Recent Developments on Introducing a Historical Dimension in Mathematics Education

Recent Developments on Introducing a Historical Dimension in Mathematics Education

Victor Katz
Constantinos Tzanakis
Series: MAA Notes
Volume: 78
Copyright Date: 2011
Edition: 1
Pages: 290
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  • Book Info
    Recent Developments on Introducing a Historical Dimension in Mathematics Education
    Book Description:

    Recent Developments on Introducing a Historical Dimension in Mathematics Education consists of 24 papers (coming from 13 countries worldwide). The volume aims to constitute an all-embracing outcome of recent activities within the HPM Group (International Study Group on the Relations Between History and Pedagogy of Mathematics). We believe these articles will move the field forward and provide faculty with many new ideas for incorporating the history of mathematics into their teaching at various levels of education. The book is organized into four parts. The first deals with theoretical ideas for integrating the history of mathematics into mathematics education. The second part contains research studies on the use of the history of mathematics in the teaching of numerous mathematics topics at several levels of education. The third part concentrates on how history can be used with prospective and current teachers of mathematics. We also include a special fourth part containing three purely historical papers based on invited talks at the HPM meeting of 2008. Two of these articles provide an overview of the development of mathematics in the Americas, while the third is a study of the astronomical origins of trigonometry.

    eISBN: 978-1-61444-300-1
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-vi)
  2. Preface
    (pp. vii-x)
  3. Table of Contents
    (pp. xi-xvi)
  4. 1 Teaching with Primary Historical Sources: Should It Go Mainstream? Can It?
    (pp. 1-8)
    David Pengelley

    I am truly honored to be asked to speak on integrating the history of mathematics in mathematics education. Advocating the teaching of mathematics using history is presumably not very controversial at this conference, more like “preaching to the choir”, as one says in English. But I wish to be somewhat provocative, perhaps even controversial, by suggesting a dream I have had for some time, that all students should learn the principal content of their mathematics directly from studying primary sources, i.e., from the words of the original discoverers or creators of new mathematics, as is done in the humanities, where...

  5. 2 Dialogism in Mathematical Writing: Historical, Philosophical and Pedagogical Issues
    (pp. 9-16)
    Evelyne Barbin

    Mikhail Bakhtin (1895–1975) was a well-known Russian literary critic and semiotician. The notion of dialogism was mainly developed in his paper entitled “The Problem of Speech Genres” and written in the years 1952–1953, which was translated into French in 1984 and English in 1986. Here, Bakhtin explained that every utterance may be considered as a rejoinder in dialogue. Firstly, every utterance must be regarded as a response to preceding utterances in a given sphere of communication. Secondly, every utterance is oriented toward the response of the others. This notion of dialogism is linked to three major notions named...

  6. 3 The Process of Mathematical Agreement: Examples from Mathematics History and An Experimental Sequence of Activities
    (pp. 17-28)
    Gustavo Martinez-Sierra and Rocío Antonio-Antonio

    Usually systematization processes are interpreted as processes beyond the processes of mathematical discovery. For example, Mariotti [9] establishes two moments for the production of mathematical knowledge: “. . . the formulation of a conjecture, as the core of the production of knowledge, and the systematization of such knowledge within a theoretical corpus.” In this same vein is to contrast the argumentation process of a conjecture with the process of a theorem proof [1].

    We proceed from the consideration that, for purposes of learning, there are propositions whose validity can be established from the outset as true to the need to...

  7. 4 Researching the History of Algebraic Ideas From an Educational Point of View
    (pp. 29-44)
    Luis Puig

    Since the early 1980s, my colleagues and I have been studying the history of algebraic ideas as a component of our research on the teaching and learning of school algebra¹. In Filloy, Rojano and Puig [6, ch. 1, ch. 3 and ch. 10], we discuss in some detail in which sense our study of the history of algebraic ideas is made from the point of view of mathematics education. What do we mean by studying the history of mathematics from the point of view of mathematics education? First, we mean that the problems of the teaching and learning of algebra...

  8. 5 Equations and Imaginary Numbers: A Contribution from Renaissance Algebra
    (pp. 45-56)
    Giorgio T. Bagni

    G.T. Bagni (1958–2009) died on 10 June 2009 in a bicycle accident, while this paper was under review. Because of his untimely death, we do not know how he would have taken into account the reviewers’ comments. However, we would have suggested a somewhat different structure of the paper and the clarification of several specialized terms, not expected to be understood by many readers to whom this book is addressed. Therefore, we provide below an outline of the rationale of the paper as it has been revised according to the reviewers’ comments. We are indebted to P. Boero, B....

