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Practice may not always be fun, but it can be purposeful. Some of my favorite tasks lately contain purposeful practice.

For instance, Dandy Candies tells students they’re going to package up 24 cubical candy boxes. It asks them, “Which of four packages uses the least amount of packaging? Which uses the least amount of ribbon?”

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This is the usual house style. Concrete imagery. No abstraction. Contrasting cases. Predictions. Students make their guesses. Then they get the dimensions from the video. They calculate surface area and ribbon length. (Ribbon length is a little bit more interesting than perimeter but not by a lot.) They validate their predictions with their calculations.

But then we ask them to find out if another package dimension will use even less material.

So now the students have to think systematically, tabling out their work so they don’t waste effort finding the surface area of a lot of different prisms.

Contrast that against a worksheet like this, which is practice also, though rather less purposeful:

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Where else have you seen purposeful practice?

I’d look to:

BTW. Given any number of cubical candies, what is the best way to minimize packaging? Can you prove it? I can handle real number side lengths but when you restrict the sides to integers, my mind explodes a little.

A workshop participant gave this algorithm. I have no reason to believe it works. I also have no reason to believe it doesn’t work.

  • Take the cube root of the volume.
  • Floor that to the nearest integer factor.
  • Square root the remainder factor.
  • Floor that to the nearest integer factor.
  • With the remainder factor, you have three factors now.
  • The smallest of all three factors is your height.
  • The other two are your length and width. Doesn’t matter which.

Note to self: test this against a bunch of cases. Find a counterexample where it falls apart.

The Chinese Room

James Greeno:

A person lives inside a room that has baskets of tokens of Chinese characters. The person does not know Chinese. However, the person does have a book of rules for transforming strings of Chinese characters into other strings of Chinese characters. People on the outside write sentences in Chinese on paper and pass them into the room. The person inside the room consults the book of rules and sends back strings of characters that are different from the ones that were passed in. The people on the outside know Chinese. When they write a string to pass into the room, they understand it as a question. When the person inside sends back another string, the people on the outside understand it as an answer, and because the rules are cleverly written, the answers are usually correct. By following the rules, the person in the room produces expressions that other people can interpret as the answers to questions that they wrote and passed into the room. But the person in the room does not understand the meanings of either the questions or the answer.

This is a pretty perfect parable.

Featured Comment

Joel Patterson, talking to commenters who’d send this parable along to their students’ parents:

I like this parable. Before you condense it, and give it to all your parents, consider whether those parents have science/engineering backgrounds. It’s a pretty complicated picture to envision. I think S/E people would grasp it (if they haven’t heard of it already) and would get your point. But if the parents have less of an S/E background, the complicated parable is likely to bore them and not convey your point.

Have an explanation at hand that is more like the guitar players who can improvise, not just repeat the 5 songs they’ve memorized. Or cooks who can put together a soup without the recipe because they know which spices and foods have good flavors together.

Previously: [Makeover] These Tragic “Write An Expression” Problems

tl;dr. I made another digital math lesson in collaboration with Christopher Danielson and our friends at Desmos. It’s called Central Park and you should check out the Walkthrough.

Here are two large problems with the transition from arithmetic to algebra:

Variables don’t make sense to students.

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We give students variable expressions like the exponential one above, which they had no hand in developing, and ask them to evaluate the expression with a number. The student says, “Ohhh-kay,” and might do it but she doesn’t know what pianos have to do with exponential equations nor does she know where any of those parameters came from. She may regard the whole experience as one of those nonsensical rites of school math which she’ll forget about as soon as she’s legally allowed.

Variables don’t seem powerful to students.

In school, using variables is harder than using arithmetic. But what does that difficulty buy us, except a grade and our teacher’s approval? Meanwhile, in the world, variables are responsible for anything powerful you have ever done with a computer.

Students should experience some of that power.

One solution.

Our attempt at solving both of those problems is Central Park. It proceeds in three phases.

Guesses

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We ask the students to drag parking lines into a lot to make four even spaces. Students have no trouble stepping over this bar. We are making sure the main task makes sense.

Numbers

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We transition to calculation by asking the students “What measurements would you need to figure out the exact space between the dividers?” This question prepares them to use the numbers we give them next.

Now they use arithmetic to calculate the space width for a given lot. They do that three times, which means they get a sense of the parts of their arithmetic that change (the width of the lot, the width of the parking lines) and those that don’t (dividing by the four lots).

This will be very helpful as we take the next big leap.

Variables

We give students numbers and variables. They can calculate the space width arithmetically again but it’ll only work for one lot. When they make the leap to variable equations, it works for all of them.

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It works for sixteen lots at once.

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Variables should make sense and make students powerful. That’s our motto for Central Park.

2014 Jul 28. Here is Christopher Danielson’s post about Central Park on the Desmos blog.

Featured Comment

Grant Wiggins:

In thinking further about your complaint about “Write an expression” I think what is also going on in this app is a NEEDED slowing down of the learning process. The text (and too many teachers) are quick to jump to algorithms before the students understands their nature and value. Look how long it takes to get to the concept of an appropriate expression in the app: you build to it slowly and carefully. I think this is at the heart of the kind of induction needed for genuine understanding, where the learner is helped, by scaffolding, to draw thoughtful and evidence-based conclusions; test them in a transfer setting; and learn from the feedback – i.e. the essence of what we argue understanding is in UbD.

Kevin Hall:

One reason I like this activity so much is that it hits the sweet spot where “What can you do with it?” and “What does it mean?” overlap.

Simon Terrell recaps his lesson study trip to Japan with Akihiko Takahashi, who was the subject of Elizabeth Green’s American math article last week:

In one case, a teacher was teaching a lesson about division with remainders and the example was packaging meatballs in pack of 4. When faced with the problem of having 13 meatballs and needing 4 per pack, one student’s solution was “I would eat the extra meatball and then they would all fit.” It was so funny and joyful to see that all thinking was welcomed and the teacher artfully led them to the general thinking that she wanted by the end of the lesson.

I can trace my development as a teacher through the different reactions I would have had to “I would eat the extra meatball,” from panic through irritation to some kind of bemusement.

BTW. The comments here have been on another level lately, team, including Simon’s, so thanks for that. I’ve lifted a bunch of them into the main posts of Rand Paul Fixes Calculus and These Tragic “Write An Expression” Problems.

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Two quick meta-items about blogging from the last week:

  • I attended Twitter Math Camp 2014 in Jenks, OK, in which 150 math teachers who generally only interact online get together in person. I gave a keynote that could probably best be described as “data-rich,” in which I downloaded and analyzed details on 12,000 blogging and tweeting math teachers. Here are links to my slides and speech as well as the CSVs if you want to analyze some data yourself. (Who doesn’t!)
  • A doctoral student in Canada is interested in blogging as “unmediated professional growth” and sent me a survey about my blogging. Here is a link to my responses. How would you have answered?

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