History of mathematics in mathematics education. Recent developments

This is a survey on the recent developments (since 2000) concerning research on the relations between History and Pedagogy of Mathematics (the HPM domain ). Section 1 explains the rationale of the study and formulates the key issues. Section 2 gives a brief historical account of the development of the HPM domain with focus on the main activities in its context and their outcomes. Section 3 provides a sufficiently comprehensive bibliographical survey of the work done in this area since 2000. Finally, section 4 summarizes the main points of this study.


Introduction
This is a survey describing the state-of-the-art on the themes of the ICME 13 Topic Study Group (TSG) 25: The Role of History of Mathematics in Mathematics Education.It gives a brief account of the developments since 20001 on the relations between History and Pedagogy of Mathematics, in order to illuminate and provide insights on the following general questions: • Which history is suitable, pertinent, and relevant to Mathematics Education (ME)?

• Which role can History of Mathematics (HM) play in ME?
• To what extent has HM been integrated in ME (curricula, textbooks, educational aids/resource material, teacher education)?• How can this role be evaluated and assessed and to what extent does it contribute to the teaching and learning of mathematics?
These are the key issues explicitly addressed in and/or implicitly underlying what we call the HPM perspective as detailed below.

The HPM Perspective
Mathematics is a human intellectual enterprise with a long history and a vivid present.Thus, mathematical knowledge is determined not only by the circumstances in which it becomes a deductively structured theory, but also by the procedures that originally led or may lead to it.Learning mathematics includes not only the "polished products" of mathematical activity, but also the understanding of implicit motivations, the sense-making actions and the reflective processes of mathematicians, which aim to the construction of meaning.Teaching mathematics should give students the opportunity to "do mathematics."In other words, although the "polished products" of mathematics form the part of mathematical knowledge that is communicated, criticized (in order to be finally accepted or rejected), and serve as the basis for new work, the process of producing mathematical knowledge is equally important, especially from a didactical point of view.Perceiving mathematics both as a logically structured collection of intellectual products and as processes of knowledge production should be the core of the teaching of mathematics.At the same time, it should be also central to the image of mathematics communicated to the outside world.
Along these lines, putting emphasis on integrating historical and epistemological issues in mathematics teaching and learning constitutes a possible natural way for exposing mathematics in the making that may lead to a better understanding of specific parts of mathematics and to a deeper awareness of what mathematics as a discipline is.This is important for ME, helping to realize that mathematics: • is the result of contributions from many different cultures; • has been in constant dialogue with other scientific disciplines, philosophy, the arts and technology; • has undergone changes over time; there have been shifting views of what mathematics is; • has constituted a constant force for stimulating and supporting scientific, technical, artistic and social development.
This helps to improve ME at all levels and to realize that although mathematics is central to our modern society and a mathematically literate citizenry is essential to a country's vitality, historical and epistemological issues of mathematics are equally important.The harmony of mathematics with other intellectual and cultural pursuits also makes the subject interesting, meaningful, and worthwhile.In this wider context, history and epistemology of mathematics have an additional important role to play in providing a fuller education of the community: not being a natural science, but a formal science closer to logic -hence to philosophymathematics has the ability inherent in itself to connect the humanities with the sciences.Now that societies value and want young people educated in the sciences, but have a hard time finding out how to get people to "move" from humanities to the sciences, integrating history and epistemology in ME can make this connection visible to students.This is most important, especially today when there is much concern about the level of mathematics that students are learning and about their decreasing interest in mathematics, at a time when the need for both technical skills and a broader education is rising.

Summary of the content of the present survey
The rationale underlying this perspective has formed the core and main concern of the approaches adopted towards integrating History and Epistemology of Mathematics in ME (the HPM domain), especially in the context of the ICMI affiliated International Study Group on the Relations Between the History and Pedagogy of Mathematics (the HPM Group) since its formation in 1972.

