How to Start Something In a #MTBoS

Why You Should Listen To Me

You shouldn’t.

OK Now Here’s The Post

Is it helpful or annoying to publicly muse on a community’s ability to get things done? Keep reading to find out!

The other day I was chatting on twitter, and out of the conversation came an idea to start a local math circle.

The other other day I was chatting on twitter, and we had the idea to start an organization that would help conference organizers find not-just-white-men to speak at their meetings.

What happens to these ideas? What do you need to do for them to actually happen?

The MTBoS has created and organized a lot of impressive things. Of these, Twitter Math Camp is objectively (objectively!) the most impressive. There are, of course, many others. I’m biased by my own involvement, but I think the Global Math Department is pretty impressive as well.

Every once in a while, these days, I hear about a new curricular website that somebody has put together. The single-purpose sites are collaborative, but not in the way that Twitter Math Camp is. Estimation180 or Open Middle or Visual Patterns or Math Mistakes all take ideas and materials from the community. There’s one or two people at the core, and then a community at the periphery.

To run an annual conference requires coordination of an entirely different sort. A significant number of people need to get their act together, together.

It seems to me that much of the ambitious coordination happening in the MTBoS right now is at the corporate level. A lot of the relational capital (so what if I made up a term) is being drawn into large organizational structures like Desmos, Illustrative Mathematics, NCTM. This is great — I’m a fan of each of those organizations.

My guess, though, is that this has slowed the pace of community organization at other levels. It takes a lot of people who trust each other and can get stuff done to make something happen. It seems to me that those people are getting busier and busier with their other highly-collaborative projects.

We’ve reached the part of the post where I speculate on what it takes to get a collaborative project off the ground in the MTBoS right now.

Step One: Decide that you actually care about this enough to make sure the ball doesn’t drop and the conversation doesn’t end. When conversation happens through asynchronous chat there’s always the chance that you’ll get ghosted. Every project that I’ve been part of lives or dies depending on whether there’s someone like this, someone who never says “sorry I’ve been crazy busy lately.” If this idea is going to happen, it might as well be you.

Step Two: Find one other person who you think is pretty close to your degree of commitment. Of course, you’ll never know, and people always are more committed at the start than they actually will turn out to be. This is normal, in my experience.

Step Three: Have a video chat or phone conversation ASAP. A phone conversation is best. The reason is because (a) having a phone conversation with a stranger is uncomfortable, and doing something mildly uncomfortable helps build trust and (b) you’re giving them your actual phone number and that (as ridiculous as it is to say) counts as an trust-building act in our near-dystopia too.

Step Four: I’m out of advice. Try to meet IRL if you can. Try to get more people involved, if you can. But the benefits of adding more people to your team, at first, are low. The main thing you need a collaborator for is to get past the part of your head that’s saying this idea is stupid and you’re an ego-maniac for thinking it was worth doing in the first place.

Two people is enough to start.

Step Later: Once things get going and you have a bunch of people involved, frequently test your group’s ability to keep working without you. Plan to step down, or to work on a side project for a year or two while someone else takes the lead. It’s sort of this sad thing where an organization takes a single high-energy person to get it going, but unless that person reduces their energy input the organization will never produce their own energy and become energy self-sufficient. (Yep, took that metaphor too far.)

I don’t know if this is true, of course, but it’s what I’ve seen so far.

It occurs to me that maybe I’m taking twitter too seriously again. True, I see a lot of ideas that never come to fruition. It’s true that the ephemeral nature of the Stream makes organization hard. But maybe that’s precisely what allows so many ideas to pop up. People would share far fewer ideas if they actually were intending to commit to them, and that’s fine.

But I think there’s room for more organization in the MTBoS than I currently see. And while I don’t think we should force anyone to do anything with their online experience — if people are happy with the way things are, that’s fine too — I suspect that people would take pleasure from forming the deeper relationships that come from doing something good together.

We have so much energy for talking about social justice, and yet there are no groups (that I know of) committed to promoting social justice in MTBoS. I think about this a lot. I don’t think it’s hypocrisy. I think we’ve communally lost the knack for organizing ourselves.

We aren’t aided by our tools. The technology we’re working with has been designed to get us to keep us using that technology. We need to fight that tendency if we want to form lasting groups online.

Back to writing a homework for tomorrow. Good night!

