Category Archives: Reflection

The Twin Cities Shapes Tour

I recently put out a call for K—2 classrooms in which I could talk shapes with students. As a result each of the next several Mondays (Presidents’ Day excluded), I will be in a different early elementary classroom somewhere in the Minneapolis/St Paul metro area.

Last week I was at two schools: Dowling in Minneapolis and Echo Park in Burnville. I talked with one kindergarten class, three first grade classes and four second grade classes. I have learned a lot.

In particular…

Young children find composing and decomposing shapes to be much more compelling than adults tend to. They nearly all saw the bottom-right figure here as being a square and four circles. Adults can see that, of course, but we are more likely to think “not a polygon”.


On that note, I am now quite certain that we spend way too much time having young children sort polygons from non-polygons. That bottom-right shape has many more interesting properties than that of not being a polygon.

For example, a class of second graders on Friday were variously split on the number of “corners” that shape has. Is it 0, 4 or 8? Second graders can understand each other’s arguments for and against these possibilities.

These arguments can lead to the reason that mathematicians use vertex instead of corner. “What exactly is a vertex?” is a much richer and meatier mathematical question than “How many vertices does this shape have?” But if that latter question only comes up with respect to convex polygons, then it is unproblematic and not interesting for very long.

So imagine for just a moment that the lower-right figure has 8 vertices (and it wouldn’t be too difficult, I now believe, to get a classroom full of second graders to agree to this perspective, whether it agrees with the textbook definition of vertex or not).

Now kids can work on stating exactly what makes a vertex.

And what makes a vertex is going to be awfully close to what makes a point of non-differentiability (large point at apex of figure below).

Screen Shot 2015-02-08 at 4.35.50 PM

I’m telling you: in twenty minutes with second graders, we can get very close to investigating things that are challenging for calculus students to describe. My point is that second graders are ready to do some real mathematics, and that sorting polygons from non-polygons is not the road to it.

Other things I found interesting:

• When kids give us something close to the answer we expect, it is easy to fool ourselves into thinking they understand. Example: on the page below, one boy said about the lower left shape that “if you tip your head, it’s a square.” A couple minutes later, it occurred to me that there might be more to the story. I asked whether the shape is a square when your head isn’t tipped, or whether it only becomes a square when you tip your head. He confirmed that it’s the latter.

2• Another second grade class was unanimous that the one in the lower right doesn’t belong because it’s not a square. When I asked “is the lower left now a square, or does it only become a square when you tip it?” the class was evenly split. This was surprising to both me and the classroom teacher.

• Diamondness is entirely dependent on orientation in the mind of a K—2 student.

• The 1:1 correspondence of sides of sides to vertices in polygons is not at all obvious to young children. I sort of knew this but saw it come up again and again in our work.

• A first grader said that the spirals below didn’t belong with all the other shapes we had seen that day because “you can’t color them in”.


Even the unshaded ones that had come before could have been colored in, you see. These spirals you cannot color in even if you try. What a brilliant and intuitive way into talking about closed figures—those that can be colored in.


Standard algorithms unteach place value

I found a page full of computations sitting around the house this evening. Naturally, I picked it up and gave it a look.

Griffin (10 years old, 5th grade) had been doing some multiplication in class today. Somehow his scratch paper ended up on our couch.

Here is one thing I saw.

37 times 22 with the standard algorithm. Wrong answer: 202.

Naturally I wanted to ask the boy about it. He consented.

Me: I see you were multiplying 37 by 22 here.

Griffin (10 years old): Yeah. But I got it wrong so I did it again with the lattice.

Me: How did you know you got it wrong?

G: I put it in the answer box and it was wrong.

It turns out they were doing some online exercises. There is an electronic scratchpad, which he found awkward to use with a mouse (duh), plus his teacher wanted to be able to see their work, so was encouraging paper and pencil work anyway.

I was really hoping he would say that 37 times 22 has to be a lot bigger than 202. Alas he did not.

Anyway, back to the conversation.

Me: OK. Now 37 times 2 isn’t 101. But let’s imagine that’s right for now. We’ll come back to that.

G: Wait. That’s supposed to be 37 times 2? I though you just multiplied that by that, and that by that.

He indicated 7 times 2, and then 3 times the same 2 as he spoke.

Me: Yes. But when you do that, you’ll get the same thing as 37 times 2.

A brief moment of silence hung between us.

Me: What is 37 times 2?

G: Well….74.

Let us pause to reflect here.

