The Meaning and Understanding of Mathematics

Batanero, C. y Díaz, C. (2007). Meaning and understanding of mathematics. The case of probability. En J.P Van Bendegen y K. François (Eds), Philosophical Dimmensions in Mathematics Education (pp. 107-128). New York: Springer, ISBN: 978-0-387-71571-1. THE MEANING AND UNDERSTANDING OF MATHEMATICS The Case of Probability Carmen Batanero and Carmen Díaz University of Granada, Spain Abstract: We summarize a model with which to analyze the meaning of mathematical concepts, distinguishing five interrelated components. We also distinguish between the personal and the institutional meaning to differentiate between the meaning that has been proposed for a given concept in a specific institution, and the meaning given to the concept by a particular person in the institution. We use these ideas to analyze the historical emergence of probability and its different current meanings (intuitive, classical, frequentist, propensity, logical, subjective and axiomatic). We furthermore describe mathematical activity as a chain of semiotic functions and introduce the idea of semiotic conflict that can be used to give an alternative explanation to some widespread probabilistic misconceptions. Key words: Probability, history of probability, proof, semiotics, misconceptions 1. INTRODUCTION The teaching of probability has been included for many years in the mathematics curriculum for secondary school. There is, however, a recent emphasis on the experimental approach and on providing students with stochastics experience (e.g., M.E.C. 1992; N.C.T.M. 2000; Parzysz 2003). As argued in Batanero, Henry, and Parzysz (2005) these changes force us to reflect on the nature of chance and probability, since the analysis of obstacles that have historically emerged in the formation of concepts can help educators understand students’ difficulties in learning mathematics. Moreover, a well-grounded mathematics education researcher or teacher requires a wide view of understanding, so that understanding probability, for 108 Carmen Batanero and Carmen Díaz example, is not simply reduced to the student’s ability to define the word. Vergnaud (1982; 1990) suggested that psychological and educational researchers should consider a concept to include not just the set of invariant properties that make the concept meaningful, but also the situations (problems, task, phenomenology) and representations associated with the concept. Godino and Batanero (1999) take from Vygotski (1934) the suggestion that the meanings of words are the main units to analyze psychological activity, since words reflect the union of thought and language, and include the properties of the concept to which they refer. As such, one main goal in mathematics education research is finding out what meanings students assign to mathematical concepts, symbols, and representations; and explaining how these meanings are constructed during problem solving activities and how they evolve, as a consequence of instruction, and progressively adapt to the meanings we are trying to help students understand. Of particular relevance for the teaching of probability are the informal ideas which children and adolescents assign to chance and probability before instruction, which can affect their subsequent learning. For example, Truran (1995) found substantial evidence that young children do not see random generators such as dice or marbles in urns as having constant properties, but believe that a random generator has a mind of its own or may be controlled by outside forces. 1.1 Components of the Meaning and Understanding of Mathematics In trying to develop a systematic research program for mathematics education at the University of Granada, Spain, we have developed a theoretical model to carry out these analysis (Godino and Batanero 1994; 1998; Godino 2002; Godino, Batanero and Roa 2005; Godino, Contreras, and Font in press), which has been successfully applied in different works of research in statistics education, in particular in some Ph.D. theses carried out at different universities in Spain. In this paper we will use the concept of probability as an example, although the theory we are describing is also useful for other types of mathematical objects, such as theorems (e.g., the central limit theorem) or even a complete part of mathematics or statistics (e.g., variance analysis). In this model we distinguish five interrelated components in the meaning of the concept, as described below: x The field of problems from which the concept has emerged. One such problem was posed to Galileo by the Great Duke of Tuscany (about 1620): although 9 and 12 can be made up as the sum of the eyes of two dice, in as many different ways as 10 and 11, and The Meaning and Understanding of Mathematics x x x x 109 therefore should be expected to have the same frequency, the observation of long series of trials makes players prefer 10 and 11 to 9 and 12. In spite of its simplicity both Leibnitz and D’Alembert were unable to solve this problem (Székely 1986). Many other problems related to chance games