“I can teach them more math than they have done in their entire lives”

Here are two conversations for which I have no patience at all…

  1. “They should know X”
  2. “You should teach X in manner Y.”

Conversation number 1 is the reason they are students, and the reason you are a teacher. What they should know is not relevant here; only what they do know is. So let’s factor that into our instruction.

But I need to focus on conversation number 2 today. EdSurge reposted an article today that struck me the wrong way. An excerpt:

Of course, the problem is deeper than a handful of students who accidentally say ironically stupid things. The problem is that American high school students are taught something named “math” for four years which is not even close to math.

Pretty sweeping generalization here. But I don’t disagree with the basic premise, which is that we aren’t doing the job of bringing mathematics to students (and students to mathematics) that we should be doing. I do disagree that the K-12 system is the only place this problem exists, but let’s get back to the matter at hand.

I fear my rant may disguise my true intentions: the problem is not the content. Geometry and calculus and algebra are very fine subjects of mathematics. The problem is that they’re taught in a way that strips out all the math and leaves a vapid husk of an education.

Now things are starting to spin a little bit out of control. Vapid husk of an education? Wow.

And the solution?

[I]f you give me an hour with a group of disillusioned but otherwise motivated high school students, I can teach them more mathematics than they have ever done in their entire lives. I can give them a dose of critical thinking and problem solving like no algebra problem can.

Child, please.

I teach at the college level these days, so I am accustomed to this sort of bravado. I try (perhaps unsuccessfully) to avoid it in my own writing because it is (a) unproductive, and (b) false.

But my beef isn’t so much with the author (although…) No, my beef is with EdSurge.

Why not feature the vibrant work that is going on in K—12 math education?

Why not republish Fawn Nguyen’s brilliant reformulation of a crappy textbook problem?

Why not post Andy Schwen’s video of a kid talking about the relationship between slope and rate of change while working on Function Carnival?


Why not feature the work of people trying to bring real mathematics to young children? Moebius Noodles, Math in Your Feet, Talking Math with Your Kids, Math Munch—these are projects where people are working on a daily basis to help parents, teachers and caregivers to support meaningful mathematical thinking for children. No bravado. No blame. Just hard working, thoughtful people working to solve a problem.

Because there is a problem. For sure there is a problem.

But an hour with Professor Awesome isn’t going to solve it.


Geometry and language

Interesting conversation on Twitter today with Bryan Meyer, Denise Gaskins and Justin Lanier. It began with these tweets on my part, the result of grading some student work.

Things quickly got too nuanced for Twitter.

An example of something my students struggle with is answering a question such as, Is a square a rectangle?

This type of question asks about class inclusion. Is an element of a subset also an element of the larger set?

Many useful and interesting questions in geometry have to do with whether one class is a subset of another class. Do all isosceles triangles have a pair of congruent angles? Are all quadrilaterals formed by connecting midpoints of other quadrilaterals parallelograms? Are all Stacys concave?

I am trying to sort out the extent to which my students’ struggles with questions of this sort are linguistic, and the extent to which they are about struggles with the idea of class inclusion.

Justin suggested this wording, which I will investigate:

Is a square an example of a rectangle?

Or, more generally:

Is an X an example of a Y?

My suspicion is that this will be helpful for some students when asked in this direction. But I also suspect that asking it in the other direction will be problematic.

Is a rectangle an example of a square?

See, part of what I wonder about is whether class inclusion—and the fact that it doesn’t have to be symmetric—is at the heart of a particular kind of struggle in geometry, and whether this is also related to the ways students think about and use language.

I hope these three (and others) will weigh in here where we have more space to work than we do on Twitter. The ideas are really useful. If you’d like to follow the prior discussion, you can follow this link.

Twitter Math Camp

I want to use this space to make a pitch for a conference session.

See, there is this thing called Twitter Math Camp. It is professional development by teachers, for teachers—nearly all of us connected through Twitter. It takes place this summer near Tulsa, OK.

I am presenting with Malke Rosenfeld. Our official description is copied below.

Malke and I have developed a really productive collaboration this year. You can browse both of our blogs to see the kinds of questions and learning this collaboration has developed for each of us.

