Teaching “Chance and Data” this week to a group of Year 8 boys gave me an opportunity to use a strategy that, over the years, has become a much more visible part of my teaching practice – making mistakes.
Not just any mistakes, but strategic mistakes that bring to the fore common mathematical misconceptions.
In this example, I was working with the boys on furthering their understanding of two-way tables and tree diagrams – key aspects of the Australian Mathematics Curriculum that underpin much of the later work that is done with data and statistics (tables, more so than tree diagrams!)
When planning the class, I was aware that a number of the boys had ‘seen’ these concepts before so I wanted to drill down a bit deeper into their understanding and challenge their ability to think through a process as opposed to jumping straight into an answer.
To set up the class, I introduced a bit of under-appreciated Australian history, the two-up ‘kip’ and pre-decimal pennies. Following a brief discussion about the history of gambling in the Defense Forces and who George VI was (!), we played – without wagering – a practice round of the game.
Tossing two coins is a staple of introductory probability, bringing the historical ‘hook’ was intended to engage a group of boys who were quickly coming to the end of their school year. It certainly appeared to work, with each student actively involved with proposing the different probabilities of the outcomes. Ideas were flowing thick and fast.
When I asked the boys to talk me through creating a two way table of the possible outcomes from one spin of the coins, I added in my visible mistake.
As you can see, the sample space of outcomes shows that only a tail/head combination is possible.
The response I wanted to hear was not a shouted out, “Hey, that’s wrong!” – even though it is. With Mathematical skills that are, at first blush, seemingly simple, I wanted to demonstrate the importance of understanding the process completely before rushing ahead.
It started with a few blank stares. Looks that said, ‘Huh? Wait a minute…’.
Then one of the hither-to quiet boys stuck his hand up and said, ‘I don’t think you should have the heads together and the tails together.’ Bingo!
With this response I was able to use some higher-ordering questioning skills to elicit to all of the students the importance of having all possible categories on each edge of the two way table.
As a teacher, it is too easy to only focus on teaching students what the ‘right’ answer looks like. My experience has been, however, that students are too often unable to recognize when they have made an error and hence cannot go back and fix it. For all the times that we tell students who finish an assessment early to ‘go back and check their work’, have we actually shown them what errors look like so that they can identify them independently?
Additionally, role-playing fallibility as supposed ‘expert’ is an important social learning opportunity for boys and girls to learn. When identifying my error, I did not seek to avoid owning it. I did not shift blame or propose an excuse. I acknowledged the student who politely identified my mistake, sought his thinking on how I could correct the mistake and then made the change with thanks. The more that our young men and women can see this type of behavior, the better!