I've been debating on whether or not to write about this for the past week, but I keep thinking about it so I figure I might as well record it somewhere. My class is in the middle of a unit on probability and last week we were working on this problem from the IMP Year 1 "Pig" unit:
Students played around with it for a bit and we even did some experimental trials before trying to tackle the theoretical probabilities...which is when things got tricky. They started listing all of the possible ways to get a sum of 2, 3, 4, etc. when rolling a pair of dice (pair of die?? not sure? well, you would say "pair of shoes" and not "pair of shoe" so I'm sticking with dice). Anyways, there was a big controversy about whether or not we should count 1+2 AND 2+1 as two different options or if we should just count them together as one option (which, in all honestly, I tried to intentionally bring out by giving students pairs of dice that were different colors).

The group that said we should count them as two different options ended up with 21 total possible outcomes and the following theoretical probabilities:

2            3            4            5            6            7            8            9            10            11            12
4.8%    4.8%      9.5%      9.5%     14.3%     14.3%    14.3%      9.5%       9.5%       4.3%        4.3%

The group that said we should count them together as one option ended up with 36 total possible outcomes and the following theoretical probabilities:

2            3            4            5            6            7            8            9            10            11            12
2.8%    5.6%      8.3%      11.1%    13.9%    16.7%    13.9%     11.1%      8.3%        5.6%        2.8%

Neither side was willing to budge, so I suggested we conduct a HUGE experiment with LOTS of trials across all three of my classes so that we could put the results together and see what conclusions we could draw. We did something like 3,500 trials...and here is what we found (experimental probabilities in brown on the far right):
In the end, there were a few people from the "21 camp" that decided to change their mind and join the "36 camp" but most people stuck with their original idea.

I was reminded of a quote from Les Steffe's work that I read a while back:
"A particular modification of a mathematical concept cannot be caused by a teacher any more than nutriments can cause plants to grow. Nutriments are used by the plants for growth but they do not cause plant growth."
I'm curious what you think and what you would do in this same situation. I let it go. I felt I did my job by helping students test their ways of thinking, not by telling them what to think.


02/24/2013 8:16pm

I've been thinking a LOT about reasoning from definitions lately. I think the definition of sample space is the number of *equally likely* outcomes. From that perspective, I think it can be logically argued that the 21 possible rolls of a pair of dice can't be a sample space, because rolling a sum of 3 (1,2 or 2,1) is more likely than rolling a sum of 2. Although maybe the whole point of students' arguments is that they think rolling a 2 and rolling a 3 are equally likely?

My counter-thought-experiment to that is, if you are going to bet on rolling a 3, how many outcomes will make you happy for the first roll? How about for the second roll? What changes if you bet on rolling a 2? Which feels like a safer bet? Why?

I would have done exactly what you did in this instance (at least I like to think I'd have been that wise), and I think that there are many more opportunities for students to encounter and problematize this idea. Better to leave it as a question than do some telling by which you think you've cleared it up and never return to it again! They'll learn this when they encounter the right pile of ideas at the right moment when they have the right questions in their head!

But since I was thinking of definitions, I thought I'd share what came to mind for me about using definitions to make a mathematical claim everything else (like common sense) doesn't clear things up.

02/24/2013 9:15pm

Hi Max...

Thanks so much for the thoughtful comment. Funny that many of the questions and scenarios your bring up also played out in our discussions (moreso in their ideas about justifying their respective stances). Many students were convinced that 2 and 3 would be equally likely. One student, in return, asked the very question you did about "how many outcomes for the first roll? the second?" Of course, this led many students to question whether or not rolling two dice at the same time was comparable to rolling a single die and then another single die at separate times (another one with varying arguments from students).

I think, for me at least, you nailed it when you said "they'll learn this when they encounter the right pile of ideas at the right moment when they have the right questions in their head." I suppose we just keep trying to help unveil the complexity in seemingly simple ideas and create chances for them to confront their own ideas. Thanks again!

02/25/2013 7:04pm

Wow, your students sound like great arguers and justifiers. I remember having this same conversation as a senior in college with a professor of probability and feeling less prepared to justify my ideas than your students sound. I certainly never thought of the "how many outcomes for the first roll? the second?" until I was teaching this myself for the 3rd time. I'm curious how you've seen their justification skills get better over the course of the year and what kinds of things you've done to work on that?

I also think it's important to note that this might not actually be a "misconception" if they're asking questions like "is rolling two identical dice at the same time comparable to rolling two single die at separate times"? That means they're getting pretty deep into what it means to think about probability.

I liked Joshua's question about "just how far off are our two different theoretical models from the experimental model?" It seems like the one thing Team 21 hasn't fully convinced me they're accountable too is the discrepancies between the theoretical and experimental models. I'm not convinced it's small enough to chalk up to chance (what a cool way to accidentally launch an investigation into hypothesis testing!)

02/26/2013 8:35pm

Yeah, I agree that they haven't been able to explain the discrepancy between the theoretical and experimental probabilities. It reminds me, I guess, that we bring things to their (students') attention that we think MIGHT disrupt their way of thinking....but ultimately it matters whether or not the students themselves accept the scenario as a disruption. While some did, many refused to accept it as a disruption. Again, maybe not a "misconception" unless the student buys into to other underlying ideas about probability (law of large numbers, quantifying chance, theoretical vs. experimental, etc.).

