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


Does Understanding the Equal Sign Matter?

Six years ago I was so worried about students’ mathematical conceptions and misconceptions that I created a website to get better at identifying them. This project quickly began to question its own legitimacy. Are conceptions stable? What do we mean by “conception” or “misconception”? Are there other causes of student mistakes?

Lately, I’ve been coming to doubt a lot of the power of conceptions to explain mathematical thinking. But this is definitely not something I’m sure about, and a recent conversation with Kent and Avery made me even less sure of myself.

In wondering how I might test my ideas — or make more sense to Kent and Avery — I thought of a wonderfully clear piece of research by Eric Knuth and others about conceptions of equality. The researchers here argue that students don’t properly understand the equal sign, and that this poor conception is responsible for troubles they have with solving equations and algebra, in general.


I wondered: what would happened if I read this piece through the lens of my new worries? Would I find my worries resolved? Or would they find renewed strength?

The researchers call the trouble-inducing notion of equality the “operational view.” This isn’t nearly as evocative as their other name they have for this, which they call seeing equals as a “do something signal.” This view is limiting. Much more useful for algebra is the “relational view,” the more sophisticated idea that what’s to the left and right of an equal sign must be of the same numerical value.

So, why does seeing the equal sign as a “do something signal” matter for algebra? This passage summarizes two possibilities from Kieran and Carpenter.

Screenshot 2016-12-19 at 12.31.30 PM.png

It’s worth digging in here. Kieran says that the misconception surrounding equality makes it difficult for kids to make sense of algebraic expressions. Why? Because it seems to students that you must do something with a variable like a. After all, if we give meaning to expressions via equations and students see equations like a + 3 = 10, then a + 3 means “do something to a and 3 to get 10.”

If students had a relational view of equality, this would be better. How? Then these students would see a + 3 = 10 as saying that “a + 3” has the same value as “10.” They would treat “a + 3” as a complex algebraic object, not as a variable with an operation.

Undoubtedly, it’s true that students would benefit from seeing a + 3 as a composite object. But it seems to me a bit like cheating to say that this has to do with seeing the equals sign as the “do something signal.” The issue isn’t so much the equals sign as a signal, the issue is that the way kids are reading a + 3 makes the ‘do something’ interpretation natural. The reason why kids read the equality as “do something” is, I think, plausibly explained by their reading of “a + 3” as “a mystery number plus 3” instead of “the composite expression a + 3,” which would indicate the more sophisticated understanding

And what of Carpenter? He says that kids with a “do something” conception should have trouble with solving 5x + 32 = 97 by subtracting 32 from both sides. “What kind of meaning can students who exhibit misconceptions about the equals sign attribute to this equation?” he asks. “Virtually all manipulations on equations require understanding that the equal sign represents a relation.”

I’m stumped as to why he says this. If you think of equality as “do something,” then there’s a perfectly fine way to understanding subtracting 32 from both sides. The equation states, if you’re of a “do something” mind, that if you multiply some number by 5 and then (do something) add 32 to it then you get 97. You should subtract 32 from both sides, then, because if you didn’t add 32 then your answer would have been 32 less.

In other words, I see no reason why “backtracking” (or “undoing the steps”) wouldn’t be available to a “do something” student. What about seeing the equals sign as “do something” would get in the way of backtracking?

As with Kieran, there is something else going on. Students often have trouble seeing equations as objects as opposed to processes. In other words, a student who sees the entire equation as describing a process (“you get 97 when you multiply some number by 5 and add 32 to it”) it’s hard to take a step back and say, OK, how would this process have gone different if you didn’t add that 32? That step back is a meta moment; it’s when you’re able to start talking about the process, and that’s a big change for kids.

But is that change all about the equal sign? I don’t think that it is. We’re putting too much into the equal sign. The conception of equality is made to stand for much bigger and more complex shifts in understanding.

(Update, 12/20: It seems to me that I might have not properly described the “do something” signal. I think they’re saying that the equal sign literally tells you to do something to a and 3. That’s why it’s impossible to solve an equation like a + 3 = 10 — it’s nonsense, or it’s a signal to combine a, 3 and 10 in some way, like 13a or something. It’s true — if kids literally think that = is a command to do something then they can’t do algebra. They also can’t do ___ + 3 = 10 or 3 + ___ = 10. I’m saying there’s a nearby conception of = that allows kids to make sense of these equations. It’s the “pick a number and then do something” conception.)

And what about the research article itself? It aims to give evidence to support Carpenter’s contention. They ask students questions about the equal sign, and then ask those same students to solve equations. Does one’s interpretation of the equal sign impact one’s ability to solve equations?