  9. 6 The Multiplicity of Viewpoints in Elementary Function Theory: Historical and Didactical Perspectives
    (pp. 57-66)
    Renaud Chorlay

    From 2002 to 2006, the “history of mathematics” group of Paris 7 IREM¹ contributed to a research project funded by theInstitut National de la Recherche Pédagogique(INRP). We chose to work on the multiplicity of viewpoints on functions. In spite of the fact that some didactical and some historical research work was available on this topic, we felt the relevant connections still needed to be pointed to and explored. We also made use of fresh historical research work, namely R. Chorlay’s doctoral dissertation on the emergence of the concepts of “local” and “global” in mathematics [7, 8].

    We borrowed...

  10. 7 From History to Research in Mathematics Education: Socio-Epistemological Elements for Trigonometric Functions
    (pp. 67-82)
    Gabriela Buendia Abalos and Gisela Montiel Espinosa

    We propose to investigate the construction of the trigonometric function from a perspective that challenges what the educational system ’teaches’ and, consequently, how the student learns; this implies questioning not only how we teach, but what we teach. By means of the theoretical perspective ofSocioepistemology[5, 6, 2] we intend to recognize theusesandsignificationsassociated with the trigonometric functions in a particular historical setting.

    This theoretical perspective problematizes the knowledge confronting the mathematics of the educational system with the uses of knowledge in different settings, like historical, professional, or even school settings when experiencing non-traditional pedagogical proposals....

  11. 8 Harmonies in Nature: A Diaglogue Between Mathematics and Physics
    (pp. 83-90)
    Man-Keung Siu

    In school it is a customary practice to teach mathematics and physics as two separate subjects. In fact, mathematics is taught throughout the school years from primary school to secondary school, while physics, as a full subject on its own, usually starts in senior secondary school. This usual practice of teaching mathematics and physics as two separate subjects has its grounds. To go deep into either subject one needs to spend at least a certain amount of class hours, and to really understand physics one needs to have a sufficiently prepared background in mathematics. However, such a practice deprives students...

  12. 9 Exposure to Mathematics in the Making: Interweaving Math News Snapshots in the Teaching of High-School Mathematics
    (pp. 91-102)
    Batya Amit, Nitsa Movshovitz-Hadar and Avi Berman

    Beyond its glorious past, mathematics has a vivid present and a promising future. New results are published on a regular basis in the professional journals; new problems are created and added to a plethora of yet unsolved problems, which challenge mathematicians and occupy their minds.

    Movshovitz-Hadar [13] suggested a classification of mathematical news into five categories which we bring here with examples, many of which can be made accessible to high-school students:

    (i) A recently presented problem of particular interest and possibly its solution.E.g.Herzberg and Murty’s paper concerning the mathematical problems related to Sudoku puzzles [10] and Murty’s...

  13. 10 History, Figures and Narratives in Mathematics Teaching
    (pp. 103-112)
    Adriano Demattè and Fulvia Furinghetti

    This article relates to a project that has been developed in various classrooms of secondary school. The main assumptions inspiring this project are the following:

    students’ motivations to learn mathematics are enhanced if they develop an image of mathematics that encompasses the ‘social sense of mathematics’ (that is they see mathematics as a human process embedded in culture)

    the history of mathematics may provide them with a context suitable to develop such an image.

    The concern about introducing a human and cultural dimension in the image of mathematics is explicitly stated by many teachers and researchers who plan to use...

  14. 11 Pedagogy, History, and Mathematics Measure as a Theme
    (pp. 113-122)
    Luis Casas and Ricardo Luengo

    Interdisciplinary research in history is a resource that can provide knowledge in breadth and in depth not only of the subjects that have traditionally been studied—historical facts—but of others related to different areas of knowledge.

    In this sense, the History of Education can contribute valuable understanding of the phenomena, institutions, and academic disciplines of schools. Research in this field has included analyses of the development and evolution of particular concepts throughout history, and studies of the pedagogical approaches that have been taken in their teaching and learning. Other work has reviewed, and in many cases retrieved, the materials...

