Bodily Experiments, Metaphors, Gestures and Artefacts in Grasping the Meaning of a Motion Graph: A Case Study

Bodily Experiments, Metaphors, Gestures and Artefacts in Grasping the Meaning of a Motion Graph: A Case Study
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  BODILY EXPERIMENTS, METAPHORS, GESTURESAND ARTEFACTS IN GRASPING THE MEANINGOF A MOTION GRAPH: A CASE STUDY Francesca FerraraDipartimento di Matematica, Università di Torino, Italia,ferrara@dm.unito.itTERC, Cambridge (MA), USA,  In light of recent findings in neuroscience, psychology, and linguistics, this paper  points out the relevance of the phenomenological viewpoint in analysing a learningsituation by taking into account the contextual ingredients that shape and constrainthe mathematical activity of 14 year-old students. Within this perspective, the focus ison the process through which four students interpret a motion graph. Attention isdrawn to their use of specific words, gestures, and functions of the artefacts in use (asymbolic-graphic calculator and a motion sensor). INTRODUCTION AND THEORETICAL FRAMEWORK Recent neurophysiological research on monkeys points out the non -symboliccharacter of the representational dynamic of the brain in the process of abstraction,and considers it “as a particular type of self-organization, with body action playing amajor role in specifying the informational routines characterizing self-organization.[…] The brain, a brain wired to a body that constantly interacts with the world is, atthe same time, the vehicle of information and  part of its content” (Gallese, 2003,p.1233; emphasis in the srcinal). As a consequence, “representational content, andthus – a fortiori – conceptual content, cannot be fully explained without considering itas the result of the ongoing modelling process of an organism as currently integratedwith the object to be represented, by intending it. This integration process betweenthe representing organism and the represented object is articulated in a multiplefashion, for example, by intending to explore it by moving the eyes, intending to holdit in the focus of attention, by intending to grasp it, and ultimately by thinking aboutit” (Gallese, ibid., p.1236; emphasis in the srcinal).Results of such a kind have to be regarded as significant for Mathematics Educationresearch, since they shed light on the important role of the body as a whole, and of the environment one is acting upon, when one grasps a concept. The influence of these factors can be observed in mathematics classes, where students have to dealwith abstract entities, and where, to make sense of them, they constantly recur todifferent forms of  representation s in the act of thinking about them: bodily actions,gestures, manipulation of materials or artefacts, acts of drawing, and even eyemovements, gaze, tones of voice, and facial expressions (briefly, the so-called  perceptuo-motor activities). In fact, there is evidence that: “while modulated by shiftsof attention, awareness, and emotional states, understanding and thinking areperceptuo-motor activities; furthermore, these activities are bodily distributed acrossdifferent areas of perception and motor action based in part, on how we have learnedand used the subject itself. […] the understanding of a mathematical concept ratherthan having a definitional essence, spans diverse perceptuo-motor activities whichbecome more or less significant depending on the circumstances” (Nemirovsky,2003, p.108). The role of sensory-motor action in mathematical activity has also beentaken into account in cognitive science by the theory of  embodied cognition , whichargues that abstract and formal mathematics is rooted in concrete sensory-motorexperiences via metaphorical thinking (Lakoff & Nùñez, 2000). In particular,according to this embodied approach, mathematics is grounded on cognitivemappings, the conceptual metaphors, that allow two ontologically different domainsbeing thought of as having the same inferential structure. Nevertheless, metaphoricalthinking does not explain everything; in particular, the “reduction of abstractconcepts to more concrete ones through metaphor fails to explain the fundamentalprocesses involved in acts of abstraction” (Schiralli & Sinclair, 2003, p.82). Thesensory-motor experiences “might variously be structured by those commonneurophysiological predispositions human beings genetically possess and  mightvariously be mediated by environmental factors including those developing culturaland symbolic systems into which specific human beings and groups of human beingsare variously and progressively initiated” (Schiralli & Sinclair, ibid., p.90; emphasisin the srcinal).All these claims highlight the significance of the context  in which mathematicalactivities are carried out. Moreover, they accord to the new naturalising paradigmrecently pursued by neurologists, mathematicians and philosophers (for an extendedstudy on the topic, see Petitot et al. (eds.), 1999). As a consequence, they force to awide reflection on the brightness of the  phenomenological point of view in embracingthe cultural, personal, and physical variables shaping the experience of the subjects.Setting the study within this wider and complex perspective and given the wave of the recent studies on gestures (see e.g. Arzarello & Robutti, forthcoming; Kita, 2000;Goldin-Meadow, 2003), I analyse the mathematical activity of 14 year-old studentswho are using a motion graph for the first time and trying to make sense of it 1 . THE CASE STUDY This paper reports on the activity of a group of four 9 th grade students who areengaged with a motion detector (CBR) to perform a back and forth uniform motion infront of it 2 . A red line on the floor marks the point where they have to changedirection. After the experiment, they elaborate on certain questions about the motion  graph