What follows consists of four sections:
Section 2 gives a brief historical account of the development of the HPM domain with focus on the main activities in its context and their outcomes since 2000 ( §2.1); a short presentation of journals and newsletters ( §2.2); and an outline with comments on the key issues mentioned in section 1 and references to the literature for details ( §2.3).Section 3 constitutes the major part of the survey.It provides a sufficiently comprehensive bibliographical survey of the work done since 2000: Collective works in §3.1 (collective volumes, special ME journal issues, conference proceedings, resource material); individual works in §3.2 (books & doctoral dissertations, papers in scientific journals, collective volumes and conference proceedings).Though the emphasis is on research results of an as broad as possible international interest, due attention is paid to nationally-oriented implementation of the HPM perspective as well.
Finally, section 4 summarizes the main points of this study and section 5 contains all references given in section 2.
Remarks: (a) In section 3, next to each reference the TSG 25 themes to which it is related are indicated, numbered as in the Appendix below.
(b) Collective works exclusively on the HPM perspective (i.e.those in §3.1) contain several important contributions.However, in order to keep this survey to a reasonable size, these contributions are not included as separate items in §3.2, though some of them are quoted in section 2 (hence, they appear in section 5).Instead, these collective works are annotated briefly.Also note that several abbreviations are used for the titles of journals, conferences etc., are explained at the beginning of section 3, and many lengthy URL have been integrated into the title of the reference to avoid making the text difficult to follow and non-appealing.These URLs are not displayed in a printout of this document, so the reader is advised to use its electronic version available at http://www.clab.edc.uoc.gr/HPM/HPMinME-TopicalStudy-18-2-16-NewsletterVersion.pdf.
(c) For several references (especially those in the annotated bibliography in section 3.1) hyperlinks are provided, where one can find online additional information in the form of an abstract, review, outline of contents etc. APPENDIX: Main Themes of TSG 25 T1: Theoretical and/or conceptual frameworks for integrating history in ME.T2: History and epistemology implemented in ME, considered from either the cognitive or/and affective points of view: a. Classroom experiments at school, the university and teacher pre-& in-service education; b.Teaching material: textbooks, resource material of any kind.T3: Surveys on: a. Research on the HM in ME; b.The HM as it appears in curriculum and/or textbooks.T4: Original sources in the classroom and their educational effects.T5: History and epistemology as a tool for an interdisciplinary approach in the teaching and learning of mathematics and the sciences by unfolding their fruitful interrelations.T6: Cultures and mathematics fruitfully interwoven.

An outline of the historical development of the HPM domain
Integrating HM in ME has been advocated since the second half of the 19 th century, when mathematicians like De Morgan, Poincaré, Klein and others explicitly supported this path and historians like Tannery and later Loria showed an active interest on the role HM can play in ME.At the beginning of the 20 th century, this interest was revived as a consequence of the debates on the foundations of mathematics.Later on, history became a resource for various epistemological approaches; Bachelard's historical epistemology, Piaget's genetic epistemology and Freudenthal's phenomenological epistemology, at the same time stimulating the formulation of specific ideas and conclusions on the learning process (Barbin & Tzanakis, 2014, p. 256 and references therein;Fauvel & van Maanen, 2000, p. 202).
This interest became stronger and more competitive in the period 1960-1980 in response to the New Math reform, when its proponents were strongly against "a historical conception of ME," whereas for its critics, HM appeared like a "therapy against dogmatism," conceiving mathematics not only as a language, but also as a human activity.In 1969, the NCTM in USA devoted its 31 st Yearbook to the HM as a teaching tool (NCTM 1969) and in the 1970s a widespread international movement began to take shape, greatly stimulated and supported by the establishment of the HPM Group at ICME 2 in 1972 and its scope in 1978 (HPM Group 1978).Thus, during the last 40 years, integrating HM in ME has evolved into a worldwide intensively studied area of new pedagogical practices and specific research activities and a gradually increasing awareness has emerged of what was described in §1.1 as the HPM perspective (Fasanelli & Fauvel (2006) for a historical account and references prior to 2000; Barbin (2013), Barbin &Tzanakis (2014), andFuringhetti (2012), for a concise outline of later developments, and references therein).
The rising international interest in the HPM perspective and the various activities of the HPM Group worldwide, led to the approval by ICMI in 1996 of launching a 4-year ICMI Study on the relations between HM and ME.After a Discussion Document written by the Study cochairs (Fauvel & van Maanen, 1997) and a Study Conference in 1998, at Luminy, France, the Study culminated in the publication of a 437-page volume written by 62 contributors working together in 11 groups (Fauvel & van Maanen, 2000).This was a landmark in establishing and making more widely visible the HPM perspective as a research domain in the context of ME and greatly stimulated and enhanced the international interest of the educational community in this area.
Below we give an account of the main regular activities and their outcomes concerning educational research and its implementation in educational practice, relevant to the HPM domain and mainly realized in the context of the HPM Group.