  15. 12 Students’ Beliefs About the Evolution and Development of Mathematics
    (pp. 123-132)
    Uffe Thomas Jankvist

    Beliefs about the history of mathematics is a topic which is touched upon from time to time in the literature on history in mathematics education,e.g., in Furinghetti [8] and Philippou and Christou [23]. However, when scanning these samples, one soon finds that these concern the beliefs of in-service or pre-service teachers. Studies on students’ beliefs about the history of mathematics seem to be rather poorly represented in the literature, if not altogether absent¹. One reason for this that I can think of is that, in general, studies of beliefs in mathematics education are conducted with the purpose of improving...

  16. 13 Changes in Student Understanding of Function Resulting from Studying Its History
    (pp. 133-144)
    Beverly M. Reed

    Professional mathematical societies are deeply concerned about the mathematical education of our teachers and are continuing to search for effective means to deepen students’ understanding of fundamental mathematical concepts [6]. National reports call for better preparation of our mathematics teachers [3, 9].

    The purpose of this study was to discern if students learn mathematics by studying its history. In particular, it investigated the changes in pre-service secondary school teachers’ thinking about functions resulting from their studying the history of the concept. This study addressed the following questions.

    Does studying the history of the concept of function deepen a student’sunderstanding...

  17. 14 Integrating the History of Mathematics into Activities Introducing Undergraduates to Concepts of Calculus
    (pp. 145-164)
    Theodorus Paschos and Vassiliki Farmaki

    The history of mathematics may be a useful resource for understanding the processes of formation of mathematical thinking, and for exploring the way in which such understanding can be used in the designing of classroom activities. Such a task demands that mathematics teachers be equipped with a clear theoretical framework for the formation of mathematical knowledge. The theoretical framework has to provide a fruitful articulation of the historical and psychological domains as well as to support a coherent methodology. This articulation between history of mathematics and teaching and learning of mathematics can be varied. Some teaching experiments may use historical...

  18. 15 History in a Competence Based Mathematics Education: A Means for the Learning of Differential Equations
    (pp. 165-174)
    Tinne Hoff Kjeldsen

    In a series of papers, Michael N. Fried has discussed a dilemma in historical approaches to mathematics education arising because “mathematics educators arecommittedto teachingmodernmathematics . . . ” and he continues “However, whenhistoryis beingusedto justify, enhance, explain, and encourage distinctly modern subjects and practices, it inevitably becomes what is “anachronical” [. . . ] or “Whig” history” [6, p. 395, italics in the original]. Whig history refers to the kind of history that is written from the present, i.e., a reading of the past in which one tries to find the present....

  19. 16 History of Statistics and Students’ Difficulties in Comprehending Variance
    (pp. 175-186)
    Michael Kourkoulos and Constantinos Tzanakis

    Although variation is central to statistics (e.g., [28, 29]), before the end of the 90’s not much attention was given to it in didactical research, as Bakker [1, p. 16], Reading [33] and others have remarked. Only recently have there been some systematic studies on the development of students’ conception of variation (e.g., [43, 33, 34, 3, 14, especially pp. 382–386] ). However, although didactical research on variance and standard deviation (s.d.) is very limited, it still points out that students face important difficulties in understanding these concepts. Mathews, Clark and their colleagues [25, 5, 14, pp. 383,...

  20. 17 Designing Student Projects for Teaching and Learning Discrete Mathematics and Computer Science via Primary Historical Sources
    (pp. 187-200)
    Janet Heine Barnett, Jerry Lodder, David Pengelley, Inna Pivkina and Desh Ranjan

    A discrete mathematics course often teaches about precise logical and algorithmic thought and methods of proof to students studying mathematics, computer science, or teacher education. The roots of such methods of thought, and of discrete mathematics itself, are as old as mathematics, with the notion of counting, a discrete operation, usually cited as the first mathematical development in ancient cultures [7]. However, a typical course frequently presents a fast-paced news reel of facts and formulae, often memorized by the students, with the text offering only passing mention of the motivating problems and original work that eventually found resolution in the...

  21. 18 History of Mathematics for Primary School Teacher Education Or: Can You Do Something Even if You Can’t Do Much?
    (pp. 201-210)
    Bjørn Smestad

    In this article, I will describe my context (Norwegian pre-service teacher education) and give some examples of different ways I work on history of mathematics. A major part of the paper will be spent discussing some of the materials I have used with students.