Here is my pitch for our session…

We are planning a session that will force our groups (including ourselves) to wonder about the origins of mathematical knowledge. We will question our assumptions about terms such as concrete, hands-on and kinesthetic.

We will participate in mathematical activity both familiar and strange—all in the service of better understanding the relationship between the physical world and our mathematical minds.

We will dance.

We will make math.

We will laugh and possibly cry.

Below is an example of Malke’s work. When I participated in a workshop last summer, my head was spinning with math questions as a result. It’s great stuff and we will use it as a launching point for inquiry into our own classroom teaching.

So if you’re coming to Tulsa, please consider joining us for our three 2-hour morning sessions.

Of course you’ll miss out on other great people doing other great sessions. But you won’t regret it. I promise.

And if you choose a different session (perhaps because you’re leading one of them!), I have a hunch there will be after hours percussive dancing in public spaces. Come join in!

Our session description

This workshop is for anyone who uses, or is considering using, physical objects in math instruction at any grade level.  This three-part session asks participants to actively engage with the following questions:

  1. What role(s) do manipulatives play in learning mathematics?

  2. What role does the body play in learning mathematics?

  3. What does it mean to use manipulatives in a meaningful way? and

  4. “How can we tell whether we are doing so?”

In the first session, we will pose these questions and brainstorm some initial answers as a way to frame the work ahead. Participants will then experience a ‘disruption of scale’ moving away from the more familiar activity of small hand-based tasks and toward the use of the whole body in math learning.  At the base of this inquiry are the core lessons of the Math in Your Feet program.

In the second and third sessions, participants will engage with more familiar tasks using traditional math manipulatives. Each task will be chosen to highlight useful similarities and contrasts with the Math in Your Feet work, and to raise important questions about the assumptions we hold when we do “hands on” work in math classes.

The products of these sessions will be a more mindful approach to selecting manipulatives, a new appreciation for the body’s role in math learning, clearer shared language regarding “hands-on” inquiry for use in our professional relationships and activities, and public displays to engage other TMC attendees in the conversation.


A little gift from Desmos

Last summer, the super-smart, super-creative team at Desmos (in partnership with Dan Meyer, who may or may not be one of the Desmos elves) released a lovely lesson titled “Penny Circle“. It’s great stuff and you should play around with it if you haven’t already.

The structure of that activity, the graphic design, the idea that a teacher dashboard can give rich and interesting information about student thinking (not just red/yellow/green based on answers to multiple choice questions)—all of it lovely.

And—in my usual style—I had a few smaller critiques.

What sometimes happens when smart, creative people hear constructive critiques is they invite the authors of the critique to contribute.

Sometimes this is referred to as Put your money where your mouth is. So late last fall, I was invited to do this very thing.

I have been working with Team Desmos and Dan Meyer on Function Carnival. Today we release it to the world. Click through for some awesome graphing fun!


It was a ton of fun to make. I was delighted to have the opportunity to offer my sharp eye for pedagogy and task design, and to argue over the finer details of these with creative and talented folks.

Go play with it.

Then let us know what we got right and what we got wrong (comments, twitter, About/Contact page).

Because I just might get the chance to work on the next cool thing they’re gonna build.

A few quick words on a function context

Geoff Krall did us all the favor of preserving a brief Twitter conversation about a lovely applications of functions example found by Taylor Belcher.

Go have a look. Won’t take you long.

An interesting story about research and assumptions

Nature v. nurture. Age-old debate on relative importance. Not gonna settle it here. Not even in the limited context of factors influencing mathematics success.

There is lots of interesting research going on, of course. I want to tell you a quick story about a very small subset of that research.

A few years back, a group of educational psychology researchers published a study that phys.org headlined, “Math ability is inborn“.

The study investigated the ability of 4-year olds to choose the larger of two sets of dots when these sets were viewed briefly (too briefly to allow for counting).

They found that children who were better at this task also knew more about numeration and counting.

A quote from one of the researchers, Melissa Libertus:

“Previous studies testing older children left open the possibility that differences in instructional experience is what caused the difference in their number sense; in other words, that some children tested in middle or high school looked like they had better number sense simply because they had had better math instruction. Unlike those studies, this one shows that the link between ‘number sense’ and math ability is already present before the beginning of formal math instruction.”

