ISSN 0919-1720

HIROSHIMA JOURNAL
OF
MATHEMATICS EDUCATION
Volume 16 

2023

Japan Academic Societyof Mathematics Education

HIROSHIMA JOURNAL
OF
MATHEMATICS EDUCATION
Editor-in-Chief: Tatsuya Mizoguchi, Tottori University, Japan (Vice-president of JASME)
Associate Editor: Takeshi Miyakawa, Waseda University, Japan
Editorial Board: Takuya Baba, Hiroshima University, Japan
Keiko Hino, Utsunomiya University, Japan
Takashi Kawakami, Utsunomiya University
Akio Matsuzaki, Saitama University, Japan
Hiroki Otani, Otsuma Women’s University
Yusuke Shinno, Hiroshima University, Japan
Advisory Board: Masataka Koyama, Hiroshima University, Japan
Masakazu Okazaki, Okayama University, Japan
International Editorial Board: David Burghes, Plymouth University, UK		Berinderjeet Kaur, National Institute of Education, Singapore		
Tad Watanabe, Kennesaw State University, USA		Gaye Williams, The University of Melbourne, Australia
International Advisory Board: Hee-chan Lew, Korea National University of Education, Korea
		Luis Radford, Laurentian University, Canada		
Heinz Steinbring, Universitat Duisburg-Essen, Germany

AIMS AND SCOPE
Hiroshima Journal of Mathematics Education (HJME) is the official English-language journal of theJapan Academic Society of Mathematics Education (JASME), and is devoted to research that seeks to
improve the mathematics education. It is an annual peer-reviewed and open-access international journal thatpublishes original papers written in English. HJME was founded by the Department of Mathematics
Education at Hiroshima University. Articles have been published by JASME since Volume 12 was issued
in 2019.
The journal is dedicated to the dissemination of research findings on mathematics education at alllevels (e.g., from preschool to university, professional development, lifelong education) on a variety ofissues related to the teaching and learning of mathematics (e.g., students’ understanding, classroom
teaching, curricula, policy, teacher education). It is open to any type of research (e.g., theoretical, empirical,
methodological, qualitative/quantitative). Papers dealing with Japanese mathematics education issues or
issues of special interest to the Japanese mathematics education community are particularly welcome.
Paper submissions are welcome from contributors from different sectors (e.g., researchers, educators,
and practitioners) who are interested in contributing to the development of mathematics education and its
research. HJME is expected to promote reflection on mathematics education and communication amonginternational researchers, educators, and practitioners in mathematics education. Papers are published ineither of two HJME sections: original paper and lecture notes. The original paper reflects the main topics ofthe journal. All papers submitted to this section are peer-reviewed, with an emphasis on papers of an
excellent level. Lecture note includes records or materials of scientific events (e.g., invited talks, symposia)
organized by JASMECOPYRIGHT INFORMATION©2021 Japan Academic Society of Mathematics Education
All articles published in this journal are protected by copyright. However, JASME does not challenge or
prohibit the author’s use of his/her article in the form of a copy or translation. In this case, the author must
gain permission from JASME and specify JASME as the source in the copies or translations.

SUBSCRIPTION
Hiroshima Journal of Mathematics Education is published once per year.
ISSN 0919-1720
The journal is presently distributed free of charge. Requests for permission to distribute the journal should
be made to the following address.
Japan Academic Society of Mathematics Education,
Graduate School of Education, Hiroshima University,
1-1-1 Kagamiyama, Higashi-Hiroshima, Hiroshima, 739-8524 Japan
E-mail: office@jasme.jp
Journal Website
Please visit the journal’s website for more information.
https://www.jasme.jp/hjme/

Editorial for Volume 16
Tatsuya Mizoguchi
Editor-in-Chief of the Hiroshima Journal of Mathematics Education,
Japan Academic Society of Mathematics Education
Domestic and international conferences have gradually reverted to face-to-face mode. Again, we enjoy thepleasure of meeting the faces that we used to encounter on a screen. Discussions in each community naturally
produce positive research outcomes. In many fields, the number of papers submitted has increased. We lookforward to your contribution to the Hiroshima Journal of Mathematics Education (HJME).
As each journal has its own peer review system, HJME operates on the system shown in Figure 1 (the actual
process is more complex, but the necessary procedures for authors are simplified).

