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.

So you have here a fairly straightforward carnival estimation game, which I decided to complicate by filling up a smaller container with the same kind of (horrid) malted eggs and making that quantity known.

I surveyed my students, my math-department colleagues, some of their students, my principal, and the central office staff. A little over 100 guesses all told. I tagged each guess with the following metadata:

• name,
• guess type (gut check, visual estimate, math computation),
• job description (student, math teacher, staff member, principal),
• current math class (eg. Algebra 1, Geometry, AP Calculus, etc.),
• grade level (freshman, sophomore, junior, senior),

I showed my students the raw data and asked them what they wanted to know. I wrote their questions on the board.

1. who won?
2. who guessed worst?
3. what was the ranking of everyone in between?
4. what type of people used math computation for their guesses?
5. were there any tied guesses?
6. what was the highest/lowest guess?
7. which grade level guessed the most?
8. which grade level guessed the best?

Define “Bounty”

I said I was offering a “bounty” for answers to those questions and asked them to define the term. Some kids had seen Dog the Bounty Hunter and explained it from that angle. I assigned each question a point value that corresponded roughly to a) the difficulty of the question and b) its relevance to my objective – how are absolute value and percent error useful for calculating accuracy? I offered 20 points for a picture of an interesting fact. (See “Interesting Pictures” below.)

They had to scrape together 100 points for the day and I offered extra credit for initiative, divergent thinking, etc.

What Happened

Students worked in pairs on laptops. They downloaded an Excel sheet with all this data, including the real name of every guesser. Naturally, they were into that.

The great part about a sample size of one hundred guesses is how easy it was to determine which groups were taking a tedious, manual approach to these questions and which were using Excel’s built-in capability for sorting and calculating. I circulated the classroom and could tell that a group was ready to learn more about Excel because they were using hash marks to count up every freshman, sophomore, junior, and senior. Those students were wandering the desert on foot, ready for the water, compass, and camels I could offer them.

Likewise, I saw another group of students subtracting all one hundred guesses from the actual answer (1831) one at a time on cell phones. It didn’t take much to convince them to experiment with another approach.

The Constructivism Multiplier

My favorite conversations with students centered around a definition of “accuracy,” as in, “who were the top ten most accurate guessers?” Our earlier trick of just subtracting the guesses from the answer messed with Excel’s sort mechanism, unhelpfully stacking positives on top of negatives, when, really, we didn’t care if you guessed 100 eggs too high or 100 eggs too low. For our purposes, those two people tiedThis is a good companion exercise, in that sense, to How Old Is Tiger Woods?.

Two students were so close to constructing that operation themselves I had to bite off my tongue to keep from spelling the whole thing out (“ABSOLUTE VALUE! SUBTRACT AND THEN TAKE THE ABSOLUTE VALUE!!”) and then the bell rang. We didn’t graph anything. We didn’t get to percent error. Half the groups got to absolute value.

Off moments like this, I have determined my constructivism multiplier to be four, which is to say it takes me four times longer to bring a student to conceptual understanding through conversation and questioning in a social situation the student helped create than it does to get up in front of the class and simply give it to them straight, no chaser, through direct instruction and a handout of questions I wrote.

What I find maddening about conversations with committed constructivists (cf. the conversation here) is the reflexive assumption that educators choose direct instruction because they’re either power-drunk or self-obsessed or because they lack faith, courage, or high expectations. I can’t, personally, wave so dismissively at the massive institutional impediments to student-constructed learning.

Interesting Pictures

Percent Error By Grade Level

Percent Error By Guess Type

It’s worth pointing out here that “Math Computation” isn’t the same thing as “Correct Math Computation.” The most accurate guessers verified their correct math computation with a visual estimate.

Percent Error By Math Class

Percent Error By Job Description

That last graph is what I meant at TEDx when I said that math gives your intuition a certain vocabulary. The math teachers have a more descriptive vocabulary for expressing their own intuitions than the students do. This is also a fair answer to the question, “when will I ever use math?” You might not. You can live without it. But it makes a lot of intuitive tasks a lot easier. And you should also understand the risk that you’ll one day be fleeced by or passed over for those who know how to speak with that vocabulary.