ICME Satellite Meetings of the HPM Group
These quadrennial meetings are a major activity that bring together individuals with a keen interest in the relationship between the HM and ME; researchers in ME interested in the HM in relation to mathematical thinking, mathematics teachers at all levels eager to gain insights into the HPM perspective, historians of mathematics wishing to talk about their research, mathematicians wanting to learn about new possibilities to teach their discipline, and all those with an interest in the HPM domain.The books published as a result of these HPM meetings are listed below using the abbreviations introduced in section 3: Swetz et al.,1995 (after ICME-6); Calinger, 1996(after HPM 1992); Lagarto et al., 1996(during HPM 1996); Katz, 2000(after HPM 1996); Horng & Lin, 2000(at HPM 2000); Bekken & Mosvold, 2003(before ICME 10 & HPM 2004); Horng et al., 2004(before HPM 2004); Furinghetti et al., 2004(at HPM 2004;revised edition Furinghetti et al., 2006;see §3.1.3);Cantoral et al. (at HPM 2008); Barbin et al., 2012(at HPM 2012; revised edition in progress).

The European Summer University on the History and Epistemology in Mathematics Education (ESU)
The initiative of organizing a Summer University (SU) on the History and Epistemology in Mathematics Education belongs to the French ME community in the early 1980s.The French IREMs organized the first interdisciplinary meeting in 1984, in Le Mans, France, followed by another three in France.The next one was organized in 1993 on a European scale; the 1 st European Summer University on the History and Epistemology in Mathematics Education, (a name coined since then, abbreviated as ESU since 2004), though many participants come from outside Europe.Since 2010, ESU is organized every four years to avoid coincidence with the HPM meetings.
Since its original conception, ESU has been developed and established into one of the major activities in the HPM domain.It mainly aims to: provide a school for working on a historical, epistemological, and cultural approach to mathematics and its teaching, with emphasis on actual implementation; give the opportunity to mathematics teachers, educators, and researchers to share their teaching ideas and classroom experience related to a historical perspective in teaching; and motivate further collaboration along these lines among teachers of mathematics and researchers on the HM and ME in Europe and beyond, attempting to reveal and strengthen the HPM perspective.Below is a list of the ESUs: The following works were published after the ESUs: Lalande et al., 1995;Lagarto et al., 1996;Radelet-de-Grave & Brichard, 2001;Furinghetti et al., 2004;Barbin et al., 2008;Barbin et al., 2011a;Barbin et al., 2015.

CERME is a regular activity of the European Society for Research in Mathematics Education
(ERME), organized every two years in the form of presentations, discussions, and debates within thematic working group (WG).Though relatively new, the HPM perspective has exhibited great potential at CERME and is expected to play a central role in the future: The WG themes were modified considerably, becoming more specific.This reflects further deepening of research in this area, with emphasis both on empirical work and its assessment on sharpening theoretical ideas, and developing conceptual frameworks adequate for describing and understanding phenomena relevant to the HPM perspective (see §3.1.3)

Convergence: Where Mathematics, History, and Teaching Interact
Since 2004, the MAA has published Convergence: Where Mathematics, History and Teaching Interact, a free online journal in HM and its use in teaching.
Aimed at teachers of mathematics at both the secondary and collegiate levels, Convergence includes topics from grades 8-16 mathematics, with special emphasis on grades 8-14.Its resources for using the HM in mathematics teaching include informative articles about the HM, translations of original sources, classroom activities, projects and modules, teaching tools such as its Mathematical Treasures, reviews of new and old books, websites, Problems from Another Time, and other teaching aids that focus on utility in the classroom.

The Bulletin of the British Society for the History of Mathematics (BSHM Bulletin)
The BSHM Bulletin aims to promote research into the HM and to encourage its use at all levels of ME.Articles on local HM and the use of HM in ME are particularly encouraged.It was originally published as a Newsletter, until 2004 when its 50 th issue became Bulletin 1.Under the influence of the late J. Fauvel, president of BSHM (1992-94), editor of its Newsletter (1995)(1996)(1997)(1998)(1999)(2000)(2001) and chair of the HPM Group (1992-96) and his successor, the late J. Stedall, the Newsletter changed from providing information to members into a scientific journal with a regular Education Section directly related to issues relevant to the HPM perspective since 2002 (issue No 46).

