Ferdinando Arzarello Dipt. di Matematica - unipa.itmath.unipa.it/~grim/dott_HD_MphCh/arzarello_febbraio_08.pdf · processi di apprendimento ... Ferdinando Arzarello Dipt. di Matematica

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Osservazione dei processi di apprendimento

nella ricerca in didattica della matematica

Ferdinando Ferdinando ArzarelloArzarelloDiptDipt. di Matematica . di Matematica UniversitUniversitàà di Torinodi Torino

Ferdinando ArzarelloMateriali Corso Dottorato “Storia e Didattica delle

Matematiche, della Fisica e della Chimica”, Febbraio

2008, Palermo

My talk addresses some recent results of the Turin research teamthat I coordinate and is dedicated to the memory of Carlos Garcia.

The group is made by the colleagues Luciana Bazzini and Ornella Robutti, by some doctoral and post-doc students, like Francesca Ferrara and Cristina Sabena, and by many teachers (from the elementary to the higher schoollevel) that participate actively to our researches: Riccardo Barbero, Emilia Bulgarelli, Cristiano Dané, Silvia Ghirardi, Marina Gilardi, Patrizia Laiolo, Donatella Merlo, Domingo Paola, KettySavioli, Bruna Villa and others.

Overview0. The basic issue in math edu

1. The What? Problem & the multimodalityparadigm

2. The Why? Problem & the APC space

3. The How? problem & the Semiotic Bundle

4. The Hence… problem & the role of the teacher

5. Conclusion

“Indeed, didactics itself is concerned with processes. Most educational research, however, and almost all of it that is based on or related to empirical evidence, focuses on states (or time sequences of states when education is to be viewed as development)….”

Hans Freudenthal:

“...the use of and the emphasis on processes is a didactic principle.”

“... States are products of previous processes. As a matter of fact, products of learning are more easily accessible to observation and analysis than are learning processes which, on the one hand, explains why researchers prefer to deal with states (or sequences of states), and on the other hand why much of this educational research is didactically pointless.”

(H.Freudenthal, Revisiting Mathematics Education: China lectures, 1991 p. 87, emphasis in the original)

A big issue: how to describe-interpret the learningprocesses that happen in the mathematics classroom?

Enhance

knowledgepractice

Our issue puts forward four types of intertwined problems for the researcher:

Researcher

Research Design

Research Paradigm Theoret. Perspective

Research Questions Research Strategy

Research Methods

Res. Strategy

Res. Paradigm

1.What?

2.Why?

3.How?

4.Hence…

Theor. perspective

Res. Design

Res. Methods

Res. Questions

If one looks to the phenomenology of learning processes in the class of mathematics, one sees a variety of actions and productions activated by the students and by the teacher using simultaneously different resources:• words (orally or in written form); • extra-linguistic modes of expression (gestures, glances, …); • different types of inscriptions (drawings, sketches, graphs, ...); • different instruments (from the pencil to the most sophisticated ICT devices),

Such resources are (also) used as communicationtools.

The language, which exploits the character of the sensory-motor system, is inherently multimodal in the sense that it uses many different modalities linked together: sight, hearing, touch, motor actions, and so on.

The stance of multimodality implies that “the understanding of a mathematical concept rather than having a definitional essence, spans diverse perceptuomotoractivities, which become more or less active depending of the context.”(Nemirovsky, 2003).

QuickTime™ and aSorenson Video decompressorare needed to see this picture.

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A major consequence of this approach is that the construction of mathematical concepts must be re-elaborated according to the new paradigm.

Hence the issue of the multimodality in learningprocesses is not only an answer to the What? problembut has important consequences also for the Why?How? and Hence…problems.

Namely it has theoretical and pragmatic consequenceson the way we:

- conceive the mathematical objects to teach (Why?)

- plan our researches and collect our data (How?)

