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
    [Pólya]
  • 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.”

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