Month: April 2012

Total 13 Posts

Ten Design Principles For Engaging Math Tasks

My work with the Pearson Foundation has changed. They still include some of my three-act tasks (all of which are available for your non-commercial use at this page) but more often lately I review units for engagement. “Dan thinks like a child,” said one of the authors, which I chose to take as a compliment. The bottom line is that engagement is incredibly tricky to nail to the wall. At one point I was asked to draft a document outlining some guiding principles for designing engaging math tasks. I’ll reproduce that document below.

  1. Perplexity is the goal of engagement. We can go ten rounds debating eggs, broccoli, or candy bars. [references a debate, long since settled — dm] What matters most is the question, “Is the student perplexed?” Our goal is to induce in the student a perplexed, curious state, a question in her head that math can help answer.
  2. Concise questions are more engaging than lengthy ones, all other things being equal. Engaging movies perplex and interest you in their first ten minutes. No movie on this list took more than twenty minutes to set up its context, characters, and conflict. The same is true of engaging math problems, either pure or applied. Use a short sentence or simple visual to “hook” the student into the space of the problem. Use later sentences to expand on it. This order is often inverted in problems that fail to engage students.
  3. Pure math can be engaging. Applied math can be boring. The engagement riddle isn’t solved by taking pure math problems and shoehorning them into contexts that don’t want them. It’s hard to argue that two trains traveling in opposite directions from Philadelphia at different speeds is more engaging than “How many ways can you think to turn 20 into 10?”
  4. Use photos and video to establish context, rather than words, whenever possible. Rather than describing the world’s largest coffee cup in words, show a photo or a video of it. Not only because our words fail to capture what’s so engaging about the coffee cup but because we should find ways to lower the language demand of our math problems whenever possible.
  5. Use stock photography and stock illustrations sparingly. The world of stock art is glossy, well-lit, and hyper-saturated and looks nothing like the world our students live in. It is hard to feel engaged in or perplexed by a world that looks like a distortion of your own.
  6. Set a low floor for entry, a high ceiling for exit. Write problems that require a simple first step but which stretch for miles. Consider asking students to evaluate a model for a simple case before generalizing. Once they’ve generalized, considered reversing the question and answer of the problem.
  7. Use progressive disclosure to lower the extraneous load of your tasks. This is one of the greatest affordances of our digital platform: you don’t have to write everything at once on the same page. While students work on one part of a problem, there’s no need to distract them by including every other part of the problem in the same visual space. Once they answer the first part of the problem, progressively disclose the next. This technique has far-reaching applications.
  8. Ask for guesses. People like to guess, speculate, and hypothesize. Guessing is engaging. Before disclosing all the abstractions of parabolic motion on the basketball court, just show a video and ask the question, “Do you think the ball will go in?” Once they’ve answered, continue the rest of your unit, lesson, or problem, now with more engaged learners. They’ll want to know if they’re right or not so be sure to pay off on that engagement later by showing them.)
  9. Make math social. More engaging than having a student guess whether or not the ball goes in is showing her how all of her classmates guessed also. Summarize the class’ aggregate responses with a bar chart. Students will enjoy seeing each others’ short answers and opinions but we can also use the same social interactions to engage them in pure math. Have your students a) select three x-y pairs and b) check if they’re solutions of x + y < 5. If everyone in the class sees the results of everyone else’s investigation, a visualization of linear inequalities will emerge on the class’ composite graph.
  10. Highlight the limits of a student’s existing skills and knowledge. New mathematical tools are often developed to account for the limitations of the old ones. You can’t model the path of a basketball with linear equations — we need quadratics. You can’t model the growth of bacteria with a quadratic equations — we need exponentials. Offer students a challenge for which their old skills look useful but turn out to be ineffective. That moment of cognitive conflict can engage students in a discussion of new tools and counter the perception that math is a disjointed set of rules and procedures, each bearing no relationship to the one preceding it.

What would you add? What would you subtract?

Featured Additions From The Comments

  • When possible, reveal information only when requested. Current word problems will have 3 numbers given and they will all be used and nothing more is necessary. Knowing what is necessary to solve a problem and what is possible to measure is key to real-world application problems. [CalcDave]
  • Once the problem has been completed, explain the cultural and historical context of this problem, if it exists. [David Wees]
  • Go crazy. You know how high 5 cups would be? What about 5,000? You can factor this trinomial? Try this octnomial. What would happen if we composed these functions 100 times? 200? Asking these sorts of questions empowers students by making them aware of just how robust the abstractions they’ve earned are. At the same time, they humble students who think that they deserve a cookie for directly measuring the height of 5 cups. [MBP]

2012 May 19: Here’s a predecessor of this document that I totally forgot I wrote.

Best Of 101questions [4/14/12]

A few of my favorite listings on 101questions this week:

  1. A Fistful of Quarters (and Dimes), Nathan Kraft. Provokes the comparison of the value of a coin against its weight, which seems at first like a useless ratio. But remember the nickel thieves? If someone let you carry as much change away as you could lift, which kind of coin would be the smartest pick?
  2. Pennies, Friedrich Knauss. Provokes the comparison of the value of a coin against its surface area which, again, seems like a totally useless ratio until you see a photo like this. If you were going to carpet your floor with a particular kind of currency, which would be the smartest pick?
  3. Handshakes, Craig. Love the clip. I find the question, “How long would it take to infect the whole office?” irresistible.
  4. Coins, Steve Phelps. I’ve noticed these kinds of first acts are difficult to pull off. (Check, please, also uploaded last week, is struggling, for instance.) They’re often too cluttered or they place students too high up the ladder of abstraction too quickly. Steve Phelps strikes a nice balance here. Moreover the task is open to several correct answers, which is unusual for material you’ll find on 101questions.
  5. 1982 Osborne Executive vs. 2007 iPhone, Carl Malartre. I tried a similar approach with Evolution. I like Carl’s more.

