kapsch alpha beta gamma

Colleen Kapsch ED 130 December 16, 2013

Alpha, Beta, and Gamma Compensations: Implications for Teaching I. Introduction In an interview with Jean-Claude Bringuier (1980), Jean Piaget says of children, “Oh no! They’re alive, it’s wonderful! They’re new! Oh no; they’re remarkable!” (p. 46). It is with this sense of Piagetian wonderment that I approach my study of alpha, beta, and gamma compensations. What follows is an analysis of these compensations and their connections to a set of conservation tasks. After studying examples of beta- and gamma-behavior, I consider the implications such an analysis has for teaching. II. Alpha, Beta, and Gamma Compensations Defined Before defining alpha, beta, and gamma compensations, it is imperative to explain what causes these compensations. According to Piaget, a subject who interacts with his environment may encounter resistance. When such resistance arises, the subject’s reaction to resistance is called a regulation (Boom, 2009, p. 139). Boom (2009) notes in his article on equilibration, “Disturbances do not result from discrepancy with some absolutely given external reality but derive for discrepancy between what is observable (e.g, as indicated by changes to the object) and knowledge and expectations derived from the actions of the subject on the other hand” (p. 141) . In more simple terms, a perturbation is a discrepancy between a subject’s expectations and his observations. According to Boom (2009), successful regulations are seen by Piaget as compensations, and these compensations have a larger impact on the cognitive structure involved. Boom notes, “From the point of view of the cognitive structure, [resistance can] be described as a perturbation 1


or disturbance of that structure” (p. 139). There are three successive levels of compensation described by Piaget: alpha-behavior, beta-behavior, and gamma-behavior. Alpha-behavior is characterized by the attempt to cancel perturbations. Piaget uses the example of a child rolling a ball to illustrate alpha-behavior: When the child rolls ball A in the direction of the center of ball B, he knows that B will move away along the prolongation of the trajectory AB. If we now ask him to aim at a side of B, he believes the effect will be the same, and when he sees B take off from the side opposite B’, he interprets this fact as merely an annoying perturbation; he begins again, for example, rolling A with greater force at B’, as if this would cancel the perturbation (Piaget, 1995, p. 807). Essential to understanding alpha-behavior is the notion that the subject sees perturbations as “annoying” and attempts to cancel them rather than take them into account. Unlike in alpha-behavior, in beta-behavior the subject attempts to reconcile the perturbation with his previously accepted notions and predictions (Piaget, 1995, p. 807). Indeed, Piaget notes that it is the “integration of the perturbation that characterizes beta-behavior (Piaget, 1995, p. 807). In more simplistic terms, the subject who exhibits beta-behavior experiences disequilibrium as a result of the perturbation and ultimately “integrates the perturbing element that has sprung up in the system” (Boom, 2009, 141). In so doing, Boom (2009) writes that in beta-behavior new connections are formed and the perturbation becomes a variation in a reorganized structure (p. 142). Gamma-behavior is characterized by the subject’s ability to anticipate possible variations that could otherwise be perturbations (Boom, 2009, 142). Piaget writes, “In gamma-behavior, finally, the subject no longer considers rolling the ball at the point B’ as a perturbation but as on of the possible variations intrinsic to the system” (Piaget, 1995, p. 807). What distinguishes

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gamma-behavior from alpha-behavior and beta-behavior that, in gamma-behavior, the subject has accounted for possible variations such that they are no longer perturbations but instead essential parts of the structure. Boom (2009) notes of gamma-behavior, “If every possible transformation is full compensated by an inverse or converse transformation, and every possible affirmation by a corresponding negations, then variations are no longer perturbations” (p. 142). When a subject exhibits gamma behavior, then, he can anticipate the result of an action without having to carry it out. III. Our Interview Tasks: Conservation To better understand how alpha-, beta-, and gamma-behavior help make sense of our interviews, it is important to give a brief overview of our tasks. In the interviews, three second graders were presented with conservation tasks. All of the students worked on four tasks over two interviews. Interview 1: In the first interview, subjects worked on conservation of number and conservation of area tasks. I will provide more detail about the tasks from the first interview, as my examples are taken from them. Task 1: Conservation of Number: In the conservation of number task, which was given first, subjects were asked to compare two rows of candy, one that belonged to the interviewer and a second that belonged to the interviewer’s brother. We initially showed the subject the “candy” with the pieces touching end-to-end, like this:

