Geometry is the area where the visual representation of shapes and soiids, the conceptual analysis of their intrinsic spatial relationships, and the creative power of deductive reasoning seem completely merged together. This is why geometry is the touchstone for a semiotic approach to mathematics. Let us take the "geometric figures*’ that are like symptoms of this cognitive merger. How to analyse them? To manage an efficient method we have to carry out three necessary dissociations.
Separating the visual aspect from the properties stated
The so-called "geometric figures” are visual representations that are coded with letters or marks indicating given properties. But letters and marks refer to a statement. Also, we always have, be it explicitly or implicitly, a dual repre
sentation.
AC and EJ are parallels. AB and El are parallels. CB and 1J are
parallels.
Prove that E is the middle of CB
Figure //..•! dual representation articulated by anchor marks
In any dual representation, letters and marks anchor the reference points of the statement in the visual representation, but they belong to the statement and not to the visual representation. The visual representation works in a way completely independent of what the given encoded imposes, because for the same statement we have several possible visual variations.
if (I) is given as the starting dual representation then (11) must be recognised as a subfigure of (I) for solving the problem, and (II) (III) can also be given as starting dual figures within one dual representation. Analysing any geometric figure requires that we analyse the visual variations in which it appears as a transitory foreground amongst many others.
Discriminating figural units in any geometric figure
It is the requirement whose complexity is seriously misunderstood because, without this discrimination, any figure, even the most simple, becomes misleading or opaque. This complexity is due to the possibility of quite different ways to split up the content of a geometric figure into figural units. For distinguishing them, we have to take into account the number of dimensions. Thus, we get these two basic ways of splitting up.
The figural units have a number of dimensions equal to that of the starting figure: 2D —» 2D in the framework of plane geometry. In this case, the figural units correspond to the shapes, which are closed outlines. For example, in Figure 12 above, we can discriminate in (I) two figural units (the two parallelograms (!I) or three triangles). This is the spontaneous perceptive interpretation.
The figural units have a number of dimensions less than the one of the starting figure: 2D —» ID (or OD). In this case, the figural units are straight lines free of any closed outlines. For example, in Figure 12, we can discriminate in (1) six Figural units which are the straight lines making up the network (Hi) underlying (I) and (II). This discrimination goes against the immediate perceptive organisation.
It is through this double possibility of splitting up that geometric figures constitute a particular kind of semiotic representation with a powerful potentiality for visual treatment (Duval, 2005a).
Two visual treatments mutually incompatible
The visual treatments depend first on the kind of figural units with which you work.
The treatment by reconfiguration works with figural units with the same number of dimensions as the one of the starting figure. It is this one which is highlighted in most teaching studies, because it provides a non-formal justification for formulas or geometric properties. Its function is mainly heuristic. But it is less obvious and
EIGHT PROBLEMS FOR A SEMIOTIC APPROACH
natural than most studies assume, because it depends on several factors triggering or inhibiting the various reconfigurations (Duval, 1995b).
The treatment by dimensional deconstruction works with figural units with a smaller number of dimensions than in the starting figure. This is the relevant visual treatment from a mathematical point of view. The reason is obvious. The geometric properties being relations, they can be visually represented only by two visual units. For instance, the definitions of polygons, polyhedra, are based on this dimensional deconstruction of shapes and solids. More generally, this amounts to work with the network of straight lines underlying a geometric figure, in which many other different figures can be recognised.
A
[ L
V
Let ABCD be a rectangle. Is the area of the rectangle ACEF greater, equal to or less than the area of ABCD?
Explain your answer.
... '.*4
Treatment by reconfiguration:
overlapping of 2D figural units
Treatment by dimensional deconstruction, focusing on dotted fines as diagonals and not as edge of
2D shapes.
Figure 13, Two visual treatments
This poor example can help us to get at the semiotic and cognitive complexity of geometry. We can condense it into three proposals:
(1) Dimensional deconstruction is a semiotic operation which goes against the spontaneaous perceptive identification (Duval & Godin, 2005). Most students have no inkling of this. No deep progression can be expected as long as geometry teaching misunderstands this.
(2) Each of these two visual treatments turns specifically towards one of the two kinds of discursive justification: either descriplion/explanation or deduction.
(3) The ability to manage these two visual treatments is a prerequisite for analysing a geometric figure according to the given encoded.
VIL DOES THE SEMIOTIC APPROACH LEAD TO DEVELOPING A METHODOLOGY PROPER TO RESEARCH IN MATHEMATICS EDUCATION?
First, the main thing is not to confuse two questions. One concerns the identification of variables that determine the development of comprehension in mathematics. The other concerns the analysis and interpretation of students’
productions that are collected either in classrooms, through individual observation or experiments. The first is too often ignored or reduced to the second, and yet, it is crucial because it involves a model of the functioning of mathematical thinking.
