Beyond motivation: history as a method for learning meta-discursive rules in mathematics

Beyond motivation: history as a method for learningmeta-discursive rules in mathematics

Tinne Hoff Kjeldsen & Morten Blomhøj

Published online: 21 September 2011# Springer Science+Business Media B.V. 2011

Abstract In this paper, we argue that history might have a profound role to play forlearning mathematics by providing a self-evident (if not indispensable) strategy forrevealing meta-discursive rules in mathematics and turning them into explicit objects ofreflection for students. Our argument is based on Sfard’s theory of Thinking asCommunicating, combined with ideas from historiography of mathematics regarding amultiple perspective approach to the history of practices of mathematics. We analyse twoproject reports from a cohort of history of mathematics projects performed by students atRoskilde University. These project reports constitute the experiential and empirical basis forour claims. The project reports are analysed with respect to students’ reflections aboutmeta-discursive rules to illustrate how and in what sense history can be used in mathematicseducation to facilitate the development of students’ meta-discursive rules of mathematicaldiscourse.

Keywords History as a didactical method . Mathematics education . Mathematicaldiscourse . Meta-discursive rules . Project work . History of mathematics . Historiography

1 Introduction

There have been an increasing number of articles and activities that advocate the use ofhistory in teaching and learning mathematics,1 but not much research has been done

Educ Stud Math (2012) 80:327–349DOI 10.1007/s10649-011-9352-z

1For literature on using history in mathematics education, see these references, for example (Arcavi & Bruckheimer,2000; Bruckheimer & Arcavi, 2000; Fauvel, 1991a, b; Fauvel & van Maanen, 2000; Furinghetti, 2004; Jankvist,2009a; Jankvist & Kjeldsen, 2011; Katz, 2000; Katz & Michalowicz, 2004; Kjeldsen, 2010, 2011a, b). For specialissues on history in mathematics education journals, see For The Learning of Mathematics (vol. 11(2), 1991; vol.17(1), 1997) as well as Educational Studies in Mathematics (vol. 66(2), 2007).

T. H. Kjeldsen (*) :M. BlomhøjIMFUFA, NSM, Roskilde University, Roskilde, Denmarke-mail: [email protected]

M. Blomhøje-mail: [email protected]

regarding whether—and if so, in what sense—history actually supports students’ learningof mathematics. Recent research on the use and/or the integration of history in teaching andlearning mathematics is beginning to address this issue. Fried (2001) has pointed out adilemma where teachers have the choice of either taking a genuine approach to history—inwhich case, they end up spending time on things that are not part of the mathematicscurricula—or trivializing history or taking an anachronistic (whig) approach in order to usehistory as a tool for learning relevant mathematics.2 Kjeldsen (2011a) has argued anddemonstrated how this dilemma can be resolved by employing a genuine approach tohistory to develop students’ mathematical competence; Jankvist (2009b) has investigatedempirically how students’ insights into the development of concrete historical episodes canbe anchored in the mathematical content of the episodes; Kjeldsen and Blomhøj (2009)have analysed learning potentials of integrating history and philosophy of mathematics intomathematics education at university level; and Kjeldsen (2011b) presents an outline for atheoretical framework for a systematic analysis and discussion of uses of history forteaching and learning mathematics.3

In this paper, we continue this line of research by proposing that, and demonstratinghow, the history of mathematics can play a part in teaching and learning of mathematics.The difference between this and previous approaches is that we suggest that history canhave a profound, perhaps even indispensable, role to play in teaching and learningmathematics from the point of view of learning proper meta-discursive rules, and we give atheoretical argument for this indispensability hypothesis based on the theory of Thinking asCommunicating of Sfard (2008).

We present our theoretical argument in Section 2, together with aspects of Sfard’stheory of thinking with a focus on mathematical discourse. This theoretical approachpresupposes a genuine approach to history where the so-called whig4 history, whichearlier played a dominant role in the writing of the history of mathematics, must beavoided. In Section 3, we present an outline of a multiple perspective approach to history,which we suggest as a suitable method by which history can be studied from actualpractices of mathematics. By using theoretical tools from historiography of mathematics,such as the idea of epistemic objects and techniques, original sources can be used inmathematics education to have students investigate and reflect upon meta-discursiverules. The experiential and empirical basis for our claims in the present paper is a cohortof project reports dealing with historical aspects of mathematics, conducted by students aspart of the mathematics programme at Roskilde University during the last 15 years. InSection 4, we analyse the reports from two such projects. The analyses of the two reportssupport our theoretical argument, by providing empirical evidence for our workinghypothesis that episodes from history of mathematics can provide occasions for revealingmeta-discursive rules and turning them into explicit objects of reflection for students. Thefirst project deals with the history of differential equations, while the second projectinvestigates Fourier’s influence on the development of the concept of a function. Thereports are analysed with respect to students’ reflections about meta-discursive rules toillustrate how and in what sense history can facilitate learning of meta-discursive rules ofmathematical discourse.

2 See also Fried (2007) for further discussions about the difficulties of integrating history of mathematics intomathematics teaching and learning.3 In the paper by Jankvist and Kjeldsen (2011), two new avenues for research on using and integratinghistory in mathematics education are proposed based on Kjeldsen (2011a) and Jankvist (2009b).4 The term whig history was introduced by the British historian Herbert Butterfield in the 1930s. It isexplained in Section 3 of the present paper.

328 T.H. Kjeldsen, M. Blomhøj

Finally, the paper ends with a concluding section in which we also touch upon how theseideas can be implemented in teaching and learning of mathematics in upper secondaryschool.

2 Mathematical discourse—object-level rules, meta-discursive rules and metalevellearning

In this section, we explain why Sfard’s theory of thinking can provide a theoreticalfoundation for our proposal that history is, if not indispensable, then at least an obviouschoice for learning meta-discursive rules in mathematics.

Sfard’s theory of Thinking as Communicating begins with a claim made by Vygotskythat collectively implemented activities, historically established, are prior to thedevelopment of human skills. Thinking is such a skill, and Sfard (2008) defines itaccordingly, as the “individualized version of interpersonal communication” (p. xvii, italicsin original). To stress the combination of communication and cognition inherent in thisdefinition, she coined the term commognition. Mathematical thinking is a human activity;hence, it is subject to Sfard’s theory of commognition. Mathematics is then viewed as atype of discourse, as a well-defined form of communication. In this theory, mathematicsemerges as an autopoietic system, which means that it is self-generative and produces itsown conceptual referents. These objects are themselves discursive constructs, and as such,they constitute a part of the discourse (Sfard, 2008, p. 129). Learning is viewed as“becoming a participant in a certain discourse” where discourse “refers to the totality ofcommunicative activities, as practiced by a given community” (Sfard, 2000, p. 160).Learning mathematics means to be socialised into a mathematical discourse.

