dy/dan

Here is one of my favorite quotes on task design from one of my favorite math educators:

A good problem seems natural. A good problem reveals its constraints quickly and clearly.

Is it possible to pose a task so quickly and clearly that it would fit in a tweet?

What's the best math task you can pose in a single tweet? Mine's next.

— Dan Meyer (@ddmeyer) February 8, 2013

I asked and lots of you gave it a shot. Here's mine as well as a few of my favorites:

You're standing at one wall in a room. You have to touch every other wall in order and return to your first spot. What's the shortest path?

— Dan Meyer (@ddmeyer) February 8, 2013

@ddmeyer @dandersod: @ddmeyer which grows faster, x^10, 1.1^x, or x!?

— Dan Anderson (@dandersod) February 8, 2013

Hold out your hands.Make your middle fingers touch, and thumbs touch.What's the largest possible area of the enclosed space? @ddmeyer

— Bob Lochel (@bobloch) February 8, 2013

@ddmeyer At what exact time does the minute hand cross over the hour hand. Find all occurrences to the nearest second.

— Nico Rowinsky (@NicoRowinsky) February 8, 2013

@ddmeyer How many ways can you uniquely arrange the letters and spaces in this tweet?

— Mr Pett (@mrpettmaths) February 8, 2013

.@ddmeyer 1st one that came to mind: Create a metric to define squareness, ie how close to a square a shape is.

— Avery (@woutgeo) February 8, 2013

@ddmeyer how much would it cost to cover our sand/dirt catastrophe of a field with field turf vs grass?

— Pete Hanson (@hansonpt) February 8, 2013

@ddmeyer you have two squares, each with area = 1. How can you cut them up to form a single square with area = 2?

— John Meerse (@jmeerse) February 8, 2013

@ddmeyer Compare the chocolate to peanut butter ratio for various sizes & shapes of Reese's.What is the optimal ratio?

— Jennifer Lawler (@jenniferklawler) February 9, 2013

@ddmeyer your watch says 3:22 pm, but its set to London time, and the sun is directly overhead. Where are you?

— Mark Betnel (@markbetnel) February 9, 2013

@ddmeyer take any map. What is the best method to define fair districts? (School, voting, electrical service, ect,)

— Molly Olson (@mathdancer) February 9, 2013

Extra merits for roping in your personal life:

@ddmeyer family game night makes me want to ask my kids: How many scenarios do you think are possible in the confidential case file in clue?

— Joy Coates (@JoyCoates1) February 10, 2013

Extra demerits for trolling:

@ddmeyer Melissa & Joe r playing a game w/ complex #'s.If Melissa has a score of 5-4i & Joe has a score of 3+2i, what's their total score?

— Eric Newman (@newman_eric) February 8, 2013

@ddmeyer You are folding a bandana to tie around your dog's neck. Which special right triangle should you use? #TrollSoHard

— Joe Lower (@lowerjos) February 8, 2013

I'll depart from Sallee briefly and say that it's nice, sometimes, when the constraints aren't fully revealed. I'd like the task to be clear, but in life the constraints often require clarification. When you ask yourself, "What extra information do I need here?" you're doing the work of mathematical modeling.

@ddmeyer where are you starting? The middle of the first wall? The corner of the room next to two walls? Does it matter?

— Kira Christensen (@MissCWHS) February 8, 2013

@ddmeyer: What do you mean by "in order"?

— Javier Moreno (@bluelephant) February 8, 2013

Feel free to play along in the comments, but you'll have to constrain yourself to 140 characters.

Featured Tasks

Jonah:

Two points A and B on a paper, 13″ apart. You have a pencil and a 12″ ruler. Construct the line segment AB.

Caitlin Browne:

How many squares are on a standard checkerboard?

Bowen Kerins:

Is it really possible for Steven Seagal to have “millions of hours” of weapons training?

Elizabeth:

Am I imagining it, or are the participants (posters and respondents) mostly male? I’d love to be wrong about this. If I’m not wrong, then why would that be the case? And more importantly, has anyone noticed whether there is there any difference in class participation between female and male students when these are used in class?

I don't ask for your gender during the registration process so it's hard for me bring any data to bear on the question. But if I allow myself some conservative guesses, it seems that at the time of this writing:

1. the top ten most perplexing users are all male,
2. nine of the top ten most perplexing first acts were uploaded by males.

So help me, I can't figure out how the interaction on the site (ask a question and click "skip") or the nature of the tasks (a context and a question) preferences men. The reviews are all blind, too. I'm looking at a photo. Maybe it was uploaded by Candice Director. Or maybe by Dan Anderson. It's impossible to know until later.

I'm highlighting Elizabeth's comment to see if anyone can help me figure this out. I'd rather this didn't turn into a general complaint window, though. I'm interested in locating the source of any gender bias, not in airing out any other grievances.

