# 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.

I have this same realization about cards myself every few years, as I regularly forget that not everyone is a card-lover like I am. I think these kinds of experiences are valuable for a few reasons.

First, this is a great opportunity for student-to-student teaching. Chances are many students understand the card system, and you can use that to help advance the collaborative culture of your classroom.

Second, it's a great opportunity to reiterate the importance of asking questions. People are afraid to admit they don't know things, but when students know

the teachercan make mistakes (like forgetting to explain what certain things mean), it can make it easier for them to speak up.I love cards now, but I actually did fail a math test (in 6th grade) because I didn't know anything about a deck of cards and so couldn't answer anything correctly despite knowing the probability content just fine.

I really struggle with how to teach probability in a way that doesn't oversimplify it. I teach AP Stats as well and my students are convinced they know probability because they've been answering questions about marbles in bags or dice for years. They assume they've got it and it takes them awhile to realize we've moved past that and they'll need to start looking at things in a deeper way.