Read more at: http://phys.org/news/2011-08-math-ability-inborn.html#jCp

Let’s pause for a moment to think, shall we?

If a child has not had formal instruction in mathematics, is the only remaining possibility that her mathematical performance is due to innate skill?

Of course it isn’t.

There is also the possibility that the child has absorbed some mathematical knowledge from her environment, and that different environments might provide differential input.

Maybe the child who is better at discerning the larger set has more practice doing just that. Maybe that child’s parents have been asking her how many? how much? and which is more? for the last two or three years.

Maybe that child’s parents have been Talking Math with Their Kids.

Sunshine shenanigans

If you need context for the following, go ahead and search “Sunshine awards”. Or just read along or skip to the next blog in your reader. It makes no matter to me.

Facts about me:

  1. My inbox is where chain letters go to die. I never forward them on.  I do not “Like” photographs that have charming children (or puppies) holding signs asking for 1 million likes. I will not apologize for this.
  2. I make a mean beer can chicken.
  3. The wings on these chickens pretty much never make it to the table. Seriously, have you ever eaten the wings off a well made beer can chicken right off the grill? (Sorry, vegetarians—much love, but this is about me.)
  4. My family never did nicknames growing up (OK, I can think of two exceptions—my mom called my sister Pumpkin and everyone called me Keefer). My wife’s family is rich with them. Our married life has taken on her family’s tradition. Among the nicknames in our house are these (very small sampling—you can have fun at home guessing which applies to whom): Dog, bird, pigeon, Boo, hompish, EP, LP, hound, rabitsu.
  5. If there is no urinal available, I prefer to sit.
  6. If your band has an accordion, I will gladly come hear you play. I cannot explain this.
  7. I am a daily newspaper reader. Paper copy. Electronic is no substitute.
  8. I am a huge introvert. Not shy, but being among lots of people wears me out. As a consequence, I relish my quiet down time at home.
  9. You will need to tear my Mac from my cold dead hands. There is no other technology about which I feel so passionately. My laptop and I are a team. We need each other to get stuff done.
  10. I am a much better person for having met my wife. Rachel is in very important ways my polar opposite and I have learned a lot from her about empathy and humanity. We also, of course, share essential core values.
  11. I make pickles. House specialty is a half-sour dill, which is fermented for about a week and doesn’t keep more than another week or two. The result is that they are seasonal. And seriously delicious.

I got nominated three times for this silliness. Fortunately, only two of these involved questions. I am compiling the master list of 22. Copy-and-paste, and hope there is some overlap. Here goes…

