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.
Kate NowakNovember 16, 2011 - 3:34 am -
Yeah but see you are still making us all better. These are both gems. You’re in a ton of classrooms.
Daniel PetersNovember 16, 2011 - 5:26 am -
One of the first problems you run into when teaching the expected value is that ‘expected value’ is a misnomer.
For a die, the expected value is 3.5, so one should think we expect the outcome to be 3.5. In fact, a die never shows the number 3.5.
We should get rid of that expression.
RyanNovember 16, 2011 - 6:15 am -
Dan, thought of this article when I read your expected value example above:
Also the job interview question a friend in the financial sector had a few years back: There are two envelopes, one you know has $1000 in it, the other has either $2000 or $500, but you don’t know which? Which envelope do you choose?
Owen eltonNovember 16, 2011 - 8:33 am -
I agree that “expected value” could be misleading for students. Why not use the term “expectation” instead? Somehow I find that it sounds more feasible: The possible outcomes are 1, 2, 3, 4, 5, and 6 while the expectation is 3.5.
Sander ClaassenNovember 16, 2011 - 9:12 am -
Probably, you’re right that spinning once a day for a month or so would lead to a lot of curiosity and many surprised students in the end, and probably a better learning result, but about what do you teach for the rest of the time during that month? Not about the expected value, I suppose. Because in that case, the effect of this game is probably much less. Or do you expect students to be so stupid they still don’t get the picture? So, in a more pragmatic approach, why not spend one complete lesson to this game? Let them place their bets in the beginning only and then spin it for a few dozen times. The repeat a similar experiment (eg the one with the envelopes in one of the comments) a week later.
Alex OttoNovember 16, 2011 - 9:53 am -
I love this. Just finished using your “Graphing Stories” project in class (thanks for your response to my email too about technology). I would love to try this idea too!
JoshNovember 16, 2011 - 11:00 am -
The timing of this could not be better as I am introducing slope TODAY! Love it.
Quick question on the spinner problem, do you print out a spinner for everyone, or have a “Class spinner”? Also, do you post each classes result on the board, and then also have them keep track in a math journal?
Matt ENovember 16, 2011 - 11:33 am -
Plus the spinner would probably have to be spun while lying flat on a table or desk, rather than hanging on a wall, in case one end of the spinner is slightly heavier than the other, which would skew your results. (The pictured spinner, for example, would DEFINITELY be biased towards the big-money amounts!)
Peter RobertsonNovember 16, 2011 - 3:05 pm -
@ Daniel and Owen
I describe expected value as the “amount you expect to win” per play of a probability game.
The numbers on a die are nothing more than 6 different markers signifying 6 different and equally probable outcomes. When I roll a (fair) die I expect each outcome to come up with equal probability. That is my expectation. Substitute a die with each side represented by a differnt colour, or picture (why do the sides of a die need to be numbers?).
Expected value is only relevent once you place numerical value on the outcomes of the event. The expected value for a die is 3.5 only if as part of the probability event, the outcome was to pay out the value of the number that turned up on the die.
Peter PriceNovember 16, 2011 - 3:39 pm -
I totally agree with the other comments posted. Dan, you’re a gem, and in lots of classrooms, as pointed out by @Kate Nowak.
Just yesterday, my preservice teacher students did their final exam (we finish the school year in Nov/Dec in Australia), and one of their questions was about their response to new ICT-enabled ways to teach math, including Dan Meyer’s. I love the way you push the boundaries of what we all think math education is about. Please keep it up.
luke hodgeNovember 16, 2011 - 3:53 pm -
I like the spinner a lot. I also think it adds much more insight than a computer simulation.
If you did spin it once a day for a month, there is about a 35% chance you won’t get any hits for 5k or 12k. If this happens, maybe it is a tougher sell on why 5k really is the best bet in the long run or maybe it makes for a good discussion on risk.
I like the suggestion of having the students spin the spinner a bunch of times in one sitting. You could turn this into a gambling game. How much should the house charge per spin? Low bid gets to be the house.
Expected value can be thought of as the average over many many trials, so playing a bunch of times and seeing the average amount the house pays out is the right idea to have. To get the theoretical value, it might be useful, in this case, to talk about your best guess for what would happen if you played 360 times. I would avoid fractions or decimals like the plague when introducing expected value.
Christopher DanielsonNovember 16, 2011 - 6:01 pm -
Totally agreed. Totally and completely agreed. If the spinning is going on inside the magical box, I don’t know that it’s not fixed. Thanks for saying it, Dan.
Dan MeyerNovember 16, 2011 - 7:01 pm -
No no. The spinner would be ritualized. Something a student would do in the first few minutes of class. For the rest of the period, you’d carry on with whatever you had planned.
You’d want only one spinner. You’d also want some public record of everyone’s bet and each day’s winning bet. Not so much because students will cheat (like they’d care that much) but because students lose paper. If they don’t have their total at the end of the month, they’ve defeated our purpose.