were used to develop the first ideas of expectation and probability. The representations of the concept. To solve problems we need ostensive representations, since concepts are abstract entities. For example, in his letter to Fermat dated July 29, 1654, Pascal used the words value, chance, combinations, as well as numbers, symbols (letters), fractions, and a representation of the arithmetic triangle to solve a problem proposed by the Chevalier de Meré. In another letter to Fermat dated August 24, 1654, he used a tabular arrangement to enumerate the different possibilities in an interrupted game (Pascal 1963/1654). Modern representations of probability include density curves, algebraic expressions, distribution tables, or dynamical graphs produced by computers. The procedures and algorithms to deal with the problem and data, to solve related problems, or to compute values. Primitive probability problems were solved by simple enumeration or using other combinatorial tools. Today we have a wide variety of mathematical tools to help us solve probability problems, including combinatorics, analysis, algebra, and geometry. Distribution tables, calculators, and computers have also reduced the gap between the understanding of a problem and the technical competence required to solve it (Biehler 1997). The definitions of the concept. These will include its properties and relationships to other concepts, such as the different definitions of probability, the idea that a probability is always positive, the sum and product rule, the relationships between probability, expectation, frequency, and odds, and limit theorems. The arguments and proofs we use to convince others of the validity of our solutions to the problems or the truth of the properties related to the concepts. Galileo gave a complete combinatorial proof of the solution of the two dice problem and showed by enumeration that there are 25 different possibilities for both 9 and 12 and 27 for both 10 and 11. This same proof would be understandable by secondary school students; although the teacher might first allow the students to get some experience with randomness by organizing a classroom experiment in which students would throw the dice, record the results, and compare the relative frequencies of the different sums after a long series of trials. Computers and Internet applets possibly 110 Carmen Batanero and Carmen Díaz could improve the simulation and increase the students’ possibilities of exploring the experiments. This experimental confirmation is, however, very different from mathematical proof, although it can play a main role in the probability classroom, especially when we are dealing with complex ideas, such as sampling distributions. Our previous discussion suggests the multifaceted nature of even apparently simple concepts, such as probability and the need to take into account the different elements of meaning in organizing instruction. It is also important to notice that different levels of abstraction and difficulty can be considered in each of the five components defined above, and that the meaning of probability is thus very different at different institutions. In primary school or for the ordinary citizen, an intuitive idea of probability and the ability to compute simple probabilities by using the Laplace rule would be sufficient, using a simple notation and avoiding algebraic formulae. A probability literate citizen (Gal, in press) would also need to understand the use of probability in decision making situations (stock market, medical diagnosis), sampling, and voting, etc. In scientific or professional work, or at university level, however, a more complex meaning of probability, including knowledge of main probability distribution, limit theorems, and even stochastic processes would be needed. We therefore distinguish between personal and institutional meanings to take into account these varieties of meanings for the same concept at different institutions and also to differentiate between the meaning that has been proposed or fixed for a given concept in a specific teaching institution, and the meaning given to the concept by a particular student in the institution. 2. MEANINGS OF PROBABILITY These ideas are particularly relevant in analyzing the historical emergence of probability and its different meanings (laplacian, frequentist, subjective, axiomatic, etc.). The concept of probability has received different interpretations according to the metaphysical component of people’s relationships with reality (Hacking 1975), and the progressive development of probability has been linked to a large number of paradoxes that show the disparity between intuition and conceptual development in this field (Székely 1986, Borovcnik and Peard 1996). Below we summarize these different meanings, using some ideas from Batanero, Henry, and Parzysz (2005). The Meaning and Understanding of Mathematics 2.1 111 Intuitive Meaning No one knows when humans first started to play games of chance. Nor is it clear why probability theory only started recently as compared to other branches