Highlights of a Great Interview with a Mathematician

I have a new mathematical hero. What a fantastic interview!

Sylvia Serfaty on mathematical research:

One of the first pieces of advice I got as I was starting my Ph.D. was from Tristan Rivière (a previous student of my adviser, Fabrice Béthuel), who told me: People think that research in math is about these big ideas, but no, you really have to start from simple, stupid computations — start again like a student and redo everything yourself. I found that this is so true.

On the nature of mathematical knowledge:

It’s really beautiful to observe, as you progress in your mathematical maturity, how everything is somehow connected. There are so many things that are related, and you keep building connections in your intellectual landscape. With experience you develop a point of view that is pretty much unique to yourself — somebody else would come at it from a different angle

On women:

I’m not super-optimistic, in terms of women in the field. I don’t think it’s a problem that is going to naturally resolve itself. The numbers over the last 20 years are not a great improvement, sometimes even decreasing.

On proof:

We prove theorems because there is an audience to communicate it to.


There is a lot of ego in this profession, let’s be honest.

What’s My Function, Rule, or Formula?

Kent just wrote a great post about his 8th Grade functions unit. He and I share a lot of the same activities and priorities, but I have an observation and a few related ideas to share related to “What’s My Rule?,” a game both he and I play with kids.*

I learned about it from the CME Algebra 1 text. 

The function concept is famously difficult to pin down. The trickiness of the definition follows its history. There’s even a fun article to read that traces the concept’s historical development (thanks Christopher!). Here’s a preview:

Screenshot 2017-02-16 at 2.30.10 PM.png

Tricky stuff, right?

One conceptual issue I’ve seen again and again in my teaching has to do with the relationship between rules, machines, formulas and functions. Not all rules that we commonly talk about are functions; not all functions are rules; not all formulas have rules; not all rules have machines…pick two: not all of one is like the other.

A major goal of my functions unit to help kids separate these ideas. So the very first thing I do is poke at it.

Three Rounds of What’s My Rule?

For the first round of “What’s My Rule?” I started with a function that has a rule that you could write a formula for: minus 1 from your input and then square it. This is to make sure we all know what’s going on, and Kent does this with his class as well (see his Day 6).

my rules_001.jpg


(I also prompted kids from the very start to notice that different inputs might have the same output.)

Next up, though, I want to move them towards rules that they don’t know how to write formulas for. I showed them a piecewise rule: double if it’s odd, halve it’s even.

my rules_002.jpg

Notice the “No”? That’s a domain limitation. My rule refuses to handle numbers that don’t make sense for it.

Now, this is a very similar second move to Kent’s. Kent’s second move is to use the rule “less than the input.” It’s a good one! But I’m doing something slightly different. Kent is showing them a non-function, but I’m trying to show them functions that don’t have formulas. I’m also establishing that functions don’t have to accept any inputs.

That brings us to Round 3. Now I want to show a non-function. I want to emphasize that non-functions are mean, inconsistent places where what’s true one moment is not true the next. See if you can figure out what this rule is:

my rules_003.jpg

Stumped? My rule is that I get to decide whether to halve the input or multiply it by 10. I choose based on whim, my mood, whatever I feel like. I do what I want.

Perhaps — if you’ve studied too much math or have been teaching for a while — you would suggest that this is not a function because there is no rule. But very clearly there is a rule! The rule is I get to choose one of two procedures, whichever I want at the moment. This is a rule, but it has no formula and it certainly feels different than the other rules.

A rule is not a function is not a formula. I give a very rough definition on the board of function: a function is a rule that’s fair. This is not the definition they’ll have by the end of the unit.

What about machines?

By the way, I’ve left out machines.

Kent brings up machines early, but I think that machines emphasize consistency in effect. If I were talking about “What’s My Rule” in terms of machines, I might get stuck. Could a machine have a “Do whatever you want?” rule? I’d rather emphasize the human choice in these early examples before bringing in mechanical metaphors.

Three Good Questions

At this point, I made a lucky call: I asked kids to take out their notebooks and write a question they could ask that would test their understanding of “function.” Each question that we talked about was golden.

Question 1: What about a rule that says, “the first time an input is called, do _____ to it; the second time an input is called, do ____ to it,” and so on? Would that be a function?

Question 2: Does a function need to have a formula?

Question 3: Does every formula have a function?

These kids were thinking about the right stuff!

Make Your Own Rule

I asked kids to play a few rounds of this game with each other, making up their own rules. Two kids came up with interesting rules that seemed worth sharing with the class. I typed them up and handed them out as the warm up for our next session.

The first really pushes on the relationship between rules and functions. His hint is to think of this table in chunks of 4 rows:

Screenshot 2017-02-16 at 3.02.57 PM.png

The second rule uses a different formula then what kid are used to seeing:

Screenshot 2017-02-16 at 3.04.37 PM.png

Who cares what a function is?

We’ve all seen those questions that show you one indiscriminate blob with arrows to some other blob and ask if it’s a function or a relation. These questions are dumb and shouldn’t be what we’re aiming for with our function unit.

