Mathematics Education Library – serie

A significant driver of recent growth in the use of mathematics in the professions has been the support brought by new technologies. Not only has this facilitated the application of established methods of mathematical and statistical analysis but it has stimulated the development of innovative approaches. These changes have produced a marked evolution in the professional practice of mathematics, an evolution which has not yet provoked a corresponding adaptation in mathematical education, particularly at school level. In particular, although calculators -- first arithmetic and scientific, then graphic, now symbolic -- have been found well suited in many respects to the working conditions of pupils and teachers, and have even achieved a degree of official recognition, the integration of new technologies into the mathematical practice of schools remains marginal. It is this situation which has motivated the research and development work to be reported in this volume. The appearance of ever more powerful and portable computational tools has certainly given rise to continuing research and development activity at all levels of mathematical education. Amongst pioneers, such innovation has often been seen as an opportunity to renew the teaching and learning of mathematics. Equally, however, the institutionalization of computational tools within educational practice has proceeded at a strikingly slow pace over many years.

This book is a product of the BACOMET group, a group of educators-mainly educators of prospective teachers of mathematics-who first came together in 1980 to engage in study, discussion, and mutual reflection on issues in mathematics education. BACOMET is an acronym for BAsic Components of Mathematics Education for Teachers. The group was formed after a series of meetings in 1978-1979 between Geoffrey Howson, Michael Otte, and the late Bent Christiansen. In the ensuing years, BACOMET initiated several projects that resulted in published works. The present book is the main product of the BACOMET project entitled Meaning and Communication in Mathematics Education. This theme was chosen because of the growing recognition internationally that teachers of mathematics must deal with questions of meaning, sense making, and communication if their students are to be proficient learners and users of mathematics.The participants in this project were the following: Nicolas Balacheff (Grenoble, France) Maria Bartolini Bussi (Modena, Italy) Rolf Biehler (Bielefeld, Germany) Robert Davis (New Brunswick, NJ, USA) Willibald Dorfler (Klagenfurt, Austria) Tommy Dreyfus (Holon, Israel) Joel Hillel (Montreal, Canada) Geoffrey Howson (Southampton, England) Celia Hoyles-Director (London, England) Jeremy Kilpatrick-Director (Athens, GA, USA) Christine Keitel (Berlin, Germany) Colette Laborde (Grenoble, France) Michael Otte (Bielefeld, Germany) Kenneth Ruthven (Cambridge, England) Anna Sierpinska (Montreal, Canada) Ole Skovsmose-Director (Aalborg, Denmark) Conversations about directions the project might take began in May 1993 at a NATO Advanced Research Workshop of the previous BACOMET project in VIII PREFACE

Mathematics is generally considered as the only science where knowledge is uni­ form, universal, and free from contradictions. „Mathematics is a social product - a 'net of norms', as Wittgenstein writes. In contrast to other institutions - traffic rules, legal systems or table manners -, which are often internally contradictory and are hardly ever unrestrictedly accepted, mathematics is distinguished by coherence and consensus. Although mathematics is presumably the discipline, which is the most differentiated internally, the corpus of mathematical knowledge constitutes a coher­ ent whole. The consistency of mathematics cannot be proved, yet, so far, no contra­ dictions were found that would question the uniformity of mathematics" (Heintz, 2000, p. 11). The coherence of mathematical knowledge is closely related to the kind of pro­ fessional communication that research mathematicians hold about mathematical knowledge. In an extensive study, Bettina Heintz (Heintz 2000) proposed that the historical development of formal mathematical proof was, in fact, a means of estab­ lishing a communicable „code of conduct" which helped mathematicians make themselves understood in relation to the truth of mathematical statements in a co­ ordinated and unequivocal way.

