Teaching Probability for Conceptual Change La Enseñanza De La Probabilidad Por Cambio Conceptual

CESAR SAENZ CASTRO

TEACHING PROBABILITY FOR CONCEPTUAL CHANGELA ENSENANZA DE LA PROBABILIDAD POR CAMBIO

CONCEPTUAL

ABSTRACT. This work presents a theoretical proposal for a methodology for the teachingof probability theory. The theoretical proposal has a dual inspiration: (1) the epistemologicalapproach of Lakatos (1978b) regarding the quasi-empirical nature of mathematical theories;(2) the perspective of conceptual change for the teaching-learning process, as formulated byStrike and Posner (1992). The scientific content taught and the didactic methods used in theclassroom should, according to this proposal, respect and conform to this dual inspiration.

We also present an evaluation of the methodology in a real context: six Spanish highschool classes of students aged 14–15. The main purpose of the research was to answerthis question: Is our didactic proposal more effective than traditional methodology? Weoperatively identify the concept of traditional teaching, and establish several indicators ofeffectiveness: the mastery of elementary probability calculations, the quality of intuitivereasoning in probability, and the conceptual and attitudinal change produced. We foundsignificant differences on all indicators, except for attitudinal change, in favour of the groupthat followed our proposal.

LA ENSENANZA DE LA PROBABILIDAD POR CAMBIOCONCEPTUAL

En este trabajo presentamos una propuesta teorica y experimental de una metodologia deensenanza de la teoria de probabilidades. La propuesta teorica tiene una doble inspiracion:(1) asume el planteamiento epistemologico de Lakatos sobre la naturaleza cuasi-empiricade las teorias matematicas; (2) asume el enfoque de cambio conceptual para el proceso deensenanza-aprendizaje, tal como lo formulan Strike y Posner. Los contenidos cientificosque se imparten y los metodos didacticos que se utilizan en el aula, siguiendo esta propuesta,han de respetar y ajustarse a esta doble inspiracion.

Tambien presentamos la experimentacion de esta metodologia en un contexto real: 6aulas de bachillerato espanol con alumnos de 14–15 anos. La investigacion tenia comoobjetivo principal responder a la pregunta ¿Nuestra propuesta didactica es mas eficazque una metodologia tradicional? Identificamos operativamente el concepto de ensenanzatradicional y fijamos varios indicadores de eficacia: dominio del calculo de probabilidadeselemental, calidad del razonamiento probabilistico intuitivo y cambio conceptual y actitud-inal producido. Encontramos diferencias significativas en todos los indicadores salvo elcambio actitudinal, a favor del grupo que siguio nuestra propuesta.

Educational Studies in Mathematics35: 233–254, 1998.c 1998Kluwer Academic Publishers. Printed in the Netherlands.

GR: 201007292, Pipsnr.: 150537 HUMNKAPeduc665.tex; 22/04/1998; 8:00; v.7; p.1

234 CESAR SAENZ CASTRO

1. DESIGN OF THE METHODOLOGY

The traditional approach to the teaching of probability theory is epistemo-logically based on the hypothetical-deductive character of the theory. Fromthis starting point, it proposes that teaching should follow to the logicalstructure of the discipline in a linear way, without taking into accountthe cognitive processes of learning. In spite of the poor academic resultsyielded by such a perspective, there is no doubt that it has two essentialqualities which explain, in our opinion, its continued use:

(1) There is a profound cohesion between the epistemological con-ception of probability and the teaching contents and methods used.

(2) By virtue of their training and tradition, teachers identify withsuch a perspective, and thus find the teaching content and methods whichare deduced from it natural and without any possible alternative.

Often, the didactic approaches which endeavour to overcome the lowacademic achievement obtained by a high percentage of students are basedon a series of methodological innovations (use of computers, problemsolving, etc) that try to take into account the students’ learning process.However, these innovations fail time and time again, and most teachersreturn to the traditional teaching approach. Why?

We think the reason is that these new didactic approaches are mereminor adjustments to traditional educational activities, and do not questionthe hypothetical-deductive nature of mathematical theories and the sub-ordination of the teaching-learning process to the axiomatic structures ofthose theories. The new strategies form a series of stimulating, but subsi-diary, classroom activities, so that ultimately, the important issue is still tomake consistent mathematical relationships explicit. Teachers get a senseof the epistemological-didactic imbalance, and, at the first sign of difficultyin the educational process, return to traditional activities with which theyfeel more comfortable.

Our didactic proposal aims to overcome the deficiencies in the tradi-tional approach, but attempts to conserve the two primary qualities of thisapproach: the existence of an explicit epistemological framework, and thecohesion of teaching content and methods within that framework. Withthis in mind, we define our teaching model according to three criteria inthe teaching-learning process: the epistemological framework, the psy-chopedagogical framework, and the content and teaching methods of themodel.

We now turn to briefly analyse the three criteria. A more extensiveanalysis of the teaching content and methods, and of the application of themethodology in the classroom, including a presentation of all the curricularmaterials used, can be found in Saenz (1995).