So, are these related questions?

Here is the figure showing the relationship between correctly solving an equation and relational/non-relational views on equals:


Here is the figure that shows the relationship between use of an algebraic strategy for solving the equation and relational/non-relational takes on equality:


In sum, if you think about the equal sign in a shallow way then you probably weren’t solving an equation using algebra in 8th Grade. And you were less likely to correctly solve equations across all middle school grades.

(Random aside: when writing about research I always prefer the past tense “you were less likely” rather than “you are less likely.” The latter supports the idea that a particular study provides a universal truth, the former allows you to focus on the current study.)

The authors of this study take this as evidence that conceptions of the equals sign impact your ability to solve equations. “These findings suggest that understanding the equal sign is a pivotal aspect of success in solving algebraic equations.”

What confuses me is how they know the arrow points in that direction. It seems just as plausible to me that students’ ability to solve equations in productive ways drives their understanding of equality. In other words, maybe how you think about equality and how you solve equations co-develop. Why should we think that how you think about the equal-sign is a gatekeeper?

The authors ask, “Does understanding the equal sign matter?” My answer is a strong yes, but I seem to interpret the question differently than the authors do. They are asking, “Is understanding the equal sign a prerequisite for learning to solve equations and do algebra?” They answer yes, but I’m unconvinced. But I certainly think that understanding the equal sign matters. It matters because as you learn algebra your understanding of the equal sign should change. It should change because the strategies that you’re learning to solve an equation often require you to think of equations in new ways, and maybe even there is no other way to gain this conception of equality than to learn algebra.

This last claim of mine — it feels like that might be going too far. But remember the terms of the discussion. The conception of equality it supposed to enable students to successfully solve equations and do algebra. Is it possible to offer students a sturdy enough conception of equality for algebra without doing algebra? I’m not sure at all.

One thing I’ve been telling Kent is that I prefer to focus on strategies rather than conceptions. The above gives a sense of where I’m coming from. The hope of Knuth seems to be that if you give kids a great way to understand equality as elementary students, they’re going to have a much easier time learning algebra. The conception will be strong enough to guide future work.

It doesn’t seem like that to me. It seems to me that a conception of equality that can support algebra develops alongside a student’s grasp of algebra. In other words, there are no shortcuts and no inoculation possible in the elementary years. If you’re interested in doing work with elementary students that would support their future algebra work, I think you have to do some algebra with them.

(Project LEAP — from many people related to this article! — is doing this research work for equations. Project Z is doing this for negative integer work. It’s a promising route, I think.)

In short: if you want to improve kids’ future conceptions, you can help them by helping them come to strategies they can use and understand. You can’t take a shortcut through conceptions.

(Random aside: is there a difference between a conception and a belief? It seems to me that conceptions are types of beliefs, and that beliefs are confusing and unstable and maybe not good things to focus on in explaining human behavior. I know Lani Horn has written about this.)

Here are some questions I’m left with:

  • Am I misreading the statistics?
  • Why did 7th Graders have a spike in relational understanding (43%), compared to 6th (32%) and 8th Graders (31%) in the study?
  • Is a strong conception of equality without algebra a resource for learning algebra? What’s the path — it feels like there should be one, but I can’t name it.


What can I get good at?

I’d like to get good at something. What can I get good at?

I’d like to get good at teaching.

But it’s totally normal desire to want to get good at something that you can share with other adults. Some call this ambition, and that’s a fine name. You could also call it a desire to participate in an intellectual community. (It’s probably best not to over-analyze this desire, right?)

Maybe I should become an expert about teaching. I’d ask questions about teaching like “what’s the best way to teach math?” and then I’d go around with my answers: “this is the best way to teach math.” There are people who do this. People sometimes even pay these people to do this.

There are problems, for me, with the “tell people how to teach math” plan. Mainly, to do this in an intellectually honest way would require me to see many more math classrooms than I currently do. How can you tell people how to teach math if you only have your own experience to go from? How can you know that your ideas aren’t just your ideas?

Researchers and PD people tell people how to teach, but if they’re any good they’re only doing that on the basis of many, many individual observations. Recommendations on the basis of one case? That’s no good.

I could leave the classroom and start looking around at other classrooms, of course, but I don’t want to. So that’s fine, and maybe the cost of that is that you don’t get to make generalizations about teaching. I’m fine with that.

So, what’s left? If others are better-positioned to tell people how to teach math, what can teachers hope to contribute?

