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.

### 58 Responses to “Redesigned: Follow That Diagonal”

1. on 01 Dec 2011 at 8:52 amMBP

I worry that there isn’t enough processing time in between the first two examples that you show. If I were doing this with a class, I’d leave up those images for a lot longer.

2. on 01 Dec 2011 at 8:53 amMBP

And by “first two” I really just mean, after the second and after the third.

3. on 01 Dec 2011 at 10:02 amJohn Golden

Pretty slickety animation!

How does the final rectangle communicate any rectangle? Why not an (n,m) vertex to echo Schoenfeld and your first examples? Or a variety of rectangles as the final image?

Why 7×10 for the first non trivial? Are you consciously suggesting classes or scaffolding approaches?

After 7×10 I could imagine a quick montage or 3-7 other rectangles.

When you want to ask a specific question, why not ask the question?

4. on 01 Dec 2011 at 10:20 amGert

First off, I don’t think there should be any more time between the first two examples. At this stage the students don’t know what they are examples of or what the problem is. At the start of the video one is just counting squares – and the viewer reaction might be “hey, this is just plain counting, this is easy, I can do this”. When the last rectangle comes on screen the reaction might be “this might not be so trivial after all, I can count all those squares but there must be a faster way”. That is one of the things I like about Dan’s treatment: finding the general result is not a question that is asked by directly by the teacher, but the viewer formulates the problem seeing the video. This might also be a good starting point for discussing with students why we generalise in mathematics.

Dan’s approach shows some of the flexibility that is possible using video. In Schoenfeld’s treatment, the problem is written down on a test so it becomes hard to motivate the problem (and the midterm is probably motivation enough to tackle the problem). In Nowak’s approach which seem to be based on a dynamic worksheet, it becomes natural to consider different cases but it is not so clear why one would want to find a general result. In Dan’s video, the sudden inclusion of the huge example generates a need to find a general result.

Second, there are many different approaches to which examples to show. Schoenfeld chooses two relatively prime numbers, Nowak chooses a number that is a multiple of the other, and Dan includes both. The examples chosen has relevance for the exploration the students engage in and what initial conjectures they make. It depends on what the aim of the problem is. If students are not used to making conjectures I guess I would start with some examples of relatively prime numbers, which rather easily leads to an initial conjecture. This would then give opportunities at testing and modifying conjectures. If student are used to making conjectures I guess I would start with some examples similar to Dan’s, then a main aim of the students’ exploration would be what type of exploration is needed to be able to make fruitful conjectures. If one on the other hand is mainly interested in the underlying mathematics, then I would choose examples such as (7,10), (14,20), (21,30) to introduce the problem. The correct conjecture is more easily arrived at from these examples (that is if the underlying mathematics one is interested in is the greatest common divisor, if one wants the students to practice slope the other approaches might be better).

Third, I tried to find an algebraic reformulation of the problem that could be used with students, but I only arrived at the longwinded (many tables of multiples of two inverse fractions, with the essential rows coloured), the obtuse (“For a fraction a/b how many pairs of numbers c and d exist such that c*a/b=d and d*b/a=c?”) and the mundane (“Given a fraction a/b, how often is a/b=c/d where c is less than a?”). So I am still without a formulation that I believe will engage most students. The last formulation however has potential in combination with the above geometric problem. The geometric problem could come after one has looked at equivalent fractions, or a part of the follow-up to the geometric problem could be a discussion on the connection to equivalent fractions.

5. on 01 Dec 2011 at 10:21 amKate Nowak

I like the video too, but I guess I’m puzzled by what it adds to experiencing the problem statement that is not provided by my three successive questions (that include an invitation to explore any number of rectangles – and the implication that the learner will need to decide what different cases are interesting) + the ggb to play with to take the imprecision out of hand-drawing? It would be nice if the intersected squares could light up on the ggb. Hm.

Is it just that the video provides a way to beg the interesting question instead of explicitly asking it?

And is that really such a pedagogical advantage, that it would be worth my time to learn how to produce such a video?

I swear I’m not being defensive; I’m thrilled that I’m famous on dy/dan today! Woo! I’m sincerely curious about these questions.

6. on 01 Dec 2011 at 10:22 amdan goldner

I’m don’t know whether watching the diagram being drawn increases the allure of the problem. It doesn’t for me but it might for others.

