit at the outset. Then, maybe you have a better shot of diving into what would otherwise be “advanced” “challenge” problems.

I usually make that second choice, and part of why is because I think good mathematical thinking can happen with the example-based materials I shared. After that warm-up (where I asked kids to notice as many details about the graphs as possible and didn’t really push the “wonder” question) I covered up the “student solution” and showed my students the “problem” I had created. Then, I uncovered the student work. There was a pause — followed by “oh!” and “ah that makes sense.” There’s a little mathematical thrill you get every time you figure something out — a few kids got that when I revealed the work.

Then I asked kids to talk about it with partner, and then to solve a similar problem with neighbors. I listened in on conversations and was able to figure out if kids were understanding the example or not. A few times I inserted myself into conversations to help. And then I led a discussion about the example where kids shared the following ideas:

• That the ratio between the heights of points on each line that are directly atop each other stays equal.
• That you could also compare points that are at the same horizontal.
• That if the axes were swapped the top line would represent the faster racer, because then the top point covers more distance in the same time.

And the kids most eager to share were not the ones who usually solve problems with the most confidence (and therefore least likely to share if all of this had come through problem solving).

And then we did the activity that had given my kids trouble each year of the past, and they were able to be struggle productively i.e. they had the “compare points on the same vertical/horizontal” strategy. And they got to extension problems.

I’m going to keep looking for chances to do make this same trade.

What we’re debating when we debate “misconceptions”

Is ‘misconceptions’ a bad word? I’ve had the conversation about misconceptions a number of times, most recently when I wrote this post. Here is a bit from the conclusion:

We see misconceptions in children because it really is true that there’s stuff that they don’t yet know. Noticing this doesn’t have to be an act of violence — in fact, I don’t think that it usually is. Usually it’s like me playing with my son and noticing there’s stuff he doesn’t yet know how to do, even as my mind is blown because oh my god my son is into puzzles! When did our baby turn into a kid?

Is it good pedagogy to ask people who don’t already see their pedagogy as abusive to forswear from using words that they use all the time? Isn’t this exactly the sort of “intellectual violence” that we’re being urged to refrain from? Shouldn’t we start with the way people actually see the world, rather than asking them to use language that is not their own?

That excerpt did not convince anybody at all, but my goal here isn’t to convince. Really all I want to do is bring up something I learned about the constraints of this argument.

There are a couple people I’ve met who have flirted with the idea of cutting out all evaluative language from discussions of teaching, but it’s largely an unsustainable position. You can’t cut out value from teaching, and the thought that you can is a bad mistake. Even if you don’t talk of “misconception” you’re still in need of language to describe thinking that isn’t yet what it could be. Maybe there are no misconceptions, but there is thinking that is e.g. inflexible, procedural, memorized, additive-but-not-yet-multiplicative, trick-reliant, stage one, whatever it is you want to say.

Plus, the math education community very clearly want to be able to understand problematic language and ideas for what they are. We want to be able to call ideas or patterns of thought racist, sexist, colonialist, etc. That’s very different than the “all thinking is just thinking” position.

And so the discussion is only ever about what is particularly harmful (or not) about the term “misconception” and its popular usage. Though people frequently talk about the issues with evaluative language in general when discussing misconceptions, that argument just confuses things. We need to be able to talk about thinking in terms of what it could, even should ideally be.

So there are really just two questions that are relevant for this discussion. Is the term “misconception” particularly harmful, compared to other evaluative language? And even if the term is intrinsically fine, is it used in particularly harmful ways?

I’ve shared my answers, but I’d make the case that those are the right questions.

YouCubed, Reviewed

This exponents activity is neither original nor at all an interesting version of the idea. It’s no better than what most teachers would make on their own, if they wanted to teach exponent rules inductively.

Better versions of this are readily available in practically any textbook, but Illustrative Math has a totally free and online unit on exponents that does this activity better. It’s less tedious and repetitive and it asks questions to push students towards generalizations, rather than asking kids to churn out rows and notice the structure at the very end (“discovery”).

Yes, it’s at a Grade 8 level, but this lesson is pretty much there too. And if you can wait a few months, you’ll have the high school version available too.

