I took precalculus as a sophomore, which was was unusual in my unified school district but not very. Halfway through the year — over winter break to be exact — I borrowed a calculus textbook and taught myself the first semester. After break I joined up with a class of seniors and set their first semester final exam curve.

Until that point I had enjoyed math in much the same way I enjoyed hearing those chimes upon completing a level of Super Mario Bros. Precalculus was satisfying but calculus was engrossing. That textbook fed me like only my favorite novels have. I’m realizing today — now — how insidiously and completely calculus has infiltrated my teaching.

If you have never found joy in an indefinite integral, you’re pre-emptively forgiven for scratching your head over this post. I hope you’ll extend me the same courtesy when I say: you’re missing out.

Calculus, on some fundamental level, says that we can understand even the most complex systems if we cut them into thin enough slices and stare long enough at the pieces.A cylinder, for example, is just many many really thin circles stacked on top of each other. A slightly more complex example is an Egyptian pyramid which is just many many really thin squares stacked on top of each other, squares which get predictably smaller the higher up the pyramid you go. And so it is with teaching, which is, outside of romantic relationships, the most complex real-life system I have ever encountered.

It is so complex, in fact, that many teachers have given up on its inherent calculus and embraced it holistically, emphasizing care over technique, choosing to believe (sometimes passively) in a mystical force (sometimes referred to as “passion”) that endows some people with the ability to teach, when, in fact, it isn’t mystical. It’s calculus.

My recurring frustration with this flaccid line of inquiry and my persistent campaign to de-mystify this job are both derivations of my enthusiasm for calculus. I just can’t get into this idea that some people are “called” to teaching when I am struck in the face every day, every class period, by the obvious slices of our job.

To wit, if you slice a great teacher into a hundred parts you’ll find a(n):

  • clear writer
  • discriminating sartorialist
  • confident speaker
  • commanding leader
  • lucid thinker
  • pop-culture scholar
  • efficient people manager
  • savvy organizationalist
  • technical craftsman
  • entertainer
  • eloquent communicator
  • graphic designer
  • nurturing caregiver.That was kind of hard to write.
  • multi-disciplinary, lifelong learner
  • emotionally secure individual
  • discerning psychologist
  • imperturbable emergency responder[Update: Dig this one courtesy of Lori]

I’m wagering $20 with myself that the first commenter on this post comes by to point out a gross oversight, a glaring redundancy, or a superfluous inclusion in the list. True, we could debate the effect of matching socks and pressed pants on our students’ learning outcome but that would be missing the larger point here:

There is a list.

That was a quick brain-dump back there and, though I gladly welcome additions and exceptions in the comments, the crucial point here is that we can cut teaching into slices, that every one of those items positively affects our classroom in varying degrees, and, most importantly, that we can take practical steps to strengthen every one of them.

For example:

In my first year, I spoke too quickly. I was nervous. I realized I was dusting my ELL students by jabbering ceaselessly and so I worked at slowing down, at speaking with my pauses.

In my first year, I realized I often phrased my instructions as questions, tossing out weak mandates like “Would you guys please get down to work now?” because I wanted to be liked. I worked at balancing kindness and firmness (“I need you guys to work quietly now.”) so that we could work and learn more.

In my second year, I realized that the clothes I wore affected my students’ respect for and perception of their teacher, which directly affected how much time we had for learning.

Here, in my third year, I’ve realized that graphic design is somewhere near the most neglected slice of our profession, that our students — voracious consumers of magazines, professional web sites, and television promos, each and every one — have subconsciously come to expect better information presentation than we’ve been delivering. Every one of us realizes they’re understimulated. Few know how tightly it hinges on graphic design. Fewer still know what to do about it. But graphic design is nothing special — just another slice. We can work on it.

There is nothing mystical or mysterious about teaching. There is only the calculus. There are only the questions.

What are the slices?

How can I improve them?

We have two options as I see them. Any rational individual will find this list overwhelming, even in such an incomplete form. I sure do, and my options then fork.

I can dismiss the list — as well as any idea of a list — and go about my business holistically as usual, keeping my motives pure and my mind open, relying on an emotional trickle-down effect to further my teaching, trusting on the goodness and purity of my intentions to push my professional development. This isn’t a terrible or uncommon option.