I think dynamics could help students solve it, though. In my own thinking about this problem, I imagined visualizing moving the upper right corner around, seeing the diagonal move along with it, watching the occupied squares light up … and seeing different patterns light up as the diagonal passes through one or more nodes of the graph.

A minor thing: the lighting at the end made the surface look curved (which got me thinking ….)

7. on 01 Dec 2011 at 10:24 amDavid Cox

Using Schoenfeld’s or Kate’s treatment of the question dumps the kid into the open ended abyss that can be very overwhelming. Using two simple examples where counting is quite easy and then opening it up to a much larger example where counting is still possible puts the kid in a position where she says, “yeah I can count, but dang, it’s gonna take a long time. I wonder if there’s a shortcut.”

I kinda favor your treatment for that reason. I’m still struggling to reconcile the message of your previous post of “just ask the question” with this one. That is, unless your intention is to start with a closed question (ie. How many squares does the diagonal pass through in a 67×35 rectangle?) and eventually open it up to an open ended create-a-rule type question.

8. on 01 Dec 2011 at 11:58 amMichael

Isn’t there a significant difference if nxm is a perfect square or not? So, shouldn’t there be “a special case” video for the perfect squares or non-perfect squares, or do we want students to come up with that on their own?

Maybe, show the perfect squares video first, then show the non-perfect square video after students get a general understanding and background knowledge of what to look for so that they can have an intuitive guess for any nxm non-perfect square rectangle.

9. on 01 Dec 2011 at 12:12 pmJohn Berray

I favor Dan’s video, if students are not given the chance to explore their own virtually created rectangles in Geogebra or GSP. Kate’s prompt seems to be one of directions to a hands-on experience, which I think wins best learning experience nearly every time.
But I would ask the question, and in this way: “IS there a pattern to the number of squares the diagonal passes through in a rectangle?” after showing them Dan’s video. I’m not sure a typical high schooler would be convinced there was a pattern. Then follow it up with, “So what’s the pattern?”

I’d use Kate’s method to get them creating rectangles of their own and have them log their findings.

Schoenfeld’s version is dizzying it moves so quickly from specifics to generalization. There has to be an invitation to join the square counting party, which it lacks.

10. on 01 Dec 2011 at 12:41 pmDebbie

I also favour Dan’s video. Firstly, articulating the context and asking the question are both done with my words. I don’t have somebody else’s language as a barrier.

My favourite aspect is the zooming out.

One distraction I had comes from having seen the original question. Where I’d expect to see (10, 7) I actually read (7, 10). I wonder if that’s deliberate and possibly too misleading.

11. [...] the way, there’s an interesting problem kicking around Dan Meyer’s blog.  But now for something completely [...]

12. on 01 Dec 2011 at 1:36 pmTim Hunt

@Michael nothing special about perfect squares, but it does matter if the sides are co-prime, I think …

13. on 01 Dec 2011 at 2:23 pmKate Nowak

Not to be argumentative Dave but its not an open-ended abyss if the teacher is RIGHT THERE. Isn’t that like half of our job? To know how to make good problems like this less abyssy for the ones who need us to?

14. on 01 Dec 2011 at 2:51 pmDavid Cox

Keep in mind, Kate, that I’m looking at this through the lens of a middle school teacher who has 2/3 of his kids less than proficient. As soon as I say, “develop a rule…” they’re gone. Dan’s version let’s the student decide whether or not to create the rule.

And yeah, what you’re talking about is at least half our job.

15. on 01 Dec 2011 at 3:18 pmDan Meyer
MPB: I worry that there isn’t enough processing time in between the first two examples that you show. If I were doing this with a class, I’d leave up those images for a lot longer.

Two days ago I swapped out a video that was about eleven seconds longer with the one you see here. I cut all eleven seconds from between the examples. I felt like the video dragged. I might have been wrong. I think these detail-oriented discussions are important to have. Mainly I’m curious about the limitations and affordances of video, generally speaking.

John Golden: When you want to ask a specific question, why not ask the question?

David Cox: I’m still struggling to reconcile the message of your previous post of “just ask the question” with this one.

I need to mind my p’s and q’s on this. I would absolutely ask the question “How many squares will the diagonal of the large rectangle cut through?” (Since added to the body of the post).

Gert: In Dan’s video, the sudden inclusion of the huge example generates a need to find a general result.

David: … counting is still possible puts the kid in a position where she says, “yeah I can count, but dang, it’s gonna take a long time. I wonder if there’s a shortcut.”

