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Reasoning on the Balance Beam in a Classroom Setting

Educational psychology is particularly concerned with the changes individuals undergo when they participate in formal educational settings. The studies of reasoning on the balance beam reported in the literature generally were conducted in laboratory rather than school settings. One of our own studies, a design experiment, took place in a mixed sixth and seventh grade, with students 11-12 years, just about the time that developmental psychologists use as a marker for the onset of adolescence (Roth et al. 1999). This setting allowed us to use clinical interviews before and after an intervention that provided many opportunities for students to learn and develop over the course of a 4-month curriculum in which they conceived and built prototypes of machines that would facilitate accomplishing certain tasks (e.g. lifting weights).

Reasoning in Context

A first remarkable phenomenon was noted during the pretest, where students were given balance beam tasks in two conditions. In the first condition, the lever arm was unmarked, that is, there were neither notches nor numbers on it: there was simply a pivoting bar. In the second condition, the beam was sectioned by means of equidistant markers, and unit distances were indicated starting with the first mark next to the fulcrum {1, 2, 3 ...}. The weights were in the form of hardware nuts. The analysis of the clinical interviews yielded responses that varied widely within and between students (Roth 1998). Most surprisingly, not one student used the same solution strategy and reasoning in the two conditions. About 15 % of the students did use the form of reasoning that Inhelder and Piaget had described as emerging during the critical age. Other students used the numbers on the beam as ordinals marking different positions farther and farther away from the pivot point. Yet others used the numbers, but not as a quantity of distance. The numbers were things to be used in arithmetical operations—like adding and subtracting. On the other hand, when there were no distance markers and numbers, one quarter of the students used the thumb and index in a caliper configuration to measure the distance of the weights from the center of the beam (fulcrum). An equal amount eyeballed the distances providing qualitative indications thereof (e.g. “about half’).

We learned in this investigation that the students were not at all geared towards the entities weight and distance that previous research presupposed. Therefore, the things that students were processing were neither qualitatively nor quantitatively the things that scientific perception and reasoning uses to predict how a balance will behave under specific conditions once released. In other words, the students reasoned in and in terms of a different world, a world populated with different entities than those appearing in the world of the investigators. In philosophy and other fields, the ensemble of objects that make a (segment of the) world constitutes an ontology. To distinguish the world (environment) as described by science and the world (environment) as experienced and acted upon by individuals, the term life- world is frequently used. In the following, we characterize lifeworlds in terms of sets of entities that subjects differentiate by means of language or actions; each set constitutes an ontology that goes with a particular lifeworld. Thus, for example, the ontology {A, B} is characterized by two entities. If another entity C is added, the ontology expands to {A, B, C}. This expansion also means that the lifeworld, in the given context that the ontology describes, has become more complex. In this approach, we take into account the suggestion that the relevant environment to be considered in educational and developmental psychology is not the objective environment, not the one that scientists describe and theorize, but the environment in the way that it appears and is salient to the individual, in herperezivanie [experience].[1]

We suggested in the past that the students’ lifeworlds differ from those of researchers, teachers, or science and that therefore the very form of analysis that have been applied to developmental tasks is inappropriate. The preceding results show that not only are the students’ lifeworlds different and populated with different entities, but also these lifeworlds change as a function of the context. This suggests that to understand performance we need to take into account person and experienced environment. In the following we see how one student, Don, during the clinical interview built up the qualitative measure (i.e. variable) “distance to fulcrum” as a relevant attribute to be used in reasoning together with its quantitative measurement. Initially, he did not have experiences with the device so that it is not surprising that the observed behaviors were at Level 3 and Level 4 in terms of cognitive complexity (Fig. 6.1). During the posttest situations, behaviors at the Level 6 and Level 7 (Fig. 6.1) were observed.