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TEACHING PROBABILITY FOR CONCEPTUAL CHANGE 235

1.1. Epistemological framework: mathematics is quasi-empirical

According to the orthodoxy of logical empiricism, while science isaposteriori, substantive and fallible, mathematics isa priori, tautologicaland infallible. Nevertheless, mathematical philosophers such as Kitcher(1983) or Lakatos (1978b) reject this and defend a radical assimilation ofmathematics into science showing that mathematical empiricism, not onlywith regard to the origin and method of mathematics but also with regardto its justification, is more alive and extended than one would think.

Let us demonstrate these ideas in relation to probability theory, andoutline a probabilistic education based on Lakatos’ epistemological per-spective.

Kolmogorov’s probabilistic model is a Euclidean theory, the first axiomsof which are:

(1) 0� P (A) � 1,(2) P (A [B) = P (A) + P (B) if A \B 6= � ,(3) P (U) = 1.Many theorems are inferred from these axioms following a strictly

deductive method. This theoretical model is justified for the student interms of the properties of the frequencies, which only operate as intuitiveevidence for the formal theory defined. Three phases of the Euclidean the-ory are reflected in this process: 1) intuitive pre-scientific state, which inthe case of probability theory can be Laplace’s model and the frequencymodel; 2) foundation period, with Kolmogorov’s axiomatics, and the the-orems deduced from it; 3) assignment of probabilities in a given problem,following symmetry criteria (Laplace’s model) or repeating the randomexperiment, taking into account the law of stability of relative frequencies(frequency model).

In a quasi-empirical perspective of the development of probabilistic sci-ence, we would say that, historically, the initial problems could be solvedwithin Laplace’s model or the frequency model, and that Kolmogorov’smodel was reached after rigorous tests and refutations of former mod-els which allowed the consideration of new problems. Thus the informalLaplacian and frequencial models co-exist with Kolmogorov’s formal mod-el, where elementary probability problems are solved. Only because thereare potential logical falsifiers for the informal models, and problems thatcannot be considered within them, was it necessary to reach Kolmogorov’smodel. This, in turn, presents falsifiers of a logical and of a heuristic kind,and thus the need to create a new formal model arises. To sum up, there isa continuous review of theories, rather than a lazy stagnation in one theory.

The characterization of probability in the classroom defended by Stein-bring (1991) has many points in common with the quasi-empirical per-

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spective of mathematics defended by Lakatos. Indeed, it is deduced fromboth perspectives that the meaning of concepts in probability is not com-pletely defined by the initial axiomatic structure, but that meaning dependson the level of development of the probability theory.

The acceptance of the thesis that mathematics in general, and probab-ility theory in particular are quasi-empirical in character, generates deepdidactical implications. As opposed to the formalist perspective, whichtends to identify mathematics with its axiomatic abstraction, in such a waythat mathematics would not have a history as such, but would consist ofan ever-growing set of eternal and immutable truths, the quasi-empiricalperspective highlights counter-examples, refutations and critiques as essen-tial and natural components of the mathematical endeavour. Deduction isnot the only heuristic pattern for mathematics, and the establishment ofdifferent rival theories for problematic facts is admitted and promoted.

Kitcher (1983) maintains that the history of mathematics is epistemo-logically relevant (we would add that it is didactically relevant); in this heagrees with Lakatos although his theory about the nature of mathematicalknowledge differs in many aspects from that of Lakatos. Kitcher’s theoryseems to us highly relevant in the following sense: it explains the mathem-atical knowledge of today’s individual with reference to a chain of priormathematical expertise. At the closest end of the chain are the mathem-atical authorities, the teachers and textbooks of today. At the other end,the end furthest away in time,is the most fundamental mathematical know-ledge which came from the ordinary perception of the daily experience ofour ancestors.

Therefore, it seems to us that to take Kitcher’s ideas to the process ofteaching and learning of probability theory implies, amongst other con-sequences, taking into consideration the ideas and conjectures of the stu-dents about random phenomena (which come from everyday experience) asauthentic predecessors of/alternatives to formal theories. We cannot ignorethem a priori, but they must be considered and submitted to a processof tests, refutations and evaluation of the heuristic potential which theycontain. The students must establish that the mathematical theories have ahistory, one of conflict and intellectual adventure, and that such a historycan be repeated in their minds when they reconstruct the theories duringlearning. This idea conforms with the psychopedagogical framework ofour didactic proposal.

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1.2. Psychopedagogical framework: the process of mathematicsteaching and learning seen as a process of conceptual change

In the last few years, in the context of the many studies published on theexistence of spontaneous misconceptions with regard to scientific phenom-ena in students, various theories of scientific learning have started to appearwhich conceive it as a process of conceptual change or transformation ofthose spontaneous concepts into scientific ones.

Posner, et al. (1982, reviewed in 1992 by Strike and Posner) construct atheoretical model of the process by which a set of concepts changes underthe impact of new ideas and information, becoming another set of concepts,often incompatible with the former one. They establish two different waysof acquiring knowledge. There are occasions when students have adequateconcepts (even if they are not fully developed) to confront the situation;in this case, the new knowledge is included in the existing cognitive struc-ture and the way in which it is acquired is called assimilation. On otheroccasions the existing concepts are inadequate to satisfactorily understandthe phenomenon in hand. In this situation, students must reorganize theirconcepts; a more radical change is thus produced, called accommodation.A model of learning through conceptual change should explain how thereorganization of the current concepts of the student is produced, or, inother words, how accommodation occurs.