Instead, we should be worried about trigonometry. sin(x), cos(x) and tan(x) are very, very different feeling for kids than the functions they’ve seen before in algebra. They hardly seem like they have formulas at all, they’re just rules.

And we should be worried about the idea that in computer programming you can define functions that lookup information and don’t have anything to do with numbers at all. And you can use cases to define those functions, and they can even involve randomish parameters.

The function vs. non-function perspective is interesting, but there is a lot of depth in the function vs. rule vs. formula (vs. machine) comparisons. That’s where I like to make sure my kids spend a lot of time.

A Few Other What’s My Rule? Rules That Are Fun

  • Take a book up to the board. Have kids call out numbers. Turn to those pages and then call out the first word from the top of the page as an output.
  • Kids call out colors, and you look at your watch and call out the time.
  • Kids call out numbers and you just repeatedly say “NO” unless they say “5,” in which case you say “5.”
  • Kids call out years and you look up the world population for that year.
  • Kids call out numbers and you respond with words that are that many letters long. (And you say “NO” if they say numbers above 10 or so.)

And, as always, the question is whether this is a rule, does it have a formula, and is it a function.

What are some other fun prompts for this game that could press on the relationship of these related ideas? Would love to hear your thoughts.

Asking Questions

I think I’m starting to understand with increasing clarity that one of my projects in my blogging and tweeting is to represent public amateurism more faithfully. While I find it difficult to articulate precisely what I mean by this, I get close with “learning in the view of others.” Another pass: “expressing the products and emotions associated with not-knowing that are part of learning.”

Part of what I’m interested in is the difficulty of competently representing incompetence. Which is why I’m having fun asking smartly stupid questions (stupidly smart?) about poems. It’s also why my most recent essay was about not knowing why a kid was struggling and not knowing how to help her — and how the not-know passes into a sort of knowing.

(I’ve also come to think that Twitter, as a medium, encourages the performance of expertise. The limited format encourages users to make declarations. There’s just enough room to give an opinion without any reasons, which IRL is what you do if you are (a) an important person who everyone trusts and believes or (b) a busy individual who just doesn’t have time for this shit. This is what the vast majority of tweets sound like to me. Trying to find ways to represent the exact opposite of this attitude within the constraints of the medium is fun and challenging to me.)

I’m not promising that I’ll never act like I know a lot about something. But for now, playing with amateurism is a blast.


There’s a way all this is coming through in my teaching: I’m just begging kids to ask questions in class these days. You finished a problem? Try to ask a question. You don’t know how to do something? Ask a question. You’re stuck? Ask a question.

Is this new? Don’t all teachers encourage kids to ask questions? I’m worried that I’m in strawman territory, but it seems to me that there’s a line of teacher talk that discourages a certain type of question. I’m talking about being proud about not answering questions; “my kids ask me for help, but I just grin at them and walk away”; talk about kids persevering through difficulty and not just screaming for help at the first sign of challenge.

I don’t know about you, but if a kid is screaming for help then I’m happy. It’s when the questions dry up that I start getting nervous.

Of course, learned helplessness is a thing. I mean, of course it is, right? I’m not entirely sure that I’ve seen it. Maybe once or twice. In general the questions my kids ask seem entirely reasonable to me. Is this right is a great question to ask when you’re getting the hang of a new procedure — especially when you’re used to getting things wrong in math, and you’re finally getting a foothold.

You know what it means to persevere? Perseverance often means asking a question, even if it’s amateurish and doesn’t make you sound smart, because asking questions is how you learn new stuff. And a lot of the time you learn new stuff by getting answers to your questions.

Don’t get me wrong. There are lots and lots of times in my classes when I decline to answer a question. If kids are doing good thinking, I’m happy to decline to interrupt it. If we’re going to have a debate, I don’t want to deflate it. I think that the source of talk about not answering questions must come from our desire to subvert the expectation that answering is the only appropriate response to a question.

(By the way, this seems to happen all the time in teaching. Group A only talks about Practice X. Group B objects to the focus on Practice X, instead boosts Y. Group A replies: of course we only talk about X! The profession only talks about Y. Evidence: you only talk about Y. Group B replies accordingly and the cycle never ends.)

You know what’s the best part of begging kids to ask you questions? They ask a lot more questions.


There are rewards — tangible and not — for being an expert about teaching. Amateurism has its rewards too. Like that part of you that’s always been a bit afraid to feel stupid? You get to slowly chip away at it. Publicly declaring what you don’t know is powerful, maybe more powerful than your ability to grit through a problem. People these days have so many answers, and I just want more questions.

Three Questions About A Poem

Screenshot 2017-02-13 at 5.25.43 PM.png

(from here)

  • Why is it called “The Storm”? She could have called it “The Snow” or “New Snow” or “Chill Out Dog” or whatever. Why focus on the storm (and, come to think of it, doesn’t all this happen after the storm)?
  • In the first stanza: “white”, “snow”, “with”, and “wild”. It’s certainly lovely to hear all those words along with each other. But then why not fill the poem with similar-sounding words? Why turn it on in the first stanza, but not in the next two?
  • Is there significance in the dissimilarity between “written…in large exuberant letters” and “could not have said it better myself”?

