The Learning and Teaching of Geometry in Secondary Schools

A Modeling Perspective

Pat Herbst, Taro Fujita, Stefan Halverscheid, Michael Weiss

• Publisher: Routledge
• Year: 2017
• Print ISBN: 9780415856904
• eBook ISBN: 9781351973526
• Topic: Didattica
• Subtopic: Didattica generale

IMPACT (Interweaving Mathematics Pedagogy and Content for Teaching) is an exciting new series of texts for teacher education which aims to advance the learning and teaching of mathematics by integrating mathematics content with the broader research and theoretical base of mathematics education.

The Learning and Teaching of Geometry in Secondary Schools reviews past and present research on the teaching and learning of geometry in secondary schools and proposes an approach for design research on secondary geometry instruction.

Areas covered include:

• teaching and learning secondary geometry through history;
• the representations of geometric figures;
• students' cognition in geometry;
• teacher knowledge, practice and, beliefs;
• teaching strategies, instructional improvement, and classroom interventions;
• research designs and problems for secondary geometry.

Drawing on a team of international authors, this new text will be essential reading for experienced teachers of mathematics, graduate students, curriculum developers, researchers, and all those interested in exploring students' study of geometry in secondary schools.

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1

THE DISCOURSE OF TEACHING AND LEARNING SECONDARY GEOMETRY THROUGH HISTORY

1.1. Introduction

On a winter morning in 1959, one of the twentieth century’s greatest French mathematicians stood before the leading mathematicians, educational policy makers, and secondary mathematics teachers of 18 nations, and declared war on a text that had stood at the center of the traditional mathematics curriculum for over 2,000 years. At the Royaumont Seminar, convened in Asnières-sur-Oise by the Organisation for European Economic Co-operation (OEEC), Jean Dieudonné—a key member of the pseudonymous Bourbaki collective, who also wrote in the late 1950s on the teaching of geometry at school—announced: “A bas Euclide! Mort aux triangles!” (“Down with Euclid! Death to triangles!”) (Roberts, 2006, p. 157; see also Fried, 2014). In the subsequently published version of Dieudonné’s address (Dieudonné, 1959) this call was presented in the slightly more moderate form “Euclid must go!” (p. 35), but the critique was no less pointed. Despite claiming to hold the “deepest admiration for the achievement of the Greeks in mathematics”, Dieudonné nevertheless described Euclid’s geometry as little more than “a chaotic heap of results with no significance except as scattered relics [. . .] or an obsolete approach” (p. 35). The school curriculum, argued Dieudonné, needed to be purged of the “dead-weight of ‘pure geometry’” (p. 38) to make space for a “modern” curriculum.

Dieudonné’s call for revolution influenced much of the discourse in mathematics education throughout the “New Math” era of the 1960s, and the ways in which students studied and learned geometry did indeed undergo profound transformations over the subsequent decades. Hansen (1998) refers to the ensuing changes in the curriculum (particularly in Europe) as a “downgrading of the teaching of geometry at all school levels” resulting from “the algebraization and formalization of mathematics in school” (p. 235). Notwithstanding this significant change in the role of geometry in school though, Euclid—perhaps needless to say—has not gone away. Despite decades of curricular innovation and multiple waves of reform efforts both within and across national boundaries, the geometry of Euclid continues to play a significant role in the secondary mathematics curriculum of virtually every country.

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What was the background to Dieudonné’s call for revolution? And why does his proposal still resonate today, more than half a century later? This chapter tries to identify the forces that have shaped the geometry curriculum, instruction, and learning from a historical and an international point of view. The discourse of teaching and learning geometry over the centuries provides a rich, at times unexpected, source of examples and indicators of why and how geometry can be taught and learned, and document which decisions regarding the teaching and learning of geometry can and must be made.

1.2. Overview of This Chapter

To structure the discussion in this chapter, we make use of how historians and mathematics educators have identified different reasons for curricular developments. Hansen (1998) starts from the observation that curricular debates, in general, must take into account four different perspectives:

• the epistemological perspective, which sketches the impact of the development of geometry into a scientific, logically coherent discipline (Glymour, 1977) on its teaching and learning;

• the pedagogical perspective, which focuses on central developments in the history of teaching geometry (Barbin and Menghini, 2014), from the design of textbooks to assuming the perspective of individual learners;

• the technological perspective, which includes such subjects as the influence of dynamic geometry on mathematics curricula; and

• the political perspective, which considers the political, cultural, and societal settings in which school curricula are developed (Damerow and Westbury, 1985).

