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.

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.

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.

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 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.

Treadmill WCYDWT from Alexander Eckert on Vimeo.

An e-mail from reader Kara Monroe:

I had a moment on the bike at the gym last night thinking of all the different questions student would naturally form just from looking at the display on a stationary bike.

I’m obliged to Kara and Alex for the inspiration here. A few remarks on Alex’s video to preface my redesign:

1. I’d like my students to look at the world and formulate their own mathematical questions. Therefore I’d like to show them as good of a facsimile of the world as digital video will allow. This means no artifice like a soundtrack or text on the screen. This also means I prefer fixing a camera to a tripod so the students aren’t distracted by this third party holding the camera.
2. Part of formulating and solving a question is deciding what information is important. So I removed the part where Alex tips them to the percent and the time elapsed: “Watch closely: 31:30 … 90% complete.”
3. I don’t include this in my redesign (which I faked from elements of Alex’s video) but you’d want to film the rest of the exercise session in order to show students the answer.

After:

Redesigned: Alex Eckert from Dan Meyer on Vimeo.

Academic Green Circumference and Area Problem from Kyle Webb on Vimeo.

First, let’s pay respect to how fast the video moves, how it sets a scene and establishes a problem in just 14 slides and 57 seconds. Webb knows his audience and its attention span. Also, none of this is stock photography. Every photo selected is of high bandwidth and relates directly to the problem. After 12 seconds, we have three different views of the lawn. After 15 seconds, a panoramic shot. I’ll begin my redesign 23 seconds in, when he mentions the lawn is 75 steps across.

This is really, really close to my textbook’s own installation of the problem. The text would ask a question like “how far is it around?” or something with a real-world spin like “how large would the ice rink be?” (standing in for “what is the area?”) and then it would explicitly define the only variable we need: 75 steps. My students would identify the formula and then solve.

This kind of instructional design puts students in a strong position to resolve problems the textbook draws from the real world but in no position to draw up those problems for themselves. This kind of instructional design also yields predictably lopsided conversation between a teacher and his students.

The fix is simple but difficult: be less helpful.

Let’s start here: is circle area just something math teachers talk about to amuse themselves or do other people care? If they care, why do they care? How do we convey that care to our students? Maybe someone needs to fertilize the lawn. Maybe someone wants to spray paint the dead lawn green in the winter. Without this component, the answer to the question “how far is it around?” is little more than mathematical trivia to many students.

So put them in a position to make a choice, a tough choice that’s true to the context of the problem, a choice that math will eventually simplify.

For instance: “how many bags of fertilizer should I buy to cover the entire lawn?”

Or, a little weirder: “how many cans of spray paint should I buy to cover the entire lawn?”

In both cases, we’re putting every student on, more or less, a level playing field. They are guessing at discrete numbers (ie. “fifty bags – no – sixty bags.”) and drawing on their intuition, which, from my experience, is a stronger base coat of for mathematical reasoning than the usual lacquer of calculations, figures, and formula.

This approach also forces students to reconcile the fact that the problem is impossible to solve as written. This is an essential moment. They need more information, but what? What defines a circle? Would it be easier to walk across the lawn’s diameter or around the lawn’s circumference? Which would be more accurate? Why is the radius difficult to measure? Did Kyle really walk through the center of the lawn or does he just think he did?

When you write “75 steps” on a photo, that conversation never happens.

My thanks to Kyle for jogging my thoughts here.

1.

Tony Alteparmakian is a 2009 Leader in Learning enacting Chris Lehmann’s vision of classroom inversion (though I don’t doubt they came to the idea separately). Their idea is that we should send our students home with what used to constitute classroom time – the lecture – and spend classroom time on labs and teacher-led enrichment of that material.

Obviously, that vision comes fully loaded with complications but Tony is resolving them one-by-one in a how-to series that has only just started.

Also, I dig his redesigns. It’s hard to argue with slide transformations like these.

Before:

After:

2.

