Thursday, July 15, 2010

Minimally Defined Problems, Chapter 2

There is understandable skepticism out there (and some not so far away in the comments of previous posts) about students' ability to engage and come up with rich mathematical problems starting with what I am calling minimally defined problems. First, an anecdote of success with this approach that happened yesterday with some real, live 2nd, 3rd, and 4th graders at Math for Love.

These kids were given a chessboard and asked to come up with some math questions. Link on over if you want to see the great things the kids came up with.

Now don't get me wrong, I'm sure I could find an anecdote showing how a bear dancing on the back of a goat can lead to better proficiency with multiplication tables. One anecdote does not make a theory and should not send everyone running to rewrite the curriculum.

On the other hand, this doesn't mean that this anecdote should be ignored...

Anyway, I first want to address some of the issues/concerns about minimally defined problems and student posed problems that were brought up in earlier posts (comments that actually hit on many of the concerns that I am addressing in my proposal). Concerns/questions addressed here can be loosely separated into the following five questions:

1. Can this approach (or any approach for that matter) increase intrinsic motivation?
Ms. V: “I was a good math student, and I like puzzles, but I have very low "task-commitment"...Certain types of problems in certain contexts will totally get me working doggedly long past when anyone else cares. But other types of puzzles bore me (once I realize there's a trick/strategy/etc)....when I hit a wall and have run through everything I can think of...[So] how do you build stamina for such things in your students? Or is it just a mismatch between problem and personality? Or are these actually not such great problems, and the really great ones DO promote commitment? Or am I just lazy?”

2. How do you change the dominant culture of explanation-example-exercise-answer-satisfaction and should this culture be changed?
O: “I think I would be initially frustrated because the question is so open ended and there doesn't appear to be a right answer." 
Perdita: “If I'd had the gumption I have now I'd have written "What comes after 5? Answer: 6". Not sure what I'd have done then, but you'd have lost me for ever.”
Perdita: " you were apparently targeting it at middle schoolers, but my 6yo answered it instantly, and I'd certainly expect any 11yo with reasonable number sense to do so, if not distracted by wondering what teacher wanted today...”

3. Are students (specifically, sixth grade students) capable of engaging in this kind of work?
Max: “It's very hard for me to see how a student without any background or context could approach that assignment in a productive way.”
Perdita: "the space of mathematical problems that are solvable with effort by a given person is very, very narrow by comparison with the space of all problems that that person can formulate. Preduction: if you ask children to make up their own problems, almost all of them will be either trivial, or insoluble."

4. What does this look like on a day to day basis?
Ms V: “What you do when they arrive in class could drastically change the results on future similar assignments, though. What *would* you do?”
O: "I think it can be frustrating to do work which you are uncertain if it is in the right direction because you can feel the work is pointless.”

5. What should we value in a mathematics education?
Perdita: "[This site] seems to be an exemplar of a trend to favour trivial creativity over solving hard problems, and it seems likely to encourage teacher-pleasing rather than mathematical work. "

Answers Responses forthcoming...


  1. This is a very exciting conversation.

    I've done some experimenting with having students generate the problems. On the basis of this limited experience it seems to me that many of the concerns of folks you quoted above are real and substantial, but none of them are deal-breakers. Challenges to bear in mind when planning, in order to make the educational experience more powerful.

    I don't think everyone would necessarily come to the same conclusion, but my experiments have been most effective when I posed the first question, the one that gets everybody moving in the first place, but invited people to move on to their own questions if they got curious about something, and then listened for this and directed people to take on their own questions when they came up. The questions that people asked when they were already working were more focused, more mathematical, and people were more already-bought-into-them, than the questions people came up with when I just defined the objects of inquiry and then said, "what questions could you ask here?"

  2. I think my current comfort level with this ind of thing is to begin with a situation that is pretty clearly crying out for a certain family of questions, so you're pretty sure to get someone asking the interesting and productive ones that they're ready to attack and learn something from. I also tend to use this sort of open-ended thing in a discussion, not on a paper the student would take home with them.

    For instance, in the chessboard example: "Here's a chessboard, what would you like to count?"

    As for your points above...
    1. Motivation? In a class discussion, keep the questions attached to the kids' names. There's a lot of motivation there to ask a good question. I often end up with X's proof for Y's conjecture about Z's question, and those kids at least are staying motivated.

    Also I often try to toss out a few different puzzles that get at the same basic idea in different directions, and then let people pursue what interests them. Maybe the ultimate in this sort of thing is Harold Reiter's famous problem set with 10 problems whose answers are all 7 choose 4, even though at first they don't seem to have anything to do with each other. That makes for a great class discussion afterwards! I'm pretty sure you can find his set as a "Stretch" in a handbook from the early 2000s at the MATHCOUNTS web site, though my quick search fails to turn it up.

    2. Change the culture? Absolutely. You can start in small ways, by beginning with a question instead of an explanation. "What is the length of the third side of this triangle?" and then by the end of class you have picked up the law of sines or cosines or both, using standard tools like "make right triangles". Then instead of memorizing formulas, it becomes a problem-solving process. (There's still plenty of cultural obstacle to get kids to value the whole journey instead of just ignoring you until the last five minutes, writing down the summary, and memorizing it.)

    3. Sixth graders? Sure, just don't leave it TOO open-ended at first, and give them some guidance and structure. Show them how to take a too-easy question and extend it, and how to take a too-hard question and find a related simpler one. Those are probably even more valuable strategies than anything else you'd teach, anyway. For Perdita's example, "How many squares are in the next row?" is probably too easy, and so is "how many squares are in the nth row?" although there you'd have at least some interest in showing them how to express their description well in English and in math. So, the natural next question is "How many squares in total in the first n rows?" or "How many rectangles in the nth row?" and then ... wait a minute, what did we just discover, and why does that work!?!?

    4. Day-to-day? Point out that some days are for skills and some days are for problem-solving, and that both are valuable just like doing drills and scrimmages in soccer or basketball. Show them how skills help them solve problems and how problem-solving helps them develop skills without it seeming quite so boring (again, the athletics analogy probably works). Getting in the habit of introducing a new idea with a question, and drawing the answer out of the students as much as possible, might help too.

    5. I like solving hard problems, but surely the art of problem-posing is at least as important as the art of problem-solving, and our curriculum has been way too over-balanced away from problem creation. I mean, how many classrooms have you seen where the students have ANY time spent coming up with the questions?

    Any other ideas?


  3. I am a senior, elementary education major at the University of South Alabama. I have never been so excited to learn and teach math in my whole academic career until now. This new excitement culminated from taking Math for Elementary teachers 1 and 2. I got very lucky by enrolling with two teachers who don't care about the right answer, but instead focus on the means for discovering a way to find the right answer. They both encourage us to think about real applications of geometry and algebra. They help us see illustrations of what is actually happening-- and they do this by giving us chessboards, or the appropriate equivalent so that we can notice patterns and make predictions. I promise that my hatred of math would have been lifelong had I not experienced it as a problem solving experience versus a memorized way of following instructions to get the desired answer. I can't wait til I have more time to come back and explore this blog. What a killer resource!

    Thanks your thoughts and strategies.