As we enter into the 21"Century, the presence of technology, particularly techno-science, in everyday life is overwhelming. Institutions in the modem world are affected by this presence. Like Janus, the ancient Roman god whose double-faced head signified his knowledge of the present and the future, education has always been a two-faced enterprise. The past establishes goals and methods of Education, and the other face tries to capture the future and suggests and proposes new directions of thought and new styles of behavior for the generation which, in a few years, will take over both routines and societal innovation. History tells us that this face of Education has always been sensitive to emerging technologies. Technologies of communication and information have been particularly influential in new directions of society, in particular of education. The transition from orality to writing marked a new role for the teacher. From the sole repository of accumulated knowledge, the teacher became a guide and interpreter of registered knowledge. The emergence of hardware, in the form of documents and books, initiated a companionship between teacher and hardware. It is also remarkable how the emergence of writing strengthened individual memory, contrary to the concerns of Thamus when Theuth explained to him the discovery of writing. The conservative king was afraid that the new invention would implant forgetfulness in the souls of men.

Exploring Probability in School provides a new perspective into research on the teaching and learning of probability. It creates this perspective by recognizing and analysing the special challenges faced by teachers and learners in contemporary classrooms where probability has recently become a mainstream part of the curriculum from early childhood through high school. The authors of the book discuss the nature of probability, look at the meaning of probabilistic literacy, and examine student access to powerful ideas in probability during the elementary, middle, and high school years. Moreover, they assemble and analyse research-based pedagogical knowledge for teachers that can enhance the learning of probability throughout these school years.With the book’s rich application of probability research to classroom practice, it will not only be essential reading for researchers and graduate students involved in probability education; it will also capture the interest of educational policy makers, curriculum personnel, teacher educators, and teachers.

As we enter into the 21"Century, the presence of technology, particularly techno-science, in everyday life is overwhelming. Institutions in the modem world are affected by this presence. Like Janus, the ancient Roman god whose double-faced head signified his knowledge of the present and the future, education has always been a two-faced enterprise. The past establishes goals and methods of Education, and the other face tries to capture the future and suggests and proposes new directions of thought and new styles of behavior for the generation which, in a few years, will take over both routines and societal innovation. History tells us that this face of Education has always been sensitive to emerging technologies. Technologies of communication and information have been particularly influential in new directions of society, in particular of education. The transition from orality to writing marked a new role for the teacher. From the sole repository of accumulated knowledge, the teacher became a guide and interpreter of registered knowledge. The emergence of hardware, in the form of documents and books, initiated a companionship between teacher and hardware. It is also remarkable how the emergence of writing strengthened individual memory, contrary to the concerns of Thamus when Theuth explained to him the discovery of writing. The conservative king was afraid that the new invention would implant forgetfulness in the souls of men.

Linear or proportional relationships are a major topic in mathematics education. However, recent research has shown that secondary school students strongly tend to apply the linear model also in situations wherein it is not applicable. This overgeneralization of linearity is sometimes referred to as the "illusion of linearity" and has a strong negative impact on students’ reasoning and problem solving skills."The Illusion of Linearity: From Analysis to Improvement" presents the reader with a comprehensive overview of the major findings of the recent research on the illusion of linearity. Although the empirical study of students’ improper linear reasoning clearly constitutes a new line of research, it owes a great deal to prior work in mathematics education research community (such as the work of the scholars of the Freudenthal Institute on realistic mathematics education). Based on both quantitative and qualitative research, "The Illusion of Linearity: From Analysis to Improvement" discusses the following issues: (1) how the illusion of linearity appears in diverse domains of mathematics and science and how it is conceptually related to other more general misconceptions identified in the research literature, (2) what are the crucial psychological, mathematical, and educational factors being responsible for the occurrence and persistence of the phenomenon, and (3) how the illusion of linearity can be remedied by appropriate instruction."The Illusion of Linearity: From Analysis to Improvement" is essential to those working in mathematics education, particularly teacher educators and curriculum research and development.