The HPM Newsletter
The Newsletter appears three times per year since 1980.Originally it was available by contacting the regional distributors; however, for the last 13 years it is also available online from the HPM Group website and its Newsletter webpage It includes a calendar of upcoming events, a guest editorial, a 'Have You Read?' section, short reviews and announcements of meetings and activities.Furthermore, for the last 13 years it has also included short articles, reports on research projects and PhD theses, book reviews, lists of relevant websites, and particular themes that are suggested for further research.

Comments on some key issues
From what has been presented so far and will become clearer from the bibliographical survey in Section 3, the last two decades have generated considerable research activity related to the HPM perspective of great variety: doing empirical research based on actual classroom implementations; designing specific teaching units; developing various kinds of teaching aids; exploring and understanding students' response to the introduction of the HM in teaching (including teacher education); designing, applying, and evaluating interdisciplinary teaching; drawing and/or criticizing parallels between the historical development and learning in a modern classroom;2 mutually profiting from theoretical constructs and conceptual frameworks developed in the context of other disciplines, especially philosophy, epistemology and cognitive science; and evaluating the effectiveness of all this in practice.
The key issues mentioned in section 1 permeate all these activities as recurring themes that form the leitmotif of the HPM domain.Though impossible to present all the work done and the not always mutually compatible opinions and results, below a few general ideas are outlined with reference to the literature for details.
Whether HM is appropriate, or even relevant at all to the teaching and/or learning of mathematics, is an issue that, despite the extended research and the many insightful and sophisticated applications in the last few decades, has not reached universal acceptance even today.In fact, a number of objections against the HPM perspective have been raised (Furinghetti 2012, §7;Siu 2006;Tzanakis, Arcavi et al. 2000, p.203;Tzanakis & Thomaidis 2012, §3.4): A Objections of an epistemological and methodological nature (a) On the nature of mathematics 1.This is not mathematics!Teach the subject first; then its history.2. Progress in mathematics is to make difficult problems routine, so why bother to look back? 3. What really happened can be rather tortuous.Telling it as it was can confuse rather than enlighten!(b) On the difficulties inherent to this approach 1.Does it really help to read original texts, which is a very difficult and time-consuming task? 2. Is it liable to breed cultural chauvinism and parochial nationalism?3. Students may have an erratic historical sense of the past that makes historical contextualization of mathematics impossible without having a broader education in general history.

B Objections of a practical and didactical nature (a)
The background and attitude of teachers 1. Lack of didactical time: no time for it in class! 2. Teachers should be well educated in history: "I am not a professional historian of mathematics.How can I be sure of the exposition's accuracy?" 3. Lack of teacher training.4. Lack of appropriate didactical and resource material.
(b) The background and attitude of the students 1.They regard it as history and they dislike history class! 2. They regard it just as boring as mathematics itself.3.They do not have enough general knowledge of culture to appreciate it.
(c) Assessment issues 1.How can you set questions on it in a test or exam? 2. Is there any empirical evidence that students learn better when HM is made use of in the classroom?
Each of these objections addresses one or more of the four key issues mentioned in Section 1. Below we comment briefly on them in the light of these objections.