The knowledge baggage of an individual is similar to the ‘hard core’of a science, in Lakatos’ (1978a) terminology. This baggage originatesin previous experiences, images or models that were created in the mindof the individual. People tend to apply it to new phenomena and it is notdiscussed; it only progresses or degenerates depending on whether it worksor not to explain new situations. Only when this baggage does not work,is an accommodation produced, for which certain conditions must apply:

– There must be dissatisfaction with existing conceptions. The usualtendency is not to make any great changes; multiple situations which cannotbe solved with the available concepts must accumulate.

– The new conception must be intelligible. The student must under-stand the terms and symbols used, and must be able to construct a coherentrepresentation of the new concept.

– The new conception must be initially plausible. It must be consistentwith other knowledge, and it must solve the problems generated by theformer conception.

– The new concept must suggest a new ‘research program’, expressedin Lakatos’ (1978a) terms. It must open up new problems and researchareas.

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It must be made clear that, in spite of the fact that the accommodationof a scientific conception is described as a radical change in the conceptualsystem of a person, it is not an abrupt change, but is usually a process ofstages of gradual adjustment.

We have chosen the conceptual change approach as the psychopedago-gical frame of our didactic proposal for two reasons. Firstly, because themodel which has been taken for the process of transformation of students’alternative conceptions into scientifically accepted ideas is epistemologyitself, which analyzes the conceptual changes undergone in the history ofthe science. This epistemological origin of the conceptual change perspect-ive fits perfectly with our premise of epistemological-didactic coherence.Secondly, these theories, although they are cognitive theories of the acquis-ition of scientific concepts, have a clear instructional perspective, in thatthey try to identify the didactic strategies that will encourage conceptu-al change from students’ intuitive or natural science into formal science(Hewson and Hewson, 1984).

1.3. Classroom work: content and didactic methods

From the epistemological and psychopedagogical framework which wehave just established, the two principal operatives that conform with ourdidactic proposal can be deduced: (1) the contents and didactic methodsshould respect the quasi-empirical nature of probability theory. (2) Thecontent and didactic method should adapt itself to the teaching-learningprocess as one of conceptual change.

Based on these two principles we solved two essential problems of theapplication of the methodology in the classroom: the construction of thecurriculum (on a macro level) and the organisation of the teaching-learningactivities of each curricular topic (on a micro level).

1.3.1. Construction of the curriculumThe design and construction phase of the curriculum was carried out overa period of 100 hours with the teachers of the groups that were to use theconceptual change instruction model. This included familiarisation timewith the methodology since it had not been previously met by any of theteachers. Furthermore, given that the role of the teacher becomes muchmore complex than that of transmitter or communicator of knowledge, asin traditional teaching (for example, as mediator between the knowledgeof the sciences and students’ understanding), the teacher is required to beable to diagnose the thinking of the students (in this respect we have usedmany ideas of the ‘diagnostic teaching’ proposal of Bell, 1987).

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Figure 1. General structure of the sequence of teaching.

We organised the list of topics which comprise the official Spanish cur-riculum on probability in secondary education into the following didacticunits:

0. Posing the problem: random vs deterministic experiences.1. Language of chance. Representations of uncertainty.2. Random experiences of a trial.3. The laws of large numbers.4. Random experiences of several trials.5. Intuitive statistical reasoning.6. Random variables.7. Binomial distribution.8. Normal distribution.9. Re-posing the problem: chance and necessity.

1.3.2. Organisation of the teaching-learning activitiesAs to the organisation of the teaching-learning activities, we structuredeach one of the above didactic units around a sequence of activities whoseobject was to produce a conceptual change in students’ thinking. In Figure Ithis general structure of the sequence of teaching is presented, accordingto the constructivist model of Driver (1988).

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As an example, we go on to present the sequence of teaching-learningin didactic unit 4 (random experiences of several trials):

Didactic structure of the unit:it is adapted to the sequence of teaching-learning from a conceptual change viewpoint (see table I). The focus ofattention is centred firstly on clarifying students’ ideas (drawing out phase),secondly, on carrying out random experiments to encourage cognitive con-flict and in explaining the normative content which explains the experimentresults (restructuring phase), thirdly, on applying the new ideas to new con-texts (application phase), and fourthly on revising the previous ideas in thelight of the new knowledge (revision of the conceptual change phase).

Implied probabilistic contents: these are shown in the correspondingcolumn of the didactic unit (see table I). This unit introduces the randomexperiences of various trials which allow tree diagrams as appropriatemeans of representation. Tree diagrams and paths are frequently used tocalculate combined probabilities and to put into operation the concepts ofindependent and dependent events.

Role of the teacher: the teacher is not only the transmitter of informationbut is the co-ordinator/director of the teaching-learning process. Thereforethere is a wide array of teacher activities which go much further than theexplanation of contents and correction of exercises: he or she moderates,motivates, summarises, criticises and supervises the activities of the stu-dents, with the principal objective that they decide if their previous ideasare useful to explain certain random phenomena or need to be refined orsubstituted for other more appropriate ones (see table I).