Given its abstract nature and the highly syntactical competence required by the use of symbolic algebra, research on its teaching and learning must rely on approaches that include semiotic concepts and analyses that recall the history of algebraic ideas, among others. Educational Algebra: A Theoretical and Empirical Approach deals with a theoretical perspective on the study of school algebra, in which both components (semiotics and history) occur. This perspective runs opposite to general theoretical models, since it submits components for the design of local frameworks for theoretical analysis. The Methodological design allows for the interpretation of specific phenomena and the inclusion, within such interpretative frameworks, of evidence not included in more general treatments. Such is the case of phenomena observed in subjects who are initiating the study of symbolic algebra, involving the production of personal sign systems at the intermediate level or the level previous to the mathematical sign system which is to be learned.Disciplines such as Linguistics, Logic, Psycholinguistics, Semiotics, general Cognitive Psychology, Mathematics Psychology, Mathematics Epistemology, History of Mathematics, and others have carried out research on the same topics approached by Mathematics Education and have redefined their results within the framework of their respective fields. Specifically, theorists in Linguistics, Information Processing and Didactics of Mathematics have done important work on the notion of code. Today, this notion is a key element to interpreting the idea of representation in the new explanatory models of cognitive problems placed by alternative teaching approaches, including those involving a technological environment. Additionally, Psycholinguistics and Artificial Intelligence in procedural models of human abilities have intended to explain how and why users of mathematical language naturally and commonly make mistakes in syntactical procedures.Educational Algebra: A Theoretical and Empirical Approach adds to previous developments with priority given to a pragmatic perspective on "meaning in use" over "formal meaning". The bulk of these approaches and others of similar nature have lead to a focus on competence rather than on a user’s activity with mathematical language.Such a shift in perspective has fundamental implications on the way mathematical language is studied. Essentially, Grammar—the abstract formal system—and Pragmatics—the principles of the use of language—are complementary domains in this volume. Both are related to different teaching models, whether new or traditional, used in helping students to become competent users of Algebra. Because of this, Educational Algebra: A Theoretical and Empirical Approach will be of interest to researchers and practitioners within the mathematics education field.

This book brings together diverse recent developments exploring philosophy of mathematics in education. The unique combination of ethnomathematics, philosophy, history, education, statistics and mathematics offers a variety of different perspectives from which existing boundaries in mathematics education can be extended. The ten chapters in this book offer a balance between philosophy of and philosophy in mathematics education. Attention is paid to the implementation of a philosophy of mathematics within the mathematics curriculum to become a philosophy in mathematics education. In doing so, many chapters provide ideas for actual practice and some practical examples directly usable in teacher training and in mathematics classrooms.

New Directions for Situated Cognition in Mathematics Education represents the maturation and expansion of the situated cognition theories applied to mathematics education. All of the situations on which the chapters of this book are based exemplify activity which would be described as mathematical, whether they are classrooms, workplaces, homes or the street. In identifying mathematical activity, this book examines the ways people talk, what they talk about, what they focus on, how they classify experience, what levels and kinds of generality occur to them, what is varied and what is fixed, what relationships they observe or construct and how they express them—much they way music, musicality, and a musician are recognized.In this book a dynamic view of knowledge is taken by all the authors. Although knowledge is considered what is produced in learning environments, each chapter offers a different perspective on its relationship to the individual, group, activity, historical conventions, and authoritarian views of meaning.New Directions for Situated Cognition in Mathematics Education provides a resource for educators, researchers and students to approach situated cognition through an organized and diverse source.

Abstract: An outline of the structure of the book is presented, making the argument that the language we use for everyday mathematical ideas presents us with valuable evidence and insights into the nature of mathematics. Keywords: mathematical discourse, nature of mathematics I begin the book by looking at the way people speaking different languages talk about mathematical ideas in their everyday conversation. I end up questioning some common beliefs about mathematics, its history, and its pedagogy. The way we (English speakers) use numbers, the way we give directions, the way we express relationships, are all so commonplace that it is hard to imagine any other way of expressing these ideas. We take for granted the structures of the following sentences: There are four people in the room. The book costs forty-five dollars. Two and three are five. Turn left. Go straight on. The sun rises in the east. A dog is a mammal. He is not my father. I will either go shopping or read my book this afternoon.