The Creative Feedback Loop Of Teaching

Where else can you get this? In all of the creative fields that have ever tempted me professionally – I’m talking about graphic design, screenwriting, and filmmaking – ideas often take months to generate and refine, years to produce, and, in many cases, you can’t do anything with the feedback except hope it’s good enough to get you your next job.

With teaching, you can get any old harebrained idea on Friday, challenge your students with it Monday morning, then adapt it for your afternoon class based on feedback from the morning. The feedback loop is fast enough to give you whiplash. It’s so much fun, this job, it seems impossible sometimes that anyone could ever walk away from classroom teaching.

The Grand Prize

Not those horrid malted eggs, that’s for sure.

When it comes to [Facebook’s] online chat function, 1.6 billion messages are sent every single day and 1.4 million photos are uploaded a second.

Not so Nat Friedman who crunched some numbers in an utterly classroom-appropriate exercise in unit conversion and calculated that this means everyone on Earth is uploading approximately 20 photos per day.

Which means I had better hurry up and get a Facebook account.

[via daring fireball]

[N.B. The Internet has been pretty generous today, right?]

[1] http://www.census.gov/ipc/www/popclockworld.html

Appeal To Their Intuition

“How much cash is this?” Take guesses. The student risks nothing with a guess but that investment pays off huge for the teacher over the life of the exercise because the student wants to know who guessed the closest.

Build Slowly

Again, ask “how much cash?” but also ask “how heavy?” Show them the weight. (I zeroed out the jar from every weight measurement you’ll see here. Don’t worry about it.) Spitball some ideas for determining the value of those coins. You’re trying to motivate the idea that the weight of the coins ties directly to how much the coins are worth. Pull up the relevant Treasury website.

Then mix in some nickels. Scoop out a small sample. Play with that. Set up a proportion between value and weight.

Iterate

Now you have pennies, dimes, nickels, and quarters. I took nine sample scoops, everything from small to big.

I formatted these at 4×6 so I could print them out at our local one-hour shop for a few bucks and put one in front of every student.

Throw A Curve Ball

Some will finish quickly. You tell them you have a jar of coins that weighs 5,500 grams. You reach in and pull out 14 nickels. How much is the jar of coins worth?

They’ll run these calculations and come up with an estimate of $55. You tell them it was really $34, which is huge error. Ask for sources of error. Then toss this up and talk about it.

Confirm The Answer

$84.00, if you were curious.

It’s essential to give some kind of visual confirmation of the answer, both so we can give credit to good initial guesses and so we can talk about sources of error. (ie. “who was off by the most? did sample size matter at all?”)

Miscellaneous

1. Show them CoinCalc, the backend of which does exactly what we’ve done here.
2. This activity follows-up nicely on the goldfish activity, where we used a small sample of fish to determine the total population of a lake.
3. We yield the floor to Jason Dyer and anybody else who would like to debate the question, “why are we doing this digitally?”

Download

Here’s the entire learning packet [62MB].

Personally, I think that this particular image lacks opportunities for inquiry. Perhaps if it was presented with other kinds of door locks leading students to come up with and answer the question, “which is the most secure lock?” [emph. added]

This is exactly right. The latest WCYDWT? installment has provoked the usual litany of Really Interesting Bite-Sized Questions, the sort of prompts that will play great in the Applications & Extensions & Assorted Mindblowers section of your lesson plan but which, on their own, aren’t a lesson plan. Those questions don’t provoke the kind of iterated, increasingly difficult practice that students need for skill development.

Again, this image on its own is insufficient. With some creative modifications, however, it will carry you through permutations. Here is that lesson plan in its broadest strokes.

Start with the image.

