Tuesday, November 17, 2009


Today I tried something I'd never done before. I did it with my 9th graders, the class that is made up of all these awesome minds, hard workers, smarties, funnies, goodies, but somehow doesn't feel like a working group. We made towers of cubes out of marshmallows and toothpicks, looking for patterns (edge & vertex counting), making functions.

I gave them almost no direction at all, just showed them a couple of small starts that I'd made and they went nuts. I should have brought my camera. It was awesome! They worked so well together, building stuff, talking about what the building blocks were (these dangly things, not fully formed cubes) and didn't eat anything or poke each other at all.

Once they were supposed to start actually counting stuff, finding patterns and rules, the whole thing deteriorated. I left class feeling sort of defeated but determined, sure that I should feel some sense of accomplishment. After all: trying something new, messy, unpredictable. That's success right there, in a way.

So next, I'm trying to figure out how to get it to feel like that more often, how to connect the work they do in their minds to the work they do with their hands. How to get them to really bite into the concept of the variable, which was my motivation to do the lesson in the first place.

Also, what to do tomorrow. Ah.

Wednesday, November 11, 2009

Good Teaching

Yesterday in class, my coteacher Phil Dituri (amazing, amazing, amazing) was asking our kids to be silent for the start of the class. We've been trying this out for the first three minutes of class the last few weeks just to get them settled in and working, so that when group work starts they can really bring themselves fully into that conversation.
Yesterday proved to be on the difficult side. The kids had had a rough morning (last day of the quarter, NYPD metal detector surprise that morning, typical season affective disorder type blues) and weren't getting quiet. After a couple of minutes of reiterating instructions, and individually trying to help kids focus, Phil went around with big blue paper tape, drew smiling mouths, some with tongues out, and gave it to kids to tape their mouths closed, which they all wanted to do.

This looked ridiculous to anyone who came in at that moment. Comical, maybe even inappropriate. But the thing is, Phil could have yelled, made the kids feel bad, expressed his frustration, disappointment, he could have sent kids outside, called home, whatever. Instead kids were "smilingly" (both on the tape and underneath) trying to be quieter than their neighbor. They ate it up.

The tape didn't exactly turn our kids into models of focus and undivided attention. But kids got some individual attention, which they seemed to need, they got a structural intervention to help them follow instructions and get quiet, and the room stayed positive and focused on the math.

I realized after I watched it happen that this kind of thing is something they didn't teach me in grad school. I've never even (consciously) heard anybody talk about it before. But it seems like a vital part of that magic that good teachers have: to be able to keep the forward momentum, the mathematical focus, the positive energy, in the face of whatever the kids throw at us. It can be so easy to take a momentary failure personally, to get frustrated, feel like the kids aren't listening, whatever. Or just to run out of ideas. What Phil did in that moment was magic because he didn't drop the expectation: the kids were still supposed to be quiet. But he didn't get nego either, on them or himself. He gave them a new reason to follow directions, perhaps clarifying the directions for some kids in the process, and then back to the math.

Phil transformed what could have been a power struggle into a collaboration. He turned what could have been rule-following into a game. He was able to honor the kids, acknowledge their state and their need, give them total respect, and still help them transition and move forward towards our mathy goals. It was beautiful.

The whole thing took maybe 60 seconds. It wasn't such a big part of classtime. But I think it's those moments, the decisions we teachers make in those moments, that set the tone and culture of the class. We are in this together. We teachers aren't going to pull the authority card just because we can. We want to have fun with you while we work. For me, cultivating this feeling in the classroom is the best leverage I've got, for getting a class working, and it's way more fun to be in a room where that's the vibe.

I love teaching with Phil. May you all be so lucky in your teaching collaborations.

Inspiration: Kate Nowak

Kate Nowak's posts are optimstic, pragmatic, generous and immediately useful. I used her row game in class yesterday and it was the best new group work structure I've tried since I started teaching.
My whole class is (ideally) based on group work. My students should nearly always be talking to each other, comparing answers, helping when someone needs help, asking questions when they're confused, helping each other stay on task, etc. Sometimes this is natural, and mostly I see that this is how kids want to do math. Some training is helpful though.
While doing the row game yesterday with order of operations and signed arithmetic, I heard the best conversations I've heard in my new 9th grade classes. I witnessed serious debate, authentic and sustained engagement, passionate investment in each other's work, and evidence of a compelling motivation to deeply look at one another's work, since they hadn't themselves already done one another's problems.
Simple to create and direct, this activity is going to change my class. THANK YOU KATE!