The discursive interpretation of learning and knowing underlines the social nature ofintellectual activities. Communicating is a rule-regulated activity, and discursive patternsresult from such rule-governed processes. Sfard distinguishes between two types ofdiscursive rules, both of which are important for learning a discourse, namely discursiverules concerning the content of the discourse and discursive rules about the discourse.These two sets are referred to as object-level rules and meta-discursive (or meta-) rules,respectively. In the present paper, we focus on meta-discursive rules. They are rules aboutproper communicative actions, and the discourse itself is shaped by these rules. They areimplicitly present in discursive actions where they govern “when to do what and how to doit” (Sfard, 2000, p. 166; 2008, p. 201–202). Meta-discursive rules are historically given;they are not necessary but contingent.

In mathematics, object-level rules regard the properties of mathematical objects such as“the line segment between two points in a convex set lies in the set itself”, which is amathematical narrative of a particular geometric shape. The meta-rules of mathematicaldiscourse govern and

manifest their presence…in our ability to decide whether a given description cancount as a proper mathematical definition, whether a given solution can be regardedas complete and satisfactory from a mathematical point of view, and whether thegiven argument can count as a final and definite confirmation of what is beingclaimed. (Sfard, 2000, p. 167)

The distinction between object-level rules and meta-rules is not absolute, since meta-rules in one mathematical discourse will turn into object-level rules when the meta-discourse becomes part of mathematics itself. As an example, we can consider statements of

History as a method for learning meta-discursive rules in mathematics 329

impossibility such as the three classical ones: the impossibility of squaring the circle, ofdoubling the cube and of trisecting the angle, using Euclidean tools. In Antiquity, thesewere meta-statements. Only much later, during the seventeenth to the nineteenth century,did such statements become part of the content of mathematical discourse, requiringrigorous proofs.5

Meta-discursive rules affect how participants in the discourse interpret the content of thediscourse. Therefore, developing proper meta-discursive rules is indispensable for learningmathematics. Hence, Sfard (2008, p. 202) argues, one of the educational goals of learningmathematics is to gradually modify the meta-discursive rules that govern a student’smathematical discourse. As a consequence, an essential aspect of mathematics education isto create teaching and learning situations where meta-discursive rules are exhibited asexplicit objects of reflection for students—and here, history of mathematics seems topresent itself as an obvious strategy, precisely because meta-discursive rules ofmathematical discourse are historically established, and as such, they can be addressedand treated directly at the object-level of history discourse. By having students ask andinvestigate historical questions about the development of practices of mathematics, usinghistorians’ tools, meta-rules can be exhibited as explicit objects of reflection. This cannot bedone within mathematics itself, except in specialised courses on the foundation ofmathematics. Hence, history can have a significant role to play in learning mathematics.

Meta-level learning is difficult, and as pointed out by Sfard, because of the contingencyof meta-level rules, it is not likely that learners by themselves will begin a meta-levelchange. Such a change is most likely to happen when (or if) the learner experiences another(new) discourse that is governed by meta-rules other than those the learner so far has beenregulated by. An experience like that constitutes what Sfard (2008, p. 256) calls acommognitive conflict, which she defines as “a situation in which different discursants areacting according to different metarules”.

According to our argument history can be used in mathematics education to reveal meta-discursive rules and make them explicit objects of reflection and—ultimately—to provokecommognitive conflicts. A prerequisite for this to happen is to implement a genuineapproach to history. In the next section, it will be suggested that studying the history ofmathematics from practices of mathematics will be a suitable method.

3 History of mathematics from its practice: the pitfalls of whig history

As has been suggested above, the reason why history is a strong candidate as a method formeta-level learning is that meta-discursive rules are contingent. They develop and changeover time, and as such, they can be investigated at the object level of history discourse. Byexamining original sources in their historical context and analysing the work of pastmathematicians, their views on mathematics, the way they formulated and argued formathematical statements, and so on, historical texts can play the role of “interlocutor”. Byusing historical sources, teaching and learning situations can be created where students canexperience differences in the way interlocutors communicate. As active learners, studentscan become aware of their own meta-discursive rules by identifying the meta-rules thatgoverned the mathematics of the past and comparing them with meta-discursive rules

5 For a history of such statements and their role in mathematics, see Lützen (2009).


governing the mathematics of their textbook and instruction. In this way, opportunities forstudents to experience commognitive conflicts are provided and proper changes can beinitiated.

This, of course, presupposes that the pitfalls of whig history must be avoided. Whighistory is a term introduced into historiography by the British historian Herbert Butterfieldin the 1930s, and it refers to an approach to history where one tries to find the present(Butterfield, 1931). The ‘whig fallacy’ lies in its so-called present-centredness, which, inthe words of Wilson and Ashplant (1988, p. 11), leads to a history that is:

… constrained by the perceptual and conceptual categories of the present, boundwithin the framework of the present, deploying a perceptual ‘set’ derived from thepresent.

From this quote by Wilson and Ashplant, we can infer precisely why whig history mustbe avoided if one wants to use history to create teaching and learning situations wheredifferences in meta-discursive rules of mathematical discourse of the past and the presentare elucidated and made into explicit objects of reflection. If one’s reading andinterpretation of historical sources are constrained by the way mathematics is perceivedand conceptualised in the present, the historical text cannot play the role of an“interlocutor” that can be used to create commognitive conflicts, as explained above, whenstudents “communicate” with the text, since differences in the way of communicating in thepast and in the present will have been “washed away” by the whig interpretation.

The reason why one needs to be especially aware of the pitfalls of whig history whenone wants to use history in mathematics teaching is that until recently, history ofmathematics was primarily written by professional mathematicians in the whig tradition.This has also had consequences for translations of sources. Teachers need to be aware ofthis methodological problem when they use historical sources. As pointed out by Schubring(2008), published translations of well-known historical classics exist that, contrary to theirclaim of being accurate, have distorted the sources.

Hence, in order for history to function as a method for meta-level learning inmathematics education, we need to take a genuine approach to history—but what doesthat mean? It is important for professional historians to place past episodes into theirhistorical contexts. Their historical awareness is conceived of as an interpretation of thepast, where they maintain a critical distance to past events emphasising differences betweenpast and present.

A methodology, where the historical development of mathematics is studied frompractices of mathematics, is characteristic of many recent studies in the history ofmathematics.6 Here, historians investigate how and why mathematicians decided to discussand work on particular problems, introduced certain concepts and definitions and employedparticular strategies of proofs, and they thereby identify changes in the understanding ofmathematical entities, notions and approaches. On one hand, such investigations concernmeta-discursive rules of the past, and hence, by identifying the meta-discursive rules that

6 See e.g. Leo Corry’s introduction (Corry, 2004) as well as the rest of the papers published in Sciencein Context, 17(1/2), 2004. See also (Daston, 1988; Dahan-Dalmedico, 1996; Epple, 1999; Kjeldsen,Pedersen & Sonne-Hansen, 2004; Kjeldsen 2006) to name just a few where also further references can befound. In history of science, there has been a trend to approach the development of the experimentalsciences from such a perspective emphasizing the local nature of research practices; see Buchwald andFranklin (2005, p. 1). For methodological discussions of this approach in history of mathematics, see e.g.Epple (2004, pp. 131–164).


governed the mathematics of the historical text, opportunities for turning meta-discursiverules into objects of reflection can be created. On the other hand, such investigations alsoprovide a foundation for understanding issues that are more related to history ofmathematics in its own right, such as: what influenced the development of mathematics?What driving forces can be identified behind the development, and what are they dependenton? What kinds of factors and actions have modified the course of mathematics?

