What are the chances?

This past week I was in a couple of different Grade 6 classes doing some probability tasks. Watching the kids tackle these led to some huge insights as to what they actually are thinking when they do probability questions. It got us as teachers thinking if students make mistakes in a probability task, is the misconception to do with the concepts of chance or the concepts of number sense.
But the only way we could get these insights was by listening to the kids' conversations!
So here is the first question we tried, and the first misconception...

To be honest, you can see this student's point! The middle bag has a circle close to the top so that's the one to choose if you want to get a circle... just make sure you pick the top shape (especially if it  feels round!) This got me thinking that the question will be better if I use the same shape with different colours (e.g. just black and white circles). Anyway we addressed this misconception by putting some shapes in a bag and shaking them up and down and asking "Do you know where the shapes are now?" It did the trick!
The more common misconception though showed that the kids were thinking additively and not proportionally.

Essentially what this student (and some others) were saying was 'Since the first bag has the least number of squares, you are less likely to pick a square... therefore you are more likely to pick a circle.' Conversely, there were other students who picked the third bag as it had more circles. The interesting thing was that when we asked the students to write the probability of getting a circle they were good at this: most were able to say 2/5, 6/15, and 10/25. So they might have understood the chance of getting a circle but they misunderstood the concept of equivalence. We challenged these ideas by representing the fractions using a virtual manipulative:

It was clear that students will still need to see concrete representations of fractions for them to move from additive thinkers to proportional reasoners.
We then stumbled across another misconception; we asked students what will happen if they flip a coin. They wrote their answers on post-it notes which allowed us to quickly display the results:

 

Here, the majority of students don't clearly understand chance. Some students thought that if you begin with the heads 'up' then it will land heads 'up'! Of course, we tested these ideas straight away.

What was also apparent was that if students only ever experience situations in which all the possible outcomes are equally likely then they might that in every situation, every possible situation is equally likely. Here is the second question we tried (in a different class):

 



 

We know that some students have the misconception that as there are three outcomes (i.e. you could pick a green, a yellow, or a blue) each one is equally likely. Again, a quick experiment usually is enough to challenge such notions. Or, in this case, say (as my colleague Chad did) "How about we play a game; every time a yellow is drawn, you give me $10 and every time a blue is drawn, I give you $10." They don't usually want to put their money where their mouths are!

Students began by representing their ideas with tiles as shown:



They could even write the probabilities correctly:

 However they had the misconception that another yellow was being added, the probability was increasing. In other words they couldn't see that the original 4/8 probability was the same as the final 5/10 probability. We asked them to think of how they could use the tiles to convince us that in fact 5/10 was larger than 4/8 (rather than us show them that in fact they are equivalent). Light bulbs began to go on when they used representations such as these:

We also showed equivalence using the virtual manipulatives above for further proof (always a good thing).

 

So for students to experience success in probability we concluded that they must have the opportunity to use represent it in a variety of ways (e.g. spinners, tiles, number lines etc.) The nlvm site has a superb applet for spinners by the way in which you can design your own spinner and have it spun for up to a thousand spins and have the results displayed in a live bar graph.. Here is one such bar graph which begs the question 'What did the spinner look like?'

But we also wondered if students would also improve their understanding of probability by playing board games, card games, and dice games like Yahtzee. Maybe this could be the perfect homework assignment!