3rd Graders Tell You Whether or Not Fractions are Numbers

I’ve been thinking lately about what it means to be a number, for kids. My guess is that, for kids, something is a number if you can perform arithmetic operations with it. So, for example, when fractions are first introduced they aren’t seen as numbers, since you can’t count by them, add, subtract, multiply or divide them. Being a number involves living the life of a number, so to speak, and that life involves counting and arithmetic, mostly.

(Related: maybe measuring should be in that mix.)

If this idea is true, it would go counter to the conventional math education wisdom. When math education speakers or thinkers want to help kids realize that fractions are numbers, they usually recommend number line activities. The thinking seems to be that if we can show that fractions mingle with whole numbers, this will lend them legitimacy.

I think that kids find this unconvincing, though. I think being a number has more to do with whether the actions you can perform with numbers can also be performed with this new thing.

Or maybe it’s a 50/50 proposition? And both are important? Anyway, I wanted to look into this.

I asked my 3rd Graders, who have been studying fractions, whether they thought fractions were numbers. I asked them to write a few sentences explaining their thought. (I had to work to get those sentences. A bunch of them were very unhappy about writing.)

So, are fractions numbers?

Here are some of my favorite responses:

  • Yes, because to make a fraction you need numbers.
  • Fractions are a certain category of numbers because without numbers fractions would just be lines.
  • I do not think this is a number (1/2) when the 2 and the 1 are numbers, but not together, for 1 + 2 = 3 number, but this is not a number it is a half.
  • Yes a fraction is a number because a fraction is a number of pieces of something.
  • Fractions are numbers because I don’t know to explain it.
  • Fractions are numbers because you could add up to a number like 1/2 + 1/2 = 1 and a whole is a different way of saying 1.
  • NO a fraction is not a number a fraction is only part of a number.
  • Yes, because a fraction is a part of a whole number.
  • I think fractions are numbers because all math is numbers.
  • NO I think fractions are half of numbers. 1/2 is a fraction. 1 is a number.
  • A fraction: a problem of numbers in groups. Therefore not really a kind of number, but closely related.
  • Are fractions numbers?! Yes because 1/4 = 6 ’cause fractions are numbers but the bigger the number the less it is.

I have no idea what that last one means either.

So, how did my hypothesis hold up?

One kid used the fact that you can add fractions to make a whole number to decide that fractions are numbers. Though it wasn’t the mere fact of addition that helped this kid — it was the way that fractions could relate to whole numbers.

Reading these responses leads me to revise my hypothesis. Kids decide that fractions are numbers when fractions/whole numbers co-mingle in equations and problems.

The number line still, I think, is not necessarily relevant unless you count along the number line. Meaning, it’s not about the number line, it’s about the number line as a representation of counting that involves both whole numbers and fractions.

After asking my students to write their sentences, I led the class in some counting that I marked up on the number line: 0, 1/4, 2/4, 3/4, 1, 5/4, 6/4, 7/4, 2…

This, along with other arithmetic operations that involve both fractions and whole numbers, will help my kids increasingly see fractions as numbers.

UPDATE: I talked to my 3rd Graders about this again today. I just reminded them of the question, and then the room sort of exploded with arguments. Here are the arguments I heard.

When two parents have a kid that kid is still a person. So that’s like when you take a 1 and cut it up into thirds it’s still a number.

You can make numbers out of fractions like 1/2 + 1/2 so those have to be numbers too, because otherwise how could they make a number?

I like that these two arguments are sort of the flip sides of each other. I told my class that to me one of these sounded like “fractions are parents of numbers” and the other “fractions are children of numbers,” which a couple kids found hilarious.

One girl said that it really depended on the fraction. People talk about having half of something all the time, so half is a fraction. Other things — like 7/13 — that wouldn’t be a fraction because nobody talks about those. Later, I tried to put pressure on this position: what if I made a recipe that called for 7/13 of something? She said it really depended on what everybody did, not just me.

Her take reminds me of debates between prescriptivists and descriptivists about language.



[NBCT] Student and Professional Needs

So, you did it. You’ve gathered information about your kids from a lot of people. And then you used that knowledge to assess the class formatively and analyzed those results. Then you had kids self-assess and used a summative assessment to show that they actually learned something from this unit.