and the number table obtained on a symbolic-graphic calculator (TI92 Plus)linked to the sonar (see Figure 1). They use a worksheet we had given them.In the following excerpt, the students are coping with the problem of describing thechanges of position over time (“Describe how space changes with respect to time(increases, decreases, etc.)”), and then of interpreting the shape of the graph(“Analyse the graph. Is it a straight line? Is it a curve? Does the curve increase? Doesthe curve decrease?”).The context in which students are working is rich in variables to be taken intoconsideration: the physical bodily experiment, the use of technological artefacts, thesocial construction of knowledge, and so on. ANALYSIS To answer the question about the changes of position, the students use the numbertable. Early Gabriele conceives of time as an increasing quantity: 155Gabriele: Time is increasing, space is also increasing (…)177Gabriele:  If you look at these seven [with a pen he is scrolling the first seven timevalues on the table] … time is always increasing, all right… space: 42,43, 50 [he is reading some digits of space values] At the beginning the other group mates do not pay attention to the number pattern.They have difficulties in distinguishing space and time values in c1 and c2 columnsin relation to Fabio’s motion: 257Filippo:  But, what is the first? c1? 258Gabriele: c1 is time 259Filippo:  Hmm… 260Gabriele: c2 is space 261Giulia: c1 is time… So, in… hmm 262Fabio: Time… 5 seconds [he is pointing at the corresponding value of time inc1]263Giulia:  Listen, isn’t the third the highest? Figure 1  264Fabio: You need to verify if I obtained… that is, keeping always the samedistance [with the thumb and the medium finger of his left hand opened,he is indicating a segment he is beating on the desk several times] hmm,it always took me the same time 265Giulia:  Hmm Fabio’s gesture (# 264) is relevant. On one hand, it matches his speech, because itexpresses the same information contained in words (see Goldin-Meadow, 2003): tokeep the same distance. Time is beat by the repetitive action of covering always thesame distance (the virtual segment shaped by the opened fingers). The match reflectsthe physical resemblance of the gesture to a space interval; but more information isembedded in its kinaesthesia, preceding the later words in speech. Therefore, ratherthan exhausting its content in the match, in the same time the gesture is nonredundant  (Kita, 2000) in that it hides more representational content  , namely theregularity of uniform motion (that means constant velocity), allowing Fabio to re-enact his run with his hand: always equal changes of space with respect to equalchanges in time 3 . It is an iconic-representational gesture (see Arzarello & Robutti,forthcoming), in which the iconic ingredient comes from the physicality, whereas therepresentational function arises from the kinaesthesia. However, this is not enough tounderstand the changes of position over time in relation to motion. The students thenrecur to a function of the calculator, the scrolling function , activated through the useof a physical instrument, the cursor: 266Fabio:  Look at, if we see, here [with his finger he is pointing to a row of thetable] he made it more or less, more or less, each 1/10 of second theinstrument revealed, it gathered how much space I ran, because there is5 and 49, 5 and 59 [he is going on to scroll the table using the cursor] The scrolling function leads Gabriele to return to his conjecture (about time andspace as increasing quantities, #155), making explicit the covariance (in the sense of Slavit, 1997) of the two variables: 267Gabriele: Yes, anyway, they are both increasing, they should increase in a regular way, but  268Giulia: Can’t we consider the initial value, the medium value and the line? [Fabio is looking at Giulia]269Gabriele: Time is always increasing 270Filippo: Yeah 271Giulia: Ok… and the space doesn’t  272Gabriele: The space increases until 4 meters, then it decreases, but not in aregular way (…)279Gabriele: They both increased until a certain point  280Giulia: Then, time…  281Filippo: Yes, but not in a regular way 282Giulia: Time is always increasing (…)296Fabio: Time is always increasing, but not until a certain point  [he is going on toscroll c1]297Giulia:  In fact… anyway it’s always an observation… space… 298Gabriele:  But they increased together until a certain point! 299Giulia: Then space started to decrease, while time continued to increase 300Gabriele: When you were, you arrived to 4 meters and more, space started todecrease All the students now acquire and share the same language: the verb “to increase” bothfor space and time data. This verb is typically used for a changing quantity: a quantitywith spatial features, whose length or height may increase or decrease. Therefore, forthe students time and space behave similarly, as if they were the two domains of a metaphor  : Time Is Unidimensional Space ; time, as the target domain of the metaphor,is grounded on the structure of space that represents the source domain (in the senseof Lakoff & Nùñez, 2000). The preservation of the inferential structure is shown infigure 2.The metaphor features the discussion up to the end, even when the students interpretthe shape of the graph: 441Gabriele:  Because if you consider a straight line [he is raising his pen in a verticalposition] ... if time is increasing [he is raising his pen further, bothvertically and horizontally]  , if time is increasing and space is increasingtoo [again his pencil up in a slanting position] ... we cannot get a curve [he is drawing a curve in the air] ... for me it is a straight line  442Giulia:  And, if… 443Gabriele:  Because they are both increasing Space Time front future behindpastour placepresent Figure 2
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