- design the teaching sessions in the classrooms (Hence…)

The cognitive space of action, production and communication

It consists of an environment for cognition, developed and shared in the classroomamong students (and possibly by the teacher) while working together.

It is constituted by different components and by a variety of relationships among them.

The APC-space frames the learningprocesses in the mathematicsclassroom within a Vygotskyanperspective.

It is an integrated dynamic system, in which cognitive processes develop through social interaction.

The components of the APC space are:

the body,

the physical world,

the cultural & institutional environment,

…

That is, the students themselves along with the context where they are acting and learning.

When students learn mathematics these and other components (e.g. the emotional ones) take an active part in the learningprocesses, interacting together.

The interaction comes from the students’work, the teacher’s mediation and possiblythe use of artefacts.

The three letters A, P, C illustrate the maindynamic relationships among its components, namely students’ and teacher’s :

Actions (physical or mental);

Productions, e.g. answering a question, posing other questions, making a conjecture, introducing a sign to represent a situation;

Interactions (Communication), e.g. when the discovered solution is communicated to a mate or to the teacher orally or in written form, using suitable representations.

The APC space is not immediately active, neither it exists only because the componentsare present in the classroom. As such, itcomes from the blending action of the teacher, who coaches their integration and triggers the activation of their relationships.

The APC space is a typical complex system, which cannot be described in a linearmanner as resulting by the simplesuperposition of its ingredients (i.e. the components and their mutual relationships).

The APC-space frame allows properlystudying the perceptuo-motor multimodalway of learning (from What? to Why?).



The experiment

• A written story was given to the pupils (5th grade) from the legend of Penelope’s cloth in Homer’s Odyssey (research within the Comenius Project DIAL-Connect: Barbero et al., in press).

• We modified the original text to get a problem-solving situation that allows facing some conceptual nodes of mathematics learning (decimal numbers; space-time variables).

On the island of Ithaca, Penelope had been waiting twenty years for the return from the war of her husband Ulixes. However, in Ithaca a lot of men wanted to take the place of Ulixes and marry Penelope. One day the goddess Athena told Penelope that Ulixes was returning and his ship would have employed 50 days to arrive to Ithaca.

Penelope immediately summoned the suitors and told them: “I have decided: I will choose my bridegroom among you and the wedding will be celebrated when I have finished weaving a new piece of cloth for the nuptial bed. I will begin today and I promise to weave every two days; when I have finished, the cloth will be my dowry”. The suitors accepted.

The cloth had to be 15 spans in length. Penelope immediately began to work, but one day she wove a span of cloth, while the following day, in secret, she undid half of it…Will Penelope choose another bridegroom? Why?

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ORNELLA

SIMONA

MARIA

DAVIDEEDOARDO

QUESTIONS

A

(Videoclip PENELOPE)

1. Use the APC space frame to interpret some parts of the 2 videoclips.

2. Compare Clip 1 and Clip 2: what do youobserve? How the APC space (or anotherframework you like better) can be useful toframe your argument?

10 minutes

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To give sense to the story, the students focus on the action of weaving and unravelling a span of cloth, which is represented by a basic initial gesture: two hands displaced parallel on the desk.



EDOARDO


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ORNELLA

1. ACTION (the basic gesture)



SHARED







The gesture becomesa notch

2. PRODUCTION (of written signs)

The drawing is also imitated and reproduced by the others: even these signs contribute to the growth of a shared space.




Numbers and words are added to the drawings, and fingers are used to compute.


COMMUNICATION





Gesture, speech, written signs and arithmeticrepresentations grow together in an integrated way.

They generate a richer and richer shared environment, which supports the students’ communication processes.

1. The question “how many days for a span?”2. The answer “4 days for a span” (local rule)

becomes the basis of an iterative process(“always -4”) towards the global rule.

3. The computation using a table, and findingthat 60 days are needed for 15 spans of cloth.

APC space (building the global rule)


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APC space can be investigated through differentlenses: semiotic, didactic, psychological,…

In the example of Penelope I have stressed the semiotic aspects, commenting the relationshipsbetween students’ gestures and inscriptions.