Methodological suggestions: the work develops in an experimental con-text and with the solving of problems using games and betting (see Table I).As an example, we present 3 problems of the worksheet 1 (this worksheetconsists of 11 random problematic situations)

(7) Consider the dice a, b, c and d whose nets are

The first player chooses a dice and the second, another (of the 3 that are left). Theythrow them and the one with the highest score wins. Would you prefer to be the first orsecond player? Why?

(8) Two players each bet 20,000 pesetas in the following manner: the player who correctlyguesses the toss of a coin six times wins all the money. When one of them is winning 5:3,

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the game has to be stopped. How should the 40,000 pesetas be divided between the twoplayers? (Version of a classic probability problem: Pascal-Fermat, 1654).

(9) There are three cards: one of them is black on both faces, another red on both facesand the third has one red face and one black face. The cards are put into a hat and withoutlooking, a card is taken out so that only one of the faces of the card can be seen. Suppose itis red. A player suggests that you both bet the same amount of money; he bets against youthat it is the card with red on both sides. Does this seem to you a fair bet?

The didactic setting of these three situations is the following:Experiment 7Three pairs are formed, each one with the four dice shown on worksheet

9. They are permitted to play for a while. Each pair must answer thequestion and justify their answer. Later, they must report to the class.Afterwards, the teacher can challenge any student to play and demonstratethat there is a winning strategy.

Experiment 8Students form ten or more couples, that begin from the hypothesis that

one of the couple is winning 5:3. Each couple plays the game 16 times.Players see which of the two wins the game. Then the results of the tencouples are collated in a table. The students have to guess from the datawhich is the probability of each player to win the game.

Experiment 9The teacher takes one card from the bag 30 times, showing one of the

faces. Students must guess the colour of the other face. One student bythe side of the teacher notes the colour of the hidden face and, at the end,shows the class the result. The students are encouraged to pick a strategy toplay the game again, if it does not occur to them, offer them the following:

(a) Choose at random the colour of the hidden face.(b) Choose for the hidden face the same colour as the visible face.(c) Choose the alternative colour.

The game is played again to see which of the strategies is most success-ful.

This is a very deceptive random situation where the majority of studentsaccept that it is a case of a fair bet. If the teachers try to solve the problemformally (with the help of a tree diagram, for example), they will find thatthe students will not accept their explanations. It is necessary to create ascenario of games of strategy so that students are convinced that there is astrategic winner, that is, precisely what is offered by probability theory.

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2. EVALUATION OF THE INSTRUCTION METHODOLOGY

We now turn to an evaluation of the didactic methodology whose designwe have just analyzed. With this methodology we wanted to solve twoproblems: that of assessing the efficiency of the proposed teaching method,and that of the interaction of this method with certain subject variables(cognitive development, performance in mathematics) with regard to thelearning of probabilities.

Here we are going to concentrate on the first problem: whether ourdidactic proposal promotes students’ learning to a greater extent than tradi-tional teaching in the classroom. We are aware that the concept of traditionalteaching is not a well defined one, and that under this umbrella are includedthe teaching styles of a wide variety of teachers in a great diversity of schoolcontexts. However, in the previous section (p. 2) we outlined the elements,positive and negative, that characterize what is generally understood astraditional teaching. We identify its fundamental parameters: (a) epistem-ological framework: mathematics has hypothetical-deductive character;(b) psychopedagogical framework: the internal logic of the discipline dic-tates the planning of the teaching; (c) application in the classroom: activ-ities are organised based on reproductive learning (Ausubel, Novak andHanesian, 1978); it induces and reinforces the activity of revision until thestudent is capable of repeating the content presented to her or him, and ofcarrying out exercises in the application of that content.

If we had to summarize the differences between the teaching stylesunder comparison, we would say that traditional teaching is teacher-centred, where the teacher explains material from a textbook and assigns(many) exercises and (few) problems for students to solve, while our didact-ic proposal is student-centred, where the student must construct knowledgefrom his or her previous ideas.

Having established the objects to be compared, we have to set or selectthe comparison criteria. Although every educational process attempts tocover many aspects of learning (conceptual structures, algorithmic pro-cedures, problem solving, attitudes, etc.), we will divide them into three,which will be taken as comparison criteria:

– The first refers to the students’ performance in calculations inprobability, which is usually the main objective of traditional instructionin probability.

– The second refers to the degree of conceptual change produced,which is an essential objective of our didactic perspective.

– The third refers to the level of attitudinal change obtained.We think that this selection of comparison criteria avoids a bias which is

very often produced when an innovative proposal is evaluated with respect

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to a traditional proposal: the assessment criteria are usually the objectivesof the innovative proposal, so that the latter has an initial advantage inthe comparison. In order to avoid this bias, we have selected an essentialobjective of each of the two teaching styles: performance in calculations inprobability includes the command of conceptual, theoretical and proced-ural structures, as well as problem-solving skills. The level of conceptualchange produced includes the modification, inhibition or substitution ofprevious ideas, and bias in reasoning in probability which operate duringthe process of knowledge acquisition.

With this selection criterion, it would seem prudent to hypothesise thattraditional methodology will be superior to our didactic proposal withregard to performance in calculations in probability (which is its explicitobjective), and it will be inferior with regard to the degree of conceptualchange produced (which is our declared objective). Nevertheless, we ven-ture to suggest that our didactic proposal not only favours a more scientificreasoning in probability, but also improves skills in probability calculationsto a higher degree than the traditional teaching style. Why?

