WHY PLANETQHE?

PLANETQHE.COM is the latest instalment of a personal journey, concieved as a response to these critical incidents from my teaching:

Critical Incident #1 My Calculus student Hany Salem described my 'Find the Volume of an American football' activity during his valedictorian speech at the American International School in Egypt graduation ceremony, 1993. Sitting in the audience, I remembered the buzz of focused creativity this activity sparked in my classroom, and thought "How did this come about and how can I do this more often?"

Critical Incident #2 In Argentina, my class of 17 year olds had to complete 13 exercises on probability such as "You choose the name of a girl from a bag containing 10 girls names and 5 boys names written on pieces of paper. You throw away the paper and choose another from the bag. What is the probability that it is the name of a girl?" The students were able to answer these questions by forming simple fractions and did so with ease and little interest. I thought "Who ever draws names from a bag? Or counters or coloured balls? Can't probability be made more interesting?"

It wasn't until I arrived in Bristol in 1999 to study for an M Ed that I managed to find the time to try to do something about I improving my teaching of probability. I formed the research question

I had a gut feeling that 'real-world' applications would help answer my question - I had been aware of an interesting application called Be a GeneticCounsellor written by Usha Kotelawala. However, after spending long hours in the Graduate School of Education library, I eventually hit on some writing by John Mason (1988:202) that really jumped out and punched me on the nose:

The punch on the nose came from the first higlighted section. I distilled the second highlighed section into this bit of 'logic':

I thought about this for some time and figured that (1) I reckoned I could make a 'relevant' application of probability to Genetics which was not "doomed", and (2) Mason's implications could be explored further. I felt that the processes in the two implications would be different: the same word - relevance - was being used as a catch-all for various constructs that students would have for relevance, either as what I called Primary Relevance and Secondary Relevance for Implications One and Two respectively. I made a classroom study into this, using Be a GeneticCounsellor and wrote up my findings in a paper entitled A 'Real World' example of Probability: An Investigation into Students' Constructs of Relevance. This work was eventually published by the ECIS in the November 1999 International Schools Journal.

I'm getting there! During the Be a GeneticCounsellor study, I recorded the following discussion between three students.

S 1: Ok, if Shalini and Niraj have two children, what is the probability that at least one will have Smith's disease? [Reads Question 3]
S 2: Again, it adds up - it's 1/4 plus 1/4 - a half again - there's a half chance that one of them would have it, yep
S 1: 'Cos the father has a 50% chance so...
S 2: '...exactly, and the child will have a 25% chance, but they are having two childs now, so they are taking twice the risk, so there is a 50% chance that at least one of the kids will have Smith's disease" [Accepted by group as correct answer to Question 3]
S 2: "If Shalini and Niraj have three children, what is the probability that at least one will have Smith's disease? [Reads Question 4]
S 3: "75%"
S 2: "No, it can't be 75, because..."
S 1: "It's 25 plus 25 plus.... yeah it is"
S 2: "You add up for ever and ever, so like if it is 5 children it is 125%"
S 3 and S 1 laugh.

It struck me then that my 'real-world' approach was only part of the answer to my original question of "How can I embed Probability in a more meaningful, relevant and interesting context?" The ease with which S1, S2 and S3 were able to throw around probabilities which were at times loosely connected with the problem led me to wonder if the application was misguided not because of the concerns of Mason, but because it had grasped the 'wrong end of the stick' in some sense - asking students to 'apply' probabilistic skills which were not firmly grasped in the first place. I returned to the library to find out more about the learning of Probability.

Authors such as Fischbein , Shaugnessey, Green, Konold and Kahneman and Tversky have written about the difficulties that even the 'educated' have in coping with the probabilistic world. (See the bibliography of my M Ed dissertation below if you are interested in find out more of these authors' works) The fact is, as a focus of Mathematical study, Probability is unique in that it seeks to describe and quantify a world of random events that are unpredicable and irreversable. Moreover, mathematians have different ways in which they view Probability; Classical, Frequentist and Subjective.

But most facinating are the times when the results of probability theory run contrary to our expectations and intuition. Let me give you a quick example, quoting from Darell Huff's excellent How to Take a Chance. Huff tells the story of a run on black in a Monte Carlo casino in 1913:

This tale illustrates one of the most familiar probabilistic misconceptions - the Gambler's Fallacy or Recency Bias. It's the same fallacy that makes the Coin thrower think that Tails is more likely after you he has tossed three Heads in succession.

There are many other baises and misconceptions in the world of probability. Another is the Representative Bias which leads one to believe that, for example, the outcome BBBGGG for a family of six children is more likely than GGGGGG because it appears to represent the 'typical' member of the distribution more than GGGGGG, which seems 'unusual' and hence less probable. This is analgous to the misconception that 9, 14, 29, 32, 39, 43 is more likely to be chosen as the winning outcome for the National Lottery than 1, 2, 3, 4, 5, 6. And don't forget the conversation I showed you earlier between my students in which they showed how easy it is to miscalculate with the measure of chance.

The long and the short of it is that after reading so much about probabilistic misconceptions and biases, I turned my reading towards formulating an approach to answering my research question. The feeling I had in Critical Incident #2, that there was a whole lot more to teaching and learning probability than I felt I was giving my students was echoed in Shaugnessy's (1993) claim that "Students need to be actively involved in both statistical investigations and probability experiments. A straight lecture approach to probability and statistics is highly inappropriate for our students"

What I needed was a framework. One alternative is offered by Hatano and Inagaki (1987), who advocate a "Hypothesis-Experiment-Instruction" model for learning mathematics, described by Brown and Campione (1990:115) as:

Students are given a problem with three or four alternative answers. the alternatives include some choices based on common misconceptions or 'bugs'. Students make independent choices, and consider alternatives in whole class discussions. They may change their choices. Finally, students' choices are tested by observing an experiment: After presenting the problem, the teacher's role is to act as a neutral chairperson; the students take control of the discussion and experimentation." Hatano and Inagaki claim that "Peer interaction, or dialogical interaction in general...invites students to 'commit' themselves to some ideas by asking them to state their ideas to others, thereby placing the issue in question in their domains of interest. In addition, the social setting makes the enterprise of comprehension more meaningful." (p115)

Now the pieces of the P-L-A-Net concept started to come together. Let's recap for a second. My research question was:

Well, I reckoned that the Hypothesis-Experiment-Instruction method was one way to go - for three compelling reasons. Firstly, it seemed to offer a powerful way to confront probabilistic misconceptions head on, through probabilisitic experiments. I'd like to think that in doing this, my students will have concrete experiences to recall in later life whenever they have to make probabilistic judgements which are counter-intuitive. Secondly, as any IB Math teacher will tell you, the misconceptions and puzzles in the Planetqhe.com website can all be solved with a clear head and application of the probabilistic content of any of the Group 5 subjects. Thirdly and perhaps most exiting is the potential for linking with our friend Mason (above) and his second implication:

In other words, could I answer my original research question not by reaching for 'real-world' applications, but by engaging student interest through exposing them to counter-intuitive probability problems and challenging them with finding their solutions? Would this deliver the authentic and rich sense of involvement (and hence relevance) that I felt lacking in my Critical Incident #2 senario?

As I write, the P-L-A-Net website is being completed and my main study methodology is being planned. So watch this space for the findings!

P-L-A-NET: Closing the circle

One current P-L-A-NET project is the WEB LINKS section. I aim for this section to become an on-line, cross-curricular resource based on how probabilistic themes and methods occur in the 'real-world'. I invite teachers to submit contributions to this section of P-L-A-NET. Read also the call for papers on this site.

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© David Kay Harris 1999-2003