Of course, the teaching and learning of geometry goes back much further than compulsory school attendance and curricula in mathematics. The study of geometry in its practical applications to agriculture, construction, and astronomy is indeed virtually as old as civilization itself. In this book, we are interested primarily in the teaching and learning of geometry in settings that resemble schools as we know them, and for that reason our narrative will focus largely on developments since the nineteenth century. Looking more specifically at the case of school geometry, it seems appropriate to follow variations of these perspectives according to three historical periods, demarcated by the late nineteenth century’s so-called “foundational crisis” on the parallel axiom and by the twentieth century’s introduction of the computer in the classroom. As we will see below, the foundational crisis was more of an extended period than a single event: Even before geometry was reestablished on a rigorous foundation in the early twentieth century, new, influential geometry courses had already begun to appear and shape the way geometry was taught and understood. The presence of computers in the classroom makes the third historical period reflect more recent discussions on curricular developments in view of the options that dynamic geometry software provides.

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In their account of the history of teaching geometry, Barbin and Menghini (2014) identify two different aspects of geometry: The “deductive/rational” and the “pseudo-practical/intuitive” approach, which built up a “tension” that “manifested itself many times throughout history” (p. 473). These aspects structure our considerations of the teaching and learning of geometry till the resolution of the crisis on the parallel axiom. Curricula in mathematics, in the modern sense, began to appear in the second half of the nineteenth century. Hansen (1998) identified five objectives as most important to the shaping of geometry curricula: To “establish knowledge”, to “prepare [. . .] for applications”, to present classical “milestones in the development of geometry”, to “develop skills and abilities”, and to “strengthen logical thinking and deductive reasoning” (p. 236).

Apart from the interest in milestones, these objectives correspond approximately with the four modal arguments identified by González and Herbst (2006), who studied the justifications put forward in the United States between 1890 and 1980 for why geometry should be studied in secondary schools: (1) a mathematical argument, which stressed that geometry could provide a context for students to experience the kinds of activities that are typical of mathematicians; (2) a formal argument, based on the claim that geometry trains students in deductive logic; (3) an intuitive argument, which emphasized the power of geometry to provide vocabulary for naming, and experiences for exploring, the real world; and (4) a utilitarian argument, which stressed the geometry that is especially useful in students’ future working worlds.

With respect to more recent curricular developments of geometry, Galuzzi and Neubrand (1998) point to the following influential areas: (1) changes in the development of mathematics itself, (2) greater emphasis on applications and modelling, (3) the debate over fundamental ideas, (4) constructivist ideas of learning, (5) the focus on mathematics as a human activity, (6) geometry seen by students as an empirical theory, and finally (7) the impact of the computer. In the fourth section of this chapter, these areas are investigated from an international perspective by re-examining nine countries whose geometry curricula have been previously compared (Hoyles et al., 2002). Since all these countries—Canada, Great Britain, France, Germany, the Netherlands, Japan, Poland, Singapore, and Switzerland—have revised their curricula since Hoyles et al’s (2002) comparison, some of those more recent developments are described and the main differences among them are articulated. In the context of that comparison, we give special attention to the options dynamic geometry software affords for the teaching and learning of geometry by discussing its impact on the development of the curricula of the countries mentioned above.

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1.3. The Development of Geometry up to the So-Called Foundational Crisis

What is geometry? The subject of geometry is vast, and related to many phenomena both inside and outside mathematics. This section takes stock of major themes and developments in the subject matter as they concern in particular its teaching and learning. The chapter does not aim to cover the long history of geometry; we refer to Barbin and Menghini (2014) for a systematic review of the history of the teaching and learning of geometry. But for their central role in this book, we deal in particular with the history of reasoning and figural concepts.

1.3.1. The History of Reasoning and Figural Concepts

Applications in craftsmanship and astronomy were the source of pragmatic, hands-on geometric rules since the beginning of highly developed civilizations in ancient Egypt, Arabia, and India. Several facts and rules of geometry are attributed to the ancient Greeks, probably because of their elaborate educational system. According to Eudemus, Thales (around 580 BCE) is said to have discovered and formulated important geometric results: “(1) that a circle is bisected by its diameter; (2) that the angles at the base of an isosceles triangle are equal [. . .], (3) that when two straight lines intersect, the vertically opposite angles are equal; (4) that if two triangles have two angles and one side equal, the triangles are equal in all respects.” (Dicks, 1959, p. 302). Over the course of the next few centuries, philosophical schools linked research, tradition, and education. For geometry, Hippasus of Metapontum (a follower of Pythagoras’ school between 530 and 450 BCE) codified results that had already been partly known to the Babylonians (approximately 1600−1300 BCE) as utilitarian facts; of those results, the Pythagorean theorem is surely the most important example (see Eves, 1990). The idea of a demonstration or proof gradually became an integral part of philosophical reasoning and ultimately led to the development of important mathematical concepts, such as commensurability and irrationality. However, Szabo’s thesis that the idea of a proof is an invention of “Greek mathematics” (Szabo, 1978) has been contested, for instance by Waschkies (1989).

