Deborah Loewenberg Ball, the full quote, so solid, knocking me back into line a bit:
It’s interesting because these problems – speaking of context – aren’t really contextualized. We’re not making them into fake cakes or breads or anything. They really are just shaded rectangles. And the kids are finding them very challenging and interesting. So I do think, on the question of context, it’s worth remembering that mathematics itself is a context and that puzzle-like problems are often both very engaging for kids and good equalizers because kids looking at those diagrams aren’t shaped by some of those same inequities about kids’ experiences. There are certainly differences in their experiences but they’re not the same as problems about how cakes get shared or other kinds of real-world things.
DavidCAugust 18, 2010 - 6:29 pm -
I like it! Thanks.
Katie WaddleAugust 18, 2010 - 8:05 pm -
Deborah Ball is really awesome. And yes! Mathematics is a context. I have always been driven by math for math’s sake, and I steadfastly refuse to believe that my students can’t get excited about this also. Haven’t figured out how it works yet (being a first year teacher and all), but I’m working on it.
Matt EAugust 19, 2010 - 6:23 am -
Speaking of math being its own context–Dan, if you’ve already done this, I missed it, but I would be curious as to your take on Paul Lockhart’s “A Mathematician’s Lament” (either the original PDF or the derivative paperback). He very much believes that math is its own context, and that the very thing that makes math fun is its total irrelevance to our lives!
Mary Jude SchmitzAugust 19, 2010 - 8:55 am -
Love this quote! I will pass it around to all the math teachers in my school and I whole heartedly agree that math is a context in itself. Many of my students get excited when they are able to finish a proof just because they solved a puzzle and feel a great sense of accomplishment just for that simple reason.
Sam CritchlowAugust 19, 2010 - 4:02 pm -
Or, as we put it in our department-wide essential understandings: “Mathematics is a realm of inquiry in addition to being a tool.” A question ahead of us is to what extent do we pursue “pure” mathematical topics vs. applications (both as ends in and of themselves and also as skill-building tools). We are lucky to teach in an independent school that leaves most of these questions open to us.
Matt E: recently reread “A Mathematician’s Lament”. Some interesting contrasts to the ideas on this blog, but many complementary ones as well:
“A good problem is something you don’t know how to solve. That’s what makes it a good puzzle, and a good opportunity. A good problem does not just sit there in isolation, but serves as a springboard to other interesting questions. A triangle takes up half its box. What about a pyramid inside its three-dimensional box? Can we handle this problem in a similar way?
I can understand the idea of training students to master certain techniques– I do that too. But not as an end in itself. Technique in mathematics, as in any art, should be learned in context. The great problems, their history, the creative process– that is the proper setting. Give your students a good problem, let them struggle and get frustrated. See what they come up with. Wait until they are dying for an idea, then give them some technique. But not too much.”
Sounds a lot like “less helpful.”
BreedeenAugust 19, 2010 - 9:00 pm -
I’m a little bit in love with Deborah Ball. In fact, just about every math teacher I know is a little bit in love with her. And I don’t fully trust the ones that aren’t.
Dan MeyerAugust 20, 2010 - 7:14 am -
I read it on the plane en route to NYC for TEDxNYED. I finished rehearsing my talk silently to myself over Nevada and then read Lockhart over Colorado. Afterward, I could only think, “… this guy already said it all.”
I concede the dissonance between our work – I love finding math in the world around me; he emphasized math as an end to itself – but where we line up, I think, is behind a) the power of a visual, and b) the importance of problems that announce their constraints quickly and clearly.
Also, Sam has a sharp eye, especially with the last bit: “Give your students a good problem, let them struggle and get frustrated. See what they come up with. Wait until they are dying for an idea, then give them some technique. But not too much.
Brendan MurphyAugust 20, 2010 - 4:11 pm -
Katie Waddle your students will be excited when they feel your passion, after they stop laughing that is.
As a title I math resource I like to give problems like these to students in the hall. Most just blow it off, but for one or two kids a year they come back every couple of days or so and try to give an answer just so they can get a new question.
Clint ChanAugust 21, 2010 - 7:49 pm -
That is a great quote! I completely agree – puzzles and thought problems are things that I think people have fun thinking about, provided that the question is understood. I think a lot of times, especially with advanced abstract math problems, students give up or lose interest because the question doesn’t make sense. When the context makes sense, then students can engage with it. Last year in my precalc class during our unit on trig identities, I was a little worried because it’s something that can be a bit dry. However, we spent lots of time not only doing the abstract manipulations with trig functions, but also doing examples with numbers. We got to the point where students could have a productive discussion as a class and in their groups about some pretty involved manipulations. It felt successful when the discussions got a bit heated because it showed that students had confidence that they knew what the problem was asking to be able to think about and engage with it, arguing and defending their steps in a discussion with their peers.
ClimeGuyOctober 5, 2010 - 2:26 pm -
Math E writes: “He [Lockhart] very much believes that math is its own context, and that the very thing that makes math fun is its total irrelevance to our lives!
That’s of course assuming that the student buys into it. What makes it pseudo is not whether math is its own context or not, but whether the student buys into it and wants to do it.
With all due respect to DLB, the problem that Dan start’s with is interesting only if the student thinks so. Which means the teacher has to use it in a context that is appropriate for his/her student(s).
I think there are contexts that are more intrinsically interesting to more kids than others. (That’s why I didn’t think the pouring water into the tank problem was all that much.) The trick is to come up WCYDWT ideas and build a pedagogical context with good math engagement that will work for most of your kids.
Dan MeyerOctober 5, 2010 - 3:01 pm -
ClimeGuyOctober 5, 2010 - 3:16 pm -
Dan, The secret of your success: Great context!
After the students made their predictions, I abandoned the water tank problem and moved on to something completely different. In each one of my classes, eventually a few of the students made a comment along the lines of “you never told us how long it took to fill the tank.” Sometimes the comment came only a few minutes after we had moved on. Other times, it came much later. More convincing evidence of the students’ level of engagement in the exercise came at the end of the lesson when I played the rest of the video.
That reminds me of a time (back in the early 70’s) when I boldly told my class that there were more VW bugs on the road today than another specific brand/model of car. Of course, I was wrong, but…
Later during lunch I watched as several of my students skipped lunch and were counting cars to prove me wrong!
Imagine telling kids today that their assignment is to count how many times a particular brand of car passes during their lunch hour? I would have to threaten with fire & damnation to get them to do it.
PS VW bugs were quite popular in those days. So they did count quite a few.
JordanOctober 23, 2010 - 4:42 pm -
Does anyoen actually answer any of these questions?!
Its quite frustrating!
JordanOctober 23, 2010 - 4:42 pm -
Does anyone actually answer any of these questions?!
Its quite frustrating!