In this chapter we interrogate the debate between mathematicians and mathematics educators. This debate is widely recognized in the United States, for instance, where the term ‘math wars’ is often used in reference to the heated disagreements between mathematicians and mathematics edu- tors. Such recognition might suggest that this topic itself has relevance both for the academy and for educational policy pertaining to mathematics edu- tion. We propose to locate this debate, unavoidably, whenever and wherever mathematics education is written. Our aim is to show how the debate itself gives rise to the inscription mathematics education. Indeed, we contend that ‘mathematics education’ cannot be defined outside of, or prior to, the debate, for it is precisely through the debate that it is constituted as a domain for 1 deliberation. The debate is more primordial than any epistémè or domain implied by it. In other words, we argue that mathematics and mathematics education do not exist epistemically prior to the debate. In order to engage this issue we consider a transcription of a panel debate 2 between mathematicians and mathematics educators held in Brazil in 1998 . There is at least a double meaning for the word ‘debate’: it may mean a meeting having a beginning and an end, well-localized in space and time and with a specific proposal or focus, as in the following sentence ‘I am going to see on TV the debate between Bush and Blair tomorrow at 5 pm’.

Abstract: An outline of the structure of the book is presented, making the argument that the language we use for everyday mathematical ideas presents us with valuable evidence and insights into the nature of mathematics. Keywords: mathematical discourse, nature of mathematics I begin the book by looking at the way people speaking different languages talk about mathematical ideas in their everyday conversation. I end up questioning some common beliefs about mathematics, its history, and its pedagogy. The way we (English speakers) use numbers, the way we give directions, the way we express relationships, are all so commonplace that it is hard to imagine any other way of expressing these ideas. We take for granted the structures of the following sentences: There are four people in the room. The book costs forty-five dollars. Two and three are five. Turn left. Go straight on. The sun rises in the east. A dog is a mammal. He is not my father. I will either go shopping or read my book this afternoon.

Mathematics is the science of acts without things - and through this, of things one can define by acts. 1 Paul Valéry The essays collected in this volume form a mosaik of theory, research, and practice directed at the task of spreading mathematical knowledge. They address questions raised by the recurrent observation that, all too frequently, the present ways and means of teaching mathematics generate in the student a lasting aversion against numbers, rather than an understanding of the useful and sometimes enchanting things one can do with them. Parents, teachers, and researchers in the field of education are well aware of this dismal situation, but their views about what causes the wide-spread failure and what steps should be taken to correct it have so far not come anywhere near a practicable consensus. The authors of the chapters in this book have all had extensive experience in teaching as well as in educational research. They approach the problems they have isolated from their own individual perspectives. Yet, they share both an overall goal and a specific fundamental conviction that characterized the efforts about which they write here. The common goal is to find a better way to teach mathematics. The common conviction is that knowledge cannot simply be transferred ready-made from parent to child or from teacher to student but has to be actively built up by each learner in his or her own mind.

Mathematics is in the unenviable position of being simultaneously one of the most important school subjects for today's children to study and one of the least well understood. Its reputation is awe-inspiring. Everybody knows how important it is and everybody knows that they have to study it. But few people feel comfortable with it; so much so that it is socially quite acceptable in many countries to confess ignorance about it, to brag about one's incompe­ tence at doing it, and even to claim that one is mathophobic! So are teachers around the world being apparently legal sadists by inflicting mental pain on their charges? Or is it that their pupils are all masochists, enjoying the thrill of self-inflicted mental torture? More seriously, do we really know what the reasons are for the mathematical activity which goes on in schools? Do we really have confidence in our criteria for judging what's important and what isn't? Do we really know what we should be doing? These basic questions become even more important when considered in the context of two growing problem areas. The first is a concern felt in many countries about the direction which mathematics education should take in the face of the increasing presence of computers and calculator-related technol­ ogy in society.

