The hard truth is that the outcomes and inequities lamented over in Principles to Actions and previous documents are precisely the outcomes that our educational system is designed to produce. Equity-oriented slogans, statements about idealized outcomes, and tweaks to teaching or curricular practices within this system do not change this fact. -Danny Bernard Martin

Martin’s entire discussion is a must-read. (I found this via Beth Burroughs). This effectively opens up the same questions for me that Grace’s synthesis does. Too many of my kids aren’t buying what I’m selling and I’m increasingly thinking it’s because they are tacitly (and likely, for some, unconsciously) calling BS on the whole system.

What should kids do when only game in town is rigged? Refuse to play? Play anyway? What’s the third option?

And, note to self, what role should I play in all this that’s truly best for all?


It was a privilege to work with 22 educators at #EduCon today. I’m particularly grateful that they chose to attend the session considering that Chris Lehmann and Zac Chase were presenting at the same time.

The teaser:

If extended, self-propelled, challenging learning experiences are critical preparation for life, what do we mean by “preparation”? How can we know whether students have learned how to learn–enough to thrive in the next stage of their education? How can we help them document and demonstrate their readiness?

The work:

  1. Please sit with 3 people you did not know before yesterday.
  2. Think of a significant experience from your youth that helped prepare you for later life.
  3. Now think of an occasion when you saw a student have the experience that you got into education to help students have.
  4. Private work time: Please answer these three questions about the experiences you selected.
    • What elements of that experience made it good preparation for later life?
    • For which aspects of “later life” did it prepare you (or the student)?
    • What changes in you (or the student) could be seen by others at the time?
  5. Team task: Please Organize the team’s collection of responses into groups or sets that go together.
  6. Team task: Consider the sets you made, and as a group, consider: What models and milestones can we hold up for students to help them envision, demonstrate, and document their preparedness?
  7. Individual task: Write three sentences about what’s on your mind as a result of the past hour of conversation.

A lot of intense thinking and talking went on in those groups. Not all the responses were recorded, but several were, and you can read them here. In my opinion the best stuff is in the final thoughts at the bottom.

The following problem set is being given to our department before we talk about how we want to teach problem-solving. 

For the following problems, please use our 5-step annotation:
(i) note the context,
(ii) underline the question,
(iii) circle useful information,
(iv) write a sentence frame for the answer,
(v) “build a bridge” from the useful information to the answer (ie, solve).

Problem 1: A 20.5 gallon fish tank is 4/5 full. How many more gallons will it take to fill the tank? [Kelemanik]

Problem 2: Imagine a triangle inside a rectangular box. How much of the box does the triangle take up? [Lockhart]
Lockhart Problem

Problem 3: Are the following lists the same list, or not?

  • Guess and check
  • Make an orderly list
  • Eliminate possibilities
  • Use symmetry
  • Consider special cases
  • Use direct reasoning
  • Solve an equation
  • Look for a pattern
  • Draw a picture
  • Solve a simpler problem
  • Use a model
  • Work backward
  • Use a formula
  • Use your head
  • Look for patterns
  • Tinker
  • Describe
  • Visualize
  • Represent symbolically
  • Prove
  • Check for plausibility
  • Take things apart
  • Conjecture
  • Change or simplify the problem
  • Work backwards
  • Re-examine the problem
  • Change representations
  • Create
    [Park School]
  1. Make sense of problems and persevere in solving them
  2. Reason abstractly and quantitatively
  3. Construct viable arguments and critique the reasoning of others
  4. Model with mathematics
  5. Use appropriate tools strategically
  6. Attend to precision
  7. Look for and make use of structure
  8. Look for and express regularity in repeated reasoning
    [Common Core]

Problem 4: Describe in detail what went on in your mind to “build the bridge” in problems 1 and 2. (Extra credit: same for Problem 3.) (Extra extra credit: same for Problem 4.)

Problem 5: Describe the relationship between the bridge-building you did in previous problems, and the list(s) from Problem 3.

Problem 6: To what extent do you relate to the following?

Mathematicians frequently report that often one of the most helpful things they can do to solve a problem they’re stuck on is step away from it. Jacques Hadamard (1949) examined his own experiences and also talked to many of his colleagues to work out what the common structure of this experience was, and determined that there seems to be a fairly predictable sequence to it:

  1. Intensely focus on the problem, working through every permutation you can think of that’s likely to produce an answer.
  2. Walk away from the problem and think about something else.
  3. The magic genie in your head might eventually, and often unexpectedly, yell a possible insight into your awareness.

For instance, Henri Poincaré reported struggling to work on Fuchsian functions over the course of several weeks and then being forced to walk away from the proof he had been stuck on due to a planned vacation. One day he was stepping onto a bus with his mind certainly not on mathematics, and suddenly the key insight he needed to finish the proof appeared in his mind. It was as though a part of his mind had been secretly working on the problem and then brought the finished product into his awareness. [Valentine]

Problem 7. Watch Grace Kelemanik’s Ignite talk, which starts around 20:25. (The other talks are also terrific.)

Problem 8. What experiences will help children become more confident in, competent at, and excited about solving problems?

Problem 9. How can we give those experiences to kids in the confines of a school schedule and in the context of required content and skills?

Last summer I gave a short talk at the Academy of Inquiry-Based Learning conference in Denver. At the time I promised Michael Pershan I’d write it up. I never did, but it was recorded. The video (~20 min) is below, and here’s the handout.