Which history is suitable, pertinent and relevant to Mathematics Education?
This has been a permanent issue of debate among historians and mathematics educators with an interest in the HPM perspective.As early as 1984 at ICME 5, d'Ambrosio stressed the need to develop three separate histories of mathematics: history as taught in schools, history as developed through the creation of mathematics, and the history of that mathematics which is used in the street and the workplace.To deal with these differences he introduced the concept of ethnomathematics as compared to learned mathematics (Booker, 1986).
In fact, implicit to the objections A(a1), A(a2), A(b1) is the idea that the term "history" is the same, whether used by historians, mathematicians, or teachers and mathematics educators.That this is not so lies at the heart of Grattan-Guinness' early refutation of some of these arguments (Grattan- Guinness, 1973; see also Kjeldsen, 2011aKjeldsen, , pp. 1700Kjeldsen, -1701;;2011b, pp. 166-167;Kjeldsen & Blomhøj, 2012, §3, and references therein for a different recent approach).On the other hand, it is undeniable that quite often the historical development was complicated, followed a zig-zag path, led to dead ends, included notions, methods and problems that are no longer used in mathematics as we know and work with today, etc. (A(a2), A(b3)).Thus, its integration in ME, on the one hand is nontrivial, and on the other hand poses the question why it must be done at all.Therefore, integrating HM in the teaching and learning of mathematics, may force history "…to serve aims not only foreign to its own but even antithetical to them" (Fried, 2011, p. 13).In other words, the danger of either unacceptably simplifying or/and distorting history to serve education as still another of its tools is real by adopting what has been called a "Whig" (approach to) history, in which "…the present is the measure of the past.Hence, what one considers significant in history is precisely what leads to something deemed significant today" (Fried, 2001, p. 395).
In this connection, an important step was Grattan-Guiness' distinction between what he called History and Heritage trying to clarify existing conflicts and tensions between a mathematician's and a historian's approach to mathematical knowledge, and paying due attention to the relevance of HM to ME (Grattan-Guiness, 2004a, b).In the context of the HPM perspective, this is a distinction close to similar ones between pairs of methodological approaches; explicit & implicit use of history, direct & indirect genetic approach, forward & backward heuristics (Tzanakis, Arcavi et al., 2000, pp. 209-210).Hence, this distinction is potentially of great relevance to ME (Rogers, 2009(Rogers, , 2011;;Tzanakis & Thomaidis, 2012), serving, among other things, to contribute towards answering the recurrent question "Why and which history is appropriate to be used for educational purposes?"(Barbin, 1997).

Which role can History of Mathematics play in Mathematics Education?
Perhaps, this is the question that has been discussed and analyzed most on the basis of both a priori theoretical and epistemological arguments and of empirical educational research.At least implicitly, such analyses try to refute some of the above objections, especially those concerning the barriers posed by the complexity of the historical development (A(a2), A(a3), A(b1)) and/or by students' predisposition to and general knowledge of both mathematics and history as taught subjects (objections (B(b1), B(b2) and B(b3), A(b2), respectively).This question has been extensively discussed from several points of view, especially in relation to the appropriateness and pertinence of original historical sources in ME.In this context, HM can play three mutually complementary and supplementary roles (Barbin, 1997;Furinghetti, 2012, §5;Furinghetti et al., 2006Furinghetti et al., , pp. 1286Furinghetti et al., -1287;;Jahnke et al., 2000, §9.1;Jankvist, 2013, §7): A replacement role: Replacing mathematics as usually understood (a corpus of knowledge consisting of final results/finished and polished intellectual products; a set of techniques for solving problems given from outside; school units useful for exams etc.) by something different (to emphasize not only final results, but also mental processes that may lead to them; hence to perceive mathematics not only as a collection of well-defined and deductively organized results, but also as a vivid intellectual activity).
A reorientation role: Changing what is (supposed to be) familiar, to something unfamiliar; thus challenging the learner's and teacher's conventional perception of mathematical knowledge as something that has always been existing in the form we know it, into the deeper awareness that mathematical knowledge was an invention, an evolving human intellectual activity.
A cultural role: Making possible to appreciate that the development of mathematics takes place in a specific scientific, technological or societal context at a given time and place; thus becoming aware of the place of mathematical knowledge as an integral part of human intellectual history in the development of society; hence, seeing mathematics from perspectives that lie outside its nowadays established boundaries as a discipline.
Considered from the point of view of the objective of integrating HM in ME, there are 5 main areas in which the HPM perspective could be valuable: The learning of mathematics; The development of views on the nature of mathematics and mathematical activity; The didactical background of teachers and their pedagogical repertoire; The affective predisposition towards mathematics; and The appreciation of mathematics as a cultural-human endeavor.
These are analyzed in detail into more specific arguments in Tzanakis, Arcavi et al., 2000, §7.2, describing in this way the role of history in the educational process.
From the point of view of the way HM is accommodated into this perspective, a distinction was made by Jankvist (2009); namely, history serving as a tool for assisting the actual learning and teaching of mathematics, and history serving as a goal in itself for the teaching and learning of the historical development of mathematics (see also Jankvist & Kjeldsen, 2011).A similar distinction between history for reflecting on the nature of mathematics as a socio-cultural process and history for constructing mathematical objects was made by Furinghetti (2004;2012, §5).
In this way, a finer and more insightful categorization of the possible roles of HM in ME resulted, reflecting the variety of their possible implementations in practice.
A small selection appears below (many more are given in section 3).-Fauvel & van Maanen (2000): Chapters 7, 8 provide a variety of examples of possible classroom implementations, for several mathematical subjects; chapter 9 gives examples of using original sources in the classroom and specific didactical strategies to do so.- Katz &Tzanakis (2011), Chapters 9, 10, 13, 14, 16, 19, andSriraman (2012), Chapters 2, 7, 14 provide particular examples, most of them emphasizing empirical results of actual implementations.- Katz et al. (2014): Rich on recent work in the HPM domain, including a sufficiently comprehensive old and recent bibliography in the editors' introduction and in its 12 papers.The papers concern theoretical issues on the history, philosophy and epistemology of mathematics, and on empirical investigations both in school and teacher education.-Doctoral dissertations with considerable work on both the theoretical issues of the HPM perspective and on empirical investigation and evaluation of actual implementations: e.g., Clark (2006); Glaubitz (2010); Jankvist (2009a); Su (2005); van Amerom (2002).