This book is a product of love and respect. If that sounds rather odd I initially apologise, but let me explain why I use those words. The original manuscript was of course Freudenthal’s, but his colleagues have carried the project through to its conclusion with love for the man, and his ideas, and with a respect developed over years of communal effort. Their invitation to me to write this Preface e- bles me to pay my respects to the great man, although I am probably incurring his wrath for writing a Preface for his book without his permission! I just hope he understands the feelings of all colleagues engaged in this particular project. Hans Freudenthal died on October 13th, 1990 when this book project was well in hand. In fact he wrote to me in April 1988, saying “I am thinking about a new book. I have got the sub-title (China Lectures) though I still lack a title”. I was astonished. He had retired in 1975, but of course he kept working. Then in 1985 we had been helping him celebrate his 80th birthday, and although I said in an Editorial Statement in Educational Studies in Mathematics (ESM) at the time “we look forward to him enjoying many more years of non-retirement” I did not expect to see another lengthy manuscript.

In the first BACOMET volume different perspectives on issues concerning teacher education in mathematics were presented (B. Christiansen, A. G. Howson and M. Otte, Perspectives on Mathematics Education, Reidel, Dordrecht, 1986). Underlying all of them was the fundamental problem area of the relationships between mathematical knowledge and the teaching and learning processes. The subsequent project BACOMET 2, whose outcomes are presented in this book, continued this work, especially by focusing on the genesis of mathematical knowledge in the classroom. The book developed over the period 1985-9 through several meetings, much discussion and considerable writing and redrafting. Our major concern was to try to analyse what we considered to be the most significant aspects of the relationships in order to enable mathematics educators to be better able to handle the kinds of complex issues facing all mathematics educators as we approach the end of the twentieth century. With access to mathematics education widening all the time, with a multi­ tude of new materials and resources being available each year, with complex cultural and social interactions creating a fluctuating context of education, with all manner of technology becoming more and more significant, and with both informal education (through media of different kinds) and non­ formal education (courses of training etc. ) growing apace, the nature of formal mathematical education is increasingly needing analysis.

Advanced Mathematical Thinking has played a central role in the development of human civilization for over two millennia. Yet in all that time the serious study of the nature of advanced mathematical thinking – what it is, how it functions in the minds of expert mathematicians, how it can be encouraged and improved in the developing minds of students – has been limited to the reflections of a few significant individuals scattered throughout the history of mathematics. In the twentieth century the theory of mathematical education during the compulsory years of schooling to age 16 has developed its own body of empirical research, theory and practice. But the extensions of such theories to more advanced levels have only occurred in the last few years. In 1976 The International Group for the Psychology of Mathematics (known as PME) was formed and has met annually at different venues round the world to share research ideas. In 1985 a Working Group of PME was formed to focus on Advanced Mathematical Thinking with a major aim of producing this volume. The text begins with an introductory chapter on the psychology of advanced mathema- cal thinking, with the remaining chapters grouped under three headings: • the nature of advanced mathematical thinking, • cognitive theory, and • reviews of the progress of cognitive research into different areas of advanced mathematics.

This book has been written to fIll a substantial gap in the current literature in mathemat­ ical education. Throughout the world, school mathematical curricula have incorporated probability and statistics as new topics. There have been many research papers written on specifIc aspects of teaching, presenting novel and unusual approaches to introducing ideas in the classroom; however, there has been no book giving an overview. Here we have decided to focus on probability, making reference to inferential statistics where appropriate; we have deliberately avoided descriptive statistics as it is a separate area and would have made ideas less coherent and the book excessively long. A general lead has been taken from the fIrst book in this series written by the man who, probably more than everyone else, has established mathematical education as an aca­ demic discipline. However, in his exposition of didactical phenomenology, Freudenthal does not analyze probability. Thus, in this book, we show how probability is able to organize the world of chance and idealized chance phenomena based on its development and applications. In preparing these chapters we and our co-authors have reflected on our own acquisition of probabilistic ideas, analyzed textbooks, and observed and reflect­ ed upon the learning processes involved when children and adults struggle to acquire the relevant concepts.