I figured this went without saying, but let’s let it stand, just in case someone’s wondering how they should orient the spinner.
Scott EliasNovember 17, 2011 - 5:43 am -
I love both of these, Dan. I shared them with our 8th grade math folks since expected value and slope are among two of our weakest areas as a school.
Thanks for sharing and I hope you have a wonderful holiday!
JerzyNovember 17, 2011 - 9:51 am -
I really really like the storytelling/suspense-building aspect of doing the spinner once a day for a month. I imagine the long buildup would make the payoff stick in their minds that much more.
But… Luke is right that it’s not unlikely to get no hits on 5k or 12k over the course of one month. Dan, this isn’t meant to be stupid nitpicking — it could substantially change the outcome of the lesson. If the point of the lesson is to show off “How on Earth did the math teacher know the best bet in advance?” then you want to be sure your bet *will* be the clear winner after only one month.
As far as I can tell from the figure, your long-run expected values per spin are $100/2 = $50; $300/3 = $100; $600/9 = $67ish; $5000/27 = $185ish; and $12000/54 = $222ish. So after MANY spins, say after a few thousand spins, the $12000 should be the best bet hands down.
But there are only roughly 20 classroom days in a month, so 20 spins of the wheel. Within a *particular* set of 20 spins, it’s quite likely that the $12000 will not come up at all. Even if it does, one of the other amounts might come up often enough to beat it.
So for use in the classroom, we have to think in these terms instead: let’s say event A is that “the $100 bet turns out to be the winner in a particular set of 20 spins”; event B is that “the $300 bet turns out to be the winner in a particular set of 20 spins”; etc… (C=$600 wins, D=$5000 wins, E=$12000 wins.) After a month of spinning once a day, one of these five events will happen. Which of them is most likely? And is it near-certain, or is it a toss-up?
Judging just by expected values, we’d guess E should be the most likely. But…
I simulated this in R (I’m happy to share the code if anyone wants), and it seems to me that on a given set of 20 spins, $12000 has only about a 30% chance of being the best bet; $5000 has about a 35% chance of winning out on a given set of 20 spins; and $300 has about a one-fifth chance of being the best bet.
So even though $12000 is the long-run best bet, $5000 is the likeliest winner if you’re only playing for 20 spins. If you’re trying to convince your kids that long-run expected value is the best way to choose your bets… think again!
Now, if you planned to spin it 100 times instead of only 20 times, $12000 has slightly over a 50% chance of being the winner; and $5000 has about a 40% chance of being the winner.
If you spin it 1000 times, $12000 has about a 75% chance of being the winner. So even then it’s not a guaranteed win.
So! I really like the idea behind it all. But the design question remains: What’s a good spinner to use such that (1) there’s a very high chance that the highest-expected-value bet will actually *be* the winner after only 20 spins, and (2) it’s not blatantly obvious to the kids that this will be the winner without doing the expected-value calculation?
James C.November 17, 2011 - 12:04 pm -
I love both exercises and will definitely incorporate them. As much as I love your ideas Dan, they’re sometimes humbling. How many times have I walked up sets of stairs and not once have I thought to use it as a lesson in calculating slope?! :)
MaxNovember 17, 2011 - 12:31 pm -
If I were going to do this, I’d use a spinner from the National Library of Virtual Manipulatives, but I’d have it only spin once at a time… at first.
As the month progressed, I’d check in with kids about what they noticed and were wondering about the running totals, about the spinner, etc.
Let some high roller say, “let’s spin it a million times! Then I’ll win for sure!” and we can talk about that.
I don’t think a virtual spinner feels rigged when it’s actually animated to spin. But I definitely agree with Christopher that just watching the totals for a million spins add up requires suspending disbelief.
The reason for the virtual spinner is to let number of spins be a variable that changes too. I’d know at least one kid was getting EV on a deeper, intuitive level if I said, “sure let’s spin a million times a day” and someone said, “whoah! Can we change our bets then?” [See Jerzy’s math above].
And I like the broader idea of having “teasers” for lessons beginning a month in advance! What a cool way to hook kids and know they’ve had the intuitive experiences we’ll hope they rely on when the math gets mathier.
MaxNovember 17, 2011 - 12:32 pm -
PS — what if kids had to “pay” to place their bet each day? How would that chance the game?
JoeNovember 17, 2011 - 1:34 pm -
The formulation can be generalized:
I don’t think people who understand _____________ understand how hard it is for other people to understand ____________.
I would submit a slight variation:
I don’t think people who understand _____________ understand what it is like to not understand ____________.
(try it with “angles”)
Paul WolfNovember 17, 2011 - 1:51 pm -
I launched the slope contest this morning in my Algebra 1 classes. Can’t wait to do this with my “slow” 8th graders I am inheriting next semester.
One block, we actually had the agrument about steepness, the other block didn’t.
luke hodgeNovember 17, 2011 - 3:06 pm -
Perhaps spin more than once a day. Also it might be nice to have two bar graphs updated each day with the number of wins & cumulative winnings by bet ($100, $300, etc). Maybe segment the bars to indicate the amount won by day as well – or create a slide show of the progression by day.
