Designing Learning Environments for Developing Understanding of Geometry and Space

Richard Lehrer, Daniel Chazan

• Publisher: Routledge
• Year: 2012
• ISBN:
  9781136490583
• Edition: 1
• Topic: Education
• Subtopic: Education General

This volume reflects an appreciation of the interactive roles of subject matter, teacher, student, and technologies in designing classrooms that promote understanding of geometry and space. Although these elements of geometry education are mutually constituted, the book is organized to highlight, first, the editors' vision of a general geometry education; second, the development of student thinking in everyday and classroom contexts; and third, the role of technologies.Rather than looking to high school geometry as the locus--and all too often, the apex--of geometric reasoning, the contributors to this volume suggest that reasoning about space can and should be successfully integrated with other forms of mathematics, starting at the elementary level and continuing through high school. Reintegrating spatial reasoning into the mathematical mainstream--indeed, placing it at the core of K-12 mathematics environments that promote learning with understanding--will mean increased attention to problems in modeling, structure, and design and reinvigoration of traditional topics such as measure, dimension, and form. Further, the editors' position is that the teaching of geometry and spatial visualization in school should not be compressed into a characterization of Greek geometry, but should include attention to contributions to the mathematics of space that developed subsequent to those of the Greeks.This volume is essential reading for those involved in mathematics education at all levels, including university faculty, researchers, and graduate students.

Part II

Studies of Conceptual Development

From the questioning of basic assumptions of how we teach traditional geometric concepts to the increased interaction of reasearchers and teachers in creating research-based, teacher-designed activities, researchers with their colleagues, classroom teachers, have begun to rethink what happens in mathematics (in particular, geometry) classrooms.

In this section, we sample that range of work, looking at models of student thinking; studies of interactive, hands-on and computer-enhanced methods of exploring geometric concepts; and the emergence and growth of students’ reasoning from the primary grades through high school. In chapter 5, Pegg and Davey graft the SOLO taxonomy of modes and levels of thinking (Biggs & Collis, 1991) onto the main trunk of the van Hiele cognitive model. They examine what is perhaps the predominant theory in K–12 geometry education—the model of “levels” of teaching and learning proposed by the van Hieles—and take the position that although the van Hiele theory offers a broad framework in which to view cognitive growth in geometry, the theory is not sufficiently fine grained to account for individual differences. The synthesis they propose suggests a way to fine-tune how we look at cognitive reasoning and conceptual development.

Lehrer, Jenkins, and Osana, in chapter 6, report on a 3-year longitudinal study of elementary-grade students that questions (as do Pegg and Davey in chap. 5) the adequacy of the van Hiele model as a description of the progression of children’s thinking. Their broad-scale portrait of children’s emergent spatial reasoning skills suggests that current curricular practices in elementary school promote little conceptual development and that, for many children, the opportunities to develop and integrate spatial reasoning skills may substantially diminish before they leave elementary school.

In the next chapter, Lehrer, Jacobson, et al. suggest both that geometry education begins in students’ informal knowledge and grows in a classroom culture that depends on skilled teachers, teacher-developed models of student cognition, and a comfortable interface between research (and researchers) and teachable moments (and teachers). This particular collaboration between researchers and elementary-grades teachers worked to design activities that stimulated children’s development of spatial reasoning and their ability to conjecture. The authors trace the growth and development of children’s conceptions of area and its measure as teachers structure instruction, based on children’s informal knowledge of space and reallotment, to allow children to experience the conflict between appearance and reality. In closing, Lehrer, Jacobson, et al. note the key roles that invented notations, classroom conversation, and teachers’ practices and understanding of children’s thinking play in the development of conceptual understanding.

Many of the contributors to this section explore ways to use students’ informal knowledge as scaffolding for more abstract concepts like structure, stability, measurement. Clements, Battista, and Sarama report on a curriculum development project that focused on spatial reasoning, in particular children’s conceptual development of measurement. They propose that children’s conceptions of space are based in physical motions like walking, measuring by hand, and so on. They trace the abstraction of length and turn and their measure from situations involving paths (especially the paths made with Logo), noting that this action-based perspective affords ready integration of space and number.