Or: I can admit right now that success in every slice is an impossible feat, a fact which can be said of very few professions. I choose to revel in that fact, to let that fact keep me bolted to this job in the moments when I start glancing at grad school brochures. Because in teaching, like few other jobs, I know I will never be bored.

Can we talk about the slices now?

I'm Dan and this is my blog. I'm a former high school teacher, former graduate student, and current head of teaching at Desmos. More here.


  1. Scott Elias

    March 19, 2007 - 9:25 am

    Sorry to cost you $20, Dan, but I think your list is pretty darn good. I especially liked “pop culture scholar” and “savvy organizationalist.” I think if more teachers took an interest (or pretended to take an interest!) in what kids are into these days, it would be an eye-opening experience to say the least.

    As a former math teacher who DOES see the beauty in an indefinite integral, your Calculus analogy is quite insightful.

    Thanks and have a great week!

    — Scott

  2. Dan Bowling

    March 19, 2007 - 10:48 am

    This is the first time I visited your website, the first article I read, and quite possibly the first page of writing that truly hit a chord with me about teaching in quite some time. Bravo.

    I can just imagine an administrator saying those same words at an in-service to introduce the days work of developing the list and creating simple actions each educator can take to improve on a few of them in the next week.

  3. dan

    March 19, 2007 - 12:01 pm

    Of course there’s a weighting here, Scott. My superficial knowledge of which rap set is hot right now and which album they just dropped doesn’t affect student learning outcomes as much as, say, my savvy organizational skills. But you’re right: at the very least, it “helps.” At this point in my career, I don’t care how little the effect, I want to know everything that helps. For reasons of time, priority, and money (in the case of the “discriminating sartorialist” up there) I won’t be able to tackle all of them but at least it won’t be for obliviousness.

    Good to have you around, Dan. I fear most of my long-time readers have become exhausted by these manifesto-style postings, which drop near the rate of nineteen per week. The Most Read sidebar contains most of them.

  4. Scott Elias

    March 19, 2007 - 1:31 pm

    Agreed, Dan. I wasn’t thinking only about the new P-Diddy album (or is it just “Diddy” now?).

    What struck me – and what continues to strike me the more I read your blog – is how tightly the way you work and teach is aligned with the way kids play and learn OUTSIDE of school. Truly a model of best practices for new and pre-service teachers. I can only hope that I’m able to hire some with a fraction of your dedication and commitment.

    From reviewing some of your Keynote decks, I would venture to say that there are times where kids are not even aware that they’re “learning” anything at all. “We did some cool stuff in Mr. Meyer’s class today…” Your students are lucky to have you.

    — Scott

  5. dan

    March 19, 2007 - 3:21 pm

    Just got my pink slip, Scott, so, uh, be careful what you wish for and all. They’ve got the Pacific Ocean in Colorado, right?

  6. TMAO

    March 19, 2007 - 5:37 pm

    Screw Colorado. Cruise up the 101 until you hit 408 (Story Road East), and you can have your pick of 8th grade algebra classes.

  7. Lori Jablonski

    March 19, 2007 - 5:55 pm

    Wow, calculus is about staring at slices? Why didn’t anyone ever tell me? The whole trajectory of my life might have been different.

    Anyway, I know I’m too late to cost you 20, but I’ll add imperturbable emergency responder thanks to today’s gun-related school shut down (still unverified whether a gun has actually been recovered)—the third this year.

    I’m not sure I’ll ever master the savvy organizationalist, but feel I’ve made some strides this year as a graphic designer. Earlier this year, one of students called me hyphy, as the rest of the class nodded vigorously. I had to call my 22-year-old son, for guidance on that one. As long as I can get him quickly on the phone, I can somewhat feign the pop culture scholar thing.

    I’m sharing this with my student teacher tomorrow morning. We’ll read it together. This is the stuff!

    P.S. I’m hoping the pink slip is of the supremely annoying we will pink slip you in March and bring you back in the summer variety.

  8. dan

    March 19, 2007 - 7:10 pm

    Getting back to my Title I roots sounds awfully appealing and all, but what would that make my blog? Teaching in the 408 Part Deux? Teaching in the 409? Son of Teaching in the 408? Nah. Too too complicated. And by all appearances this is just a nuisance: real enough to annoy me, not real enough to worry me.