Later in the unit you could use a virtual spinner. Have students come up with strategies for different spinners & other gambling games, then run the virtual spinner, dice, etc. a million times to see the results.
JerzyNovember 17, 2011 - 4:03 pm -
Looks like you’d have to spin it about 200 times a day for a month to be near-certain your bet would turn out to be right:
Of course you can still have a great discussion even if the teacher’s bet is wrong. But I’d love to figure out a spinner that *does* give a high chance of correct prediction after only a month, and have it be a non-obvious prediction without doing the math!
louiseNovember 17, 2011 - 5:01 pm -
Every child in every class spins it once a day for a month(say,when they enter the room), and records it on the tally sheet next to the spinner. Someone fabulous enters the totals on the handy dandy computer each day. In our school that gives us 300×20 spins in a month. Probably enough.
Raising another series of questions – does it matter when you spin (entering the class or leaving it? Before or after a girl/boy? if you leave to got eh the bathroom and then come back?)
Dan MeyerNovember 18, 2011 - 7:55 am -
This thread got away from me but a quick word of affirmation for the concerns Jerzy and others have raised. We could tune this thing up a lot of different ways, from spinning more often to concluding with a computer simulation to altering the cash payouts and the likelihood of receiving them. All of these have upsides and downsides. Exercises for the reader, etc.
Also, if anyone messes with these in class, I’m always interested to find out if the version of classrooms that’s in my imagination when I write these things in any way matches reality. In other words, will these be as productive as I’m imagining them to be?
JerzyNovember 18, 2011 - 9:28 am -
(Sorry for taking the thread so far in this direction; I’ve just really enjoyed this particular exercise for the reader! The other suggestions are all great, and the stairs contest is awesome as well.)
My last two cents: if anyone does try the spinner exercise, it should be almost 90% reliable if you spin it 60 times, and use the following spinner:
1/2 chance of $100, 5/16 chance of $200, 1/8 chance of $400, and 1/16 chance of $2500.
I think that’s a good balance of “the best answer feels risky / isn’t obvious” and “not too much data to track.” And you can do this by marking up a Twister spinner, since the sectors are all in 16ths.
Paul SchonfeldNovember 18, 2011 - 1:58 pm -
I think it’s important to consider that the mathematical ‘expected value’ payout may not equal with the ’emotional’ payout one gets by winning (or in other words, ‘not losing’).
If this were an activity with real money, I can completely understand why someone would bet on $100 for each of the 20 class days in the month. Winning $100 (on average, every other day) would feel great!
Betting on $12,000 and ‘losing’ day after day would feel horrible. And say you did win $12,000 on the last day, how much better would you feel than the person that won $2,000?
My point is that it feels pretty good to win, and it feels horrible to lose (I think Daniel Pink wrote about this). So if you’re trying to find a way to bet and be happy, you might not use the expected value as the basis for how you bet.
DMTNovember 20, 2011 - 7:06 am -
I used a protractor to check the angles on Dan’s spinner and I get that 5000 is the best bet. In any case, I think it is more interesting to not have one the “extremes” be the best bet when making a point to students.
In my opinion, it makes a stronger point of the value of doing the calculations if the best bet is a “random bet in the middle somewhere”.
Paul WolfNovember 20, 2011 - 8:22 pm -
A kid asked of the steepest stairs contest, “What if I take a picture of a ladder standing straight up?”
I told ’em ladders weren’t allowed, but I thought the suggestion was gold.
Paul WolfNovember 20, 2011 - 8:30 pm -
That is to say, I told them it was a contest ahead of time…don’t know how that will affect it.
DonNovember 21, 2011 - 2:03 pm -
A kid asked of the steepest stairs contest, “What if I take a picture of a ladder standing straight up?”
I told ‘em ladders weren’t allowed, but I thought the suggestion was gold.
I admire your restraint in not answering that a straight vertical line wasn’t defined in your slope assignment.
Paul WolfNovember 22, 2011 - 7:06 am -
Well, I sort of teach it that way, that “undefined” means something like “too big to measure” in slope terms. I know that’s a massive oversimplification, but it makes sense visually and sets the stage for vertical asymptotes.
Good outside-the-box suggestions for shallowest: say you have just no step between the two levels (like a split-level house, from the back door to the patio, or a curb for that matter) does that count? What about ramps? I said neither are truly stairs, but we did talk about the implications.
Angie BDecember 15, 2011 - 10:42 am -
I’ve been behind on my Reader, but I love that I’m reading this two days after my 4th grade son asked me who had a better chance of winning a game of battle ship if someone just placed the destroyer (2 peg) and played against someone who places all of their ships on the board. Probability tells me onw thing, but having read the post about human ability to be random and knowing that I play battleship with a certain strategy makes me reconsider my initial thought process.