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.

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:

- Show how this new, difficult problem arises from an old, easy problem.
- Make an appeal to student intuition.
- 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**

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.

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.

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.

What program do you use to construct this video?

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.

## 30 Comments

## Dan Anderson

August 29, 2011 - 7:05 am -No kidding. Getting these type of kids to

careabout a topic which they have failed at for years is simply awesome. I’m eternally grateful to have been given the kids who struggle at math. The experience has opened my eyes to math in a whole new way. Like most math teachers, I was the honors kid in high school. Seeing math from a different viewpoint has made it a much more accessible and beautiful topic to me.## Andy Rundquist

August 29, 2011 - 9:31 am -I find it ironic that, in a post where you say you’ll mind your manners moving forward, you take the time to point out to everyone that Dr. Shaughnessy misspelled a word in a blog comment.

## gmt

August 29, 2011 - 9:36 am -I wouldn’t have even noticed if you hadn’t pointed it out,

Andy. (Even with the “sic”)## Steve G.

August 29, 2011 - 12:42 pm -@Andy:

I don’t think that “[sic]” is impolite. It’s a factual note to the reader that the original quotation had a misspelling. It’s only fair for the one doing the quoting not to have to ‘take credit’ for a spelling error. If someone [sic]-ed a quote from me, I would be annoyed with myself but not the person quoting me.

## Andy Rundquist

August 29, 2011 - 12:54 pm -out of courtesy, I would have either fixed the error, not written [sic], or not put in the direct quote. All of those would have accomplished what you wanted without calling attention to the misspelling.

## Dan Meyer

August 29, 2011 - 1:49 pm -This conversation is pointless. Steve is right. Andy misunderstands both the citation “sic” and the context of my remark about manners. Maybe the definition of “ironic,” too. I’m not sure. Any more comments on the matter get iced.

## Telannia Norfar

August 29, 2011 - 4:27 pm -I really do believe that anyone who is great at math should be required to teach a class to students who have had no success. I also believe teachers should be exposed to students whose native language is not English. I believe you really understand the power of teaching when you are exposed to this type of situation.

## Brian Frank

August 30, 2011 - 3:47 am -Dan, I enjoy your detailed attention to argument and your search for critics who engage at the level of arguments.

## Alex Eckert

August 30, 2011 - 4:59 am -“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?”

That’s three questions. Sorry, have to poke a little fun at our hero every now and then :)

I don’t know if “purist” is the right word, or if it’s a word at all, but I can understand how the purists would be compelled to…you know what? Scratch that, I can’t understand it. Redesigning this problem doesn’t make it any less beautiful or interesting. It makes it several things. It makes it more beautiful and more interesting. It makes it more accessible. It makes it more understandable. Not redesigning this problem would be like Dan not writing a blog and instead handwriting a letter every now and then, sending it in to a newspaper, and twenty readers handwriting their comments and sending them in and the comments being posted under the original letter two weeks later. No one benefits from that.

No one benefits from Dan not redesigning this problem.

## Climeguy

August 30, 2011 - 6:51 am -Alex writes:

“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?”

You could have chosen a better (alternative) problem that would have worked for even more students.

-Ihor

## Dan Meyer

August 30, 2011 - 7:13 am -Alexdidn’t write that. I did.I try not to treat renovations like demolitions. I’m trying to redesign the problem on its own terms, not say, “It sucks. Here’s an entirely different problem students will like more.” What can you do with

this?## Climeguy

August 30, 2011 - 7:39 am -That’s Ok as a curriculum writer, but as a teacher how much effort are you going to put in to make this problem better? Renovations are OK if it gets you more bang for the buck, but sometimes you have to stop and say “why bother”. That doesn’t mean you (Dan) can’t do magic with it. You probably can tell that this is just sour grapes for me… I didn’t like the problem very much.

-Ihor

## AnotherDan

August 30, 2011 - 7:45 am -“I try not to treat renovations like demolitions. I’m trying to redesign the problem on its own terms, not say, “It sucks. Here’s an entirely different problem students will like more.” What can you do with this?”

I think this is an important point. It seems to me that all too often the two approaches are:

1) not change anything about the problem/task/lesson next time we want to use it or

2) scrap the problem/task/lesson we’ve used and start over from scratch

The issue with the first approach is we don’t make intentional progress. The issue with the second is we can’t learn from how it went in the past. It seems to me that there’s a lot of benefit to thinking about what isn’t so great about problems/tasks/lessons we’ve used (or are considering using) and renovating them rather than continually trying something different.

My understanding of Lesson Study is that this idea is at the heart of it – renovation.

## morrowmath

August 30, 2011 - 7:55 am -I have to comment on this:

“If I cut my teeth teaching honors kids in Fairfax County, I imagine this would be a very different blog.”

We all have taught these students at some point in our career: they just happen to be concentrated in this location. I have sent middle school students to “that school” to which you probably refer, but more importantly, I send my own daughter there.