One way of answering such questions and providing explanations for historicalprocesses of change is to adopt a multiple-perspective approach to the historical study ofmathematical practices.7 Hereby, we mean to analyse and consider the practice ofmathematical activities at a specific time and place from several points of observation—or contexts. The perspectives can be of different kinds and the mathematics can beconsidered from different angles, such as various sub-disciplines of mathematics,techniques of proofs, applications, “nature of mathematics” positions, other scientificdisciplines, sociological institutions, personal networks, genders, religious beliefs and soon. This approach to history raises the question of which perspectives to choose, since, ofcourse, not every perspective that one can think of is necessarily interesting (or accessible)regarding a particular historical analysis. A way to handle this difficulty when using historyin mathematics education is to adopt a problem-oriented approach, that is, to have clearlyformulated historical research questions and then to focus on perspectives and to choosehistorical episodes that explicitly address and relate to the issues one wants students toreflect upon. In situations where teachers want to use history to facilitate meta-levellearning, they must have the students analyse the sources from perspectives that address themeta-discursive rules of the mathematical discourse of the text.

Outstanding tools for analysing mathematical historical sources in this respect are thenotions of epistemic objects and epistemic techniques. These tools have been accommo-dated into historiography of mathematics by Epple (2004) from a methodologicalframework developed by Rheinberger (1997) for the study of experimental sciences. Thesenotions are used to capture what has been called “the intellectual workplace” of (groups of)mathematicians. Epistemic object(s) refers to the mathematical object(s) that is/are treatedin the text, i.e. the object(s) the author (the mathematician) is studying. Epistemictechniques refer to the mathematical methods, which the mathematician applied toinvestigate the object(s). These concepts are constructed to distinguish between problem-generating and answer-generating elements of specific mathematical research episodes.Hence, they can be used to gain insights into the functions and the dynamics of problemsand techniques in developments of pieces of mathematical knowledge (Kjeldsen, 2009). Ifstudents work with the sources in ways that mimic historians’ questions about the texts anduse historians’ tools such as epistemic objects and techniques to elucidate issues thatinvolve meta-discursive rules of the discourse of the text, history might be a promisingcandidate for facilitating meta-level learning in mathematics education.

In the next section, two project reports that deal with issues from the history ofmathematics will be analysed within the methodology outlined above with respect to howand in what sense meta-rules of mathematical discourse were made into explicit objects ofreflection and whether possibilities for meta-level learning were present. The reports werewritten by students within the problem-oriented and student-directed project work at themathematics study programme at Roskilde University in Denmark.

7 This is inspired by the Danish historian Bernard Eric Jensen. We have borrowed the term multiple-perspective approach, which is a direct translation from Danish of his term “flerperspektivisk tilgang”, fromhis work (Jensen, 2003).


4 Analyses of students’ history of mathematics projects with respect to developmentof meta-discursive learning

We have many years of experiences with supervising student-directed and problem-orientedprojects at the science and mathematics study programmes at Roskilde University (Blomhøj &Kjeldsen, 2009; Kjeldsen & Blomhøj, 2009). Over the years, we have observed that historicalperspectives on mathematics and its applications often seem to support students’ reflectionson mathematics as a subject as well as students’ development of important mathematicalcompetences (Niss, 2004), such as, for example, becoming competent in thinkingmathematically and using and reflecting upon symbolic and graphical representations inmathematics.

Similar experiences have come out of our developmental in-service courses where uppersecondary teachers have developed, implemented and evaluated courses related tomathematical modelling (Blomhøj & Kjeldsen, 2006) and/or the historical developmentof mathematics (Kjeldsen, 2011b). In general, our approach to researching the learningpotentials of the use of history in mathematics teaching can be characterised as theorisingpractice that works. Kjeldsen (2010, 2011a) explains and illustrates how the ideas of amultiple perspective approach to the history of practices of mathematics and a competence-based understanding of mathematics education can be used to analyse student-directedprojects dealing with the history of mathematics. In these papers, the analyses served toexemplify how mathematics can engage in interdisciplinary work with history to contributeto general educational goals and to learning mathematics and how history can be used as ameans for learning (parts of) traditional core curriculum mathematics without distortinghistory, respectively. Hence, the motivational benefits of taking up history in mathematicsteaching with the purpose of providing students with an understanding of mathematics as acultural and historical product of human intellectual activities can go hand in hand withdevelopment of (parts of) students’ mathematical competence. These theoretical ideas andconstructs have made it possible for us to capture and conceptualise learning potentials thatwe have experienced in these contexts.

In this paper, we have gone a step further by suggesting that history might have a muchmore profound role to play for learning mathematics based on Sfard’s theory ofmathematics as a discourse. In the two previous sections, we have outlined the theoreticalargument for history as an obvious, if not indispensable, tool for revealing meta-discursiverules and turning them into explicit objects of reflection for students. The purpose of thissection is to offer some empirical evidence for this suggestion.

The method we have applied in the following is an in-depth analysis of two projectreports written by two groups of students. The two reports belong to a huge cohort ofproject reports in, with and about mathematics, which have been produced and accumulatedsince 1972 in the mathematics bachelor’s and master’s programmes at Roskilde University.Characteristics of all these reports are (1) that they are written by self-formed groups ofstudents (of 3–8 in average) who study mathematics as one of two subjects at the level of abachelor’s or master’s degree; (2) that they are problem-oriented in the sense that whatdetermines and guides the students’ project work, and what decides the content of theproject, is a formulation of a problem, or a complex of problems, of the students’ ownchoice. There are no restrictions on content and no requirements of a particular curriculumthat should be dealt with in the projects. The only restriction is that the problem must bedeveloped in accordance with the so-called project constraint. In the mathematicsprogramme, each student participates in three problem-oriented project works, belongingto three different project constraints; in daily talk referred to as “the modelling” project,


“the mathematics as a discipline” project and “the profession” project. The projects that areinteresting for the questions dealt with in the present paper are performed under the“mathematics as a discipline” or “the profession” constraint; (3) that they run for onesemester, during which the students are supposed to use half of their study time on theirproject work. The other half of the time, the students are occupied with more traditionallyorganised course work in mathematics; (4) that the students themselves identify andformulate the problem they want to investigate and answer during the project work; theyfind, select, read and analyse relevant literature; they identify the theoretical core of theirproject; if necessary they design and perform experiments; they solve their problem, anddiscuss strengths and weaknesses of the method they chose, their analyses, and theirfindings. All project groups have a professor assigned as their supervisor; preferably aprofessor who is a specialist in the area of the students’ work; (5) all the reports areproducts of such a project work.8