Would love your help in understanding this better! Speculations and answers are invited.

It’s my blog and I’ll write about poetry if I want to

Goldfish Aquarium Ornamental Fish Pets


(Source: Poetry Foundation)

I used to find line and stanza breaks in poems to be entirely mysterious. I still, basically, do find them mysterious, but I’m trying to get better at noticing them and taking them seriously.

For instance, the first stanza features no punctuation to stop the flow of images. There is a brown fish — my attention drops for a moment as I find the next line — and it’s hanging on the wall and it swims along — another pause and a tug of attention — in his frame, and finally I can take a breath at we are eating dinner.

It strikes me that the line breaks in this stanza effectively create a sensation of being tugged along by the sentence across breaks in the line. The language swims in little bursts that mirror the swimming of that little brown fish.

Some questions:

  • Why a brown fish? Why not red? [Maybe any other color would be a distraction. If red, we would focus on the color. If gold, we’d assign meaning to that. Brown is nice and neutral, allowing us to focus on the fish’s movement.]
  • Why “in his frame”? [Perhaps we’re contrasting the fluttery movement of the fish with the stasis of the frame?]
  • Why “dinner”? [It mirrors “dining room” in a pleasing way. And I guess people tend to gather around the table for dinner more often than lunch? Not sure.]

The next stanza has a lot of commas. Why?

I enjoy the slower, more reflective pace of the second stanza. It matches the mood of the poet, I think: we first are asked to notice this interesting phenomenon, the dynamic fish in the static image, and then we’re asked to reflect on it. The lines here come with reflective pauses for the reader to dwell on the import of the fish.

And what is that import? That he is on display (“in candlelight for all to see”), that because its motion is projected on it by the viewer that motion has a timeless, eternal feel to it. It’s a motion that is entirely in thought: the fish was moving “even in the darkness of the ink before someone thought…”

Why break the stanzas here?

Maybe to emphasize that the motion of the fish — which is entirely a product of thought, imagination — is the motion of thought and imagination itselfany thought. The line, with the stanza can be read as describing the darkness of the ink before ANY thought at all.

But we are talking not just about thought in general, but about a particular imaginative ability, and so the next stanza picks up where the last left off — the thought of this particular fish, this particular image.

In this third stanza, the poet once again describes the motion of the fish and so once again we get a pair of lines without comma or pause to slow down the tug of the fish’s movement: “to draw him and the thin reeds waving in his stream…and the clear pebbles strewn upon the sand.”

I find the stop of this motion to be surprising, and somewhat jarring. Why stop this movement before the end of the stanza? [Perhaps to give us a moment to collect ourselves, and to notice that the poem is wrapping up, and that we’re getting ready for another thought?]


I have no idea how to read this second half of the poem. I’ll do my best, but I’d appreciate any help.


So, now we get a pair of thoughts (“No wonder…No wonder…”) broken across three stanzas.

Is it broken to remind of us this projected motion?

“No wonder he continues his swimming deep into the night, long after we have blown out the candles and gone upstairs to bed.”

This seems wrong to me. The fish swims only because we imagine it to be swimming. It seems to me that if we go to sleep, it rests.

Two possible answers:

  • Its movement isn’t about my imagining it so, it’s about the meaning that, collectively, we assign to the image. Even when I sleep, the fish’s image still moves under the spell of the meaning we have collectively imbued it. (And because of this, one can still imagine that meaning to persist when any one human is asleep, or dead. Meaning persists.)
  • Maybe my whole read is wrong, and this isn’t a poem reflecting on how people assign meaning to inanimate tokens. Maybe the poem is about some sort of fate or destiny. The molecules were always destined to become ink that was always destined to become the image of this fish. It’s not about people at all — the opposite — it’s about the universe and fate.

“No wonder I find him in the pale morning light, still swimming, still looking out at me with his one, small, spellbound eye.”

I don’t know why the lines break how they do. I don’t find them mysterious, I’m just not sure what to make of them. All I can say is that when we talk about swimming in this poem, we’re likely to have that image broken across the line to drag us along in that motion, and this continues in the last few stanzas.

And then what’s with that mysterious last image? What is that one, small, spellbound eye?

Is it that art — the images and symbols we imbue with meaning — ultimately peer back at us? We look at them, and imagine them in motion, but we also imagine them starting at us just as we stare at them? That is to say, that we imagine the fish as real and then we further imagine that fish seeing us as real?

Is this a poem about how the act of creating meaning actually does assign meaning to us? Is this what is suggested by “spellbound”?

(What are all those s’s in that last stanza? What function do they serve?)

A 3rd Post about NBCT AYA Component 4

I’ve been trying to blog about this NBCT portfolio for two reasons. First, because there is not a ton of information out in public about what goes into these portfolios. Second, the actual write-up of the portfolio requires a careful eye for concision, brevity, and more generally for careful consideration in your lexical choices, an ability to get the point across in as few words as possible and to make painful — but necessary — cuts from your prose in order to maximize the evidence/word ratio, avoiding at all costs writing that is overstuffed, repetitive, repetitive and indulgent.

But here, I write how I like.


