HISTORICAL IMAGES, NARRATIVE AND GOALS IN MATHEMATICAL THINKING
Adriano Demattè
Liceo “Rosmini”, Trento
GREMG - Dipartimento di Matematica Università di Genova (Italy)
Within this activity students engaged in interpretation of original historical images. Their protocols
contain stories in order to organize knowledge. Every story has got a goal which completes the
sense of the narration. Reflections regard didactical relevance of both historical images and
narrative. The presence of a goal in students’ narrative is also meaningful in other mathematical
situations and it appears as a symptom of a real understanding.
Dans cette activité les étudiants ont interprété des images historiques originales. Leurs protocoles
contiennent des histoires
finalisées à l’organisation de la connaissance. Chaque histoire a un
but qui complète le sens de la narration. Les réflexions concernent l’importance didactique soit des
images historiques que des narrations. La présence d’un but dans la narration des étudiants est
significative même dans d’autres situations didactiques et apparaît comme le symptôme d’un réel
apprentissage.
INTRODUCTION
At CIEAEM 57 I presented an activity that was implemented in a secondary school: students were
asked to interpret a historical picture that showed a scene in which some people are working, using
instruments or numbers. In the present paper I would like to start from this experience to introduce
some didactical reflections about the use of narration in different didactical activities.
Narrative vs. scientific knowledge: the American psychologist Jerom Bruner (1996) highlights that
narrative and scientific knowledge are the two branches of our culture. At the same time, he
illustrates many possibilities to create a connection between them: narrative is also an interpretation
tool with respect to situations that are traditional objects of scientific inquiry.
The use of original sources in the classroom is “the most ambitious of ways in which history might
be integrated into the teaching of mathematics” (Fauvel and van Maanen (eds.), 2000, p. 291). In
16th and 17th century books many images illustrated mathematical activities in the context of
historical period. Painters enriched the scenes with many details regarding buildings, dresses,
natural environment and so on. These kinds of images enclose both mathematical and historical
value.
In producing a global interpretation the most relevant outcome has been the fact that students have
written stories. A story shows the presence of a goal and is structured with respect to this goal. The
students produced conjectures in order to obtain a probable story, beyond the information that
images showed.
THE IMAGE AND THE GOAL
Elena interpreted the scene illustrated in fig. 1. and wrote:
The scene takes place in an army’s encampment: on the left, by a tent, some soldiers are refreshing themselves
(one of them has a straw covered bottle) while another is cleaning a cannon inside.
Near there, on the right, stands a man who looks like a scientist and seems to be painted two other times in the
picture, always with the same instrument (a kind of telescope, considering the lens in the middle of the rod and
the two small holes at both extremities of the instrument).
By means of that instrument and the help of a compass, the scientist seems to be able to measure the target’s
distance: by means of two different angles he carefully measures the distance of the highest tower in the picture,
which is located in the background (trigonometry).
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The encampment scene in the foreground ends with a river and beyond it, in the background, a city stands out
(objective of the previous measurements).
(Elena, aged 17)
Fig. 1. Johannes Caramuel, Mathesis biceps vetus et nova, Campania, 1670. Source: La matematica su CD-rom. Una
collezione di volumi antichi e rari di matematica e scienze affini. IL GIARDINO DI ARCHIMEDE – Un Museo per la
Matematica, CD #2 [A collection of original documents].
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An analysis of this writing leads us to point out the clear presence of a goal that Elena has
established: the persons in the picture act with respect to this goal. Elena refers to measuring
instruments but she doesn’t use mathematics in order to implement a reasoning. She produces a
detailed description and her awareness about the existence of a goal inside the scene takes a
backseat.
The role of a goal is ‘locally’ relevant with respect to the structure of a story, but is relevant in
broader sense as well. A student constructs narrations in order to organize knowledge. The goal
provides the ‘direction’ in the student’s activity and this fact is relevant from both a cognitive and
an affective point of view. Interpreting historical images can become a paradigmatic situation with
respect to other situations in the didactic of mathematics because it poses questions about key
concepts: narrative, of course, and also the learning situation, goal and metacognition. What
characteristics does a story constructed by students in a mathematical learning situation have? Does
it include aspects about the students’ self consciousness (metacognition)? Starting from other
protocols inherent to algebra I am going to broaden the reflection in order to find possible answers.
Historical images are real objects for students. In their written descriptions they interpret, and
conjecture about, the documents by means of their basic abilities. Students use mostly linguistic
abilities, while they use mathematical knowledge only in its elementary form. Radford (2002)
examines “the students’ formula as a painting” and he observes that
the students’ formula (n+ b)+2 tells a story. A classical story of traditional problems with pattern generalization
that the students have previously encountered in school arithmetic but recounted now in a fantastic way.
This is a didactical suggestion to help student to include formulas in the set of real mathematical
objects which can be interpreted.
A GOAL IN A MATHEMATICAL REASONING
Let’s see how students produce a narration regarding another historical situation and another visual
representation. Students’ protocols show different approaches to the same mathematical content or
procedure. Luca and Sara engaged with Al-Khwarizmi’s quadratic equation “concerning squares
and roots equal to numbers”, interpreting the following figures about the equation x2+10x=39 (fig.
2). One lesson was followed by individual work: students were requested to write an explanation,
hypothetically addressed to their mates who didn’t know the procedure.