We have personal experiences with many of the projects performed during the past15 years, and in particular with the two selected projects, from being the coordinatingteacher for the project work during many of the semesters, from being a supervisor(especially for the first project analysed below, which was supervised by the first author ofthe present paper), from being an opponent supervisor, and from informal discussions withthe students and the other supervisors (especially for the second project analysed below). Ingeneral, the supervisor meets with the students in a project group once a week. In the idealsituation, the students have provided an agenda before each meeting and written somepages with thoughts about their project, such as outlines of chapters for their report,problems they are having with reading a text they do not understand or finding somespecific literature or deciding how to structure their report, dealing with the internal groupdynamic or a specific group member, organizing and planning their work and so forth. Themeetings with the supervisor normally take place in the groups’ “group room”, which is asmall office with a table in the middle, a computer, a blackboard, some bullet boards and alocker. During the meetings, students discuss the issues about which they want theirsupervisor’s advice. They can also ask the supervisor to prepare a lecture for their nextmeeting on some specific topic they need to be aware of, and, vice versa, the supervisor candemand that the students provide and prepare essays or something else for an upcomingmeeting. These meetings are quite informal. They are planned by the group and thesupervisor in collaboration. Sometimes the group will cancel a meeting if they feel that itwill be a waste of time, either because they already know what to do and do not need helpat the moment, or else they did not progress as far as they had planned since the lastmeeting. Students are encouraged to keep minutes from these meetings, but it does notalways happen. At the end of the semester, the project work is evaluated through an oralfinal with an external evaluator, normally a professor from another Danish university. Eachstudent is evaluated individually. Three days in advance, the student draws a questionrelated to the project work formulated by the supervisor. At the examination, the studentpresents an answer to the question, followed by a discussion with the supervisor and theexternal evaluator of the entire project.

The two projects treated below are analysed primarily on the basis of the written projectreport that was submitted by the students at the end of the project work, but for the first

8 For further details on the project-organised study programmes at Roskilde University, see Niss (2001). Fora presentation and discussion of how history and philosophy of mathematics and science is integrated in the2-year introductory science and mathematics programme, also through problem-oriented project work, seeKjeldsen and Blomhøj (2009).


project below, the analysis also draws on minutes and reports from meetings between thegroup of students and their supervisor.

Both project reports belong to the “mathematics as a discipline” constraint. Toparaphrase from the study regulations, in this project work, students should work with aproblem through which they will gain experiences with the architecture and the nature ofmathematics as a scientific discipline, such as its methods, concepts, theories, foundationand so on. This should be done in such a way that aspects of the historical development ofmathematics, and/or of its “theory-of-science” status and/or of its place in society arerevealed. Not all projects performed under this constraint are history projects, but many ofthem are, and the two reports that will be analysed below are representative of the cohort ofhistory projects.

As should be clear from the above description of the problem-oriented project work as it isperformed and organised within the mathematics programme at Roskilde University, we aredealing with very complex learning situations. Winsløw (2006, p. 223) identifies fourdimensions to characterise advanced student activities for learning: (1) the scientific content andscope of the work—how advanced is the scientific level of the students’ work? (2) Perceptionand production of scientific language—do students move beyond passive conception of simplepropositions and concepts or do they actively produce and engage in scientific discourses? (3)Autonomy—do students solve closed problems with predetermined techniques and methods ordo they formulate and investigate open problems independently and evaluate and criticize theirresults? (4) Affective positioning—do students move from an external positioning with respectto the subject to an internal positioning where they experience themselves as dealing with thesubject voluntarily and competently? As mentioned by Winsløw (2006, p. 224), the mostradical interpretation of these four dimensions of advanced study is that students should beengaged in research-like activities. This is exactly what happens in the problem-orientedproject work at Roskilde University. The complexity of the project work is also reflected inthe fact that the students’ project reports can be analysed from several perspectives withrespect to learning outcomes and possibilities: development of mathematical competence incombination with development of overview and active insights into the historicaldevelopment of mathematics (Kjeldsen, 2011a; Kjeldsen & Blomhøj, 2009); developmentof interdisciplinary and of general educational competencies in combination with develop-ment of mathematical competence (Kjeldsen, 2010); and below, the potentials for meta-levellearning through project work in history of mathematics.

The two project reports have the titles “Physics’ influence on the development ofdifferential equations and the following development of theory” and “Fourier and theconcept of a function—the transition from Euler’s to Dirichlet’s concept of a function”,respectively. They have been selected because they represent two types of history projectsdone under the “mathematics as a discipline” constraint: one where the problem thestudents want to investigate relates to how external factors, in this case from anotherscientific domain (physics), have motivated and influenced the development of mathemat-ics, and the other where the problem relates to what could be called internal developmentsof mathematical concepts. As is well known, the concept of a function was developed in adynamical process involving problems from physics, but in this project work, the studentswere not interested in understanding the influence from physics. Instead, they focused onthe internal mathematical development of the concept of a function. The projects have alsobeen chosen because they deal with mathematical theories and concepts that are alsopresent in mathematics curricula of upper secondary school—and in the next section, webriefly discuss how the possibilities of using history for meta-level learning might beimplemented in upper secondary school.


4.1 Project 1

4.1.1 Physics’ influence on the development of differential equations and the followingdevelopment of theory

This project report was written by five students who formulated the following problem-complex9 that they wanted to investigate:

How did physics influence the development of differential equations? Was it asproblem generator? Did physics play a role in the formulation of the equations? Didphysics play a role in the way the equations were solved? (Paraphrased from work byNielsen, Nørby, Mosegaard, Skjoldager and Zacho, 2005, p. 8)

The students studied three original sources from the 1690s: the solutions of thebrachistochrone problem by Jakob Bernoulli and Johann Bernoulli, respectively, and JohannBernoulli’s solution of the catenary problem. In order to answer their questions, the students hadto read and understand the sources within the mathematical discourse of the time. On one hand,this is a difficult task because the students’ points of departure in dealing with the sources aretheir own mathematical discourses, which are different from the discourse of the authors of thesources. On the other hand, this is exactly the reason why history, and working with originalsources, can serve as an effective method for meta-level learning. In the following, we will givesome examples from the students’ work with Johann Bernoulli’s text on the problem of thecatenary where meta-discursive rules were exposed and reflected upon.

Consider a flexible chain that is hanging freely between two points. The catenaryproblem is to describe the curve formed by such a chain. The students read Bernoulli’s textsupported by English translations (Bos, 1975, p.36) of parts of the text, and, as we will seebelow, the students’ explained how Johann Bernoulli derived his differential equation

dy : dx ¼ a : s

geometrically by using the length of the curve s, the infinitesimals dx=gG and dy=Ha (seeFig. 1) (here and in the following quote from the students’ report, the reader needs to beaware that the letter a is used in three different ways (1) to denote the force F0, (2) to denotethe generic angle in sina=sin (180°−a) and (3) to denote the letter facing H on the diagram)and five hypotheses obtained from statics. We do not present those five hypotheses here, eventhough in the extract below from the students’ report, there are references to the fivehypotheses, but the reader should just ignore them, since they are not essential for theargument we want to make. Figure 1 was drawn by the students and is similar to a figure inBernoulli’s text, except for the cosine–sine circle which was added by the students. Thestudents wrote (Nielsen et al., 2005, pp. 21–23)10.