To what extent the History of Mathematics has been integrated in Mathematics Education (curricula, textbooks, educational aids/resource material, teacher education)?
Considerable work has been done over the last 15 years on understanding better and formulating more sharply the methodological issues raised by the integration of HM in ME, on producing appropriate educational aids of various types (B(a4)), and on designing and implementing teaching approaches to specific subjects and instructional levels in this context, with special emphasis on teacher education (B(a2), B(a3)).
According to the classification of the various approaches to integrate HM in teaching and learning mathematics given in Tzanakis, Arcavi et al. (2000), there are three broad ways that may be combined (thus complementing each other), each one emphasizing a different aim: To provide direct historical information, aiming to learn history; To implement a teaching approach inspired by history (explicitly or implicitly), aiming to learn mathematics; To focus on mathematics as a discipline and the cultural and social context in which it has been evolving, aiming to develop a deeper awareness of its evolutionary character, its epistemological characteristics, its relation to other disciplines and the influence exerted by factors both intrinsic and extrinsic to it.
From a methodological point of view, Jankvist (2009b) classified the teaching & learning approaches in three categories: Illumination approaches, in which teaching and learning is supplemented by historical information; Module approaches, in the form of instructional units devoted to history, often based on specific cases; History-based approaches, in which history shapes the sequence and the way of presentation, often without history appearing explicitly, but rather being integrated into teaching.
Approaches may vary in size and scope, according to the specific didactical aim, the mathematical subject matter, the level and orientation of the learners (A(b1), B(b3)), the available didactical time (B(a1)), and external constraints (curriculum regulations, number of learners in a classroom etc.).
Though accommodating the HPM perspective in an essential way into the official national curricula does not seem to have attained wide applicability, 3 intensive efforts have been made to train teachers and explore changes in their attitude and/or teaching, and to design, produce and make available didactically appropriate resources.Some indicative examples (more are given in section 3): Teacher training: Arcavi & Isoda, 2007;Bruckheimer & Arcavi, 2000;Clark, 2011;Liu, 2003;Mosvold et al., 2014;Povey, 2014;Smestad, 2011;Waldegg, 2004.

Resource material & educational aids:
The need for didactical resources along the lines of the HPM perspective has been satisfied to a considerable extent in the last 15 years, so that such material is available nowadays in a variety of forms.Some examples: -A wide spectrum of resource material can be found in Convergence; e.g., (i) HPM Newsletter, No90/2015, pp.10-12 for a recent sample; (ii) Clark, 2009 (detailed description of a teaching module).- Katz & Michalowicz (2004): didactical source material within 11 mathematical modules.-Siu (2007): a useful survey of the literature and available resources.
- Pengelley et al. (2009): Didactical material for discrete mathematics based on original sources.- Pengelley & Laubenbacher (2014): A website with many references to published work and material available online.- Barnett et al. (2014): Extensive information on teaching with historical sources and bibliography on its theoretical framework and available resource material.-Books with material that can be used directly and/or inspire teaching; e.g., Barbin, 2015(review in HPM Newsletter No 89/2015, pp. 13-14); Stein, 2010(review in HPM Newsletter No77/2011, pp. 8-9).

