Abstract
We offer a new approach to emergent knowledge in processes of conceptual change in dyadic interaction by drawing on Pickering’s (The Mangle of Practice, The University of Chicago Press, London, 1995) Mangle of Practice theory, which theorizes the emergence of new scientific knowledge as occurring due to material resistance and human accommodations to such resistance. We use Commognition (Sfard in Thinking as communicating, Cambridge University Press, New York, 2008), which conceptualizes learning as a change in discourse, and conceptual change as change in meta-discursive rules, to examine the interaction between several dyads working on proportional reasoning tasks. These dyads were selected for close scrutiny based on a large previous experimental study designed to examine the most productive constellation for pairing students of different levels with or without a hypothesis testing device. The Mangle helps us theorize the emergent, unpredictable aspects of learning and conceptual change. We provide three criteria for examining the productivity of the interaction: Availability of an alternative discourse (proportional vs. additive), Resistance of material/disciplinary agency and Positioning of students to make meta-discursive shifts. We discuss this model as an aid for monitoring and designing learning situations that enhance conceptual change.
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Notes
The study of conceptual change is an extremely vast domain of research. We have here mentioned one camp in this domain (very partially)—the camp of knowledge-as-theory perspective, and totally ignored the knowledge-as-elements perspective. According to the knowledge-as-theory perspective (e.g., Chi 2005; Ioannides and Vosniadou 2002; Wellman and Gelman 1992), naïve knowledge is highly organized (in theory, schema, or frame), coherent, involves the creation of mental models, and its replacement is revolutionary. In contrast, the knowledge-as-elements perspective (e.g., Clark 2006; diSessa 1993) is characterized by naïve knowledge constituted as a collection of quasi-independent knowledge elements, and high contextual sensitivity. Conceptual change according to this perspective involves evolutionary revision and refinement. The explanations involve the p-prims and other elements that are most strongly cued by the context. We intentionally unified the two perspectives in the body, to stress resistance to change, as naïve knowledge is fueled by everyday experience. This simplified presentation enables us to introduce more smoothly the “solution” proposed by the Genevan school to overcome resistance to conceptual change.
For the present task, which includes physical objects (cubes and blocks) that have physical attributes (weight), it may have been necessary to add a fifth characteristic: that of tactical/concrete or physical mediators. However, since eventually this mediator is not discussed in the present analysis, we chose to omit this characteristic from our table.
The quaternions started out as an extension of the complex number system. Hamilton’s goal was to generalize the mapping between algebra and geometry, as can be seen in the mapping of complex numbers to the Cartesian two-dimensional plane, into three-dimensional space. He started out by searching for “triplets” that would map to 3D space in a similar way to x + iy mapping onto the plane. This was done by adding an imaginary “j” and constructing the triplet t = x + iy + jz in the hope that this “j” act in algebraic calculations similarly to the familiar i. However, a series of “resistances”, in the form of geometrical meanings not aligning with the algebraic calculations, prohibited Hamilton from attaining his goal. After a series of attempts to adapt his new system to previous mathematical rules (disciplinary agency), Hamilton’s finally came up with another imaginary: k = ij. He named the set a + ix + jy + kz a “quaternion” and rapidly came up with a series of equations that satisfied both algebraic rules and geometrical meanings. However, with that, Hamilton effectively abandoned his original intention of mapping algebra onto 3 dimensional space. Pickering uses this case as another example in which human intentions are “mangled” with material (or disciplinary) agency, leading to unexpected outcomes and cultural extensions.
Pickering (1995) did not use the term “conceptual agency”, but rather the term “conceptual practice”. His use of the term “concept”, however, seems to agree with that of ours. Following, Sfard (2008 p. 111) we define a concept as “a symbol together with its uses”. Pickering suggests moving away from a “representational idiom” of science, where science is regarded as a representation of reality, to a “performative idiom” of science, where science is “regarded as a field of powers, capacities, and performances, situated in machinic captures of material agency” (p. 7). Though this performative idiom does take into account material agency, the “conceptual” part of science is always regarded by Pickering as social or “cultural” as he terms it, thus being in harmony with a “discursive” definition of concepts.
The dyads’ interactions were initially reported in Cohen-Eliyahu (2011).
A fully proportional correct answer would yield that D was necessarily bigger than C. One possible explanation would be that a B cube is 1.111 times an A cube and therefore 28 C cubes times 1.111 equals 31.11 which is necessarily bigger than 31 D cubes. Algebraically this could be written as 10x < 9y ⇒ 31y < 31.1111y < 28x.
We are aware that the experimenter’s role in this situation could be thought of as critical for the advancement that Dina made, and that this could be seen as strengthening the socio-cultural view insisting on the necessity of an expert’s discourse for the occurrence of conceptual change. However, we claim that this contribution, was, after all, quite minor as it only strengthened existing ideas. Thus, the expert did not play the role of introducing new meta-discursive rules but rather strengthened the position of the students to make the meta-discursive shift. As important as this may be, positional strength (or strong mathematical identities) may be achieved by other means, not just adults’ assurances.
In other domains, such as science education, increasing attention has been given to “hot” conceptual change, including motivation, emotions and beliefs (cf. Sinatra 2014). Similar attention has not been given, to the far of our knowledge to conceptual change in mathematics.
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The writing of this paper was partially supported by the Golda Meir postdoctoral fellowship of the Hebrew University.
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Heyd-Metzuyanim, E., Schwarz, B.B. Conceptual change within dyadic interactions: the dance of conceptual and material agency. Instr Sci 45, 645–677 (2017). https://doi.org/10.1007/s11251-017-9419-z
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DOI: https://doi.org/10.1007/s11251-017-9419-z