Gert and David articulate this aspect of my aesthetic really well. Both Schoenfeld and Nowak explicitly assign the general rule. I’d like the general rule to emerge from a question that demands it.

The larger rectangle also lends itself to guessing, which is important to me. “How many squares do you think it’ll cut through? Write down your best guess. While you’re at it, write down a number you think is too high, and too low.”

Kate Nowak: I guess I’m puzzled by what it adds to experiencing the problem statement that is not provided by my three successive questions

The appeal of video is more in subtraction than addition. The video does so much heavy contextual lifting for me that all I have to do is say twelve words to start the investigation. Many students wouldn’t even need me to say anything. (Which doesn’t mean I’d ask them to guess the question in my head.)

The language demand of Schoenfeld and Nowak, meanwhile, is higher. The language demand will close some students out of participating in a problem they’d be curious about otherwise. The video codes much of that language visually.

Kate Nowak: It would be nice if the intersected squares could light up on the ggb. Hm.

David, you’ll include that in your upcoming post, right? I’ll post it here otherwise.

Kate Nowak: And is that really such a pedagogical advantage, that it would be worth my time to learn how to produce such a video?

Definitely not. Do not learn to produce video on account of any (theoretical) gains posited here. I’m asking the question “Which is better? And for what purpose?” in the abstract. Given the video’s existence, let’s talk about it.

16. on 01 Dec 2011 at 3:37 pmDan Anderson

Better or worse is a difficult question. I think it depends on the audience. I think David is right that Nowak’s (you’re published now, so last name it is) or Schoenfeld’s approach wouldn’t work for lower level students as well as Dan’s video. But for upper level students, I think it is important that they are able to translate the problem on their own without the frosting of a video. I think they’ll hook onto the problem just as well (or even better) using Nowak’s approach.

17. on 01 Dec 2011 at 3:50 pmBowen Kerins

Detail-oriented baloney:

- I don’t think 1-by-2 is the best choice for the first example in the video. Since the diagonal goes through all the squares, it’s not clear what is being counted. I’d prefer 2-by-3 as the first example.

- Similarly, the 5-by-5 doesn’t make it totally clear what is being counted, although I think it’s smart not to go directly to 7-by-10. Maybe 3-by-6 as the second one? It’s probably fine.

- I like the choice of a “big” rectangle as the last, and wonder how many students would proceed to count off its dimensions before doing anything else. Perhaps giving the coordinate of the big rectangle would help, but at the same time, I think it’s deliberately intended to represent a general case, which is very nice.

- What if you drew in the diagonal first each time, then lit up the squares? The advantage is that when it gets to the general case you can draw the diagonal, then stop the video, keeping you from having to ask those 12 words that now accompany the video.

18. on 01 Dec 2011 at 3:56 pmPhil Smith (@liketeaching)

I like the visual progression of larger and larger rectangles in the video. To me it suggests that if I managed to count the squares touching the diagonal then there would be an even larger rectangle next. I’m not sure if that’s just because I know the general question is coming though.

Also, having taught many kids with EfL recently, I definitely agree with your point about language demand.

To me, the benefit of Nowak’s approach is that the kids will spend a little more time thinking about the specific cases and will be able to use them for reference when they get to the general case. Possibly that would be an example of making decisions for them on how to solve the problem, however(?)

19. on 01 Dec 2011 at 3:57 pmDavid Cox

David, you’ll include that in your upcoming post, right? I’ll post it here otherwise.

I was just waiting on you, boss. Here you go.

20. on 01 Dec 2011 at 4:31 pmLL

I like the fact that Dan & Alan quickly get to the point of the puzzle, though Alan’s would require an audience that is comfortable with that level of abstraction. The shading on Dan’s video could make for some interesting patterns on bigger rectangles. I also like Kate’s approach, but would rather use verbal/visual instructions – maybe just project the first diagram, count the squares together, then ask them to make other rectangles & try to develop a rule. I don’t see video adding or subtracting value in this case. Depending on the class, and the amount of time you wanted to spend on this, I could see any of the three being the best approach.

21. on 01 Dec 2011 at 4:39 pmDan Meyer
Phil Smith: Also, having taught many kids with EfL recently, I definitely agree with your point about language demand.

