Organizing Principles

As I lock picture on season six of this incoherent teevee drama called “Teaching,” I am very interested in articulating some general themes and principles around which I’ve tried to organize my instruction. Problem is: the more time I spend creating math problems, the more I detach myself from those interesting generalizations. I am grateful, then, to a couple of edubloggers who have done some of that heavy lifting for me.

1. The Skills Aren’t Arbitrary

Tim Childers:

The original Greek texts [of the New Testament] were written in all capital letters with no spacing and no punctuation. I wondered what would happen if I gave kids the note below on the first day of class?

It is exceptionally easy for me to treat the skills and structures of mathematics as holy writ. My default state is to assume that every student shares my reverence for the stone tablets onto which the math gods originally etched the quadratic formula. It is a matter of daily discipline to ask myself, instead:

  1. what problem was the quadratic formula originally intended to solve?
  2. why is the quadratic formula the best way to solve that problem?
  3. how can I put my students in a position to discover the answers to (a) and (b) on their own?

That’s hard.

And the same mandate goes for any hapless ELA teacher reading this blog. Why spaces? Why apostrophes? Why different words for “happy” and “ecstatic?” Why hyphenated compound adjectives?

2. Great Problems Are The Coin Of The Realm.

Avery Pickford offers a five-bullet definition of great problems. It’s excellent and concise. Here is the first bullet:

The problem should be accessible. It should minimize vocabulary and notation, have multiple entry points, and include ways to collect data of some sort. It should have multiple methods that promote different learning styles and celebrate different ways of being smart.

Dr. Tom Sallee, math professor and president of College Preparatory Mathematics, gave two of the best conference sessions I have ever attended (recapped here and here) and said this in one of them about good problems:

A good problem seems natural. A good problem reveals its constraints quickly and clearly. Developing good problems is not at all an easy task. I have a lot of experience with it and I have failed many times.

The best part about this particular currency is that as I get richer, you do too. When you create and post a great problem about Applebee’s, that’s money in my pocket as well.

I find myself dazzled daily by the great problems y’all share. We’re just printing money lately.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.


  1. So…I hate that I’m gathering momentum with this “making things more problematic in math” at the end of the school year. I want more time! (Did I really just say that when there are only 6 days left?!). ::Sigh:: Guess I can use some of this greatness when I teach math intervention in summer school :)

    Most recent victory: I was to teach a math lesson to my little darlings (8 yo) introducing multiplication as repeated addition. Now, the curriculum does a pretty good job of bridging these concepts to make them developmentally appropriate, but I decided to take a stab at being “less helpful.” (This was HUGELY inspired by your TED talk, Dan. Thanks for modeling how to cut out too much help in a textbook….very practical skill.)

    Cut to me abandoning the assigned journal page and improvising on the fly when I saw a golden opportunity.

    Instead of: “There are two ladybugs. Each ladybug has six legs. How many legs in all?” (alongside a picture of a ladybug, with “six legs” written underneath it)…

    We did: “There are four _____” (student suggested cars). How many…wheels? Windows? Seatbelts?

    This generated LOTS of discussion about what kind of car (lamborghini vs. minivan), whether there was a middle seat in front, etc. GREAT stuff. Some of my kids made up their own, to the point where two boys came up and asked:

    “Do you know how many gears are in a gear shift?”
    Then, “How many gallons of gas does a Ferrari tank hold?”

    Eight. Years. Old.

  2. Judith Diamond

    June 2, 2010 - 6:41 am -

    I like to say that I am not teaching my students how to do each problem, but I am giving them the paths to get there.

  3. Laura: We did: “There are four _____” (student suggested cars). How many…wheels? Windows? Seatbelts?

    Super fun right there. Way to go.

  4. Sorry to quibble, but if you write about the necessity of grammar rules, how can I resist pointing out that “spelling and grammar” is a compound subject and requires “are” instead of “is”?

  5. @Andy, if ever there were an appropriate place to quibble, that’s it right there.

    @MPG, my impression is that Anglophiles refer to all corporations in the plural form.

    “Simon and Garfunkel are …. ”
    “Phish are ….”
    “Microsoft are ….”

    Etc. This is perhaps exemplary of an arbitrary quality to certain rules.

  6. Fun combinatorics problem I had one time: I ran into a friend at the airport, and it turned out that we were on the same flight (from NYC to LA). Then later, when boarding the flight, it turned out that I was sitting next to him! I can’t remember exactly now, but let’s say there were 23 rows on the plane, and each row had 8 seats. What were the chances of us sitting next to each other? :)

    Another fun fact: My friend and I were both math teachers. We did the problem on the back of a puke bag. He and I each used different ways to calculate the probability, but got the same answer.

  7. Mimi: We did the problem on the back of a puke bag. He and I each used different ways to calculate the probability, but got the same answer.

    That’s what it’s all about right there.