We have checked that many mistakes in calculations in probability madeby students are due to failures in conceptual understanding and to a lackof adequate cognitive and metacognitive learning strategies. Traditionalteaching does not take these issues into account, and attempts to solvethe problem by means of repetition and memorisation of algorithms. Aswe have seen, our methodology is structured in multiple activities withdifferent but complementary objectives, and which incorporates learningtechniques and strategies, allowing enough algorithmic tasks to be per-formed whose meaning is understood by the student.

As well as the need for a conceptual change to be produced in thestudent along the teaching-learning process, an attitudinal change towardsmathematics must also be produced. The low performance observed inmathematics at school, and the generalised attitude of rejection towardsmathematics in students of all educational levels, require that we analysethe features of the teaching process which lead to these results. Gairin(1987) claims that there is an association between performance and attitude,but a causal relationship between those variables is yet to be proven. Aiken(1976) highlights the relevance of the teacher’s attitude and the teachingmethods in students’ attitudes towards mathematics.

One of the essential principles of our didactic proposal consists ofinsisting on the attitudinal contents of the teaching program, and takingcare of the affective issues in its application in the classroom. The modelof learning through conceptual change avoids the ‘mindless imposition’ ofscientific theory and proclaims the progressive assimilation by the student

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of her or his correct intuitions, but also of the limitations of her or hispersonal model, and the need to overcome it.

2.1. Hypotheses of the experiment

(1) The instruction model based on conceptual change will improve thestudents’ performance in elementary calculations in probability in relationto the traditional expositive instruction method.(2) The instruction model based on conceptual change will improve thestudents’ performance in reasoning in probability, diminishing the biasesand mistaken conceptions of students about chance and probability.(3) The conceptual change method will improve students’ attitudes towardsmathematics compared with the traditional method.(4) The level of conceptual change produced with the conceptual changeteaching method will be higher than that produced by the traditional meth-od.

2.2. Method

2.2.1. ParticipantsAlthough we designed teaching-learning materials and activities whichcover the probability curriculum of all the Spanish secondary school age-range (14–18 years), we only did the experiments with students in the firstyear (ages 14–15) (didactic units 0 to 4), so that the sample was establishedin the following way:

EXPERIMENTAL GROUP (Learning through conceptual change group:CCG): 75 students belonging to three first year groups, each from one sec-ondary school in Madrid, taught by three different teachers.

CONTROL GROUP (Traditional teaching method group:TRG): 61 stu-dents belonging to three first year classes, from the same secondary schoolsas the experimental group, and with different teachers.

2.2.2. DesignIt is an experiment with two previously formed groups (CCG and TRG),from each of which measurements were taken before and after instruction.

Independent variables: there is one manipulated inter-subject variable,which is the teaching method, with two levels: Conceptual change andTraditional teaching approach. There are four dependent variables: (1) per-formance in probability calculations, (2) performance in probability reas-oning, (3) attitude towards mathematics, (4) level of conceptual change.

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2.2.3. Procedure(1) In order to control the extraneous variables, we proceeded in the fol-lowing way:The length of the instruction for both groups was 15 sessions lasting onehour for each group; before and after, each group followed the standardmethodology.

The control of subjects’ age is ensured by the fact that students belong tothe same academic year group, the first year in high school; the experiment-al group was composed of students belonging to three day high schoolsin Madrid, from middle class areas, and the control group was composedof students from the same schools as in the experimental group, whichensures the homogeneity of groups with respect to the socio-cultural levelof students.

One of the most important variables to be considered in research of thissort is undoubtedly the profile of the teachers, as it determines, to someextent, the quality and effectiveness of the teaching-learning process. Theeasiest and most convenient thing to do in order to control this variablewould be to have the researcher him or herself impart the instruction toboth groups, but this has at least three serious disadvantages:

– the anomaly that would come into the research by the introductionof a hidden variable: the confirmatory bias, the tendency to prove, ratherthan disprove, the research hypothesis, which would lead the researcherto take special care over the conceptual change model as opposed to thetraditional expository model;

– the unavoidable contamination of one method by the other; it isvery difficult to keep the purity of two methods used simultaneously bythe same teacher; either the standard method or the preferred one wouldtend to impose over the other;

– the impossibility of confirming the generalisability of the instruc-tion model based on conceptual change. The fact that an instruction modelworks in a laboratory environment, with a few students, and under theperfect control of the researcher, does not guarantee its general validity,i.e. its efficiency in normal classrooms with their usual teachers.

A second possibility would be to have the same teachers who follow theconceptual change model in the experimental group, also teaching the con-trol group, following the expository model. But the first two disadvantageswould remain.

Bearing all these arguments in mind, it was decided that different teach-ers would give the lessons with each method. The teachers from the controlgroup were mathematics teachers from the same secondary school as thosein the experimental group, and they showed a disposition to apply the

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same probability teaching process, except for the method followed; in oth-er words, they adapted to the same teaching period, they covered the samesyllabus and they applied exactly the same tests applied in the experimentalgroup.

