Probability is one of the most important mathematics concepts to understand, being applicable to everyday contexts like weather forecasts, games and lotteries, and even human behaviour. We need to be able to judge and understand likelihoods in all areas of life as well as being able to understand the differences between theoretical and experimental probability, as likely events sometimes do not occur and unlikely ones do.
Probability has its own special counting system where you can calculate the number of combinations or permutations of events. These systematic counting concepts have links to real life applications in manufacturing, computer programming, the sciences, and more.
We wondered if it was possible for our primary-aged students to be able to conduct simple probability experiments in order to find differences between theoretical and experimental probability. We were also curious to see if our students could link their primary knowledge to the more complex concepts they will be learning in later grades.
There is a vein of probability running throughout the Ontario mathematics curriculum from kindergarten to grade 8 and in the grades 11 and 12 college level courses, but it is only present in university level courses in the optional grade 12 Math of Data Management course. This absence in high school means that some students might never learn more probability than what they learn in grade 8. From relating simple probability statements to everyday events in kindergarten and grade one, students begin to experience theoretical probability in grade two. This mathematical calculation of the chances of an event occurring has a direct link to the study of combinations and permutations in later high school.
Grade 2 Probability: e.g., “I tossed 2 coins at the same time, to see how often I would get 2 heads. I found that getting a head and a tail was more likely than getting 2 heads.”
Grade 8 Probability: Toss a fair coin 10 times, record the results, and explain why you might not get the predicted result of 5 heads and 5 tails.
Grade 12 Mathematics of Data Management: e.g., “If I simulate tossing two coins 1000 times using technology, the experimental probability that I calculate for getting two tails on the two tosses is likely to be closer to the theoretical probability of ¼ than if I simulate tossing the coins only 10 times”
Conceptually, there is a lot of similarity between the study of probability in elementary school and in high school. There is a steady increase in the level of precision of the students’ ability to mathematically justify their probabilistic thinking, but the ideas are essentially the same throughout. Is an event likely or not? How likely/unlikely? What decisions might you make based on this knowledge?
The students experienced the mathematics of probability through a story about the rematch between Tortoise and Hare. We followed the associated lesson plan which starts by looking at a completely random effect. The students had a number cube race with one number cube. “Each of you pick a number. The first to the finish line wins.” This reinforced what the students already believed: that rolling a number cube is random, you cannot really predict the outcome. This was to serve as a discrepant event as the students went on to experience events that were not so random. The use of discrepant events helps to focus the students’ thinking as they try to reconcile the reasons behind them.
Working through the mathematics experiences in the story, the students find that, naturally, Tortoise wins again. The students soon discovered that she wins not by being slow and steady, but by knowing and using mathematics - the mathematics of probability. There seems to be something happening when they rolled two number cubes that was not happening when rolling one - most of the time Tortoise won in their simulations, and Hare never won. This idea that certain numbers will come up more often than others when rolling two number cubes was in direct conflict with their prior knowledge which asserted that rolling number cubes results in a random outcome.
We wondered out loud why this might be so and encouraged the students to talk to each other to see if they might find something different about the outcomes of rolling two number cubes as opposed to one. To help this along the students had been given different coloured number cubes, and it did not take long for someone to notice that there were multiple ways of rolling numbers. It was then quite easy to have the students systematically count the different combinations of number cube rolls that would result in certain number outcomes. The students now had the big idea of theoretical probability, and that you can actually calculate the likelihood of outcomes mathematically.
This served as the second discrepant event for the students in this series of math experiences. There is a predicted probability for an event, but it is still possible for something else to occur. The students tested this theory using a graph of the expected outcomes and rolling number cubes to see how different the actual outcomes were. Even though the individual outcomes were all different they could easily see that there was a trend towards certain outcomes that matched their predictions. The students really had no difficulty with the idea that even though there is an expected outcome, there is still a possibility that it might not occur. They were comfortable with the idea of uncertainty in a way that we, as adults, had not predicted they would be.
