Selçuk J. Appl. Math. Selçuk Journal of Special Issue. pp. 73-82, 2011 Applied Mathematics
Risk in Science and Society: Towards New Pedagogies of Probability Bayes Rules
Ramesh Kapadia
Institute of Education, University of London, UK e-mail: ram esh.kapadia@ virgin.net
Abstract. The core ideas of probability are taught in schools and universities in a rather abstract sense, particularly since probability is usually taught as a branch of mathematics and rarely linked to realistic contexts. This paper will discuss these ideas and ways to develop new pedagogies of probabilit
Key words: Risk in science and society, new pedagogies in probability, prob-abilistic thinking.
2000 Mathematics Subject Classification: 62C10, 47N30. 1. Introduction
The project was Promoting [T]eachers’ [U]nderstanding of [R]isk in [S]ocio-Scientific Issues, directed by Professor Pratt and Dr Levinson, with Dr Kent as the Research Officer. The work was undertaken with mathematics and science teachers, fulfilling an ambition to promote cross-curricular approaches, and is reproduced in sections 7 and 8 below. This explains the main title of this paper, Risk in science and society: towards new pedagogies of probability. The sub-title about Bayes is a reference to subjective probability, which is too often ignored in school and college courses, yet is often used in daily life, including by probability and statistics teachers, lecturers and professors.
1.1. Some Examples
We start with an example (1a,1b), devised by Huerta (2009) as part of a series of questions on conditional probability for use with both school students and student teachers. It is typical of the sort of artificial question which may be set in any class. Huerta has done some ground-breaking research to classify all such problems according to level, context and complexity, which are characterised by the symbols in brackets The question is phrased in two ways — one as a question about percentages and the other about probabilities, yet the underlying question is the same. The query remains, subject to further research, as to whether there is a difference between how students tackle such questions and whether the use
of the word ‘probability’ makes a difference. One key aspect is that it is hard to predict an intuitive estimate of the answer such as whether there are more boys than girls, since the question involves manipulation of fractions and inverse operations.
Example 1a ( L3, C0, T2, algebraic setting, social statistics)
From the female students at a department, 37.5% wear glasses. From the male students, 28.6% wear glasses. Among those who do not wear glasses, 50% are male. Among the students of the department, what percentage is female? Example 1b
The chance of a female student wearing glasses in a department is 0.375, while the chance of a male wearing glasses is 0.286. Of those who do not wear glasses, the chance of being male is 0.5. What is the chance of a student being female in the department?
In a forthcoming chapter on Modelling in Probability and Statistics (Borovcnik and Kapadia, 2011), two examples are discussed as noted below. One is an example (Example 2) from a question which appeared in an examination in Germany and led to a protracted and heated debate between academics as to its legitimacy and implicit assumptions of independence, whilst the other is innovative (Example 3) from a realistic situation requiring careful analysis, modelling and use of a spreadsheet.
Example 2 (Nowitzki task). The German professional basketball player Dirk Nowitzki plays in the American professional league NBC. In the season 2006-07 he achieves a success rate of 90.4% in free throws. (For the original task, which was administered in 2008, see Schulministerium NRW, n.d.)
Probabilistic part. Calculate the probability that he a. scores exactly 8 points with 10 trials;
b. scores at the most 8 points with 10 trials;
c. is successful in free throws at the most 4 times in a series. (translated from German)
Example 3. Donations of blood have to be examined as to whether they are suitable for further processing or not. This is done in a special laboratory after the simple determination of the blood groups. Each donation is judged — inde-pendently of each other — ‘contaminated’ with a probability p = 0.1 and suitable with the complementary probability of q = 0.9. Determine the distribution of the number of non-suitable donations if 3 are drawn at random.
3 units of different donations are taken and mixed. If one of those mixed was already contaminated then the others will be contaminated and become useless. One unit has a value of € 50. Determine the loss for the various numbers of non-suitable units among those which are drawn and mixed.
While the sporting question features routinely in courses on probability and should continue to be used there is also a wider question, which is specially relevant in the school situation. To what extent do such examples prepare
students to use probability in everyday situations? This is a complementary goal of teaching mathematics with regards to the vast majority of students who complete their mathematics education at the end of compulsory school. An example of such a real-world situation featured in an advertisement (Example 4, in common with many others) as reproduced below, relating to the World Football Cup in 2010, when Rooney was seen as one of the star players, often scoring goals in the England side.
Example 4. World Cup FREE Gear4 Speaker Worth £39.99 If Rooney Scores A Hat-Trick!
Spend over £20 at iWorld this week and if Rooney scores a hat-trick in the England vs. Algeria game on Friday we’ll send you the Gear4 HouseParty 60i Speaker System worth £39.99 for FREE! There’s no catch - spend over £20, use offer code RHTEG at the basket before midnight Thursday and if Rooney scores a hat-trick you get a FREE speaker!
