Friday, November 2, 2012

Student Posed Problems

While not nearly as sexy as iPads 
and definitely not as lucrative as Khan Academy
I for some reason made the choice to lead a 90 minute session this morning on student posed problems. By this I mean creating a culture where students are producers of mathematics, not just consumers of mathematics.

I began by talking about classroom values that I try to instill which foster students creating good mathematical problems of their own. I'm sure there are more, but I highlighted the following:
  • Promote collaboration 
  • Focus on process, not just answers
  • Build comfort with unfamiliar problems
  • Start with good problems
  • Create intentional vagueness
  • Assess as check-ins, not evaluations
We unpacked each of these values and did some math. I then tried to make a pitch for why it is important to let students create their own math problems. I first reached out to the romantics, claiming that we make music in music class and art in art class, and math should be no different. I then gave the non-romantics a list. Non-romantics love lists.
  • Checks for deeper understanding 
  • Differentiate
  • Model mathematicians
  • Teach higher order thinking
  • Determining parameters is oftentimes the “real” mathematics 
  • Meaningful sharing
We unpacked these ideas and did some more math (a pattern you see?). If this was the skin and bones and hair of the talk, we now delved into the meat. 

How do you get students to ask good mathematical questions?

Of course, this can be simply answered with a list.
A. Creates variations
B. Creates generalizations
C. Creates extensions
D. Looks at simpler examples when necessary
E. Looks at more complicated examples when necessary/interesting
F. Creates and alters rules of a game
G. Invents new mathematical systems that are innovative, but not arbitrary

Ok, not actually simple. Required unpacking. And math to keep it fun. And most likely lots more practice long after this session. Finally, I talked specifically about The "Invent Your Own" Project, a 5th grade project we do at the end of the year where students deeply explore a topic of their choice. If you're interested in the nitty gritty details of the project, feel free to explore what we hand to the kids. Please use this! All I ask is that you cite where it came from and let me know what worked, what didn't work, and what you made better. For example, I'm planning on completely revamping the rubric I used for this project so if someone else chooses to do this before I get around to it in the spring, whippee. 

I next did a little student bragging and some examples of past projects. Here are three.

Model of a hypercube
Fractal made on a laser cutter
Exploration of cuts made from different paper folding
People were just begging for one last list, so I shared some non-visual examples of past topics:
  • A variation on RSA Encryption
  • Workbook to teach Egyptian Fractions
  • Scaled Barbie
  • Map projections
  • Efficiency of product packaging
  • Different ways to calculate pi
With the last list out of the way, we spent the last 25 minutes brainstorming ways people could start to incorporate student posed problems into their curriculum. Good times. Good times.

To those of you visiting here for the first time, welcome. Please continue the conversation in the comments. To my long time reader(s), I'm back from a pretty long break...errr, hiatus...actually, I like sabbatical. Yes, let's call it a sabbatical. So to you, welcome back.


  1. It's amazing what happens when students begin to realize the creative power of mathematics.

    I recently gave a very similar talk at a Math for America workshop here in NYC. Our bullet lists are virtually identical.

    The only other thing I have in the "Why?" list is the value of becoming a good technical writer. If you can learn to express complex mathematical ideas clearly, you can write clearly about anything.

    I, too, am often asked for lists of starter projects, but it almost defeats the purpose, which is for students to find something unique to them.

  2. Patrick: Thanks for the comment. If you have anything you think would be worth sharing, I'd love to see how you've been both conceptualizing and implementing this idea.

    1. Hi Avery-

      As a follow up, I've put together a few "Math Research Project Outlines", which are really just sets of questions built on a particular seed question. For example, here's one on Grid Walking.

      As loose as they are, this is pretty much how I mentor student projects: I help them construct interesting, relevant questions, and then help them put the pieces together. I'd love some feedback, if you're so inclined.

      I've put these together as an attempt to get more teachers and students involved in the Greater New York Math Fair, a student math research/writing event:

      Hopefully these can shed some light on how students might start an investigative project.

  3. The paper folding mobile reminded me of a beautiful theorem that some of my math circle colleagues in SF showed me: the "one cut" theorem. What it says is that if you draw anything on a piece of paper -- even a concave thing -- even a disconnected collection of things! -- there exists a way to fold the paper such that with one straight cut all the way across, precisely the region(s) you outlined will fall out. Well, it doesn't quite exactly say that, but I'll leave it to your students to find out what kinds of things might be uncuttable, and what the rules might be for folding.

    In my own practice, I often present a situation that leads to a sequence or pattern, and then ask the participants what they see, what they wonder, whether what they wonder is always true, and how we might find out. So there's some of the feeling of creating their own questions but within the framework of a shared mathematical context. Sometimes the context seems to push people to ask pretty much the same questions, but every once in a while I'm really surprised by the patterns people see and the questions they ask!

    1. A talk by Erik Demaine got me into the one-cut challenge a few years ago. I put together a few resources like this:

      It's the perfect problem in many ways--easy to state, easy to get into, something you can explore physically, and surprisingly deep!

  4. I've posted some thoughts about your wonderful session on my math education blog:

    Who knows, I may start blogging regularly!

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