Tag Archives: wcydwt

My Oreo manifesto, part 2

So you read my previous blog entry in which I paid homage to the culinary sleuthing of Al Sicherman and alleged that Double Stuf isn’t really double stuf.

When we left off, we had been working with the assumption (soon to be demonstrated false) that Double Stuf is double stuf. We had learned that a single unit of stuf has about 17 calories and a single wafer has about 19 calories.

Enter the Triple-Double.

Again, we use Nabisco’s own data. One serving Triple Double Oreos has 1 cookie and 100 calories.

Translation: One serving of Triple-Double Oreos is 3 wafers (that’s the triple part), 2 stufs (that’s the double) and 100 calories (that’s the OMG! part).

Comparing the Triple Double to the Double Stuf, we can see that the only difference between one Double Stuf cookie and one Triple-Double cookie is 1 wafer and 30 calories.

What can we conclude? That a wafer has 30 calories. But wait! That’s over 50% more calories than we got under the assumption that a Double Stuf has double stuf. We have reached a contradiction, which means we need to reject our initial assumption.

Double Stuf ain’t double stuf.

QED.

My Oreo manifesto

Math is powerful.

No one has demonstrated this more frequently and more convincingly than Al Sicherman of the StarTribune. In his Tidbits column, Sicherman keeps tabs on prices and variations of processed foods. A favorite category for him recently has been Oreos. Writing recently on the topic of the new Peppermint Creme Oreos (which he describes as toothpastey), Sicherman says:

All of that is not to say that there is no significant difference between Cool Mint Creme Oreos and Peppermint Creme Oreos: They are the same price on the shelf, but the package of Cool mint Oreos is 15.25 ounces (30 cookies), and that of Peppermint Creme Oreos is 10.5 ounces (20 cookies). So Peppermint Creme Oreos cost 45 percent more per ounce — or 50 percent more per cookie.

Whoomp! There it is! Math, baby!

Gotta love this guy. (But seriously, would it kill the Strib to dispatch a photographer with a camera that has more than 1 megapixel?)

Inspired by his example, I’m prepared to turn up the heat on Nabisco’s Oreo division. Are you ready?

Double Stuf isn’t double stuf.

Let me say that again…DOUBLE STUF IS NOT DOUBLE STUF!

How do I know? Math.

How should we do this? Reductio ad absurdum? Let’s assume it is double stuf. Then, according to Nabisco’s own data (found on the Nutrition Facts), one serving of Oreos is 3 cookies and 160 calories. Double Stuf is 2 cookies and 140 calories.

Let me translate that for you. One serving of regular Oreos has 6 wafers, 3 stufs and 160 calories. Meanwhile one serving of Double Stuf Oreos has 4 wafers, 4 stufs and 140 calories.

That’s hard to compare. So let’s consider two servings of regular Oreos (6 cookies, 12 wafers, 6 stufs, 320 calories) and three servings of Double Stuf (6 cookies, 12 wafers, 12 stufs and 420 calories).

Those Double Stuf Oreos account for 6 additional stufs and 100 additional calories (same number of wafers, remember). We conclude that each stuf is approx. 17 calories. And from this we can conclude that each wafer has approx. 19 calories.

What, you prefer the formal mathematical solution? Fair enough:

Let w be the number calories in a wafer and f be the number of calories in a unit of stuf. Then Nabisco’s calorie claims yield this system of equations:

\begin{cases} 6w+3f=160 \\ 4w+4f=140 \end{cases}

We can rewrite this equivalently as:

\begin{cases} 12w+6f=320 \\ 12w+12f=420 \end{cases}

And then solving by combination, we get:

6f=100 or f=16 \frac{2}{3}

You’re comfortable with that. No problem, right?

Tune in tomorrow for the dénouement.

What CAN you do with this?

My son (Griffin, nearly 7) saw Hex Bugs in a store months ago and was instantly smitten. His dream came true recently when my wife bought him one.

The Hex Bug now does shows in his home/stadium. See video.

There is something quite lovely about this guy’s random walk. But I can’t quite figure out what scenario I can put him to generate a real probability problem. So I ask, What can you do with this?

Larry and Stu (annotated edition)

Aaron B. says:

You’ve got me interested in the opinion that you aren’t explicitly stating.

And Dan Meyer says he was:

Mostly confused. Those two [Larry and Stu] are strange. (via Twitter)

So forthwith I offer a deconstructed, annotated version of the conversation. All links from the original post have been deactivated in order to avoid a second round of massive pingbacks. My comments include a few new links; see the original post for the original references.

Larry and Stu have been teaching high school math for 10 years. They take a load off in the teachers’ lounge at lunchtime.