How can this role be evaluated and assessed and to what extent it contributes to amend the teaching and learning of mathematics?
Evaluating the effectiveness of the HPM perspective on improving ME from the point of view of both teaching and learning mathematics is an issue clearly stressed in objections B(c1), B(c2).Those who oppose, or are reserved about the role of HM in ME rightly ask for sufficient empirical evidence about its effectiveness.Quite early it has become clear that this is a key issue (e.g., Siu & Tzanakis, 2004, p. 3;Jankvist, 2007), and that any such evaluation is a complex process relying more on qualitative, than quantitative methodologies: to consider changes induced in teachers' own perception of mathematics; to examine how this may influence the way they teach mathematics; and to explore if and in which ways this affects students' perception and understanding of mathematics (Barbin et al., 2000, particularly § §3.1, 3.2).
Additionally, any such evaluation goes together with actual classroom implementations, in school teaching and teacher pre-and in-service education.Therefore, many, if not all, works referring to such implementations necessarily address evaluation issues about the effectiveness of the approach considered in each case (e.g., those listed in § §2.3.2, 2.3.3).This is an area of currently active research with no established results of universal acceptance because of several reasons: (a) Such a complex process is not expected to lead to spectacular changes in a short time interval.Preconceptions, misconceptions, predispositions either of the teachers or the students are too stable to be easily and/or quickly modified.Therefore, one should expect to see such changes after a considerable time exposure to an approach adopting the HPM perspective; often this time is not available.
(b) There is strong dependence on the instructional level (primary, secondary, tertiary) and orientation of the students, teacher-students included (science or humanities; elementary or secondary school teachers etc.), as well as, on their entire previous educational path, which has determined their knowledge of, attitude towards, and preconceptions about mathematics.
(c) There is influence by external "technical" factors that may favor, impede, or even prevent the implementation of an approach based on the HPM perspective: the curriculum and the corresponding regulations; the number of students in the class (e.g., a small number facilitates group work and teacher's effective supervision); the structure of the educational system (e.g., in a centralized system, teachers have less freedom, hence fewer possibilities to apply an innovative teaching approach not necessarily falling into the official curriculum regulations).
(d) Not all mathematical subjects are equally accessible or appropriate to be taught and/or learned in a historically motivated/driven context.All of this constitutes a complex network of factors interfering with each other, so that empirical findings of different research works are not easily comparable.Therefore, despite many thoughtfully designed and carefully applied empirical investigations, much work is still needed to evaluate the effectiveness of the role of HM in ME in an undisputable way.

A bibliographical survey in the HPM domain since 2000
This section provides a comprehensive bibliographical survey of work related to the HPM perspective since 2000, indicating the TSG 25 themes to which each item is related and relevant, except those collective works that practically touch upon all themes (cf.remark (a) and Appendix in §1.2).Within each subsection, items appear by publication year and for each year by authors' alphabetical order.For those works included in Section 5 only author names and a note "see section 5" are given.
Throughout this topical study, names of journals, proceedings, and conferences are abbreviated as follows: This volume consists of two parts: On an epistemological analysis of the historical development of vector space theory; and on didactical issues addressed and actual implementations at the undergraduate level, to which this analysis is related.

2003
Bekken & Mosvold (see section 5).T1, 4, 2a, 5 A continuation of Swetz et al. (1995).Of its 27 chapters, 9 are related to the HPM perspective; the others concern either the HM or the history of ME.Hanna, G., Jahnke, N., & Pulte, H. (Eds.).Explanation and proof in mathematics: philosophical and educational perspectives.New York, NY: Springer.T1, 5 The 17 chapters assemble perspectives from ME, its history and philosophy to strengthen mutual awareness and share recent findings and advances in these interrelated fields.By a variety of examples, the authors explore the role of refutation in generating proofs, the varied links between experiment and deduction, the use of diagrammatic thinking in addition to pure logic, and the uses of proof in ME.
An all-embracing outcome of activities within the HPM Group during 2007-2009; to

