Abstract
In mathematical interactions, young learners express their ideas in multiple ways to interact with each other and to come in contact with the provided culture-based mathematical environment—to construct in common mathematical meaning, so to say. To deal with the complex multimodality seen in these interactions, this chapter investigates the interplay between gestures and speech used by second graders while they are occupied with a geometrical problem in pairs. In the chapter, gesture and speech are analyzed with an interaction analysis, and a detailed reconstruction of the semiotic process on a microscopic level. The main research question is: How and in what kind of modality—in gesture and/or speech—will mathematical ideas be introduced, adopted, developed, and/or refused by the children during their occupation with the given mathematical problem?
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- 1.
The term mathematical ideas can be understood as any kind of expressed contribution of the second graders, which contain any suggestion to solve the given mathematical problem. What can be described as a mathematical idea emerges and is constructed in the interaction by negotiations of the participants. By dint of a detailed analysis, these mathematical ideas can be reconstructed.
- 2.
Please note that the interaction theory is the leading approach according to the learning of mathematics in the present chapter. In some sections in the chapter, other approaches will be described to clarify the current state of research according to the theme of the chapter. In some of these other approaches, the conception of the learning of mathematics is quite different from the interactional point of view. For example, the mismatch theory according to Goldin-Meadow (2003) in the section about gestures in the learning of mathematics is ascribed to the psychological view and brings into focus the individual rather than the social constitution of learning.
- 3.
To be a part of means that learners orient themselves on the behavior of others. To take part means that the own behavior is used as orientation for others. Whereas the former can be described as a receptive behavior, the latter is rather an active participation (cf. Krummheuer and Brandt 2001, 17 f.).
- 4.
Interaction patterns emerge in interactions and are a kind of routines or structures which can be reconstructed by dint of a detailed analysis of the interaction processes. These routines contain implicit and, for the interlocutors, rather unconscious rules which determine the process of interaction (cf. Voigt 1984, Krummheuer 2011). The benefit of those patterns is to stabilize the progress of the mathematical interaction and to guarantee the functionality. Mutual coordination of the interlocutors (cf. Krummheuer 1992, 40 ff.) can be realized. Interaction patterns are no (teaching) methods which are available or can be applied consciously. Both, the rules and the patterns emerge in these interactions between the interlocutors.
- 5.
- 6.
This argumentation, although rather from a perspective of psychology, is in line with the above-mentioned aspects of the relation of individual and social learning in interaction as described by Krummheuer (2011) and Sfard (2003). Maybe the examination of gestures can thus provide a bridge between approaches of psychology and psycholinguistics and the theory of interaction in the learning of mathematics.
- 7.
The specific interplay of gesture and speech leads to the here described research focus. This does not mean that other expression modes as well as the influence of the given material on mathematical interactions are ignored or disregarded. Those aspects are considered analytically by dint of the interaction analysis and are integrated in the interpretation of gesture, speech, and their relation to each other.
- 8.
Spontaneously produced gestures are differentiated from those gestures which have any kind of standardized well formedness and/or can be understood, e.g., within a language community without accompanying speech, e.g., like it is described for emblems (cf. McNeill 1992, 36 ff.).
- 9.
These gesture dimensions are often described as “categories” (McNeill 2005, p. 41). The different features of these categories are often mixed in the same gesture. The word categories seems to imply a hierarchy, which is why McNeill (2005, p. 41) recommends to use dimensions instead of categories.
- 10.
Please keep in mind that those studies are not necessarily compatible with the introduced interaction theory, especially with regard to the conception of mathematics learning. First, regardless of these differences, the approaches will be described. At the end of this section, these approaches will be discussed with regard to the interactional approach used in the present chapter.
- 11.
The children had to solve equation problems of the following kind: 3 + 7 + 4 = __ + 4. A match was observed, when a child said: “I add 3, 7 and 4” accompanied with a pointing gesture from the left to the right on the 3, the 4 and the 7 on the left side of the equation sign. A mismatch was observed when a child uttered the same words in speech but simultaneously showed a pointing gesture to 3 and 7 with an extended index and ring finger. In this case, the 4 on the left side of the equation sign was gesturally excluded (Goldin-Meadow 2003, p. 44).