By the way, neither in the talk nor in the handout did I ever credit Tony King, the instructor of a class I took on teaching ELL’s. I stole the premise of Problem 1 from him.

Enjoy, Michael!

STUDENT: You know that i to n power just keep repeating itself- which creates a circle, BUT THAT CIRCLE IS A UNIT CIRCLE.

Now, take that point on the complex plane that I gave you, which the angle is pi/4  or 45 degrees.

If you use Euler formula, and you put e^ipi/4= pi/square root of 2 + pi/ square root of 2 i

BUT, that is just the point on the complex plane. Why that happen knowing that I just put e^ i pi/4 ?????/ but why e to the i angle gives me the exact point on the plane???? why the base must be e??? It is just a simple plane, whyy base e to the i angle????

I hope you understand what I mean, because I do not even understand myself.

ME: I think I know what you mean.

A point on the unit circle is just (cos(theta), sin(theta)). If that unit circle is on the complex plane, we write that point as cos(theta) + i sin(theta). So far, that’s no big deal: it’s just how it is, because of what we mean by sin and cos and i.

But Euler’s formula says, that same point on the unit circle on the complex plane, cos(theta) + i sin(theta), is also e^(i theta)!!

I think you are asking, how the hell (excuse me) did e get into it?

That is a VERY good question.

STUDENT: YES THAT IS MY QUESTION- but you are not answering it??

ME: What fun would that be?

-Summer correspondence with my most inquisitive student.

I think “whyy” should be a new word.

I was just figuring out how to rotate 21 kids through 7 stations in such a way that each kid does each station with a different group. And it just dropped in to try this: 1/3 of the kids advance by 1’s. (If you’re at station 2, move to 3). 1/3 advance by 2’s (e.g. if you’re at 2, move to 4). 1/3 advance by 3’s (if you’re at 2, move to 5.) And it works! I wonder if any group reconstitutes itself along the way somewhere … and what other combinations of kids/stations this or similar patterns work for ….

Figuring out a problem yourself is better than copying someone else’s down because then it stays processed in your brain. And once it’s processed in your brain, you can’t lose it because you did it on your own.

-A student, reflecting on her first term.

Since September I’ve been dealing with two things that are forcing me to grow. First, I’ve been doing what I’ve done for a year or two, only this year, the response is different. Students are more restless, with shorter attention spans, so giving them long periods of independent work time results in little work from them and little coaching from me because I am spending all my time redirecting. So it’s back to table groups and roles and more, shorter tasks and putting things in front of them to complete instead of writing a broad direction on the board.

Second, I’ve been trying to learn how to develop fluency with some basic skills. My current experiment is in learning how to teach graphing quadratics. There’s lots of great material out there. (Check out James Tanton’s course at G’Day Math). But I thought they would be able to do it as easily as me walking them through it. The skill that common core calls “Seeing Structure in Equations” seems to take time and repetition to build. It’s something I do without realizing I’m doing it. How did I not know this? Seriously, I thought we’d do one day on graphing from vertex form and maybe one additional day to review, then on to the next standard. What’s required, at least for this group this year, is one day on where’s the vertex. One day on width. One day on finding intercepts. One day on synthesis of the above. So good: that’s what it takes. But I’m inventing as I go. It feels a lot like I am just now figuring out how to do what every other teacher already knows how to do.

The student quoted above came to see me after school to get coaching on a problem because she didn’t present any original solutions last term and wants to start 2nd term off by presenting. She announced to me as she arrived that she’s “really horrible at math.” As we worked through some background problems and re-discovered what her classmates had discovered, she lit up (“Whoa! That’s really smart!”), and as she got into the new problem she wants to present, she was really enjoying it. She told me she’s been doing what a lot of students do: waiting for other people to present and then just writing reports of their solutions (which I allow, though I will taper that off as winter turns to spring). Then out of the blue, she said what’s quoted above. I said, “Wait, I want to write that down. You just reminded me what I’m doing here.”

Twenty-six of our strongest sophomores from last year have been given to me as juniors, to take algebra 2 and pre-calculus in one year. About twenty are serious and ready to work. About six are boisterous and unable to hold their attention on listening to one person for more than about 90 seconds—but they do quality work when they work. All twenty-six would clearly (to me) be bored in our regular algebra 2 class. The knee-jerk reaction: the ones who know how to stay focused can stay, the other six have to go. But that’s the reaction that has put thousands of students, primarily low-income students of color, in courses below their ability level for decades. So the right thing to do is keep them all, and work explicitly on classroom behavior skills. And the only time that doesn’t feel good is when the first twenty students keep telling me they can’t concentrate, that the six don’t belong there, that I should get rid of them.

And there you have it: the single most influential tension in the US education system. It’s why we have private schools and charter schools and parochial schools and good schools and bad schools.

Announcing a Global Math Department webinar on Problem-Based Course Design, Tuesday, 10 September, 9pm Eastern.

Following up on a #TMC13 session, we’ll have an open discussion on ideas and questions people have about designing their own problem-based courses. All are welcome – if you didn’t catch the TMC13 session, you can catch up by looking over the materials. Questions or comments? Leave them here or #PBCmap on twitter. See you then!

We had a super session at #TMC13 on designing problem-based courses. Materials and notes are posted for those interested.