Distance, the feature of the balance beam that appears in the literature as the variable students act upon and use in reasoning, in the beginning was not an aspect of the student’s lifeworld and the associated ontology. Initially, Don answered the interviewer’s questions about where to put a weight to balance the lever in terms of locations by using the deictic term “here” and then also included the term “there,” which maps onto Level 2 of our category scheme (Fig. 6.1). In each case, the term coincided with a pointing gesture to a specific aspect of the material configuration of the balance. The use of indexical terms, therefore, left it to the interviewer to find in the material configuration any relevant fact.[2] That is, in the beginning of the clinical interview, the salient entities were specific locations so that his lifeworld may be described by the set {here, there, PoiNT[position]}, where an action is denoted by means of small capitals.

Distance, as a qualitatively new discrimination function, only became reality for Don (and his peers) in the course of the clinical interviews or in the subsequent lessons. Thus, for example, during the interview Don changed the position of the weights along the arms of the beam. He thereby operated on specific aspects, thereby changing what he oriented to and acted upon. His lifeworld now included a new movement, MovE[weight], resulting in the ontological set {here, there, poiNT[position], MOVE[weight]}. This behavior would be characteristic of Level 3 (Fig. 6.1). In response to further interview questions, the replies changed to include the verbal descriptor “from here to there,” thereby expanding the ontological set describing the balance beam-related world to {here, there, PoINT[position], MovE[weight], from here to there}. The two position markers “here” and “there” came to be connected by means of the action MovE[weight] (from one location to the other) to form a more complex description. This description was not merely verbal but included material aspects. The task-related world expanded and became more refined with the generation of a new discrimination function that was the result of connecting already existing features. Importantly in theoretical terms, the new feature arose from the combination of a verbal and a material action. Thinking here literally was handwork, and we observe neither a gap between body and mind nor a possible reduction to mental properties. But the result was not yet the variable “distance”; it was a change of position.

The next addition to the lifeworld was the distance of the weight from the fulcrum, which is characteristic of Level 4 (Fig. 6.1). This new discrimination was observed in the justification of an answer to one of the interview problems.[3] Don suggested basing himself on the comparison “between ‘here’ [position left weight] and ‘here’ [position fulcrum] and then between ‘there’ [right end of beam] and ‘here’ [position of right weight].” Simultaneously, his thumb and index finger formed a caliper configuration while saying “between.” Thus, the lifeworld now included “between here and there,” which resulted from the action M0VE[weight] that linked the two positions “here” and “there” so that the entire set of things evolved to {here, there, poiNT[position], M0VE[weight], from “here to there,” between “here and there,” caliper sPAN[here, there]}. On the balance beam with distance markers and measures, Don also included these to name the positions (e.g., “put it on five”) or to provide a measure for “between here and there.”

In the course of the clinical interview, existing features were integrated to produce new ones. These features were either of the same type (e.g., “here” and “there” or M0VE[weight] and sPAN[here, there]) or of increased complexity (e.g., the construction of “from here to there” from its constituents “here,” “there,” and M0VE[weight]). That is, the lifeworld not only became more differentiated but also came to include new discriminating functions of a higher complexity—not, we insist, as if it were the result of something in his mind, but in this particularly configured setting and the relation with the clinical interviewer (see next subsection). During the interview prior to the unit on simple machines, Don did not talk about distance as a measure, which includes a particular quality and its associated measurement—even though the interviewer can be seen asking him about it. However, later during the 4-month unit, immediately after completing specific teacher- designed tasks with levers, he began verbally referring to this newly emergent quality. During the tasks, he added “close” and “far” from some reference point (e.g., fulcrum, scale that measured effort) resulting in the ontological set {here, there, poiNT[position], M0VE[weight], from here to there, between here and there, caliper sPAN[here, there], close(r), far(ther)}. This lifeworld included the marker for magnitude “between here and there.”