This is a general fact:

analysing the story of the different semioticresources used by the students allows us toenter into their processes of solution and tofocus the richness and complexity of the multimodal shared cognitive environment, where their resources develop (APC space).

But to do that a broader notion of semioticsystem is necessary.

In fact the classical semiotic approaches place strong limitations upon the structure of the semiotic systems they consider.

They generally result too narrow for interpreting the complexity of didactical phenomena in the classroom.

As a consequence, the classical semioticlens is blind with respect to a lot of semiotic resources that are active in the classroom.

It pictures the learning processes onlypartially.

Also the resulting frameworks are seriouslylimited.

The basis of the semiotic analysis of the APC-space is a theoretical structure that I have called Semiotic Bundle.

It enlarges the classical notion of semioticsystem within a Vygotskyan perspective.

We have already used it in describingPenelope’s story as a story of “signs”.

It reveals particularly suitable for grasping allthe learning processes and for setting themwithin the dynamics of the APC frame, according to the multimodal paradigm.

Now let us enter into it through anotherexample.

The Semiotic Bundle

QUESTIONS

B (Videoclip of CIRO)

1. List the semiotic resources that are presentin the Videoclip.

2. Stress the main relationships that you seeamong them.

3. Do you find analogies with the Clip of Penelope?

15 minutes

CIRO V-CLIP 1



PART 1 (after 13’)1. Ciro: Practically one could say to explain that this straight line…2. Teacher: Uh3. C: That is it must join, ok, the X axis the interval...is [always?] the same4. Teach.: The X interval is the same; delta-X [∆x] is fixed5. C: Delta…eh, indeed, however there are some points where… to explain it

one can say that this straight line must join two points on the Y axis, which are farther each other, hence it is steeper towards...

6. Teach.: Yes7. C: Let us say towards this side. When, here, …when …however it must

join two points, which are farther, hence there is less distance8. Teach.: More distance?9. C: Less far [he corrects what said in #7] 10. Teach.: Eh11. C: On the Y axis I am saying12. Teach.: Yes13. C: It slants softly from this side; in fact here is the point, let us say...we

may call it zero, since this is a real line like so.

A

B

That is it must join, ok, the X axis the interval...is [always?] the same

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THE BASIC SIGN (A)

The inscriptions are extracted from the screen and embodied in Ciro’sgesture. The evolution of the gesture from Fig. 1 to Fig. 4 illustrates a concept not present in the words: the line is joining two points whose x-coordinates are very close, but it is not the same for the corresponding y-coordinates. Ciro’s words refer only to the (quasi) tangent line and express the fact that the interval is always the same. For this reason, we call the gesture in Fig. 1 the basic sign: in fact it triggers a semiotic genesis of signs, which will span over all the other episodes of this activity and beyond.

1 2 3 4

The semiotic bundle is made of: • the inscriptions on the screen (the two dynamic

graphs and the moving tangent line); • Ciro’s gestures and words;• the teacher’s words.

The synchronic analysis of the bundle points out some different ingredients introduced by Ciro to explain how the tangent is varying: namely the fact that for equal and small ∆x’s the corresponding ∆y is different (e.g. bigger and bigger when the slope increases). Gesture and speech convey different information (it is a case of non-redundant gesture): this information will be explicated verbally in the following Episode B.













THE COVARIATION BETWEEN ∆x and ∆y (B)



1 2 3

4 5 6 7

In this episode Ciro explains his previous insights: the semiotic bundle is enriched with more visible relationships among its different components. We need both synchronic and diachronic analysis to grasp what is happening: the meaning of Ciro’s speech and gestures completely appears if one compares them to his previous speech and gestures.Moreover, as the photos clearly show, the inscriptions on the screen are always present in the semiotic bundle activated by Ciro.

