This paper discusses findings from a three-year qualitative case-study of 12 middle school teachers participating in a social semiotics lesson study group. The paper focuses on teacher use of gesture and diagrams in exploring, developing and implementing lessons that target complex word problems involving a range of verb tenses and temporal (durational) events. We focus on how teachers leveraged multi-modal resources to tackle complex temporal relationships described in word problems. Although research has shed light on students making representations of spatio-temporal processes in Cartesian or proto-Cartesian graphs (Bremigan, 2001; Radford, 2004; Sherin, 2000; Sinclair & Armstrong, 2011; Stylianou et al, 2005), this work has not addressed the specific challenges of word problems that involve complex verb conjugations and grammatical structures that are linked to the logical structure of the given problem. Gerofsky (2004) has shown how tense and verb conjugation in typical word problems create stumbling blocks for students, and suggests that more research must be completed on how students and teachers leverage semiotic resources in such situations. This paper recommends particular instructional interventions to address this demand, as well as showcasing effective professional learning experiences for teachers.

Theoretical Framework

Social semiotics draws on a wide range of theoretical traditions to study the way that people use “semiotic resources” defined broadly to include all “actions and artifacts” used to make meaning (Van Leeuwen, 2005, p.3). Instead of separately treating distinct semiotic modes (language, image, gesture), social semiotics explores how these modes are integrated or combined in multimodal artifacts and events (Kress, 2004; Morgan, 2006). Many of the analytic concepts in social semiotics are derived from Halliday’s (1978, 2004) Systemic Functional Linguistics (SFL) and have been extended, elaborated and amplified so as to study modes other than language. O’Halloran (2000, 2003, 2005, 2007a, 2007b, 2008) has shown how social semiotics can shed light on the “multimodal grammaticality” that binds visual, symbolic and linguistic elements together in mathematics texts (O’Halloran, 2007c, p. 205). Rivera (2011) has drawn on O’Halloran’s multi-semiotic or multi-modal approach, and on the work of others who have extensively studied visualization in mathematics (Presmeg, 2006), to show how a more visually oriented curriculum might develop students’ use of diagrams as strategic implements and less as mere illustrations or secondary representations of other processes.

Social semiotics also draws on analytic concepts from Peircian semiotics (Van Leeuwen, 2005). According to Peirce’s tripartite theory of signs, diagrams are “complex signs” composed of icons, indices and symbols (Bakker & Hoffmann, 2005). Icons are those signs that have some physical resemblance to that which they signify, indices are the traces/imprint of the referent, and symbols signify through custom or habit. The task of moving from the iconic to the symbolic in diagramming while decoding word problems is a complex semiotic skill central to the doing of school mathematics. Indeed, such diagramming skills are essential in problem solving.

Methods

During lesson study sessions, teachers engaged in a series of activities that focused on diagrams (as objects) and diagramming (as activity) and on the multi-modal mediation of diagrams through gesture and verbal and written language. Lesson study sessions explored diagramming through such prompts as: What do you notice about this diagram? If this diagram had a title, what would it be? What does this diagram prove? How would you make this diagram? All of these activities were meant to develop creative habits around diagramming in general. Participants also studied transcripts and created “interaction maps” (de Freitas & Zolkower, 2010) to determine what role a given diagram was playing in examples of classroom interaction. In year two of the project, we focused on a series of word problems consisting of complex verb forms and pertaining to spatio-temporal processes, and studied the semiotic affordances of algebraic and diagrammatic approaches to these problems. Teachers then collaboratively developed lessons around these word problems and implemented them in their classrooms.

In this paper, we focus on three key research questions:

(1)  How do students and teachers jointly operationalize diagrams, gesture and language in a whole-class conversation as they grapple with complex word problems consisting of multiple verb forms?

(2)  How do students and teachers map correlations between the semiotic (verbal, diagrammatic, etc) and the logical relations of the problem?

(3)  What are the semiotic affordances and constraints in diagramming word problems that pertain to spatio-temporal processes?

All lesson study sessions and subsequent experimental lessons in teacher classrooms were videotaped and transcribed. Transcripts and videos were analyzed for evidence of diagram use, and further coded in terms of the function ascribed to the diagram (or diagramming) within the classroom discourse, drawing on Peirce’s tripartite theory of signs as well as O’Halloran’s grammatical categories. Both teachers and researchers will co-present this data during the presentation, showing and discussing transcript and video excerpts.

Results

Data from the lesson study sessions indicates that strip-diagrams functioned particularly well in developing teachers’ capacity to visualize spatio-temporal processes and to decode complex grammar. We argue that these kinds of diagrams allowed the teachers - and later the students - to position themselves within the problem in relation to the various spatio-temporal durational events described. This act of positioning the problem-solver in relation to the described events (before, after, during) points to the linguistic concept of aspect. Aspect broadens the concept of tense, so as to better recognize the felt-experience of time. Aspect conveys duration, completion or frequency of an event (Bhattacharya & Hidam, 2011), amplifying the ways in which the past, present and future are experienced. The aspect of a verb defines its temporal flow (or lack thereof). We argue that the concept of aspect sheds light on the way semiotic resources can be effectively leveraged when working with word problems that involve time. In addition, we argue that the concept of aspect can be used to map the correlations between the logical relations of the given problem and the complex conjugated verbs found therein.

Scientific Importance

This research contributes to the ongoing work on multi-modality and language-use in mathematics education.

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