I am always thinking about what my own students (whether the context is middle-school, college or professional development) take away from the experiences we have in the classroom. How does what I say match what they hear? And vice versa.

In a particularly vivid example, I recently worked with a group of teachers on a Connected Mathematics implementation. Several of these teachers had been to Michigan State and had met the authors of the curriculum. One of these teachers claimed definitively that Betty Phillips had told her that the units in the curriculum could be taught in absolutely any order. Problem is, Betty is our loudest voice on the importance of the order of mathematical ideas and the sequence of units in the curriculum. Surely what Betty said is not what this teacher heard.

So I’m always curious about the relationship between what is said and what is heard. I wonder about the takeaway message, in common business parlance.

Larry and Stu are two typical high school math teachers with enough experience to have settled in to a standard American mode of mathematics teaching, but also not so deep into their careers that they aren’t interested in what else might be out there. I imagine them being good at what they do, caring about students and proud of their work.

Larry: Y’know Stu, I’ve been doing a lot of reading on the Internet. There’s some great stuff out there. And I’m convinced that Math Class Needs a Makeover.

Surely someone has emailed Larry and Stu a link to Dan Meyer’s TED talk. I didn’t see it until February of this year but most of the people I talked to afterwards had already. In May, a friend posted it to Facebook.

Stu: What do you mean, Larry?

L: We gotta make math real-world.

S: Huh?

This is not new language. Many people have burned many calories thinking about what it means to make math “real-world”. The Dutch Realistic Mathematics Education program has done the best job of defining the “real” part of “real-world”. Something is “real” if it can be brought to life in the imagination of the student. If students can operate on it in their minds, it matters little whether the thing actually exists.

Dan recently was on the cover of an Education Week supplement with the phrase, “Real-world learning” in large type next to his photograph. He expressed discomfort with this phrase (calling it “a phrase I try to avoid”). Meanwhile, Karim over at Mathalicious unambiguously defends the real world as “an underused launching point in most math classrooms”.

I understand Dan’s hesitance to adopt the “real-world” label as it has been used and abused to justify all sorts of instructional decisions. It is perhaps only a bit more meaningful and a bit less bullying than “good for kids,” as in “I just want to do what’s good for kids.”

L: OK, see there are these Russian Dolls. And, oh yeah, these escalators. And there’s a whole mess of these graphing stories. Kids watch these videos and they make a graph showing the relationship between time and height, or time and distance, or heck-there’s even one on time and time!

I have indicated on this blog and in the comments on Dan’s that these are all brilliant contexts and great contributions to the field. After I met Dan back in February, my mind was buzzing for weeks. I spent a lot of time sorting out what exactly I could learn from Dan’s work. Ask my colleagues; I wouldn’t shut up about it.

S: Sounds interesting, Larry. Tell me more.

L: Right, so you show this video and kids ask the question. You don’t have to give them some phoney-baloney task, the video leads them right to it. (By the way, do you know what exogenous means?)

I am all about purging the mathematics curriculum at all levels of phoney-baloney tasks. But here is an interesting tension I have come across in following Dan’s work.

Dan is working very hard to help math teachers to understand what it means to pose a problem in a compelling way to students who are not predisposed to seeing the world mathematically. I’m 100% on board with that agenda, and I have learned a lot from it.

I am more critical than he is of the accepted, enacted curriculum in the U.S. I love his visual reformulation of the house painting problem as a counting beans problem, for instance. But I’m not willing to give the math curriculum a pass on the value it assigns to the work equation. There’s the tension for me.

My office mate and I have debated many times the value of this very equation. I argue that the work equation unnaturally emphasizes time per job, rather than the more natural job per unit of time. So if I’m painting a house, I am more likely to be operating on the “fraction of a house I can paint in an hour” than “the fractions of an hour it takes to paint a house”. And if I think about houses per hour, I can’t use the work equation.

I am more interested in eliminating the work equation from the standard curriculum than I am in retooling the problem we use to get at the work equation. Lounge chairs, deck of the Titanic, etc.

And the “exogenous” thing is a good-natured jab at my man, the erudite Karim who wrote the lovely sentence on Dan’s blog, “I think the prohibition on exogenous questions may be a bit too strict.”

S: No idea. But, OK, kids ask the question. Cool. Then what?

L: Well, then you give ‘em whatever information they need to solve the problem. And you let them work, or maybe some days you lecture. But the point is, they’re motivated to do the work because they asked the question and they identify with the context.

Here’s another tension for me, and it has been a common topic of discussion over on Dan’s blog. Let’s say we craft this perfect mathematical problem, kids ask the question and really, badly want to know the answer. Then what?