Ciro’s attention is concentrated on the relationshipsbetween the ∆x and the corresponding ∆y variations.

Gesture and speech convey the same information: they express the covariation between ∆x and ∆y, underlining the case when bigger variations of ∆ycorrespond to small values of ∆x.

This description involves gestures and words, simultaneously used in the activity.

Ciro’s actions have a multimodal feature: the student isspeaking and simultaneously gesturing conveying the same meaning in two ways. But gestures (and alsowords) run an evolution from Episode A to Episode B, which can be described by a diachronic analysis.

The gestures in Episode A have a genetic role withrespect to the words in Episode B: the latterexplicates the former within a different semioticregister.

Moreover, the first gestures in Episode B, which are very similar to the previous ones, now are redundantwith respect to the speech. Using a Vygotskian frame, the external sign (represented by the inscriptions on the screen) has triggered an interiorisation process.

(Smthg similar but not identical in Penelope story)

speech

gestures inscription-1speech

gestures

inscription-2

inscription-1speech

gestures

inscription-2

inscription-1

tSemiotic bundle

Some relationships between the resources areshown by a synchronic analysis.

speech

gestures



Some others are shown by a diachronicanalysis in the short or in the long period.

Short period:

There is a geneticprocess of gestures fromthe inscriptions on the screen.

Through it the studentsappropriate the meaningof what they are seeing.



COMPARING DIFFERENCES:∆y w.r.t. ∆xLong period:

From the past the culture of the classroom (finite differences) appears.It is expressed through words and gestures

QUESTION C

(Videoclip CIRO 3)

How can you frame the role of the teacher in this Clip? (use the frameyou like better)

10 minutes

CIRO V-CLIP 3



The interventions of the teacher can be analysedfrom a semiotic point of view.

However the semiotic lens cannot explain all that ishappening.

It is necessary the APC space frame.

PART 3 (after 1h 12’)18. Teach.: Hence let us say, in this moment if I understood

properly, with a fixed delta-X, which is a constant,…19. C: Yes20. S: Yes21. Teach.: It… is joining some points with delta-Y, which are

near22. [overlapped with #21] C: In fact, while they [the points on the

graph] are approaching each other23. C:…they [their ordinates] are less and less far. In fact, I do

not know how to say it,…the slope is going towards zero degrees. 24. Teach.: Uh, uh25. C: Let us say so26. Simone: Ok, at a certain point here delta-Y over delta-X

reaches…27. C: The points are less and less far 28. Teach.: Sure29. S:…a point, which is zero.

C

D

4

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18.Teach.: Hence let us say, in thismoment if I understood

properly, with a fixed delta-X, which is a constant,…19.C: Yes 20.S: Yes21.Teach.: It… is joining some points with delta-Y, which are near

Simone

1 2 3

Ciro

The teacher uses:

1. the gesture registerto be in tune with hisstudents: he “echoes”their gestures;

2. the verbal register toexpress the rigorousmathematical meaning, different from that usedby the students.

1 2

4

3

18.Teach.: Hence let us say, in thismoment if I understood properly, with a fixed delta-X,which is a constant,…19.C: Yes 20.S: Yes21.Teach.: It… is joining somepoints with delta-Y, which are near

C

Through the semiotic game the teacher pushes the personal senses that the students have attached to the initial sign(on the screen) towards the shared mathematical meaning.

S EM I O T I C GAME







The episode shows that an important ingredient of APC space, communication, is particularly active in thismoment.

We have entered into it through the semiotic lenses, butthe real sense of what is happening is given consideringthe interaction between the teacher and the students and the way the teacher is coaching it, within his didacticstrategy and design.

The semiotic game is revealed through the semiotic lensbut it is a phenomenon that belongs to didactics. It isnecessary to frame it into the APC space to grasp itsnature.

The semiotic lens is similar to the use of an economic lens in order to interpret some historical event: it is very useful but it may not explain all the complexity of the historical phenomenon.

