Redesigned: Follow That Diagonal

Which is a better treatment of that problem with the rectangle's diagonal? How are you defining better? Better for what purpose? Help me out here.

Schoenfeld

From Alan Schoenfeld's 1994 Math 67 midterm:

The diagonal of the 3 x 5 rectangle below passes through the interiors of 7 of the 15 squares that comprise it. In general, consider an N x M rectangle. Through how many of the NM squares that comprise the N x M rectangle does the diagonal pass?

Nowak

From Kate Nowak's blog:

Draw a 9 by 3 rectangle on a square grid. Draw one diagonal. How many squares does the diagonal pass through? Draw some non-similar rectangles with one diagonal. How many squares does the diagonal pass through? Develop a rule to determine the number of squares a diagonal passes through for any rectangle of any size.

Meyer

My own treatment, submitted for review, correction, and debate:

How many squares will the diagonal of the large rectangle cut through? [This question added because it wasn't clear I'd ask it - dm]

I'll follow up in the comments at some point on the decisions that went into my redesign.

2011 Dec 1. Check out David Cox's parallel investigation of this problem, leading to an incredible Geogebra applet.

NCTM President Michael Shaughnessy Responds To My Revision Of His Geometry Task

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.

[3ACTS] Some Really Obscure Geometry Problem

At the NCTM Institute last month, we broke into task groups to discuss reasoning and sensemaking (the conference themes) in content focus groups. I slipped into Geometry a little late and found a seat. The group was discussing approaches to this problem:

This was the session immediately following my keynote and the difference between the tasks I had described and the task they had just finished was stark. Someone asked, "How would we apply Dan Meyer's approach to this problem?"

I ducked.

It isn't fair. It's apples and oranges. Paper is a great medium for a lot of math problems. Paper is a terrible medium for representing how people apply math to the world outside the math classroom. My techniques for one problem type have limited use for the other. My enthusiasm for one problem type shouldn't be mistaken for a lack of enthusiasm for the other.

That said, I don't find myself terribly enthusiastic when I think about assigning this problem to Geometry classes I have taught. As a challenge problem or extra credit, sure, but in its current form — with the abstract mathematical language and symbology smacking you right in the face — students are going to wonder, "Who comes up with these problems, seriously?"

If we make a better first act, though, we can engage, I dunno, 17.2% more students without any cost to the math. That's empirical, friend.

Here's the redesign:

  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.

Act One

Some Really Obscure Geometry Problem from Dan Meyer on Vimeo.

  1. Start with a square.
  2. Draw the diagonals of the square.
  3. Ask students to tell you what percent each of those regions is of the whole. This is insultingly easy and that's the point.
  4. Drag the endpoint of one diagonal halfway down the side of the square.
  5. Ask them, "How about now?"
  6. Ask them to guess the percents again.

Watch the video. Basically, we're applying pressure to their confidence, which is how I try to approach pure math problems. Start from what they know. Then mess with it in some trivial way (eg. we just dragged the endpoint down a little) that requires math that is anything but trivial.

Act Two

You and your students will begin to find it very difficult to talk about all these different segments and regions without labels. So add them. A recurring point around here is that if you want to disengage a lot of students who might otherwise be engaged in the math, simply start the problem with as much abstraction as possible. If you want to engage those students, don't introduce that abstraction until students know why they should care about it.

Act Three

You've been walking around and taking note of different solution strategies, right? Have students come up and explain those different strategies. Then show use this Geogebra applet to show the percentages changing, in case anyone still needs convincing.

Sequel

The sequels here are really, really great.

Suppose M cuts side CD so that MD = n • CM. What are the ratios of the areas of the four regions?

Send n to infinity and watch the fireworks.

Again, though: print-based media require you to keep everything on the same page — the sequels in the same visual space as the original problem. I realize that math teachers by nature don't mind that. Do students?

Featured Comment

J Michael Shaughnessy, President of the National Council of Teachers of Mathematics and designer of the problem under discussion in this post:

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.

2011 Aug 29: My response here.

Redesigned: John Scammell

So John Scammell uploaded this #anyqs, which captured an interesting moment. In his tweet, he wrote, "When I was a kid, I'd grind other kid's pencils down to nothing."

John Scammell — Original from Dan Meyer on Vimeo.

Some things I'd like to accomplish in the redesign:

  1. Get the camera lens parallel to the pencil, an angle that makes it easier to see the length changing.
  2. Convey to the student visually what John wrote in his tweet: that this pencil is about to get ground down to nothing.
  3. Postpone the pencil measurements until the second act. The moment where John measures the pencil is useful and necessary but the first act (the #anyqs) should focus exclusively on curiosity and context. The math introduces itself later in act two to help resolve that curiosity.

Act One

Pencil Sharpener – Act One from Dan Meyer on Vimeo.

Act Two

Pencil Sharpener – Act Two from Dan Meyer on Vimeo.

Act Three

Pencil Sharpener – Act Three from Dan Meyer on Vimeo.

The Goods

Download the full archive. [10.8 MB]

Toaster Regression

David Cox has WCYDWT by the throat. He used digital video, Adobe AfterEffects, and MovieMaker to export a clever visualization of toaster times versus toaster settings.

Toaster Question from David Cox on Vimeo.

Not that he asked, but I wouldn't change a lot here. I'd rather see the data for settings one through four and use those to regress the eighth setting. By providing the seventh setting and asking for the eighth, he's made it easier for students to jump right into the math which makes it less likely that my remedial students will invest a guess.

I would have also sped up the first four videos (even more) because I want my students' impatient toe-tapping aligned to the question, "when will it end?" not before.

It's really strong work, though, and you're only going to see more of it from David because it just gets easier and easier to clear the annoying technical hurdles of video production. Soon he won't even notice them and it'll be as if there isn't anything in between the curriculum he can imagine and the curriculum he can create.