The advertisement appeared on June 14th before the World Cup football game in South Africa on 18 June. What are the risks to the company iWorld and to the customer? Ostensibly, there is no risk to the customer and only the company will lose in the admittedly unlikely event of a hat-trick in a supposedly easy fixture. Of course the reality is more subtle and is an interesting current example to analyse and discuss.
The mathematical elements would include estimates of various probabilities: Rooney actually getting a hat-trick —based on various elements of information (has he ever scored any before, has he suffered any injuries lately, is he definitely going to be picked, etc); the chances of Algeria having a weak defence (again based on prior knowledge); another more subjective element is the degree to which the customer wants to believe this will happen; the gain to the company is extra purchases made; the advertisement also builds on underlying interest amongst customers; it may lead to discussion of the offer with others, a further gain to the company in terms of marketing. Of course this is really no different to any other form of betting; the key distinction is that some may not see this as betting (as no actual loss is made)
The question is whether students can use ideas learnt at school in probability to understand the basis of the advertisement. Indeed firms which used such advertisements often took out an insurance against the risk of the hat-trick actually occurring (which, in practice, did not happen)
2. Causal to Probabilistic Thinking
The Nobel prize-winning work of Tversky and Kahneman, Judgment under Un-certainty (1980s) and our subsequent research has confirmed some underlying issues in moving from a causal to an uncertain domain of thinking. People’s intuitions are very weak even in simple and basic situations. Even with correct information, people make relatively simple errors and mistakes, when judged against the accepted mathematical framework. There are a number of famous examples such as the accuracy of identification of a taxi-cab and the gender of a
bank clerk. In the first, people ignore base rates whilst in the second a conjunc-tion of two events is often seen as more likely than either single event. There are long and detailed analyses of both problems over the years and these are discussed elsewhere. It is crucial for education to find a way of helping students to overcome such difficulties not only in artificial problems in the classroom but also in daily life.
Recent work to address such difficulties has been undertaken by Gigerenzer. The term ‘satisficing’ has been coined to denote the conjunction of the ideas of satisfy and suffice, important ideas in areas of extreme uncertainty and low probabilities, such as health issues. Sometimes, medical experts have insufficient experience to make correct judgements. It reminds one of the old joke made by a doctor about an operation with a failure rate of 90%, that since the last nine people have already died, the next operation will be safe.
3. Science and Uncertainty
Issues of chance have always been part of science even though the goal of science is to find causal links and certainty. Nevertheless, with the diversification of science into many individual disciplines, there was a Royal Society Conference on “Handling uncertainty in science” of the Royal Society, London in 2010. In the same year, the President of the Society presented the prestigious Reith Lectures. Lord Martin Rees, the President of the Royal Society (founded 350 years previously in 1760) spoke with the title of Scientific Horizons, with the two main themes of The Scientific Citizen, and Surviving the Century. He noted the motto of the Royal Society ’To accept nothing on authority’. Nevertheless, with the rise of the internet, authority is now more frequently questioned compared to 50 years ago. He noted that science is self-correcting, but is now more prone to corruption, and not always objective, because of the varied sources of funding and interest groups. The main defences to such issues are openness and debate. He pointed out that science is a global culture, impinging on everyday life. Moreover, the public want and expect the right answers now. Yet, science is organised doubt and many key questions have not been not solved. In particular he discussed various examples on risk. The acceptable level of risk for volcanic ash has been known scientifically for some time, yet a more realistic policy (than requiring zero volcanic ash) was only implemented after huge commercial losses were made with the Icelandic situation in 2010. Earlier in the United Kingdom, with BSE (mad cow disease), the banning of beef on the bone proved wrong in retrospect. There are certainly very high stakes in the debate on climate change: it is known scientifically that CO2 is very high, which will lead to long-term warming, but the timescale is not known. There is often a mismatch between public perception and actual risk. There can be a denial relating to low probability and high impact (positive or negative) events, such as the eventual financial crisis in 2008, with the avowed subsequent aim to avoid such catastrophes in the future: the adverse negative consequences of an approach which is too stringent, and leading to stunted or even negative growth, are too easily forgotten. Lord Rees noted that whilst science helps
us deal with uncertainties, the subsequnet decisions still involve emotions and various interested parties, which are not only scientific questions. A debate in England initiated by a prominent scientist about the higher dangers of horse-riding to ecsatsy led to his sacking from an influential official committee on the dangers of various drugs.