Providing for her mathematical education has been a continuous challenge! The abstract mathematics provided for it’s own sake is the norm, and because students perform well, the approach is validated. I have said that students perform well *despite* instruction, not because of it. There are exceptions, as there always are, of course. I have heard a running joke from the kids there.

Here’s a great real life problem: “Mr. So and So is walking down the street, looks down and sees a piece of paper on the ground. He picks it up and reads this problem: insert abstract problem of your choice here.” Haha. Not funny.

So, as she debates taking AB or BC Calculus up to the last minute before senior year (which is required, as well as the lowest level math classes offered to seniors), I will again write her counselor an email and ask, “Which teacher do the students dislike because s/he actually makes them think about the mathematics?”

That’s where I want my daughter to be!

Your blog could have been about the lingering guilt you feel that your students are learning content, but are not learning to *do mathematics.*

## Bowen Kerins

August 30, 2011 - 1:42 pm -About redesigning the problem… I don’t think of myself as a purist, but long-term I want students to come up with these redesigns, rather than providing students with them. I want students to look at that original geometry problem (with less notation, perhaps) and think “It would be interesting to see what happens when this point moves”.

Another recent topic (“I Don’t Hate This Very Much” was it? Nah…) showed something from a textbook: a problem that asked students to sketch a diagram, write a verbal model, assign labels to the model, write an equation that represents the model, solve the equation, then answer the question. I think students need to be able to do all of those things, but not all at once! They should get chances to judiciously choose representations or solving paths based on the problem, rather than being told to do them in ABCDE order. Learning ways to think about problems and when they might be useful takes a lot of time and energy, but the payoff is huge.

The visuals in these videos are powerful and give access to mathematics in ways that are very difficult otherwise. But they are scaffolds: over time, stripping away the scaffolding could help students reach a point where they can construct visuals independently and conduct their own thought experiments. (They’ll have to do that in their college or career life…)

What about the prospect of giving students some “dry question” then asking them to create a video or other representation for the question? Has anyone tried this?

## Jason Dyer

August 30, 2011 - 4:21 pm -I am a little confused why tech is necessarily a part of the counterargument. The rewording could just be:

Here is a square with both diagonals drawn in

(diagram)

The ratio between the four areas created inside the square is 1:1:1:1. Now take the end of one of the diagonals and move it to the midpoint of an adjacent side.

(diagram)

What is the ratio between the areas now?

Fiddling with percent instead of ratios or whatnot aside, this strikes me as stronger presentation of the puzzle, and turns what would only show up inside a geometry textbook into a Martin Gardner puzzle (which have the common attribute of elegantly relating to something the reader can get a grip of).

## Nick

August 31, 2011 - 2:01 am -Great point about only introducing abstractions as a necessary tool for solving more interesting, “real-world” problem. The same strategy is effective in science as well with regards to scientific vocabulary. Instead of teaching vocabulary without context, it is better to introduce it after students have experience with a phenomenon and have a chance to use their own words to describe it. Then it’s clear why specific vocabulary (and abstraction in general) is necessary- to simplify the way we think and communicate about scientific questions/mathematical problems.

@Bowen: I hear what you’re saying about wanting students to do this kind of breakdown independently, that is the ultimate goal. But I still don’t see the necessity of starting with a “dry” problem. The only dry problems that exist in the world are printed in textbooks and standardized tests- whereas real-world problems by definition exist in a greater, “juicier” context. Unless all we care about is training students to break down problems on a test, I think it’s much more engaging and valuable to start with the juicy problem.

## Dan Meyer

August 31, 2011 - 4:49 am -Why?

I considered something like this image. It’s better than the status quo but it lacks the video’s interpolation of the two frames, which isn’t just window dressing.

This came up at least once in the other thread and I guess I’m struggling to see the arrangement from the student’s POV. It’s like the teacher says, “I know you don’t like math and you think this problem is kind of a drag, so here’s your assignment: make the problem something you’ll like. And while you’re at it, make

mathsomething you’ll like.” Isn’t thatourjob? I know this is a distortion but I’m struggling to see it another way.## luke hodge

August 31, 2011 - 6:35 am -I think “What would happen if we dropped the diagonal” turns what was originally a good problem (but mediocre handout) into an even better problem with a nice hook. In my view, you have come up with a really nice variation, but rushed through it by showing the video. Of course it depends on the class, but I believe a number of students would be interested in describing what they visualize happening — which of the four pieces would get bigger, etc. I wouldn’t want to sacrifice that opportunity by showing the video (or showing the two diagrams together) and immediately revealing what happens. That is what I think is lost by showing the video at the beginning.

