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Developments Relevant to the Use of Tools in Mathematics
Monaghan J.
This chapter explores developments in mathematics, computing, mathematics education and scholarship relevant to understanding tools from 1960 to the time of writing. This exploration is biased in accentuating influences relevant to tools and mathematics education. The chapter presents a broad landscape and focuses on selected technological advances, ideas and people that are considered important. The chapter begins with a section charting developments in mathematics, computing and education followed by a section on intellectual trends relevant to understanding tools and tool use. The final section focuses on the development of ideas in mathematics education regarding tools and tool use.
Interlude
Monaghan J., Trouche L., Borwein J.M.
These pages are, after Chaps.
6
and
11
, the third ‘space for reflection’ in this book and our focus is on issues in mathematics education related to tool use; issues that we address and issues which we have not addressed (at least as the focus for a chapter in the book). Part C focused on the curriculum, the calculator debate, mathematics in the real world and the mathematics teacher. Part D, with an eye to the future, focuses on task design, games and connectivity. We structure our reflections in two unequal sections: a short section on futurology and the selection of issues in Part D; a consideration of issues on which we have not explicitly focused.
Discussion of Issues in Chapters in Part II
Monaghan J., Trouche L., Borwein J.M., Noss R.
This chapter is a second opportunity for one of the authors of this book to question the other authors (and a guest, Richard Noss) about matters raised in Chaps.
7
–
10
. It is designed as a series of questions from John, and Jon, Luc and Richard were free to answer (or not) as they deemed appropriate. The chapter ends with a short review.
The Calculator Debate
Monaghan J.
The ‘calculator debate’ refers to an (often heated) exchange of ideas, on the ‘proper’ use of specific forms of hand-held technology in mathematics instruction and the assessment of learning, that has been ongoing for decades. This chapter considers issues in this debate in three sections. The first section positions ‘the calculator’ within ‘portable hand-held computational technology’ and reviews calculator use, the research literature on the use of the calculator and the ‘calculator debate’ itself. The second section considers the calculator with regard to Wertsch’s (Mind as action, Oxford, England, 1998) ten properties of mediated action. The last section speculates on a possible future of the calculator debate.
Doing Mathematics with Tools: One Task, Four Tools
Monaghan J.
This chapter illustrates a variety of mathematical and educational issues arising from doing a single task with different tools. One task is considered, bisect an angle. The chapter has four sections, each devoted to issues in using one tool to complete this task: a straight edge and compass; a protractor; a dynamic geometry system; and a book.
Discussions of Part I Chapters
Trouche L., Borwein J.M., Monaghan J.
This chapter is an opportunity for one of the authors of this book to question the other two authors in the light of issues raised in Chaps.
2
–
5
. It constitutes both a follow-up to discussions between authors which occurred over the writing process, and emergent issues—new discussions once the book was almost complete. Some fundamental issues are addressed, about the birth of mathematics (and its deep links with the birth of writing), the relationships between mathematics and other sciences, the interactions between conjecture and proof, and the role of visualisation and of gestures. The text is kept short in order to provoke the readers to reflect on these issues rather than for the authors to ‘provide answers’.
Constructionism
Monaghan J.
Constructionism is a hybrid theory—creed—set of principles for task design, which many scholars working with digital tools in mathematics education have appropriated. It is historically associated with the programming language Logo. This chapter presents a brief history of constructionism; positions Logo in the history of computer developments; and outlines ideas in two influential constructionist books, Papert’s 1980 book Mindstorms and Noss & Hoyles’ 1996 book Windows on mathematical meanings. The final section attempts to distill the/a constructionist view(s) of tools in learning mathematics.
Mathematics Teachers and Digital Tools
Monaghan J., Trouche L.
This chapter considers mathematics teachers’ appropriation and classroom use of digital tools. The first section considers teachers—who are they, how are they conceived in the literature and what aspects of teachers have been studied? The second section examines twenty-first century research on mathematics teachers using digital tools. This sheds light on the complexity of mathematics teachers’ appropriation and classroom use of digital tools but what we find is that our focus is too narrow and we need to consider digital tools within the range of resources use in planning and realising their lessons, which leads us to the third section, mathematics teachers using resources. We end with a review of the current state of understanding and an agenda for future research.
Integrating Tools as an Ordinary Component of the Curriculum in Mathematics Education
Trouche L.
This chapter is dedicated to the analysis of the mutual influences between tools and mathematics curricula. Mathematics learning indeed develops ‘under the umbrella’ of the ‘really used’ tools. And the development of tools depends—partially—on the curricula intended as well as implemented: the design of calculators specially conceived ‘for the test’ is a clear illustration of this influence. This chapter aims to evidence that the integration of tools in mathematics curricula is far from being a linear and natural process. It depends on a set of conditions designing new tasks, new techniques; training teachers; finding new equilibrium for teachers’ and students’ mathematical activity. This chapter is organised in three sections setting the scene: the first section proposes a vertical (historical) point of view, aiming to evidence the continuity of some issues over the time; the second gives an horizontal (international comparative) point of view, aiming to evidence, beyond the national peculiarities, some common features; the third section proposes a case study, the French policy on assessment, seen as paradigmatic. The conclusion addresses some questions and draws some perspectives for further studies.
