Piaget’s Viewpoint on the Teaching of Probability: a Breaking-off with the Traditional Notion of Chance?

situations. On the contrary, the formalistic school of mathematics uses this distinction to emphasize that the development of all branch of mathematics does not depend on a particular situation. Moreover, Probability Theory has been developed in this way by Kolmogorov (1950). Thus, the teaching of probability and statistics following Piaget’s tradition does not necessarily ICOTS-7, 2006: San Martín (Refereed) 4 mean teaching mathematical statistics or theory of probability, but requires the philosophical and mathematical notions underling probability theory (as explained by Kolmogorov, 1950; see also Gini, 1966). A consequence of statement (b) above is that Probability and Statistics are viewed as tools for Physics. As a matter of fact, the main point of Piaget’s viewpoint concerning the acquisition of the notion of chance is that it is a matter of experimenting with the physical chance. Therefore, probability and statistics are related to the physical world. Thus, probability and statistics not only deal with real data, but also become the tools to describe physical chance. CHANCE, OUR IGNORANCE ON THE CAUSAL CHAIN The relationship between probability, statistics and physics is not new. It can be traced in Huygens (1656), Arbuthnot (1710), Price and Bayes (1763), Laplace (1774) and Quetelet (1853). Nevertheless, the relationship between chance and causality seem not only to be different from that established by Cournot and Piaget, but also to be related to specific philosophical and theological aspects, which in turn were elaborated from Newton’s, Bentley’s and Boyle’s perspective; for details, see San Martín (2005). The semantic field of the notion of chance can be grasped from the arguments claimed in order to publish Bayes’ (1763) essay at the Philosophical Transaction. These arguments are contained in Price’s letter addressed to John Canton (Bayes and Price, 1763), who in turn published in the Philosophical Transaction around twelve manuscripts on experimental philosophy (i.e., experimental physics); for details, see San Martín (2005). First argument: why was a letter to Canton addressed for deciding the publication of Bayes’ essay? Experimental philosophy is closely interested in the subject developed in Bayes’ essay since both are concerned with the establishment of eventual causes from actual effects. This agreement can indeed be understood from Canton, an experimental philosopher. Second argument: why should experimental philosophy pay attention to Bayes’ essay? As pointed out above, the concern of experimental philosophy is to look for eventual causes from actual effects. The orientation of causal series is the same as time, so the problem reduces to making inferences about future events from past events: the key idea is to use available information (i.e., former instances) to deduce the likely future consequences. Moreover, the larger number of experiments we have to support a conclusion, the more reason we have to take it for granted (Bayes and Price, 1763). Nevertheless, what is relevant is to determine “in what degree repeated experiments confirm a conclusion [...] which, therefore, is necessary to be considered by any one who would give a clear account of the strength of analogical or inductive reasoning” (Bayes and Price, 1763). Bayes’ rule (i.e., the rule of inverse probability) is, therefore, considered as a tool for ensuring this type of inductive inference. The use of probability as a tool for quantifying likelihood degrees in inductive reasoning is traditional in the sense that can already be traced in Huygens’ (1673, 1690) inductive reasoning in Physics, the degrees of likelihood being previously developed in Huygens (1656); for details, see San Martín (2005). Third argument: why do causal series underly inductive inference? To evaluate the relevance of Bayes’ contribution in the context of experimental philosophy, Price considers it as the complement of De Moivre’s contribution, namely the Law of Large Numbers, the concern being to find the probability of the occurrence of an event (i.e., the probability of an effect) from a great number of trials (i.e., causes). Bayes’ contribution leads to a solution of the inverse problem (i.e., from effects to causes), namely “the number of times an unknown event has happened and failed being given, to find the chance that the probability of its happening should lie somewhere between any two named degrees of probability” (Bayes and Price, 1763). Both De Moivre’s solution and