of mathematics. Hacking (1975) analyzes and discards different reasons that have been proposed to explain this delay, including obsession with determinism, lack of collections of empirical frequencies, religious beliefs, scarcity of easily understood empirical examples, or economic incentives; and he argues that even if none of these reasons are valid, the idea of probability did not appear until around 1600. Intuitive ideas related to chance and probability appeared very early, as they appear in young children or non-educated people who use qualitative expressions (probable, unlikely, feasible) to express their degrees of belief in the occurrence of random events. These ideas were of course too imprecise, and people needed the fundamental idea of assigning numbers to uncertain events to be able to compare their likelihood, thus applying mathematics to the wide world of uncertainty. Bellhouse (2000) analyzed a 13th-century manuscript, “De Vetula”, attributed to Richard de Fournival (1201-1260), which is possibly the oldest known text establishing the link between observed frequencies and the enumeration of possible configurations in a game of chance. By counting the 216 possible permutations of all the different values of three dice, the author computes the possibilities of the 16 different sums. Hacking (1975) indicates that probability has had a dual character since its emergence: a statistical side is concerned with stochastic rules of random processes, while the epistemic side views probability as a degree of belief. This duality was present in many of the authors who contributed to the progress of probability theory. For example, while Pascal’s solution to games of chance reflects an objective theoretical probability, his theoretical argument for the existence of God is a question of personal belief. 2.2 Classical Meaning The first probability problems were linked to games of chance, and it is therefore not surprising that the pioneer interpretations of probability were expressed in terms of winning expectations. Cardano (1961/1663) advised players in his “Liber de Ludo Aleae” to consider the number of total possibilities and the number of ways the favorable results can occur, and compare the two numbers in order to make a fair bet. Pascal (1963a/1654) solved the problem of estimating the fair amount to be given to each player in an interrupted game by proportionally dividing the stakes among each player’s chances. In his “Traité du triangle arithmétique” (1963b/1654) he 112 Carmen Batanero and Carmen Díaz developed combinatorial rules that he applied to solve some probability problems arising in his correspondence with Fermat. In “De Ratiociniis in Aleae Ludo” Huygens (1998/1657) showed that if p is the probability of someone winning a sum a, and q that of winning a sum b, then one may expect to win the sum pa + qb. The first definition of probability was given by Abraham de Moivre in “The Doctrine of Chances”: Wherefore, if we constitute a Fraction whereof the Numerator is the number of Chances whereby an Event might happen, and the Denominator the number of all the chances whereby it may either happen or fail, that Fraction will be a proper definition of the Probability of happening (de Moivre, 1967/1718, 1). This definition was not without problems. Laplace suggested that the theory of chance consists of reducing all the events of the same kind to a certain number of equally possible cases, that is to say, to such as we may be equally undecided about in regard to their existence, and gave this definition: probability is thus simply a fraction whose numerator is the number of favourable cases and whose denominator is the number of all cases possible (Laplace, 1985/1814, 28). This Laplacian definition of probability was based on a subjective interpretation, associated with the need to judge the equipossibility of different outcomes. Although equiprobability is clear when throwing a die or playing a chance game, it is not the same in complex human or natural situations. As noted by Bernoulli in “Ars Conjectandi” published in 1713, equiprobability can be found in very rare cases and does not only happen except in games of chance. 2.3 Frequentist Meaning Bernoulli suggested a possible way to assign probabilities to real events, in applications different from games of chance, through a frequentist estimate (Bernoulli 1987/1713). He also justified a frequentist estimation of probability in giving a first proof of the Law of Large Numbers: if an event occurs a particular set of times (k) in n identical and independent trials, then if the number of trials is arbitrarily large, k/n should be arbitrarily close to the “objective” probability of that event. The convergence of frequencies for an event, after a large number of identical trials of random experiments, had been observed over several centuries. Bernoulli’s proof that the stabilized value approaches the classical probability was interpreted as a confirmation that probability was an objective feature of random events. Given that stabilized frequencies are observable, they