Kate's current post is also beautiful and I'm going to use it in class imminently. I'm excited about the accessibility of the visuals and the ease with which I think kids can interpret and make meaning of them.

While I'm thanking Kate, I should also say that she is the reason I know about blogging in the first place, and the primary support for me starting this here blog, which I'm so glad I did.

3 cheers.

Sunday, November 8, 2009

Nearly Quarter 2.

According to one of my 9th graders, our first unit was due the title:

"Educational Revolutionary Mathematics"

That felt like the arising of my own personal cheerleading team.
That was also about three weeks ago.
Since then, I've been negotiating parent teacher conferences, failed experiments in the classroom, incorporating a student teacher (my first), dealing with longer darker evenings, the storming phase of group dynamics, and my own personal dramas around all of the above. Truth is it's felt like such a hard couple of weeks that I didn't want to look at the reality of it in my own head, much less write about it to anybody else. However, I did learn some things.

Highlights that I'll share:
If I feel disrespected, discouraged, frustrated, like a failure...
it's worth considering how my students are feeling. Because they are probably feeling a bunch of those things, and I want to be conscious and sensitive to how I respond to such volatile and hot emotions. Also because I feel immediately compassionate for them in a way that I struggle to feel for myself, and that's a useful button to push in me.
Because when I'm feeling pissed off at myself or them or the situation or whatever, I get distracted by all those feelings and forget to actually look at what's happening, pay attention to the math and the kids who are learning (or struggling to learn it). But once I hit my compassion button, and start looking out for my kids' feelings, then I'm paying attention again, and it's just a problem solving activity, rather than an ego trip or an identity crisis.

I can't be sure, but I think it's safe to guess that I'm not the only teacher who takes stuff personally when they shouldn't, and for all of you out there, I just want to offer this:
Whatever your worst feelings, your students' are worse. Focus on that and it's easier (for me) to let go of my own head trip and get back to thinking about something I can actually do something about (tomorrow, or right now) rather than impossibly fretting about the past.

I want to be your cheerleader now, assure you that somehow it's all about the process, all about showing up and doing your best and working hard and enjoying it as much as possible and in fact doing whatever you need to do so that the hard work feels good. Remember to sleep enough. Remember to eat three square meals every day. Remember to love yourself. Remember to tell them how awesome they are, whenever they are. Look for those moments so you can jump up and down and cheerlead them. We all need it, eh?

Hope your last three weeks have been fantastic. Hope that your Mondays are awesome.

Larry Zimmerman strikes again

I wanted to share another few excerpts from Larry. He gave a talk last Tuesday at the NY Math Circle's PD for Middle School Teachers, and I was impressed again by how he asks questions to extend thinking about a problem, offer more entry points, and deepens my interest. He does it so quickly, I can't help but get interested and then my mind is thinking about 10 different but connected things at once, and whatever I get into relates to the bigger picture, and it's like magic differentiated instruction and guided inquiry. It's really awesome. We did a bunch of problems, and here are three that I wrote down his questions for:

Show that the sum of two odd numbers is an even number.

What is an odd number?

What is an even number?

Is 4 even?

Is 0 even?

Are there any even primes?

What’s the next even prime?

Are all prime #s greater than 2 odd?

Are all #s greater than 2 prime?

Is -4 even?

Is -6 even? Why?

So now, what is an odd number?

Is every even a multiple of 4?

Show that no perfect sq ends in 3.

Do you believe it?

Do any end in 6?

Did you know that every sq ends in 6 iff the number’s 10s digit must be odd?

Would anyone like to see a proof of that? Good! Go do it.

(I was curious to observe here that every sq is

- a multiple of 5

- one less than a multiple of 5

- one more than a multiple of 5)

Show that x2 – y2 = 2 has no solutions in integers

Translate this into words.