Bernoulli considered the part of the chain A to B (see Fig. 1).… Since the weight ofthe chain is evenly distributed along the chain one can choose a unity of weight suchthat the weight of AB is equal to the length of the curve s.… Furthermore, Bernoullisets F0=a, since F0 is a constant according to hypothesis 4. The tangents to the curve(the forces F1 and F0) meet in the new P here called E. This means that the mass of

9 The exact formulation of the problem that guides the students’ project work is a process that often goes onthroughout the entire semester where the problem formulation becomes more and more qualified, as thestudents become more and more familiar and knowledgeable within the scope of their project work.10 All the quotes from the students’ report have been translated into English by the authors of the presentpaper.


the piece of the chain AB is placed in E in hypothesis 2. According to hypothesis 5the equation can now be expressed in the following way:

weight of AB

F0¼ s

a¼ sinðϕ1 þ ϕ2Þ

sinϕ1

Since sin a=sin (180° – a) (see Fig. 1):

sin< ¼ sinðϕ1 þ ϕ2Þ ¼EL

EA

From the same figure we see that sinϕ1 ¼ ALEA so by combining these three expressions

we get the following expression:

s

a¼ sin<

sinϕ1¼

ELEAALEA

¼ EL

AL

We now see that the ratio between the two sides EL and AL corresponds to the ratiobetween the two smaller sides of a right angle triangle. The same ratio can beexpressed from a much smaller but similar triangle, namely the ratio dx/dy (seeFig. 1) [BG=x, GA=y, Gg=dx, Ha=dy]. Hence we get the expression used by JohannBernoulli to express the catenary:

dy

dx¼ a

s

This whole derivation began with an analysis of how the forces affect the catenary,and from there through tools from geometry a single expression could be derived.

The students found it quite difficult to understand Johann Bernoulli’s arguments.First of all, his mixed use of infinitesimals, geometry and statics (which we todayactually would characterize as mathematical modelling) was very different from the

Fig. 1 The catenary


way the students were used to working with differential equations in their mathematics courses.Second, Bernoulli’s reasoning is not in accordance with our modern standards of rigour.Together, these two circumstances created cognitive hurdles for the students. The technique ofusing the infinitesimal triangle was new to the students, and they were quite sceptical when theyrealized that Bernoulli, in using similar triangles to argue that s/a=dx/dy, actually operated as ifin Ha=dy belongs to the tangent, which it does not, as the students noted in their report, sinceit is situated on the catenary.

This problem with the infinitesimals was discussed at a group meeting with thesupervisor on the 4th of October 2005. The day before the meeting, the studentswrote the following in an e-mail to their supervisor (the first author (Tinne) of thepresent paper)11:

Hello Tinne,We will see each other tomorrow (Tuesday) at 10 am, and as last week we will “warmup” with a group meeting before we meet with you.We will discuss the following issues:

1) working paper (The problem of the Brachistochrone, the Vibrating string, theCatenary).

2) Future strategy including a “skeleton” for our project report.3)What do we want to discuss with Tinne? (This will be decided at our morning meeting).4) Concrete tasks (What we ourselves think we would work on, deadlines etc.).

I suspect that you will also be involved in our considerations regarding the differentissues, in particular our “future strategy”. Therefore I have attached some key words wehave planned to discuss tomorrow. You only need to “skim read” them—it’s just so thatyou have an idea of what we have just discussed before we meet with you. (e-mail, datedOctober 3, 2005 from the group to Kjeldsen).

In the attached file, one of the key words was ‘infinitesimals’ of which the students hadwritten the following:

Infinitesimals (1700’s) Problems with the infinitely small; doubts about thefoundation for the theory of differential equations. What significance does theinfluence from physics have, when we begin to scrutinize what kind of entitiesinfinitesimals are? When does it block the way for the development of the theory?When does it act as a driving force? (Notes attached to e-mail, dated October 3, 2005from the group to Kjeldsen)

The students’ problems with the infinitesimals and Bernoulli’s way of arguing withinfinitesimals also surfaced in their treatment of Bernoulli’s reformulation of the differentialequation. In order to eliminate s, Bernoulli rewrote the differential equation into the formdy ¼ adx

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x2 � a2p.

. In this process, Bernoulli treated ds as if it was part of the tangent(when it actually is part of the catenary) in order to derive the equation:

ds ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

dx2 þ dy2:p

11 The “contract” with the students was that they should provide an agenda for the weekly meeting with theirsupervisor a few days in advance. As a supervisor, I always try to press the students to deliver something inwriting for each meeting accompanied by a reading direction such as: you only need to skim part such andsuch; in section such and such, we need feedback on the issue of such and such; we consider part such andsuch to be almost finished, so please read it carefully, etc.


The students used three pages in their report to explain this reformulation and they addedthe following comment:

We can see that even though he [Bernoulli] had learned some new techniques, thecalculus, his way of thinking was still to a large extent geometrical. This can beexemplified through the following two examples: One of the examples is from JohannBernoulli’s reformulation of the differential equation on page 29: “Since s=adx:dy weshall have ds ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

dx2 þ dy2p

”. We had problems understanding this statement, and inorder to understand it we had to use geometrical considerations. We remind the readerthat we used one and half pages to understand and explain this derivation. (Nielsen etal., 2005, p. 41)

The fact that Bernoulli just used Pythagoras’ theorem on the infinitesimal triangle wasnot obvious for the students, probably because of the lack of rigour in this way of arguingcompared to their own mathematical discourse. The following quote from the students’report signals that the students did not master the contemporary discourse:

If Johann Bernoulli had not defined the curve to be homogenous, it, and thereby alsothe differential quotient of the curve, would not behave as they actually do. (Nielsenet al., 2005, p. 40)

Even though the students were aware of the significance of the homogenous assumption,they “translate” it into their own discourse as “the differential quotient of the curve”, whichdoes not make sense in the proper discourse of mathematics in 2005.

The students did not use the concepts of epistemic objects and epistemic techniques intheir analysis of Bernoulli’s text, but it would have been a really powerful tool, becausethese concepts are constructed to analyse what could be called the mathematical practiceconveyed by the text, thereby giving insights into the way mathematics was practiced at thetime. In a teaching situation, analyzing the text with respect to what was the object underinvestigation and what were the methods used to reach answers could give rise tointeresting historical and mathematical discussions. If, for example, the text is taken at facevalue, the object under investigation is the catenary, but by situating the text in themathematical culture of Europe at its time, one can argue that perhaps the purpose of thetext was to explore the power of the new methods of infinitesimals. The techniques usedwere a mixture of pure and applied mathematics: knowledge gained from statics,geometrical reasoning and infinitesimals.

