McClain, Kay

The Journal of Mathematical Behavior, vol. 19, nr 2, pp. 189-207, 2000.

Abstract

The classroom that is the focus of the analysis reported in this paper was the site for a year-long first-grade teaching experiment. The goal of the classroom teaching experiment was to investigate ways to proactively support students' ability to develop strategies for adding and subtracting quantities. In addition, the project also focused on the teacher's proactive role in supporting students' mathematical growth in an inquiry mathematics classroom. Within the project classroom, the teacher often redescribed and notated students' responses so that what students had done mathematically might become an explicit topic of conversation. As part of this process, she frequently introduced either informal or conventional notation to record students' explanations of their mathematical activity. The introduction of these notational schemes led to students' development of ways of notating their own reasoning. In this way, the notation emerged from the students' activity while supporting shifts in their mathematical development.

Annotatie

Een voorbeeld van een artikel uit de Cobb-school. Ik probeer het argumentatieve verhaal te volgen als voorbeeld voor het type verhalen dat ik wellicht zal schrijven.

Introductie

• Uitleg over terminologie, kernbegrip van het artikel: symbolizations. "Ways of symbolizing should therefore emerge from students' informal mathematical activity and be consistent with their developing concepts and strategies (Cognition; Lesh and Thompson). In addition, they should serve as a resource that students can use to express, communicate, and reflect on their mathematical activity (Confrey and Johnson)."
• Het doel van dit artikel (waar wil het aan bijdrage aan leveren): "The purpose of this paper is to analyze the teacher's role in the emergence of ways of symbolizing and notating in one first-grade classroom. In doing so, I will document how the ways of symbolizing served as protocols of action as the students explained and justified their mathematical reasoning (...). In this way, the teacher's role can be characterized as that of supporting the emergence of both individual and collective ways of symbolizing."
• Experimentele setting: waar en wat.
• Structuur van het artikel.
• Methodologie
• Data collectie: videotape, copies of children written work, daily field notes, videotaped interviews.
• Analyse: Cobb&Whitenack en Glaser&Strauss: "(...) constant comparative method for conducting ethnographic studies. It involves constantly comparing data as they are analyzed against conjectures and speculations generated thus far in the data analysis."
• "The theoretical framework that guided the analyses is the emergent perspective (cf. Cobb & Yackel, 1996). The emergent perspective involves coordinating constructivist analyses of individual student's activities and meanings with an analysis of the communal mathematical practices in which they occur."
  Dus twee perspectieven: psychological (individual, constructive) en sociological ("interactionist and views communication as a process of mutual adaptation wherein individuals negotiate mathematical meaning").
• De analyse
• Introductie, korte (pakkende!) beschrijving van wat gebeurd is: "The ways of symbolizing that emerged in Ms. Smith's classroom evolved from the need to clarify and communicate students' thinking. (...) During these activities, Ms. Smith often introduced ways of symbolizing in an attempt to clarify a student's thinking either for herself or for other students. In contrast, during the second part of the Structuring Numbers sequence, students completed individual activity sheets more frequently and were asked to make a record of their thinking so that others might understand their reasoning. These activities were subsequently discussed in whole-class setting during which time Ms. Smith typically redescribed and notated students' explanations of their activity."
  Met het uiteindelijke resultaat ondersteund door leerlinguitwerkingen. "As the semester progressed, the use of notational schemes in Ms. Smith's classroom came to serve as thinking devices as students began to use the written records on the board as a means of reflecting on and comparing their own and other students' mathematical activity (Wertsch & Toma, 1995). In addition, they began to incorporate elements of these schemes into their individual work. (...) The lack of conformity to any one notational scheme and the variety of schemes on any individual sheet support the notion that these students had developed ways of notating that fit with their activity. The emergence of these notational schemes offers evidence that these written records provided students with a means of expressing their mathematical reasoning as they engaged in problem-solving activities."
• Vervolgens hoe het is gebeurd.
  Eerst introductie van "number sentences" door docent.
  Anekdote.
  "(...) the number sentence provided a record for other students to use in reflecting on how Amy had solved the task."
  Anekdote.
  "The fact that grouping solutions could be easily expressed via symbols contributed to the students' opportunities to reflect on and compare different grouping ways."
  Anekdote.
  "As students continued to offer different ways, the notation served as a written description of their thinking. It is not clear what specific meanings the students may have given to the number sentences at this juncture. However, the students seemed to be able to reflect on the number sentences as a record of solution methods and to use those to determine if their method constituted a different way. This is evidenced by the fact that students rarely offered solutions that had already been explained. Further, their subsequent flexible use of number sentences indicates that they began to utilize the offered sentences as a way to think about and interpret their own strategies. The absence of individual student work and lack of direct evidence of the students' actual interpretation of number sentences prevent definitive claims. However, it is reasonable to suggest that students' participation in these classroom activities did give rise to learning opportunities that supported the development of number sentences as an accepted way of communicating mathematically. A shift in the students' thinking can be seen later in the lesson when Ms. Smith flashed the 10-frame with six chips".
  Anekdote.
  "In this episode, we see further evidence of the students' ability to make a distinction between their solution strategy and those that had been explained. They are able to clarify the differences by referring to the "groups" or collections that they had used in determining the total."
  Anekdote.
  "A second important shift occurred in that the goal or purpose of the task was no longer just determining the number of chips in the 10-frame. The purpose now entailed identifying a way to group the chips to find the total."
  