Say this in words: x2 + y2

(x + y)2

x2 + 5x + 6 = (x + 2)(x + 3) Find the value of x that makes this false.

Factor it using diff of squares: (x + y)(x – y) = 2

Note: remember given info

Integers factors must be 2 & 1

“I wanna savor that for a minute. To some people that sounds trivial, but to me it sounds profound. Never dismiss the obvious as being trivial. Even if it’s obvious it may be very very important.”

x + y = 2 and x – y = 1 (or other way around)

“Do something, do anything, and if it doesn’t work, SO WHAT?”

Thursday, October 15, 2009

quotes of the week

i don't know if any of these translate without audio and facial expressions, but here's hoping:
"jesse come look come look! i have my own rule for doing this now, let me explain it to you!"
"this class is so sexy...erhm, i mean, i just really love math class."
"when you sing, it makes angels cry. CRY!"
"there are 10 days in a week"
"there are 12 hours in a day"
and, finally, no one asked why their math teacher asked them to draw a picture of a flying monkey.

Tuesday, October 6, 2009

smarter than 2

this whole conversation started when ben blum-smith read me the preface of mariana cook's new book of photographs of mathematicians:

"Mathematicians are exception. They are not like everyone else. They may look like the rest of us, but they are not the same. For starters, most of them are a great deal smarter."
i totally nodded right along with him reading! what is that? confession: i walk around thinking that my math friends ben and justin and all these people are just smarter, more capable, better than i am.
and the thing is, it doesn't matter if that's true. the thought that it is, the possibility that it is, is so debilitating that i have felt (in the last three weeks!) that i should stop teaching, despite the fact that i know i'm doing this about as well as people do it and the people that have seen me do it would hate to see me stop. or whatever, even if i was bad at it, i won't get better if i think i'm just not smart like mathematicians are smart. or whatever. the problem is the very idea that true hierarchy exists. the notion of hierarchy is limiting because then some people are just smarter, and then the thought "i am one of them" or "i am not one of them" are equally problematic.

why do we (or a big portion of the general we) see mathematicians as different? and how is it affecting my students? kids (plus me and unapologetic too, it seems) are walking around feeling defined by either having smarts or a lack of them. i question the truth of these ideas in the first place, but definitely the usefulness of these beliefs when kids get attached to these self-definitions. i wonder how to keep the identity question balls up in the air during high school, rather than supporting kids in pinning themselves down.

i'm proposing that ALL kids can feel empowered and smart, not because all kids are the same, but because kids aren't all one thing. i want to give kids opportunities to perceive themselves in different roles so that they don't ever feel limited, in anything. i want my kids to be able to bite into the experience of success (which will look different for different kids) and struggle (a very useful thing for everybody to know how to work through). i want them to see each other as resources, sometimes surprising and unexpected. i want anyone in the room to be able to be the one that has the insight, no matter what has happened in their pasts.

good luck, right?
well i'm trying.

seriously, big thanks to you big debaters out there. i love it.

Sunday, October 4, 2009

smarter than...

two big ideas came together for me today in a beautiful way:
idea 1: from the psychoanalytic standpoint, group theory suggests that students take on different roles: monopolizers, hiders, harmonizers, bullies, etc. this is natural, even useful. problems in the group arise not when kids take on roles, but when kids are always playing the same roles! if the same person is always doing all the talking, maybe at first a bunch of folks are content to sit back and let them do the heavy lifting. but eventually the group is going to get pissed because they aren't getting their say. so it's important to keep roles in the classroom (or whatever group you're leading) fluid, and how to do that is what i've been studying and thinking about lately. just trying to watch it happen is pretty cool.

idea 2: ben blum-smith made a comparison for me today between racism and the idea that some people are better at math than others. he was pointing to both as culturally insidious ideas of inequity that have maintained their strength despite obvious uselessness and inaccuracy because the structural privilege they provide benefits too many people. i couldn't quite grab onto this parallel until he explained that the benefits of the idea of mathematical talent (let's call it) aren't just to the elite mathematical class, but to anyone who has ever felt like they were smarter than someone else. ever.


