Category Archives: Reflection

“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 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?

Reflections on teaching

I am working on a ton of interesting projects right now. Not least of these is my classroom teaching at the community college. My fingertips are sore from typing.

And yet there is always more to say. More to think about. More conversations to have. Here is a peek into one that is ongoing.

Malke Rosenfeld and I have been going back and forth about math, dance, Papert and learning for a few months now. I am learning a lot from the conversation. She asked some questions this morning.

Malke: A thought just entered my head — why are you offering TDI? Is it based on a question you are unsure of and want to see what others think? Or are you seeing a deficit in math teachers’ thinking that you want to shore up?

Me: When ranting on Twitter, I could see that some of my assumptions about baseline teacher knowledge about fraction/decimal relationships as they pertain to developing children’s thinking were unfounded. That is, I was assuming teachers knew a lot more than they seemed to. Which has implications for my Khan Academy critiques, and lots of other writing on my blog. Yet people were also curious. So I wanted to say more in a way that would draw from and build on a larger collective knowledge, so it’s not just my spouting off.

Malke: Is there a reason you offered it specifically as a course, and not a moderated discussion (which it sort of seems like right now)?

Me: When you view learning as a social process, you tend to think of courses AS moderated discussions. I mean this quite seriously. I know that it goes against the grain of online (and face-to-face) course design. But that’s not because I think of online instruction differently from others; it’s because I have a particular view of learning that runs much deeper than that. If I tell and quiz, you’re not learning very much. If I propose a set of ideas, listen to what you have to say, encourage you to interact with others and move the conversation in directions that seem useful based on those interactions, you’re probably going to learn a lot.

As long as I can keep you engaged in that process. Which is a different challenge online than in the classroom.

Malke: Is there a place you specifically want your students to get to by the end of the seven weeks?  Or are you just curious to see what develops?

Me:  I am curious to learn what I can about teaching at every opportunity. I want to produce “students” who can articulate important questions (see? learning as having new questions to ask?) about curricular approaches to decimals. Ideally, I would help them to develop a critical voice that speaks to/through them when they work with individual students, when they plan lessons and when they talk with their colleagues in a variety of settings. In short, I want to change the way teachers view the territory of decimals, fractions and children’s minds. Strange mix of lofty and specific there, huh?

Another great question from College Algebra

Here is something cool that happened in College Algebra today. We were doing a short thing to summarize our domain and range work before moving on.

A student asked, Is the only way to find range to make a graph?

This stopped me in my tracks. I had not really thought about the knowledge I draw on when identifying the range of a function, and the question cut to the heart of the matter.

My gut instinct answer was yes. But I wanted to explore that a little. I concocted a silly function to do so. \sqrt[3]{x^{5}+x^{2}}+x^{2}-sin(x). I wanted to say that I would need to graph that to know its range.

But the longer I looked at it, the more clear it was that I knew a lot about this silly thing without graphing it. The x^{2} term dominates, for instance, in the long run, so I know it goes to infinity on both sides of the y-axis. I could see that 0 is in both the domain and the range.

But I wasn’t 100% sure whether there were any negative values for the function.

Later in the day, this got me thinking about end behavior. This is why we teach that end behavior silliness, right? It’s not about end behavior, it’s about knowing what values can come out of a function, and having a basis for knowing this.

I am brainstorming here. The point is that the student question showed a sign of her learning, and it pushed me to rethink something too. Win-win.


Another cool thing happened, too. We were comparing y= x^{2} and y=2^{x}, looking for sameness and difference. I had to push to get domain and range on the table.

We agreed that the two functions have the same domain—all real numbers. We were split on whether they have the same range.

But not for the reason I expected. Not at all.

A student argued that The only time when they are the same is when x=2. Therefore they do not have the same range.

My students found this argument compelling.

Ignore the second intersection point in the left half-plane. Focus on the essence of the argument.

Do these functions have the same range? is interpreted as Do these functions intersect?



That seems like a useful insight into the mind of a College Algebra student.