Gutiérrez, A. (1996). “Visualization in 3-dimensional geometry: In search of a framework”

From MathWiki

Gutiérrez, A. (1996). “Visualization in 3-dimensional geometry: In search of a framework,” in L. Puig and A. Guttierez (eds.) Proceedings of the 20th conference of the international group for the psychology of mathematics education (vol. 1, pp. 3-19). Valencia: Universidad de Valencia.

Available through York’s eResources: (it is a large file, the article in question is on page 41 of the PDF file, but perhaps some of the other articles may be of interest as well!)

Table of contents

Visualization: A Developing Framework

In his work on visualization and spatial thinking (which, in his framework, are equivalent), Gutiérrez (1996) explores the role that geometry software can potentially play in the development of these skills. Based on a literature survey of relevant psychological and educational literature, Gutiérrez (1996) initially outlines what he means by visualization as “the kind of reasoning activity based on the use of visual and spatial elements, either mental or physical, performed to solve problems or prove properties” (p.9). He asserts that a definition is necessary since “there is no general agreement about the terminology to be used in this field” (p.4). In a coherent fashion, Gutiérrez reconciles the varying theoretical approaches to understanding visualization, and finds that many of these seemingly different perspectives actually share a lot of common ground. Thus, based on the available literature, Gutiérrez (1996) suggests that four main elements unify visualization:

  • Mental Images – “any kind of cognitive representation of a mathematical concept or property by means of visual or spatial elements” (p. 9). Interestingly, mental images include kinesthetic images, which are “created, transformed or communicated with the help of physical movements” (p.7), and dynamic images – “those images with movement in the mind” (p.7). Both of these ‘types’ of images can be relevant to a discussion of the use of physical and virtual manipulatives.
  • External Representations – “any kind of verbal or graphical representation of concepts or properties including pictures, drawings, diagrams, etc. that helps to create or transform mental images and to do visual reasoning” (p.9-10).
  • Process of Visualization – “a mental or physical action where mental images are involved” (p.10). There are two central processes of visualization – “visual interpretation of information” (used in creating mental images), and “interpretation of mental images” (used to “generate information”) (Gutiérrez, 1996, p.10).
  • Abilities of Visualization – required by students “to perform the necessary processes with specific mental images for a given problem” (p.10). The nature of these abilities is dependant on the specific characteristics of the problem being solved, but Gutiérrez (1996) identifies the main abilities as being:
    • “Figure-ground perception” – the ability to identify and “isolate” a specific figure out of a complex background (p.10)
    • “Perceptual constancy” – the ability to realize that some characteristics of an object are independent of “size, colour, texture, or position” (p.10)
    • “Mental rotation” – “the ability to produce dynamic mental images and to visualize a configuration in movement” (p.10).
    • “Perception of spatial positions” – the ability to relate figures (“object, picture, or mental image”) to oneself.
    • “Perception of spatial relationships” – the ability to relate several figures (as above) to “each other, or simultaneously to oneself” (p.10)
    • “Visual Discrimination” – the ability to compare several figures and to determine how they are similar and how they are different.

Process of Visualization

Gutiérrez (1996) concludes his theoretical treatment of visualization in this paper with a diagrammatic representation of the process of visualization in solving a mathematics problem. For copyright purposes, I cannot include the diagram here, but the paper is available through York’s eResources (the diagram can be found on page 11 of the article). The following is the accompanying textual description:

“The statement of the task is interpreted by the students as an external representation
suitable to generate a mental image. This first image initiates a process of visual
reasoning where, depending on the task and students’ abilities, they use some of their
visual abilities to perform different processes, and other mental images and/or external
representations may be generated before the students arrive at the answer” (Gutiérrez,
1996, p.10).

Research into visualization skills of students using software: A Case Study

Gutiérrez (1996) begins the research component of his paper by noting that enough research has not been done into the (optimal?) role of visualization in the learning and teaching of 3D geometry. While some research highlights students’ difficulties in moving between 3D objects and their 2D representations, Gutiérrez (1996) claims that research needs to look into the potential of computer software to enhance students’ visualization skills. Gutiérrez believes that the plethora of different representational positions possible with computer software create a rich spatial experience for the teaching and learning of visualization. Yakimanskaya’s (1991) work on the development of spatial thinking in schoolchildren suggests that the “the richer and more diverse the store of spatial representations, the more highly perfected the methods of creating representations and the easier it is to use images” (in Gutiérrez, 1996, p.12). Thus, Gutiérrez (1996) claims,

“When a person handles a real 3-dimensional solid and rotates it, the rotations made with
the hands are so fast, unconscious, and accurate, even in the case of young primary
school students, that one can hardly reflect on such actions; However, a software
package limiting the directions of rotation forces the students to devise strategies of
movement and to anticipate the result of a given turn” (p.12, emphasis added).

Case study findings

Thus, in his experiments, Gutiérrez (1996) uses several figures, including a cube with different pictures on its faces, embedded in a HyperCard stack (the use of this program reflects the age of this report!). In the HyperCard environment, students were allowed to rotate the cube 90 degrees along its axes, using pre-constructed control buttons. Gutiérrez (1996) includes accounts of a student in grade 2, and a student in grade 8. The second grader was asked to rotate the cube to match a target position. The second grader shows an ability to create mental images of the cube by predicting the location of a picture after rotation – but only with one face at a time (usually the front face). The student in grade 8 was given the same task, but with a greater degree of difficulty, by being asked to “predict the position of the cube after one or more rotations” (Gutiérrez, 1996, p.16). The most salient feature of this student’s activity was “the extensive use the student makes of her hands to show the rotations, i.e. of kinesthetic images” (p.17).

This last point is central to our discussion of the use of virtual and physical manipulatives. In her activity, the student in grade 8 repeatedly used her hands to show the rotations that she was examining. At one point, the student even rotated the piece of paper that was in front of her, in order to examine the effect of a specific rotation. This observation highlights the importance of using hands-on materials concurrently with the use of technology. As Gutiérrez contends, technology can provide a valuable and rich experience of spatial geometry for students. However, the experience of the eighth grader supports the centrality of kinesthetic mental images and the use of concrete materials.

Gutiérrez’s Conclusions: Further Research Crucial

Gutiérrez (1996) concludes that in the drive to show that “visual reasoning in mathematics is important in its own right,” (Dreyfus, 1991, p.46, in Gutiérrez, 1996, p.17) we are in need of a theory from mathematics education that attempts to answer several key questions (extracted and paraphrased):

1. How are mental images of mathematical concepts formed?

2. How can students gain mastery in creating and using mental images?

3. What role do mental images play in the understanding of mathematical concepts and in problem solving?

4. When is visualization more (or less) useful to students than analytical methods?

5. How can mental images be transmitted? (Communicated or taught)

Gutiérrez’s work is interesting and certainly relevant to our work: it provides a literature survey of relevant materials on spatial reasoning, and it would be useful to see how far the literature has come since then in answering some of the questions raised here (in fact, many of the questions that we have raised in our own study groups!). Perhaps we (there is a lot of material here, for those who are interested) can look into other relevant research from more recent International Group for the Psychology of Mathematics Education (PME) conferences (all available through York’s eResources).


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