These ideas of probability, uncertainty, theoretical and experimental probability were not comfortable concepts to some of us in our school, so we trained together before teaching it to the students. We experienced the mathematics of probability in the same way as the students would. We read a good math story, rolled number cubes, reconciled discrepant events, and generally had a lot of fun while learning the mathematics we would need to teach. The mathematical surprises we experienced while learning became focal points for our teaching. We wanted our students to experience the same, wonderful math surprises we had experienced. This idea of sharing mathematical surprises led to a chain effect. We shared surprise with our students, and our students were very excited to share their surprise with their parents.
We taught the students a game that used the probability distribution of rolling two number cubes. [insert pics: Dice Game 1.png & Dice Game 2.png] They were asked to share the game, their new learning as well as to read a copy of Tortoise and Hare: the Rematch with their parents. The feedback we received from the parents was quite amazing. Parents were asked to comment on what their child had shared with them, and to tell us what they, personally, had learned from this math experience.
Parent comments on their child’s learning:
“Math can be fun. There are many ways to solve a problem.”
“Probability thinking and not eagerness will help win the race.”
“The chances of each results by rolling one dice are different than when rolling two dice. In the latter case one shouldn't judge the outcomes as equal.”
“My child shared with me that probability means how likely it is for something to happen.”
“You can use a grid different ways: Using dice to make a hypothesis about which numbers will come up. Set up counters in such a way so that you maximize your chances of winning.”
Parent comments on their own learning:
“Middle numbers have higher probability to occur than the numbers on the edges.”
“I didn't really know that this is a lesson on probability and the normal distribution curve until I noticed the same outcome pattern from the game and the book. Then you figure out why there are many more chances of adding 6 or 7 compared to 12.”
“The number 7 is the most frequently rolled because it has the most possibilities with rolling dice. We have learned probabilities . . . leading to the explanation of odds . . . she is now booking a trip to Vegas. :-)”
“I learned that these math activities are teaching more than just math!”
“Math can be easy to understand when learned through stories. There is a moral to this story.”
The parents seemed to be as engaged in the learning as their children had been. Wrapping up these mathematical surprises in a story and the learning in a set of experiences that could easily be shared led to a high level of engagement for the teachers, students and parents. We all had something in common and it was mathematics.
We decided to document and share our learning through making a song and a video. Lyrics for a song were composed by taking parent and student comments from the feedback and turning them into a story of what we had learned. Pictures from during our lessons as well as videos of students performing probability experiments were strung together to make the video. At the end of the video, our students narrated a short summary of what we now understood about probability.
Implications for Practise
Our students were able to understand and think about complex probabilistic situations. They did not have difficulty with the discrepant idea that experimental probability (actual outcomes) can be different than theoretical probability (predicted outcomes). They were even able to understand that the probability of a chain of events can be very different from individual outcomes.
“I have just flipped a coin five times in a row and I got five heads! The probability of flipping six head in a row is one in 64 - not very likely. What is the probability that my next flip will be a head?” I asked.
“Oh, come on!” they replied, “One coin? One flip? One half. You can’t trick us.”
We need to engage our students in more complex mathematics at their level rather than simpler mathematics if we want them to learn more. The idea that we need to ‘dumb down’ the math we teach our students, or to lead them through in a series of small steps may not yield the same results that we were able to get by teaching through the big ideas of discrepant probabilistic events.
As teachers, we were surprised by just how much complex mathematics our students could understand. Our students were also surprised that they could do so much, especially our students who had not experienced as much success before. Every one of the students were able to understand and do mathematics at a much more complex level than they had before. The parents were surprised not only at their children’s mathematics learning, but several were surprised by their own learning as well.
The mathematics that we asked the students to experience is not new, nor is it just an acceleration by taking curriculum expectations from later grades. The approach we took was to deliberately expose our students to grade level concepts in a deeper, more complex manner that also revealed to them the scope of what they will be learning as they get to higher grades. In essence, the big idea of experimental versus theoretical probability was used as a context to learn from, and as a scaffold for their later learning. When they get to later grades, these students will already have experienced the big idea of what they now need to learn. They have experienced the context, and they have experienced it in a fun and surprising way that is memorable.