The perils of playing cards and probability. What do you assume your students know?

One of the topics taught in the first semester of my first semester teaching was probability. Flipping coins and rolling dice both serve to bring the kids to understand how it is used in games, but the first thing a couple teachers told me to do once they got the basics was to go to playing cards. This seemed like a natural fit to get the students excited – I figured they had seen people playing cards on the street as I had seen countless times wandering around the city. There are also so many opportunities to talk about intersection and union of sets. How many cards are hearts or face cards? How many are hearts and face cards? Sounded like a good idea to me.

When I actually did this with my class the first time, there were a couple really big issues that came up. Being a new teacher, I wasn’t as strong in terms of preventing students from calling out answers. When I did write up some fairly simple questions on the board (such as find P(red card) if a single card is selected) the enthusiasm for three or four students in answering these led me to believe that this small sample was representative of the class. If these four knew it (or so assumed my naive first year teacher brain), the rest probably knew it, but just didn’t feel comfortable answering. This was a ridiculously inaccurate assumption. In fact, I think it’s a painfully clear example of self-selection bias that all teachers should consider when asking any question of an entire class. Who is going to raise their hand for the purposes of establishing that he or she does not know what I am talking about?

Another issue appeared when I started walking around the room during independent work. I saw that the students were struggling both with the idea of probability AND with the details of the different types of cards. It was hard separating the two bodies of knowledge because I had framed the topic only in terms of these concepts. Students that didn’t understand what the various cards were couldn’t answer the questions because they couldn’t figure out which cards were desired outcomes. Students that didn’t get probability in general didn’t understand how the sample space and the desired outcomes fit together to calculate theoretical probability. Some didn’t understand either idea.

After the class, I talked to a few teachers about it. One said a phrase that makes my blood boil every time I hear it: “Come on – they really should know _______”. In this case, the phrase in the blank was “the types of playing cards.” The assertion that there is something wrong with a student because he or she doesn’t know an arbitrary fact is not an argument we should be making. The biggest reason it is a problem is this: if your lesson predicates itself on students knowing a fact, and you haven’t made any effort to establish as part of your lesson whether or not students actually know that fact, your lesson is going to backfire. Hard. It will be like pulling your own teeth while simultaneously telling your students “look, you can do it too!”

I understood more about this in talking to my mentor teacher. He pointed out that using playing cards is one of the worst ways teachers could teach probability because of the cultural bias inherent in assuming students have the required background knowledge. Reasons why:

  • Alright kids, we’re going to do some probability, but make sure you know what these words mean first, because I’m going to be using them all with the assumption you do: suit, hearts, diamonds, clubs, spades, face card, king, queen, jack, ace, joker. Don’t forget that there are red cards and black cards.
  • Wait, English isn’t your first language? OK, so spend your time learning these words in addition to the math content terms I really want you to learn: probability, sample space, and outcome. Uh…I guess that learning this esoteric set of words will be good for you because it will help you understand spoken English better. The more words you know, the better your English, right?
  • What’s that? How can you not understand that something can be a face card and a club? Face cards are jacks, queens, kings, and aces – get it? And there are four different suits, so there have to be four face cards that are also clubs – get it?. Well, yes, spades are also black, but clubs are black and have the little clover shape. Yes, the symbol tells you the suit. No, the card doesn’t actually say “spades” or “hearts”. But it’s easy because the heart is for hearts, the diamond is for diamond, and well, you might just have to remember the others. Oh wait, the spade is shaped like a shovel – did you know shovels are sometimes called spades? That will help you remember it. Get it? [By now, the student is nodding to get you out of his face.]
  • So now that we’ve covered all the vocabulary, what is the probability of randomly picking a card that has a value of 10 or greater? Oh, you don’t know about the value of cards? Sure, well that’s just fine. Obviously the jack is above the 10 because it has a guy on the front. It has the lowest value of the face cards, because the queen and king are higher. The king is of higher value than the queen because of the patriarchal culture that has dominated the globe for, well, forever. And then there’s the ace. Sometimes the ace is the highest card. Other times it has the lowest value. That’s life. Who has an answer?