(2) Pretest and PostestAll the subjects answered two questionnaires we devised for this purpose,ad hoc, in each phase. The first one is the measurement instrument forthe first dependent variable, which we have called performance in prob-abilty calculations; it includes items on concepts in probability, principlesand procedures, and problem solving tasks. The second questionnaire is themeasurement instrument for the second dependent variable, which we havecalled performance in probability reasoning; it includes items which dealwith everyday random phenomena and with deceptive situations in whichthe subjects’ intuitions enter into conflict with the formal laws of chance;these tasks do not demand a quantitative resolution, but a qualitative one.

The items in the probabilistic calculus questionnaire have been selec-ted from the most common mathematics texts in the Spanish classroom,with the criterion of covering the concepts of probability included in thecurriculum of the first year in high school. The items in the qualitative prob-ability reasoning questionnaire have been selected from the abundance ofavailable literature on probability thinking (Fischbein, Nello and Marino,1991; Godino, Batanero and Canizares, 1987; Kapadia and Borovcnik,1991; Nisbett, Krantz, Jepson and Kunda, 1983; Piaget and Inhelder, 1951;Shaughnessy, 1992; Steinbring, 1991; etc).

The selection of each of the items was done after rigorous epistemolo-gical and didactic analysis of the task which underlies each item. Let usreview a couple of tasks to clarify this:

An urn contains two white marbles and two black marbles. A friend randomly extracts twomarbles, one first, and, without putting it back in the urn, he extracts the other. You cannotsee the colour of either.

(a) Imagine that your friend shows you the marble he first extracted; it is white. What isthe probability of the marble extracted afterwards being also white?

(b) Imagine now that, without seeing the marble he extracted the first time, your friendshows you the marble he extracted afterwards; it is white. What is the probability of thefirst marble being also white?

This item describes the random situation know as Falk’s Phenomen-on (1988) which deals with the concept of statistical independence andgenerates one of the most important misconceptions of conditioning prob-ability: it arises when the conditioning event occurs after the event whichit conditions.

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Helen is 37 years old, and she is quite an energetic person. She studied Political Sci-ence, and she was among the first few in her degree. As a student, she participated inthe university’s social movements very actively, specially in the anti-nuclear and anti-discrimination struggle. What do you find more probable: that Helen will work as a bankcashier, or that Helen will work as a bank cashier and will also be an active feministmilitant? Explain your answer.

This item is a classic task of the paradigm of heuristics and biases(Kahneman, Slovic and Tversky, 1982) in which the representativenessbias makes difficult the application of the law of conjunction which seemsto us so simple(P (A \B) � P (A), P (A \B) � P (B)).

In order to measure the third dependent variable, attitude towards math-ematics, before and after educational intervention, we have used Gairin’s(1987) attitude test, which consists of a Likert scale of 22 items.

The fourth dependent variable, level of conceptual change, refers tothe number of previous mistaken conceptions which are positively over-come after the instructional period, and also to the amount of previousconceptions which are correct intuitions and which teaching maintains orreinforces but does not change into incorrect ones. In order to measurethis variable we selected the counter-intuitive and intuitive items of theprobability reasoning and calculation tests. The definitions of intuitive taskand counter-intuitive task are founded on the categorization we made informer research about the intuitive acceptance of the elementary probabil-istic concepts or laws (Saenz, 1995). The concept of intuitive acceptanceis taken from Fischbein, Tirosh and Melamed (1981).

(3) Intervention phase (A) Traditional teaching method group (TRG). Theyreceived a probability theory course lasting 15 hours, at an introductorylevel, with typical methodology and materials for first year high schoolstudents in Spain. Specifically, the books used were those which they werealready using for the other topics in mathematics, and lessons consistedof explanations by the teacher and the solving of exercises from the book(reproductive learning: Ausubel et al., 1978).

(B) Learning through conceptual change group (CCG). They received acourse on the same syllabus, of the same length and with the same dates asthe other group, but following a teaching method which is quasi-empiricaland for conceptual change, with the features described in the design of themethodology section.

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TABLE II

Comparison of the means in the probability reasoning test

Treatment group Pre-test Post-testGain

Traditional group (TRG) 3.61 4.17 0.56

Conceptual change group (CCG) 3.95 6.00 2.05

TABLE III

Comparison of the means in the probability calculation test

Treatment group Pre-test Post-testGain



2.3. Results

We have done an analysis of variance of one factor, taking as dependentvariable, in turn, the scores in the probability reasoning pre-test, the prob-ability calculation pre-test, and initial attitude test, which appear in TablesII, III and IV, with the teaching method as independent variable, and wecan see that there are no significant differences (P > 0:05) between theTRG and the CCG in relation to these variables. This indicates that bothgroups start from the same attitude towards mathematics, and from thesame level of skills and knowledge in probability (All scores in the tablesare on a scale of 0 to 10).

Given the design of this experiment (pre-test-intervention-post-test),we did a variance-covariance analysis: we took as dependent variables, inturn, the intuitive reasoning post-test, the probability calculation post-test,and attitude towards mathematics after intervention; as factors, we took theteaching model; as covariant variables, we took the intuitive pre-test, thecalculation pre-test, and attitude towards mathematics before intervention.