can be considered as approximate physical measures of this probability. The Meaning and Understanding of Mathematics 113 In the frequentist approach, moreover, probability is defined as the hypothetical number towards which the relative frequency tends when stabilizing (Von Mises 1952/1928). In this conception, the existence of the number for which the observed frequency is an approximate value is assumed. According to authors such as Gnedenko and Kolmogorov, mathematical probability would be a useless concept if it did not find this precise expression in the relative frequency of events resulting from long sequences of random trials, carried out under identical conditions. From a practical viewpoint, however, the frequentist approach does not provide the exact value of the probability of an event, and we cannot find an estimate of the probability when it is impossible to repeat an experience a very large number of times. It is also difficult to decide how many trials are needed to get a good estimation for the probability of an event. And of course we cannot give a frequentist interpretation of the probability of an event which occurs only one time under the same conditions, such as is often found in econometrics (Batanero, Henry, and Parzysz, 2005). 2.4 Propensity Meaning In trying to solve some of the problems in the frequentist meaning of probability, as well as to make sense of single-case probabilities, Popper (1957, 1959) considered probability as a physical propensity, disposition, or tendency to produce an outcome of a certain kind, or the long run relative frequency of such an outcome. This idea was also suggested by Peirce (1932/1910) who advanced a concept of probabilities according to which a die, for example, possesses would-bes for its various possible outcomes, and these would-bes are intentional, dispositional, directly related to the long run, and indirectly related to singular events. While the classical and frequentist meanings reduce the notion of probability to other, already known concepts (ratio, frequency, etc.), Popper introduces the idea of propensity as a measure of the “probabilistic causal tendency” of a random system to behave in a certain way. This idea was discussed by different authors, who distinguished long run and single case propensity (Gillies 2000). In the long run theories, propensities are tendencies to produce relative frequencies with particular values, but the propensities are not the probability values themselves (Popper, 1957, 1959, Hacking 1965); for example, a fair die has an extremely strong tendency (propensity) to produce a 5 with long run relative frequency of 1/6. The probability value 1/6 is small, so it does not measure this strong tendency. In single-case theory (e.g., Mellor 1971) the propensities are identical to the probability value and are considered as probabilistic causal tendencies to 114 Carmen Batanero and Carmen Díaz produce a particular result on a specific occasion. In this theory we consider the die to have a weak propensity (1/6) to produce a 5. Again this interpretation was controversial. In the long run interpretation, propensity is not expressed in terms of other empirically verifiable quantities, and we then have no method of empirically finding the value of the propensity. Moreover, the assumption that an experimental arrangement has a tendency to produce a given limiting relative frequency of a particular outcome presupposes a uniformity which is hard to test, either a priori or empirically (Cabriá 1992). As regards single case interpretation, it is difficult to assign an objective probability for single events. The probability will vary according different conditions, and then the probability is attached to the conditions and not just to the event itself. Since the same event can be included in different reference sets, we might introduce subjective elements when selecting the reference class in which to include the event (Gillies 2000). It is also unclear whether single case propensity theories obey the probability calculus or not. 2.5 Logical Meaning Keynes (1921) and Carnap (1950), among other authors, developed the logical theories, which retain the classical idea that probabilities can be determined a priori by an examination of the space of possibilities, although the possibilities may be assigned unequal weights. This approach defines probability as a degree of implication that measures the support provided by some evidence E to a given hypothesis H. Keynes accepts the conventional calculus of probability since it applies to his probability relations. He symbolizes certainty and impossibility by 1 and 0, and all other degrees of the probability relation lie between these limits. Since the deductive relations of implication and incompatibility can be considered as extreme cases (with degree of implication values 1 and 0 respectively), deductive logic is amplified in this approach. Carnap (1959) started by constructing a formal language and defined probability as a rational credibility function, using the technical term degree of