Year: 2011

Total 140 Posts

When Is Video Valuable?

The question that bugs me at all hours is “When is video / photo / print valuable?” This video is one minute long and gets me closer to an answer.

The intermediate value theorem says that because you picked purple when the purple slice was big and blue when the purple slice was small and because slices run continuously from small to big, there is a particular slice that makes you go, “Meh,” that’s exactly in between “I choose purple” and “I choose blue.”

I love that students have an intuition about that slice, an informal understanding of probability that we can develop into something formal. We can access that intuition with video by showing that small slice growing continuously into the big. How do you replicate that experience in print, a medium which does a bang-up job with static quantities but has something of a panic attack when those quantities change?

Featured Comment

Avery Pickford:

Know what I’d really love. For every student to be able to click their mouse (or some equivalent) when they would make the switch and to have this data show up on my screen right after the video was done.

2011 Nov 29. Evan Weinberg hacked together something that does what Avery described. The results surprised me.

Palo Alto High School Math Teachers: Some Of Our Students Objectively Can’t Learn Algebra

Last April, fourteen of Palo Alto High School’s twenty math teachers petitioned their school board [pdf] against raising graduation requirements to include Algebra II:

We live in an affluent community. Most of our students are fortunate to come from families where education matters and parents have the means and will to support and guide their children in tandem with us, their teachers. Not all of them. [..] We are concerned about the others who, for reasons that are often objective (poor math background, lack of support at home, low retention rate, lack of maturity, etc) can’t pass our Algebra II regular lane course. Many of these are [Voluntary Transfer Program] students or under-represented minorities.

Since those students objectively can’t pass Algebra II, the next appropriate step is to compile a list of those students and prevent their enrollment in Algebra II in the first place. Otherwise, you’re putting them in a position to care about passing a class we can be objectively certain they will fail. If I were a parent of one of those students, this determinism would probably drive me out of my mind.

The signatories are Radu Toma, Suzanne Antink, Kathy Bowers, Judy Choy, Arne Lim, Deanna Chute, Natalie Simison, Misha Stempel, Maria Rao, Charlotte Harris, Scott Friedland, Lisa Kim, Ambika Nangia, and David Baker.

Featured Comment

Jason Buell:

Their hearts I think were in the right places but they whiffed badly. The point isn’t can every kid take Alg 2, but should they.

2012 Jan 16: Coverage from the San Jose Mercury News.

Shoulda Woulda Coulda

Two things I’d do if I were still doing the job instead of just talking about it:

Set Up The Expected Value Spinner

I don’t think people who understand expected value understand how hard it is for other people to understand expected value.

Let’s say I roll a die. I ask if you want to bet on an even number coming up or a five. You’re bright. You pick the even number. It has a 3/6 shot versus a 1/6 shot for the five. But what if I said I’d pay you $150 if the even number comes up and $600 for the five. What if I said I’d keep on giving you that same bet every day for the rest of your life? This is where expected value steps in and puts a number on the value of each bet, not its probability. The expected value of the even number bet is (3/6) * $150 or $75. The expected value of the five bet is (1/6) * $600 or $100. The five bet will score you more money over time.

This is tricky to fathom in gambling where superstition rules the day. (“Tails never fails,” betting your anniversary on the pick six, blowing on the dice, etc.) So one month before our formal discussion of expected value, I’d print out this image, tack a spinner to it, and ask every student to fix a bet on one region for the entire month. I’d seal my own bet in an envelope.

I’d ask a new student to spin it every day for a month. We’d tally up the cash at the end of the month as the introduction to our discussion of expected value.

So let them have their superstition. Let them take a wild bet on $12,000. How on Earth did the math teacher know the best bet in advance?

BTW: You could make an argument that a computer simulation of the spinner would be better since you could run it millions of times and all on the same day. My guess is that your simulation would be less convincing and less fun for your students than the daily spin, but you could definitely make that argument.

Host A Steepest / Shallowest Stairs Competition

Tonight’s homework: Find some stairs. Calculate their slope. Describe how you did it. Take a picture.

Your students should then determine whose stairs were the steepest and the shallowest and you’ll post those photos at the front of the classroom. You’ll make a big fuss over them. Then you’ll post a bounty for stairs that will knock them off their perch.

One interesting thing about slope is that it doesn’t have a unit, so you don’t need a measuring tape or a ruler to calculate it. Anything your students have on hand will work, including their hands.

Be prepared for a contentious discussion about the difference between the tallest steps and the steepest steps. It’s possible to design steps that are extremely shallow but too tall for anyone to climb up. Wrap your students’ heads around that one.

Be prepared also for students who can’t shake the sense that math is here every time they climb up a new set of stairs.

What a cool job the rest of y’all have.

[photo credits: moyogo, vulcho]

2012 Jan 17: Useful description and modifications from James Cleveland.

Sweat The Small Things

“There are five birds and three worms.” That’s the set-up.

The pay-off is that Tom Hudson found significant improvement in achievement when he asked primary students, “How many birds won’t get a worm?” instead of “How many more birds than worms are there?”

Two things that probably go without saying:

  1. Your students with poor math achievement may be achieving poorly at something besides math. Like language.
  2. It’s hard not to love a job that rewards this kind of obsessive attention to detail.

Applies To Education, And Educational Technology, Also

Ed Begle:

Mathematics education is much more complicated than you expected, even though you expected it to be more complicated than you expected.

If anyone tries to tell you the problems of math education, educational technology, or capital-E education are simple, or that the solutions are simple, or that the people who don’t accept those solutions are simple-minded, kick the crash bar and don’t stop running. They’re wrong and none of this work would be very much fun if it were that simple anyway.

Featured Comment:

Peps Mccrea:

Ever heard of Veik’s law of commensurate complexity? He suggested that no model can simultaneously be both: simple, general, and accurate. It can be 2 of the 3, but not all 3. ‘Simple’ can be important, because complexity is difficult to manage. Particularly in a world where no one person can know enough to make an informed decision. It helps get things done. Just sometimes at the expense of the general and the accurate.