“Equity” is dead, long live equity

By the time organizations — even organizations whose work I really like — start using the language of equity to advertise their work, it’s a sign that we’ve overtaxed the latest bit of edu lingo. “Equity” is at that point in the edu fad life cycle; it’s beginning to mean just about anything.

I don’t know if there’s anything to do about this. I think this is less about education and more about the corporate world — business lingo isn’t much better than edu lingo. People want to signal that they get it, without getting too bogged down in what exactly “getting it” entails.

The thing I try to remind myself is to be specific and to use familiar, boring words whenever possible. In place of stuffing meaning into abstract terms, I try to put it into sentences. And instead of “equity” I try to talk about the particulars: unsafe classrooms, hot schools, bad water, inexperienced teachers, and so on. This is my personal resistance to the educational world’s endless desire for catchy language, as I think it’s really all we’ve got.

Some questions about the problem of teachers leaving the classroom

Is it actually a problem for kids? Would schools be more effective places if more teachers on the margins of leaving were to stay in the classroom? How do we know? Is there a correlation between ambitious and teaching skill? What is the correlation?

Do master teacher programs improve learning for a district?

How much of the stress in education about people leaving the classroom could be explained by how uniquely meaningful working with children is? After all, going into management involves a change at work across professions. (Sales managers don’t go on sales calls; you leave the regular police work to get a desk job; you still do rounds occasionally but mostly you don’t see patients, etc.) How much of the problem is that there is a huge emotional gap between teaching and higher-paying work that keeps teachers in the classroom, marginally?

Would people be more effective at their administrative jobs if they were partly in the classroom? Would they be more influential?

William Carlos Williams was a doctor by day, poet by night. No one suggests that there should be more doctor/poet jobs. How do we decide what sorts of jobs their ought to be?

I find this so confusing. What questions do you have? Comments are open.

Some of my assumptions for communicating about teaching

These all might be wrong, but I think some of them are worth exposing. Maybe you’ll help me see how I’m wrong?

1. When have something I want to say about teaching or learning, there is a temptation to coin a new word that identifies a new concept. I try to avoid this temptation.

Suppose, for example, that I get up at a conference and say “math should be sticky.” There are some risks. First, there’s the risk someone will spend a lot of time puzzling over what I mean by “sticky,” remember the phrase, and have no idea what I meant by it in context. (This happens often — people remember memorable tags but struggle to articulate what they mean.) Probably then I’ll start hearing people say that I believe that you should teach in such-and-such a way because it’s “sticky” when that’s not what I meant. There’s also a risk that my word will have connotations that I didn’t expect. (Oh, you think “sticky” is gross and bad? Oops.)

So as a rule — a writing rule, a speaking rule — I try very hard to only use words that I think everybody pretty much uses in the same way.

This is not easy, because (I associate this thought with Ilana Horn) the meanings people assign to seemingly clear words like “discover” in teaching varies a great deal. I might say “worksheet” and you might imagine “evil packet that kids work on in silence and struggle” and I imagine “a bunch of problems on a page that hit the sweet spot for kids, who are asking questions and talking together about math.”

So it’s not easy, but I do try. It helps to keep an eye out for words (like “worksheet”) that could be misunderstood, and to replace those with context and sentences that make it clearer what’s happening and what I’m imagining.

2. I try to avoid advocating for practices unconditionally. What I mean is that I never say “we should do this in class more!” without suggesting when it might be useful to do that in class. I’m thinking about this right now with worked examples. I think example-based learning is great and cool and fun, but I would never give a talk (I think) calling for greater use of examples in teaching. Instead, though, I would give a talk describing situations that especially call for worked-examples and teaching people how examples can be useful in that context. (Here are two: “examples as feedback” and “examples as models for really complex thinking.”)

Likewise, I try never to talk in general about teaching, or about teaching math in general. I try to stay conditional.

***

These two things, I think, make communication about teaching easier. As a consequence, I think it ensures that nobody thinks that I mean something I don’t mean, and nobody thinks that I have solutions to many of their teaching problems, or a message that would revolutionize math teaching.

And, as a further result of that, what I have to say is less broadly meaningful, polarizing and also less popular. That’s the tradeoff, I think. Clarity for popularity.

