ICME 11 TSG 12
HELPING ELEMENTARY TEACHERS DEVELOP VISUAL AND SPATIAL SKILLS FOR TEACHING GEOMETRY
Walter J. Whiteley / Margaret Sinclair/
Stewart Craven / Lily Moshe
/
Anna Dutfield / Melissa Seco
/
York University, Canada
This paper concerns findings from our research into the development of visual and spatial skills related to the teaching of geometry. The study involved 18 practicing elementary teachers, who, over three day-long sessions, worked on tasks involving transformational geometry concepts. The tasks focused on the use of physical and dynamic models to explore 2- and 3-D relationships. Although, we were encouraged by participant successes in re-thinking misconceptions, and in developing communication and representation skills, observations and work samples showed that participants had significant difficulties in applying visual and spatial reasoning to geometric tasks. We recognized that teachers failed to use key strategies related to: inquiry, communication, and connections. We describe a number of strategies that we believe will help teachers effectively carry out investigations involving visual and spatial reasoning, and thereby enable them to develop a richer background in the domain of geometry. We used some of these strategies to advantage during the study, others are proposed based on our observations of participants’ difficulties in completing various tasks.
There are significant questions about how to help educators build visual and spatial reasoning. We know that the ability of a learner to make visual connections is strengthened by experiences with 2D and 3D representations; however, we have found no framework for selecting and structuring these experiences for elementary school teachers.
Our research concerns elementary teachers’ development of visual and spatial abilities for teaching geometry. We are investigating this development by observing teachers engaged in co-operative experiences with 3D and 2D tools such as physical models, diagrams and interactive computer sketches. Our research goals are 1) To explore the advantages and disadvantages of multiple external visual/spatial representations in helping learners grasp, and communicate about key mathematical concepts; 2) To understand better how to support learners in developing internal visual/spatial representations of mathematics concepts; and ultimately, 3) To develop a set of sequenced tasks to provide effective learning experiences for teachers. In this paper we focus on the second goal; we describe a number of strategies that we believe will help teachers effectively carry out investigations involving visual and spatial reasoning and thereby enable them to develop a richer background in the domain of geometry. We used some of these strategies to advantage during the study; others are proposed based on our observations of participants’ difficulties in completing various tasks.
Literature
According to Goldenberg, Cuoco and Mark (1998), mathematical pictures and diagrams are difficult to interpret because they contain a great deal of information represented in a concise but “nonsequential” format. We must first organize the visual information that we receive. Also, the meaning we take from an image depends in part on what we already know about what we are looking at (Hoffman, 1998; Wheatley, 1998; Whiteley, 2004). Dimension is a factor: initial human experiences involve interpreting a 3D world but school mathematics focuses almost entirely on 2D representations. Communication is a factor: if we are working with a mathematical diagram created by someone else, we must interpret the underlying relationships between that diagram and the object, scenario or concept it represents.
Research shows that increased attention to visual and kinesthetic approaches helps students make connections between various representations of underlying mathematical concepts (cf., Clements & Battista, 1992; Goldenberg & Cuoco, 1998). For example, researchers have found that visualisation allows students to acquire a knowledge of models that is deeper and more robust (cf., Nemirovsky & Tierney, 2001; Pea, 1993; Sinclair, 2003), and the addition of motion, through technology or through body movement (e.g., gestures) offers additional visual information to help students make sense of time-space concepts (Kaput & Roschelle, 1998; Radford, Demers, Guzman, & Cerulli, 2003).
Notwithstanding these results there is a lack of information about how to help teachers develop the necessary background. The current study draws on the idea of a “hypothetical learning trajectory”; Clements and Sarama (2004) note, “…we conceptualize learning trajectories as descriptions of children’s thinking and learning in a specific mathematical domain and a related, conjectured route through a set of instructional tasks … created with the intent of supporting children’s achievement of specific goals in that mathematics domain…” (p.83). Our long term goal is to create sequences of tasks to support adults’ thinking and learning in the domain of geometry.