    And, Lori, dang. Google sez you teach at McClatchy, which, for the record, was the school all of us Florin High School teachers were scared of. And all of EGUSD was scared of us. Purple heart, girl.

  9. Scott Elias

    March 19, 2007 - 7:16 pm

    I grew up in Florida. The ocean is overrated. You need to take up skiing in a state where you can still afford to buy a house…


  10. dan

    March 19, 2007 - 7:19 pm

    Ha ha you must mean rent a house. Ha … “buy a house” … get this guy …

  11. Chris Lehmann

    March 19, 2007 - 8:12 pm

    Dude. Philadelphia.

    Our budget sucks. The murder rate is up. But SLA is a place where you will love the arguments you bring up.

  12. Lori Jablonski

    March 19, 2007 - 8:36 pm

    Hah. Florin scared of McClatchy? Maybe this ups my cred. Really, I wouldn’t want to teach anywhere else. Great kids, great staff. Beautiful WPA building and a strong tradition. The state says we are among a small handful of the most most racially AND socio-economically diverse schools in California. Plus I live four blocks away so I walk to work.

    I hate reading my previous posts…all the typos. I really need to learn to proof and edit. I certainly wasn’t comma confident in the last one.

    Have a good rest of the week.

  13. e

    March 20, 2007 - 3:44 am

    Dan, don’t get this comment wrong. You all may well be right trying to get Dan to move. He’s an exceptional writer. But what seems curious is that nobody seems to be interested in his math credentials? That’s what he’s supposed to teach, right?

  14. H.

    March 20, 2007 - 5:53 am

    If these be the slices of teaching – what would be the elements of a good credentialing program?

  15. Chris Lehmann

    March 20, 2007 - 9:45 am

    e — fortunately, I’ve had a few very long conversations about teaching and math and such… Let’s just say that if those were his interviews, I feel pretty confident in his ability to add a great deal to the SLA staff.

  16. dan

    March 20, 2007 - 4:35 pm

    Between the kind employment offers and e’s two-faced compliment lies H’s question, which has had my mind grinding at times (class) when my mind should’ve been a little more singleminded.

    So, after thinking about it, I think that a good credentailing school oughtta preoccupy itself with a specific several of those slices — particularly the efficient people manager, the eloquent communicator, and the discerning psychologist. I’m pretty sure those are the roles that define success and failure for a frosh teacher. It was that kind of training that kept me rocking through some very unhappy days a few years ago.

    At the same time, though, I’d host frequent inquiry into the other slices, on a more informal level. I’d try to inculcate the idea that really good teaching is just a strong show of various independent slices, that great teaching is their apparently effortless integration.

    To this end, I’d roll tape frequently. I’d play video of good teachers plying their trade and ask my preservice teachers to deconstruct their performance into slices. Because, see, it was only after I left my credentialing program that I figured out how to observe teachers and process my observations.

  17. Stephen Humphrey

    March 20, 2007 - 4:54 pm

    Dan, I don’t quite know what to make of your “pink slip” comment. Everyone so far seems to be taking it at face value, but when I try to do that, I’m only coming up with “WTF” (voiced loudly). I can only wish that my own children will have teachers as focused as you are at asking “what must I do to teach better?”

    I’ve been lurking here for only about a month. This is only my second comment. But the lucidity of your posts have made your site a daily of dose of insight for me, and they are largely responsible for pushing me off the fence post on which I’ve been resting while decided if I should try teaching secondary math. Last week, I submitted my paperwork to join my state’s Alternative Route to Licensure program using my 20-year-old electrical engineering degree.

    Please tell me this hiccup has more to do with your caustic attempts to unseat the status quo, and not with your abilities (and promise!) in the classroom. Otherwise, I’ll face my own “Career Crisis #1” before I even get a temporary credential.

    Dude, I know if I were you I’d be majorly depressed. Just know this middle aged math geek looks up to you as the kind of person I want to become.


  18. Eric Hoefler

    March 20, 2007 - 6:40 pm

    Wait, what? I missed something. Pink slip!?