Didactics of Mathematics as a Scientific Discipline describes the state of the art in a new branch of science. Starting from a general perspective on the didactics of mathematics, the 30 original contributions to the book, drawn from 10 different countries, go on to identify certain subdisciplines and suggest an overall structure or `topology' of the field.The book is divided into eight sections: (1) Preparing Mathematics for Students; (2) Teacher Education and Research on Teaching; (3) Interaction in the Classroom; (4) Technology and Mathematics Education; (5) Psychology of Mathematical Thinking; (6) Differential Didactics; (7) History and Epistemology of Mathematics and Mathematics Education; (8) Cultural Framing of Teaching and Learning Mathematics.Didactics of Mathematics as a Scientific Discipline is required reading for all researchers into the didactics of mathematics, and contains surveys and a variety of stimulating reflections which make it extremely useful for mathematics educators and teacher trainers interested in the theory of their practice. Future and practising teachers of mathematics will find much to interest them in relation to their daily work, especially as it relates to the teaching of different age groups and ability ranges. The book is also recommended to researchers in neighbouring disciplines, such as mathematics itself, general education, educational psychology and cognitive science.

Advanced Mathematical Thinking has played a central role in the development of human civilization for over two millennia. Yet in all that time the serious study of the nature of advanced mathematical thinking – what it is, how it functions in the minds of expert mathematicians, how it can be encouraged and improved in the developing minds of students – has been limited to the reflections of a few significant individuals scattered throughout the history of mathematics. In the twentieth century the theory of mathematical education during the compulsory years of schooling to age 16 has developed its own body of empirical research, theory and practice. But the extensions of such theories to more advanced levels have only occurred in the last few years. In 1976 The International Group for the Psychology of Mathematics (known as PME) was formed and has met annually at different venues round the world to share research ideas. In 1985 a Working Group of PME was formed to focus on Advanced Mathematical Thinking with a major aim of producing this volume. The text begins with an introductory chapter on the psychology of advanced mathema- cal thinking, with the remaining chapters grouped under three headings: • the nature of advanced mathematical thinking, • cognitive theory, and • reviews of the progress of cognitive research into different areas of advanced mathematics.

Mathematics teaching and learning have been dominated by a concern for the intellectual readiness of the child, debates over rote learning versus understanding and, recently, mathematical processes and thinking. The gaze into today's mathematics classroom is firmly focused on the individual learner. Recently, however, studies of mathematics in social practices, including the market place and the home, have initiated a shift of focus. Culture has become identified as a key to understanding the basis on which the learner appropriates meaning. The chapters in this timely book attempt to engage with this shift of focus and offer original contributions to the debate about mathematics teaching and learning. They adopt theoretical perspectives while drawing on the classroom as both the source of investigation and the site of potential change and development. The book will be of fundamental interest to lecturers and researchers and to teachers concerned with the classroom as a cultural phenomenon.

In Nineteen Eighty-Four George Orwell gives a description of different forms of suppression. We learn about the telescreens placed everywhere, through which it is possible for Big-Brother to watch the inhabitants of Oceania. However, it is not only important to control the activities of the inhabitants, it is important as well to control their thoughts, and the Thought Police are on guard. This is a very direct form of monitoring and control, but Orwell also outlines a more imperceptible and calculated line of thought control. In the Appendix to Nineteen Eighty-Four Orwell explains some struc­ tures of 'Newspeak', which is going to become the official language of Oceania. Newspeak is being developed by the Ministry of Truth, and this language has to substitute 'Oldspeak' (similar to standard English). Newspeak should fit with the official politics of Oceania ruled by the Ingsoc party: "The purpose of Newspeak was not only to provide a medium of expression for the world-view and mental habits proper to the devotees of Ingsoc, but to make all other modes of thought impos­ sible. It was intended that when Newspeak had been adopted once and for all and Oldspeak forgotten, a heretical thought - that is, a thought diverging from the principles of Ingsoc - should be literally unthink­ able, at least as far as thought is dependent on words.

Social constructivism is just one view of learning that places emphasis on the social aspects of learning. Other theoretical positions, such as activity theory, also emphasise the importance of social interactions. Along with social constructivism, Vygotsky's writings on children's learning have recently also undergone close scru­ tiny and researchers are attempting a synthesis of aspects ofVygotskian theory and social constructivism. This re-examination of Vygotsky's work is taking place in many other subject fields besides mathematics, such as language learning by young children. It is interesting to speculate why Vygotsky's writings have appealed to so many researchers in different cultures and decades later than his own times. Given the recent increased emphasis on the social nature of learning and on the interactions between student, teacher and context factors, a finer grained analysis of the nature of different theories of learning now seems to be critical, and it was considered that different views of students' learning of mathematics needed to be acknowledged in the discussions of the Working Group.