How much math content has actually been explored during this entire (imagined) dialogue? Furthermore, if we assume that playing cards is an engaging and authentic application of probability, shouldn’t understanding the math content be made easier by the presence of all of this extra knowledge? Think about the reverse situation – should a student that knows her probability, but does not know the details of the card system, get a 50% on a quiz of this topic in a math class?

I don’t know about you, but I didn’t actually play cards that much as a kid. It’s a dangerous assumption to make that all kids have. If you don’t know if your students have this knowledge or not, and don’t want to guess from looking at them (which is always good policy not to do), and don’t want to spend class time reviewing, it probably isn’t a good idea.

One of the other teachers with whom I discussed this issue gave out a reference sheet with all of the vocabulary, pictures, and cards in order of value, and let them use it for quizzes and tests that included this topic. I think that’s fine. An even better solution though? Find a topic that doesn’t require so much background knowledge. Flip a coin and roll a 20 sided die. Put numbers on index cards, throw them in a bag, and ask for probabilities of drawing a card that is even or prime. At least in that case, students need to use mathematical knowledge to classify the outcomes. That’s what you want to assess anyway, right?

Making connections to background knowledge is one of the most powerful ways to help students learn. Making assumptions about what background knowledge students have is an easy way to make a lesson a dud. Assess, don’t assume.

Testing expected values using Geogebra

I was intrigued last night looking at Dan Meyer’s blog post about the power of video to clearly define a problem in a way that a static image does not. I loved the simple idea that his video provoked in me – when does one switch from betting on blue vs. purple? This gets at the idea of expected value in a really nice and elegant way. When the discussion turned to interactivity, Geogebra was the clear choice.

I created this simple sketch (downloadable here)as a demonstration that this could easily be turned into an interactive task with some cool opportunities for collecting data from classes. I found myself explaining the task in a slightly different way to the first couple students I showed this to, so I decided to just show Dan’s video to everyone and take my own variable out of the experiment. After doing this with the Algebra 2 (10th grade) group, I did it again later with Geometry (9th) and a Calculus student that happened to be around before lunch.

The results were staggering.

Each colored point represents a single student’s choice for when they would no longer choose blue. Why they chose these was initially beyond me. The general ability level of these groups is pretty strong. After a while of thinking and chatting with students, I realized the following:

  • Since the math level of the groups were fairly strong, there had to be something about the way the question was posed that was throwing them off. I got it, but something was off for them.
  • The questions the students were asking were all about winning or losing. For example, if they chose purple, but the spinner landed on blue, what would happen? The assumption they had in their heads was that they would either get $200 or nothing. Of course they would choose to wait until there was a better than 50:50 chance before switching to purple. The part about maximizing the winnings wasn’t what they understood from the task.
  • When I modified the language in the sketch to say when do you ‘choose’ purple instead of ‘bet’ on the $200  between the Algebra 2 group and the Geometry group, there wasn’t a significant change in the results. They still tended to choose percentages that were close to the 50:50 range.

Dan made this suggestion:

I made an updated sketch that allowed students to do just that, available here in my Geogebra repository. It lets the user choose the moment for switching, simulates 500 spins, and shows the amount earned if the person stuck to either color. I tried it out on an unsuspecting student that stayed after school for some help, one of the ones that had done the task earlier.

Over the course of working with the sketch, the thing he started looking for was not when the best point to switch was, but when the switch point resulted in no difference in the amount of money earned in the long run by spinning 500 times. This, after all, was why when both winning amounts were $100, there was no difference in choosing blue or purple. This is the idea of expected value – when are the two expected values equal? When posed this way, the student was quickly able to make a fairly good guess, even when I changed the amount of the winnings for each color using the sketch.

I’m thinking of doing this again as a quick quiz with colleagues tomorrow to see what the difference is between adults and the students given the same choice. The thing is, probably because I am a math teacher, I knew exactly what Dan was getting at when I watched the video myself – this is why I was so jazzed by the problem. I saw this as an expected value problem though.

The students had no such biases – in fact, they had more realistic ones that reflect their life experiences. This is the challenge we all face designing learning activities for the classroom. We can try our best to come up with engaging, interesting activities (and engagement was not the issue – they were into the idea) but we never know exactly how they will respond. That’s part of the excitement of the job, no?