  1. Why do you teach?
    I am fascinated with how people’s minds work. Trying to think like someone else—to see the world from their perspective—is endlessly interesting to me. Mathematical thinking is where I am most skilled at this. Teaching exercises this skill.
  2. If you didn’t teach, what would you do for a living instead?
    I don’t know. I could probably be happy somewhere in the food industry. It would have to be somewhere that allowed for creativity and problem solving.
  3. Money being no obstacle, where would you like to visit? Why?
    I want to go back to Japan. Rachel and I visited in 2002 and I found Tokyo amazing.
  4. Kids always ask who your favorite student is.  Describe the characteristics of yours
    I love the ones who are trying their best to grow. The ones who are satisfied with their present selves frustrate me. The growth they seek does not need to be mathematical, but it needs to be visible in our teacher-student relationship somewhere.
  5. What is your favorite board game and why?
    Chess. I am not that good and I do not have much opportunity to play. But the complexity that arises from a simple set of rules is beautiful. As is the fact that the game is about ideas. If you do not play chess, this probably makes no sense to you. Sorry.
  6. What is the most frustrating component of education right now?
    That U.S. teachers are increasingly put in corners where they feel (rightly or wrongly) that theirs is not a creative profession, and that they have limited autonomy to make important classroom and curricular decisions.
  7. Would you rather buy a Mac or a PC?
    See fact 9 above.
  8. What is your favorite book?
    Can’t pick one favorite of all time. Recently I read Children’s Minds by Margaret Donaldson which was amazing for where I am in my own work and thinking right now. That’s my favorite recent read.
  9. If you had to choose blogging with no way to share it (ex. via twitter) or tweeting with no way to elaborate (ex. via a blog), which would you choose?
    Blogging with no way to share. For sure. I blogged for two years before finding my math nerd friends on Twitter. I have too much to say, and I work out my ideas by saying it. I have to write.
  10. Who is your hero?  Why?
    For me, hero suggests a lack of faults. We are all too complex for that which is why comic books exist. I wrote about important mentors for me in life and work a couple years back. Those people are still tops.
  11. What is the most exciting part about your job?
    The moments of engagement with students’ ideas. Those moments when ideas are on public view and the classroom community is considering them, changing them and adopting them. I live for those moments. They are more frequent for me the longer I teach and that feels like a reasonable measure of success.
  12. If you had to pick one area/concept of math that is your “jam”, what would it be?
    Fractions. Next question?
  13. To quote Rodney (Chris Rock) from Dr. Doolittle, “You can’t save them all, Hasselhoff.” True, but there’s at least one student that sticks out in my mind that I feel I failed. Do you have one? 
    Yes. Joe. My last year in the classroom. He needed more weaning from the teacher as answer key than I gave him, and this led to him shutting down too often. I needed a more nuanced approach with him and I didn’t realize it until too late.
  14. Twenty years from now, what’s something kids will probably remember about you (phrase, moment, habit, characteristic, etc.)?
    That I made them think.
  15. I nominated you because I think you’re great, but I know we are all our own worst critics. What’s something that you’d like to “fix” about yourself in your current job?
    Timeliness in responding to student written work. I am working on this. Part of it is straight-up self-discipline. Another part is being proactive about when and what I collect, and about what kinds of feedback I promise. I am working on both parts.
  16. Name a movie title that describes you and why.
    Definitely not Stand and Deliver. I will choose from the movies currently playing at my local multiplex (and I will avoid the easy Frozen!)…
    American Hustle. That was fun. There were some funny options. Why American Hustle? I am always hustling in the classroom. Coaxing, marketing, anything to get those minds open.
  17. I love TMC because at night I can hang out with my favorite tweeps over a beer or two (or eight). Which tweep would you love to have a conversation with over a beverage?
    I have been blessed with opportunities to do this many times in the last couple of years. I’ll pick a couple who I haven’t had the chance with yet. I would like to brainstorm Would You Rathers with John Stevens. I would love to talk fraction learning with Nicora Placa. I have shared a beverage, but not a real conversation with Fawn Nguyen; that needs to change. I haven’t met Andrew Stadel yet. This also needs to change. Jason Buell, Jose Vilson…all of these are people I would love to talk with over coffee or beer, and have not yet.
    Also, as a heads up: I accept all such invitations I can make time for. Invariably these invitations lead to me being described as “not as strange as I expected”. I am OK with this.
  18. If you couldn’t teach your specific subject, what else would you teach?
    Kindergarten. Can I still teach math as part of the day?
  19. Everybody has a song they car dance/jam out to. What’s yours?
    Dig the rhythms and the horns here. For me the “lovers” are my work commitments, which frequently become too numerous.
  20. TMC13 enlightened me on karaoke night. A few people completely blew my mind (I’m lookin’ at you, Pershan). Who would you love to see karaoke at TMC14 and why?
    I want an encore from Karim Ani and Eli Luberoff.
  21. What’s one thing (item, app, software, etc.) that you love so much that you can’t imagine doing your job without it?
    I mentioned my MacBook earlier, right? Seriously. When I moved from MSU, Mankato to Normandale, I asked for a MacBook and was told no. So I bought my own.
  22. If you could job shadow one tweep for a week, who would it be and why?
    Sadie Estrella. I need to get out of this cold.

Now. You remember how I said chain letters go to my Inbox to die? I can’t do the 11 nominations and 11 more questions.

I will thank Ms. Hedgepeth, Mr. Stevens and Ms. First (sorry, I tried to find a name on your blog) for the nominations. Thank you!