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With the candy in this formation, we asked the subject whether the interviewer and her brother both had the same amount of candy. The candy was then rearranged such that one row touched end-to-end and the other was spread apart. The candies might have looked like this:

Subjects were asked what they thought of this new arrangement. Did the interviewer or her brother have more candy? Did they both have the same amount? Interviewers asked a variety of follow-up questions depending on the subjects’ response to the above arrangements. Task 2: Conservation of Area: In the conservation of area task, subjects considered the arrangement of a “horse pasture” where the grass was arranged in a variety of ways. In the first question, subjects were asked if two horses would have the same amount of grass to eat if their pastures looked identical. The pastures looked like this:

As the task progressed, subjects were asked to consider different arrangements of the grass in the pasture and determine if the horses still had the same amount to eat. Some possible arrangements subjects examined included:

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Interviewers asked a variety of follow-up questions depending on subjects’ reactions to the different arrangements. Possible follow-up questions included: • Do the pastures have to look the same for the horses to have the same amount of grass to eat? • Can you think of any arrangements using 12 cubes when the two horses wouldn’t have the same amount of grass to eat? Interview 2: In the second interview, subjects worked on conservation of liquid and solid volume tasks. Task 1: Conservation of Liquid Volume: In this task, subjects considered the amount of liquid in cups of different shapes and sizes. Initially, subjects were presented with two identical cups with the same amount of “juice” (water) in each. The interviewer then explained that she was responsible for bringing cups to a party. If the subject agreed that the two cups had the same amount of water, the interviewer poured the content of one cup into a third cup, explaining that she only had two identical cups. Would it be okay to bring a third cup that looked different from the first two to the party? Would the person who received the third cup have the same amount of juice as the people who got the first two cups? Why or why not? Depending on the subjects’ responses to these questions, a fourth, differently-sized cup might be introduced.

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Task 2: Conservation of Solid Volume: This task was a variation on Piaget’s “islands task.” In an attempt to make this task more accessible to our second graders, we changed the context from buildings on islands to a chocolate factory. In our variation, interviewers gave subjects a “chocolate bar” made out from cubes with dimensions 4x4x4. The interviewer then explained that the man who made the chocolates wanted to make sure that everyone who bought his candy got the same amount of chocolate regardless of the bar’s shape. The interviewer then asked the subject if he or she could build chocolate bars that could fit into wrappers with a variety of base dimensions. For example, in one of the interviews, the subject was asked if he could build a candy bar whose base was 1 cubes by 2 cubes. Interviewers asked follow-up questions depending on the subjects’ response to the tasks. IV. Examples of Beta and Gamma Compensations from Our Interviews Throughout our set of interviews, there were many examples of beta-behavior and gamma-behavior, and I will consider these instances in the following pages. I did not see a clear example of alpha-behavior in the interview set, so I will consider a hypothetical example of alpha-behavior that I hope will serve as a contrast to the beta and gamma compensations that follow. Alpha-Behavior: Though there was not a clear example of alpha-behavior in the interviews, I speculate that a subject who had an alpha reaction to the conservation of number task might have responded as follows when the interviewer spread one row of candies apart like this:

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The subject might have responded to this new arrangement by saying something like, “The second row has more because it looks longer.” Similarly, if the candies were again placed end-to-end, the subject with an alpha reaction might say something like, “Now they’re the same because they look the same.” In this way, the subject with an alpha reaction fails to take the perturbation into account; he sees the disturbance as little more than an annoyance, and it doesn’t matter much to him that no candies were added or taken away. Rather than attending to the number of candies in each row, the subject with an alpha reaction might might react more to the overall appearance of the row of candies. Beta-Behavior: There were several examples of beta-behavior in our interviews. One example stands out because of the subject’s attempt reconcile her expectations with what she observes in the conservation of number task. What follows is an excerpt of an interview with Grace: Interviewer: We just had candies lined up like this. This row was mine and this one was my brother’s candy row.

Do we have the same amount of candies? Grace: Umm...[counting the counters with her pencil] Int: This here is my row and this is my brother’s candy row. Grace: 2, 4, 6, 8, 10, 12, 14. [Counting out loud] Int: Do we have the same amount of candy?