Bayes’ solution are understood assuming as a principle that a thing cannot occur without a cause which produces it. This is correctly considered as a principle because causal series are viewed as the actual government of a Deity: “to shew what reason we have for believing that there are in the constitution of things fixt laws according to which events happen, and that, therefore, the frame of the world must be the effect of the wisdom and power of an intelligent cause; and thus to confirm the argument taken from final causes for the existence of the Deity” (Bayes and Price, 1763). Bayes’ solution should be understood in this quote: “It will be easy to see that the converse problem solved in this essay is more directly applicable to this purpose; for it shew us, with distinctness and precision, in every case of any particular order or ICOTS-7, 2006: San Martín (Refereed) 5 recurrency of events, what reason there is to think that such recurrency or order is derived from stable causes or regulations in nature, and not from any irregularities of chance” (Bayes and Price, 1763). These considerations can directly be traced in De Moivre (1718) as Price himself pointed out; for details, see San Martín (2005). Moreover, De Moivre exemplifies these considerations with Arbuthnot’s (1710) contribution. Arbuthnot considers that the physical balance that is maintained between the number of men and women is a footstep “of the Divine Providence to be found in the Works of Nature.” This fact is “not the Effect of Chance but Divine Providence, working for a good End.” In other words, Chance is opposed to Divine Providence, but when considering Divine Providence as the government of a Deity through causal laws. Since these natural laws have a precise objective, in the particular case Arbuthnot is considering that “Species may never fail, nor perish,” a right moral behavior is precisely contrary to the attempt of avoiding, in the context of human relationships, that a natural law works. It should therefore be mentioned that De Moivre’s conception of the role of natural laws nearly follows that of Arbuthnot (this type of discussion was developed in puritan circles; see Ames, 1643). In the preface of Huygens’ (1692) English translation, Arbuthnot explicitly considers Huygens’ method in the context of physics, the advantage being that the computation of a likelihood degree means avoiding a computation of mechanical aspects underlying the observed event. In this context, chance is conceiving as the ignorance of an exact cause producing an even and, therefore, an explicit confession that events are due to causal laws. Moreover, he insists that this is applicable to both “the great Events of the World, as in ordinary Games,” the difference being just the complexity of computations. The opposition between chance and design (art) in the sense that all things occur due to a cause, and the idea that causal laws are the expression of a Deity (namely, a God who actually governs all physical and social events) are explicitly found in Newton’s conception on natural laws, as De Moivre himself acknowledged; for details, see San Martín (2005). Moreover, Laplace’s philosophy on chance and probability essentially develops the traditional one already summarized above; for details, see, e.g., Laplace (1774) and San Martín (2005). TEACHING PROBABILITY AND STATISTICS UNDER THE TRADITIONAL PROGRAM The traditional notion of chance is, therefore, related to that of inductive inference. As a matter of fact, all events, either social, moral or physical, occur with a cause. The problem consists precisely in detecting a particular cause among a set of possible causes which generates the event of interest. The experience of related phenomena provides data to induce such a particular cause, but the complexity of the causal chain, along with the (historical) limitation of human knowledge, lead to performance of a probabilistic induction (induction was always a probabilistic induction for Price, Bayes, De Moivre, Laplace, Quetelet, etc.). Thus, chance is an expression of our ignorance and, in the context of inductive reasoning, probability is the more sure complement of human knowledge. Therefore, the teaching of Statistics under the traditional approach leads to placing the accent on epistemological considerations in the sense that when analyzing complex systems under an inductive inferential viewpoint, it is necessary to measure possible inductive errors. This means that knowledge is not definitive and that experience must be accumulated to continually revise inductive procedures. Under this