Addendum: I have nobody in particular in mind with this post, but it was inspired by a lot of the tweets I saw from the NCTM conference. I’ll say that the “unconditional” thing was inspired by advocacy for a lot of the thinking prompts that don’t call for precise answers — numberless word problems, goal-free problems, estimation problems, notice/wonder, etc.

These are all incredibly useful, but (I think) far more useful when a topic is new to a student. So I think the general direction is that these more open prompts are great ways in, but you sort of want to call for more and more precision in your prompts as the learning progresses.

I was once talking to a friend who felt burned by Estimation180. Why, I asked. Well, she was trying to use it every day to improve her students’ number sense, but it hadn’t worked. She was disillusioned.

I’m not disillusioned. I know that Estimation180 tasks are useful in some situations and less useful in others. I have some thoughts about where and when they’re useful in my teaching. I try to stick to talking about that when I’m talking about teaching and estimation.

What is retrieval practice when you’re learning math?

I’ve never really carefully read the retrieval practice literature, but I think it gets confusing when people talk about retrieval practice when talking about math skills, as opposed to mathematical facts.

Here is the description from @poojaagarwal‘s website, committed to promoting retrieval practice among practitioners:

Retrieval practice is a strategy in which calling information to mind subsequently enhances and boosts learning. Deliberately recalling information forces us to pull our knowledge “out” and examine what we know. For instance, I might have thought that I knew who the fourth U.S. President was, but I can’t be sure unless I try to come up with the answer myself (it was James Madison).

But how does this apply to math skills? Can trying a problem (i.e. practicing the skill) ever count as retrieval practice? Does it make sense to use the metaphor of ‘calling information to mind’ to describe what’s going with skills practice?

I think not. But I also am finding retrieval practice useful in my lesson planning. There is a great deal of knowledge that is useful for students to know when they’re learning something new. This sort of knowledge is the sort of thing that I’d like my students to know (i.e. retrieve from memory), more than I’d like them to derive.

Often, at the beginning of class, the first thing I ask my students to do is to remember some facts that they may (or may not yet) know from memory. Some constraints:

• I don’t ask students to solve a problem and call it retrieval practice — that’s skills practice, not retrieval practice, and tickles other parts of the mind.
• I only ask students questions that I think they could remember, even if it might be difficult to recall these things. Ideally, these would be things that either students could derive if they can’t recall them.
• Because stuff from the last few days of class can often get forgotten really quickly, I often use these prompts to strengthen the memory of what we’ve recently done. (The prompt “Summarize what we did yesterday” is surprisingly difficult!)

Here are some prompts I’ve recently used with students:

“Draw a pair of ramps that are pretty close to being of equal steepness.”

“Write an equation of a quadratic, describe what it would look like.”

“What happens when you use the tan button on the calculator? Give some examples.”

“Write several pairs of decimals, and write the number that is between them.”

The truest ‘retrieval practice’ of these is the one about the tan button. Next in line is the one about the equation of the quadratic, since I’m prompting kids to remember what the features of the graph are (though it’s also skills practice). What made me think about these as retrieval practice is that they were all calling back on the previous day’s class.

Here are some purer examples of retrieval practice prompts in math:

“What’s the Pythagorean Theorem?”

(If a specific procedure is supposed to be known for converting a decimal into a fraction:) “How do you convert a decimal into a fraction?”

etc.

As I’m messing around in graph theory, I’m noticing that there are a lot of things that would be useful to remember — particular proofs that could serve as paradigms, constraints (in the form of inequalities) on possible planar or non-planar graphs, theorems, specific graphs that are useful examples, etc. If I had a teacher of graph theory, I’d want that teacher to prompt me to remember these things so that I could have more of them available as resources when I’m trying to learn something new or do some creative proving or problem solving.

(I should probably bust out some flash cards at some point…)

As an aside, I think that retrieval practice is sometimes mixed up with spaced practice, but I think these are different things. Spaced practice might be a better fit for what people are describing when they talk about intentionally building time-separated practice of skills into their courses and assignments. I think this requires a different sort of finesse than retrieval practice, though, as the problem with spaced practice is making sure students have something productive to do if they’ve actually forgotten the material.

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?”

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

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.)