EFFECTIVE USE OF VISUALIZATION IN MATHEMATICS

The term visualization has been used in various ways in the research literature of the past two decades, it is necessary to clarify how it is used in this review. Following Piaget and Inhelder (1971), the position is taken that when a person creates a spatial arrangement (including a mathematical inscription) there is a visual image in the person’s mind, guiding this creation. Thus visualization is taken to include processes of constructing and transforming both visual mental imagery and all of the inscriptions of a spatial nature that may be implicated in

doing mathematics (Presmeg, 1997b). This characterization is broad enough to include two aspects of spatial thinking elaborated by Bishop (1983), namely, interpreting figural information (IFI) and visual processing (VP).

The term inscriptions is preferred to that of representations , because the latter became imbued with various meanings and connotations in the changing paradigms of the last two decades. The difficulty in articulating an accurate definition for the term representation is worth stressing. An indication of this difficulty is that definitions for the term “representation” in the literature often include the word “represent” (Kaput, 1987). Kaput maintained that the concept of representation involved the following components: a representational entity; the entity that it represents; particular aspects of the representational entity; the particular aspects of the entity it represents that form the representation; and finally, the correspondence between the two entities. This level of detail is unnecessary for the purposes of the present case. Thus the term inscriptions will be employed, characterized by Roth (2004) as follows: “Graphical representations, which in the sociology of science and in postmodern discourse have come to be known as inscriptions, are central to scientific practice” (p. 2). Roth viewed these inscriptions as essential to the rhetoric of scientific communication. Nevertheless, the term representations is maintained in this chapter when it is used by the authors cited.

Following the usage of Presmeg (1985, 1986a, 1986b), a visual image is taken to be a mental construct depicting visual or spatial information, and a visualizer is a person who prefers to use visual methods when there is a choice. (Rationales for the use of these terms and their definitions may be found in Presmeg, 1997b.)

The title of Presmeg’s (1991) research report was “Classroom aspects which influence use of visual imagery in high school mathematics.” The aim of the complete three-year study (Presmeg, 1985) was to understand more about the circumstances that affect the visual pupil’s operating in his of her preferred mode, and how the teacher facilitates this or otherwise. She had chosen 13 high school mathematics teachers for her research, based on their mathematical visuality scores from a “preference for visuality” instrument she had designed and field tested for reliability and validity (with parts A, B, and C reflecting increasing difficulty). The teachers’ scores (on parts B and C) reflected the full range of cognitive preferences, from highly visual to highly nonvisual. Subsequently, 54 visualizers (who scored above the mean on parts A and B of this instrument) were chosen from the mathematics classes of these teachers to participate in the research. Lessons of these teachers were observed over a complete school year, and 108 of these audio taped lessons were transcribed. The teachers were also interviewed, and so were the visualizers, their students, on a regular basis (188 transcribed clinical interviews, apart from the interviews with the teachers). Teaching visuality (TV) of the 13 teachers was judged using 12 refined classroom aspects (CAs) taken from the literature to be supportive of visual thinking. (For an account of the refinement process and the triangulation involved in obtaining this teaching visuality score for each teacher, see Presmeg, 1991.) These classroom aspects included a non-essential pictorial presentation by the teacher, use of the teacher’s own imagery as indicated by gesture (a powerful indicator) or by spatial inscriptions such as arrows in algebraic work, conscious attempts by the teacher to facilitate students’ construction and use of imagery (either stationary or dynamic), teacher’s requesting students to use the motor component of imagery in arm, finger, or body movements, teaching with manipulatives, teacher use of color, and finally, teaching that is not rule-bound, including use of pattern-seeking methods, encouragement of students’ use of intuition, delayed use of symbolism, and deliberate creation of cognitive conflict in learners (Presmeg, 1991, III: p. 192).

The first major surprise relating to the teachers was that their teaching visuality (TV) was only weakly correlated with their mathematical visuality (Spearman’s rho = 0.404, not significant). This result was understandable in the light of the common sense notion that an effective teacher adapts to the needs of the students: for instance, Mr. Blue (pseudonym) felt almost no need for visual thinking in his mathematical problem solving, but he nevertheless used many visual aspects in his mathematics classroom as evidenced by his TV score of 7 out of a possible 12. The TV scores divided the teachers neatly into three groups, namely, a visual group.

Based on field notes of observations in the classes of these teachers, and on transcripts of 108 audio-recorded lessons, further classroom aspects that characterized the practices of teachers in each group were identified in four areas, namely, relating to teaching, students, mathematics, and visual methods. In a nutshell, the visual group of teachers, while sometimes using the lecturing style and other aspects characteristic of the nonvisual group, in addition manifested a myriad of additional aspects:

The essence of the teaching of those in the visual group is captured in the word connections. The visual teachers constantly made connections between the subject matter and other areas of thought, such as other sections of the syllabus, other subjects, work done previously, aspects of the subject matter beyond the syllabus, and above all, the real world. … It was a totally unexpected finding that visual and nonvisual teachers were distinguishable in terms of certain characteristics associated with creativity … such as openness to external and internal experience, self-awareness, humour and playfulness. (Presmeg, 1991, III: p. 194).

Teachers in the middle group used many of the visual methods characteristic of the visual teachers. However, whereas the visual teachers were unanimously positive about these aspects, the middle group of teachers entertained beliefs and attitudes that suggested to their students that the visual mode was not really necessary or important – that generalization was the goal, and that visual thinking could be dispensed with after it had served its initial purpose. (See Presmeg, 1991, III: pp. 95-96 for examples.) Thus the middle group of teachers inadvertently helped their visualizers to overcome the generalization problem, while allowing them to

use their preferred visual mode for initial mathematical processing. The result was that visualizers were most successful with teachers in the middle group – a counterintuitive and unexpected result! However, it was suggested that if teachers in the visual group had been more aware of the potential pitfalls relating to visualization and generalization, they might have been more successful in helping visualizers to overcome these difficulties. Visualizers in the classes of the nonvisual group of teachers tried to dispense with their preferred visual methods in favor of  the nonvisual modes used by their teachers. Rote memorization and little success were the unfortunate consequences in most cases (Presmeg, 1986. 1991).

With the exception of Presmeg’s (1991) paper on teaching and classroom aspects of visualization, most of the research reports related to visualization at this period had a distinct psychological flavor (appropriate although interest in social and cultural aspects of learning mathematics was already growing in this association). Many of these studies involved structured or semi-structures clinical interviews with individual students for the purpose of investigating aspects of their use of visualization in the service of learning mathematics, a theme that is continued ...........