TABLE IV

Comparison of the means in the attitude towards mathematics test

Treatment group Initial attitude Final attitudeDifference



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TABLE V

Percentages of change of responses from the pre-test to the post-test in the item 2

Treatment group Negative change No change Positive change

(-1) (0) (1)



Chi-squared= 6.38, p< 0:05.

We should comment on several results of the covariance analysis in relationto our research hypotheses:

1. Influence of the instruction model on performance in probability andon attitude towards mathematics.

Significant differences exist in the mean scores both of the probabilityreasoning post-test (F1;116 = 46:18, p<0.01) and of the probability cal-culation post-test (F1;116 = 26:30, p<0.01) in favour of the conceptualchange group. There are no significant differences, with regard to teachingmethod, in attitude towards mathematics after the educational intervention(P>0.05).

2. Degree of conceptual change obtained.As we have already said, in order to analyse the conceptual change

produced by the teaching-learning process, we have done two things:(1) Selecting the items which compose the probability reasoning test

and the probability calculation test, and which had been categorised aseither intuitive or counter-intuitive.

(2) Defining, for each of the aforementioned items, a variable (INC)that will take the values 0, 1 or -1, depending on whether there has beeneither: (a) no change in the response from the pre-test to the post-test,(b) whether there has been a change in the response from incorrect tocorrect, or (c) whether there has been a change from correct to incorrect.

In order to compare the levels of conceptual change between the tra-ditional teaching group and the conceptual change group, we used a chi-square test, contrasting the percentages of values of the INC variable foreach selected item (see Table V as an example).

As we can see, this table shows significant differences (�22 = 6:38,

p<0.05) in the percentages of the INC variable between the TRG andthe CCG. With this procedure, we have found significant differences in 7items (out of a total of 12); for all of them, the percentages of 1 valuesare significantly higher in the CCG and the percentages of (�1) values

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are significantly higher in the TRG. Furthermore, two interesting pieces ofinformation can be highlighted:

(1) The percentage of success in counter-intuitive items, which is verylow in the pre-test of both groups, has a significantly higher increase in thepost-tests of the conceptual change group.

(2) The percentage of success in the intuitive items, which is high in thepre-test of both groups, increases in the post-test of the CCG, but is merelymaintained, and even goes down, in the post-test of the traditional teach-ing group. This shows that traditional teaching can sometimes generatesecondary mistaken intuitions about random situations in which primaryintuition, springing from everyday experience, works well.

So that the reader is more clear about the difference in the probabilityreasoning of both groups before and after the instruction, we are going todescribe some typical replies:

The teacher empties a packet with 100 drawing pins on the table. Some drawing pinsfall ’facing upwards’ and others fall ’facing downwards’. The result is: 66 facing upwards,34 facing downwards. Then the teacher asks a student to repeat the experiment. Choosefrom the following list the result you think the student will get, and justify your choice:

(a) 34 upwards, 66 downwards,(b) 63 upwards, 37 downwards.(c) 51 upwards, 49 downwards.(d) 82 upwards, 18 downwards.(e) All the results have the same probability of occurring.

The TRG students gave on average answers (c) and (e) more afterthe instruction than before. The CCG students increased the frequency ofgiving answer b). Are the random tasks used in traditional teaching (fairdice, coins and roulette wheels) encouraging the bias of equi-probability?

A bag contains some white balls and some black ones. A girl takes a ball out, observesits colour and puts it back in the bag; she then shakes the bag. She does this four times,and she always takes out a black ball. Then she takes out another ball. What colour do youthink she will probably get? Justify your answer.

Something similar occurs to what happened in the previous item. TheTRG students tend to say that it is more likely that a white ball is obtained.The CCG students bet more often on the black ball using the availableinformation. Of course, none felt it was necessary to make a contrastinghypothesis.

Which of the following results is more likely?(a) Having 7 or more boys among the 10 first babies born in a hospital.(b) Having 70 or more boys among the 100 first babies born in a hospital.

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The typical answer both of a student of the TRG and the CCG is thatboth results are equally probable with regard to 7=10= 70=100. The ideaof sample size does not appear in spite of the fact that the CCG studentsworked on it at an intuitive level. The tendency to ‘make a calculation’with the available numbers is very strong.

3. CONCLUSIONS

The results obtained allow us to contrast the four research hypotheses, suchthat, globally, it is possible to assert the following:

(1) The instruction model based on a quasi-empirical and conceptualchange perspective significantly improves students’ skills in elementaryprobability calculation, compared to the traditional expository instructionmethod.

(2) The model of instruction through conceptual change significantlyimproves subjects’ intuitive probability reasoning, compared to the tradi-tional instruction model.

(3) The model of instruction through conceptual change does not modi-fy significantly students’ attitudes towards mathematics. This result rejectsthe hypothesis that the model of conceptual change, because of its specialemphasis on the affective issues implied in the teaching-learning process,and because of its experimental perspective, based on proofs and refuta-tions, would promote a more positive attitude towards mathematics thantraditional teaching.