    As a subtitle, I have chosen “Can you dosomethingeven if you can’t do much?” In conferences in the HPM community, we get to see wonderful examples of how rich a resource the history of mathematics can be, but often I am left with the question “Will I have time to do this with my...

  22. 19 Reflections and Revision: Evolving Conceptions of a Using History Course
    (pp. 211-220)
    Kathleen Clark

    As with the construction of any secondary mathematics education course, a course on the history of mathematics for teaching can assume many different forms. For example, if the secondary mathematics education major resides in a Department of Mathematics, the course may tend to be more of a pure mathematics course instead of one with explicit attention to pedagogical ideas. Alternatively, if the course is a College of Education offering, it may shed some of its strict mathematical content and concentrate more on biographical, anecdotal, or pedagogical information. In recent years, what constitutes a history of mathematics course has become the...

  23. 20 Mapping Our Heritage to the Curriculum: Historical and Pedagogical Strategies for the Professional Development of Teachers
    (pp. 221-230)
    Leo Rogers

    In 1998/9 the education system in England and Wales began a mathematics curriculum based on ‘core skills’ that enshrined traditional beliefs about ‘levels’ of knowledge and encouraged a utilitarian approach with a prescriptive pedagogy. Textbook design followed the syllabus, past test papers becamede factopart of the curriculum, and the emphasis on examination targets produced little serious engagement with substantial mathematical thinking in most classrooms. The Teacher Training Establishments were inspected for their conformity to the system, and consequently the situation with regard to the history of mathematics has been far from that well-organized plan described by Lingard in...

  24. 21 Teachers’ Conceptions of History of Mathematics
    (pp. 231-240)
    Bjørn Smestad

    Many researchers are convinced that history of mathematics merits a more important role in mathematics education than it currently occupies. To achieve that goal, more knowledge on teachers’ conceptions of history of mathematics is needed.

    In this paper, I will describe an interview study on Norwegian teachers’ conceptions of the history of mathematics. After giving some background, I will describe the method used before discussing the main findings of the study. These findings provide insights that should be taken into account in further attempts to get more teachers to include history of mathematics in their teaching.

    Curriculum: In the Norwegian...

  25. 22 The Evolution of a Community of Mathematical Researchers in North America: 1636–1950
    (pp. 241-250)
    Karen Hunger Parshall

    The story of mathematics in colonial North America may be said to begin in 1636 with the founding by the Puritans of the Massachusetts Bay Colony of Harvard College as a Congregationalist institution.² It is not by chance that the first colleges in the British colonies south of what would become the border with Canada were Congregationalist, and this includes Harvard and Yale (as well as Dartmouth, Williams, Bowdoin, Middlebury, and Amherst). As heirs of “rational and hierarchical Calvinsim in America” [2, p. 248], Congregationalists valued the intellect and placed considerable emphasis on transplanting from England “the apparatus of civilized...

  26. 23 The Transmission and Acquisition of Mathematics in Latin America, From Independence to the First Half of the Twentieth Century
    (pp. 251-260)
    Ubiratan D’Ambrosio

    Recent scholarship traces the occupation of the Americas, justifiably called the New World, to about 40,000 years ago. The search for explanations (religions, arts and sciences), systems of values and behavior styles (communal and societal life), the psycho-emotional and the imaginary, and the models of production and of property were developed, in these cultures, in a way which is completely different of what was developed in the so-called Old World.

    The major encounter identified as the Conquest of the Americas, initially by Spanish and Portuguese navigators, later followed by British, French and Dutch, brought European knowledge systems to the New...

  27. 24 In Search of Vanishing Subjects: The Astronomical Origins of Trigonometry
    (pp. 261-272)
    Glen Van Brummelen

    As any high school math teacher will tell you, the word “trigonometry” means “triangle measurement.” The more perspicacious teacher might even know that the word was first coined by Bartholomew Pitiscus with hisTrigonometriae[16], a study of the so-called “science of triangles” (Figure 24.1). This sounds familiar, even comfortable to modern teachers and researchers; we feel that we know what trigonometry is about, what it’s for, and where it came from. For most of us, we couldn’t be more wrong.

    By 1600, much of the trigonometry that we saw in school had been known for well over a millennium....

  28. About the Editors
    (pp. 273-273)