## Tom I.

August 31, 2011 - 10:05 am -I’m not sure that we’re really arguing about the nature or value of abstraction, abstractness, etc. What strikes me the second time around is that you haven’t made the problem any less abstract (or difficult) in its essential character. What you have done is made the presentation of the problem more effective and interesting. So you’re not really rejecting the problem, nor are you redesigning it. The key idea was to preface the problem by considering a simple square with 2 diagonals, get rid of the rebarbative labels, then ask what happens if you shift one of the diagonals. You can do that on paper, and I agree with Shaugnessy that using video vs. paper is a non-issue.

We might also consider that the original formulation would be appropriate on a test (a hard test), and students eventually need to come to terms with that type of presentation. But Dan’s presentation works better as a classroom exercise.

## Zeno

August 31, 2011 - 2:47 pm -@Dan: Why is the visual interpolation of the two frames important?

## Brian Frank

August 31, 2011 - 3:28 pm -@zeno Here’s what I’m thinking. Without the transition, I don’t think its really challenging your intuition. It’s almost like “here’s an easy problem you can intuit”, and “now here’s a hard one you can’t”. By showing the easy problem continuously morph into the hard one, I think your brain’s intuitive systems stays on the whole time. Dan’s premise is there’s value in taking intuition to the edge, and I think this betters ushers me to that edge.

## Bowen Kerins

August 31, 2011 - 7:56 pm -Good point about the “dry problem”, let me try and rephrase what I meant. I guess what I was going for is that after students work on a difficult, interesting problem (dry or otherwise) they’ll have some perspective on the ways they thought about it, or even parallels between the problem they solved and something else. I try to block off a lot of discussion time for students to describe these ways of thinking. I’ve never had kids make videos, but that’s what I was getting at.

Yeah, the other way sounds like “Here’s this pig, you put the lipstick on it, kids.” That ain’t good, and it sure ain’t math!

At some point, though, don’t there have to be some “dry problems” in a curriculum? Sticking with geometry, here’s a dry one: “What happens when you connect the midpoints of a quadrilateral?” This is a great problem with plenty of “legs” (it can be revisited several times) and students have a terrific time exploring it. I don’t see how a video introduction to the problem could work without killing the “aha”.

Good luck to everyone starting the new school year!

## Jason Dyer

September 8, 2011 - 11:29 am -Sorry for the super-belated; I responded to this, but it looks like the Internets ate my homework.

this strikes me as stronger presentation of the puzzleWhy?I should be clear I didn’t mean stronger than the video, I mean stronger than the original presentation.

It is not practical to make everything a video (at least at the moment?) So techniques to improve dead-tree problems into better dead-tree problems are helpful.

## brooke

December 30, 2011 - 1:25 pm -I’m just reading through your blog … so late to the conversation that doesn’t probably exist anymore. So, it’s really just a comment for you.

I’m a homeschool mom who has a passion that my kids learn subjects in ways that actually benefit their thinking skills. I like what you are trying to say/get across. Your goal clearly isn’t “use video”, rather your goal seems to be teaching students to think. It reminds me of the Benezet articles I read years ago. (I’m assuming you’ve heard of him, but there is a series of 3 articles online that attempts to explain his approach. I wish I had more on him.) The basis really is that crossover between intuition and exactness. I think some people, such as my nephew, are really born with math intuition that extends quite far. Others have to be coaxed towards it, such as my oldest son! The goal isn’t to be catchy. The goal is to extend that intuition, the actual understanding of a problem, so that there is little to no jump in performing the math required to gain an exact answer. Your approach, whatever media or paper or illustrations you may use to get it across, gives students the belief that they CAN understand, they CAN think, that math CAN make sense to them.

However, this all really intimidates me! I’ve made it easily to 7th grade math with my kids. I have no problem causing them to think, explaining it in an understandable way, keeping it interesting to a degree … but looking at the levels which you are teaching makes me think, there is NO way I can get that far with them unless I go back to school!

## Dan Meyer

January 2, 2012 - 8:39 am -Hi

Brooke, thanks for the comment. I’d never heard of Benezet so I’ve added him to the reading list. Also, FWIW, my mom homeschooled me up until the point (eighth grade) that she could no longer teach me math. At that point, we used a VHS series and afterthatpoint, I went to public school. I’m not saying that was the right or wrong course or action. I’m just sympathetic to parents who find themselves in the challenging (but probably rewarding) spot of facilitating their kids’ learning in every subject. Yikes.## Sue VanHattum

January 8, 2012 - 7:40 am -Also, if you’re near any math circles (check here for possibilities), that might be a way to help him continue. Or check out Art of Problem Solving.

## hodge

January 8, 2012 - 12:46 pm -The Benezet article is really worth a look, if for no other reason than to get a peek at education in the 20’s & 30’s. He argues that paper & pencil arithmetic gets in the way of logical reasoning, and experimented with abandoning all but mental arithmetic in the first few years of school. He gives a lot of specific examples and anecdotes that still ring true today, as well as transcribed exchanges with classrooms from the 1920’s and 30’s.