Bernoulli’s apparent lack of rigour (in our modern understanding) caused discussionsamong the students, as well as between them and their supervisor about criteria for rigourand how such criteria are decided. During these discussions, the students became aware thatstandards of rigour had changed over time. This is a clear example of the contingency ofmeta-discursive rules in mathematical discourse. Through the work with the historical textsof the Bernoulli brothers, situations were created where the students, among themselves andwith their supervising professor, discussed and reflected upon issues belonging to meta-discursive rules of mathematical discourse at the object level of history discourse. Thestudents drew on these discussions and reflections in answering their problem formulationin their project report. One example of this is the following:

Here we can see that physics [the homogenous assumption] gave Bernoulli a “nice”curve with respect to ds, which the differentiation depended on. Here we see veryclearly that physics functioned as a driving force with respect to the derivation of thedifferential equation for the catenary. (Nielsen et al., 2005, p. 40)


The second example we have chosen, where opportunities for meta-level learning surfaced,emerged when the students, again supported by the English translation in Bos (1975, p. 41),worked with the part of Bernoulli’s text where he solved the differential equation:

Johann Bernoulli and his colleagues in the 17th century did not know the logarithmicand the exponential functions. Instead he interpreted the integral geometrically as anarea below a curve. Hence, his solution was in the form of a geometrical construction.(Nielsen et al., 2005, p. 36)

Bernoulli drew the figure below (Fig. 2) (except for the figure with the shaded areaswhich is a figure added by the students). He took BA=a and described an equilateralhyperbola BC with centre A and vertex B. He then constructed a curve DI such thateverywhere BA is the middle proportional between KC and KD.

The students explained Bernoulli’s construction in the following way:

A hyperbola has the expression

x2

a2� y2

b2¼ 1

Since the hyperbola BC is equilateral, we have a=b. We manipulate the above and get:[omitting the students’ manipulations] y ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x2 � a2p

. Since x=x+a, we know frombefore that the above can be written as y ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x2 þ 2axp

. The curve DI is nowperformed as the middle proportional between KC and KD. This middle proportionalmeans according to [here the students have a reference; they had never met this termbefore.] that: KC/BA=BA/KD. Since … BA=a and KC is our hyperbola, we can usethis to find an expression for KD: KC/BA=BA/KD, hence KD=(BA)2/KC and then

Fig. 2 Bernoulli’s solution of the catenary equation


KD ¼ a2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x2 þ 2axp

.

We notice that KD is the expression for the curve DI with respect to the y variable.Bernoulli then reasons as follows:He constructs a rectangle GBAF the area of which is equal to the area bounded byIDKBH (the grey areas in the added picture in Fig. 2). Notice that BA is constant. Anypoint on the catenary can be constructed with a x (BK) and the corresponding value ofy (BG) determined by the line segment FG’s position. Bernoulli does not prove thatthis is correct. The correctness of the construction rests on the assumption that an areabelow a curve can be described with a rectangle. This is problematic since DI tends toinfinity when x tends to zero. … Even though the method is unorthodox (from amodern point of view) it gives the right result. Constructing the curve demands a fairamount of geometrical intuition and cannot be extended into a general method.(Nielsen et al., 2005, p. 37–39)

Bernoulli constructed the curve geometrically, which, again, was different from the waythe students perceived solutions to differential equations. In struggling to give meaning tothe text, the students were forced into discussions among themselves and with theirsupervising professor about conceptual aspects of what it means to be a solution to adifferential equation. They realized that what is understood by a solution has changed overtime; it belongs to the meta-discursive rules of mathematical discourse. And likewise, theyexamined what counts as a valid argument, about which they wrote the following:

On page 26 we can follow how Johann Bernoulli transformed his physical knowledgeinto mathematics. Approximately half of Johann Bernoulli’s account for thederivation of the differential equation of the catenary was paraphrased in that section.Whereas he later became much briefer in his derivations, he was very particular inthis derivation. […]. We interpret this as if Johann Bernoulli felt a need to documentthat precisely this transformation from physics to differential equations was wellfounded. This was done with geometry which had a high degree of validity in thisperiod. (Nielsen et al., 2005, p. 41)

Finally, the following quote from the students’ report shows yet another instance, wherethe students reflected upon meta-discursive rules of mathematics in their “dialog” withBernoulli’s text:

Bernoulli did not know the logarithmic function, so he could not describe the curve[of the catenary] analytically. Even though he showed an incredible geometrical andmathematical intuition, his construction [of the solution] did not lead to a generalsolution of similar problems. (Nielsen et al., 2005, p. 42)

Didactically, it is important to find and identify historical sources that are of such a characterthat opportunities for creating situations in teaching and learning like the ones described aboveare present when students work with the sources. As is clearly documented in the students’project report, working with the sources as presented above provided opportunities for learning,not only of meta-level rules but also of object-level rules of mathematical discourse. Thestudents were challenged to reflect upon the differences between the conceptualisations, asrevealed in the historical sources and as presented in their modern textbooks, of equationsbetween differentials, of the styles of argumentation and of concepts of a solution to adifferential equation. Regarding object-level rules, the students’ investigations of Bernoulli’s


use of infinitesimals as actual quantities and their discussions about why his argumentationresulted in the “right” answer, even though he had neither the concept of limit nor the concept ofa function, created fruitful connections between the students’ experiences with the historicaltext and their experiences with their textbook’s text. These experiences made them question andreflect upon their own understanding of the concepts of differential equations, limits and so on,on a structural level that went far beyond the mere operational level of understanding of theinvolved concepts. Hence, possibilities for learning of object-level rules were also created byworking with the historical sources.

4.2 Project 2

4.2.1 Fourier and the concept of a function—the transition from Euler’s to Dirichlet’sconcept of a function

This project was conducted by four students who were interested in the development of theconcept of a function. They began their report with a modern definition of a function:

In modern textbooks in mathematical analysis a function is often defined in thefollowing way:

Let A and B be two sets. If for every element x in A there corresponds one and onlyone element f(x) in B, then f is said to be a function from A to B.

Hence, the modern definition of a function is based on an idea of relationship betweenvariables. (Godiksen, Jørgensen, Hanberg & Toldbod, 2003, p. 1; italic in the original)

As can be seen from the quote, the students emphasised the basic idea that govern our conceptof a function. They then compared this definition to Euler’s concept of a function in the eighteenthcentury, and again emphasised the basic idea and the differences between then and now:

With Leonard Euler’s (1707–1783) treatise Introductio in Analysin Infinitorum theconcept [of a function] became central in analysis—a mathematical discipline that hademerged in the 17th century. Euler’s concept of a function, which dominated analysis,was based on algebra and the idea of one analytic expression. Hence, it was differentfrom our concept of a function in many ways. (Godiksen et al., 2003, p. 1; italic in theoriginal)

Due to the work of Fourier, Euler’s concept came under attack. The students formulatedit as follows:

In 1807 something happened that later turned out to be the beginning of the end ofEuler’s concept of a function. Joseph Fourier (1768–1830) […] showed that afunction can be represented by a certain trigonometric series. Fourier was met withgreat scepticism from the elite mathematical society. Especially Fourier’s use of thosetrigonometric series that later became known as Fourier series made a great stir. Butthe results of Fourier’s work could not be rejected in the long run, and Euler’sconcept of a function began to crumble away. (Godiksen et al., 2003, p. 1)