This boy can think about numbers. He got 37 times 2 faster in his head than I would have with pencil and paper. But when he uses the standard algorithm that all goes out the window in favor of the steps.


The steps trump thinking. The steps trump number sense.

The steps triumph over all.

Back to the conversation.

Me: Yes. 74. Good. I like that you thought that out. Let’s go back to imagining that 101 is right for a moment. Then the next thing you did was multiply 37 by this 2, right?

I gestured to the 2 in the tens place.

G: Yes.

Me: But that’s not really a 2.

G: Oh. Yeah.

Me: That’s a 20. Two tens.

G: Yeah.

Me: So it would be 101 tens.

G: Yeah.

I know this reads like I was dragging him through the line of reasoning, but I assure you that this is ground he knows well. I leading him along a well known path that he didn’t realize he was on, not dragging him trailing behind me through new territory. We had other things to discuss. Bedtime was approaching. We needed to move on.

Me: Now. We both know that 37 times 2 isn’t 101. Let’s look at how that goes. You multiplied 7 by 2, right?

G: Yup. That’s 14.

Me: So you write the 4 and carry the 1.

G: That’s what I did.

Me: mmmm?

G: Oh. I wrote the one

Me: and carried the 4. Yeah. If you had done it the other way around, you’d have the 4 there [indicating the units place], and then 3 times 2 plus 1.

G: Seven.

Me: Yeah. So there’s your 74.

This place value error was consistent in his work on this page.

Let me be clear: this error will be easy to fix. I have no fears that my boy will be unable to multiply in his adolescence or adult life. Indeed, once he knew that he had wrong answers (because the computer told him so), he went back to his favorite algorithm—the lattice—and got correct answers.

I am not worried about this boy. He is and he will be fine.

But I want to point out…I need to point out that this is exactly the outcome you should expect when you go about teaching standard algorithms.

If you wonder why your kids (whether your offspring, your students, or both) are not thinking about the math they are doing, it is because the algorithms we (you) teach them are designed so that people do not have to think. That is why they are efficient.

If you want kids who get right answers without thinking, then go ahead and keep focusing on those steps. Griffin gets right answer with the lattice algorithm, and I have every confidence that I can train him to get right answers with the standard algorithm too.

But we should not kid ourselves that we are teaching mathematical thinking along the way. Griffin turned off part of his brain (the part that gets 37 times 2 quickly) in order to follow a set of steps that didn’t make sense to him.

And we shouldn’t kid ourselves that this is only an issue in the elementary grades when kids are learning arithmetic.

Algebra. The quadratic formula is an algorithm. Every algebra student memorizes it. How it relates to inverses, though? Totally obfuscated. See, we don’t have kids find inverses of quadratics because those inverses are not functions; they are relations. If we did have kids find inverses of quadratics, they could think about the relationship between the quadratic formula:

x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}

and the formula for the inverse relation of the general form of a quadratic:

y=\frac{-b \pm \sqrt{b^2-4ac+4ax}}{2a}

Calculus. So many formulas (algorithms) that force students not to think about the underlying relationships. If we wanted students to really think about rates of change (which are what Calculus is really about), we might have them develop a theory of secant lines and finite differences before we do limits and tangent lines. We might have Calculus students do tasks such as Sweet Tooth from Mathalicious (free throughout October!). There, students think about marginal enjoyment and total enjoyment.

On and on.

This is pervasive in mathematics teaching.

The results are mistaken for the content.

So we teach kids to get results. And we inadvertently teach them not to use what they know about the content—not to look for new things to know. Not to question or wonder or connect.

I’m telling you, though, that it doesn’t have to be this way.

Consider the case of Talking Math with Your Kids. There we have reports from around the country of parents and children talking about the ideas of mathematics, not the procedures.

Consider the case of Kristin (@MathMinds on Twitter), a fifth grade teacher, and her student “Billy”. Billy made an unusual claim about even and odd numbers. She followed up, she shared, we discussed on Twitter. Pretty soon, teachers around the country were engaged in thinking about whether Billy would call 3.0 even or odd.

But standard algorithms don’t teach any of that. They teach children to get answers. They teach children not to think.

I have read about it. I have thought about it. And tonight I saw it in my very own home.

“Not all white people”

I have a very modest goal for (me and) my white colleagues:

To be able to read something like José Luis Vilson’s recent post, or Mia McKenzie’s recent post, without feeling defensive.