Why are mathematical ideas so hard? Is mathematics an unassailable peak, which only the few can ever hope to conquer? Or can mathematics be broadened to be accessible to the many? Noss and Hoyles have written a book to challenge some of the conventional wisdoms on the learning of mathematics. They use the computer as a window onto mathematical meaning-making, drawing together the threads of their individual and collaborative research over more than a decade. The pivot of their theory is the idea of webbing, which explains how someone struggling with a new mathematical idea can draw on supportive knowledge, and reconciles the individual's role in mathematical learning with the part played by epistemological, social and cultural forces.

Why are mathematical ideas so hard? Is mathematics an unassailable peak, which only the few can ever hope to conquer? Or can mathematics be broadened to be accessible to the many? Noss and Hoyles have written a book which challenges some of the conventional wisdoms on the learning of mathematics. They use the computer as a window onto mathematical meaning-making, drawing together the threads of their individual and collaborative research over more than a decade. The pivot of their theory is the idea of webbing, which explains how someone struggling with a new mathematical idea can draw on supportive knowledge, and reconciles the individual's role in mathematical learning with the part played by epistemological, social and cultural forces.

In Greek geometry, there is an arithmetic of magnitudes in which, in terms of numbers, only integers are involved. This theory of measure is limited to exact measure. Operations on magnitudes cannot be actually numerically calculated, except if those magnitudes are exactly measured by a certain unit. The theory of proportions does not have access to such operations. It cannot be seen as an "arithmetic" of ratios. Even if Euclidean geometry is done in a highly theoretical context, its axioms are essentially semantic. This is contrary to Mahoney's second characteristic. This cannot be said of the theory of proportions, which is less semantic. Only synthetic proofs are considered rigorous in Greek geometry. Arithmetic reasoning is also synthetic, going from the known to the unknown. Finally, analysis is an approach to geometrical problems that has some algebraic characteristics and involves a method for solving problems that is different from the arithmetical approach. 3. GEOMETRIC PROOFS OF ALGEBRAIC RULES Until the second half of the 19th century, Euclid's Elements was considered a model of a mathematical theory. This may be one reason why geometry was used by algebraists as a tool to demonstrate the accuracy of rules otherwise given as numerical algorithms. It may also be that geometry was one way to represent general reasoning without involving specific magnitudes. To go a bit deeper into this, here are three geometric proofs of algebraic rules, the frrst by Al-Khwarizmi, the other two by Cardano.

In Greek geometry, there is an arithmetic of magnitudes in which, in terms of numbers, only integers are involved. This theory of measure is limited to exact measure. Operations on magnitudes cannot be actually numerically calculated, except if those magnitudes are exactly measured by a certain unit. The theory of proportions does not have access to such operations. It cannot be seen as an "arithmetic" of ratios. Even if Euclidean geometry is done in a highly theoretical context, its axioms are essentially semantic. This is contrary to Mahoney's second characteristic. This cannot be said of the theory of proportions, which is less semantic. Only synthetic proofs are considered rigorous in Greek geometry. Arithmetic reasoning is also synthetic, going from the known to the unknown. Finally, analysis is an approach to geometrical problems that has some algebraic characteristics and involves a method for solving problems that is different from the arithmetical approach. 3. GEOMETRIC PROOFS OF ALGEBRAIC RULES Until the second half of the 19th century, Euclid's Elements was considered a model of a mathematical theory. This may be one reason why geometry was used by algebraists as a tool to demonstrate the accuracy of rules otherwise given as numerical algorithms. It may also be that geometry was one way to represent general reasoning without involving specific magnitudes. To go a bit deeper into this, here are three geometric proofs of algebraic rules, the frrst by Al-Khwarizmi, the other two by Cardano.