[JOB POSTING] Math Adviser @ Caine’s Arcade

Caine’s Arcade tells the story of a nine year-old in East Los Angeles who made a functioning arcade — games, tickets, prizes, etc. — out of cardboard. He’s a scrappy underdog. He also needs your students’ help with math.

1. Fun Pass Economics

Nirvan Mullick:

I asked [Caine] how much it was to play. He was like, “For one dollar you get four turns. But for two dollars you get a fun pass.” Well how many turns do you get for a fun pass. “You get five hundred turns for a fun pass.” I got a fun pass.

Your students could probably lend a hand there.

2. Fun Pass Security

Caine installed calculators on all the games. Why? In order to validate the fun passes. He has a number on one side of the card. You type that number into the calculator, press “the check mark,” and another number comes out — a number that Caine thinks is totally unpredictable. So Caine writes the output number on the other side of the card and, thus, the fun pass is validated.

Resist the urge to editorialize about how our students think all of math is a fun pass validator. Instead, have your students show Caine how his system can be fooled and then suggest alternate methods for validating the cards.

Benefits? Not great. Pay? Not great either. Apply at the manager’s office, just past the cardboard skee ball machine.

2012 April 11. Aaaaand … cue the forged fun passes.

Hot Links

Amy Gruen’s blog is a pile of fun. She’s a magpie, looking about her world for odds and ends to bring back to her classroom, then posting pictures and explanation for our benefit. Recommmended.

Bryan Meyer:

You always hear people say, “kids don’t like math!” Correction…kids don’t like feeling dumb. People don’t like feeling dumb.

Dan Goldner:

I’m flabbergasted. I have a number of students—maybe 10? 20?—who determine by division how many bills there are, then figure out by multiplying 60x60x24 how many bills are given away in a day. Fine. But then they start subtracting … after day 1 there are 9,913,600 bills left. After 2 days there are 9,827,200. Almost immediately many students lose interest, but there are a few arithmetic ox that start chugging through it (with calculators, to be sure). 9,740,800. 9,654,400. I watch in disbelief as the markerboards are filled in, line by line. 8,617,600. 8,533,000. After a while I can’t help myself. I casually mention that people sometimes use division to do repeated subtraction, and I countdown from 10 by 2′s and compare to 10/2. They are a little chagrined at not having thought of that, but they try it. Then they face confusion about handling the remainder.

[3ACTS] Bucky the Badger

W. Stephen Wilson:

The ability to communicate is not essential to understanding mathematics.

The Common Core State Standards for Mathematical Practice:

[Mathematically proficient students] are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others.

Feel free to jump straight to the task page.

The jist, if you aren’t the movie-watching type, is that whenever Wisconsin’s football team scores, their mascot has to do push-ups equivalent to Wisconsin’s total score. This screen grab is a useful talking point.

Play the first act, which ends after Maddow announces the final score of the Wisconsin-Indiana game: 83-20. “83 POINTS!

Ask your students to write down a guess. “How many push-ups do you think Bucky did over the entire game?” Ask them to write down a number they think is too high and too low.

Here’s where it gets interesting. Ask them to write down all the information they’ll need to figure out the answer. That question is controversial even among math teachers at workshops I facilitate. Some argue that all you need to know is that Wisconsin scored eleven touchdowns and two field goals. Others argue that you also need to know the order of those touchdowns and field goals. In W. Stephen Wilson’s ideal math classroom, we’re stuck. Communication is inessential to Wilson’s ideal math classroom but communication is essential to any resolution of this dispute.

The mathematical practice standards require an argument. Both sides aren’t right. How will one side persuade the other? At this point, we learn a useful technique for arguing mathematically. One side has said, “Order never matters.” All we need to sink that rule is a single counterexample. One person suggests trying [7, 7, 3] and then [7, 3, 7] — the same scores in different sequence. Another suggests an even less costly test of [7, 3] and [3, 7]. And the matter is settled.

Having established that order matters, another question then arises: “If you’re Bucky, when do you want your team to score its field goals — at the end of the game or the beginning?”

Sidenote #1: Paper Wrecks This Problem

Paper is non-neutral. It changes the student’s task. NCTM posted a similar problem featuring “Push-Up Pete.” [h/t Cathy Campbell, John Scammell]

The question that’s rarely asked in print is, “What information will you need?” That information is generally nailed to the floor, written directly on the page. NCTM has revealed in the text of the problem that the order of the scores matters when all the action is in deciding whether or not the order of the scores matters.

Sidenote #2: Opposition To The CCSS Makes For Very Strange Bedfellows

The CCSS aren’t remotely above criticism. It’s bizarre to me, though, how many edtech pundits leapt on that Fordham piece, grateful for any institutional validation of their position against the CCSS. But Wilson and Wurman, the authors, like the punditry’s technological utopianism even less than they do the CCSS. The enemy of your enemy is not your friend.

Featured Comment


My best two students disagreed on whether order mattered and I was able to convince (falsely) one of them that order didn’t matter. And sure enough one of my “average” students – who always works her butt off but is rarely rewarded publicly in class for that work – was the only one to figure out and show that order matters.