The Study
In light of research on mathematical knowledge, visual reasoning, and task design we designed/modified tasks that used images, animated models, and physical models in investigations of transformational geometry concepts. We created a sequence of these tasks, and tested them with 18 elementary teachers, most of whom had very little mathematics background, over the course of three day-long sessions, spaced approximately a month apart. Information on the tasks and timing over the three sessions is provided in Table 1.
Table 1
Task information
Day / Name / Description1,2, & 3 / Tangram activities / Shapes created with tangrams were shown briefly; participants drew what they remembered.
1 / Buildings / Participants investigated connections between structures built with linking cubes, isometric drawings, and grid representations.
1 / Reflection symmetries / Participants folded polygons to investigate lines of reflection; they used pattern blocks to create designs with one or more reflections; they then identified planes of symmetry in a (general) rectangular prism and a cube.
2 / Rotation symmetries / Participants used acetates and rosettes to investigate rotational symmetry; they then used cut-out hands on balsa rods to investigate center of symmetry.
2 / Reflections and Rotations / Participants investigated the relationship between reflections and rotations.
2 / Tessellation activity / Participants investigated rotations and reflections in infinite tessellation patterns.
3 / Dilation activity with GSP / Participants investigated the center of dilation, and the ratios of lengths and areas in similar figures using pre-constructed Geometer’s Sketchpad sketches.
3 / Filling activity / Participants filled triangular and square-based pyramids tilted at various angles and observed the shape of the top surface, its dimensions, and its area. This task also addressed concepts of scaling and dilation.
Qualitative data was collected through observation, participant work samples, and videotape; quantitative data was collected via a questionnaire, and pre- and post- tests focused on 2D and 3D mathematics concepts. The focus of this paper is the analysis of video data and observation field notes as it reveals the approaches used by participants.
Overview of Findings
The data revealed that participants had significant difficulties in three key areas: inquiry, making connections and communication. In general, participants lacked systematic inquiry strategies, and their ability to conjecture was limited by prior knowledge of procedures, and common benchmarks (e.g., angles of 30º, 60º, 90º). They frequently attempted to use formulas (even when there wasn’t an appropriate one), indicating that they either didn’t trust and/or didn’t know visual approaches. Misconceptions in interpreting characteristics of 2D and 3D objects were pervasive. For instance, we found that participants had very little experience with 3D motions and symmetries, but were also tentative about these concepts in 2D. Communication was hampered by the misuse of terms, and by the lack of common vocabulary. In addition, participants struggled with representing their reasoning verbally, with gestures, by drawing, and in writing. Connections to other mathematical ideas or to other representations – even within the activities - were seldom made.
Nevertheless, we found that the participants grew in their understanding of spatial (both 2D and 3D) relationships. All were able to move back and forth between a 3D building and a 2D grid. Working in pairs and in group discussions participants began to use appropriate language, (e.g., order of rotation), and to differentiate between, for example, plane of symmetry versus line of symmetry, and congruent versus similar. Through paper folding and use of manipulatives many participants became aware that some of their ideas, e.g., that symmetry only applies to reflection, and that objects can’t have rotational symmetry without having reflective symmetry, were misconceptions. Through opportunities to explore unfamiliar concepts with physical objects some participants experienced AHA! moments. For example, on exploring a dynamic geometry sketch on dilation, several noticed the connection to filling a tetrahedron with water, and paper folding led some to recognize that a parallelogram does not have a line of symmetry. We contend that critical experiences contributing to the growth of participant understanding were a) hands on work with a variety of 2D and 3D materials/models, b) peer interaction, and c) facilitated discussion.
Recommendations
Based on our findings, we believe that in order to fully engage in and benefit from visual/spatial experiences, teachers must possess a repertoire of investigative strategies. In particular they require approaches to: inquiry, making connections, and communicating that are appropriate for visual/spatial contexts. The focus of this section is on the teacher; that is, an adult who possesses a range of knowledge and experience, and who is able to metacognitively reflect on the learning process.
Inquiry
In many situations we noticed that the teachers, while not averse to handling the materials, did not know how to interact with the materials to see the mathematics. This could be related to the participants’ views of learning in general; that is, as adults they have come to equate learning with “book work” as opposed to physical exploration.