Dan Anderson: I think David is right that Nowak’s or Schoenfeld’s approach wouldn’t work for lower level students as well as Dan’s video. But for upper level students, I think it is important that they are able to translate the problem on their own without the frosting of a video.

Now we’re getting somewhere w/r/t “what do we mean by better, for what purpose, and for whom?” The largest variable as of yet unmentioned is the involvement of the teacher the students negotiate the terrain. Schoenfeld’s problem is explicitly an assessment task. How does an assessment differ from a classroom assignment?

22. on 01 Dec 2011 at 4:45 pmLL

Bowen: I think the choice of rectangles to use in the intro depends on the class. If you want a slam dunk start, 2×1 & 5×5 are nice choices. Kate also made a visually simple first case, though not as simple as 2×1 & 5×5. In some ways 3×2 is springing the meat of the problem on them right away.

23. on 01 Dec 2011 at 5:28 pmKatie

I’m really sorry that I don’t have to time to read all the comments to make sure I’m not being redundant, but I was so excited that this problem actually arose in my class today unintentionally! I gave a problem where I gave a super simple map of our neighborhood, literally just a grid (streets are lines, blocks are the squares), and labelled our school and two bart stations. One is clearly closer to walk to, but the crow-flying distance is a little ambiguous. This was supposed to get kids using the pythagorean theorem, but instead one group wanted to count how many squares the crow-flying lines pass through to get to the stations. They were equal, which didn’t help them solve the problem at all. It was interesting.

24. on 01 Dec 2011 at 5:54 pmSean

They’re all great.

1. I’m drawn to Schonfeld’s because of the shaded squares. I know, I know: be less helpful, not more. What can I tell you? It got me where I needed to go, faster. Plus I don’t know, I like that NxM conversation.

2. Nowak’s gets the job done, like usual. Right mix of exploration and explicit instruction. But my favorite thing she’s ever done is this. Her original materials average a 99.3 out of 100. That absolute value revelation is a 648 on the same scale.

3. Meyer’s is the sleekest (shocker), the most visually compelling (durrr), and the clear winner for ELL. Just a monster effort. If it’s too fast, pause it. If you want to run something back, rewind it.

Audience-wise, I think Schonfeld works for a particular (possibly idiosyncratic) kid, Nowak’s works beautifully if you set the stage right, Dan’s gives the teacher more room for error.

25. on 01 Dec 2011 at 6:17 pmBowen Kerins

If, in presenting the video, I get to tell students “It’s counting the number of squares the diagonal goes through” then it’s perfect. If not, I think careful selection of examples can make it work without words.

Just watching the 2×1 and 5×5 don’t give me enough information to tell what the question is. It could just be counting the width of the rectangles. The 2×1 (or really any Nx1) is too small to uniquely describe the question — is it the width? the area? the approximate length of the diagonal? We know which one, because we’ve seen the other formulations of the problem, but if I gave students the first 9 seconds of video, a lot of them wouldn’t know what was being counted.

Using a 3×2 (through four squares, still easily visible at the scale used for the 2×1) makes it a lot more clear what the counter refers to.

The 5×5 is alright but it sort of has a “small numbers” issue… the 5×5 counts off … 5. Trying to do this without words, I don’t think it’s completely clear what is being measured — a kid could think “it’s counting the largest dimension” and would be right so far, then would have to go back and rewatch after the 7×10. A 2×4, 2×6 or 3×6 would resolve this while still passing through corners which I think was the other reason to use 5×5.

26. on 01 Dec 2011 at 6:34 pmgasstationwithoutpumps

I liked Schoenfeld’s best (other than the use of the word “comprise” which seemed a little too formal). I could see the advantage of giving several examples (both with relatively prime n and m and ones with common factors), but the video seemed like a waste of time, both to produce and to watch. Doing the same thing on the board (without erasing) would give the students several examples to look at and not take any longer.

27. on 01 Dec 2011 at 6:43 pmDan Meyer
gasstations, etc.: Doing the same thing on the board (without erasing) would give the students several examples to look at and not take any longer.

Sometimes I can’t tell if this guy is pulling my leg or what.

28. on 01 Dec 2011 at 6:51 pmBryan

Or another idea for inspiring a productive investigation:

1. Have each student draw a 2×1 on a sheet of graph paper and ask them to draw a diagonal. How many squares does it go through? They all certainly agree that it is 2.