that hit me harder than anything all week. and i've had a big week. is it actually true that some people are smarter than others? certainly the power of "i'm smarter than..." rang loud and clear for me. there's a lot of meat in here for me to consider, but the thing i wanted to share was this:

what makes sense to me is that kids need to experience themselves in different ways in order to learn, in order to learn how to learn, in order to get good at living. and so it's not ok for the same kid to always be the one that's good at it. who gets it. who explains it. who aces it. it's not ok for the same kid to always feel like they're just not good at it. the good at it and the not so good at it are ok, maybe, as long as those labels don't get stuck in the same place.
this idea feels like the most radical one i've ever had. that we none of us "deserve" to feel like we're smarter all the time. or that we're less smart all the time. i don't believe it, but even if someone had research to show that it was, i'd still advocate for the structural intervention of pretending that it's not.

practical applications for tomorrow:
- see kids as changing beings. really observe, listen, watch, each day with conscious but open eyes (as opposed to permanently labeling them in my own mind as high fliers and low.)
- give them lots of different kinds of things to do, and to highlight the success and struggle of everyone in those contexts, so that it's clear that there will probably always be folks in those roles, that the roles in and of themselves don't necessarily mean anything, because those roles are changing all the time.

Wednesday, September 30, 2009


"replace ambition with curiosity" - paraphrasing nancy stark smith
let go of assumption.
my ideas of the day.

Thursday, September 17, 2009

September 2009

I want to say a word about hope here.

I want to say a word about how the most important thing is just to show up, to have faith, to be present, to fuck up and know it and still love yourself completely, and show up the next day, still with faith and a willingness to be present, to really see what is happening in this moment, now.

I want to say a word about how a person can spend 4 years teaching and feeling incompetent and frustrated and like it doesn't matter, and then out of the blue, for relatively unimportant reasons, ten kids that you taught when you were just starting, tell you without hesitation or qualification how you changed their lives, and then go back to talking about pizza and what they're going to do after school.

I want to say a word about the importance of uncomplicated friendship that rejuvenates and inspires, clarifies, answers, supports.

I want to say a word about gratitude. To be able to notice any of this is the whole point, I think.

We are revolutionaries, all of us, each of us doing the best we can, loving and hoping and serving every day, even the ones that feel like a total sham. Eat it. Enjoy it. Live it. Trust it. Take it. It's yours.

Tuesday, August 4, 2009

Why does it matter?

I just read an interesting post, addressing a student's question "Why does this matter?" and it's got me thinking.

As a teacher, being prepared to answer the question of how whatever we're teaching is relevant is important. In fact, I hope that it drives our planning, that we are riveted, fascinated, engaged in the usefulness and application of what we teach. If we are clear about the context, meaning, beauty and application of a given lesson, being transparent about the topic preempts the question.

In my experience, whenever this question does get asked it's not because they actually want to know why a lesson or topic is important. It's because they're not learning. If they have time to ask this question, either they are not experiencing enough challenge or they are not experiencing enough success and one or the other is arresting their learning.

"What's the point" is code for "I'm bored and I don't want to do this because it doesn't matter" or "I'm lost, and I've been lost, and I don't want to do this because it sucks to feel lost." In either case, an explanation of the value of the topic doesn't actually address the real concern: if actually doing the math is not interesting or engaging or challenging enough to capture their interest, no amount of verbal explanation is likely to help; if they are too confused to do the math in the first place, no amount of verbal explanation is going to get them to "get it."

I think it's our job to figure out what to do to get the kids learning again. Even with an awesome explanation for the worth of algebra, if they're asking why it matters then something more basic is missing for them. When they are learning, both feeling successful and being challenged, the question doesn't come up.