For adult learners we contend that the following strategies build the required familiarity with the objects and also with investigation processes, and prepare teachers to recognize whether/when a child is doing something sensible in an exploration: 1. Physical inquiry strategies; 2. Organizational strategies; 3. Experimentation strategies.
Physical inquiry. Unfortunately adults can be more inhibited than children; they fear doing something wrong, or just doing something different. When participants worked with The Geometer’s Sketchpad (Jackiw, 1991), the tendency to hold back or be tentative about taking the initiative to explore, was pronounced. Arzarello et al. (1998) found that secondary students who produced good conjectures moved onscreen objects purposefully. This intentional approach emerged from an earlier exploratory modality that the researchers called “wandering dragging” (p. 37) – a form of play. Our observations of the study participants suggest that they also needed a period of “play” with the materials; however, they were reluctant to engage in unstructured activity.
Another source of difficulty was exploring the mathematics within the physical models. Participants’ geometry experiences had in most cases been limited to reasoning with 2D representations and not with actual objects. As a result, the participants lacked exploration approaches. Throughout the sessions we encouraged them to a) look at objects and diagrams from different points of view; b) to systematically turn the object or diagram, c) to re-position themselves to view the object or diagram from eye level, from the back, or from the top. These simple strategies are key to gathering the information needed to “see” the mathematics.
Organizing. In the study, we found that many participants lost track of what they were doing and thinking (e.g., in counting vertices or planes of symmetry, in determining the order of rotational symmetry) or failed to grasp the essence of the whole. They did not use effective organizational strategies, the most important of which are:
1. Tracking by parts. This strategy includes mentally and/or physically taking an object or diagram apart and tracking various pieces, pairings or subgroups. In connection with this strategy “marking” (i.e., with sticky dots, markers, elastics, tape) can facilitate counting.
2. Looking at/forming the whole. Here, as noted earlier, re-positioning oneself or the object is an important strategy for recognizing overall relationships. Another strategy involves using one object as a marker while moving another, e.g., for visual comparison of initial and later positions of the object.
We contend that being able to go back and forth between seeing the whole and noticing the parts is an important goal. Glaser (1992) notes that while novices often fail to see the big picture experts are able to employ decomposing and chunking to organize information. Our findings suggest that the Tangram activities (Wheatley, 1990) helped the participants build these strategies. An image made of tangram pieces was shown for a brief time covered, sketched and shown again for editing. The teachers shared their strategies for remembering the image. Some focused on “seeing the whole” -- “a butterfly” or “a sailboat” while others noticed the parts, e.g., two small triangles, a large triangle and a parallelogram. With subsequent images many participants adopted strategies that others had mentioned.
Experimenting. Some investigations involve experimenting, that is, actively changing some feature of the situation and analysing the result. We contend that the experimenter needs to be: a) open to variation, and b) able to pose questions.
Being open to variation is especially important for elementary teachers. According to Chaille and Britain (2003) young children learn what works by experimenting, by changing some aspect of a situation (e.g., the slope of a ramp, the height of a tower). For experimenting, children and adults require a range of options for action and thought. During the study we recognized that the teachers constrained their actions and responses to a limited set of possibilities. They often stopped experimenting once they had an answer, rather than look for more. In addition to encouraging the teachers to consider other possibilities, e.g., what additional block patterns might have the same rotational or reflectional symmetries, we introduced the strategy of considering extreme cases. For instance, we suggested that participants tilt their water-filled pyramids in more directions and through a wider range of angles. This strategy was important for the participants because it forced them into an area of uncertainty where they needed to think about what they saw.
Another experimental strategy is considering a simpler or alternate situation. We encouraged use of this strategy in the study by, for instance, drawing participants’ attention to 2D analogs of 3D tasks.
Many researchers have talked about the importance of having students pose problems (cf., Lavy & Bershadsky, 2003). In the study, participants initially worked from task questions set by the research team, e.g., “How many lines of symmetry did you discover when you folded each shape?” During the investigations research team members asked additional questions to help pairs expand their investigation, e.g., “What happens if you tilt it further?” We contend that adult learners can strengthen their investigative skills by formulating such “what if?” questions throughout an exploration.