THEN, have them all draw large rectangle (let’s say 20×35 or whatever) and have them draw the diagonal. How many squares does it go through? I’m betting money they don’t all get the same answer. So, which is correct? How can we be certain? Why are we getting different answers? I think most students are now ready to figure out if theirs was right.

29. on 02 Dec 2011 at 4:19 amKate Nowak

Okay so on this one I think I’m going to argue that the really-large case is a dicey way to go for some classes. I can predict different kids having one of a number of reactions to that:
“I am just going to sit here and count the boxes because I don’t know what else to do, I don’t care how long it takes, and I want to be a good kid and please my teacher. Counting counting counting…”
“Where is she going with this? I’m going to quietly stare at this paper until she gets to the point.”
“Who cares. This lady is crazy. I’m going to chat about my weekend with my buddy.”

But I can certainly see adding some more structure by providing more examples, maybe setting up the beginning of a table to organize information. If I used this in regents Geometry, I certainly just wouldn’t give them my prompt and say “go.”

On the other hand, I can see an Honors class or my calculus class seeing the huge rectangle as a fun challenge and diving in without much help, and they’d understand the question quicker and more fully than they would with one of the static prompts.

Alot of this discussion boils down to “depends on the class.”

30. on 02 Dec 2011 at 8:25 amDan Meyer
Kate: Alot of this discussion boils down to “depends on the class.”

Maybe. Something to think about, though, is how easy it is to add vocabulary, prompts, and discussion. It’s comparatively very difficult to subtract those things after they’re already out there having given some non-trivial fraction of my class — even an advanced class — a panic attack.

31. on 02 Dec 2011 at 9:31 amCarl Malartre

I love the applet!

I would like to observe how students will use it. What kind of students will use it in what kind of way? Will some freeze in front of it?

Carl

32. on 02 Dec 2011 at 10:24 amgasstationwithoutpumps

Dan, I’m generally not pulling your leg. I think that you are sometimes so enamored of your video skills that you miss simpler and more effective low-tech solutions.

Having the students able to simultaneously see both rectangles with relatively prime edges and ones with common factors is far more important than having them watch an animation of a rectangle being drawn and counted, then expecting them to remember what they have seen.

Some of your videos have been good starting points for math problems (like the basketball parabola, which would be difficult to do on a whiteboard). But this one gains little or nothing from the video. Having a computer draw larger examples may be useful, and the Geogebra applet may be a good way to do that, if every kid gets to play with Geogebra at the same time. Otherwise, it is better just to have the kids sit down with squared paper and draw and look at a dozen or so examples.

33. on 02 Dec 2011 at 10:42 amCarl Malartre

@gasstationwithoutpumps:

The video add something to a whiteboard and geogebra in one category: storytelling. Of course the first two tools can be used by a competent teacher to add some drama. The video can be a good tool there to inspire teachers too and could be seen only by teachers.

“then expecting them to remember what they have seen.” I don’t think that’s expected.

“Having a computer draw larger examples may be useful, and the Geogebra applet may be a good way to do that” That’s where the whiteboard fail. Or succeed: it shows the student that it’s slow and painful!

Carl

34. on 02 Dec 2011 at 10:56 amDan Meyer
gasstations,etc.: I think that you are sometimes so enamored of your video skills that you miss simpler and more effective low-tech solutions.

When all you have is a hammer, etc.

I’m curious about your experience teaching low achieving students, though. As far as I know, your only teaching experience is at the higher ed level and with your son, who is apparently the exact, polar opposite of low-achieving, etc. I’m trying to figure out if you know what the video is trying do, and for whom.

35. on 02 Dec 2011 at 1:00 pmKate Nowak

I don’t think I understand what the video is trying to do, and for whom. It’s presenting the problem without spoken or written words, yes? Is that better? Why? And for whom? Better because the language of the question is formulated an uttered by a student? Is that any better, when the video makes the intended question pretty obvious?

36. on 02 Dec 2011 at 1:10 pmDan Meyer
Kate: It’s presenting the problem without spoken or written words, yes? Is that better? Why? And for whom?

The problem is presented with the video and the teacher’s question: “How many squares will the diagonal of the large rectangle cut through?” Is that better? All other things being equal, fewer words is better than more words because we’d like to assess our students’ mathematical ability, not their verbal ability.

37. on 02 Dec 2011 at 2:10 pmDan Anderson

@Dan
This blog post is asking “which version is best for *assessment*?”
There are teachable moments that happen with combining the literacy of reading and parsing a question that do not exist with the video version. Right?