confirmation. The degree of confirmation of one hypothesis H, given some evidence E, depends entirely on the logical and semantic properties of and relations between H and E, and therefore it is only defined for the particular formal language in which these relations are made explicit. This degree of confirmation is just the conditional probability of H given E and allows inductive learning from experience. Carnap reduces the problem of defining the degree of confirmation for the particular formal language in which he is operating to the problem of selecting a probability for elementary state descriptions in that language. The Meaning and Understanding of Mathematics 115 One problem in this approach is that there are many possible confirmation functions, depending on the possible choices of initial measures and on the language in which the hypothesis is stated and in which the confirmation function is defined. A change of language might then make invalid any particular confirmation of a theory. A further problem is in selecting the adequate evidence E in an objective way, since the amount of evidence might vary from one person to another. 2.6 Subjective Meaning Bayes’s formula permitted the finding of the probabilities of various causes when one of their consequences is observed. The probability of such a cause would thus be prone to revision as a function of new information and would lose its objective character postulated by the above conceptions. Keynes, Ramsey, and de Finetti described probabilities as degrees of belief, based on personal judgment and information about experiences related to a given outcome. They suggested that the possibility of an event is always related to a certain system of knowledge and is thus not necessarily the same for all people. The difficulty with the subjectivist viewpoint is that it seems impossible to derive mathematical expressions for probabilities from personal beliefs. Both Ramsey (1926) and de Finetti (1937), however, suggested a way of deriving a consistent theory of choice under uncertainty that could isolate beliefs from preferences while still maintaining subjective probabilities. The basic idea behind the Ramsey-de Finetti derivation is that by observing the bets people make, one can assume that this reflects their personal beliefs about the outcomes, and then subjective probabilities can be inferred. From this subjectivist viewpoint, the repetition of the same situation is no longer necessary to make sense of probability. The fact that repeated trials are no longer needed is used to expand the field of applications of probability theory. Today, the Bayesian school assigns probabilities to uncertain events, even non-random phenomena, although controversy remains about the scientific status of results which depend on judgments that vary with the observer. 2.7 Mathematical Meaning and Axiomatization Throughout the 20th century, different mathematicians contributed to the development of the mathematical theory of probability. Borel’s view of probability as a special type of measure was used by Kolmogorov, who applied sets and measure theories to derive a satisfactory axiomatic system, 116 Carmen Batanero and Carmen Díaz which was generally accepted by different schools regardless of their philosophical interpretation of the nature of probability. Probability is, therefore, a mathematical object, and probabilistic models are used to describe and interpret random reality. Probability theory proved its efficiency in many different fields, but the particular models used are still subjected to heuristic and theoretical hypotheses, which need to be evaluated empirically. These models also allow assigning probabilities to new events that made no sense in previous interpretations. One such example is the Cantor distribution, a probability distribution with a Cantor function as a cumulative distribution. This distribution is a mixture of continuous and discrete; it has neither probability density function nor point-probability masses. It serves to solve problems such as the following: Let the random variable X take a value in the interval [0, 1/3] if we get heads in flipping a coin and in the interval [2/3, 1] if we get tails. Let X then take a value in the lowest third of the aforementioned interval if we get heads in the second flipping of the coin, and in the highest third if we get tails. If we imagine this process continuing to infinity, then the probability distribution of X is the Cantor distribution, with an expected value of 1/2. Of course interest in this distribution is mainly theoretical. 