The effects of the semiotic game

S. uses the words previously introduced by the teacher (# 18, 21) and converts what Ciro was expressing in a multimodalway through gestures and (metaphoric) speech into a freshsemiotic system. His words in fact are an oral form of the symbolic language of mathematics: the semiotic system is notany longer the everyday language but already the language of Calculus.



Ok, at a certain point here delta-Y over delta-X reaches…

C is looking carefully to what S isdoing and saying, comments “The points are less and less far” and repeats again a gesture, which isa small variation of the basic one. Namely he intertwines the representations within his semioticbundle with those used by S.

Hence Ciro and Simone share a complex semiotic bundle (possibly a semiotic node has been created: see Radford, 2003). This allows them to grasp a complex concept and is the root of the future concept of the slope of the tangent as the limit of the incremental ratio of the function.

An evidence of this fact comes from what happens in the classroom a couple of weeks later, when this notion

is afforded in a more formal way.







hence it issteeper towards



It must join two points,…that is

there is less...less distance



since this is indeed a straight line like so



SEM IOT IC MEDIATION



speech

gestures

inscription-2

inscription-1

APC space

The timeline for describing the learning processes

The issue of complementarity (1)Didactical phenomena can be analysed according to different time-and space-scales, with different methods (e.g. qualit./quantit.), a. s. o.

My analysis sticks at the small scale level (but diachronic analysis may enlarge the time-scale).

To grasp properly didactical phenomena it is not possible to restrain to a single-grain analysis: all the scales are necessary.

Of course it is a difficult problem to coordinate the fine grain analysis of short-term processes with the analysis of long term processes.

An ecological and reasonable attitude is to assume a complementary approach (Steiner, 1985; Arzarello & Bartolini Bussi, 1998).

QUESTIONS D

Taking into account your answer to the previous questions, your own research and my last slide, please read the following quotation and answer the question below:“…”What about the above juxtaposition of methods? Are other solutions possible?

10 minutes

An analogy may be useful to focus how I see the relationship between my analysis and other frames.

In historiography the research methods of Les Annales have put forward a new method of studying the historical phenomena.

The idea is to consider also small scale phenomena (local history). It is at this scale that the life of people really happens.

Hence one can read in them the large scale (classical) history of events, provided (s)he uses also tools like sociology, demography etc. to interpret her/his data.

My framework is like the local history with respect to the largescale history.

An analogy

The semiotic lens corresponds to the use of documents in the historical research. In fact, it puts forward observed facts, that are unquestionable as such.

The point is that my interpretation of the semiotic fact (Semiotic Bundle in APC space) is small-scale based and attentive to some “documents” that are not usually considered in different frames.

An analogy

I think that the ATD-frame of Yves Chevallard is a good candidate for this complementary analysis of the same “ostensifes”, described in my APC-frame.

The issue of complementarity (2)The APC space is suitable for focussing the semiotic mediation coached by the teacher (e.g. the semiotic game): the students act and product as a re-action both to the teachers actions and to the semiotic resources that the didactical situation offers to them.

The milieu (of Brousseau) can be described more profitably as an inter-subjective space (the terminology is from Vygotsky), where the internalisation processes happen because of the actions/productions of students.

Are we satisfied of the above juxtaposition of methods? Are other solutions possible?

One might say that it is untimely to look for a coordination of methods of analysis, because of the insufficient development of theoretical reflection.

Maybe the search for methodological purity has to be given up for some time, at least in innovative research paradigms.

Yet we feel the pressure of the development of trends of research that overcome the distinction between theoretical and pragmatic relevance (Sierpinska 1993, Bishop 1998) producing results with a sound theoretical basis and with a deep impact on the practice of teaching in society.



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Ferdinando Arzarello Dipt. di Matematica - unipa.itmath.unipa.it/~grim/dott_HD_MphCh/arzarello_febbraio_08.pdf · processi di apprendimento ... Ferdinando Arzarello Dipt. di Matematica