38. on 02 Dec 2011 at 2:14 pmLL

I have watched the video a few more times. Do you have the boxes turning grey as a hint or to prevent a hint? At first I thought it was a hint. As a look at the problem more, I think seeing the diagonal cross a horizontal line is actually more helpful, and the grey boxes prevent the viewer from seeing this.

The language involved in presenting this could be minimal without using video. I am not sure I see the type of student you believe will benefit more from the video than pictures side by side. I don’t buy the “story telling” argument some have offered. The story is boring, you want to zip through it quickly to get to the interesting question.

For some classes the video might be great, for others it might actually be de-motivating and mildly insulting.

Has anyone done this on graph paper or on sketchpad? Graph paper could be a nightmore for students that struggle with motor skills, but I think having the diagonal drawn instantly makes it tougher to see some of the key ideas.

39. on 02 Dec 2011 at 2:27 pmDan Meyer

@Dan, let me amend myself: “We’d like to assess and exercise our students’ mathematical ability, not their verbal ability.”

40. on 02 Dec 2011 at 8:56 pmZeno

Why shouldn’t the teacher simply pose the question as follows:

“Consider a rectangle with sides of length M and N, divided into squares of equal size arranged in M rows and N columns. How many of these squares will a diagonal of the rectangle pass through?”

Wouldn’t the students be able to understand this?

Granted, the students will probably need to look at pictures of some examples in order to find the solution, but couldn’t they select and sketch the examples themselves?

Does the teacher really need to be more helpful?

41. on 02 Dec 2011 at 9:00 pmTony

Dan’s last comment is where it hits home for me. I have a lot of students that couldn’t get through Schoenfeld’s or Kate’s problems because of the language barrier. For that reason, Dan’s approach (not his video…his approach…for those of us that can’t create a video or script an applet) is superior. Why do we have to tell them to research non-similar triangles? Why do we have to tell them much of anything? How much more powerful is it if they develop the question on their own? I would much rather see a student realize that their might be special cases and develop the research process on their own.

42. on 02 Dec 2011 at 9:02 pmTony

@Zeno: To answer your questions (at least with regard to my students): no and yes.

43. on 03 Dec 2011 at 3:37 amSteve Phelps

Interesting discussion…

Though I appreciate the video, I think may take away some of the sense making that might happen when first approaching this problem. For example, what does it mean to pass through a square? In a 2 by 2 rectangle, does the diagonal pass through just two squares? or through all four? or is passing through a vertex of a square not the same as passing through the square? and does it matter? would the solution/answer be any less correct?

44. on 03 Dec 2011 at 3:58 amaaronthill

Tony (and others above) have mentioned that Dan’s approach is better because students get immediate access to the question without needing to go through some technical language. One can argue (as some some have in comments above) about how much of a barrier this really is, but certainly both were written in the “math dialect” that exams, assignments, etc. are typically written in. Avoiding technical language and the math dialect can make it easier for some students to understand the problem.

But there is another result of using technical language to pose a problem. It sets the stage for the discussion that will follow. If a student is not very comfortable with this math dialect, it takes real guts to join the conversation. Not so with Dan’s approach. He mentions the questions:

“How many squares do you think it’ll cut through? Write down your best guess. While you’re at it, write down a number you think is too high, and too low.”

It’s easy to jump into that conversation, and that’s where I think Dan’s approach shines brightest.

45. on 03 Dec 2011 at 6:12 amLL

Layer the language on top of the problem as it is needed, but make the entry into the problem more inviting by removing language. The vast majority of math problems & examples that I run across use way too much language to illistrate concepts or present problems. The visual & intiuitive aspects of math seem to constantly get barried with language.

46. on 03 Dec 2011 at 12:10 pmZeno

@Tony: Why wouldn’t your students be able to understand the question? Do they not understand what a rectangle is, what a diagonal is, what rows and columns are? This is all very basic. If students don’t understand these things, wouldn’t it make sense to teach them before introducing the diagonal square counting problem?

@StevePhelps: For this problem, “passes through” means “intersects the interior”. Schoenfeld makes this explicit. Including squares which the diagonal only touches at a vertex would be a different problem.

@aaronthill: Avoiding mathematical language may make it easier for some students to understand this problem. Will it help them learn to understand mathematical language? Is being able to understand mathematical language important?