There are three sections of this NBCT portfolio:

  • Knowledge of Students
  • Generation and Use of Assessment Data
  • Participation in Learning Communities

I’ve been spending the last month doing the legwork for the “Knowledge of Students” piece. This involves three little sub-tasks:

  • Making sure you’ve got good sources for your knowledge
  • Synthesizing what you gather into a profile of the kids
  • The meta-task: justifying/explaining your sourcing and synthesizing process


I like to think of myself as someone that knows his students well. I have relatively small classes, I like talking to kids and they tell me stuff about themselves. I also think of myself as someone who is pretty good at snuffing out what they’re good at and where they flop.

This is not what NBCT is interested in. They want an investigation — an active effort to gather information about kids from people who aren’t you. You need sources for the knowledge of students you profess to have. This is probably a good requirement, and it’s one that I enjoyed trying to fulfill.

Here are people I ended up talking to about the kids in my class:

  • Other math teachers who taught my kids in previous years
  • Learning specialists at my school who work with the kids in my class with learning disabilities
  • A guy who tutors one of my kids
  • Someone who observed this class for me after I reached out to them
  • Parents

(To give you a sense of the NBCT expectations, my conversations with parents are actually the weakest source for my submission, since I’m just reporting what we talked about during parent-teacher conferences. I didn’t especially arrange those conversations.)

How did I do, by the standards of NBCT? Here is what NBCT calls for on the section on sources:

Screenshot 2017-02-05 at 3.58.06 PM.png

I did a good job of talking to a lot of the “school or professional personnel” about my kids, and I also talked to families. My case would probably be stronger if I could support if with something that isn’t a conversation. My school doesn’t keep assessment data from previous years around, but we do have anecdotal reports for every kid going way back. It would probably be good for me to take a look at them…and I just got in touch with my department chair to make this happen. If I add anecdotal reports to the mix, I think I’ll be pretty solid on my sourcing.


Besides for parents, I ended up having serious conversations with about eight colleagues about the kids in my class. What did I learn? And — NBCT emphasizes — how will this affect my teaching choices?

I think my class is roughly made up of three types of kid: high-flying, seriously struggling while learning disabled, and academically serious but historically struggling students. What makes this class special, I think, is that, even though this isn’t always the case, in this class each of these sub-groups prefer and naturally seek skill-development as the primary activity in class.

High-fliers often love problem solving, or working hard on tricky situations, asking wild questions, working their way around new and sophisticated ideas. This is not what colleagues tell me about the strongest math students in my class, though:

  • “Empowered by her toolkit being filled, liked getting right answers, not so much into big problem solving without clear path forward. She likes getting techniques down.”
  • “Thrived in algebra, applied it everywhere. Not motivated by optional work, preferred to work independently unless she was well-matched mathematically and personally; not a mathematical risk-taker.”
  • “She enjoyed solving equations and ‘getting into the groove.'”

So the high-fliers in my class happen to love to chase skills.

About a third of the students in my class have been evaluated for a learning disability. Their issues cluster around attention and reading comprehension, and their best successes in math have come when they have a particular type of problem they can get good at solving:

  • “____ enjoys feelings of mastery, will even enjoy 10 identical problems; has one of the most serious math disabilities the school has seen, has reading difficulties too, he’s almost never giving full attention, is anxious about math.”
  • “She loves mastering techniques. Worked with learning specialist on study techniques, implemented them and really loves them.”
  • “He worked well with a partner on a skill-practice assignment…has reading difficulties. Enjoys getting right answers.”

And many of my students come with disclaimers about their past academic lives, even though they’re academically serious.

  • “Father says: Likes math, but if you ask her if she’s good at it she would say ‘no.'”
  • “Low-processing speed.”
  • “Disliked and did poorly in math in previous years.”
  • “Has geometry under control despite struggles in previous years.”

In a lot of classes these kids would be checked-out of math, but they aren’t in my class. They take school seriously, and they want to succeed. And if you’ve struggled in math in the past but are seeking ways to be successful? Skill-mastery came up again and again in my conversations about these kids. This totally fits what I’ve seen in class. A kid recently told me that she never gets bored by repetitive worksheets. She’s right — she doesn’t. She loves it, and to the extent that it makes sense to talk about the preferences of a collective, this entire group of kids loves it too.

The big question: how does this affect my teaching?

How does this affect my teaching?

On the one hand, you can get decently far in math by just learning skills. There’s no such thing as just learning skills, of course, if you’re really learning them. To really be able to solve a type of problem is to combine conceptual and procedural knowledge, and (if you do it right) to apply that knowledge in creative ways to new scenarios.

And experience has shown me, over and over again, that this group most naturally hums along when we’re working on skills and problem types. It’s when things just work with this class. And this fits perfectly with the picture of the kids I gathered from my conversations.

But math is more than just mastering types of skills. It’s also about trying stuff when you have no clue what to do. It’s about inventing new words or concepts to describe mathematical phenomena. It’s about mathematical modeling, which inherently involves putting out ideas for revision’s sake.

This stuff is harder for my kids; it’s not natural for the group. It takes careful planning to pull off, and even then it often flops.