One square equals to x2, 10 roots are divided in 4 parts of a rectangle (2.5). The 4 parts are put to the sides of the
square (x2). In order to find the total area of the biggest square, that is made up by the 4 parts, it multiplies 2.5
(one of the 4 rectangle’s parts) by 4, plus the 39 units, that makes 64. 64 is the total area that equals to 82. 8 is the
square’s side. To find the area of the small squares: 2.5 2 = 6.25. To find the square’s area x2 it makes 8(2.5+2.5)=8-5=3
x=3
x2 =32 =9 square’s area.
(Luca, 17)
In order to solve the equation x2+10x=39
x2 is seen as the area of a square, therefore x is a side.
10x is seen as 10·x, that is, he [Al-Khwarizmi, as Sara told me after the written work] hypothesizes it is the area
of a rectangle and 10 is the base. Dividing the rectangle in 4 parts and putting them at the square’s sides you
obtain a cross.
I connect the cross to obtain one square.
I calculate the squares’ area: 2.5·2.5=6.25 and I sum [just so] them by four: 6.25·4=25.
Now I can calculate the area of the whole picture, knowing that the area of the cross is 39 and the area of the
smaller squares is 25.
39+25=64
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With the picture’s area I can calculate the side
l=64=8
Now I can calculate how much is x taking away 2.5 and 2.5 that are the smaller squares’ measures added [just
so] to calculate the area of the larger picture.
x=8-2.5-2.5=
x=8-5=3
Substituting 3 to x in the equation x2+10x=39, it has:
32 + 10·3 = 39
9 + 30 = 39
39 = 39”
(Sara, 17)
10
x2
x
x
2,5
2,5
2,5
2,5
2,5
6,25
6,25
6,25
6,25
2,5
x2
2,5
x
2,5
x
Fig. 2. Al-Khwarizmi (9th century): quadratic equation “concerning squares and roots equal to numbers” (a remake of
the original).
Sara begins her protocol referring to Al-Khwarizmi and using impersonal expressions. After a few
lines she begins using verbs in the first person because, as she said: “I make that experiment!” In
her writing Sara poses herself inside the narration, showing this way her self-confidence.
On the contrary Luca never says “I”. Moreover his protocol contains a gap: “… it multiplies 2,5
(one of the 4 rectangle’s parts) by 4, plus the 39 units, that makes 64…” (reflecting on his error, he
reconstructed the right passage).
The conclusions are also different. Sara checks the resolution and this fact is almost pleonastic, but I
understand it as a way to underline that the goal was achieved. I can interpret Luca’s “x 2=32=9
square’s area” as a way to ‘exorcise’ the square, a mathematical object that shows its ‘unforeseeable
face’, in this context. His narrative is hesitant and not always goal-oriented but worried about
details of Al-Khwarizmi’s reasoning.
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CONCLUSIONS
In spite of the fact that Elena’s protocol was produced in another situation it shows surprising
analogies compared to written works about algebra: the reference to the goal (the people’s purpose
in the painting and the equation’s resolution), the attention to details (the behaviour of some people
present in the scene, but not directly involved in the action, and the Luca’s worry for single
mathematical objects without understanding their role in the procedure). This metaphorical
approach to protocol analysis suggests the unity of knowledge and recalls the perspective which is
illustrated in (Lakoff & Núñez, 2000) about conceptual metaphors and the “embodied mind”.
Nowadays the debate about the didactic of science is also inherent to the importance of narrative as
a basic way to elaborate knowledge. In my opinion this fact shouldn’t mean that you cannot treat
classical topics and you cannot provide students with basic tools. I understand the use of narration
as a possibility to extend access to mathematics to the most part of students. Nowadays it is
desirable that narrative gets again its forgotten role. A hermeneutic approach to the
teaching/learning process passes through the revaluation of interpretation and consequently of
narrative as subjective act that can involve students in mathematical thought.
The use of historical images has posed some questions. We saw that Elena identified “a straw
covered bottle” in the picture, but another student called it “a broach”. This fact recalls the theme of
visual abduction and Peirce’s diagrammatic thinking: I consider that a square gets different roles
(quadrilateral, second power of the root,…). Visual perception, mathematical objects, abduction,
conjectures: is there any connection among them? Different students, what differences do they show
establishing these connections? Can historical images have a metaphorical value with respect to
mathematical objects?
REFERENCES
Bruner, J.: 1996, The Culture of Education, Harvard University Press, Harvard.
Demattè, A.: (in press), ‘Narrazioni per interpretare immagini storiche’, La matematica e la sua didattica.
Fauvel, J. and van Maanen, J.(eds.): 2000, History in mathematics education: the ICMI study, Kluwer Academic
Publisher, Dordrecht.
Furinghetti, F.: 2005, ‘History and mathematics education: a look around the world with particular reference to Italy’,
Mediterranean Journal for Research in Mathematics Education, v. 3, n. 1-2, 1-20.
Lakoff, G. & Núñez, R.E., 2000, Where mathematics comes from. How the embodied mind brings mathematics into
being, Basic Books, New York.
Radford, L.: 2002, ‘The seen, the spoken and the written: a semiotic approach to the problem of objectification of
mathematical knowledge’, For the Learning of Mathematics, 22(2), pp. 14-23.
Radford, L.: (forthcoming), ‘Rescuing Perception: Diagrams in Peirce’s theory of cognitive activity’, in Lafayette de
Moraes and Joao Queiroz (Eds.), C.S. Peirce's Diagrammatic Logic, Catholic University of Sao Paulo, Sao Paulo.
Struik, D.J. (ed.): 1986, A Source Book in Mathematics - 1200-1800, Princeton University Press, Princeton.
Adriano Demattè
dematte.adriano@vivoscuola.it
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