Battista and Clements, in chapter 9, investigate students’ strategies for spatial structuring and their exploration of volume measure. They find that children’s reasoning about volume evolves through cognitive restructuring of the shape of space. In this instance, children learn to coordinate and integrate two- and three-dimensional cube representations, working with cube arrays and developing models of cubes as composite spacefilling “layers.”

Middleton and Corbett encourage children to explore not only traditional, basic geometric properties like linearity and congruence, but also the more abstract notions of stability (resistance to deformation) and force. Building on children’s informal notions of structure and stability, they develop engineering contexts that facilitate children’s understanding and abstraction of relationships between geometry and structure.

Working from the perspective that area and volume measure are essential for the investigation of nature, Raghavan, Sartoris, and Glaser explore the ways that students “mathematize” their explorations of contexts involving immersible objects that students float and sink. Reporting on the MARS curriculum project, which uses computer modeling to assist integration of mathematical and scientific concepts, they note that students need extensive hands-on activities and call for more attention to the interplay between mathematics and science.

Dennis and Confrey also investigate the interactive roles of tools and representation. They report a case study of the “methods, voice, and epistemology” of a high-school student as he investigates analytical geometry with curve-drawing devices that reincarnate forms of thought that have been abandoned for several centuries. The student’s investigations constitute an extended argument for physical exploration of geometric concepts, supported by attempts to representationally and symbolically redescribe what his hands produce with the curve-drawing tools.

One issue explored in several chapters in this volume is the development of argumentation. Koedinger proposes a cognitive model, reflecting that conjecture and argument serve not only discovery, but also problem solving and recall. In his study of secondary-school students, he finds that student arguments are often limited by excessive reliance on overly specific cases, perhaps because student conceptions of form are often rooted in visual prototypes, as noted in the Pegg and Davey and Lehrer et al. investigations. Koedinger also explores the consequences of different forms of teaching assistance and the role of tools for the arguments students develop and strongly suggests, as does Gravemeijer in Part I (chap. 2), that students be guided through their reinvention of concepts.

Collectively, the research reported in this section of the volume takes a wide-angle view of instructional environments and classroom cultures that attempt to promote the development of students’ understanding. Again we underline the emphasis in these studies on the importance of building (and valuing) students’ informal knowledge and on encouraging students to physically explore geometry in contexts that support reflection, conflict, and generalization.

REFERENCE

Biggs, J., & Collis, K. (1991). Multimodal learning and the quality of intelligent behavior. In H. Rowe (Ed.), Intelligence, reconceptualization and measurement (pp. 57–76). Hillsdale, NJ: Lawrence Erlbaum Associates.

5

Interpreting Student Understanding in Geometry: A Synthesis of Two Models

John Pegg
University of New England

Geoff Davey
Christian Heritage College

There is increasing evidence that many students in the middle years of schooling have severe misconceptions concerning a number of important geometric ideas (see, e.g., Burger & Shaughnessy, 1986; Dickson, Brown, & Gibson, 1984). There are many possible reasons for this. A clear divergence of opinion exists in the mathematics community about the methods and outcomes of geometry, and, as a result, textbook writers and makers of syllabuses have failed to agree on a clear set of objectives. Anecdotal evidence suggests many teachers do not consider geometry and spatial relations to be important topics, which gives rise to feelings that geometry lacks firm direction and purpose.

To some extent these problems may be due to the relatively small quantity of research (as compared with, say, research in number) that has been undertaken into students’ thinking in geometry at the school level, which, in turn, may stem from a perceived absence of a theoretical framework. Even though Piaget and his coworkers published two significant works relating to this area, The Child’s Conception of Space (Piaget & Inhelder, 1956) and The Child’s Conception of Geometry (Piaget, Inhelder, & Szeminska, 1960) and these have been followed by various studies in the field of spatial cognition, little impact on classroom practice has resulted. Part of the problem lies with Piaget’s “topological primacy theory,” on which it has proven difficult to build a school syllabus and about which there have been some fundamental doubts (see Darke, 1982).