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Grace: Yes. Int: How many candies does Ms. G [Cimone] have? Grace: 1,2, 4, 6, 8. No, I mean 2, 4, 6, 7! [counting out loud] Int: How many candies does my brother have? Grace: 2,4,6, Seven! [counting out loud] Int: So my brother and I [each] have seven candies. In this example, Grace exhibits beta-behavior because she does not anticipate the variation the interviewer proposes (spreading the candy in one row apart), but she does not attempt to cancel out the perturbation as a subject exhibiting an alpha reaction might. Instead, Grace counts the candies in both rows in an attempt to reconcile her expectations (when something takes up more space, there must be “more” objects) with what she knows (no counters were added or taken away, so the number of candies in each row must be the same). Because she does not yet “anticipates” this change, she must repeatedly confirm her idea by counting the candies. Gamma-Behavior: In an interview with Danny, he exhibits a gamma reaction during the conservation of number task. What follows is an excerpt from his interview: Interviewer: My brother liked to arrange his candies like this [indicating bottom row]. What do you think about that?

Danny: It’s the same. Because, like, you just moved them around. You still have the same.

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Here Danny is presented with the same variation as Grace (above), but he responds in a very different way. Rather than counting the number of candies in each row, Danny notes that the interviewer “just moved them (the candies) around.” Since no candy was added or taken away, the number in each row remains the same. By articulating a generalization that can be applied to any possible arrangement of candy (if no candy is added or removed, the number in each row is the same), Danny indicates that he can anticipate variations in the task without having to carry them out. His statements therefore provide an excellent example of gamma-behavior. Later in the interview, during the conservation of area task, Danny once again displayed a gamma reaction. When the interviewer challenged his idea that the the horses still had the same amount to eat when their pastures looked different, he said: Danny: It’s still the same amount ‘cause it’s just...just like the first one, but Interviewer: -but the horses’ [pastures] look different. Their things [pastures] look different now. Danny: They’re still the same amount, but just - it looks different. Just because it looks different doesn’t mean there’s a different amount. Once again, Danny is not phased by the different arrangement of the cubes. On the contrary, he anticipates this variation and does not see it as a perturbation. When the interviewer later asks him if he can imagine an arrangement using 12 cubes where the horses would not have the same amount to eat, he says, “Hmm...not really. I can’t think of any.” This response indicates that Danny would recognize any rearrangement of 12 cubes as conserving area without needing to confirm their total. He therefore anticipates

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the arrangements that conserve area without actually seeing them. One again, this anticipation of variation is characteristic of gamma-behavior. V. Conclusion: Implications for Teaching What do the results of these interviews help us understand about students and teaching? What can we learn from identifying student reactions like the ones described above? Perhaps most importantly, we can gain insight into how a student approaches a particular task. The extent to which a student can anticipate variation, for example, might help us think of new ways to challenge him. There are, however, limitations to the conclusions that can and should be drawn from identifying a student’s reaction to a task, be it alpha, beta, or gamma. We should not assume that students who have a gamma reaction on one day and with one task will always demonstrate gamma-behavior. The same can be said for students who exhibit alpha- or beta-behavior: what a child does on one day and in one context should not be taken as indicative of their overall capacity. Similarly, students who show gamma-behavior should not be seen as “superior” to those who show alpha- or beta-behaviors. On the contrary, we should celebrate every child’s attempt to make sense of the world around him for, as Piaget said, every child is alive, and it’s wonderful.

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References Boom, J. (2009). Piaget on Equilibration. In U. Müller, J. I. M. Carpendale, & L. Smith (Eds.), The Cambridge Companion to Piaget (pp. 132-149). New York, NY: Cambridge University Press.

Bringuier, J.-C. (1980). Structures. Their Mechanisms. Assimilation and Accommodation. Conversations with Jean Piaget (pp. 17-22). Chicago, IL: University of Chicago Press.

Piaget, J., Gruber, H. E., & Vonèche, J. J. (1995). XI: Factors of Development . The essential Piaget: an interpretive reference and guide (pp. 807-808). Northvale, N.J.: J. Aronson.

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kapsch alpha beta gamma

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