Congrats! You’re two-thirds of the way done with Component 4 of National Board Certification.

The final third has been hard for me to make sense of. It’s called “participation in learning communities,” and apparently proving that you participate in learning communities requires filling out a bunch of forms.

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(Successfully pursuing NBCT may or may not show that you’re a better teacher, but it definitely does show that you’re willing to fill out a lot of forms.)

Cynicism aside, one thing that was driving me nuts was trying to figure out what the difference between a professional and a student need is for NBCT. Especially since you’re supposed to provide evidence that addressing your professional need impacted the students. Doesn’t that mean that every professional need is also a student need?

I’ve broken the code, though. The key is this passage:

Screenshot 2017-03-14 at 2.35.03 PM

This distinction aligns perfectly with the differing requirements of the professional and student need submissions. For the professional need you are supposed to describe something you needed to learn and show how you used colleagues/others to learn it. For the student need you don’t need to learn anything — you just need to recognize and identify something that would make a difference to kids in your school, and then you’re supposed to impact your colleagues/others.

Which is why you don’t necessarily need to provide evidence that the student need impacted your kids. This is teacher as advocate and leader, affecting your colleagues. When you’re a learner you need to be affected by your colleagues, and show an impact on your kids.

(There are parts of this process that I don’t enjoy, but I won’t pretend I don’t love the exegesis. Sue me.)

So, yay, I understand what I’m supposed to do. How can I do this? I’m usually pretty deferential around the office, and “teacher as advocate” doesn’t sit well with me. That said, why not share ideas with my colleagues? It would be good for me to do more of that, especially in, oh, the next month or so.

Here’s what I’m thinking.

Teacher as Advocate: Better Middle School Geometry Experiences

I’ve taught high school Geometry at my school for the last four years. It’s the course I’ve taught the most. And while kids do alright in my classes, I think our school could be better preparing kids for their high school geometry work.

First, they often come in to geometry without having thought much about angles as rotations, or as angles being greater than 180 degrees.

Second, they have inconsistent experiences with the Pythagorean Theorem.

Third, they have had inconsistent experiences with the relationship between shared angles and similarity.

In the next month I’ll try to share some of the things I know about middle school geometry with my colleagues. There are three things I’ll do to advocate for geometry in Grades 3-8 (which is what we cover):

  • Create and share resources for various geometric topics, and ask some of my colleagues to share them with their students and tell me what they find out about their geometric knowledge.
  • We have a shared curricular space in our department. I’ll make a page to share some geometric resources that are appropriate for various grade levels, and try to better organize some of the things our department is already doing.
  • I’ll share some of this work at one of our math department meetings.

Collecting evidence is always really tricky with these portfolios, and is most of the reason why I end up submitting at the last minute. (I find that things never really work out when I try to collect evidence after the fact. I need to know what I’m doing so I can collect evidence during the process.)

What could evidence be for this? It’s tricky, but I could collect student work from any assessments I make from my 4th, 8th and Geometry classes, and I could also try to collect the student work of any colleagues who try out my assessments. I’m also thinking that maybe, if I share a collected activity page via email, I could take snippets of any emails people write back to me about the resources I’ve shared.

This plan is a B+ plan.

Teacher as Learner: Learning Disabilities and Proofs

I’ve got a few kids in my geometry classes who have learning disabilities. I don’t know how to support them with understanding and creating complex proofs. These kids have attentional issues that are related to low working memory. This makes it hard for them to e.g. keep in mind the premise of an argument, or e.g. an earlier diagram after it’s been transformed into some new one.

I might as well focus on the next thing that I’m trying to teach — proofs of the Pythagorean Theorem. Is there any way that I can help my kids with learning disabilities make sense of these sorts of arguments?

I’ll ask the learning specialists in my school, but I’ll also ask smart people that I’m connected to online. As evidence of my learning I’ll excerpt our conversations. And as evidence of the student impact, well, hopefully I’ll help more of my kids learn these sorts of proofs.

And then if I can do that, then I just have to write the 12-page written commentary and I’m set.

And then, with the permission of an anonymous Pearson grader, I’ll be NBCT.

