Comparing and Contrasting
I recently had an opportunity to partake in a bit of research as a natural experiment loomed during a Y11 revision lesson. I had recently been working with my Y10 class on conditional probability and counting methods and decided to spend a lesson looking at permutations and combinations in a variety of contexts.
I was aware the some of them would “know the formula” so I did my best to come up with different contextualised problems that would feel sufficiently different to others (despite being somewhat analogous). The time passed quickly as we considered anagrams, picking teams, routes on a grid, piles of fruit; at a certain point Pascal’s Triangle was discussed. Discussions about where the connections came from and how the different contexts were essentially the same thing.
In a following lesson I set some probability questions that had an element of counting required. The questions that caused particular problems were coin flip questions (in three coin tosses, probability of getting at least one T) and a lateness problem (if the probability of a student being late is 0.15, probability of at least one student being late in a class of 12?). Seeing the ‘negative space’ so to speak and using the addition to 1 property was challenging, but students discussed and could explain it well.
I then increased the challenge and, in the context of the previous lesson, decided to see how students would manage a harder coin flip question (in 10 coin tosses, what is the probability of getting at least 6 T?). It was fascinating to see what students did, and the ways in which some drew upon previous work to start counting the ways of getting 6 T, 7 T etc.
The next day during Y11 revision some questions about counting methods came up and so I showed students the Coin Toss and Lateness problems. It should probably be obvious but I was taken by how hard the students found the 10 coin tosses question and how little recourse they had to quantifying the ways in which 6 T can be achieved.
Whilst I do of course realise that this suggests that students are better at topics that they have only just covered, I felt that there was more going on. The first lesson where students had the chance to work on similar ideas in different contexts developed the plasticity of their understanding — I feel it was this that helped them apply the ideas to new questions so fluently.
There is still a lot to unpack here, but an exciting beginning.