47. on 04 Dec 2011 at 4:06 amaaronthill

I think each of the three versions is well-suited for the activity that it is trying to promote. How you pose a problem should clearly be influenced by the how you want the students to engage the problem.

If you want a wide open classroom discussion in which students contribute ideas and work together towards a solution, I’d recommend Dan’s approach. It does a great job of making it easy to understand the problem and join the conversation (see my previous comment).

If you want pairs or small groups to work on the problem, then Kate’s approach is excellent. Her presentation orients the students to specific tasks that will help them to figure out the problem. That orientation is really important when groups (or individuals) are working on in-class activities. Given that she must explain the problem and orient the students to specific tasks, I think Kate did an excellent job of minimizing the language barrier (using simple, clear sentences, etc).

It you want the problem as an exam question for an intro to proofs class (my best guess for math 64), Alan’s question is excellent. The language is unambiguous, there is a nice picture that helps in understanding the question, and the variables present prompt the student to replace them with specific numbers and try some examples.

48. on 04 Dec 2011 at 4:36 amaaronthill

@Zeno Yes, it is important to understand the “math dialect” that we typically use when writing mathematics. But that doesn’t mean it must be a part of everything we do (especially a classroom discussion that is primarily verbal).

49. on 04 Dec 2011 at 6:58 amRobert Hansen

I like Dan’s video, for a class that is just beginning to solve deductive problems and I am trying to teach, though I would also prefer his examples as still images for the reasons mentioned, I would want the different examples available at the same time. Dan’s video adds the modulus operation aspect of this problem. After they have developed deductive problem solving skills I prefer Schoenfeld’s version because it is has elements of what is actually required outside of the classroom. You have to see and investigate these contexts on your own, in your head and on paper.

50. on 04 Dec 2011 at 12:36 pmTim Hartman

I think the crux of this conversation is a question that hasn’t been posed: How much literacy should we teach in the mathematics classroom? In the past, Dan’s videos have been about storytelling – an approach that motivates students to want to find the answer to a math problem. In this case, I think all three of the approaches are still very abstract. The only difference in my mind is the level of language that is being used from nearly none in Dan’s case to very academic language in the original Schoenfeld problem.

While I want my (minority, inner-city) high school students to succeed in math, what I want more is for them to succeed in life. For them to do that, the first step is success on the state exam needed to graduate, and then maybe the SAT. While I appreciate the videos that engage their math curiosity, I don’t think videos that simply circumnavigate language are helpful. More important would be teaching the tools to understand the problems as they are worded on (unfortunately necessary) standardized tests. In other words, mastery of math is GREAT, but if they can’t prove to society that they’ve mastered it, they’re never going to get a leg up.

51. on 04 Dec 2011 at 12:41 pmDan Meyer
Tim: While I appreciate the videos that engage their math curiosity, I don’t think videos that simply circumnavigate language are helpful.

I agree, but this particular video intends to postpone language, not to circumvent it. Academic vocabulary and precise mathematical language is important and valued in my classroom. But that language should be used in the service of a problem that interests us. For instance, it’s really hard to talk about the video if you’re always referring to “the segment that connects two opposite vertices of the rectangle.” That’s where we introduce the term “diagonal.” The same goes for the rest of the language.

52. on 04 Dec 2011 at 3:54 pmGreg Port

Hey Dan… Like the video. This same idea was used by the Maths300 people here in Australia. The software they have does a great job of inviting students in with the questions we WANT them to ask (or at least think!). It animates the diagonal and gives a count of how many squares as it goes. You input the dimensions of the rectangle (up to 1000×1000) so you can go for relatively prime numbers first, then when they think they have it, introduce like a 6 by 4 rectangle. The students are always keen to see the software verify their guesses.
They also couch the problem in a story of an electrician laying cabling diagonally across a tiled floor.

53. on 05 Dec 2011 at 12:53 pmsimon

“But that language should be used in the service of a problem that interests us.”

Dan, I for one would be really interested if you could write a post on this exact issue.
How do we move towards getting students using the normative (‘correct’) words? What if they come up with their own word(s)?

Also do you see this as a problem particularly relevant to Maths? Or is it education in general.

I know how offputing ‘using the correct terminology’ can be. One professor laughed at me when I said you had to flip fractions upside down if there was a divide sign. With a dismissive “don’t you mean invert’.