In this introduction, the students were outlining changes in the meta-level rule of what afunction is, and in their project work, they wanted to investigate these changes historically,that is at the object-level of history. Thereby, these meta-level rules were turned into explicitobjects of reflections for the students. The students chose to focus on Fourier’s influence,


and they formulated the following problem formulation, (which is not yet a fully formedproblem formulation)12:

We wish to investigate the significance of Fourier for the development of the conceptof a function. (Godiksen et al., 2003, p. 2)

The students focused their historical investigations on relevant sources written by thethree mathematicians Euler, Fourier and Dirichlet, which they analysed with respect tochanges in, and past discussions about, the concept of a function and the proper way toargue with functions. The students used the original sources as “interlocutors”, or as theythemselves phrased it:

The strength of focusing on these three mathematicians is, that it has given us theopportunity to study their original works (sometimes in translations) in depth, whichhave given us a more direct impression of their thoughts than secondary literaturecould have given us. (Godiksen et al., 2003, p. 2–3)

The point of departure for the students’ work was Euler’s concept of a function as hepresented it and worked with it in Introductio in Analysin Infinitorum from 1748, but asthey wrote in their report:

The main elements of Euler’s conception of a function could easily be explained veryshortly, but that would not contribute to any deep understanding of the concept. Inorder to obtain this, one has to look at how Euler worked with functions. (Godiksen etal., 2003, p. 17; italic in the origin)

The students’ point here is that in order to understand Euler’s conception of a function itis necessary to study Euler’s mathematical practice, how he used and worked withfunctions, so here they are approaching the history of the practice of mathematics from theperspective of how the concept of a function was understood by different interlocutors.

They found that:

His [Euler’s] algebraic approach to analysis is revealed already in the introduction [ofIntroductio in Analysin Infinitorum]…

Euler begins with a definition of a function:

A function of a variable quantity is an analytical expression composed in any waywhatsoever of the variable quantity and numbers or constant quantities. [Euler, 1988][art. 4]

As one can see, Euler’s definition of a function is not built on the idea of relationshipsbetween variables, as in our modern definition, but on the concept of an analyticalexpression. (Godiksen et al., 2003, p. 17; italic in the original)13

12 Often students and their supervisor will discuss the problem formulation throughout the entire semesterfocusing on whether the problem can actually guide the students’ work, whether it is in accordance with whatthe students are actually investigating and so on. As has been explained (Blomhøj & Kjeldsen, 2009;Kjeldsen & Blomhøj, 2009), the students do not always manage to formulate good research questions that areconsistent with what they have researched in their project. In the present case, the problem formulation isway too broad to actually guide the students’ project work. This does not mean that the project work has notbeen successful, but it indicates that the students’ formulation of their problem is not a precise articulation ofwhat they actually searched for.13 The Euler citations in the quotes have been taken from the students’ report (Godiksen et al., 2003). Thestudents have used the English translation from 1988 of Euler’s Introduction to Analysis of the Infinite (Euler,1988).


Euler thus defined a function as a formula composed of variables and constants thatcould be written down in mathematical notation. The students gave the followinginterpretation of Euler’s conception of a function:

The definition of a function [Euler’s definition] does not contain any specific informationabout its domain and image. This is because in Euler’s theory variable quantities areascribed a property that render specifications of such sets superfluous.…

…Euler conceived a variable as an arbitrary element, quite like our conception, butno constraints are allowed. The variable should be able to take all values…(it isuniversal). Euler formulated it like this:

…Thus a variable quantity encompasses within itself absolutely all numbers, bothpositive and negative, integers and rationals, irrationals and transcendentals. Evenzero and complex numbers are not excluded from the signification of a variablequantity. (Godiksen et al., 2003, p. 18)

Euler’s analysis has been characterised as an algebraic analysis because algebraicmanipulations played a major role. It has also been characterised as global in nature,because, as the above quotations show, a variable quantity was not limited in scope; it wasuniversal. As a consequence, functions were defined for all values of their variables, andthey had the property of analytical continuation, since the continuation of a function outsidean interval I, in which the function was represented, could be obtained simply bysubstituting values outside of I for the variables. This generality of analysis is a meta-levelrule, which has changed, and which, through this historical project work, was made into anexplicit object of reflection for the students, who wrote:

This property, which […] has been named the criterion of the generality of the variable,clearly reflects the earlier mentioned paradigm of the general validity of analysis.[…]Even though the use of the methods of analysis often created weird results themethods were used frequently in Euler’s concept of a function. The reason why thereweren’t that many contradictions and paradoxes was that almost all Euler functions,which consists of analytical expressions, have all the above mentioned properties(they were nice), except maybe in isolated points. … Hence, there was no naturaldriving force that led to a clarification of the concepts of continuity, differentiability,and integration, since these properties so to speak were build into the concept of afunction. (Godiksen et al., 2003, p. 22; italic in the origin)

As pointed out by the students:

Euler … was of the opinion that the analysis had to be developed such that it was ableto describe all situations that occur in nature. (Godiksen et al., 2003, p. 23)

And in their treatment of Fourier’s work, they continued:

Fourier expresses clearly that mathematics is a tool for describing nature andmathematics had to be governed by nature. (Godiksen et al., 2003, p. 53)

This is a third meta-level rule that the students discussed in their study of originalsources dealing with functions.

Regarding Fourier’s thoughts about the concept of a function, the students’ key source wasChapter 3 of his work Théorie Analytique de la Chaleur from 1822 in which he solved theequation for heat conduction in a solid. Fourier’s text is rather complicated both regarding the


structure of the text and the mathematical level. We will not go into technical details but onlydiscuss aspects of the students’ work and their insights into how and in what sense Fourierbroke with the (loosely) established concept of a function.14 The students examined this bystudying the criticism, raised by some of the leading mathematicians of the time, of howFourier used and viewed functions. They realised that his understanding and use of functionswent against the conception of the universality of the variable and the general validity ofanalysis, which were dominant meta-level rules in the eighteenth century. Fourier found threedifferent expressions of x/2, two of them were the following:

sin x� 1

2sin 2xþ 1

3sin 3x� 1

4sin 4xþ . . .

and

p4� 2

pcos x� 2

32pcos 3x� 2

52pcos 5x� . . .

about which Fourier pointed out that:

It must be remarked that these…values of 1/2 x ought not to be considered as equal;with reference to all possible values of x, the … preceding developments have acommon value only when the variable x is included between 0 and 1/2π. (Godiksen etal., 2003, p. 45)15

The students used this quote from Fourier to argue that:

Here Fourier is delivering a direct attack against the property of analyticalcontinuation, by presenting … functions that are equal in a certain interval, butdiffer from one another outside this interval. (Godiksen et al., 2003, p. 46)

Through this project work, the students became aware that what we would call meta-discursive rules were changing during the development of the concept of a function, andthey were able to understand at least some of the reasons for these changes. Another point,not discussed above, is that the students also realised that what makes sense in mathematicshad changed. In the eighteenth century expressions of divergent series were conceived asfunctions, whereas later, such expressions became meaningless. Through their readings andanalyses of the original sources the students experienced and discussed an example of twomathematical discourses, prevailing at different times in history, that are incommensurable;what made sense for mathematicians in the eighteenth century was considered meaninglessfor nineteenth century mathematicians.16