BTW: My adviser has done a lot of work in gender and math. I should probably check in.

Featured Comment:

Too many. A really great discussion down below. Here's a link to my summary.

Hola, amigos. I'm back from Spain, back in the game after sidelining myself for a helluva comment thread. It turns out that NCTM President Michael Shaughnessy designed the task that I critiqued in a recent post and he stopped by with a few notes on my redesign.

Michael Shaughnessy:

Not all math problems have to be posed everytime in a a high tech environment. Sure, it’s ‘cooler’ that way, but i completely disagree with your comment on this one, about ‘how the problem was posed.’ It’s only boring in the beholder’s eyes, depends on how it’s pitched to a group.

The last line seems to contradict itself, though. Either boredom is in the eye of the beholder, in which case we should just pose the task however we like and accept that it simply won't engage some students or engagement depends on how the task is posed, in which case we can discuss productive ways to pose it. They both can't be true, though.

I figured there were three productive ways to pose that task, three revisions to Shaughnessy's original problem that would open it up to a few more students. I'm quoting my original post here:

1. Show how this new, difficult problem arises from an old, easy problem.
2. Make an appeal to student intuition.
3. Introduce abstraction (labels, notation, etc.) only as a necessary part of solving a problem that interests us.

What's interesting is how many critics, Shaughnessy included, saw a video and assumed I was aiming at something "high-tech," "cool," and "hip." But those are beside the point. The point is helping more students access an interesting problem. Video was the means, not an end.

Shaughnessy also reports having "gotten a LOT of mileage out of this problem with middle school kids, high school kids, perspective teachers [sic]" without anything fancier than the paper the problem was printed on. I don't doubt that's true. But if that brief video opens the problem up to even one more student, my only question is why not? Why not get a little more mileage out of the problem? What's the downside?

While most critics decided early on that I was just trying to buy off the YouTube generation with something shiny, I was grateful that Tom I. critiqued the redesign on its own terms:

… it seems like Dan is always recommending that we (more or less) apologize to our students for the abstractness of math. The abstractness makes it hard, but must we assume that it makes math pointless and uninteresting for our students?

Abstraction doesn't make math harder. Abstraction makes math possible. It's one of the most powerful and satisfying tools in the mathematician's box. The trouble is that you can't abstract a vacuum. You start with something concrete (not necessarily "real-world") and then abstract its essential features. Again: you start with something concrete and then abstract it. Over and over again, though, math curricula provide both the concrete and the abstract simultaneously, one on top of the other. This is unnatural. (R. Wright puts it artfully: "This is a charming problem when posed simply and innocently, not flayed alive by terminology, labels, and notation.") Unnatural abstraction is boring and intimidating. When we put abstraction in its rightful place as a tool for simplifying the concrete, it's interesting and empowering.

Other Featured Comments

Debbie:

By starting off with a very familiar problem-style and seeing you apply your approach to it I think I’m finally convinced that this isn’t a one-trick pony but something that can work with all sorts of maths.

Bowen Kerins:

I also want to point to some language used in the discussion here. The initial problem is “insultingly easy”, while the later problem is “trivial” (Alexander’s comment). This is in the eye of the giver of the problem, not in the eye of the recipient.

This is a strong point and I'll mind my manners going forward. Rephrasing: the goal isn't to start with a problem every student will find easy. The goal is to show how something relatively simple quickly turns into something relatively more complex.

Tom I:

I bet 9 out of 10 readers of this blog thought [Shaughnessy's original] was a fun problem and felt an itch to solve it. Why wouldn’t students feel that way?

Because there isn't a one-to-one correspondence between things math teachers like and things students like. They aren't like us. Please: do whatever you can to imagine what it feels like to walk into a math class as a high school freshman who's been convinced since fifth grade she's stupid, who's now on her third year of the same Algebra class. She isn't thrilled by the same mathematical investigations you and I are. She's threatened by them.

If I cut my teeth teaching honors kids in Fairfax County, I imagine this would be a very different blog. I'd have a very different career. As it is, they tossed me to the wolves in my third year teaching and I had to make friends in the wild. I couldn't be more grateful for the empathy that experience required.

Carlo Amato:

What program do you use to construct this video?

Dvora:

On the tech side of things… how did you create the video? What programs did you use?

All Keynote. Let me see what I can put together for Keynote Camp.

Bob Parker, in the comments, re the speed of cash transactions vs. credit:

Credit can’t be slower then cash – Visa told me so. They told me I am a social pariah if I pay in cash.

I believe he refers to this ad spot:

Jerram Froese, taking offense that anyone would walk out of a conference session after it started:

Damn. If only our students could stand up and walk out of class at any time. Wow, people.

That's a mixed metaphor right there, but an essential hypothetical for any classroom teacher and especially for core subject teachers who are (overall enrollment notwithstanding) guaranteed an audience year after year.