Perhaps the short period of instruction with the conceptual changemethodology is not enough to change beliefs and attitudes towards math-ematics which were originated through the students’ long experience withthe study of this discipline. We have seen that attitude towards mathematicsis a variable that correlates significantly with performance in mathematics,and we believe that a teaching methodology like the one we are proposingcan affect both variables positively. This research shows that the teachingmethod affects performance in a sector of mathematics like probabilitytheory, and we assert that if the conceptual change teaching method is usedfor a long period of time (an academic year, for example), not only theperformance in mathematics, but also attitudes towards it will improve.That, however, will be the subject of further research.

(4) A higher level of conceptual change is produced in the group thatfollows conceptual change methodology than in the traditional teachinggroup, because in most of the items studied (7 out of 12) there are signi-ficant differences in the change produced in both groups, in favour of the

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conceptual change group; there are no significant differences in the otherfive items.

Our didactic proposal was based on three educational imperatives whichwe find essential in the light of the results of this research:

(1) The need to take into account the preconceptions and the beliefsystem of students regarding chance and probability. It is not enough toinstruct according to the logical structure of probability theory, but we alsohave to take into account the psychological structure of the subject whenconfronting random phenomena.

(2) The need to give an empirical focus to the teaching of probabilitytheory. If games, bets and random experiments were of great importance inthe historical development of the concepts and laws of chance, they mustcontinue to have an important role in the statistical education of presentgenerations. Moreover, this empirical focus, which makes sense within thequasi-empirical perspective of mathematics, is the one that provokes thenecessary cognitive conflict in the student to perform the transition fromthe personal model to the scientific model of chance.

(3) The need to give specific training in probability reasoning. It isobvious that mere instruction in probability calculation will not guaranteethe eradication of biases and errors in the students’ reasoning. Thus, wewant to emphasise that the design of this research allows us to analyseseparately calculation skills and reasoning in probability. The interrelationbetween intuitions and mathematics is the proper basis for educationalintervention (Fischbein, 1987; Poincare, 1952).

The key issue in the educational process, and what differentiates anineffective method from one which achieves its objectives is the acquis-ition by students of balanced mathematical thinking, which includes notonly the mastery of algorithms and automatisms, but also the ability to poseproblems and to make use of appropriate and creative personal heuristics.

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Ausubel, D.P., Novak, J.D. and Hanesian, H.: 1978,Educational Psychology. A CognitiveView, Holt, Rinehart and Winston, New York.

Bell, A.: 1987, ‘Diseno de ensenanza diagnostica en matematicas’, in A. Alvarez (ed.),Psicologia y educacion. Realizaciones y tendencias actuales en la investigacion y en lapractica, Visor-MEC, Madrid.

Driver, R.: 1988, ‘Un enfoque constructivista para el desarrollo del curriculo en ciencias’,Ensenanza de las Ciencias, 6(2), 109–120.

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Falk, R.: 1988, ‘Conditional probabilities: Insights and difficulties’. in R. Davison andJ. Swift (Eds.),The Proceedings of the Second International Conference in TeachingStatistics, University of Victoria, Victoria B.C.

Fischbein, E.: 1987,Intuition in Science and Mathematics. An Educational Approach, D.Reidel Publ. Co., Dordrecht, Holland.

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Gairin, J.: 1987,Las actitudes en educacion, PPU, Barcelona.Godino, J.D., Batanero, C. and Canizares, M.J.: 1987,Azar y Probabilidad, Sintesis,

Madrid.Hewson, P. W. and Hewson, M. G.: 1984, ‘The role of conceptual conflict in conceptual

change and the design of science instruction,Instructional Science13, 1–13.Kahneman, D., Slovic, P., and Tversky, A., (eds.): 1982,Judgment Under Uncertainty:

Heuristics and Biases, Cambridge University Press, New York.Kapadia, R. and Borovcnik, M., (eds.): 1991,Chance Encounters: Probability in Education,

Kluwer Academic Publishers, Dordrecht, Holland.Kitcher, P.: 1983,The Nature of Mathematical Knowledge, Oxford University Press, New

York.Lakatos, I.: 1978a,The Methodology of Scientific Research Programmes: Philosophical

Papers. Vol 1, Cambridge University Press, Cambridge.Lakatos, I.: 1978b,Mathematics, Science and Epistemology: Philosophical Papers. Vol. 2,

Cambridge University Press, Cambridge.Nisbett, R., Krantz, D. H., Jepson, C. and Kunda, Z.: 1983, ‘The use of statistical heuristic

in everyday inductive reasoning’,Psychological Review90, 339–363.Piaget, J. and Inhelder, B.: 1951,La genese de l’idee de hasard chez l’enfant, P.U.F., Paris.Poincare, H.: 1952,Science and Method, Dover Publications Inc., New York.Posner, G. J., Strike, K. A., Hewson, P. W. and Gertzog, W. A.: 1982, ‘Accommodation of

a scientific conception: toward a theory of conceptual change’,Science Education66(2),211–227.

Saenz, C.: 1995, Intuicion y matematica en el razonamiento y aprendizaje probabilistico,Tesis doctoral, UAM, Madrid.

Shaughnessy, J. M.: 1992, ‘Research in probability and statistics: reflections and directions’.In D.A. Grouws (Ed.),Handbook of Research on Mathematics Teaching and Learning,McMillan Publishing, New York, pp. 465–494.

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Instituto de Ciencias de la Educacion (ICE),Universidad Autonoma (Cantoblanco),Madrid 28049, Spain

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