«The current situation concerning the problem studied in the dissertation Introducing the concept of fractions has a common basis the world over. A ...»
Autoreport of the dissertation
of Martha Isabel Fandiño Pinilla
In this paper we dissertation the findings of a principally bibliographical long-term
research project, concerning “fractions”. This is one of the most studied questions in
Mathematics Education, since the learning of fractions is one of the major areas of
failure. Here we present a way of understanding lack of success based on Mathematics
Education studies, rather than on mathematical motivation.
The current situation concerning the problem studied in the dissertation Introducing the concept of fractions has a common basis the world over. A given concrete unit is divided into equal parts and some of these parts are then taken. This intuitive idea of fraction of the unit is clear and easily grasped, as well as being simple to modelize in everyday life. It is, however, theoretically inadequate for subsequent explanation of the different and multiform interpretations given to the idea of fraction.
As we shall see, one single “definition” is not sufficient.
When a child of between 8 and 11 years of age has understood that represents the concrete operation of dividing a certain unit in 4 equal parts, of which 3 are then taken, it would seem that everything is proceeding smoothly. Unfortunately, almost immediately it is clear that the simple construction of that knowledge is blocking the way, it is an obstacle to subsequent real learning. This is knowledge, but inadequate to continue in the construction of further correct knowledge. If, for example, we have a unit divided in 4 equal parts, what does it mean, from this point of view, taking of it?
At times it seems that many teachers are unaware of the conceptual and cognitive complexity involved. I believe that it is necessary to dedicate a whole section to different ways of intending the concept of fraction, that we would like the pupil to acquire.
To give reliability to my work I am obliged to propose an overview of international research in this delicate field, certainly one of the most cultivated the world over. It is impossible to quote the whole of these researches, since its vastness goes beyond our imagination. I will quote only the works which have been directly influential for my subsequent choices, dropping the others. They will be rather a lot anyway. My hope is that this painstaking bibliographical research (I shall propose mainly quotations with regard to the period 1970-1990, and a more detailed bibliography with reference to the period 1990-2000) may be of use also to others who wish to pursue research in the same field. It has not been trivial constructing it.
The objectives of the dissertation
The research conducted for this dissertation aims to show that:
• the mathematical concept of fraction, generally considered as having one unique definition, in fact assumes various meanings in differentclassroom situations, not only from a didactic point of view (concerning the interpretations on the part of students) but also from a scientific point of view; this fact has already been documented in the international research literature and will be investigated in action in concrete teaching situations;
• in the face of widespread failure in the teaching-learning of fractions, attempts to find solutions are all based on conceptual reformulations; the dissertation will rather analyse the teaching approaches highlighted in the research of the last 20 years.
Research methods used for the dissertation The research has been conducted with numerous class a different school levels with
• Primary school
• Secondary school (the majority of the students involved)
• University degree course in Education for Primary School teachers as well as teachers attending in-service training courses
Given the variety of subjects and situations, the research has been conducted through:
• classroom observation
• activities proposed to participants
• interviews and discussions;
as relevent facts emerged they were immediately used for discussion and clinical interviews.
A considerable part of this research is also of a theoretical nature, in that it contains a detailed analysis of the vast literature concerning this field, a study concerning over 200 authors conducted over 6 years. This analysis has allowed me to classify the enormous quantity of research on the basis of criteria of analogy.
Theoretical points of departure The idea of fractions is formally introduced at primary school level, in Italy usually in the third year, even though it is already present in the most immediate sense of “half” an apple or “a third” of a bar of chocolate, or divide a handful of chocolates in 4 equal parts, at a much earlier age.
What schooling does is formalise the written form and institutionalize its meaning.
Roughly speaking, we can say that the universal first approach is that of taking a “concrete object of reference”, considered as unit, which should have the following
• be perceived as pleasant and thus fun,
• clearly unitary and
• already familiar, thereby not requiring further learning.
Normally a round cake or a pizza is chosen in almost all countries the world over; both these objects have the above requirements Situations are then imagined in which this given unit (a cake, a pizza or similar) must be shared between a number of pupils or people in general. In this way the pupils arrive at the idea of a half (dividing by 2), a third (dividing by 3), and so on: the “Egyptian fractions”, which are our first historical example.
For each of these fractions specific written forms are established that for the above cases are and and reading these forms as “a half” and “a third” poses few problems. Nor does generalizing from these examples the written form, which assumes the meaning n of an initial unitary object divided into n equal parts. With young pupils various examples are considered, assigning different appropriate values to n.
If then the guests, for different reasons, have the right to different amounts of the equal parts into which the unitary object had been divided, this gives rise to different written forms such as (two fifths) meaning that two of the five equal parts into which the unitary object had been divided are taken. Several new ideas thus arise and a number of
characteristics of these written forms are then established:
• the number beneath the little horizontal line is called the denominator and this indicates the number of equal parts into which the unit has been divided;
• the number above the little horizontal line is called the numerator and this always indicates the number of parts taken (in this way, the numerator expresses the number of times the fractional part must be taken, and thus a multiplication);
• to give sense to this, the fractional parts of the unit must be equal, a point much stressed and to which we will return later in a critical way.