Concluding remarks
The HPM perspective described in §1.1 emerged gradually over the last decades as a perception of mathematics worth exploring, thanks to research and teaching work done worldwide, thus establishing the HPM domain as a valuable research area in the context of ME.Launching the ICMI Study volume in 2000 was a decisive step in this direction.This highly collective work motivated, stimulated, oriented, encouraged and supported research in this area, to a large extent realized in the context of the HPM Group and the main activities related to it.At that time central issues were (and still are): -To put emphasis on pre-and in-service teacher education as a necessary prerequisite for the HPM perspective to be possible at all.-To design, produce, make available and disseminate a variety of didactical source material in the form of anthologies of original sources, annotated bibliography, description of teaching sequences/modules to serve as a source of inspiration and/or as generic examples for classroom implementation, educational aids of various types, appropriate websites, etc. -To perform systematically, carefully designed and applied empirical research in order to examine in detail and evaluate convincingly the effectiveness of the HPM perspective on improving the teaching and learning of mathematics, as well as students and teachers' awareness of mathematics as a discipline and their disposition towards it.-To acquire a deeper understanding of theoretical ideas put forward in the HPM domain and to carefully develop them into coherent theoretical frameworks and methodological schemes that will serve as a foundation for further research and applications in this area.
In the last 10 to 15 years much work has been done on these issues and more is still in progress.In this survey: • An attempt was made to provide enough evidence -mainly based on the literature -that HM is relevant to ME in several ways and may have a multifaceted influence on improving the teaching and learning of and about mathematics.More specifically: • An outline of the development of the HPM domain has been given; • The key issues in this domain have been formulated and briefly discussed; and • A sufficiently comprehensive survey of the existing literature has been included.
We hope that the present survey will serve both as a working document and as a motivation for all those who desire information about the HPM perspective and to explore further the possibilities offered for supporting and improving ME.
Hand Int RME: Handbook of International Research in Mathematics Education ICMI Study Volume: J. Fauvel & J. van Maanen (Eds.),History in Mathematics Education, The ICMI Study.Dordrecht: Kluwer, 2000.Int.Hand.Res.Hi.Phil.Sci.Teach.: International Handbook of Research in History, Philosophy and Science Teaching TMMEM: The Montana Mathematics Enthusiast Monographs Other ERME: European Society for Research in Mathematics Education HPM Group: International Study Group on the Relations between the History and the Pedagogy of Mathematics ICMI: International Commission on Mathematical Instruction MAA: The Mathematical Association of America NCTM: National Council of Teachers of Mathematics (USA) Proc.: Proceedings TSG: Topic Study Group WG: Collective volumes in this area, with research papers, reviews of work, etc. 2000 Dorier, J.-L. (Ed.).On the teaching of linear algebra.Dordrecht: Kluwer.T2, 5 Katz et al. (see section 5).T1, 2a, 5Special issue with an introduction accompanied by an extensive bibliography and 12 papers directly related to the HPM perspective, divided into 4 sections: theoretical issues in the use of the HM in teaching; direct uses of the HM in the classroom; HM in teacher education; relations between the philosophy, the epistemology, the teaching and the sociology of mathematics.Kourkoulos, M., & Tzanakis, C. (Eds.).History of mathematics and mathematics education.Education Sciences.Special Issue for 2014,5-198.T1, 2a, 3a    Bilingual issue with 9 papers related to the HPM perspective (3 in English, 6 in Greek): 3 concern general ideas, conceptual frameworks and methodological schemes and 5 refer to specific issues with focus on classroom implementations (from elementary school to the university).

2 Special issues of international journals of ME 2004
present an overview of the state of the art in this area after the appearance of the ICMI Study volume.7 chapters on theoretical aspects of HM The 9 chapters provide examples of the influence or use of HM in the math classroom.Authors work with students from upper secondary school to tertiary, including teacher training.They show how their own reading and reflection has led to direct use of historical material in the classroom, either through the use of original texts or by devising tasks based on the methods or examples of their subjects (HPM Newsletter No80/2012, pp.12-13).study of geometry involves both thinking and doing and it is through the doing (drawing, measuring, copying) that we develop a sense of what geometry is.School mathematics has today lost much of its geometry.It is in an attempt to recall this loss, and to report on the history of geometrical constructions, that provides the spur for this collection (see HPM Newsletter No 89/2015, pp.13-14).Siu & Tzanakis (see section 5).T1, 2, 3a, 5A special double issue with 10 papers originally presented at ICME 10, TSG 17 "The role of the History of special issue with 10 papers, seeking to deepen the understanding of the pedagogical role HM may play in contemporary ME.Some provide examples of the use of the HM in school practice and teacher education; others address theoretical questions that have become crucial to understanding the profound intertwining of past and present, conceptual developments on spreading new epistemologies and theories of learning.