Sunday Poem – The Boatman

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(Source: Poetry Foundation)

Click through to the link and you can hear the poet reading it, which I recommend. I’m in the middle of understanding this. Here’s where I’m at:

  • The big, juicy lines here remind me of The Odyssey and other epic poetry. It seems likely to me that The Boatman is pointing us to Odysseus, and asking us to see the refugee as not just a victim but a person of ultimate bravery.
  • I don’t know anything about meter. Does the meter of this poem more specifically suggest epic works?
  • If that’s right, then the poem doesn’t go so far as to frame the refugee as entirely heroic, I think. The poem is clear that the refugee has been forced to the sea in the absence of any other options. What, then, does their epic heroism consist in?
  • (I have no clue. Maybe the poem suggests that real bravery consists in facing the horrors you’ve been dealt as opposed to seeking them out, from some love of adventure.)
  • The line-break that surprised me the most was “we fetched a child, not ours, from the sea, drifting face-//down in a life vest, its eyes taken by fish or the birds above us.”  Why break there? Why not keep “facedown” in the line?
  • I’m definitely going to read this again. What are your thoughts?

[NBCT] Self-Assessment

In the previous episode, I had given my students a formative assessment task and analyzed their responses by strategy.


I had no idea, though, how to do a meaningful self-assessment.

I liked what I ended up doing. Here is what I asked my students to work on today:


The figures are (again) taken from this Shell Center activity.

For as long as I’ve been thinking carefully about feedback and assessment, I’ve had a hard time getting excited about self-assessment. The whole point of assessment is that the assessor can direct your attention to things that you yourself have not seen. That makes self-assessment a pretty tough tool to use while you’re learning something.

If I had asked my kids “what could you have done better on this task” that would be lame. If you knew how to do something better, wouldn’t you? Unless you were lazy or tired or careless, and I’m interesting in teaching math. That other stuff is very rarely math.

My way out, though, is to reframe self-assessment as “assessing your own stuff against some other standard.”

I had noticed that some of my kids, on the initial task, weren’t finding the area using interesting structural features. Instead they were counting individual squares. That would hurt their ability to understand what the Pythagorean Theorem is saying (they wouldn’t be able to quickly check square areas) and would also hamstring their ability to understand proofs of the Pythagorean Theorem (since the proofs use those structural features).

Here is a self-assessment response from a kid who counted individual squares the first go around:


I would have described this student’s work on the initial task as strong, and her self-assessment was strong as well:


I thought this student could have dug a bit deeper.


I’d say that this activity worked at the level of “compare your approach to these two nice approaches that use nice geometric structure” for everybody, and then the rest of the activity worked well for like half the class.

So, I didn’t fall in love with self-assessment today. But I did figure out a way to do this part of my NBCT portfolio in a way that didn’t make me puke, and that I think helped kids learn something.

That counts as success, right?

Sunday Poem

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Here’s my best read of this week’s poem: you have the frustrated artist, who finds himself completely capable of artfully representing reality but failing to apply that same art to imagined, fictional creations.

The poet contrasts painting with photography. Painting is what he aims for; the photographer’s lens merely captures snapshots. The snapshots are clearly attempts at assigning importance to the moments they capture, but frequently fail to do so effectively because they’re constrained by what actually happened (“paralyzed by fact”). The poet’s own work often feels just like photography, limited in this way.

But is this really a limitation? Vermeer is a painter who provides such realistic portrayals of reality that they seem almost photographed. And yet, they have a powerful effect. The artist — the photographer, the painter, the poet — does something wonderful, the poem concludes, by accurately representing actual moments. Our entire lives are composed of such short, delicate moments, and the photograph snapshot allows us to assign each of those unimportant moments some importance.

Thoughts and questions:

  • Why is the poem called “Epilogue”? Does it come at the end of some other work?
  • What does “all’s misalliance” mean?
  • Oh, wait, here’s a short essay about the poem.
  • From that essay: “the poem also recalls the classical recusatio (refusal), in which the speaker claims he is unable to write the kind of poem the occasion calls for.” That’s cool! I want to read more recusatios.
  • So is this poem imagined or recalled?
  • What does it mean to “tremble to caress the light”?
  • I like that line break — “I want to make // something imagined, not recalled.” But read the line break and it’s not just a felt inability to write imagined poems, but an inability to make anything at all. (As opposed to just recalling, I suppose?)


How Medicine is Like Education

Everyone’s favorite edu-game is comparing education to medicine. Do teachers garner the same respect as doctors? Should teachers be more like doctors? Is education like medicine?