NBCT wants to know how my knowledge of kids affects my teaching. For this group of kids, I know that learning will happen if we’re working on getting better at solving a particular type of problem. But the wider vocabulary of mathematics is harder to bring to this class, and I have to think very carefully when I try to share it with this group.

So, NBCT: how did I do on this one?

Screenshot 2017-02-05 at 5.09.18 PM.png

I’m doing OK, I think, though I haven’t yet identified an area for future information gathering. Since the growth area I’m looking at is sharing non-skill development math with this group, that would be a good place to gather more info. (And maybe it involves talking to people from beyond my school who might have smart things to say about this — there’s room for that in NBCT’s world.)

Reflecting on All This

NBCT gives you reflection prompts for the written commentary.

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Q: What guided you in selecting those particular sources of information?

A: Things were going ok for this class, but I wanted to know what made them hum. The kids seemed generally content, but I wanted to know what about math got them excited. So I sought out people who knew them well — their past teachers, tutors and the learning specialists who worked them regularly. The sources were appropriate for helping me figure out what these kids enjoyed about math.

I also had a bunch of kids with learning disabilities, so the learning specialists (and the kids special out-of-school tutors) were a no-brainer, source-wise.

Q: How did you determine the relative importance of the different kids of information you gathered?

A: This is a tough question for me to answer, and I’m not exactly sure how to go about it.

The learning specialists provided a really important perspective, but sometimes they were talking more about what they saw working one-on-one with a kid instead of the fuller, social situation. For one kid they said that the enjoys working on a lot of problems that are nearly identical — this surprised me, because he does not seem to enjoy this in class. My interpretation is that the social situation is different for him, mathematically. He worries a lot about doing math in the presence of others. He still might enjoy working on a lot of similar problems, but around other people he gets self-conscious about looking stupid by not knowing how to do a problem, and chooses to just opt-out of the whole game.

Sometimes people would tell me various instructional techniques that worked well for various kids, getting close to making recommendations. Example: he likes moving, he likes having very clear examples, he likes touching things. I rated this information as interesting, but non-crucial. Instructional techniques are my job. I wanted information about what kids were like in other settings, what they enjoyed. That’s what I listened for, even while people were recommending teaching moves to me.

I might end up with some surprises when I look through old anecdotal reports. I’ll have to decide how to situate those to the conversations that I’ve had with others.

When parents tell me what their kids are like in math, I take that as data, but I treat it as tentative. A few sets of parents told me that their kid historically struggled, but a few others told me that their kids were looking for more of a challenge. This was helpful, but not as important to me as knowing the specifics of what their kids enjoyed doing. Knowing that a kid likes math or doesn’t, that they want more challenges or they don’t — these aren’t exactly easy to act on. I needed more specifics, so the specifics I gathered from colleagues helped me figure out what to actually do. So I guess that makes them more important, though this whole investigation started from the comments I got from parents at conferences, so it’s not like they were unimportant comments. I don’t know.

Q: What are some of the trends you identified?

A: That several kids have diagnosed learning stuff that makes it harder for them to understand new contexts and problems, and as a result they tend to really like when there are a series of questions of a similar problem-type.

That a bunch of kids have baggage from previous math experiences, but they have found success with developing their skills. Even the kids who don’t have math-baggage prefer not to take big mathematical risks, and this is something that their previous teachers noted to me too.

In terms of pair/individual preferences, my class is mixed. Some kids really prefer working alone, others prefer working with partners. No consistent trend here, which is sort of a trend itself.

Q: How did you identify or confirm the trends?

A: I asked one person, but then I waited to see if others brought it up. And the skill-development theme just piled-on with each successive conversation and observation. We’ll see what happens when I go through anecdotal reports. (My answer here seems thin, and I’m not entirely sure how to beef it up.)

The observation was really helpful in firming up what I thought I was seeing with my kids.

Q: What other factors did you take into account when analyzing and reflecting on the various sources of information and why?

A: My first reaction: I have no idea what this question is getting at. That’s my second reaction too. Anyone think they know what this question is all about?

One thing I took into account is that my class — high school geometry — might be different than kid’s previous experiences in math class. Like a lot of geometry teachers, I’ve noticed that kids sometimes love or hate it because it feels different from their other experiences. And part of your job in geometry, I think, is to help kids learn that this is also math. You want to teach them that geometry isn’t so different, and hopefully use that to help everybody come to like math a bit more.

I also didn’t want to assume that any of the things that people were telling me about kids were stable. So if a kid enjoys skills work? That doesn’t make them a skills-lover. It’s good knowledge to have, but even if that’s true there’s no reason to think that’s fixed. The mathematical experiences that a kid has had impacts their preferences — those can change.

I don’t know if that response makes any sense.

Q: What are the needs of this group of students and what kind of supports to you anticipate providing in order to meet those needs in fair and equitable ways?