Sean Sweeney is an extra-value meal. In one corner of the edublogosphere you have the edtechnologists, the district IT staff, the ICT professionals, the policy wonks, etc., all asking huge, important questions about merit pay, technology integration, assessment, online schooling, etc., and posing reckless hypotheticals about limitless resources with nothing less than the future of education at stake, and all of it makes me grateful for guys like Sean who are driving 90MPH up the right lane, offering educators something they can use in the classroom right. now.

I’m talking about his quadratic catapult project. Or his Graphing Stories remix. Or his exercise in grocery store estimation. And that’s his output over two weeks.

This is math-instruction-as-artistic-expression and it’s cool as hell to watch.

Michelle Baldwin, dissenting from the comments:

In considering Dan Meyer’s arguments, I don’t really agree with him. At all. It’s all about finding the “right” photo to enhance the text.

Is that what presentation is all about? Witty aphorisms and inspiring photos?

You have a thesis. Let’s assume there are very real, really real real-world implications to your thesis. Why not cut to that chase? Why make an abstract matter like edutechnology even more abstract with dramatic photography and 140-character pullquotes from your Twitter feed?

1. Show me something real.
2. Give me a space to interact with it.
3. Let me have your thoughts on it.

In this case, if learning really is social, please show me examples of that social learning. Or show me examples of how dangerous it is when that learning is taken out of a social context. If you find it difficult to connect your thesis to video or screenshots or sound clips (“multimedia,” basically) then it’s possible you are chasing down the wrong thesis or that your thesis doesn’t lend itself to a presentation mediumI caught David Jakes’ Black Coffee presentation on Slideshare last week and was impressed that something like 95% of its 63 slides were screenshots, archival photos, YouTube videos, newspaper clippings, etc., etc. Jakes had done his groundwork..

I like that Darren modified the stock photography (adding the “Learning Is Social” placard) to connect it better to his thesis than the average stock photo slide but I wonder if we’re approaching the question, “What is presentation?” along two different vectors.

Then

Now

Something I have been completely wrong about is the best way to use slide software in a math class. A few years ago I wrote a design series explaining how I use color theory, grid systems, etc., to clarify complex procedures, but the whole thing turns out to be simultaneously a) a lot more fun and b) a lot less time-consuming than that.

My reversal in slide design reflects a shift in my math pedagogy also. Far more important to me now than “developing fluency with complex procedures” is “developing a strong framework for interpreting unfamiliar mathematics and the world.”

I’m not trying to set up a false dichotomy here. We do both. Both are important. But all too often slides like that first one, with the classroom dialogue and solution method predetermined, cordon off classroom dialogue and student reflection onto very narrow paths. That kind of pedagogy does nothing to unify mathematics, tending, instead, to position complex procedures in isolation from each other, which is a very confusing way to learn math and a very laborious way to teach it.

Instead, I want my students to focus without distraction on a) how new questions are similar to old questions, b) how tougher questions demand tougher procedural skills, asking themselves c) which of their older tools can they adapt to these tougher questions?

For example, I put six equations on separate slides, equations we have seen. I asked, “how many answers are there?” One. Two. Zero. Etc.

Then I put up an inequality, tweaking the problem slightly, and quickly.

They told me there were lots of answers. I asked my students to start listing them. “7, 6, 5, 4.2, 4.1, 4,” etc.This became tiresome quickly and made the introduction of a graph – a picture of all those answers – clear and necessary.

Slide software makes it easy to sequence these mathematical objects, ordering and re-ordering them to promote contrasts and complements. Slide software lets me sequence these mathematical objects quickly, from anywhere on the globe, from photos and videos I take, from movies my students watch, from textbooks too. Graphic design is useful to mathematics, but I am happy to have discovered certain constraints on that usefulness and, simultaneously, higher fruit hanging elsewhere.

It is the curation of this mathematical media that interests me now, though I reserve the right to return to this space shortly and reverse myself again.