    Blogs are not necessarily the proper places to discuss details of this nature, but … like, for serious?!

    Unless you finally caved and started dealing to supplement your income, I’m sure many of your readers would be happy to join a virtual march of protest …

  19. dan

    March 20, 2007 - 6:42 pm


    It’s awfully cool to see careerists stepping into teaching, particularly when I so frequently seduce myself with the reverse. You could do a lot worse than poking around the ed’blog’sphere for examples of fine methods. Glad for whatever part I could play in that.

    For whatever it’s worth to you (and Eric), this pink slip isn’t on account of any supervisory dissatisfaction in my work. (I know it’s worth a lot to me.) My district is under declining enrollment and I came to this job under extremely last-minute, temporary terms. Pink slips are how they play it safe.

    So as I wait for the opportunity to re-apply for my job, I’ve been corralling any stray admin and tastefully reminding them of my utility to the faculty. A parent e-mailed with some positive feedback yesterday and I mentioned that she could do my future at SLVHS a lot of good by forwarding her comments to our AP.

    Did I say tasteful? I meant tacky.

  20. mrc

    March 20, 2007 - 6:46 pm

    I like the metaphor: antiderivatives as a teacher-metric. Area-finding as professional development. Riemann sums of our skills. And a willingness to take the limit as n goes to infinity… while 1/n is the number of hours you sleep. Keep on rocking in the free world.

  21. dan

    March 20, 2007 - 6:54 pm


    It’s awfully cool to see careerists stepping into teaching, particularly when I so frequently seduce myself with the reverse. You could do a lot worse than poking around the ed’blog’sphere for examples of fine methods. Glad for whatever part I could play in that.

    For whatever it’s worth to you, this pink slip isn’t on account of any supervisory dissatisfaction in my work. (I know it’s worth a lot to me.) My district is under declining enrollment and I came to this job under extremely last-minute, temporary terms. Pink slips are how they play it safe.

    So as I wait for the opportunity to re-apply for my job, I’ve been corraling any stray admin and tastefully reminding them of my utility to the faculty. A parent e-mailed with some positive feedback yesterday and I mentioned that she could do my future at SLVHS a lot of good by forwarding her comments to our AP.

    Did I say tasteful? I meant tacky.

  22. H.

    March 20, 2007 - 7:36 pm

    Most of these many comments are off topic, and the reason why that is sad is that this is such an amazing topic. So, yeah, appropriate credit to Dan for bringing up such a useful, interesting and momentous theme. And yeah, this blog is wonderfully inspiring and rich in ideas, and yes, Dan seems an amazing teacher, and would be a great asset to any school. He has an incredible talent for initiating fruitful and consequential discussions of real issues. All of this is indisputable. Now, back to the topic: what are the slices of good teaching, and how – and to what extent – can they be developed?

    If a credentialing program should concentrate on teaching candidates to develop as people managers, eloquent communicators, and discerning psychologists – all of which makes sense – how can this most effectively be taught? Some of this can be learnt the usual way, by reading textbooks. Not all of it can be taught this way. I learn theory easily, and can write it back to an instructor in decent English, but that does not translate into classroom performance in a simple way. What kinds of learning experiences could do more toward developing appropriate behaviors and habits? Dan’s suggestion of doing lots of observations of good teachers and analyzing them makes good sense, too. Anything else?

    And, if your credentialing program does not do nearly enough to prepare you – what learning experiences do you seek out? Observe teachers, read widely, discuss with other teachers, read blogs (!) – anything else? Take acting classes? Study management theory? On a very practical level – what psychology texts would you all recommend? My psychology training consists of a couple weeks of preservice summer training. Our instructor was really fascinated with infant development, and we watched lots of footage of grimacing babies. We watched no footage on teenagers, and most of us were going to teach high school just a few weeks later. But I’m off topic too now; the point is not to talk about how unhelpful a lot of teacher ed is, because that’s something we all know about, but what are the best resources out there for educating and improving yourself?

  23. dan

    March 20, 2007 - 9:57 pm

    I’m trying to figure out some unifying theory here, H, and coming up short. I think it is supremely useful — though not a guarantee of good practice — to be interested in and critical of as much as possible.

