TEACHERS’ USES OF SEMIOTIC CHAINING TO LINK HOME AND MIDDLE SCHOOL MATHEMATICS IN CLASSROOMS

Matthew Hall	Norma Presmeg
Florida State University	Illinois State University
mhall@math.fsu.edu	npresmeg@msn.com

In this case study, two middle grade teachers became increasingly autonomous users of semiotic chains for the purpose of bridging the gap between their students' everyday activities and school mathematics. By using the semiotic chaining model, the teachers were able to address specific curriculum standards by selecting appropriate familiar activities of their students that would, through the creation of signs, link to the predefined mathematical goals. While the teachers exhibited some difficulties with creating appropriate linkages in the early stages, their continued participation generated increased ownership of the process resulting in the reduction of errors. The results of the study seem to indicate that teachers are able and willing to create semiotic chains that ground the teaching of mathematics in their students' realities.

Objectives or purposes

The purpose of the research described in this paper is to investigate the implementation in two middle grades mathematics classrooms of a pedagogical model that bridges situated activities of students and the mathematical concepts required to be taught in schools. In this model, teachers increasingly take the initiative in building semiotic chains that lead from activities in which their students engage, to selected topics from the school mathematics curriculum.

Theoretical framework

Literature in situated cognition (Lave, 1988) and cultural psychology (Abreu, 1998) suggests that mathematics is perceived by students to be more relevant to their lives if “realistic” situations are involved in its learning. However, “real life” activities undergo transformations when used in the classroom, which may cause them to be seen as less than real by students (Walkerdine, 1988). The most that can be said is that such activities are “realistic”, as in the situations used in the “realistic mathematics” of the Freudenthal institute (Treffers, 1993). Researchers such as Civil (1994 & 1998) and Presmeg (1994; 1998a & b) have experimented with ways of introducing activities that are authentic to the home cultures of students into mathematics classrooms. The current project is grounded in this literature, but takes a semiotic conceptual framework a step further than Presmeg did (1997 & 1998b) in asking teachers to enter a process in which they learn to construct semiotic chains from their own students’ activities and use these in their mathematics lessons.

We believe that mathematics is “the science of detachable relational insights” (Thomas, 1996), or, resonating with this conception of the nature of mathematics, “the systematization of relationships” (Ada Lovelace, in Noss, 1997). Accordingly, in attempting to use everyday activities of students in the learning of school mathematics, teachers need ways of helping students to accomplish such a systematization of relationships. The conceptual framework of our research involves Lacan’s inversion of Saussure’s posited semiotic relationship between signified and signifier in a sign. When signifiers are given free play, they may be linked in chains of increasing generality that may provide a symbolic bridge between realistic activities and classroom mathematical concepts (Cobb et al., 1997; Presmeg, 1997 & 1998b; Walkerdine, 1988; Whitson, 1997). In the current project, two teachers were introduced to this semiotic chaining model, and the research investigated their use of the model in increasingly autonomous ways in their own practice. The methodology of the research was as follows.

Methods of inquiry

In order to gain a rich understanding of how teachers might use semiotic chaining for the purpose of teaching academic mathematics from everyday activities, a case study of two middle grade teachers was employed. The development of the case study was designed in a manner so as to familiarize the teachers with the process and purpose of semiotic chaining while fostering their increased independence in creating chains. To this end, the study was designed in three phases. In the first phase, the teachers were introduced to semiotic chaining by means of provided chains. These chains were designed by the first author to develop a mathematical topic that was specified by the teachers from an activity of the students. In order to create such a lesson, a realistic activity that was experienced by the students needed to be found. The chosen activities were the result of multiple researcher interviews and observances of the students during their free-time at school. During this phase, the teachers requested a lesson on converting fractions to decimals. From several interviews, it was clear that many of the students in the classes were interested in baseball and could quote some statistics for many popular players. Among these statistics was the batting average. Thus, baseball appeared to be an appropriate everyday activity that could be linked to the predefined mathematical goal. The following semiotic chain was given to the teachers to structure the development of a lesson.