A: All the above leads to some clear moves I can make:

  • I’ve got 1/3 of my class with diagnosed learning disabilities. These kids need lots of processing time, they need the context and scenarios of problems to be carefully developed for them, to reduce their reading load.
  • About 1/2 of my class has had difficulties with math in the past, and that impacts their ability to take academic risks in class. They don’t want to look dumb. They need to feel safe that they won’t feel dumb to do math that goes beyond skills-development work.
  • That means that, if I’m going to help them try more mathematically adventurous things, they’ll need lots of time to understand the problem, and the problem will need to be non-text heavy.


I think that’s a perfectly fine answer, though I feel like maybe I’ll think of a way to go deeper. If you’ve got ideas, I’d love to hear them!

Q: What other educators, professionals, family members, or community members will you need to collaborate with to meet these students’ needs and why?

A: I’ll need to talk to some people who work with kids with learning disabilities similar to the ones my kids have.

I’ll need to collaborate with tutors and anyone else who meets with a kid out of class.


Hmm. Again, I’m not exactly sure.

Next Steps

To finish off my source-gathering, I want to look at anecdotal reports for my students from previous years. I feel good about the trends that I’ve identified and the way I’ve synthesized what I’ve learned about the kids. When it comes to the written commentary, though, there are a few questions where I’m giving answers that don’t feel like they go so deep. I need to think more about those.

I’m ready to try writing this up for submission, though. And then I need to figure out what unit I’m going to plan, and what I’m going to assess my kids’ ability to handle.

I think a good goal would be for my kids to engage more seriously in non-skills math in this next unit. A formative assessment might help all of us figure out how to do this in a non-threatening, non-text heavy way.


New Essay: On Learning What You Don’t Know

Over the weekend, I posted an essay I wrote about multiplication, struggling students and struggling teachers. I don’t worry much about what things are called, but it’s definitely not a blog post and you might be disappointed if you read as if it’s one. It’s long, and I’m trying to say and do something that I couldn’t say without the length.

You can find it here.

I hope you like it. I also have a not-secret-any-longer hope that reading it makes you want to write something. In particular, I really want to invite your critical engagement. Like really really. Like so much do I want you to write about your own differing perspective that I will suggest particular ways to do so:

  • Have you taught students like mine? How did you help them? What did they teach you?
  • Do you think differently about how students come to learn so-called “math facts”?
  • Do you think I misunderstand the significance of May and Rachel’s stories?

(I’m trying to use “critical engagement” and “differing perspective” in place of disagreement. Not that I wouldn’t value disagreement if you have it. But I aim to include asking questions and telling related stories as properly critical responses that aren’t what we usually count as disagreement.)

One of the anxieties I have while working on bigger projects is that people will see how much effort I put in to the project (it is a lot) and therefore be less likely to critically engage with the ideas, for fear of making that person (me) feel badly. People are more likely to dig in to a tweet than a book, I worry. But (like any teacher) I love provoking thoughts, and nothing makes me happier than reading them.

So: please read, enjoy and then write something.

A 2nd Post About NBCT AYA Component 4

I’m going to do some of the work of developing my submission for NBCT Component 4 here. I’ll enjoy thinking in your good company, and maybe it’ll be useful to others.

Where to start?

One thing that makes this submission tricky is identifying its core. At first (and at second, third and fourth) it appears to call for a mishmash of evidence: seeking out knowledge about your kids, talking to parents, assessing and instructing the kids, contributing to the learning community.

I’ve been trying to think of this portfolio as really, truly about the cycle of investigation/application we do in teaching. This happens at multiple levels in our work, according to NBCT:

  • You seek out knowledge about kids from outside your classroom, and you use that to create assessments…
  • …and those assessments give you more information about kids, which you use to teach ’em stuff…
  • …which necessitates a summative assessment, which lets you know that your kids have a need that isn’t being addressed…
  • …so you seek out knowledge from the broader learning community to address that student need.

It all holds together, even if it’s a bit wobbly. (Do we always discover student needs that force us to collaborate with colleagues? Do we always need to look outside the classroom for our knowledge about kids?)

Because all of this is so inter-connected, it’s hard to know where to begin the work. How can I decide what knowledge to collect without knowing how it’s going to impact the assessment? How can I pick the assessment without knowing if that’s going to lead to a student need that I can write about?

I’m sure there’s more than one way in. My plan is to start by seeking out knowledge about my kids from families and colleagues. I want to do this strategically, though, to make sure that the knowledge I seek is able to impact the design of my assessment.

Assessment Decisions

The portfolio calls for evidence that you gather knowledge about students from parents, families or colleagues, but also that knowledge needs to be used to plan the assessment. In order for this to work, then, I need to be systematic in the knowledge that I seek. Otherwise, I might ask questions that are useless for informing my assessment design.

The group that I’m writing about is a going to be studying similarity and dilations after winter break. Really any task can be a formative assessment if you do it right, so the content they’re assessed on is somewhat flexible. For the sake of thinking this through, I’ll just choose a task at random from the New Visions site.


OK, suppose that some version of this mathematics was the focus of my formative assessment task. What knowledge about my kids (that I could gather from families or colleagues) could impact how I do or design this task?

In order to answer that, it might be helpful to describe some of the choices I have to make about an assessment. With some help from the Thinking Through a Lesson Protocolhere are dimensions along which an assessment task could vary.