2.8 Summary In this brief description, we have analyzed different historical views of probability that still persist and are used in the teaching and practice of probability. We can consider these views as different meanings for probability, according to our theoretical framework, since there are differences not only in the definition of probability, but also in the related problems, tools, representations, properties, and concepts that have emerged to solve various problems (see a summary in Table 1). When teaching probability (or any other mathematical concept), these five different types of knowledge should be considered and interrelated, as research on the understanding of mathematics has shown that students find difficulties in each of these components. In the same way, the different meanings of probability should be progressively taken into account, starting with the students’ intuitive ideas of chance and probability, since, understanding is a continuous constructive process in which students progressively acquire and relate the different elements of the meaning of the concept. The Meaning and Understanding of Mathematics Table 1. Elements Characterizing Different Historical Meanings of Probability 117 118 Carmen Batanero and Carmen Díaz 3. SEMIOTIC FUNCTIONS AND MATHEMATICAL REASONING While our analysis of the elements of meaning is used to focus the attention on the different components of mathematical teaching and learning, it is insufficient to describe mathematical reasoning and mathematical activity (at a micro level of analysis). From a didactical point of view it is useful to consider the notion of semiotic function: “There is a semiotic function when an expression and a content are put in correspondence” (Eco 1979, 83). The original in this correspondence is the significant (plane of expression), the image is the meaning (plane of content), that is, what is represented, what is referred to by the speaker. An elementary meaning is produced when a person interprets a semiotic function with a semiotic act (interpretation or understanding), in which he/she relates an expression to a specific content. An elementary meaning is the content that the author of an expression refers to, or the content that the reader/listener interprets. Some examples are given below: x x x x The expression “product rule” or the symbolic representation P(M ˆ I) = P(M) x P(I) describes a procedure carried out by a student to compute a compound probability in the case that M and I are independent. The statement of a problem can represent the real situation; the simulation of random phenomena can represent the phenomena themselves; for example, we can simulate the number of girls in a sample of 10 newborn babies with coins. We can say “Pascal and Fermat’s solution of de Meré’s paradox” to refer to an argument. In expressions such as, “Let P be the mathematical expectation of a random variable”, the notations P, the expressions mathematical expectation and random variable refer to particular abstract concepts. Semiotic functions are always involved in creating mathematical concepts, establishing and validating mathematical propositions, and, as a rule, in problem solving processes; and they can produce elementary or systemic meanings. For example, when we say “computing probabilities in the normal distribution” in a general sense we refer to the whole set of procedural elements to compute these probabilities, including use of tables, standardization, computer programs, simulation, etc. When we speak of studying the normal distribution, we refer to the whole system of practices The Meaning and Understanding of Mathematics 119 associated with this distribution, whose structural elements are the five types described in section 1. These are examples of systemic meanings. In each semiotic function, the correspondence between the expression and the content is fixed by explicit or implicit codes, rules, habits, or agreements. Sometimes the teacher and the student attribute different meanings to the same expression and there is a semiotic conflict (Godino, 2002). These conflicts appear in the linguistic interaction between people or institutions, and they frequently explain the difficulties and limitations of teaching and learning mathematics. 3.1 Reasoning as a Chain of Semiotic Functions When solving a mathematical problem or carrying out any mathematical activity or reasoning, one or more semiotic functions appear among the five components described in section 1. It is then useful to consider mathematical reasoning as a sequence of semiotic functions (or as a chain of related pieces of knowledge) that a person establishes in the problem-solving process. The notion of semiotic function is used to indicate the essentially relational nature of mathematical activity and of teaching/learning processes. Let us consider, as an example, the incorrect solution given by a student to the following problem (taken from Díaz and de la Fuente 2006). Problem We threw two dice and the product of the two numbers was 12. What is the probability that neither of the two numbers was 6? The solution given by a student is the following: We have broken down this solution into analysis units in order to show the semiotic activity carried out by the student and to discover his or her semiotic conflicts (see Table 2). 