4. Theoretical Development: Rational and Behavioural Views Uncertainty is pervasive, yet choices have to be made. Given that decisions made by individuals under uncertainty affect economic outcomes, especially when those decisions interact, a key challenge for educators is to understand how decisions are made. There are two broad views on decision making under uncertainty — the standard, ‘rational’ view of ‘homo economicus’, and the ‘be-havioural’ view informed by the psychology literature. In practice, the applica-bility of each approach depends on the context. For example, rationality may be a reasonable approximation when decisions are carefully considered (perhaps because the returns to a ‘good’ decision are high).
Distinguishing between these views may be important if their predictive power differs. But, rather more fundamentally, given that they have contrasting im-plications for how decisions are made and information is processed, they may yield different conclusions on both the desirability of policy intervention and the form it should take.
‘Rational’
• The benchmark approach to choice under uncertainty is the expected utility hypothesis, under which expected utility is maximised by rational players who follow the laws of probability.
• Expected utility theory assumes that the uncertainty over outcomes is quantifiable, in the sense that the decision-maker can assign (subjective or objective) probabilities to possible outcomes. However, Knight distinguished between risk, which can be quantified by assignment of mathematical proba-bilities, and uncertainty, which cannot. The pre-eminent approach to decision making under “Knightian” uncertainty is robust control, which aims to max-imise utility under the worst conceivable outcome (hence ‘maximin’). This type of decision rule can explain changes in behaviour seen in crises.
‘Behavioural’
• Behavioural economists and psychologists contend that people often make decisions using simplistic or ‘fast and frugal’ heuristics because of limited time, information, and cognitive capacity. This ‘behavioural’ view highlights psychological traits that violate the assumptions underlying the ‘rational’ view. • This decision-making toolkit is the product of evolution, and as such may be thought to trade off optimally competing demands for the brain’s at-tention. But the modern world differs markedly from the environment in which humans evolved as hunter-gatherers.
• In practice, there is indeed strong experimental evidence that people are subject to biases when collecting and processing information under uncer-tainty, and that such processing only occurs intermittently. People are also
fre-quently incoherent in assessing, assigning and processing probabilities, even in highly artificial settings where information is complete, and find it particularly difficult to assess probabilities which are very low or very high, partly because there is less scope for learning in these cases. And in the real world when infor-mation is imperfect, such as when deciding whether to have eye surgery, people often use rough heuristics, or go on first impressions, appearances, gut instinct or intuition. This leads to four key principles.
1. Perceptions of risk are influenced by examples that can be easily brought to mind (the availability heuristic). For example: after an earthquake, demand for earthquake insurance increases at first and then declines, while the probability of the next large earthquake may grow over time; so-called ‘Depres-sion babies’ tend to take less financial risk. Too much may be spent on counter-terrorism, and too little on tsunami early warning systems, climate change and averting financial crises. (This is the ‘Today programme syndrome’).
2. People’s decisions under uncertainty are very sensitive to the way questions are presented, with choices often being influenced by anchoring or the status quo. People understand probabilities better if they are expressed in terms of natural frequencies (1 in 10,000) rather than percentages (0.001%). Status quo bias is reflected in empirical evidence that the degree of participation in pension schemes is sensitive to whether enrolment is automatic or not.
3. People tend to follow the actions of others when making decisions under uncertainty. It is known that in football goalkeepers facing penalties dive (rather than staying in the middle of the goal) more than the direction of penalties suggests that they should. These types of behaviour reflect the fact that people are unwilling to stand out from the crowd. Key players are particularly important in this dynamic. Such informational cascades can explain panics and herding behaviour. They can explain how some restaurants are successful when similar ones are not; and how some journal articles (like this one) get published, while others do not.
4. In spite of uncertainty, people have excessive faith in their own judge-ments and are subject to confirmation bias and wishful thinking. Experijudge-ments also show that people selectively process information in a way which confirms their prior beliefs..
We end with a collection of ten statements from an educational book on research in probability education published exactly twenty years ago, Chance Encounters (Kapadia and Borovcnik, 1991). These are presented as issues which should be addressed by any curriculum in probability if, as well as teaching basic ideas, an aim is to enable adults to use probability sensibly in daily life.
1. People use personal experience in assessing chance haphazardly. 2. People process information in a rather incomplete way.
3. People process information in a way biased by memorable events. 4. People find it hard to assess probabilities which are very low or very high.
5. People do not assign values of 0 for impossibility and 1 for certainty. 6. People equate certainty and impossibility with physical events. 7. People equate 50-50 chances with coin tossing.