I have no idea, and I sort of hate this game.*

In 2014 there were ~700,000 physicians and ~1,500,000 kindergarten and elementary teachers, with ~961,000 high school teachers. Which is just to say that there are a lot more teachers than doctors. (Source: BLS BLS BLS)

With the above as a caveat, a few things I recently read about medicine made me think of teaching. Not that they are definitive or representative of medicine, or anything. Just that they made me think. I have no idea how to put all the pieces together, but each made me think that in some little ways medicine might be a bit like teaching.

Doctors Don’t Have Time to Read Research and This Makes the Field Subject to Destructive Fads

I’ve been reading Dreamland (“The True Tale of America’s Opiate Epidemic”) and it’s consistently fascinating. Part of the story is about how the medical profession fell in love with opiate pain-killers after years of caution. How did this happen? Several trends converged, but an important part of the responsibility lies at the feet of the medical profession:

For many years, it was believed that pain protected against the development of addiction to opioid medications.


This false belief (med-myth?) had its source in some bad interpretations of research.


I read this page in Dreamland this afternoon, and the story felt familiar. Even more familiar was this tag to the story: “To actually look up Porter and Jick, to discover that it was a one-paragraph letter to the editor, and not a scientific study, requires going to a medical school library and digging up the actual issue, which took time most doctors didn’t have.”

Could the troubled relationship of research to practice be similar across many fields, and not just in teaching?

Medical School Doesn’t Prepare You for ‘Real Medicine’ and in ‘Real Medicine’ Often There is No Cure

House of God is a novel, a satire. Let’s start there: you shouldn’t take it literally.

It’s also from the 70s, and it is a very 70s book. Everyone’s constantly talking about Nixon. Everybody’s smoking. And everybody is having sex with everybody.

Seriously, when I told my doctor friend I was reading this book he took great pains to explain, You know it’s nothing like that anymore, for one there is way less S.E.X. (He spelled it, to save my toddler the embarrassment of hearing it I suppose.)

OK, so it’s somewhat dated and it’s satire. Still, it has passages like this:

Again, like the day before, most of what I’d learned at BMS about medicine either was irrelevant or wrong.

Sound familiar? Could it be that complaints about the relevance of professional training are more common in medicine than teachers usually think? (The question isn’t rhetorical. Could it be?)

The thing that really caught my attention, though, was this passage:

Talking about medicine, I told him with bitterness about my growing cynicism about what I could do, and he said “No, we don’t cure. I never bought that either. I went through the same cynicism…And yet, in spite of all our doubt, we can give something. Not cure, no. What sustains us is when we find a way to be compassionate, to love. And the most loving thing we do is to be with a patient, like you are being with me.

Which seems to me an entirely correct perspective on teaching as well.

Not quite sure how to do student self-assessment [NBCT C4]

As part of NBCT Component 4, you need to give a formative assessment. Last week, I gave this formative assessment to my high school geometry students:


Here was some of the work that my students came up with:


A wide variety of strategies and ideas, as well as levels of sophistication.

NBCT wants us to study and synthesize the results from the whole class.


This does not worry me. No problemo, there’s a lot to say here. I’m imagining doing something like this (or maybe this itself):


The student work analysis isn’t scaring me. The call for student self-assessment, on the other hand, gives me the jitters (the howling fantods):


The rubric idea? I hate that idea. And collecting a recording and making a transcript feels like a pain in the neck.

What other ways are there to have kids self-assess? Please, share your ideas. Here is what I’ve come up with so far:

  • Do what I always like to when asking kids to revisit their past work: do some whole-group activity that teaches them something related to the student work, and then ask them to improve their work. Usually I hand back a marked-up copy of their work — honestly, I think that’s an important part of revising — but I could just ask them to self-assess?
  • Maybe my whole-group activity could be teaching them different language for their strategies? And then they identify them in the student work?
  • Ooh, I sort of like this idea: what if I gave them a proof of the Pythagorean Theorem and asked them to compare their work to the proof diagram. Self-assess: how is your diagram similar or different from this one?


I dunno. Self-assessment isn’t something that’s an important part of my teaching at the moment, and I’m unsure whether it should be. Either way, I need to find a nice way to ask kids to self-assess for this portfolio.

Any ideas?