Tag Archives: wcydwt


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?


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?

Salt, continued

Callum writes:

Filling a container with salt, sorry but…is this kind of thing inspiring to students learning maths? Isn’t this all rather pointless – you filled a container with salt – well done!


But can we agree that my salt problem is no worse than any of the following?

These were gleaned from a quick sample of middle-school, remedial college and mathematics for elementary teachers textbooks on my shelf. I grabbed four books off my shelf that I thought might have volume problems in them and found a problem for the gallery in each of them. No cherry-picking here. And I swear I didn’t leave out any compelling applications of volume of a cylinder.

But even if we agree it’s no worse, that’s not a very strong argument in favor of the problem.

So is there any aspect in which it might be better? I think there are several.


The initial question-will it fill or spill? admits student guesses. Students will have a hunch about the answer, and an intuitive sense of why it will or will not fill or spill.

This contrasts with these other problems. Students are asked to find a volume for the sole purpose of finding a volume. Not in order to answer anything some more meaningful question. And not even I have an intuitive sense of the volume of 678 flapjacks.

Plus, the question can come from the students. They can ask whether it will fill or spill; I don’t think I’ll have to. And then we’ll need to find some volumes in order to make a good prediction.


These are real containers. Perhaps not very compelling containers (although I’m a big fan of vintage Tupperware). But real containers nonetheless. Unlike anything in the problems above, these are objects in their daily lives.

Perhaps this is a sign of my hopeless math geekdom, but I am pleasantly surprised every time I refill my salt container that it fits perfectly. No leftover salt; no space left in the container. A perfect fit. I imagine some of that enthusiasm will be contagious in the classroom. And perhaps inspire some students to look at the containers in their own homes a little bit differently and wonder which ones are “bigger” than others.

the answer

Did I mention that the salt fills the container perfectly? And that we can see it happen before our eyes?

I’m not looking to draw eyes away from the Super Bowl with this problem, nor to cause students to switch their major. But I hope they’ll be a bit more invested in the outcome than they are in the textbook problems above.

intuition again

Here’s an interesting task from the math for elementary teachers book.

from Beckmann, S. (2010). Mathematics for Elementary Teachers. Boston: Pearson.

My instinct is that, at middle school, where the salt task would be appropriate, this will still be part of some students’ intuition. It is much more abstract to run the calculations and see that they are very, very close than to run them and then see that closeness play out in the physical world.

I’m not hoping to draw students to mathematics with this problem; I’m hoping to get them engaged for a lesson on volume.

But we’ll see. I’ll be using the problem with my future elementary teachers in a few weeks. This is not a population that is already sold on math (although by this late in the semester, I’ve reeled them in pretty well). I’ll report back.

And I welcome further critiques.

How do we make volume compelling?

The pH of concentrate


I set this up and critiqued it in this blog before teaching it. The basic idea is that I needed a way into logarithms with my College Algebra students.

The questions

I have two question videos. I debated which to use in class and ultimately chose the first option below.

Option Number 1

In this option, we measure the pH of water and the pH of pure orange juice concentrate.

Option Number 2

In this option, we measure the pH of water and the pH of 100 ml of water mixed with 1 ml of orange juice concentrate.

I imagine that Option 1 will prompt questions such as, “What is the pH of regular orange juice?” and “What if we mix it 50-50?”

I imagine that Option 2 will also prompt the question, “What is the pH of the straight-up concentrate?”

In fact, this last question is the one that motivated this project. I was expecting (as will my students) that the difference between the pH of juice and the pH of concentrate will be quite large.


Juice (20 ml)

This actually doesn’t demonstrate juice (see post where I regret this oversight in my data collection). But it’s pretty close. Real OJ should be 3:1 water to concentrate. This is 5:1.

50-50 (100 ml)

10 ml

1 ml

Pure concentrate

…I got it (partly) wrong, addendum

I lamented the other day that I didn’t capture a question I was sure students would ask about my pH videos.

And today they didn’t ask it.

But they asked other stuff I had not anticipated.

I taught the lesson today. I chose the video in which I measure the pH of water and the pH of pure concentrate. We clarified what we had seen (including the lovely question Did the color of the orange juice concentrate (orange) affect the resulting color of the pH strip (also orange)?)

Then I asked them what there was to ask or wonder about here. Their questions:

  1. How are logarithms used in the calculations of pH?
  2. How much water do we need to add to the concentrate to get a pH of exactly 7 (in the video, the water measures pH=7.2)
  3. What is the concentration of hydrogen ions in the orange juice concentrate?
  4. How does the mixing of pH work? Does 1 unit of pH 1 mixed with 1 unit of pH 2 give us 2 units of pH 1.5?

Two of these questions were from students who have come to office hours many times, but who have never piped up with a question or answer in a whole class session. They are always engaged and on task, but never vocal.

I hadn’t thought of adding water to concentrate. I had thought of adding concentrate to water. So I had to tweak question 2 into one closer to something my premade answer videos could answer. To wit:

our question

How much concentrate do I need to add to 100 ml of water in order to make the pH exactly 7? (Recall that the water starts at 7.2)

Student guesses

In pairs, students wrote their answers on slips of paper I had provided, then held them up all at once. Their guesses ranged from 3 ml to 300 ml. The median was approximately 7 ml.

We watched the 10 ml answer video and I asked whether anyone wanted to change their guess. They all did.

We watched the 1 ml answer video. And students revised their guesses again.

Then I restated the definition of pH and demonstrated solving for the hydrogen ion concentration in the orange juice concentrate (question 3, recall).

On Friday, we’ll do some pH computations to verify/approximate the experimental results.

On the way out of class, a student left her final guess on my desk.