  • The mathematical goals or the precise nature of the task itself might vary (e.g. I might focus on proof or calculation)
  • The support for the task they have might vary (e.g. I might slowly build up to the task with a notice/wonder intro or I might ask them to solve for x)
  • Kids can have resources to help them or not (paper, pencils, rulers, scissors, etc.)
  • They can work solo, in pairs, with a group. Those groups can be assigned or random or not-assigned.
  • They could record their work on paper, whiteboards, on the chalkboard. They could have one record per group. They could not record it and be assessed based on their conversations.

Are there more ways in which an assessment can vary? If so, please point them out in the comments!

Knowledge To Inform Assessment Decisions

I’m used to looking inside my classroom for knowledge about kids. In contrast, NBCT is asking me to look outside the classroom for knowledge to inform my assessments. How am I going to do that?

Most of my knowledge-seeking about these kids happened at the start of the school year, and mostly for this one kid, Kid A. I had a lot of conversations with a lot of different people about Kid A. I talked to the school’s three learning specialists, this kid’s former math teachers, Kid A’s current math tutor and mother. I worked hard to set up a once-a-week meeting with this kid outside of class. I have OK-not-great records of these conversations, so it wouldn’t be impossible for me to submit them as evidence. And I could probably write a bit about how Kid A’s special set of qualities requires a special sort of assessment: one that starts with plenty of individual think time, then some pair time that I can listen-in on, and only after all that a chance to try to record some ideas on a page.

The issue, though, is that NBCT doesn’t want evidence that you sought knowledge out about one kid in your class. They want to know that you are seeking knowledge out about the entire class. I need to do more than show that I’ve got Kid A’s back.

The class that I’m writing about is quite small — about ten kids. I went through my roster and thought about what questions I still have about them. I’m going to need to find something to learn about the rest of my class for this portfolio.

For a bunch of kids — Kids B-F — I really want to know what they find challenging or fun. This is based on conversations I had at parent-teacher conferences with their parents. All of their parents mentioned that their kids felt math class could be more challenging.

Normally I’d dismiss this as parents making stuff up about their kids, except that (a) this is something I’ve heard from some of these kids myself and (b) focusing on supporting kids who need more time is persistent feedback I get about my teaching and (c) Kid A in particular needs a lot of time, and so does Kid G, and I’m definitely guilty of asking myself while planning so how much longer can we practice this in class before Kids B-F stage a mutiny?

Back to NBCT: I have the notes that I took at conferences scanned, and I could submit this itself as evidence. I don’t know if that’s enough evidence, because I’m not sure how I would take this desire for challenges into account while planning the assessment. Maybe…making sure there’s an optional challenge that’s part of the assessment? Or a choice of two possible paths after kids solve that initial task together?

Any ideas here, people?

I’m also not sure what I could ask parents/colleagues to tell me about the kids that would help me get a better picture of what they find challenging. Hey Parent B, just checking in. Has Kid B mentioned anything about which topics they find challenging or interesting? This is an insane question to ask a parent, instead of asking the kid themselves.

Maybe I could go to these kids’ Algebra 1 teacher and ask what they found interesting last year? I’m not sure what their teachers could say that would help me. Maybe: “Kid C likes to draw”; “Kid D loved solving equations.”

That sounds doable.

The Plan

The details matter for NBCT. (Oh lord do the details matter.) Here is what you need to show NBCT for your knowledge of the childrens:

  • Show that you gathered information from at least two of the following sources: families, colleagues, professionals in the district or in the field, and/or other community members.
  • A description of the information about the group of students you collected from multiple sources and how you collected it. [i.e. show that you’re not pulling this out of your butt and lazily being like a parent called me with a question, no, you’re supposed to seek out the info]
  • Write a detailed profile or description of the group of students you selected to feature in this portfolio entry based on the information you gathered. [i.e. all kids not just Kid A]
  • How did this knowledge inform the kinds of assessments (formative and summative) you planned to use and any modifications that would be necessary given students’ learning modalities, social and emotional growth, exceptionalities, abilities, interests, etc.?


Based on all the above, here’s my plan:

  • I’ll show that I sought out info from learning specialists about Kid A and from parents about the degree of challenge. This doesn’t make the strongest case that I actively seek out this info (since the parent-teacher conference thing just falls in your lap) but I’ll reach out to my kids’ past and current teachers to try to get some more perspective. If that works out then that would make a strong case that I seek out this info systematically.
  • I’ll need to think some more about what should be part of the detailed profile, but just going on what I have, I could write a bunch now.
  • Based on what I currently know about the kids, my submission is going to be about supporting Kid A (and Kid G) who have IEPs and issues that make it hard for them to dive into a tough task on their own while maintaining challenge for Kids B-F. In practice, this might mean multi-layered assessments that include pair/individual work, accessible/work that feels challenging, talking/writing math. That’s sort of a mishmash but I’m generally having a tough time meeting all the kids in my class. (That could be my learning need for the other part of this portfolio, come to think about it.)

Next up: reaching out to colleagues and designing my assessment.

Other People’s Posts from 2016 (+ my stuff)

I made this list by going through my twitter feed and looking for pieces that I retweeted. Problem: only the second half of the year would load in my browser. If you published an awesome post between January and June, it is not on this list.