Tasks and Digital Tools
Monaghan J., Trouche L.
This chapter considers scholastic tasks with digital tools. The first two sections consider tasks in ‘ordinary’ classrooms (tasks for learning) and issues relating to tasks using mathematical software. The first section presents examples of tasks with digital tools to highlight potential problems and opportunities for learning. The second section considers issues arising from the literature on tasks design with and without digital tools. The final section looks at task-tool issues in larger-than-the-individual classroom research and in assessment; it also comments of avenues for further development.
Tools and Mathematics in the Real World
Monaghan J.
This chapter considers, with special regard to tool use, mathematics in out-of-school practices and attempts to replicate these practices in school mathematics. Both foci are important and problematic issues in mathematics education. This chapter has four sections. The two central sections address the two main foci. The opening section sets the scene with an historical account of ways that mathematics has been subdivided with regard to its application(s). The last section considers problem issues.
Games: Artefacts in Gameplay
Monaghan J.
This chapter reviews the past and looks to the future of the potential for games and gameplay to provide opportunity for engaging in mathematical activity. This review a glimpse into a possible future is conducted with a specific focus on the role of artefacts in gameplay. The chapter is in four sections. The first section considers the range of games; the second section considers artefacts in games and gameplay; the third section addresses games in mathematics education; and the final section looks to future development.
The Life of Modern Homo Habilis Mathematicus: Experimental Computation and Visual Theorems
Borwein J.M.
Long before current graphic, visualization and geometric tools were available, John E. Littlewood (1885–1977) wrote in his delightful Miscellany:
A heavy warning used to be given [by lecturers] that pictures are not rigorous; this has never had its bluff called and has permanently frightened its victims into playing for safety. Some pictures, of course, are not rigorous, but I should say most are (and I use them whenever possible myself). (Littlewood, . London: Methuen, 1953, p. 53)
Over the past 5–10 years, the role of visual computing in my own research has expanded dramatically. In part this was made possible by the increasing speed and storage capabilities—and the growing ease of programming—of modern multi-core computing environments. But, at least as much, it has been driven by my group’s paying more active attention to the possibilities for graphing, animating or simulating most mathematical research activities.
Activity Theoretic Approaches
Monaghan J.
Activity theory is an approach to the study of human practices—including mathematical and educational practices—in which mediation, including mediation by artefacts/tools, is a central construct. The chapter is of four sections. The first section provides an overview of AT. Section 9.2 traces early influences of AT in mathematics education research. Section 9.3 considers foci of a set of mathematics education papers recent at the time of writing. Section 9.4 explores emphases and tensions in papers considered in Sects. 9.2 and 9.3.
Tools, Human Development and Mathematics
Monaghan J.
This chapter raises a number of issues from pre-history and history that one mathematics educator considers ‘worthy of mention’ with regard to tools and mathematics. These issues are: tool use in the development of the human species (phylogenesis); tool use in a mathematical culture, ancient Greek mathematics that goes beyond the obvious tools; an example from ancient Indian mathematics that bears some resemblances to Jon’s experimental mathematics described in Chap.
3
; the mutual support of hand, mind and artefact in expert use of an abacus; a consideration of a period (sixteenth-century Europe) where there was a rapid advance in the development of mathematical tools.
The Development of Mathematics Practices in the Mesopotamian Scribal Schools
Trouche L.
This chapter proposes a view on a particular moment in the development of mathematics and the learning of mathematics, 2000 bce in Mesopotamia: a particular moment regarding the medium, with the development of writing and of systems of signs; particular regarding the development of mathematics, with the development of a sexagesimal positional numerical system and of associated algorithms; particular regarding the places dedicated to learning, with the development of scribal schools; and, last but not least, particular regarding the ‘supports’, with the use of clay tablets ‘still alive’ today. It aims to evidence the complex system of artefacts supporting mathematical practices, and mathematics teaching and learning in scribal schools.
Connectivity in Mathematics Education: Drawing Some Lessons from the Current Experiences and Questioning the Future of the Concept
Trouche L.
The concept of connectivity, following the development of Internet resources, is more and more widely used, in the society in general, and in the mathematics education community in particular. This chapter aims to question the different meanings, and the potential, of this emergent concept. For this purpose, it lies first on the experience of the author, considering both connecting students as a support of their mathematics learning, and connecting teachers as a support of their professional development. Then it considers the views expressed in the connectivity panel occurring in the 17th ICMI study, dedicated to technology in mathematics education. Finally, it discusses the dynamics of the concept itself for the future of mathematics education.
Didactics of Mathematics: Concepts, Roots, Interactions and Dynamics from France
Trouche L.
This chapter analyses specificities of the French field of ‘didactics of mathematics’, questioning its reasons, tracing the geneses of concepts related to artefacts and following influences on, and interactions with the international communities of research. This complex genesis is traced in four sections: a first section on the roots of the didactics of mathematics in France, a second section on two founding theoretical frameworks (the theory of didactical situations of Brousseau, and the theory of conceptual fields of Vergnaud), a third section on the anthropological approach of Chevallard, a fourth focusing on specific approaches dedicated to artefacts and resources in mathematics education. Beyond historical and cultural specificities, the chapter aims to evidence the potential of interactions between teachers and researchers, as well as interactions between researchers in mathematics and mathematics education for improving our understanding of learning and teaching issues in mathematics.
