240 Sinikka Kaartinen and Timo Latomaa
4.1. A Case-Based Description
This case-based description highlights the collaborative processes of one teacher group
when factoring polynomials with the help of algebra tiles. The extract characterizes the
teachers‘ discourse as they negotiate and apply the usage of algebra tiles in collaborative
problem solving.
Table 2 shows the discourse data of the teacher group. The extract consists of 24
conversational turns in total, from a 3-minute continuous working period. The data presented
in Table 2 will be discussed here by firstly summarizing the findings from the analyses of the
teachers‘ social interaction within the group.
Special attention will be paid to the identification of problem-solving episodes in the
teachers‘ discourse. This is followed by a micro-level investigation of three interaction
episodes in the teachers‘ discourse.
The analysis of the teachers‘ discourse reveals altogether three thematic episodes in the
construction of an application for the usage of algebraic tiles. The themes for episodes are
problem solving with the help of algebra tiles (Episode 1), clarification through
mathematizing (Episode 2), and clarification through hands on activities (Episode 3). The
analysis of discourse moves shows that the thematic episodes started from the initiation,
questioning and tutoring moves, leading to several conversational turns which took the form
of problem solving elements such as questioning, extending, evaluating and tutoring. The
analysis of the cultural focus of the teacher participation reveals the interplay of procedural,
identity and semiotic modes of interaction.
In the procedural mode of interaction, the mathematical activity included problem solving
and problem posing and the symbolic nature of the interaction was grounded in the pictorial
and procedural application of algebraic tiles.
In the identity mode of interaction, the prior learning experiences were reflected upon
through the application of new cultural tools. In the semiotic mode of interaction the problem
was clarified by negotiating the nature and meaning of algebra tiles and the mathematical
activity was approached through mathematizing the situation either verbally, symbolically or
pictorially. The material mode of interaction was seldom present and it was used for
organizing the working space.
4.1.1. Episode 1
In Episode 1 the teacher group started the activity by posing the problem. The episode
suggests that the usage of algebra tiles was new to all of the participants. Maritta was eager (6
turns of 11) in participating and tutoring the others. Annikki (turn 2) expressed here
unfamiliarity with the usage of cultural tools and Maritta and Liisa made their thinking visible
in their turns so that Annikki had the possibility to follow the joint problem solving. Karra
was mainly silent but when participating (turns 6 and 10) he supported the group‘s problem
solving by questioning and evaluating. In this episode the group reached the solution for the
problem. Episode 1 clearly shows the evolving role of the sign in the shared meaning making
process.