Tape Diagrams, Big Feelings and other Predicaments of Teaching by Kim Van Duzer

I still don’t know whether tape diagrams are worth the Big Feelings they bring up.  But what happened on Wednesday reminded me that all teaching and learning, math or otherwise, is emotional business.

  • Online eyeballs are tough to capture, though, and I got more interaction and responses from a blog post that used ideas from the essay than the essay itself. I don’t know what this means for the future of longform writing about teaching, but I was proud of the post: Cognitive Load Theory and Why Students are Answer-Obsessed.

This is a blog post about NBCT Math Adolescent/Young Adult Component 4

I’m not trying to be interesting in this post. NBCT Mathematics AYA Component 4 is very difficult to make heads or tails of. I wished there were something to read to help me make sense of it, but I found nothing so I wrote this.

What’s the big picture?

NBCT is great at giving you all the details in a long jargon-riddled list without any structure. Here is some three-syllable word salad describing your Component 4 submission.

Screenshot 2016-12-20 at 3.17.29 PM.png

Having trouble? I am! Luckily I’m here to simplify this for me:

  1. You submit two assessments, one formative and one summative. This is the core of the portfolio. You have to show that you do this well. “Doing this well” is defined in a bunch of different ways. You have to show that kids learned stuff. You have to show that the assessments were fair and helped learning. NBCT also thinks that students should be self-assessing, so you have to show that kids are using your assessments to figure out what they know on their own. (Whatever.) Most importantly, however, you have to show that the assessments were based off of knowledge of students…
  2. You show that you go out and get knowledge about students from families and colleagues. This knowledge of the kids influences your assessments. Since the assessments are supposed to be tailored to the kids, you have to show that you went out and gathered info about the kids. The evidence that shows that you did this needs to be evidence of your communication with parents or colleagues, so notes, emails, transcripts, past info from earlier teachers. Remember: this info needs to be used to explain how you custom designed assessments for this group of kids.
  3. Finally, you show evidence that assessment and learning about the kids created problems that you need to solve. Two problems, actually: a professional learning problem and a student problem. You show where these problems came from and that you solved them effectively. Seriously? There’s a “Description of a Student Need” form? A “Description of Professional Learning Need Form”? Yes, there is. This is NBCT at it’s most bureaucratilicious. You have to show how meeting both your learning need and the students’ needs helped the kids, with evidence and stuff.

OK super-quick summary: The core idea is that you’re showing that you’re effective at getting relevant info and using it to help kids. Relevant info about kids comes in two forms: info from inside your classroom (assessments) and from outside your classroom (colleagues, families). You show how the outside info impacts your assessments. And then you show how all this creates the need for more outside info, and that you can get that outside info and turn it into results inside your classroom.

Is this impossible? Yes this sounds very difficult.

What else are you thinking? I’m thinking that it could be helpful to look at the rubric for grading your submission. What are the verbs here?

Screenshot 2016-12-20 at 3.43.55 PM.png

The verbs: collaborates; applies knowledge; understands assessment; selects/creates assessments; analyzes data; helps students apply feedback; encourages self-assessment; reflects; expands professional knowledge.

(OK so I included adverbs and direct objects sue me.)

What’s your plan? I really have no idea. This is an overwhelming submission in a way the other components of NBCT are not.

OK but seriously you must have a plan? I think it makes sense to start with the assessments and work backwards and forwards.

I’m going to look at my next unit and start writing a formative assessment. And I’m going to make a list of ways that the assessment could be different, depending on things that I don’t know about kids. Then, I’m going to find out those things about the kids. That should take care of 2/3 of the submission.

The really tricky part seems to be part 3. I need to show that the assessment/knowledge of kids pointed me to something that I need to learn and some student need that required collaboration. Then I need to show that I learned this thing/addressed this need. Then I need to show pre- and post- evidence that shows that my learning and my dealing with the student need impacted the kids positively.

I can easily imagine how I’ll address the professional learning needs. The assessments might raise aspects of student thinking that I’ll need to better understand. I could ask colleagues, or I could ask people on twitter to help me out. I also have kids with IEPs or behavior issues that are tricky to deal with, and I could imagine telling NBCT about how I learn more about that.

What’s a student need? Seriously, what are they imagining here?




So, I guess the assessments will raise issues. I mean not to be stupid but if I notice that something isn’t being particularly well-taught and then I blog about it and then I get evidence from someone who reads my blog that they changed their teaching…would that be what the NBCT is looking for?

Or if I realize that kids need more practice on some skill because the parents are worried about the SAT, so I get in touch with parents and point them to resources, and then the parents use those resources and I can show that kids do better on SAT problems, would that be addressing a student need?

Or…I’m stumped on this.

Or I give a webinar presentation on something that comes up in this assessment, and then I very nicely beg people to share evidence from their classrooms that shows how what I shared helped their kids?

Lalala this is fun.

OK wrap it up: In summary, this is a lot of work. The end.

Also I made some docs to help me plan for the written commentary. I don’t know how helpful they’ll be.

Link 1

Link 2