We shall see how the understanding of these elements, and in particular those marked by italics, end up being an obstacle to the construction of the concept of fraction.
The dissertation analyses over 200 research publications subdivided on the basis of criteria of chronology and analogy in terms of the objectives of the research.
1. From the 1960s to the 1980s
2. From the 1990s to the present day Preparation and conduct of the experiment The preparation of the dissertation consisted principally of an analysis of the vast literature related to the field.
Moreover, in order to guarantee that all interpretations of the idea of fraction,
contrasting and real, were considered, I also conducted:
• classroom observations of teaching practices
• an analysis of the differences between what teachers taught and required of students and the contents of teaching manuals
• clinical interviews with teachers after lessons;
in certain cases I also used in-service teacher training courses to propose:
• reflective dialogues
• personal reflective writing.
This approach has given rise to a new area of research and the already-published article:
Campolucci L., Fandiño Pinilla M.I., Maori D., Sbaragli S. (2006). Cambi di convinzione sulla pratica didattica concernente le frazioni. Una learning story basata su una ricerca-azione di gruppo e sua influenza sulle decisioni relative alla trasposizione didattica delle frazioni. La matematica e la sua didattica. 3, 353-400.
During the experiment it became immediately clear that the teachers themselves needed
to reflect on:
• preceding mathematical aspects
• mathematical aspects after the didactic trasposition
• the effects of their teaching action in order to subsequently consider
• possible solutions offered by current research literature.
Conclusion of the results of the experiments (a posteriori analysis) The experiment based on the classroom observation of teachers has led to a number of
various considerations briefly illustrated in the following points:
Different ways of understanding the concept of fraction Something which often strikes teachers on training courses is how an apparently intuitive definition of fraction can give rise to at least a dozen different interpretations of the term.
1) A fraction as part of a one-whole, at times continuous (cake, pizza, the surface of a figure) and at times discrete (a set of balls or people). This unit is divided into “equal” parts, an adjective often not well defined in school, with often embarrassing results such
as the following, concerning continuous situations:
or discrete ones: how to calculate of 12 people.
Providing students with concrete models and then requesting
reasoning, independently of the proposed model, is a clear indicator of a lack of didactic awareness on the part of the teacher and a sure recipe for failure.
2) At times a fraction is a quotient, a division not carried out, such as, which should b be interpreted as a:b; in this case the most intuitive interpretation is not that of part/whole, but that we have a objects and we divide them in b parts.
3) At times a fraction indicates a ratio, an interpretation which corresponds neither to part/whole nor to division, but is rather a relationship between sizes.
4) At times a fraction is an operator.
5) A fraction is an important part of work on probability, but it no longer corresponds to its original definition, at least in its ingenuous form.
6) In scores fractions have a quite different explanation and seem to follow a different arithmetic.
7) Sooner or later a fraction must be transformed into a rational number, a passage which is by no means without problems.
8) Later on a fraction must be positioned on a directed straight line, leading to a complete loss of its original sense
9) A fraction is often used as a measure, especially in its expression as a decimal number.
10) At times a fraction expresses a quantity of choice in a set, thereby acquiring a different meaning as an indicator of approximation.
11) It is often forgotten that a percentage is a fraction, again with particular characteristics.
12) In everyday language there are many uses of fractions, not necessarily made explicit, e.g. for telling the time (“A quarter to ten”) or describing a slope (a 10% rise”), often far from a scholastic idea of fractions.
In this respect the studies of Vergnaud are illuminating. I am personally convinced that conceptual learning is the first stage of mathematical learning. So many different meanings for the concept of “fraction” require an attempt to find some unifying principle. Following Vergnaud, we can consider a concept C as three sets C = (S, I, S)
• S is the set of situations that give sense to the concept (the referent);
• I is the set of the invariants on which is based the operativity of the schemata (the signified);
• S is the set of linguistic and non-linguistic forms that permit symbolic representation of the concept, its procedures, the situations and treatment procedures (the signifier).
Thus it is evident that the choice of a single meaning of fraction cannot conceptualize the fraction in its multiple features.
As we have seen:
• Behind the same term “fraction” are hidden may different situations which give sense to the concept
• Each of these situations contains invariants on which are based the operativity of the schemata,
• Various linguistic forma can be used to represent the concept.
Thus it is necessary to conceptualize the fraction via all of these meanings and not just through one or two of them, a scholastic choice that would lead to failure.
Vergnaud proposes also a theory of conceptual fields: “a set of situations, concepts and symbolic representations (signifiers) closely interdependent which cannot be analyzed separately” … “a set of problems and situations the handling of which requires different concepts, procedures and representations which are strictly interconnected”.
On the one hand, it is impossible to imagine an approach to teaching fractions in isolation from the mathematical context which gives them sense: fractions, ratios, proportions, multiplications, rational numbers, are but a few of the emerging features from all of that which gives sense to fractions. These concepts must not be separated, but rather should flow together in one sole learning process. On the other, all the different acceptations of fractions must be explored and put in relationship between one another, since there are considerable differences between some of them.