    The kind of deconstruction I’m talking about has become — for better and often worse — a part of how I process everything in my life. It never shuts down.

    Like, I love movies and think they’re just priceless for how many varied character interactions and personality clashes they pack into 90 minutes. For several years I really dug moments in movies where two guys stepped to each other to see who’d back down. At that time I was teaching in an environment that thrived on machismo and watching those movies with sharp eyes helped me get a hold on how exactly I could diffuse those kind of confrontations.

    Too many words to detail one example. Trying harder:

    I don’t know how any of this fits into a tradition ed school model, but I’ve made it a matter of instinct to stop and reflect whenever my mind registers strong approval or disapproval about anything.

    If someone is a lousy speaker, like this pastor who always stares at the same spot on the back wall when he addresses his congregation …

    If someone is a great comedian, like recent-discovery Todd Levin, who makes magic out of genial insecurity and precise enunciation,

    If someone makes a handout or a computer-based presentation that strikes me as beautiful or transcendently clear somehow like Khoi Vinh (to cite one of many amazing graphic designers) …

    … I make sure I can describe that lousiness or greatness to myself in replicable steps. For several years now, it’s been impossible for me to passively enjoy or dismiss anything without first tearing it apart to see what went right or wrong. And then make it mine.

    I’m occasionally insufferable at parties.

    But whatever good I am as a teacher comes in large part from this personality trait that, though strengthened by UC Davis Teacher Ed, wasn’t built there.

  24. H.

    March 20, 2007 - 10:31 pm

    Thank you, Dan – and I apologize for the somewhat impatient tone in the previous post. I was only worried that this topic would be buried under more entries before it was discussed properly – and this thread promised to be satisfying like a neat epsilon-delta proof, which, it seems, it still will be.

  25. Miss Profe

    March 21, 2007 - 4:54 am

    I concur with D when he says, “But whatever good I am as a teacher comes in large part from this personality trait that, though strengthened by UC Davis Teacher Ed, wasn’t built there.” I would also venture to say that what D knows and is able to do in terms of math content wasn’t built there, either. If we are completely honest, many of us would admit thay we did not receive much of an education via our respective Schools of Ed. Even the nationally-recognized Schools of Ed.

    Dig it: Teaching is a profession largely developed on OJT (On the Job Training). And, many times, it draws on the resources – personal and financial – of the teacher him/herself to build, develop and maintain. This is unlike law or medicine, where the education time is spent developing a foundation on which to build. Schools of Ed, on the other hand, have failed miserably in this regard.

    Why? Unlike schools of law and medicine, which are built on a common core of knowlege, and which doesn’t differ too much from state-to-state, or, from country to country, for that matter, Schools of Ed develop courses and theories and practices per the research whims of the faculty member. If we can put men and women on the moon, one would think that we could devise a common core of knowledge for teachers which is comparable to law and medicine in academic rigor and effectiveness. And, yes: I do believe that the training of teachers can and should be to a level which is comparable to that of legal and medical training. Once a person leaves the training, iti s up to him/her what he/she does with it. After all, like teachers, there are excellent doctors and lawyers, and mediocre doctors and lawyers.

    Perhaps the most effective credentialing course in which I was enrolled was taught by an middle school principal. It was a course on classroom management. Actually, the course focused little on discipline, but more on what makes an effective teacher in terms of lesson planning, organization, taking care of oneself outside of the classroom, and dealing with parents.

    Teachers don’t need Schools of Ed. They need a solid theoretical and pragmatic foundation in their content area(s). Then they need education courses which actually train them to become effective teachers. I think if we examine the base of knowledge used to educate and train doctors and lawyers, we may get some ideas which will help us to raise the level of attainment in the professional education and training of teachers. And, as far as discipline is concerned, I suggest that we bring in a group of old-school grandmothers. I’m actually not kidding here. We have certainly lost our way with respect to home training, which is why things have fallen apart at school, and reinforced by faulty School of Ed. theories. But, as they say, that’s another discussion.

    BTW: As cool a flyguy as D is, I, too, want to return the discussion to its original intent. :) So, I’m with H on this one.