I'm sure my students would appreciate your comment about their ability to "argue." :) I would agree, they are often good at arguing (although we are still working on understanding other perspectives). Don't be fooled by what I write, I struggle a lot with finding ways to get more students involved with thinking, debating, etc. I DO think, however, that the "habits of a mathematician" as a central feature of our class has been helpful. One of them is "justify and support" and, because we refer to these things often, I think (some/most) students have accepted that this is part of what mathematicians do and also part of how our class runs. In a somewhat circular fashion, it also brings me back to my initial post here...do we step in as a teacher or not? I guess I think that the more we choose to step in, the less students will rely on their own thinking and discussion to sort things out. Again, not sure....

Thanks for such an interesting conversation here, Max.

02/24/2013 11:15pm

I would be uncomfortable leaving it with such a big misconception. I agree that there are a lot of questions and answers waiting, and that pushing it too hard would just make me feel better while not actually having the kids learn anything.

On the other hand, maybe doing a similar game with HH, HT/TH, and TT, would make it more clear more quickly what's really going on. You could flip a penny and a nickel to make the analogy to two different colored dice. More than two coins might work even better: with HHH vs HHT/HTH/THH vs HTT/THT/TTH vs TTT, the 3:1 ratio camp would show up as different from the 1:1 ratio camp pretty quickly, unlike with the dice where the only big 2:1 ratio is among outcomes that are already pretty unlikely.

I also wonder if asking them to come up with a way to measure how far off the data were from each of the two sets of predictions would show how hugely much better the correct answer is at predicting the data you have.

02/25/2013 12:49pm

Hi Joshua...

Thanks for your honest comment. I keep going back and forth with your suggestions in my head. As a class, we had done some different investigations with coins and there didn't seem to be any hang up with HHT being distinct from HTH (or some other combination) but somehow things change when you're talking about the sum versus the outcome. I guess the equivalent question would be something like "is it more likely to get two heads/one tail or ALL heads?" Then, trying to make the connection back to the dice problem....why is it different in one case but not the other. Maybe I'll try that as a warm-up this week and see what happens?

At the same time, I wonder when it becomes "guess what the teacher wants us to think" if we keep giving them the same (or similar) situation to consider. At which point, I would imagine they surrender their ideas to authority versus making conclusions on their own. What do you think? Great suggestions, Joshua....thanks for your input.

03/04/2013 12:19pm

Yeah, I agree that you can end up in the "authority" mode where the kids give up on understanding and start trying to predict your behavior instead. It is a big danger.

On the other hand, there's a big danger in having them think they've got it figured out for themselves when their beliefs are still the opposite of reality.

I'd be curious to find out what happens if you do a similar game with coins. Maybe you can get "sum" into the picture by asking for the "total number of heads" or some kind of phrasing like that, to make the analogy more clear. Or, you can write 0 for tails, 1 for heads, and then literally sum them.

03/02/2013 9:55am

I don't think that giving the students "the answer" is the same as telling them what to think. They've clearly done a great deal of thinking about this problem, but some have come to false conclusions. I think it's important not only for students to make conjectures & justify them but also to grapple with their own misconceptions.

I remember a couple years back we explored the Monty Hall problem in my advanced 8th grade math class. Students played the game a few times, made their own conjectures about what strategy was best, and then simulated the two strategies (switching vs. staying) using spinners. We compiled everyone's results like you did and came up with a pretty accurate experimental probability. Then students debated whether one strategy was preferable and had to justify why their conclusion made sense. I remember two of our most out-spoken young mathematicians vehemently debating their positions until one was convinced by his peer. But I now wonder whether his understanding was an enduring one.

One question: you seem to do a great job allowing students to be involved with the process of mathematizing. How do you support their gradual progression from intuition to conjecture to formal structure? For example, do students use "outcomes" or "outcome space" to describe all the possibilities? I think providing the words to describe the math can be a big help for students coming to grips with very abstract mathematical concepts.

02/27/2013 6:23am

One thing that's going on in this story that feels extra neat to me is that you've created a situation in which this is a community concern. It's not just a matter of you thinking about one student and their ability to get the right answer to a probability problem individually. It's a situation where a group of mathematicians are working on a problem together and getting different results. I'm curious what you've heard from the students about how that feels? Are they happy to agree to disagree? Or does it feel weird to either group not to be able to come up with a model for dice probability that everyone can accept? I was thinking that if I were home-schooling one member of Team 21 I would feel compelled to keep offering more and more probability experiences to find out what they had/lacked in their mental model that didn't match mine, and see if I could get us on the same page. In a classroom setting where I wasn't one of only two voices I think I would feel less like "I need to keep probing this" and more interested in how this community of mathematicians would cope with discrepancy in their midst. I think.

