Online Problem Solving – Problem Statement

A nice problem: Given n lines in general position, how many regions do they break the plane into? How many are bounded, and how many are unbounded? (In this picture, there are six lines, 10 bounded regions, and 12 unbounded regions.)  What if we work in 3 dimensions, with planes instead of lines? Can we generalize to k dimensional space?

One of the big discussions we’ve (myself, David Stern, Toni and Alan Beardon, and occasionally David Minga) been having these last couple weeks is, ‘How can we develop online materials that do a good job of teaching problem solving?’  In a lot of ways, a good problem solving course is one of the most important parts of an education in mathematics.  One gains a flexibility in approaching problems well beyond trying to reproduce an answer on an exam, and encounters numerous techniques and ideas that will motivate later coursework which might otherwise seem really dull.  (Linear algebra comes to mind: it’s stupidly important, but can seem really obtuse if you encounter it in a void.)  General problem solving skills also translate to a wide variety of contexts outside of mathematics: How do I approach this issue flexibly and adapt it into something I can address with the tools available to me?  Furthermore, can I solve bigger problems with my tools than the one immediately in front of me?

The best solving courses take the form of a conversation between students and teachers.  It’s about developing the skills to get started, to actually act on a problem creatively, rather than reproduce what a teacher tells you.  So a good problem solving course typically focuses on getting the students to actually solve problems, with a relatively small amount of guidance and advice from the instructor.

But this method is heavily reliant on reactive, non-linear instructor interaction.  Generally, it’s agreed that this is at the core of why it’s hard to put high-quality math courses on line.  How do you foster creativity with a computer interaction?

Because this is such a hard problem, it’s an important problem to try to find a solution to!  Anything we come up with should be helpful in other math courses and other contexts, even if the solution to the original problem of online problem solving isn’t entirely solved in itself.

Towards this end, the solution I’ve envisioned is an adaptive, interactive textbook built around leading a student through individual problems.  Each problem stands more-or-less on its own as a module, perhaps with links from one to another when they make sense.  There are a few immediate questions to try to answer here:

  1. What kind of problems should we include?  [EXPAND Click here.]
    There’s been a pile of discussion about this.  Essentially, we think the problems should be approachable, generalizable, and not fully solvable.  The problem pictured above is a good example:You can make progress on the initial problem, then generalize it in interesting ways, and get to new versions of the problem which aren’t easy to find answers for.  That particular problem leads to piles of interesting mathematics, including algebraic geometry (intersection of curves and surfaces) and combinatorics (hyperplane arrangements).[/EXPAND]
  2. How will students interact with the book?  [EXPAND Click here.]
    I was initially imagining something like a piece of interactive fiction (example).  (Note for nerds: I quickly decided it didn’t make sense to extend the z-machine to include support for mathJax, calls to Sage and html5 rendering.)  But the idea persists: imagine a transcript of a conversation with the book.  It says a few things to get you started (probably with some video, too, to get oriented and motivated), and asks a series of questions of the student.  Based on the student answers, hints are proffered, the questions modified, and a direction of attack on the problem is explored.  There’s a very crude mock-up of the idea here. Where you see the orange questions, imagine a response box that the student types something into. (Credit where credit is due: The problem there is a direct adaptation of text from alan Beardon’s forthcoming  book, ‘Creative Mathematics II.’  The problem has also been considered on the NRich site, which collects lots of information for secondary students and teachers.[/EXPAND]
  3. No, really, how will that interaction work?  Isn’t that like the hard problem of AI?[EXPAND Click here.]
    Uh, kind of.  I’m imagining two different sorts of questions, both of which are apparent in the mock-up.  First (and easier) are closed questions, which ask the student to perform some small computation in order to show that they are actually following the text.  (Specifically, a closed question has a specific answer that we’re looking for.  Like ‘What’s 5+3?’)  Let’s make sure the reader understands how to do X before we start telling them about Y.  Lots of dead-tree textbooks do this, but the electronic format gives an opportunity to enforce interaction with these small exercises, and provide hints and help when the student is having trouble moving forward.  In fact, getting a good, flexible implementation of handling closed questions will be generally useful for all kinds of mathy e-leaarning.  I’d love to have something where I can drop a problem into anything from a moodle quiz to a random webpage.[/EXPAND]
  4. What’s the second kind of question?[EXPAND Click here.]
    The other kind of question are open questions.  These are questions that don’t have a single right answer that we’re looking for.  A nice example is “What do you notice about this picture?” Different people could notice lots of things…  But as you work with many people, common answers emerge, and we can cluster answer-types together and respond to them dynamically.  Getting a good system together for this will require a combination of natural language processing and symbolic computation; it’s a genuinely interesting problem, which I’m looking forward to working on.[/EXPAND]
  5. Static text is static.[EXPAND Click here.]
    David Stern is really interested in including video components, too.  I think it makes sense to have video for initial problem statements, as well as occasional short clips for guidance and response.  Imagine typing in something and instead of getting big red letters saying “NO.  You are WRONG.”, you instead get a little five-second clip of our ‘instructor’ saying “Good try, but maybe we can do better.”  One gets a lot of assurance from voice contact, after all.  (Of course, one would have to be a bit careful; getting something wrong ten times and hearing exactly the same spoken message would be kind of annoying.) Video as a learning medium is something I could go on about for a while; it has its ups and its downs.  But it’s great for injecting a bit of humanity into a discussion, and that’s important in this context.[/EXPAND]

This is obviously a lot to work on!  There’s a lot to learn and a lot to do, and I’m feeling like I’ve written enough to get the idea out for now…  Obviously, this will be a topic to come back to.