Batting averages for each player

Success fractions for each player

Chart of hits vs. at bats for each player

Baseball Game

This type of chain came to be known as an intra-cultural chain since the culture of baseball enveloped the whole chain. That is, the semiotic chain did not move away from the culture it was grounded in, but rather moved within the culture toward the specified goal. The type of chain that was created (intra-culture or inter-culture) appeared to depend upon how explicit the target mathematical concept existed within the foundational culture.

To increase the teachers’ autonomy in creating semiotic chains for their own purposes, the second phase of the study permitted the researcher to provide the everyday activity that might lead to a specified mathematical topic, but required the teachers to develop the links in the semiotic chain. This not only allowed the teachers to become more familiar with the intricacies of the chaining process, but also allowed them to take greater ownership of the chain by working their own ideas into the chain itself.

Finally, the third phase required the teachers to develop their own semiotic chains from their observations of the students’ everyday activities towards desired mathematical topics without any direction from the researcher. In order to teach basic algebra, the teachers created a semiotic chain that moved the students from music to algebraic equations.

Algebraic equations

Written music

Beats of the music

Music

The teachers began this lesson by listening to and conducting informal discussions about different popular songs. Once they concluded that the differences in the songs had a lot to do with the beat, the teachers reviewed how the students could represent the beat of a song using different notes and time signatures. Next, they used the beat values of different notes to determine which single note was needed to fill a measure under a specified time signature. Finally the teachers moved the students away from music toward more formal algebra by using equations to represent that which they were doing. As an example, when the time was four-four (four beats per measure) and a quarter note (one beat) and a half note (two beats) were present in a measure, the following equation was used to determine the missing note:

1 + 2 + ____ = 4.

Data sources

The data for the project were collected in a university lab school. In order to gain a better understanding of the everyday activities of the students, the research began with interviewing several children in each class. The interviews were videotaped and transcribed for the purpose of ascertaining appropriate beginning points of instruction. Before each semiotic chain was used in the mathematics classrooms, the teachers and the researcher met to discuss the instructional process that was to be undertaken. Field notes were taken during and after these meetings. During the lesson using the semiotic chains, the teachers were videotaped and field notes were also taken. After their presentations, the teachers and the researcher met again to discuss the semiotic chain and suggestions for improvement. Finally, the teachers’ links in phase two and semiotic chains in phase three were collected.

Results and conclusions

The data from the study indicate several results that are important for mathematics education. First of all, it seems that following a path of increasingly abstract signs needs to be undertaken carefully. When a semiotic chain is developed as a pedagogical tool from a student’s everyday activity to school mathematics, a gentle transition from one link to the next is most appropriate. It was observed that if the abstraction from one sign to another is too great, then students appear to lose the link to reality that is fundamental to this process. Once this was done, it was difficult to undo the lost connection without retreating to the realistic situation and beginning the building process again.

Furthermore, the logical progression in a semiotic chain appears to be important for similar reasons. If a mathematical topic is not developed from the everyday activity through the process outlined by the semiotic chain, then links to previous signs are lost. On several occasions during phase one, the teachers confused the signs in the chain and presented them out of order. This created many problems with children not realizing why or how the current activity related to the previous one. Here again, once the damage was done, the teachers had to return to the everyday activity and progress through the semiotic chain in order to correct the problem.

Finally, it appears that teachers are able to create semiotic chains that link their students’ activities to desired mathematical topics. Initially, it was unclear as to the impact that semiotic chaining should have at the classroom level. That is, would it be more of a teacher initiated tool or one usable only by curriculum developers? However, from the chains that were produced by the teachers, it appears that they were not only able but willing to participate in this type of lesson development. This is most assuring since the everyday activities of children are far from being universal. Thus, it seems most appropriate that teachers who witness their students’ activities be the developers of their mathematical curriculum. The issue of ownership is also important here. An increased sense of ownership on the part of teachers and students may enhance students’ learning of mathematics (Presmeg, 1988a), and teachers are in a position to judge activities of their own students that have the potential for ownership of related mathematical constructs.

References

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