5 Discussion and conclusion

Historical studies of strategies of proofs, and of the emergence and stabilization ofmathematical concepts, provide a window into how meta-rules of mathematical discourseare established. In the first case discussed above, by reading the sources dealing with the

14 It is important to be aware that this was a time when the concept of a function was not clearly defined.Even though there are some explicit, general definitions of a function around in the eighteenth century, therewas a gap between the way mathematicians used functions and how they defined them (Lützen, 2002).15 The following Fourier citations have been taken from the students’ report (Godiksen et al., 2003). Thestudents have used the Dover version from 1955 of Freeman’s English translation from 1922 (Fourier, 1955).16 See e.g. (Lützen, 2002).


catenary problem, the students experienced that strategies of proofs have changed overtime. In the second case, by reading the sources to understand the debate that was stirred byFourier’s work, the students were able to follow and study discussions, where differentdiscussants were acting according to different meta-rules, at a point in time where amathematical discourse was changing and new meta-rules were about to emerge and benegotiated. Opportunities for students to gain insights into such contemporary processes areextremely rare in mathematics education at levels below Ph.D. studies, but as this exampleshows, history can be used in mathematics education to give insights into how meta-discursive rules of mathematical discourse are established and why they change.

Because of the contingency of meta-discursive rules, they belong to the object levelof history discourse, and as such, they can be investigated, discussed and reflectedupon in historical narratives. This is why we suggest that history has a profound roleto play in mathematics education as a method for meta-level learning. Based on theanalyses of the students’ project reports, we conclude that by studying historicalsources and using proper historical methods to answer historical questions, studentscan become engaged in discussions and reflections about meta-rules of mathematicaldiscourse. This is clearly documented in the analyses of the students’ reports. Whetherthese discussions and reflections cause changes in the students’ mathematicaldiscourse towards developments of proper meta-discursive rules is another question,which is much more subtle and complex and also much more difficult to document.According to Sfard, possibilities for meta-level learning occur when studentsencounter experiences with a discourse that deviates from their own. This happenswhen they try to make sense of historical texts, interpreted within the context of thetexts that is in the mathematical discourse of the texts. Opportunities forcommognitive conflicts can be created in learning situations where students, throughhistorical investigations of original sources and discussions of the sources with theirprofessor, become aware that: first of all, there are meta-rules that govern the narrativesof mathematical texts; second, that such meta-rules have characteristic properties; andthird, that rules of the discourse of the sources are different from those that govern thenarratives of contemporary textbooks versions of mathematical analysis. In this complexweb of relations between meta-discursive rules of historical texts, of modern textbooks’accounts and the students’ own mathematical discourses, students can experienceincommensurability not only between the discourse of the historical text and the textbookbut also between their own discourse and their teacher’s and their textbook’s, resulting incommognitive conflicts and opportunities for students to develop proper meta-discursiverules.

In this paper, we have argued and illustrated how history can be used in mathematicsteaching and learning as a method for turning meta-discursive rules into objects ofreflections—and suggested that history can be used for learning meta-discursive rules ofmathematical discourse. We have also documented that the advanced student activities forlearning that are implemented through the problem-oriented and project-organised studyprogrammes at Roskilde University provide possibilities for this to happen in practice,when students work with the history of the practice of mathematics from perspectives thataddress issues that involve meta-discursive rules.

Meta-discursive rules are contingent, and as such, rational justifications cannot be given.This raises a problem for teaching of school mathematics. As has been pointed out by Sfard(2000, p. 184), teachers face an inherent dilemma if “we insist on teaching mathematics thatcannot be easily incorporated in, or derived from, everyday discourse”, since this requiresthat students are introduced to meta-discursive rules of mathematics even though these


cannot appear meaningful within everyday discourse. But there is no way out—to view therelaxation of meta-discursive rules and avoid any kind of mathematical rigour does notsolve the problem. There is a limit to how much the rules can be bent, renegotiated and/orrelaxed in the classroom if we want students to learn mathematics, since if we deprivestudents the access to meta-discursive rules, we also deprive them the opportunity to learnmathematics. Again, working with history in the ways exemplified in the present paper canbe an effective tool in dealing with this problem, since meta-level rules are discussed andreflected upon, not with respect to everyday life and rationales, but with respect tohistorically determined changes.

This is not an easy task, and teachers who want to pursue our line of thought in theirday-to-day teaching practice do not find much help. We think, however, that theoreticalarguments combined with empirical evidence from teaching practices, showing that historyof mathematics can function as a means for teaching and learning mathematics that extendsbeyond mere motivation, provide possibilities that can facilitate mathematics education.The theoretical argument for the learning possibilities as outlined in this paper providesguidelines for how to proceed in practice; hence, examples like the ones given here canhave a positive effect on the actual teaching of mathematics.17 Didactically, it is importantto find and identify historical sources that are suitable for provoking discussions inclassrooms among students and with their teachers about different meta-discursive rules.Likewise, it is important to perform research about how this can be done, how teachingactivities that support such discussions and reflections can be designed and how theeffectiveness of such teaching and learning situations can be evaluated in practice. Ourpaper provides the theoretical argument and some empirical examples that can be taking asa starting point for such investigations within the research field of ‘the use of history inmathematics education’ in mathematics education.

History can also be integrated into upper secondary school so that its benefitsinvolving meta-level learning can be exploited, while at the same time students canacquire genuine knowledge about the historical development of mathematics. This canbe done through matrix-organised group work, where groups of students investigatedifferent mathematical and cultural historical questions from multiple perspectives,related to the chosen issues, sources and/or modern versions of the narratives of themathematical discourse in question. Through such organisation, broader historicalpictures can be painted and shared in classrooms without taking too much time awayfrom curriculum teaching. A scheme for such a practice for inquiry-based teaching ofmathematics is under development.18

17 In Jankvist and Kjeldsen (2011), the lack of impact of history in mathematics education research isdiscussed and new avenues for history in mathematics educations are proposed together withexplanations of theoretical constructs that can be used as guidelines for practice. In Kjeldsen (2011b),a theoretical framework is outlined that can be used to analyse specific implementations that have beenrealized and as a tool to orient the design of future implementations of history of mathematics inmathematics education.18 This suggestion was outlined in at the workshop Does history have a significant role to play for thelearning of mathematics? Multiple perspective approach to history, and the learning of meta level rules ofmathematical discourse at the The 6th. European Summer University on the History and Epistemology inMathematics Education (ESU 6) in Vienna, July 19–23, 2010, and interested readers are referred to theforthcoming proceedings of the meeting, see Kjeldsen (2011c). A matrix organised design for using historyin mathematics education to elucidate meta-rules of past and present mathematics, to have students reflectupon those, to develop students’ mathematical competence, and general educational skills of independenceand autonomy is being tried out in a Danish upper secondary class at the moment. Results from the study willbe published in forthcoming papers.


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Beyond motivation: history as a method for learning meta-discursive rules in mathematics

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