It was a combination of these problems and their own classroom experiences in the Netherlands in the 1950s that caused husband-and-wife team van Hiele and van Hiele-Geldof to put forward a theoretical perspective for the teaching and learning of geometry. This theory is referred to universally as the van Hiele theory, and the reader is referred to Structure and Insight: A Theory of Mathematics Education (van Heile, 1986) for a detailed account of this work.

In summary, the van Hiele theory is directed at improving teaching by organizing instruction to take into account students’ thinking, which is described by a hierarchical series of levels. According to the theory, if students’ levels of thinking are addressed in the teaching process, students have ownership of the encountered material and the development of insight (the ability to act adequately with intention in a new situation) is enhanced. For the van Hieles, the main purpose of instruction was the development of such insight.

Reaction to observed inadequacies in Piaget’s formulations also inspired Biggs and Collis to explore ways of describing students’ understanding more deeply than was offered by current quantitative and qualitative methods. Their work focused attention on students’ responses rather than on their level of thinking or stage of development. This focus arose, in part, because of the substantial décalage problem associated with Piaget’s work when applied to the school learning context, and the need to describe the consistency observed in the structure of responses from large numbers of students across a variety of learning environments in a number of subject and topic areas. Their research resulted in the development of a categorization system referred to as the Structure of the Observed Learning Outcome (SOLO; Biggs & Collis, 1982). Although the SOLO taxonomy has its roots in Piaget’s epistemological tradition, it is based strongly on information-processing theories and the importance of working memory capacity. In addition, students’ familiarity with content and context plays an influential role in determining the response category.

Although at first glance there may appear to be irreconcilable differences between the two theoretical stances (namely, van Hiele was concerned with underlying thinking skills and SOLO with observable behaviors), a closer examination reveals that the two stances have much in common and that the models are complementary. A synthesis provides a fresh perspective in considering student growth in understanding. The last part of this chapter takes up this theme. First, we give a brief but inclusive review of both models. Second, we consider related research to help develop these ideas further by enabling a comparison of common features. Finally, implications of the findings are explicated and discussed.

THE VAN HIELE THEORY

Since the early 1980s, the van Hiele theory has been under extensive investigation in the western world (for a detailed summary see, e.g., Clements & Battista, 1992; Hoffer, 1981). Although there are two fundamental aspects to the theory, namely, levels of thinking (a hierarchical series of categories that describe growth in student thinking) and five teaching phases (which help guide activities that lead students from one level to the next), the main focus of the research has been on the nature or existence of the levels and the assumptions that underpin them. The initial conception envisaged five levels (see van Hiele, 1986), described here as they relate to two-dimensional figures:

Level 1. Students recognize a figure by its appearance (i.e., its form or shape). Properties of a figure play no explicit role in its identification.

Level 2. Students identify a figure by its properties, which are seen as independent of one another.

Level 3. Students no longer see properties of figures as independent. They recognize that a property precedes or follows from other properties. Students also understand relationships between different figures.

Level 4. Students understand the place of deduction. They use the concept of necessary and sufficient conditions and can develop proofs rather than learn them by rote. They can devise definitions.

Level 5. Students can make comparisons of various deductive systems and explore different geometries based on various systems of postulates.

Although these descriptions are content specific, van Hiele’s levels are actually stages of cognitive development: “the levels are situated not in the subject matter but in the thinking of man” (van Hiele, 1986, p. 41). Progression from one level to the next is not the result of maturation or natural development. It is the nature and quality of the experience in the teaching/learning program that influences a genuine advancement from a lower to a higher level, as opposed to the learning of routin...

Table of contents

• Cover Page
• Half Title Page
• Series Page
• Title Page
• Copyright Page
• Contents
• Preface
• I Why Teach Geometry?
• II Studies of Conceptual Development
• III Defining A New Semantics of Space Computers, Software, and the Electronic World
• Epilogue
• Author Index
• Subject Index