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Teaching probability

My colleague found an activity to do with his 5th grade class, similar to this one. Basically, he gave the students 10 coins each, and asked them to put the 10 coins on a number line (with numbers from 1 to 12) with a partner. Each round they roll 2 six-sided dice, find the total, and remove a coin from their number line if it matches the roll. They keep going until one of the two students has no coins on the number line.

 

At first, most student’s starting positions looked like this:

Student 1 - flat distribution

or this:

Student 2 - another flat distribution

 

At the end of the lesson, we played a 5 coin version of the game, and one student’s paper (after 1 round) looked like this:

Student 3 - All 4 coins on number 7

 

Unfortunately, although most of the students did notice that some numbers came up more frequently than others, as evident by their distributions looking a bit less flat, and bit more centred on 7 on the number line, many of the students still had obvious misconceptions of probabilty. Students made comments like:

"If I spin around twice before rolling, I get a more lucky roll."

"I got a few 11s last game, so I’m going to put a few more coins on 11."

"8 is my lucky number! I’m going to put 3 coins on 8."

"I need to spread out my numbers so I have more chance of getting a coin taken on each roll."

Our plan for next class is to have students switch up groups, discuss insights they’ve had on the game, play a couple of rounds, switch them up again, while we walk around and see what strategies they use. Hopefully we’ll see less students spinning around in order to improve their dice luck…

I also asked for resources on Twitter for using a probability game as part of a lesson on probability, and had the following 5 games recommended (all of these look good because they are relatively easily produced and used in an elementary school classroom).

 

Update:

I wrote a simulation to test to see what distribution of coins is the best. I am somewhat surprised by the results. Check it out here.

Why am I surprised by the results? My intuition about how this game works failed me. I thought that the likelihood of each number coming up was the most important factor in deciding a good strategy for the game. It turns out that one has to balance out the knowledge of the likelihood of each number being rolled with having a selection of numbers available. It may be that the probability of a 7 being rolled is 6/36 but the probability of a 6 or a 7 being rolled is 11/36.

In fact, it turns out that having a selection of different rolls available is fairly important. I updated the simulation so I could choose the number of coins to place, and with 3 coins placed, choosing 4, 5, 6 is better than having all three coins on the number 7.  With 2 coins, it is better to choose 6 and 8 than to put both of the coins on 7 (although 6, 7 is better than 6, 8 – but only slightly).

There are three messages I get from running this simulation.

  1. One should, in general, not make too many intuitive assumptions about how probability works, particularly with somewhat complicated examples.
  2. One should be careful how one uses even simple games to teach probability. All of the students saw that 7 was the most common number, but their intuition of "choosing a wide spread" is a valuable one for this game, but it doesn’t help us get at the idea of some numbers being more likely than others. I think we’d need a different game for that.
  3. It is probably a good idea to build the simulation before you play the game with students, if at all possible.

Introducing Probability Using Settlers

This past week I was looking for a way to introduce probability to my 9th grade students.  One of the problems students have when they are first learning probability is developing some intuition about what to expect.

I decided that one of the best ways to develop intuition about probability is to have some strong emotions associated with the results of their initial probability experiments, so I decided to teach my 9th grade students how to play Settlers of Catan.  I didn’t give them any information about best strategies to play the game, I just taught them the basic rules and set them loose.  Here are some rules for your reference.

Settlers Map

The basic idea is, each hexagon produces resources, but only when the number shown on the hexagon is rolled as the total of 2 six-sided dice.  If you have a settlement located at one of the vertexes of a hexagon which has just produced resources, you gain 1 of those resources.  You can then save up these resources, trade them with other players, or then use them to buy more settlements, cities, etc… Essentially if you gain enough resources of the right type before your opponent, and you win.

The actual system we used to play is called JSettlers, and it is an open source Settlers of Catan server.  I hosted it on my laptop with no difficulty and shared the link to my students to play it.  This way I didn’t have to pay for a class set of expensive Settlers of Catan games.

It only took about 10 or 15 minutes of playing for the kids to realize when they had made poor choices, or when someone had an obvious advantage.  The question I had once we had played for enough time that they had gathered some data (I required them to keep track of what was rolled as they played), which starting settlements were poorly placed, and which were in the best locations.  Students looked at the following situation and decided that this intersection of hexagons was a good place to put a settlement.

Good choice of settlement

They looked at an intersection like the following and decided that this was a poor place to put a settlement.

Poor settlement location

I asked them why they liked the first spot and didn’t like the second?  One of them said it perfectly, "well, the numbers 8,9, and 10 are WAY more likely to come up than 2, 4, and 11."

We followed with a discussion of why each number was not equally likely to come up using a typically sample space table, and then we kept playing, having both put some context on the probability we were learning, and developing some intuition about which numbers were more likely to come up.  I was able to extend their thinking quite a bit, as there were several different games being played, none of which had exactly the same set of numbers rolled.  It really worked well, and I’ll continue to use an example like this in my practice.