perspective, the teaching of Statistics and Probability should be developed in parallel with Philosophy and eventually with Theology. In Philosophy, the subject to be discussed is the relation between empiricism and inductive inference. Illustrations are needed to show how inductive inference can be incomplete and erroneous. In Theology, the relationships between natural theology (mainly developed from puritan theology) and the rise of experimental philosophy should be explained; see Hooykaas (2000). Consequently, Statistics and Probability need to be introduced as epistemological disciplines helping to formalize inductive inference in all discipline: Physics, Social Sciences, Biology, etc. This tradition would explain why Statistics is widely used in these disciplines; see also Gini (1966). From this traditional perspective, Mathematics is not the main related discipline since a differentiation between probability and measure of a probability is made; see, e.g., Laplace (1774): Mathematics is related to the second aspect, whereas the first one is related to a procedure of reasoning which can be applied even in ICOTS-7, 2006: San Martín (Refereed) 6 fields in which computation is eventually impossible (due to its complexity). Thus, the teaching of Statistics and Probability precede the teaching of Mathematics and could be in parallel with Physics. FINAL REMARK The teaching of Statistics and Probability involve, in a certain sense, the teaching of specific philosophical views of nature, causality and epistemology. Once a viewpoint is chosen (the ones developed in this note), Statistics becomes either a transversal discipline (because it is viewed as an epistemological branch) or a complementary one (because it is viewed as the complement of Physics and Mathematics). ACKNOWLEDGEMENTSThis research was partially supported by three FONDEDOC Teaching ProjectsDGP/075/2003, DGP/074/2004 and DGP/169/2005 from the Pontificia Universidad Católica deChile. The author acknowledges several stimulating discussions with K. Verduin (LeydenUniversity), C. Friedly (PUC), A. Cofré (PUC), R. Espoz (U. de Chile and CEMDAL) and A.Monares (CEMDAL). The author acknowledges María Gabriella Ottaviani, Carmen Batanero andthree anonymous referees by their careful reading of this manuscript. In particular, the authorwants to acknowledge one anonymous referee’s help improve the English language. REFERENCESAmes, W. (1643). The Marrow of Sacred Divinity. London.Arbuthnot, J. (1710). An argument for divine providence, taken from the constant regularityobserv’d in the births of both sexes. Philosophical Transaction, 27, 186-190.Ascher, E. (1984). The Case of Piaget’s Group INRC. J of Math. Psychology, 28, 282-316.Bart, W. M. (1971). A generalization of Piaget’s logical-mathematical model for the stage offormal operations. Journal of Mathematical Psychology, 8, 539-553.Bayes, Th. and Price, R. (1763). An essay towards solving a problem in the doctrine of chances.Philosophical Transaction, 53, 370-418.Beth, E. W. and Piaget, J. (1961). Épistémologie Mathématique et Psychologie. Essai sur lesrelations entre la logique formelle et la pensée réelle. Paris: PUF.Cournot, A. A. (1843). Exposition de la Théorie des Chances et des Probabilités. Paris: Librairiede L’Hachette.De Moivre, A. (1718). The Doctrine of Chances: or, A Method of Calculating the Probability ofEvents in Play. London.Gini, C. (1966). Statistical Methods. Roma: Biblioteca del “Metron,” Università degli Studi diRoma.Hooykaas, R. (2000). Religion and the Rise of Modern Science. Vancouver: Regent.Huygens, C. (1656). De Raciotiniis in Ludo Aleae. Huygens, C. (1673). Letter to Pierre Perrault. Huygens, C. (1690). Traité de la Lumière.Huygens, C. (1692). Of the Laws of Chance, or, A Method of Calculation of Hazards of Game Translated by J. Arbuthnot, London. Kolmogorov, A. N. (1950). Foundations of the Theory of Probability. New York: Chelsea.Laplace, P. S. (1774). Essai Philosophique sur les Probabilités.Piaget, J. (1950). Une expérience sur la psychologie du hasard chez l’enfant: Le tirage au sort des couples. Acta Psychologica, 7, 323-336.Piaget, J. and Inhelder, B. (1974). La genèse de l’idée de hasard chez l’enfant. Paris: PUF.Quetelet, A. (1853). Théorie de Probabilités. Bruxelles: Societé pour l’emancipation intellectuelle.Rabinowitz, F. M., Dunlap, W. P., Grant, M. L. and Campione, J. C. (1989). The rules used by children and adults in attempting to generate random numbers. Journal of MathematicalPsychology, 33, 227-287.San Martín, E. (2005). Todo es efecto de un diseño, no del chance. Santiago de Chile: Editorial