SL Probability of events
Possibly the most relevant, useful and appropriate part of any mathematics course in schools! Everyone has an intuitive understanding of probability at its most simple level, but it is amazing how short lived that is. Many intelligent, able and influential members of our society have fundamental flaws in their understanding of this topic. Get ready for thought provoking mathematics!
Please find below some slides to help present and teach these ideas to students. You can click on each image to make it fill the screen and then click through them. you might also consider using presentation mode for using these slides.
The following is a list of teaching ideas that relate to the teaching of this topic. Some will be links to activity pages and some sill just be ideas. Teachers will be the first visitors to these pages, but it is anticipated that these will be given student access and can be set as assignments for students.
'Use what you already to know to see just how and why probability tree diagrams work' Probability is a topic that moves very quickly from that which is easily understood and intuitive to something seemingly...
'Fairground games are not fair!!! - Play with experimental probabilities 'Design a fairground/carnival game so that it is possible to win but the odds are heavily stacked in your favour. People have got...
OK, so we need to get past the sinister angle of the play on Russian Roulette and just take this as a bit of fun that will help us to think about and explore the different conditions that can affect probability....
'Switch or stick? What would you do? Does it make a difference? Play with this classic probability problem'This is a fantastic problem arising from a real game show 'Lets Make a Deal' whose host was named...
'A sentence as an attention grabbing tag lone for the activity, in quotes, centered, size 16 and Navy blue'The False Positive Paradox is a good, if not somewhat alarming example of counterintuitive Probability....
You are all going in to business against each other selling Christmas trees for the month of December! You'll need to apply your knowledge of probability, formulae, percentages, general arithmetic and...
This section will offer teachers advice about the unit planning element of teaching this sub topic. This will cover references to ToK and the ATTL sections
Theory of Knowledge
As with so many areas, there is so much rich potential here for addressing ToK. this topic has possibly even more than others. Perhaps in time, I will add more to this section as it is so charged, but for now there are a couple of key areas that I recommend focussing on...
Counter Intuitive Results
Considering the above activities, Monty Hall and the False positive Paradox (There are of course many more) - These provide excellent examples of how problems can be seriously counter intuitive. As such the present excellent examples of where the limits of Intuition as a way of knowing are. In turn this gives the opportunity to explore the extent to which we do rely on intuition against the amount we should! How do we know the limits of our intuition. I have written this post about 'The Intuition Line' in which I suggest that, with probability, point at which intuition stops serving us well is abrupt and brutal! In the False Positives activity, there are some links to articles that demonstrate that this can be as true of doctors and lawyers as well! This is a perfect opportunity to draw this out.
Practice and Theory
We are all know that this plays with our minds. How often do we need to repeat a strategy before the Practical outcome, begins to look like the theory. The endless question. When do we use probability to hep us make decisions. This too is a lovely ToK moment. We can calculate with certainty what the theoretical probability of a given event happening is. We can collect evidence to conclude relative frequency as well. That does not mean it is going gto happen. Og 5 possible events, one maybe the ore likely than all the others, but all the others put together are more likely than this one. Much, much fun to be had and the activities suggested above can bring all of this out nicely ring the ToK bell loud and clear.
Teaching and Learning
This section is intended to help teachers and students understand when opportunities arise during this unit to develop the skills, attributes and behaviours that are set out in the IB core.
The topic of probability lends itself very well to experiments. The activities above, eg Monty Hall, all involve a degree of experimenting. This provides plenty of opportunity for inquiry and collaboration. By collaboration, we don't mean just working in groups - when these activities go well, the groups that are working with them will get in to a debate about the issues at hand and end up reasoning with each other and having the kind of discussions that help people clarify their thoughts. The debate where students try to convince each other - and themselves that sticking or switching is better can be one of the more entertaining and fruitful conversations of the year! The activities all demand a level of critical thought as well. For example, the probability trees activity invites students to make critical links between representations which can help them to move from ones that appear to make sense intuitively to ones that don't. The context of the False positives activity allows us to consider how important it is that counter intuitive probability is thoroughly understood in global contexts. The activity points to a number of significant contexts where this misunderstanding has had serious consequences in the fields of medicine and law.
This section will offer relevant links that teachers may be interested in. In each case these links will have some commentary from the site authors to suggest how it might be of use.
The Problem with Monty Hall - A post about the difference between learning about the problem and experiencing the problem.
'The Intuition Line' in which I suggest that, with probability, point at which intuition stops serving us well is abrupt and brutal!