It seems to me that what’s left for me is the art of learning about teaching. Asking particularly good questions about teaching in particularly good ways. Understanding, in a particularly good way, what the experts are saying and how to use that. There is a kind of expertise in being a non-expert about teaching, and it seems to me that this is a knowledge that I could strive for. Not knowledge about teaching, but knowledge about learning about teaching.

Maybe, though, the premise of this post is wrong? Maybe it makes sense to try to generalize about teaching from the position of a teacher? Maybe generalizing from the position of a researcher and generalizing from that of a teacher are complementary perspectives?

I really don’t know.

Bringing Addition to a Multiplication Party

I just realized that two things I had thought to be quite different might, actually, be really similar.

First, a series of mistakes my 4th Graders make when they use addition thinking for multiplication problems:


Second, my 4th Graders’ thought that multiplication by a negative would make a number positive, but smaller:

Yesterday a couple of 4th Graders asked, “Wait can you multiply by a negative?”

Any guesses as to what prompted this question?

Kids had been working on a multiplication puzzle and (accidentally) gotten themselves into a position where they needed to solve ___ x 20 = 10. If positive numbers make multiplication bigger, then shouldn’t negative multiplication make things smaller?

What is this mistake? Why should multiplication by a negative make a number smaller, but positive?

Here’s what I’m realizing: it comes from the thought that positive/negatives have opposite effects in multiplication/division. Which isn’t true, but it is true that positives/negatives have opposite effects in addition/subtraction.

The relevant opposites when it comes to multiplying aren’t positives/negatives, but instead numbers greater/less than 1. To draw the contrast really clearly, when it comes to adding the relevant opposites are numbers greater/less than 0.

This is not some out-there and abstract idea, though. When kids work with negative numbers they regularly reveal an understanding that positives and negatives should have opposite effects, as with 3 – (-5):


We talk a lot about opposite operations, but do we talk enough about opposite numbers? We talk a lot about negatives as opposites to positives, but do we talk enough about numbers less than 1 as opposites to numbers greater than 1? How much of learning is trying to figure out the limits of thinking like addition?


What’s Going On in Pershan’s Classes This Week

A few weeks ago, I brought up negative numbers in my 4th Grade class. I framed this in terms of “giving kids permission” to bring up negatives in class.

Yesterday a couple of 4th Graders asked, “Wait can you multiply by a negative?”

Any guesses as to what prompted this question?

Kids had been working on a multiplication puzzle and (accidentally) gotten themselves into a position where they needed to solve ___ x 20 = 10. If positive numbers make multiplication bigger, then shouldn’t negative multiplication make things smaller?

This is very smart and interesting, and that’s all I have to say about that.


On a totally different note, there’s a dead pigeon right outside the window of my 9th Grade class. It’s becoming a distraction. (“Someone should really call animal control.”)

I was trying to talk about congruence and stuff and kids started getting fascinated and freaked out by the dead bird. So I climbed up on the window sill (there’s no draw string) to pull the shade down. And then the whole curtain contraption got ripped off the wall and missed a few kids’ head by an inch.

I get reminded about this daily. And I still haven’t called anyone about the bird.


I made a commitment this year to work on the relational side of teaching. To an extent, I am, but I’m realizing that it’s been entirely haphazard. Is there a way to be more systematic about this?

Another question: working on curriculum and learning can be serious intellectual work. Can working on relationships be serious intellectual work? How?


I was going to work on fractions in 4th Grade yesterday, but the multiplication practice that my kids dug into swallowed them up for the whole period. They loved it! (It was from these books.)

Multiplication practice can be fun for kids.

Then again, I’ve been trying to get one of my Geometry classes to work on flowchart proofs and that is a slog. I’d love for them to get great at this, but it looks like we’re going to move on without having everyone on board.

Sometimes skills practice sucks, and sometimes it doesn’t.

Chatting with @benjaminjriley about “discovery” talk.

Benjamin and I were having one of these enormous and confusing twitter conversations about this (probably not-quite-accurate) Galileo quote, and what educators mean when we talk of discovery more broadly.

First, Benjamin’s take, as I understand it.

Ben thinks that this quote is badly wrong. It is possible to teach people things! And there is plenty of knowledge that you simply cannot just discover all on your own. Even worse: discovering knowledge within yourself? Show me precisely where it is, within yourself, that your knowledge of my middle name is. The answer is that it’s not within yourself, obviously.

This quote is incredibly wrong, Ben says. And this makes it problematic for professional educators to share it, because it completely undermines the importance of the profession. If teachers don’t teach, then who needs teachers? Kids don’t need school if they can’t be taught. If it’s all about discovery — even worse, discovery about yourself — then you can’t really be taught. Teachers and schools, then, are useless.