  Vervolgens de ontwikkeling van "notational schemes" door de leerlingen.
  Enkele anekdotes over hoe de docent klassediscussies begeleid met notaties op het bord.
  Het blijkt dat leerlingen aspecten hievan overnemen, maar ook zelf notaties ontwikkelen.
  "In examining the written record the students made on the activity sheets for the task of eight plus six, it is important to note that there was diversity in the students' notational schemes (see Fig. 18a¯d). Only the first student's way of notating is entirely consistent with that of the teacher. (...) Consequently, although students might understand ways of talking about their activity that were expressed in terms of the teacher's notational scheme, they continued to solve tasks using a range of different, personally meaningful notational schemes."
  Anekdote.
  "... the notational schemes served as thinking devices that supported reflective shifts in the students' activity. By the end of the sequence, individual students were utilizing a variety of notational schemes. This can be seen in the samples of student work taken from late December and January (see Fig. 1, Fig. 2, Fig. 3 and Fig. 4). Note the diversity present on a single sheet. The flexible nature of their work implies that they had not simply imitated a given way of symbolizing. While components of Ms. Smith's notational scheme can be seen in the students' work, it is clear that the students had developed personally meaningful ways of notating that still fit with the taken-as-shared understandings."
  
  
• Conclusie
• De docent speelde een cruciale rol in het starten van de ontwikkeling van notaties door de leerlingen: "... it appeared that Ms. Smith's use of ways of symbolizing offered a way for the students to organize and reflect on their activity ¯¯ providing an opportunity for shifts in thinking to occur. These ways of symbolizing enabled reflection on and analysis of the students' prior mathematical activity. This is evidenced by the fact that on their own initiative, students often referred to the notation to explain their thinking to other children during whole-class discussions. In addition, students began to use the records as a means of comparing solutions, thereby initiating shifts in the discourse such that features of their reasoning became explicit topics of conversation. As such, the students' participation in these conversations supported their reflection on and mathematization of their prior activity."
• Vervolgens een slag om de arm: "It might be argued that the ways of symbolizing that emerged in Ms. Smith's classroom were imposed by the teacher to fit with her interpretation of the students' solution processes. (...) Nonetheless, Ms. Smith's proactive role in guiding the development of ways of notating appears to have been critical in supporting her students' mathematical development. The students increasingly notated flexibly on their own initiative as they solved problems while working both individually and in groups. (...) Consequently, the use of notation contributed to the productiveness of whole-class discussions by helping to make individual students' contributions explicit topics of conversation that could be compared and contrasted."
• Tot slot een inbedding van de conclusies in een breder debat: "The students' ability not only to communicate their mathematical thinking, but also their ability to generate and refine notation schemes points to their development of what diSessa, Hammer, Sherin, and Kolpakowski (1991) call meta-representational competence (diSessa et al., 1991). (...) In arguing that students in Ms. Smith's classroom developed meta-representational knowledge of notational schemes, it is important to highlight the fact that (1) the students were able to use each other's representations, (2) the understandings of the mathematics improved because of the use of the notational schemes, and (3) the meanings of the notational schemes became taken-as-shared in the classroom. Because the students were developing meta-representational knowledge of the notational schemes, they were able to develop a "richer and more flexible sense" of the use of notational schemes (diSessa et al., 1991, p. 150). Consequently, the notational schemes became a tool that facilitated the communication of the students' mathematical problem-solving strategies. It was as they participated in the development of these notational schemes that the teacher guided her students' transition from informal, pragmatic problem solving to more sophisticated yet personally meaningful mathematical activity."