Thursday, July 30, 2009

Larry Zimmerman Plus

Larry Zimmerman is an extraordinary man, teacher, mind; one of those math teachers that infectiously inspires creativity and enthusiasm. As a teacher, he is alert, industrious, sensitive, clear, direct. He presents an entirely different model of teaching than anything I’ve seen before, and it made me wonder, if a little desperately, how my life would change if he had been my coach. He has recently retired from a long career at Brooklyn Tech and is pretty involved in the New York Math Circle (http://www.nymathcircle.org/)

My notes and quotes on him came from a workshop he taught on problem creation yesterday. My take on this part of his teaching is that it taps into students' (& teachers) metacognition (thinking about thinking, thinking about the fact that they are doing math while they're doing math), pattern seeking abilities and natural curiosity. I'm trying to approach planning by simply stating the questions I want to be prepared to ask during a lesson, as well as return to through the course of a unit. Here's an example of the list of questions that came from one initial problem (which I didn't initially even think was all that interesting.) I found the questions we came up with fascinating and pretty surprising.

“Theme”: How many distinct positive integer factors has the number 36?

“Variations” in no particular order:
- How many distinct positive integer factors has the number 37?
- List the factors of 36.
- What is the smallest positive integer which has the same number of factors as 36 (including 1 and itself)?
- How many distinct positive integer factors has the number 40?
- How many distinct positive integer factors has the number 49?
- How many distinct positive integer factors has the number 944?
- If x < y, and x divides y, can the number of factors of x be greater than or equal to the number of factors of y?
- Is there a positive integer for which the integer is less than the number of its factors?
- Can you find the number of factors of a number without enumerating them?
- Which positive integers have an even number of factors?
- Which positive integers have an odd number of factors?
- How many distinct positive integer factors of 36 are even/odd/perfect squares/multiples of 6/etc.?
Other inspired questions, not particularly related to the theme:
- Is zero an even integer?
- Is zero a factor of 5?
- Is 5 a factor of zero?
- Is 2 prime?
- What is the next even prime?
- Define a definition.
- What is the purpose of a definition?

Wednesday, July 29, 2009

Larry Zimmerman Quotes & Notes

“It is perhaps more important to be able to compose problems than to solve them.”
“Problem creation is the essence of mathematics.”
“The first effort is rarely if ever the final product.” (we must emphasize editing)
“Creating problems is theme and variations.”
“Determine what you are looking for and WRITE IT DOWN.”
“I don’t know what I want them to realize.”
“Beware of reinforced suspicion,” which is not a substitute for problem solving.
“If I’m 100% I will not forget it, I write it down.”
“There is nothing wrong with clarity by redundancy.”

Types of Problems
• Surprising
• Novel
• Fruitful
• Charming, imaginative, alluring
• Historically significant
• Haunting, musical
• Beautiful, elegant, sublime

Elements of Problems
1. a goal (construct, prove, maximize, minimize, classify, compare, compute)
2. given information
3. special rules (sometimes)

Some Big Ideas
- Plant a seed and then walk away until someone says something. Students and teachers struggle with silence, but it’s important and it saves time.
- Keep a notebook (both teachers and students) of interesting questions.
- We are well versed at turning a lot of words into symbols. Do the reverse.
- There are 3 varieties of equations: identities (always true), conditionals (sometimes true), and contradictions (never true)

Sites of the Day

Arbitrary digits of pi in base 16

Online Encyclopedia of Integer Sequences: http://www.research.att.com/~njas/sequences/Seis.html

Mary O'Keeffe Best Games Ever

Number Bracelets Game (adapted from Marilyn Burns at

Imagine that you have lots of beads, numbered from 0 through 9, as many as you want of each kind.

Here are the rules for making a number bracelet:

  • Pick a first and a second bead. They can have the same number.
  • To get the third bead, add the numbers on the first and second beads. If the sum is more than 9, just use the last (ones) digit of the sum.
    To get the next bead, add the numbers on the last two beads you used, and use only the ones digit. So to get the fourth bead, add the numbers on the second and third beads, and use the ones digit.
    Keep going until you get back to the first and second beads, in that order.
  • How long (or short) a bracelet can you make?
Other Questions:
How long is the longest?
How short is the shortest?
How many number bracelets are there?
- How many different starting points?
- How many different bracelets?
Does it necessarily repeat?
Change the mod/cutting number
Change the rule (from Fibonacci to some other sequence)
Why no bracelet of 6?
Why no bracelet of 5, 10, other factors of 60?
What if only use even numbers between 0 and 9?