in the end, Don explicitly used the notion of distance in the context of lever problems. His lifeworld relative to the property of distance had expanded to the set {here, there, poiNT[position], M0VE[weight], from here to there, between here and there, spAN[here, there], close(r), far(ther), distance, cHANGE[distance], MEASURE[distance]}. During the posttest Don explained how he found the corresponding weights and distance by saying, “distance is ... ” and moving his index finger from the weight to the fulcrum on the drawing in front of him. The set {sAYs[distance], MEAsuREs[distance], GESTUREs[distance]} was associated with the movement of weights along the beam. Even though he already used the multiplicative rule to balance the beam[4], his explanation included material actions. This situation thereby was analogical to the ones we show in Chap. 2, where petrified movements become things. Here, the movement from one position to another, “MOVE[weight] ‘from here to there’” became the thing “distance[weight, fulcrum].” There is no need, however, to attribute this qualitatively new shift in the reasoning to the individual, as if it was his construction. Instead, the action of moving the hand between two points and the associated talk was for the clinical interviewer and as the second turn in a {query I reply}, which constituted each part of the pair as social rather than individual action (see Chap. 4). The petrified version also is part of a {query I reply}, inherently presupposed to be intelligible and therefore a manifestation of communicative possibility. In this regard, the example also is reminiscent of the case presented in Chap. 3, where a group of students develop new perceptual and conceptual differentiations as they engage with, modify, and talk about a digital animation . Along with those differentiations emerge new possibilities for talking about aspects that no longer needed to index the specific situation. This allows for the attribution of higher levels of cognitive complexity. In the case of Don, the change from reasoning without distance to reasoning with distance was of the qualitative kind, which arose in a context of quantitative expansion of the lifeworld. Those changes could be accounted for in terms of individual construction but took place in and through the {thinking I communicating} dialectic.

Many scholars have accepted the notion that cognition is situated—as had been described for the differences in performance when adults were making best buys in the supermarket, in a simulation with some products arranged on a table, or in paper-and-pencil format (Lave 1988). But very few scholars have drawn the consequences of that notion. One educational psychologist who did integrate person-situation transactions into his theory was R. E. Snow. He called that union of person and situation aptitude; we may also denote the unit as {person I situation} or {person I environment}. Snow suggested that “to understand the effects of person characteristics on performance, one must specify the performance situation” and conversely, “understanding situation effects on performance requires knowing the personal qualities capable of influencing response” (Corno et al. 2002. 216). Transaction means that we cannot undo the {person I situation} unity/identity and reduce it to smaller elements. This is so because understanding one part (person, situation) requires understanding the other (situation, person). In our account of Don engaging with the balance beam, we observe changes in the relation between the person and the experienced environment (lifeworld). That is, Don, like the students in Chap. 3, did not act in a constant world but in a world (i.e. situation) that became more differentiated and expanded, thereby affording more and more complex actions. As a result, the entity marked “Don” is not a constant but continuously changing aspect of the {person I environment} unit. Even though some actions were more complex, up to this point we did not see a change that would qualitatively change the form of reasoning, an example of which would have been if Don had used the schema of proportions that Inhelder and Piaget were observing among their research participants. One important aspect that was left out in the present account, however, is the role of the social relation and how it made possible forms of reasoning that are observed for the first time in a particular individual. That is, what has not yet been shown in our account is how a social relation is at the origin of the observed development. What has been shown so far, on the other hand, suggests the need to consider the relation as a {person | environment} unit for understanding the reasoning practices observed.

  • [1] The category perezivanie was developed during a lecture only a month preceding Vygotsky’sdeath (Vygotsky 1994). It found relatively little attention in the research literature. While writingthis book, we also contributed to a special issue of Mind, Culture and Activity entirely devoted tothis category, particularly focusing on how to use it in empirical studies (Roth and Jornet 2016).
  • [2] This is not unlike the situation observed in Chap. 3 where the students working on the heat pumpproblem refer each other to the display, inviting others to “see.”
  • [3] The effect of such calls for justification is described in Chap. 5.
  • [4] The multiplicative rule in fact calculates the torques on each side, which have to be the same forthe beam to be balanced: w1 • d1 = w2 • d2. Empirical studies show that subjects are more likely arriving at this rather than at Piaget’s (INRC-based) ratio rule.
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