A modest goal, for sure. But a necessary one, and one that will allow us to move forward.

Each of these posts is by a Black (/Latino) writer, with teachers as (at least) part of the intended audience, and each calls out racism in schools. (And sexism—for which I have an equivalent goal for my male colleagues—it shouldn’t be hard to reread this post replacing race with gender wherever it appears.)

When white people read this writing, there is an instinctive reaction that begins and ends with Not all white people. That is the defensive response I hope we can do away with.

Here’s the problem with that response: Racism is not about white people’s understanding of the nuances and varieties of white people. It is about the lived experience of people of color.

“Not all white people” is a racist response.

“Not all white people” denies the experience of the writer.

“Not all white people” cuts off further conversation about race.

This leads me to a second claim.

Refusing to discuss race is a racist act.

There is a certain brand of white liberalism, for example, that believes noticing race to be a racist act. This view makes it impossible to talk about race.

In such a climate, asking a colleague what he knows about Somali culture in a quest to better understand a classroom incident is called into question as an act of racism because some white people engage in the same behaviors, and therefore there should be nothing to ask about. In such a climate we cannot speak of the vastly differential racial demographics of developmental math courses and College Algebra courses at the college level. To do so is seen as racist. Because—after all—we give the same placement tests to everybody.

Now a question for my white colleagues: Why is “racist” that rare varitey of action that we allow the power to define us?

We can live with duality in other areas of our lives: I did/said a ___ thing, but this does not make me a ___ person.

I have done many stupid things in my life, and I accept the potential for doing more stupid things in the future. Yet I am not a stupid person. I am comfortable owning that something I did was stupid. I can wish that I hadn’t done that stupid thing. But I don’t let the stupid thing define me.

Furthermore, it is OK to talk about how stupid something I did was, and the goal in talking about it is to ensure that I don’t do something that stupid again—or at least to eliminate this particular brand of stupidity from my repertoire.

But we treat racism differently. We pretend that only racists do racist things. (Again, do only stupid people do stupid things?) Therefore, we cannot own our racist actions. If we admit that we have done, thought or said something racist, we become racists.

This mindset—this inability to speak of our racist actions; to name them (even the inadvertent ones) as racist—keeps us from being able to talk about our mistaken ideas and actions. But talking about them would help us to avoid perpetuating and repeating them.

You don’t need to own the racism of your fellow white people. You don’t need to identify as a racist because someone else has done something racist, nor even because you have.


You need to (I need to) honor the experiences of others. When a racist incident is brought to your attention, you need not to explain that “not all white people…” or that you have not experienced this. Doing so puts the focus back on you as a white person (which, again, is a racist act; and which, again, you—I—can own as an act without needing to own the title racist).

See, you don’t need to explain the experience of others away. Instead you need to listen. You need to acknowledge that racist acts are committed in the world, and that our goal is to reduce and ultimately to eliminate their incidence. Pretending—through denial or through silence—that racist acts do not exist is itself a racist act. Pretending—through denial or through silence—that racist acts have no relevance is a racist act. Pretending that racist acts can only be committed by people who are racists through and through—this is not an effective means to the end.

I understand that my goal is modest: Reading accounts of racism, written by people of color, without becoming defensive. But we have ample empirical evidence that the goal has not yet been attained, and it is clear to me that moving forward to really dealing with racism is impossible in its face.

Achieving this goal allows us to listen.

And listening—to our own hearts, and to the hearts and experiences of others—is where learning begins.

Teacher Appreciation Week

I am grateful for Ronald Webb, my English teacher at Dearborn High School.

He taught me to write.

I didn’t really have anything to say yet. But I learned grammar, structure, passion, and the value of just getting words on the page from Mr. Webb.

I draw on those skills in everything I do professionally; whether it is curriculum writing, blogging or conversing.

My words flow more easily. My ideas are more clear. My thinking is better. I owe these things to him.

Thank you for that, Mr. Webb.

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.

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!

Where does math come from?

As math teachers, we need to stay vigilant about how we represent our subject to our students.

At her recent workshop for teaching artists here in the Twin Cities, Malke Rosenfeld said to the gathered group,

It would be fair to say that most of think about math inside a textbook context.

She paused.

Heads nodded, eyes wide with recognition. Malke prepared to demonstrate that this is not the full story.

Math comes from, and lives within, textbooks. I am not OK with this.

So what can we do in every lesson every day to represent mathematics as a subject that comes from, and lives within, the minds (and bodies) of our students?