120 Carmen Batanero and Carmen Díaz Table 2. Semiotic Analysis of the Student’s Solution Analysis Unit [U1] 3.4=12; 4.3=12 [U2] 1 1 P(31 ˆ 4 2 )= x 6 6 Semiotic Functions Established by the Student The expressions 3, 4 (symbols) refer to the numbers 3 and 4 (concepts) and these refer to the possible outcomes for each die (phenomenological object). The expression 12 (symbol) refers to the number 12 (concept) and this refers to the fixed value of the product (operation). The student identifies the favorable cases in the event (applies a concept to a practical situation). The student is able to distinguish that order (concept) is relevant. The expression P(31 ˆ 4 2 ) refers to compound probability and at the same time to the 1 intersection of two events (concepts). 36 [U2] refers to the product rule (proposition) in the case of independent events (property). The expression [U4] > 2 36 1 refers to the product 36 of fractions (rule) and its result (procedure). The student is able to perceive the independence of the two dice. The student distinguishes the order of outcomes. Similar to Unit 2 [U3] 1 1 P(41 ˆ 32 )= x 6 6 1 1 x 6 6 1 36 The arrows (graphical representation) refer to the sum of the two probabilities. The symbol 2 refers to the value of this sum. 36 The student is correctly identifying and applying the union axiom to compute the probability of the union of two incompatible sets. The student does not restrict the sample space. The Meaning and Understanding of Mathematics 121 We can see from this example the complexity of solving even elementary probability problems. This student was able to identify the problem data, apply the sum and product rules, assess independence and the relevance of order, use complex notation, and identify the favorable cases. He has not, however, restricted the sample space in computing the conditional probability (none of the numbers is 6, given that the product is 12) to the relevant cases (only 4 possibilities). A semiotic conflict appears when the student computs the compound probability “that the product is 12 and none of the numbers is six,” instead. 4. PROBABILITY MISCONCEPTIONS AND SEMIOTIC CONFLICTS Probability misconceptions have been widely documented, although recently some researchers are wondering if they can be explained only in terms of psychological mechanisms (e.g., Callaert, in press). Below we analyze some of these misconceptions and suggest that semiotic conflicts are alternative explanations for some of them. 4.1 Equiprobability Bias Pratt (2000) reports interviews with two children who declared that the totals 2 and 3 for the sum of two dice were equally easy to obtain. He explained this response in terms of Lecoutre’s (1992) equiprobability bias, by which children consider all the events in a random experiment to be equiprobable. Callaert (2003) analyzed these results and remarked that the final outcome for which the probability is being determined is not at all related, in a simple and direct way, to any underlying experiment that can easily be conceptualised by the children. In the experiment there are exactly 36 different ordered pairs (x,y) of numbers and 21 different unordered pairs of numbers. When the children in Pratt’s (2000) experiment are interviewed about their response, one of them (Anne) replies: “Well, 1 and 2 and 2 and 1 are the same ...; they come to the same number”. As suggested by Callaert, children were paying too much attention to the mathematical operation of adding numbers. Since this operation is commutative, it invites children not to pay attention to order when computing probabilities. Here a semiotic conflict appears when children assign a non–existent property (commutativity) to the sample space, assuming the outcomes (1, 2) and (2, 1) are identical when they are not. The students and the teachers are, moreover, assuming different sample spaces for this experiment (see Table 3, a and b). 122 Carmen Batanero and Carmen Díaz Table 3. Different Sample Spaces Assumed by the Teacher and the Student a. Sample Space Assumed by the Teacher 1,1 1,2 1,3 1,4 1,5 1,6 2,1 2,2 2,3 2,4 2,5 2,6 3,1 3,2 3,3 3,4 3,5 3,6 4,1 4,2 4,3 4,4 4,5 4,6 5,1 5,2 5,3 5,4 5,5 5,6 6,1 6,2 6,3 6,4 6,5 6,6 b. Sample Space Assumed by the Student 1,1 1,2 1,3 1,4 1,5 1,6 2,2 2,3 2,4 2,5 2,6 3,3 3,4 3,5 3,6 4,4 4,5 4,6 5,5 5,6 6,6 4.2 Conditional Probability and the Fallacy of the Time Axis Falk (1979; 1999) proposed the following problem to eighty-eight university students and found that while students easily answered part (a), they were confused about part (b). Problem An urn contains two white balls and two red balls. We pick up two balls at random, one after the other without replacement. (a) What is the probability that the second ball is red, given that the first ball is also red? (b) What is the probability that the first ball is red, given that the second ball is also red? Students typically argued that, because the second ball had not been drawn at the same time as the first ball, the result of the second draw could The Meaning and Understanding of Mathematics 123 not influence the first. Hence, the students claimed that the probability in part (b) is 1/2. Falk suggested that these students confused conditional and causal reasoning and also demonstrated the fallacy of the time axis. That is, they thought that one event could not condition another event that occurs before it. This is false reasoning, because even though there is no causal relation from the second event to the first one, the information in the problem that the second ball is red has reduced the sample space for the first drawing. In essence, there are now just one red ball and two white balls for the first drawing. Hence, P (B1 is red/ B2 is red) =1/3. Our hypothesis is that there is a semiotic conflict in the use we make of the words dependent/ independent in statistics and in other areas of science and mathematics, where this word has a causal meaning that students bring to the study of statistics (Díaz and de la Fuente 2006). 