This question is outside the domain of Dan’s WCYDWT work. He’s working on setting up the task. That’s good. And yet it is a very real problem of practice to identify how students are to learn some mathematics from the task. It requires an entirely different set of skills to move students forward mathematically than it does to pose the problem in the first place.

I think Larry and Stu could very well show the escalators video, get everyone to the point of wondering how long it takes Dan to go up the down escalator, and then not really know where to go from there. If they are going to implement these ideas in ways that support student learning, they’re going to have to improve other aspects of their practice as well.

S: Good. Go on….

L: OK. Then after they answer the question, you don’t just look in the back of the book for the answer, you show the answer in the video. They see the results of their calculations. If they’re right, they get validated. If they’re wrong they know they’re wrong. They get that math solves real problems.

This is a lovely feature of WCYDWT, I think. If the problem has been posed with multimedia, the solution to the problem can be found there too.

By the way, I am frustrated that I cannot get my escalators answer to agree with reality within 3 seconds. Even making the most generous assumptions about things we can’t quite see in the video I’m off by at least 3 seconds. I want to be right!

S: Nice. Do we do videos every day?

L: No. That would get old. Once every couple of weeks or so.

S: What do we do in between?

L: Well, what we’ve always done.

S: Oh, so not all of Math Class Needs a Makeover? Just 10% of it?

OK, so here’s where I tip my hand and get Dan’s goat (see comment on original post). Of course I know that he doesn’t think only 10% of Math Class Needs a Makeover. Of course I know that his laser focus on one aspect of teaching doesn’t imply endorsement of everything else.

But the use of “Math Class” is ambiguous and I have played with that ambiguity in my mind as I’ve thought about Dan’s work. Does “Math Class” refer to a single class session? Or does it refer to the mathematics curriculum in a more general sense? Larry and Stu are sorting through this ambiguity in their conversation.

L: Maybe. Or maybe we flip the classroom; I’ve read about that, too.

There are others on the Internet working on changes in the ways we teach math. Perhaps you have run across Khan Academy?

Most of the discussions of “flipping the classroom” that I have come across have been founded on a set of faulty assumptions about the kind of makeover math class needs. Dan’s scope is limited to one class period, one problem at a time. I agree with his assumptions about how people learn-by being interested in the topic at hand, by being puzzled by something and wanting to figure it out, by connecting new knowledge to prior knowledge, etc.

But KA and much of the larger “flip the classroom” rhetoric is founded on assumptions that students learn best when they are told clearly and repeatedly and by practicing what they have been told. So Khan tells clearly in his videos and he can tell repeatedly because students can watch the video as many times as they like. No need for the teacher here. Then the teacher becomes the tutor who helps students with their practice in real time. If you buy that this is what learning mathematics is about, then flipping the classroom is a substantive improvement. But if you question the curriculum, or if you question the effectiveness of telling, then KA and flipping don’t have much to offer you, I’m afraid.

For me, there’s not much interesting in these ideas. I’m not opposed to them, I just disagree with the assumptions they represent and I don’t see that they offer much for me to learn. So I’m not gonna put in the kind of work I did for Dan.

fin

The main thing to understand from all of this is that I respect Dan’s ideas and work enough to spend time critically questioning them. Not dissenting so much as asking hard questions.

An imaginary conversation

Larry and Stu have been teaching high school math for 10 years. They take a load off in the teachers’ lounge at lunchtime.

Larry: Y’know Stu, I’ve been doing a lot of reading on the Internet. There’s some great stuff out there. And I’m convinced that Math Class Needs a Makeover

Stu: What do you mean, Larry?

L: We gotta make math real-world.

S: Huh?

L: OK, see there are these Russian Dolls. And, oh yeah, these escalators. And there’s a whole mess of these graphing stories. Kids watch these videos and they make a graph showing the relationship between time and height, or time and distance, or heck-there’s even one on time and time!

S: Sounds interesting, Larry. Tell me more.

L: Right, so you show this video and kids ask the question. You don’t have to give them some phoney-baloney task, the video leads them right to it. (By the way, do you know what exogenous means?)

S: No idea. But, OK, kids ask the question. Cool. Then what?

L: Well, then you give ‘em whatever information they need to solve the problem. And you let them work, or maybe some days you lecture. But the point is, they’re motivated to do the work because they asked the question and they identify with the context.

S: Good. Go on….

L: OK. Then after they answer the question, you don’t just look in the back of the book for the answer, you show the answer in the video. They see the results of their calculations. If they’re right, they get validated. If they’re wrong they know they’re wrong. They get that math solves real problems.

S: Nice. Do we do videos every day?

L: No. That would get old. Once every couple of weeks or so.

S: What do we do in between?

L: Well, what we’ve always done.