What are the variables that need to he taken into account for organising a learning situation?
The core variables must be determined from the unique epistemological situation of mathematics (above I (Q.3)), the cognitive paradox it creates (III) and the nature of mathematical activity as a change of semiotic representations (II, V). This means that we cannot start first from the particular mathematical contents to be taught, but through them, we must focus on the conditions that will allow' students to understand and use them as expected from a mathematical point of view.
We have seen that there are two fundamental cognitive processes intrinsically linked to the mobilisation of multiple semiotic systems of representation: the conversion and treatment. This means that we must start by separating the variables related to the conversion and those related to the treatment. But to make this separation operational, we must have a precise classification of the various registers of semiotic representation, i.e. systems used not only for communication or objectivation, but also for the purpose of treatment.
The principle of classifying all possible kinds of representation, both the semiotic and non-semiotic, is very simple: there are as many kinds of representations as systems producing representations. If we confine ourselves to semiotic systems, there is the classic distinction between languages and all the various kinds of pictures (drawings, figures, diagrams, etc.). But this distinction must be crossed with another essential for mathematics: those whose the treatments can be made into algorithms and those for which this is not possible when it comes to the native language or geometrical figures (V above). In this way, we get the four classes of registers used in mathematics (Duval, 2006a, Figure 1, p. 110).
Now, it is easy to identify the variables related to conversion. The cognitive distance underlying any conversion depends on the kinds of source and target registers. So, there are as many kinds of conversions to investigate and take into account, as there are various couples of registers.
The variables related to treatment depend on two things specific to each register:
- the means of producing quite new representations (for instance, by constructed designation for the discursive registers)
- the means of changing semiotic representations in a continuous and creative way by composition, substitution, reconfiguration, dimensional deconstruction, according to the possibilities specific to the register chosen
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EIGHT PROBLEMS EOR A SEMIOTIC APPROACH
These two kinds of cognitive variables must be taken into account to organise, as well as experiment, devices as activities for learning in the classroom.
How can we analyse and interpret the students’ productions?
This issue is the main challenge and a flimsy point in the current research in mathematics education. The vast majority of studies are validated by reproductions in extenso or much too lengthy quotations of what very few students have told, written or drawn. We are in fact faced with a mass of raw data, whose interpretation is half clinical, half assessment and, in any case, difficult to rebuild or check. It is like a validation by testimony.
One semiotic approach allows describing at least one analysis procedure. It comprises three stages:
(1) The observations being made in the context of a problem; it is essential to begin by making the representational map of the whole representational working field (Figures 1, 6 and 9 above) in which a search for solving can be managed by students. This does not depend on what the students have done, but on what is given to them or expected of them.
(2) This working field is a tool for breaking each student’s production into segments or interpretable units according to:
- the passages he/she does or does not move to (i.e. the ways he/she finds or does not find) between the different registers of representation
- the register chosen by the student to perform a treatment So, we obtain two kinds of significant observations.
(3) Finally, on this basis, a verifiable comparison can be carried out between the various productions collected. Naturally, this comparison can be correlated with the level of mathematical performances, without being confused with them. And, this comparison can also be extended to productions collected over long periods of lime, in order to observe whether the comprehension evolves in depth or not.
Besides a diagnosis of students’ particular difficulties, this procedure highlights, under the global performance of a group or classroom, the individual disparities of comprehension and preparation for further acquisitions.
Conversion as a method for discriminating meaning
The representation conversion is the most powerful tool that the semiotic approach provides. But this is beyond the scope of this overview. 1 will confine myself to the next question which is a recurring problem for most students and ... teachers! How do we discriminate-or make students discriminate-various kinds of statements that look alike linguistically because they use the same words? For instance, how do we discriminate a proposition from its converse, a word problem from another word problem whose solution is different-as is already the case with additive problems with only one operation-and more generally, a valid reasoning from a non-valid reasoning?
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Teachers have long since noted the need for auxiliary or transitory representations, and the educational innovation possible in this area seems inexhaustible. The controversial issue is whether all supports devised provide the means for truly distinguishing the difference in meaning between two statements that look alike, and determining which statements’ meanings are mathematically relevant. Most innovations often take that which precisely causes trouble for students for granted, namely, converting representations.
In reality, it is not the auxiliary' representations that are important for understanding, but the conversion activity that leads to their production. The auxiliary' representations must therefore meet some semiotic specifications:
- highlighting the discursive operations which underlie statements - providing a non-iconic visualisation of their organisation - being easy to produce and check by the students themselves
Graphs on a semantic bi-dimensional representation seem to be the best candidates and have been shown to be effective with students (Duval 1992, 1995, 2007b).
VIII. DOES THE ANALYSIS OF COMPREHENSION PROCESSES IN MATHEMATICS