The internet has failed me.
In spite of following 150 creative math teachers on Twitter and subscribing to 750 creative math teacher blogs (including one blog that’s dedicated exclusively to creative math), I’m only now learning about Gordon Hamilton’s Unsolved K-12 Project. It’s creative. It’s math. It’s almost three years old!
Better late than never.
See, Hamilton convened a bunch of creative math types in Banff in November 2013 to a) select unsolved math problems and b) adapt them for use at every grade in K-12. Not a simple task, and I’m enormously impressed by their results. You can watch videos introducing the problems at this page or read about them in these slides.
Here are two of my favorites. (Click for larger.)
Grade 3: Graceful Tree Conjecture
Grade 10: Imbedded Square
These two problems have the capacity to develop fluency just as well as any worksheet or worksite. In working out their solutions, students will perform the same operation dozens of times — subtracting whole numbers in the third grade task and calculating slope and distance in the Cartesian plane in the tenth grade task. But these problems ask students to think strategically and systematically in addition to practicing efficiently and accurately. That’s no easy feat, but Hamilton and his team pulled it off thirteen times in a row.
Mike LawlerJune 23, 2016 - 8:44 am -
Here are a few dozen more (probably with a more than a few overlaps) if you like:
“Amazing math from mathematicians to share with kids:
“Reacting to a wonderful blog post from Lior Pachter”
“Sharing math from mathematicians with the Common Core”
David WeesJune 23, 2016 - 9:14 am -
The Internet (or rather the small subset of it you are currently following) has failed you even more than you know, since this work actually dates back to at least 2006. Gordon’s been working on this idea for a long time, the Banff Conference (to which I was invited but unable to make it) was more of a culmination of his work.
Gordon’s website is filled with other kinds of activities to develop fluency while problem solving.
His basic idea though seems super sound to me. As kids become fluent in the procedures they use to solve these problems, more and more of their brain power will be available to think about the patterns and generalizations that emerge, which means these problems can both lead to fluency with some set of mathematical procedures or ideas AND also allow children to learn something new from the activity.
I have some of my own problems/projects in my head that I really need to find some time to share. They aren’t as cool as his though.
Michael PershanJune 23, 2016 - 9:29 am -
Can I begin by apologizing for ignoring the fantastic math that you’re sharing and focusing on not-even-a-tangential point that you made?
OK, I’m sorry.
I think it’s worth thinking about our theories of fluency development. I think I disagree with the “high repetition” theory.
Here’s a juicy quote about the relationship between high repetition practice and fluency:
The conventional mode of mathematics teaching is stereotyped. New material is presented and one or two worked examples using the new materials are demonstrated, followed by a reasonably large number of problems or exercises…Solving many conventional problems may not be the best way of acquiring this knowledge.
(~John Sweller, source)
The other issue is that if students aren’t yet fluent in some skill, I have a hard time seeing how they’ll get much better at it through high repetition in the context of such beautiful math. A dilemma: either they’re devoting attention to the beautiful math (then unlikely to be thinking much about the skill) or they’re working on the skill (not devoting much attention to the beautiful math). We can only attend to so many things at once, sadly.
(That said, I can imagine using this task in class in the hopes that some students would get some beautiful math and others would focus on skill fluency. That would be sad, because everyone should be able to focus on the beautiful math, but I can see myself making that decision.)
I can’t offer you Pershan’s Theory of Fluency Development, but I can say that ideally I want students to give as much attention as possible to the thing they are developing. Otherwise, development gets sketchy, I find.
Dan MeyerJune 23, 2016 - 9:55 am -
No apology necessary. Interesting issue you’re raising. One possible solution:
“Put the odd numbers anywhere you want. Now put their differences in the middle.”
“Okay, have any of the differences repeated? Yes? Okay, you can either shuffle the odd numbers randomly and repeat the subtraction or shuffle them systematically and repeat —Â your call.”
That’s a continuum — from random to systematic. It allows students to dial up their systematicity as their working memory resources allow.
I’m generally skeptical of Sweller et al’s skepticism that children can walk and chew gum simultaneously. And I know you’re skeptical that students convert from novices to masters abruptly, at a certain threshold of skill practice. These problems allow students to shift gears gracefully rather than abruptly.
Michael PershanJune 23, 2016 - 10:09 am -
I like your solution, but I see that as a solution for how to give everybody access to the math in the problem. Which is great, that’s an issue I raised in my comment and I think you’re right, there are moves to make the problem more accessible.
What I’m still having trouble seeing is how this will help students get better at subtraction.
timfcJune 23, 2016 - 10:23 am -
MP… Have you read Yuichi Handa’s book?
Or his ESM article?
You might really enjoy–the ESM piece is:
Teasing out repetition from rote: An essay on two versions of will.
I can get it to you if you’re interested and don’t have easy access.
Dan MeyerJune 23, 2016 - 10:38 am -
That’s the first math ed abstract I’ve ever seen to quote Martin Buber. Got my interest piqued.
Gordon HamiltonJune 28, 2016 - 9:25 pm -
The number one challenge of being a teacher is the spectrum of student ability. How do you engage the top students without losing the bottom kids? How do you engage the bottom students without boring the top kids?
Curricular unsolved problems are one answer.
Scott HillsJune 28, 2016 - 11:27 pm -
Gordon Hamilton….I agree 100%
My only real disappointment in looking at the link is that the project seems to have been a 1-time thing. (or I missed something)
The top students are already engaged, for the most part. The bottom students are the ones for whom “buy in” is necessary. I want my students working on thinking of
1. What is happening?
2. What else works like this?
3. What is necessary?
4. How can I guess ahead?
5. How can I describe what I’m seeing to everyone?
Even if not right, the answers to #5 are great for discussion.
Gordon HamiltonJune 29, 2016 - 4:30 am -
The project is not a 1-time thing. The objective is to get one unsolved problem in front of all students world wide once per year. It has failed to get to 1% adoption in any country.
Although top students are bought in – seldom is the problem worthy. Boredom is an insidious enemy that creeps up on too high a fraction of these top students.
Students do 1-4 naturally. They do not need to articulate these connections for them to be happening.
Occasionally I do extroverted math like you are suggesting in 5, so I agree that it belongs in a teacher’s arsenal. I especially do this when weaker students get a good idea and I want them to celebrate the success with a wider group.
However, most of the time I want sharing to be done with a single partner – the default for nearly all MathPickle activities.
Unlike most teachers, I usually do not wrap up lessons in the final 5 minutes. If the students are 90%+ engaged I can do nothing better than to let them keep working. Until proved wrong, I consider “reflection” time at the end of a class a useful, but overused tool.
I leave students with dangling questions and irritatingly unfinished work – not a comfortable wrap-up. Both approaches have merit. Teachers should experiment.