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 "Who ever draws names from a       bag? Can't            probability be       made more          interesting?" About this site Planetqhe.com is the latest instalment of a personal journey, a response to these critical incidents from my teaching: Critical Incident #1 My Calculus student Hany 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 was completing 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 little interest. I thought "Who ever draws names from a bag? 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 these thoughts. I formed the research question "How can I embed Probability in a more meaningful, relevant and interesting context?" Punch on the nose..... 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 Genetic Counsellor 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: When pupils ask 'Why are we doing this?' they are rarely satisfied by a reply of the form 'It is important in the steel industry' In fact, they are usually not asking a question at all, but rather making a plea for help, because they have lost contact with the content of the class. It is not 'relevance' in the utilitarian sense that is sought, but rather a statement that they are no longer coping. I claim that most attempts to 'make mathematics relevant to pupils' are misconceived and doomed to failure, because relevance is not a property of mathematics, nor of its application to a particular physical context. Relevance is a property of an appropriate correspondence between qualities of some mathematical topic and qualities of the perceiver. Relevance is a relative notion: it describes a 'ratio' between aspects of the content and the pupil..... I suggest that if pupils become involved in some topic or question, then they see that what they are doing is relevant to them at the time, and conversely, if something seems relevant, then the way is open for involvement. I distilled Mason's words into this bit of 'logic': "I eventually hit on some writing by John Mason (1988:202) that really jumped out and punched me on the nose" (Implication One)   If RELEVANCE then INVOLVEMENT (Implication Two)   If INVOLVEMENT then RELEVANCE 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 Genetic Counsellor 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. Be a Genetic Counsellor - lessons learnt. "..the child will have a 25% chance, but they are having two childs now, so they are taking twice the risk.." During the Be a Genetic Counsellor 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. "the application was misguided not because of the concerns of Mason, but because it had grasped the 'wrong end of the stick" The Problem with 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 unpredictable and irreversible. Moreover, mathematicians have different ways in which they view Probability; Classical, Frequentist and Subjective. But most fascinating 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: ..black came up a record twenty-six times in succession. Except for the question of the house limit, if a player had made a one-louis (\$4) bet when the run started and pyramided for precisely the length of the run on balck, he could have taken away 268 million dollars. What actually happened was a near-panicky rush to bet on red, beginning about the time black had come up a phenomenal fifteen times...players doubled and tripled their stakes (believing) that there was not a chance in a million of another repeat. In the end the unusual run enriched the Casino by some millions of francs. "There are many other biases and misconceptions in the world of probability." 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 biases 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 analogous 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" Towards a framework and closing the circle 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." 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: "How can I embed Probability in a more meaningful, relevant and interesting context?" "After presenting the problem, the teacher's role is to act as a neutral chairperson; the students take control of the discussion and experimentation." I reckoned that the  Hypothesis- Experiment- Instruction  method was one way to go - for three compelling reasons. 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 probabilistic 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: If INVOLVEMENT then RELEVANCE 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 scenario? © David Kay Harris 1999-2005