8. People assign equal likelihood in unknown situations.
9. People are incoherent in assigning and in processing probabilities. 10. People are supra-additive.
5. Education: Schools and Colleges
The use of language in this context is vital and it would be particularly helpful to develop definitions of terms in common use. Some language is carefully defined and generally accepted in probability but some terms are undefined and used in a rather loose sense. For example, it is commonly accepted that probability is measured on a scale from 0 to 1; it is less clear for many people (despite commonly agreed definitions in probability) that 0 represents impossibility and 1 certainty, partly because of the difficulties of understanding very low or very high probabilities — indeed this also occurs with mathematical terms involving very high or very small numbers. There are also ideas such as uncertainty, likely, and risk, which are used in common language and yet not well defined in probability. There are similar problems in all languages and so collaborative research is needed to investigate such words. In probability, language links precise words in probability with everyday life; the probability of getting a six on a die is 1/6 is only based on a model not the reality, unlike in measuring length of a table where no model is needed, only a common unit of measurement. 6. Deborah’s Dilemma
As noted above, the author has been involved with others on a project, funded by the Wellcome Trust to explore and develop new pedagogies of teaching risk. In connection with this, a problem was developed using a software platform for teachers to explore what became known as Deborah’s Dillemma. The platform allowed exploration of the simulating the operation and showing the results, as well as (fictional) reports from various expert doctors, which were inconsistent (as happens in normal life). Results of this work are described in papers by Pratt et al (2009) and Kapadia et al (2010), as noted below. Here a few key points are reproduced below.
Lindsey and Aden (not real names), who were two teachers, of science and mathematics respectively began with a focus on the operation and by exploring the models with probabilities. They noted that failures were relatively rare and complications even less common. The operation seemed quite safe and they adopted a pro-operation stance. When their attention changed to the lifestyle modelling, Lindsey and Aden began to see making lifestyle changes as less threatening and less risky, more at the instigation of Lindsey, the science teacher who also happened to be female. The shifting of position seemed to relate to where their attention was directed, often by affective reactions to the context of the scenario at any particular point in time. Discussion between the two people was a crucial component of the approach taken. Four aspects emerged, as well as four pedagogic components.
a) Co-ordinating impact and likelihood seemed a considerable challenge The teachers often flipped from consideration of impact to consideration of likelihood.
• Mechanisms for trading off one with the other were needed; • For this reason, a mapping tool was added.
b) The reliability of data was a regular concern
The teachers often referred to whether the data could be taken as reliable. • The source of data was important to them.
• The amount of data was occasionally salient to them.
• A common strategy in the face of uncertainty was to take the view of whichever source was regarded as the most authoritative.
c) Understanding the problem context
The teachers drew extensively on their knowledge about the problem context. Knowledge of the context enabled the teachers to
• Draw inferences;
• Empathise with Deborah.
d) Knowledge of the context is not however unproblematic as teachers might: • Be unable to distance themselves to make analytical judgements; • Appreciate the rules of the game that might suggest when it is in-appropriate to make use of contextual knowledge and how to handle personal experience.
Four Pedagogic Theory Components
From these ideas, four pedagogic components emerged.
1. Risk is a multi-disciplinary topic that can be addressed within conventional school structures.
2. Risk is multi-dimensional, embracing at least the elements of likelihood, impact and value-laden ethical considerations.
3. A modelling approach that encourages making explicit the dimensions of specific contextualised socio-scientific dilemmas in executable models supports recognition of and discussion about those dimensions, as well as awareness of the consequences of their characterisation of the dilemma.
4. Expressive tools can be designed that support the co-ordination of the di-mensions of risk.
7. Research Development in Probability Education
Further research envisages collaborative work across several countries on the teaching of risk which is becoming increasingly important in schools. Indeed the teaching of risk features in the current curriculum requirements in several subjects in school such as mathematics, science and citizenship. Yet materials for teaching risk remain limited in availability and scope. Links should be made to the standard philosophical approaches, including the theoretical, frequentist and Bayesian positions on probability.
Previous collaboration has led to the IEJME (October 2009) publication on the teaching of probability across the world. This special issue included key, influential research papers from across the world, representing current thinking in many countries. The challenge is to collate and develop these ideas into material which can influence teachers in schools, especially in the teaching of risk. The special issue included a number of papers which are relevant for this discussion. Ideas from two of the papers are presented here, as points on which to build.
1 Parallel discussion of classical and bayesian ways as an introduction to statistical inference
Ödön Vancsó
Do students understand probability and statistical methods better by focussing on subjective and objective interpretations of probability throughout the course? Do they understand classical inferential statistics better if they study Bayesian ways, too? There is evidence that they understand the concepts better in this way. The results also support the thesis that students’ views and beliefs on mathematics decisively influence the work in their later profession.
2 Hands-on activities for fourth graders: a tool box for decision — mak-ing and reckonmak-ing with risk Laura Martignon & Stefan Krauss
The intention of this work is to exhibit how children can be provided with a kit of elementary tools for judgment under uncertainty, for good decision making and for reckoning with risk. Children, it is claimed, can acquire this tool kit through a mosaic of simple, play-based activities which are devised to make them aware of the characteristics of uncertainty. This research is guided and inspired by empirical results on human decision making in the medical and financial domain.
References
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