S: Oh, so not all of Math Class Needs a Makeover? Just 10% of it?

L: Maybe. Or maybe we flip the classroom; I’ve read about that, too.

fin

50/50

Sign at entrance to the exhibitThe setup

The Walker Art Center in Minneapolis has an exhibit titled “50/50″. The premise is that half of the exhibit was selected by public vote online and the other half of the exhibit was selected by museum curators in response to the public selections.

The questions

In the photo below, we see the public half of the exhibit on the left. The curators’ side on the right is masked in black. In fact, you can only see about 2/3 of the public’s side; the remainder is past the left-hand edge of the photograph.

The exhibit-public on the left, expert masked on the right.

(1) How many works are in the exhibit?

(2) How many did the public choose?

(3) How many did the experts choose?

Answers

Click here to see the unmasked image.

The museum claims approximately 200 works in the exhibit. Now how many do you think the public chose?

NOTE: I shall return to gather the necessary missing data-photographs of the entire exhibit. I don’t count anywhere close to 200 total in these images.

Michael Phelps v. Morton Salt girl: The grudge match

An article in the latest issue of Mathematics Teacher advocates fighting fire with fire in the competition between math class and the media for students’ attention. We are exhorted to Present Examples in High-Resolution Video, to Connect to Students’ Interests, to Show Appealing Faces and to Hold Students’ Attention

As examples of Connecting to Students’ Interests, H. Wells Wulsin offers,

Monitoring a breeding bunny population would show the process of exponential growth. Baseball batting averages could introduce percentages. Such applications of mathematics to daily life would help students build important links between new ideas and the concepts they already understand.

The article continues. Under Showing Appealing Faces, Wulsin advocates,

What if Michael Phelps calculated the volume of an Olympic swimming pool or Beyoncé computed the time delay needed for speakers at an outdoor concert? Why not let Danica Patrick figure the monthly payment on an auto loan?

I have to confess that I find the whole piece a little depressing for two reasons.

(1) Dissemination of the 1989 standards

I can cite examples of rabbit population contexts for exponential growth going back to the original NSF-funded curricula of the early 1990’s. But I doubt the example below is the first time bunnies appeared in an algebra book.

Bunnies in Connected Mathematics, circa 1998

And baseball batting averages date back much, much farther back in Algebra I curriculum.

Batting averages in Algebra I, Modern Edition, circa 1970. Note the annotation at the bottom of the page.

Is this the progress we have made in disseminating the vision of the 1989 NCTM Curriculum and Evaluation Standards? Have we made so little progress that these examples still make for a viable opinion piece in Mathematics Teacher?

(2) Lipstick on a pig

The present guru of high-resolution video in math class is Dan Meyer. I worry that when people look at Meyer’s work, they see the glitzy exterior, not the instructional techniques the technology supports.

Dan is a handsome enough guy (Show Appealing Faces) and his escalators problem has high-enough production values (Present Examples in High-Resolution Video). But that’s not what his work is really about. It’s about storytelling-about putting students in a situation where they ask the mathematical question that we know will be productive. High-resolution video makes this easier to do, but it’s not the core of the idea.

The core of the idea is that we are engaged with the story. We see Dan about to go up the down escalator and it makes us wonder (1) whether he’ll do it without falling flat on his face and (2) if he does do it, how long will it take? The viewer is invested in the narrative.

So here’s a question that points out the distinction I think is important.

Why does Michael Phelps care what the volume of the swimming pool is?

There is no story here; no narrative. If I don’t believe Michael Phelps cares about the volume of the pool (and I do not believe that-just to be clear), then why should I care about the volume of the pool?

Making Michael Phelps (or Beyoncé, or Danica Patrick, or even Danica McKellar) the handsome spokesman for the same old word problem hardly advances an agenda of real and engaging mathematics for real students, nor one of turning students into believers that a mathematical perspective on their everyday world might be (1) useful or (2) intellectually stimulating. It’s just lipstick on a pig.

You may or may not be bored by my salt problem. But there’s a reason for computing the volumes. There’s a story-someone is going to fill that container by opening a brand-new box of salt. It’s unclear at the outset whether it will all fit in there-the new container is shorter but wider.

While we don’t need to compute the volume to find out whether the salt will fit, we can. And the more carefully we measure and compute, the better we should expect our prediction to be. And in a middle school classroom, this is likely to get some students’ competitive juices flowing.

Why do we want to know the volumes of these containers? Because we want to correctly predict how the story is going to end, and because we want to be more right than our classmates.

I would gladly put the Morton Salt girl up against Michael Phelps in a year-long curricular grudge match. Which will engage students with more mathematics over the course of 180 school days: filling Tupperware with salt or pep talks from Olympians?