Month: September 2011

Total 11 Posts

Bloggers In The Media

  1. John Burk on the physics of Angry Birds. [Charlotte Observer]
  2. Frank Noschese on Khan Academy. [MSNBC]
  3. Me on applied math problems using multimedia. [NBC]

Featured Comments


Check and mate.

Richard Hake:

The anonymous “Jay” links to the totally vacuous paper “Why Minimal Guidance During Instruction Does Not Work: An Analysis of the Failure of Constructivist, Discovery, Problem-Based, Experiential, and Inquiry-Based Teaching” [Kirschner et al. (2006)] and then states “Check and Mate.” [..] Anyone who thinks s(he) can even Check, let alone Mate, with that paper must be woefully ignorant of the literature — see e.g., Hmelo-Silver et al. (2007), Kuhn (2007), Schmidt et al. (2007), and Tobias & Duffy (2009).


Holy Cow! Richard Hake just commented. There is physics education research royalty lurking around here.

Where Sal Khan Loses Me, Frankly

Sal Khan:

Imagine learning to ride a bicycle, and maybe I give you a lecture ahead of time, and I give you that bicycle for two weeks. And then I come back after two weeks, and I say, “Well let’s see. You’re having trouble taking left turns. You can’t quite stop. You’re an 80 percent bicyclist.” So I put a big C stamp on your forehead and then I say, “Here’s a unicycle.” But as ridiculous as that sounds, that’s exactly what’s happening in our classrooms right now.

How else would people learn to ride a bicycle if you didn’t give them a lecture about it ahead of time?

Partial Product

Imagine you’re at a store that lets you pull products apart and pay for as much or as little of them as you want. What will your total grocery bill be for these three items?

If a student has no idea where to start, you can prompt her to list a price that sounds fair to her for the three sodas or, failing even that, a price that seems unfair to her. (You’re basically asking her to give a wrong answer. It’s a lot easier to give wrong answers than right answers because there are a lot more of them.)

You’ll find students who divide all the way down to the unit rate (ie. each egg costs 19 cents) and then multiply back up. You may also find students who set up a proportion, which will disguise the unit rate in an interesting way.

You’ll find students who set up different but equivalent unit rates. (ie. 19 cents per egg and .05 eggs per cent.) You’ll find students who set up different but equivalent proportions.

One of your many challenges during this activity will be to select students to show work that highlights a) the different ways to find the unit rate, b) the different ways to set up the proportion, c) the equivalence within those methods, and d) the equivalence between those methods (ie. ask your students to help you find the unit rate within the proportions).

The Goods

Partial Product

Featured Comment

Larry Copes:

I’m with Christopher and, I think, Dan here: Toss it out with the understanding that students can use any method that makes sense to them. Then not only share those methods but compare them to see why they yield the same result. Love the word “catharsis” in this context.

Cornered By The Real World

Samuel Otten catalogs different responses to the “when will I ever use this?” question and points out their shortcomings:

I believe that thinking and acting as if the justification for teaching and learning mathematics is found solely in everyday applications can be dangerous. Mathematics does not exist only to serve other professions, nor is it merely a collection of algorithms and procedures for dealing with real-world situations. Such a mind-set essentially paints our discipline into a weak and lonely corner and leaves undefended many of its greatest aspects.

A fantastic piece and a quick read. His closing recommendation is dead on. When kids students ask that question, they aren’t really asking that question, right?

Featured Comment

Carl Malartre:

Not yet sure what to think. I’m trying to remember when I was 12 how I would react to How would your students react?

2012 Aug 28. Damon Hedman offers a great example of the teacher getting cornered by the real world:

Student: When am I ever going to use this?
Hedman: [mentions a gazillion real life uses]
Student: But those don’t apply to me!
Hedman: [bangs head against the wall]

No one wins at that game.

Eric The Sheep

Last Wednesday at UC Berkeley in Alan Schoenfeld’s class on mathematical thinking and problem solving, Kim Seashore wrote the following paragraph on the board:

Eric is standing at the end of a line of fifty sheep, waiting to be sheared. He is hot and impatient. Each time a sheep is sheared and Becky, the sheep shearer, turns to put the wool away, Eric sneaks around the next two sheep in line.

“What question am I going to ask next?” Seashore asked us. We thought for a moment and then shared out responses. Here are a few:

  • How many sheep will be behind Eric when it’s finally his turn to be sheared?
  • How many sheep were sheared before Eric?
  • How many kilograms of wool will the sheep yield?
  • How many sweaters can you make out of them?
  • What’s the significance of skipping two? Why two instead of three?
  • Will the other sheep get mad?
  • What if the 49th sheep had the same idea after seeing Eric skip ten more sheep and started skipping three every time? Who gets sheared first?

“Great,” Seashore said. Then she had us categorize those questions:

  • Which can we answer?
  • Which can’t we answer?
  • Which need more information to be answered?

Then she asked us to work for awhile on a question that interested us and was answerable. One person took up “how much wool?” and she asked him to be explicit about his assumptions. After ten minutes we grouped ourselves and explained our work to other people.

A Few Notes On This Scene

  • “What question am I going to ask next?” isn’t the same question as “What question interests you here?”
  • Why fifty sheep? How was that number chosen? Fifty sheep was short enough that some students determined how long Eric would wait to be sheared by simulating the entire problem. What is gained or lost by describing a line of 1,000 sheep?
  • Asking students to generate their own questions is risky. Seashore encouraged us to pursue our every whim even though the “kilograms of wool” question was going to involve very different mathematical thinking than any of the others. I don’t know how she planned to reconcile that difference. ¶ My approach is to sample the room for questions and take +1’s for each. (ie. “Is anybody else interested in Sam’s question?”) This reveals a hierarchy of student interest which we handle in order. ¶ Meanwhile, I am in contact with teachers who ask their students to generate questions only to coerce them down to the one they (the teachers) originally wanted to pursue. This interaction will only pay off negative dividends, as far as I can tell. These classes would be much improved if the teacher would simply ask a concise question that she knew in advance would be of some general interest to her students. Most questions asked in math class are neither concise nor of much interest to the students so we’re already way ahead of the game.
  • Abstraction was nine tenths of the work. In answering, “how long will it take Eric to get sheared?” I had to represent the problem with variables and build a model out of them. This was, by far, the hardest work of the problem. Moreover, no one I spoke with chose the same independent variable that I did.
  • Your textbook would abstract the problem for your students.

Be less helpful.

2011 Sep 20: Bowen Kerins locates the original text of the problem, which mercifully leaves the hard work of abstraction to the student.

Featured Comment


A line of 50 sheep makes me wonder why I would ever have to use variables to represent the problem.

A line of 1,000 sheep makes me wish I had an easier problem — one I could actually act out.

What number of sheep will motivate me to model a simpler case and look for patterns? What number of sheep will force me to generalize and move from concrete models to abstract thinking, without stepping over the boundaries of the story?

Shearing a line of 1,000 sheep? Eric will be waiting a very long time.