54. on 07 Dec 2011 at 10:30 amAndy

I hadn’t done this stuff in a while, so rather than think of this as a presentation issue, I tried to figure out how to solve the problem. The way _I_ ended up figuring it out required two insights that can be illustrated with two rectangles.

First, if you take a rectangle whose side lengths are relatively prime, you might see that the number of squares crossed is really the number of lines making up the squares crossed (except for that pesky place at the end where you cross both the horizontal and vertical line at the same time).

With that, a second rectangle, which had, say, some common factor to the side lengths could help one to see that the pesky place at the end isn’t really a special case any more than the other places where you cross at a vertical and horizontal line at the same time.

Anyway, to me, providing these two steps could allow one to see what was going on without getting embroiled in really big rectangles with lots to count. The point, I think, is to help students to look at the fundamental issues, rather than get into a fruitless counting exercise. And, while brute force can provide some information, one needs to always brought back to generalization and simplification – that’s what mathematics is about.

I like the paper (over something online), because I can draw on it as I’m thinking about the problem. Dan’s treatment has the rectangles get bigger fast, but at least for me, getting bigger didn’t provide any insight, and really made things harder to solve. It was the nature of the rectangles, not their sizes, that was important for me.

55. on 11 Dec 2011 at 6:04 amLinda Fahlberg-Stojanovska

Just reading briefly so excuse my ignorance. MD showed me this version http://www.ohiorc.org/pm/math/richProblemMath.aspx?pmrid=56. a couple of years ago and I thought it atrocious.

But my mind works very methodically. So I – like David Cox – like the “start small and look for the pattern process”.

But I understand that everybody does not and in fact everybody seems to hate the “other” approach. I think that is the reason for all of this discussion. I once had a teaching assistant who taught in a completely different style than I do. I thought – this is never going to work – but actually it worked out well. Each student got at least something he could understand well.

BTW: Zeno – the questions Tony, Steve, … ask are EXACTLY the questions I asked when reading the original problem. I would never assume that everyone processes text in the same way and comes to the same conclusions and has the same pictures in their head. English is my first language and I am constantly bewildered by what people consider “completely clear”.

My one comment is a general comment. I think we should have problems we re-examine every couple of years in school. (Why do we invert and multiply, why is the area of a circle pi*r^2, this problem…?) I need “real” time to process things.

56. on 11 Dec 2011 at 8:21 amLL

“I once had a teaching assistant who taught in a completely different style than I do. I thought – this is never going to work – but actually it worked out well.”

Linda, what a breath of fresh air it is to hear you say that.

57. on 18 Dec 2011 at 12:37 pmJ

I like the idea of a video for this problem, but compared to your usual videos, I find this to be on the confusing side for a couple of reasons:
1) The origin moves. Tricky when I’m supposed to be gleaning the dimensions of the rectangle just from the single labeled point.
2) The process of drawing, shading squares, and un-drawing the figures is kind of distracting–I’m not sure what to focus on. The problem has less to do with the process of drawing a rectangle or its perimeter than the video makes me start to think.
3) The figures disappear too quickly, and the first time through I felt like I immediately forgot what I had just been looking at. Maybe each example could just be shrunk and moved over to the side? Maybe it’s just the graph paper background as opposed to a real life scenario, but I just feel my eyes glazing over as I watch this.
But even so, cool problem, cool video.

58. on 28 Dec 2011 at 8:17 pmGavin

I think a worksheet with a few examples followed by an invitation to find the answer for some larger (printed) rectangles would suffice, ending with the open-ended question. The back page of the worksheet could be covered in grid paper so students can draw some rectangles of their own.

The video would be a nice introduction to the worksheet. I feel that it’s not necessary and the effort required to create it is not worth it, but *given* it exists, I would use it enthusiastically. If it didn’t exist, I think the worksheet would still do the job quite nicely.

What can I say? I like a nicely conceived and executed worksheet.

I spent some time last night solving the general case, and having access to that Geogebra applet was great for my motivation and exploration. I’d happily give students access to that. Is the file available for download? I want to learn how it’s constructed.

Final comment on the video: I am gobsmacked at its production quality, offer my congratulations at your skills, Dan, and would love to know how it’s done. What software, and how long does it take when you’re reasonably experienced at using that software.

Many mathematical concepts are crying out to be visualised. Geogebra and friends are great — we are so lucky to have them — but sometimes a well-produced video would be fantastic.