4.3 Ruling out Semiotic Conflict Does Not Always Solve Probabilistic Misconceptions Of course the conjecture that the above difficulties might be explained by the participants’ semiotic conflicts in interpreting the tasks should be empirically tested. On the other hand, there are many other probabilistic misconceptions and errors which seem to defy semantic explanations. One such example is the conjunction fallacy. In related research, many students and professionals were presented with different variations of the following problem (Kahneman and Tversky 1982). Problem Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply involved with issues of discrimination and social justice, and also participated in antinuclear demonstrations. The task is to rank various statements by their probabilities, including these two: (a) Linda is a bank teller (b) Linda is a bank teller and is active in the feminist movement Many students and professionals ranked response (b) as more probable than (a). This judgment apparently violates the probability rules, since sentence (a) refers to the probability of a single event P (B), while sentence (b) refers to the probability of the compound event P (B and F), which at best is equal to P(B) assuming P(F)=1. 124 Carmen Batanero and Carmen Díaz Some researchers organized experiments to ensure that participants were not misled into interpreting (a) as “Linda is a bank teller and not active in the feminist movement” and did not therefore assign an incorrect meaning to the term probability (Sides, Osherson, Bonini, and Viale 2002). In spite of these precautions, conjunction fallacies were still very frequent. Attempts to explain the fallacy in terms of misinterpreting the word and in sentence (b) to refer to the conjunction of two events (even though the word possesses semantic and pragmatic properties that are foreign to conjunction) were still unsuccessful (Tentori, Bonini, and Oshershon 2004). 5. IMPLICATIONS FOR TEACHING PROBABILITY This discussion shows the multifaceted meanings of probability and suggests that teaching cannot be limited to any one of these different meanings because they are all dialectically and experientially intertwined. Probability can be viewed as an a priori mathematical degree of uncertainty, evidence supported by data, a propensity, a logical relation, a personal degree of belief, or as a mathematical model that helps us understand reality. The controversies that historically emerged about the meaning of probability have also influenced teaching. Before 1970, the classical view of probability based on combinatorial calculus dominated the secondary school curriculum. Since combinatorial reasoning is difficult, students often found this approach to be very hard; moreover, the applications of probability in different sciences were hidden. Probability was considered by many secondary school teachers as a subsidiary part of mathematics, since it only dealt with games of chance. In other cases it was considered to be only another application of set theory (Henry 1997; Parzysz 2003). With increasing computer development, there is growing interest in the experimental introduction of probability as a limit of the stabilized frequency. Simulation and experiments help students solve some paradoxes which appear even in when teaching simple probability problems. A pure experimental approach, however, is not sufficient in the teaching of probability. Even when a simulation can help to find a solution to a probability problem arising in a real world situation, it cannot prove that this is the most relevant solution, because the solution will depend on the hypotheses and the theoretical setting on which the computer model is built. A genuine knowledge of probability can be achieved only through the study of some formal probability theory, and the acquisition of such theory should be gradual and supported by the students’ stochastic experience. The Meaning and Understanding of Mathematics 125 On the other hand, much more research is needed to clarify the fundamental components of the meaning of probability (and in general of every specific mathematical concept) as well as the adequate level of abstraction in which each component should be taught, since students might have difficulties in any or all the different components of the meaning of a concept. 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