- 12.
Please note that the concept of instruction in the here-described approach is understood rather in a narrow sense and as a kind of very clearly defined teaching sessions.
- 13.
Goldin-Meadow (2003) describes a continuum of matches and mismatches based on the degree of overlap of information in gesture and speech (cf. ibid. p. 26).
- 14.
Mismatch of space (with relation to the gesture space): In an ongoing narration, an actor is sited in a certain area of the gesture space, e.g., on the left side of the gesture space. Then the narrator uses another area as the space of reference for the same actor, which was already established for another actor. The gesture shows a shift of space, whereas the speech implies continuity of reference, e.g., by using the same pronoun “he.” Mismatch of form: A narrator uses verbs that refer to a motion but do not convey any information about the manner of this motion, e.g., come. In gesture then the form of motion is shown, e.g., by bouncing up and down with the hands (cf. McNeill 1992, p. 135).
- 15.
Instruction in the here-described study of Cook and Goldin-Meadow (2006) is understood as clearly structured teaching sessions with planned gesture and speech instructions.
- 16.
Among equals means that both interlocutors were equal concerning their role in the mathematical interaction: No explicitly and previously defined role model or more advanced interlocutor of adequate mathematical reasoning participates. Both pupils take part in the interaction with their mathematical way to interpret the given problem. There is no knowing professional, and no inexperienced and unknowing novice. It is a more symmetrical interaction between peers in which, of course, these asymmetrical roles can possibly emerge and can be negotiated between the interlocutors.
- 17.
Each situation is accompanied by an adult who presents it to the children and gives spare impulses if needed. The concept of instruction in the didactic design patterns is understood in a broader sense and allocates a set of those impulses. In the planning of this set of impulses thought has been given to describe possible ideas of the children in the situation. The accompanying person can use this set of impulses to choose them adequately according to the mathematical ideas of the children.
- 18.
In the present paper the analyses will not be described in detail, but portrayed as summarized interpretations.
- 19.
With regard to an adequate number of pages, the transcript of the chosen and described sequence is not portrayed in the given paper. The produced utterances in speech and gesture, the actions of the interlocutors, as well as the whole process of interaction can be seen in the semiotic process card (cf. Fig. 10.5).
- 20.
The interaction analysis is based on a sequential proceeding to reconstruct the development of the theme in the progress of the interaction. In this proceeding from turn to turn, the interaction analysis leans among others on the conversation theory (cf. e.g. Eberle 1997). Aspects of the conversation analysis are adopted to focus not only the organizational aspects but mainly the development of the theme of the interaction (Krummheuer and Brandt 2001, p. 90).
- 21.
In the following analyses, the LEGO DUPLO bricks will be signified after the numbers of knobs they have on their upper side, e.g., the 4 × 2 brick is called 8-brick. Furthermore, in the buildings most of them are numbered, e.g., brick 2, brick 15, etc. (cf. Fig. 10.2).
- 22.
There is a triad for gesture (on the right) and a triad for speech (on the left). When there is no speech utterance at all, there is only one triad for gesture. The triads are numbered. Parallel utterances are portrayed by parallel triads which are marked with indices a, b,….
- 23.
Please keep in mind that instruction here is understood in a very broad sense and not exclusively as a kind of teaching by a knowing expert like a teacher. Rather instruction here is understandable as an impulse which can be used to foster one’s own insights and further development.
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The preparation of this chapter was funded by the federal state government of Hesse (LOEWE initiative).
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Huth, M. (2014). The Interplay Between Gesture and Speech: Second Graders Solve Mathematical Problems. In: Kortenkamp, U., Brandt, B., Benz, C., Krummheuer, G., Ladel, S., Vogel, R. (eds) Early Mathematics Learning. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4678-1_10
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