  26. tang

    March 21, 2007 - 9:44 am

    Like dan bowling, the first time i’ve been here and the first post i read and i found it marvelous! I love the calculus metaphor! (i’m a math major)

  27. dan

    March 21, 2007 - 5:13 pm

    I’ve been itching to get to Miss Profe’s comment all freaking day, if only because her first paragraph is distinctly un-representative of any point I was trying to make about teacher ed.

    Me: But whatever good I am as a teacher comes in large part from this personality trait that, though strengthened by UC Davis Teacher Ed, wasn’t built there.

    By which I meant that, independent of my education as a teacher, I hammer away at my personality in ways that make teaching easier. So I take exception with both of the following:

    Miss Profe: I would also venture to say that what D knows and is able to do in terms of math content wasn’t built there, either. If we are completely honest, many of us would admit thay we did not receive much of an education via our respective Schools of Ed. Even the nationally-recognized Schools of Ed.

    The argument against teacher ed echoes strangely against the Unschool movement, where those who had negative, functionally useless experiences in public school argue that it’s pointless and should be razed and disbanded.

    The refutations of both arguments are the same. Some teacher educators aren’t any good (just like some teachers are bad teachers) but that’s no reason to disband the system. Just like teachers need good training and a strong work ethic, teacher educators need the same.

    I mean, you say, “Teachers don’t need Schools of Ed.” and then immediately give, “… education courses which actually train them to become effective teachers …” as the prescription.

    Unless I’m missing some semantic difference here, that’s exactly what I got at my ed school. (Which more and more seems deserving of a check from this alumnus.)

  28. Miss Profe

    March 21, 2007 - 8:18 pm

    Ok, D. Then, I did mis-interpret the point you had made. My apologies.

  29. H.

    March 21, 2007 - 9:04 pm

    What both Dan and Miss Profe seem to agree on is that a strong autodidactic streak is a great asset, maybe a necessity, in a teacher. Dan furthermore breaks down the autodidactic inclination into particular approaches to movies, speeches, and the like – in neat accordance with his overall theme of breaking down the complex into possibly replicable parts.

    This reminds me that I once planned to read “An inquiry into Autodidacticism” by Joan Solomon, in order to see whether it contained any wisdom about how to teach (or design learning experiences that might prompt) students to become self-motivated and self-directed learners. (Checking the reference again I realize that the reason why I never acquired the book was probably that it is heinously expensive.) Anyway, while there seems to be something pretty paradoxical about teaching anyone autodidacticism, it seems that’s what ed schools should be trying to develop in teacher candidates, and what we should develop in ourselves, of course. And here we have some particulars about how to go about it. Please keep it coming.

    I’d still be interested in a recommendation for a good text in adolescent psychology.

  30. H.

    March 21, 2007 - 9:22 pm

    I think one way of focusing my question would be to say that while I know a lot about how to learn – and to teach – facts, ideas and theories, I know much less about learning – and teaching – skills, habits and attitudes. Credentialing classes are largely words and ideas, and I give my students words and ideas, but the kinds of learning experiences that would make me a more skilled teacher, and my kids better students, have to do with knowing-how rather than knowing-that, and with feeling and deciding in certain ways. And that’s where I’m seeking further input on how to bring about the necessary changes. There have been quite a few useful suggestions so far, and I’m hoping there’ll be more about how to acquire the kinds of skills for which (d teaching)/(d skills) > 0. (‘scuse that)

  31. Miss Profe

    March 22, 2007 - 7:00 am

    H, it seems to me that what you are talking about is teaching someone how to fish, which comes from teaching kids how to learn – metacognition. It also means teaching kids how to construct knowledge – constructivism. Of course, in order for kids to change how they learn, teachers would need to change how they teach.

    And, H, thank you for rescuing the point I was trying to make :).I knew that there was something valuable in my long commentary.

  32. dan

    March 22, 2007 - 5:05 pm

    Right. It’s the metacognition I find so difficult to teach, especially to a population that has enough trouble with cognition. Re ed schools and my own metacognitive development: my supervisor observed us frequently and asked us probing questions about how we knew a student knew what we had taught. Then, in the second semester, probably realizing he wouldn’t be around to ask the questions much longer, he started asking us to predict the question he was about to ask next. Which I didn’t recognize as a deeply metacognitive game until now.