Introduction to the Book
Monaghan J., Trouche L.
From Mathematical Problem Solving to Geocaching: A Journey Inspired by My Encounter with Jeremy Kilpatrick
Lingefjärd T.
The activity of problem solving is probably as old as mankind. Within mathematics education, there is a vast amount of books, textbooks, general articles, and research articles published concerning the use of and learning of problem solving. There are many different views and opinions regarding what a problem really is, and sometimes problems are divided into closed or open problems. An open-ended problem is a problem that has several or many correct answers and several ways to the correct answer(s). One activity developed by people using new technology together with open-ended problems is the activity geocaching. In this paper, I will try to give a flavor of how intriguing and almost addictive mystery solving in geocaching might be.
Fifty Years and Counting: Working with Jeremy Kilpatrick
Wilson J.W.
This is a personal reflection about Jeremy Kilpatrick by a friend and colleague. I attempt to portray him as an outstanding scholar in mathematics education but from a personal perspective as well as professional. We worked together as graduate students and research associates at Stanford, and our collaboration continued as he joined the faculty at Teachers College, Columbia, and I joined the faculty at the University of Georgia. In 1975, he joined our faculty in Georgia, and I was his department head for the first 18 years he was here. Jeremy’s mentors in graduate student days included George Polya, Edward G. Begle, William Brownell, and Lee J. Cronbach. I submit that Jeremy’s link to these four scholars is a unique and rich intellectual heritage, but he also earned the admiration and support of each of them. More than any other person, Jeremy Kilpatrick has transformed the field of mathematics education and led us into the development of our field as an emerging discipline. His vision and influence are recognized worldwide, and he has brought an international perspective to mathematics education.
Problem Solving, Exercises, and Explorations in Mathematics Textbooks: A Historical Perspective
da Ponte J.P.
This paper analyzes the tasks proposed in several Portuguese mathematics textbooks from the nineteenth to the twenty-first century. A look at the nature and intended purpose of these tasks raises interesting issues about school mathematics teaching and learning. Has the meaning of terms such as “problem” and “exercise” been always the same? What other terms have been used in textbooks to designate mathematics tasks? What were the reasons for the changes? The analysis of the evolution that occurred in the terminology as well as in the nature of the tasks proposed to the students provides elements to reflect about what are the changes that have occurred in mathematics teaching and learning and how some changes are more apparent than real.
Toward a Profession of Mathematics Education: Guidance from Jeremy Kilpatrick’s Words and Deeds
Silver E.A.
With reference to a list of attributes of professions suggested by Shulman—the obligation of service to others, the need for understanding of a scholarly or theoretical kind, a domain of skilled performance or practice, the exercise of judgment under conditions of unavoidable uncertainty, the need for learning from experience as theory and practice interact, and a professional community to monitor quality and aggregate knowledge—this chapter considers how those attributes apply to the professional field we call mathematics education. Using specific examples of contributions Jeremy Kilpatrick has made in his writings and professional activities, the paper argues that his leadership by example has mapped a pathway along which mathematics education can advance toward achieving the status of a full-fledged profession.
Ruminations on the Generated Curriculum and Reform in Community College Mathematics: An Essay in Honor of Jeremy Kilpatrick
Mesa V.
In this essay I elaborate on a notion of curriculum, put forward by Jeremy, that highlights the tensions that emerge from the curious interplay between reform, teaching, learning, and culture. I use the setting of American community colleges, to illustrate some of these tensions. I provide vignettes of teaching mathematics at community colleges in different levels of mathematics, to illustrate these tensions, which result in an anticipated stability of the curriculum: the stability is a consequence of the current historical, societal, political, and cultural conditions that surrounds this particular institution.
What’s Involved in the Work of Dissertation Advising? An Interview with Jeremy Kilpatrick and Some Personal Reflections
Herbst P.
This chapter addresses the mathematics education doctorate and, in particular, the dissertation or thesis. The core of this chapter is an interview with Jeremy Kilpatrick in an effort to document Jeremy’s own advising style. The interview is situated within an attempt to bring out more general issues that are in play in how a mathematics education scholar might advise or direct a doctoral dissertation, fueled mostly by my own introspection into this role. After brief descriptions of dissertation experiences in other fields and personal descriptions of my own experience working on my dissertation, I document Jeremy’s responses to the question of how he sees his style as dissertation advisor. I then propose that the dissertation work develops in response to four stakeholders: (1) the student, (2) the advisor, (3) the field, and (4) the institution. In an exercise of speculation, I examine those stakeholders in terms of how they could conceivably be invested in the successful completion of a doctoral dissertation. I go back to the interview with Jeremy Kilpatrick to see how Jeremy responds to the question of how and how much each of those stakeholders matter in his own dissertation advising style and how he views the dissertation.