Prime Number Monopoly
Materials: 2 dice, 5 cards (one for each of the first 5 prime #s) with the following costs and deed contracts:
Prime Number Deed:
2 – $250
3 – $167
5 – $100
7 – $71
11 – $45
The owner of this deed
splits the winnings
when another player gets a multiple of:
(prime number here)

Game: Roll two dice and can choose the amount of either number formed by those two digits. Can buy prime numbers. Then whenever another player chooses a number that is a multiple of your number, they split the $ with you.

Goal: Get as much money as possible (there will be times when it will be to your advantage to choose the smaller number). Play for a set amount of time, clearly stated in advance.

How did I work out the pricing?
What’s the best prime real estate buy?

Mary O'Keeffe Quotes

“Math can and should be a social experience.”
“Get people comfortable with making mistakes and taking intellectual risks in public.”
“Give people powerfully compelling activities they will want to share with others, who will want to share them with others, etc.”
“Best way to learn is to share things with others.”
“I like to blurt out things when I’m not at all sure that I’m right.”
“We respect respect and we correct mistakes.”

Mary O'Keeffe Book Recommendations

About Teaching Mathematics by Marilyn Burns
Aha! Gotcha by Martin Gardner
Aha! Insight by Martin Gardner
The Annotated Alice by Martin Gardner
Feynman and Asimov
Melisande by E. Nesbit
Mathematical People (interviews)
More Mathematical People (interviews)
Raymond Smullyan logic stories
Calculus By and For Young People by Don Cohen http://www.mathman.biz/

Tuesday, July 28, 2009

David Hankin Questions, many of them good & big ones

• What is geometry?
• When we study geometric figures, what are we concerned with?
• When we study geometric figures, what are we not concerned with?
• What kinds of motion (transformation) keep a figure “the same”?
• What is the maximum number of these motions (isometries) are necessary to get from one figure to another if they have the same orientation? If they have different orientation?
• What minds of motion change a figure?
• What is an angle?
• Given a diagram, what do you think might be true? (rather than given a diagram and some true things, prove this other thing)
• What is symmetry?
• Why do we study almost exclusively symmetric figures in geometry?
• Is a line segment symmetric?
• Is an angle symmetric?
• Are triangles symmetric?
• What can we say about the points on a perpendicular bisector?
• What happens when we find the perpendicular bisectors of the sides of a triangle?
• When is the circum center ON (/in/out) a triangle?
• What happens when we find the angle bisectors of a triangle?
• How many lines of symmetry does a quadrilateral have?
• Is it possible for a figure to have more than four lines of symmetry?
• How many lines of symmetry does a circle have?
• How do I draw a symmetry line through a circle?
• What happens when we draw the diameter through a point on a tangent line?
• What happens when we draw perpendicular chords through a diameter?
• What happens when we draw progressively smaller and smaller perpendicular chords?
• Problem: Given a triangle and side lengths, find the lengths of the segments formed by the incircles.
• Problem: Given a quadrilateral and side lengths, find the lengths of the segments formed by the incircles.
• Must a quadrilateral have a circumcircle?
• Reflection Problem: Given two points A & B, and a line (below them), what is the shortest path from point A to the line and then to B?
• Reflection Problem: In pool/minigolf, where do you aim so that A bounces once (twice, 3 times, 4 times back to A!) before hitting B? When are these possible/not?
• What has translational symmetry?
• Translation Problem: Two towns on opposite sides of a 1 mile wide river. Where should we put the bridge (to minimize the distance)? What if there are two nonparallel rivers?
• What letters are preserved under 180 rotation (/half turn/point reflection)?
• What geometric figures are preserved under point reflection? (which have point symmetry?)
• What geometric property follows from the fact that the letter Z has point symmetry?
• Point Reflection Problem: two circles intersect. Draw through the point of intersection a line which creates congruent chords in each circle.

David Hankin Quotes

1. Majority rules in math right?
2. Did we prove it or did we just all agree?
3. Math is not about computation, but about proving things.
4. When something works well in math, you should milk it.
5. Geometry is something dynamic, not a bunch of stationary objects. Transformations help us see this. If you’re good at geometry, chances are you moved stuff around in your head.