That said, I love coming up with investigations and so I couldn't help but think of two things I've thought of recently about experimental probabilities. One was in a course I took... Using coin flips to model if a baby would be a boy or a girl and testing an alternative to China's one baby policy which is to let families have babies until they have a boy. Would that increase the population rate? What would it do to the number of girls? I made a prediction then did the experiment, found my prediction was way off, then looked at te theoretical probability to confirm my experimental. I learned a lot... And the issue has been in the news recently b/c of the lack of women in many rural villages leading to whole towns of just bachelors (and also bride stealing and selling).

I also thought of something I overheard recently which was that even though theoretical probability suggests it's more likely you'll get struck by lightning while winning the Nobel prize than win the lottery people do it (win the lottery) every day. So the person was saying that this is the difference between experimental and theoretical probability. That seemed weird to me, like it suggested you shouldn't trust either probability or like they had nothing to do with each other. So it seemed like a neat investigation: if the lottery is so hard to win, why do people win? Is that a problem for theoretical probability?

Bob Miller
03/01/2013 12:46pm

I have used a similar activity with third through 8th graders with similar results.

The method I have used that has been most successful in clarifying the problem to seed student thinking is the following.

I always give students (usually pairs of students) two dice of different colors. If the dice are the same color, the mathematics seems to be harder to see. When the issue you have described arises, as it always does, and when the members of the table group has exhausted themselves in explanations, I ask if I can talk with them to get clear about what the problem is. First I arrange the dice (off the table) so that the two single dots (one on the red, one on the blue) are aligned, and then ask:

"So you are saying that there is only one way for the dice to come up so that there will be a sum of two, right?" There is always agreement.

I then arrange the dice so that the 1 one the red die is aligned with the 2 on the blue die, AND at the same time the 2 on the red die is aligned with the 1 on the red die. I rock the dice back and forth so that they can see that there are two different sets of faces that total 3.

"So, are you claiming is that there are the same number of ways for dice to come up to make a three, as there are ways to have dice come up to make a two?" and then I leave it to them again.

The students who argue that "reversing the order" doesn't make a different result, seem to me to be confusing "the number of ways the faces of dice can come up" with "the number of addend pairs that sum to a certain value". They are right about the number of addend pairs, and wrong about the number of ways the dice can arrange themselves.

We have obviously done a reasonable job of teaching the commutative property but they need a new insight to make sense of the new situations.

All but a few find the little dice demonstration gives them a way of beginning to talk about what is happening.

Bob Miller
03/04/2013 1:19pm

Another thought I just had was to use a rectangular area to model the "sample space". Create a grid that looks like an empty multiplication with the numerals 1-6 down one edge (marked Blue Die) and also across the top (marked Red Die). Have the kids help you fill in the sums. They will notice that there is a single instance of 2 as a sum, and that there are two instances of a sum of 3 (see my earlier post).

An area model is a good strategy for them to have in their back pocket
to model other probability situations, too.

03/09/2013 2:18pm

So, Drs. Terezinha Nunes and Peter Bryant are trying to understand student's construction of probability. They published a very thorough Lit Review (link below). They are now trying to understand "teaching" strategies that may cause kids to think about probability "correctly."


03/14/2013 10:17am


You said: "I let it go. I felt I did my job by helping students test their ways of thinking, not by telling them what to think." Does that mean you did not explain to the "21 camp" why they were wrong?

07/17/2013 2:56am

Inspiring, Bryan! I'm an engineering college teacher; I once jotted down that "The learning process should leave the learner with an innate, ever-improving 'filter' of quality, to be able to evaluate for oneself, when encountering some piece of information, whether it is reliable, new, significant, important, or useful... [however] what is taught [in most classrooms] is often only what is already well understood, processed and neatly packaged. This makes it harder for students to think outside the box or understand the current limits of our knowledge, let alone how they can be extended... The loss of autonomy and the presentation of processed information deprives students of the confidence to develop their own filter..." It's so clear from your blog that your class is very different, close to the ideal. I am thrilled to have chanced across your blog (highlighted it on mine just now: http://academicsfreedom.blogspot.com/2013/07/teaching-mathematics-as-way-of-thinking.html

Keep up the great work and this awesome blog!

07/30/2013 8:18pm

Help me out. How do you get 21 possible outcomes rather than 42?

07/31/2013 5:32am

Uh-oh. Maybe it's that I haven't had my coffee yet, but doesn't counting them both as different get 36 possible outcomes overall?


08/07/2013 1:54pm

I'm with you Brent. Didn't the experiment actually yield the correct result? 36 possible outcomes yields the percentages shown with 21 & vice-versa. 7 is the most common & as you noted, the basis for the game.

Bryan Meyer
08/07/2013 3:25pm

Brent and Chris...

Could you say more? I'm not sure how your concern is different from what was expressed in the post?

08/07/2013 1:58pm

Oops, I meant the "together' & 'separate' are reversed, not the 21 & 36. That is, if you count them as separate you will have more combinations (possible outcomes).

08/02/2013 8:10am

What a great activity! Games and math - always a good combination.

My feeling to the 21 problem is that we have to think in experiments, just as you inspired your class to do. Not even real ones, a thought will do.

Finally, the best challenge comes at the end. What is the best way to distribute the coins? All on the highest probability? Or rather following the shape of the density?


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