Tell them the code is 1 digit long. Tell them the code is 2 digits long. Tell them it’s as long you want it to be. I respected the rule of least power here, which meant that when I took this photo I tried to stay out of the way of your lesson planning. Have them write down all the possible codes for n=1, n=2, n=3, etc. The increasing obnoxiousness of the task will motivate a formula for the general case. That’s arrangements.

Tell them the lock is a 4-digit lock. Now turn on the blue light.

Ask them to list the possible codes. You can iterate this a bunch of times until they have discovered on their own this tool that mathematicians call a factorial.

Remind them it’s a 4-digit lock. Then put up this image. It will be confusing, but only for a second. Ask them to list every possible code.

Iterate this with two and three buttons until they have generalized permutations. Then maybe you iterate the entire thing with another keypad lock.

Then maybe you dip into the comments of the original WCYDWT? post and help yourself to some very-interesting follow-up questions. I recommend Alex’s.

Let me close by saying how shocked I am at how little all of this costs.

[Update: Bruce Schneier has a good follow-up on information leakage. Two photos.]

[Update II: due to the peculiarities of many car door locks punching in “123456” tests both “12345” and “23456.” Consequently, there is a number string 3129 digits long that will test every five-number comination.]

[Update III: more information leakage.]

[Update IV: more information leakage.]

I added it to the What Can You Do With This? segment featuring The Bone Collector, which seemed like an obvious pair to me. In trying to find the best classroom entry point for this program, I can only think of the question, “How can we break this thing – trick it into giving an incorrect measurement?” I imagine someone can do better.

Session Title

Games And Puzzles That Develop Sequential Reasoning

Better Title

OMG MICHAEL SERRA!!1!

Presenter

MICHAEL SERRA!!1!

Narrative

A structure not dissimilar to Megan Taylor’s yesterday, where Serra debuted games and puzzles and gave us time to tease them out.

I sat with two former colleagues in the back – all of us now at different schools. One teacher enthused over Sudoku puzzles. They challenge kids. Kids like them. It gets them comfortable with numbers. The other enjoys Serra’s games and puzzles, like Lunar Lockout. Both cite improved student disposition toward math and improved deductive reasoning.

I disagreed with them. In general, I find it dangerous to put too much distance between “fun time” and “math time” preferring, instead, to have that cake and eat it too, creating as many challenges as I can that are both fun and mathematically rigorous. (Which Sudoko, to put it plainly, isn’t.) My task is harder, I think, and I know I fail at it more, but I’m more satisfied on balance.

It was a good conversation. Feel free to interrupt us.

Serra’s best offering for my money was Racetrack Math:

It’s like this:

1. Draw a racetrack on graph paper, however crude.
2. You and your opponent start anywhere on the starting line.
3. You travel along vectors. You may increase or decrease either the x-value, the y-value, or both, but only by one unit per turn.
4. First person to the finish line wins.
5. (P.S. No crashing.)

This gets very interesting very quickly. You start out with tiny vectors which lengthen by one unit every turn. If you fail to notice the side of the track off in the distance, though, and fail to slow down in time, you crash. (Which I did in the example above.)

I hereby toss all of my battleship exercises in the recycling bin. This is a much more straightforward introduction to positive/negative coordinates since each new turn is relative to the last turn rather than relative to this strange coordinate axis thing.

Plus, your students can create racetracks of their own, of infinite complexity, within seconds. Serra cited some kids who created a pit lane, which you had to enter on your second lap, and oil slicks, on which you could not adjust your vector at all. I’m impressed.

Visuals

PowerPoint. Which is tough when you’re asking people to solve a puzzle. If someone suggests an alternative route to the one you have programmed into your slide, you have to dodge their answer a bit.

Handouts

Blank puzzles and games to draw on. Again, paper is not dead. How do you do this digitally? Load each picture one at a time into Skitch and pass a stylus back and forth? Moderation, please.

Homeless

• “There is no research that demonstrates these games improve outcomes in other mathematical procedures like two-column proofs,” Serra admitted reluctantly. “It has to be there. I know it is.