Here was my question to Ben, though: if this is so incredibly and badly wrong, and if it so terribly misunderstands the work of teaching…then why does it appeal to professional educators? Shouldn’t professional educators be the first to understand how badly this rhetoric misrepresents the work?

I didn’t get a chance to hear Ben’s full answer on Twitter — he said it had to with the culture of teaching — but either way, his take is that professional educators have fallen into a characterization of their own work that is badly, badly wrong.


Now, my take.

I get suspicious of arguments like Ben’s. Are teachers badly misunderstanding their own work? I guess it’s possible, but why would we do something so self-destructive? And how come this idea, which is so obviously and clearly wrong, appeals to so many teachers?

I’m not saying that this is a knock-down argument against Ben’s take. But it makes me start to think — is there another way to interpret something like the quote Tracy shared?

I think that there is, and it has to do with the various meanings that “teach” and “discover” have.

What does it mean to teach? It could mean a few things:

  1. “I taught it to him” <–> “I explained it to him”
  2. “I taught it to him” <–> “I directly caused him to understand it”
  3. “I taught it to him” <–> “I set the conditions of his understanding”
  4. “I taught it to him, but he didn’t understand it” <–> “I attempted to cause learning, but failed”

(See here for more on the various meanings of “teaching.”)

What does it mean to discover?

  1. “I discovered it” <–> “I was the first to know it”
  2. “I discovered it for myself” <–> “I was the cause of my learning of it”
  3. “I discovered it for myself” <–> “I discovered the meaning for myself”


The quote that Tracy shared is

“We cannot teach people anything. We can only help them discover it within themselves.”

My most generous read of it is

“We cannot directly cause anyone to understand something. We can only help them discover the meaning for themselves.”

Which is absolutely true.

In fact, maybe it’s so obviously true that the statement becomes banal. Why would teachers make a big deal out of this statement? I think it’s because so many people outside education think that teaching involves directly causing kids to know stuff. Why don’t you just teach it? is something every educator has heard.

People who haven’t taught don’t typically understand the things that can get in the way of directly causing kids to know — their prior knowledge, their personalities, their attention, their peers, their priorities.


If my read is correct, it also is a pedagogy-neutral observation. Kids need to discover meanings. When I was in college, I often made sense of a teacher’s lecture — that was me discovering ideas for myself, while sitting in a lecture.

Ben and my conversation is one that has been had over and over about constructivism. For example, here is a passage from a really fantastic piece called “A Reformulation of Telling”:

Screenshot 2016-12-04 at 4.05.33 PM.png

When teachers say that “you can’t teach” it’s just saying this — that all meaning has to come from the student, at the end of the day. Though the entire point of teaching is to help kids make this meaning — that’s the work!

I see no reason why this understanding of teaching is incompatible with public esteem. It’s serious work, and the public could understand that this is serious work.


It seems to me that Ben is not convinced by my rereading. If I understand him correctly, he thinks that when teachers share these quotes online or talk this way, they really mean that you can’t cause kids to learn stuff and that kids need to discover knowledge on their own.

I disagree. I think that this reading is both possible and likely, given that it doesn’t involve teachers badly mischaracterizing the nature of our own work. I’ve literally never met a teacher or seen teaching that was centered on “not teaching” as Ben reads it, and in fact that’s impossible to do while meeting external learning standards. Why would teachers engage in talk that directly contradicts what they’re doing in classrooms? It doesn’t make sense to me.

It would be great to do a bunch of interviews with teachers to try to gauge their meaning, if they assent to “discovery” talk. Has anyone looked into this? It might be a helpful thing for someone to study.


Here is where I’ll agree with Ben, though: there is a cost to teacher talk that emphasizes discovery and “not teaching.” It’s a line that is just waiting to be misheard, and whether that’s fair or not is irrelevant.

This sort of talk is easy to misunderstand, because of the various meanings of “teach” and “discover.” (I might as well have written a long post explaining to those who engage in discovery talk how that quote could be misunderstood in the other direction!)

Teachers know how important it is to know the landscape of potential misconceptions when they’re teaching math. If you care about helping people understand what the work of teaching is really about, we also need to take people’s misconceptions about our work into consideration.

Talking about teaching and learning is hard. It’s really really hard. It’s partly hard because the work is so invisible, and because the physical manifestations of that work (explaining, sitting, writing) are poor stand-ins for it. There is no agreed-on language, and many of the fiercest debates in education are best seen as desperate calls for clarity, each person silently begging the other to learn how they mean. That is what I think is happening here.