Geometry was considered particularly important in the philosophical schools of ancient Greece. Plato’s theory of ideal forms shows the influence of geometry on philosophical thought more broadly. Geometry was also closely connected to teaching and learning: The dictum “Let no one ignorant of geometry enter here” is said to have been inscribed over the door of Plato’s Academy (Smith, 1999, p. 131). In Plato’s Meno, Socrates engages Meno’s slave in a discussion of the problem of doubling the area of a square as an illustration of Socrates’ view of the origins of knowledge. The episode is a famous example of what has come to be known as the Socratic method, and also shows some features that are typical in teaching and learning of geometry. In particular, Plato’s dialogue depicts the activity of hypothesizing in the course of defining and solving problems (Wyller, 1964).

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According to Plato, geometric “shapes had an independent life in the world of ideas” (Hansen, 1998, p. 12). By focusing further attention on the distinction between the world of ideas and the world of experience and sensation, Aristotle played a crucial role in the development of a foundation for mathematics. The philosophical concept of the world of ideas made it possible to develop an abstract concept of shape: “The concrete figures live in the real world and the abstract in the world of ideas” (Hansen, 1998, p. 12). That is to say, the triangles and circles whose properties are studied in geometry were understood to be distinct from the physical and diagrammatic representations of them; the latter were regarded as imperfect realizations of the pure abstract forms that were the proper focus of mathematical study. This marked the beginning of mathematical thinking about objects that exist solely in the world of ideas, and the ontological status of such abstract objects was assured solely by the soundness of the reasoning that was applied to them:

From the perspective of ontology or of epistemology the question may be asked: ‘What is the nature of mathematical objects?’ To ask the question ontologically amounts to asking for the real subjects, the things in the world, with which mathematics deals. Epistemologically the question is more likely to be directed at mathematical reasoning: ‘What is mathematical reasoning about?’

Plato seems to have given the same answer to the question from both perspectives. lt is even possible that his answer to the ontological question was inferred from his epistemological analysis of mathematics in some way like the following: Mathematicians reason as if they were dealing with objects that are different from all sensible things, perfectly fulfill given conditions, and are apprehensible by pure thought; mathematics is correct; therefore, there are such objects. Argument of this kind is also characteristic of the modern mathematical Platonist.

(Mueller, 1970, p. 156)

Aristotle launched a systematic programme of organizing mathematics as a system based on definitions and postulates, and theorems deductively derived from that basis. About a century later, Euclid’s Elements appeared as a compilation of previous works in geometry and number theory that had originated in the philosophical schools. The Elements is the oldest conserved systematic text on mathematics, and its impact in both shaping and following the ideal of thoroughly axiomatic reasoning cannot be overstated (Mueller, 1991). The Euclidean practice of deducing everything from a relatively small set of axioms, postulates, and definitions is often regarded as the preeminent example of Aristotle’s methodology. However, there are significant differences between the use of terms in Aristotle and Euclid (Szabo, 1974) and in the logical organization of the Elements (Torretti, 1978, p. 5), which might indicate that the philosophical school or schools from which Euclid’s Elements originated did not follow Aristotle’s concepts from the beginning. In addition, the Elements can be (and indeed has been) critiqued as an imperfect execution of Aristotle’s vision, insofar as it contains numerous tacit assumptions and other logical gaps. Some of these gaps may stem from the fact that the Elements drew on independent earlier sources whose contents and organization did not always fit neatly together; others may have their source in the inherent difficulty of studying the properties of geometric figures without relying on visually obvious features of those diagrams. Indeed, it was not until the early 20th century that a fully rigorous theory of geometry redressed the perceived flaws in Euclid’s Elements.

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Table of contents

• Cover Page
• The Learning and Teaching of Geometry in Secondary Schools
• IMPACT (Interweaving Mathematics Pedagogy and Content for Teaching)
• Title
• Copyright
• Contents
• IMPACT—Series Foreword
• Acknowledgments
• Introduction
• 1 The Discourse of Teaching and Learning Secondary Geometry through History
• 2 Geometric Figures and Their Representations
• 3 Students’ Thinking and Learning in Geometry
• 4 Teaching Practice and Teacher Knowledge in Geometry Instruction
• 5 Improving the Teaching and Learning of Geometry in Secondary School Classrooms
• 6 A Conclusion and a Beginning: Doing Research on the Teaching and Learning of Secondary Geometry
• References
• Index