Figure 1. HJME peer review system

Along with peer review systems, the type of peer review is often discussed. This journal currently employs
a double-blind peer review process. Each type of review has its own advantages and disadvantages. In
general, double-blind peer review prevents any type of bias or mutual pressure. Not knowing the author’s
background emphasizes the content of the paper rather than the assessment of the individual; whereas,
excluding knowledge of the context of the research topic makes it difficult to obtain the necessary information
to peer review.

The Special Issue of this volume has been peer-reviewed independently of the journal. An open-review
design was adopted. Open-review is considered to ensure transparency in the review process and make thetension for reviewers, while simultaneously maintaining authors’ satisfaction and restraining negative
criticism. In recent peer reviews of applications for presentations at international conferences, the open-type
has often been adopted. This creates a community around the research topic and enriches discussions during
conferences. Peer review systems and review types are important issues at the heart of journal operations. At
every opportunity in the future, we, the editorial team, would like to have the courage to be proactive inmaking improvements.
In addition, because HJME is an online journal, the editorial team discussed copyrights. Consequently, we
decided on a Creative Commons [CC BY] licence, starting with Volume 16 to maintain alignment with otherinternational mathematics education journals. We thank the authors whose contributions appear in Volume
16 for their agreement in this regard.
Volume 16 features one contributing paper and five Special Issue papers. The contribution by Louise MeierCarlsen analyses activities related to pupils’ study of equations using CAS with praxeology from theanthropological theory of the didactic. It draws the interesting conclusion that students learn to use equations
not only as tools but also as objects, and that this has a positive effect on technology and theory, that is, thelogos part of praxeology.
The introduction of Special Issue papers will be in the Editorial for Special Issue by the guest editors. The
HJME editorial team would like to thank Angelika Bikner-Ahsbahs, Ivy Kidron, Yusuke Shinno and Takeshi
Miyakawa for their great contributions as guest editors.
Finally, as I repeat the same thing every time, please note that HJME is published annually. The accepted
papers are published online before the publication of the next volume. Submissions are accepted at any time.
We look forward to receiving your contributions.

HIROSHIMA JOURNAL
OF
MATHEMATICS EDUCATION
Volume 16

2023

Tatsuya Mizoguchi
Editorial for Volume 16
Research Articles
Louise Meier Carlsen
Designing Activities for CAS-based Student
Work Realising the Lever Potential

1

Special Issue: Rethinking the Diversity of Theories in Mathematics
Education. Contributions Related to the Topic Study Group 57 of ICME 14
Angelika Bikner-Ahsbahs, Ivy Kidron,
Yusuke Shinno, and Takeshi Miyakawa
Guest Editorial

21

Research Articles
Michèle Artigue
Facing the Challenge of Theoretical Diversity: The Digital Case27

Ivy Kidron
The Role of a Priori Analysis in Theories

45

Luis Radford
Ethics in the Mathematics Classroom

57

Yusuke Shinno and Tatsuya Mizoguchi
Networking Praxeologies and Theoretical Grain Sizes inMathematics Education: Cultural Issues Illustrated byThree Examples from the Japanese Research Context

77

Angelika Bikner-Ahsbahs, Estela Vallejo-Vargas, Steffen Rohde,
Thomas Janßen, David Reid, Dmitry Alexandrovsky,
Anke Reinschluessel, Tanja Döring, and Rainer Malaka
The Role of Feedback When Learning with a Digital Artifact:
A Theory Networking Case on Multimodal Algebra Learning

95

HIROSHIMA JOURNAL OF MATHEMATICS EDUCATION 16: 1-19, 2023

1

DESIGNING ACTIVITIES FOR CAS-BASED STUDENT WORKREALISING THE LEVER POTENTIALLouise Meier Carlsen
IT-University of Copenhagen

Abstract
This paper explores two types of lever potentials of CAS and a tool for designing suchactivities. In the first activity, CAS is used for students to study the concept of equations. In thesecond activity, CAS is used to strengthen the relation between traditional algebraic paper-andpencil manipulations of equations and the theory that the solution of the equation must remain thesame. As the design tool for CAS-based activities that capitalise on the potential of CAS, thenotion of praxeology from the Anthropological Theory of Didactical is employed.
Keywords: Computer algebra system; school algebra; task design; lever potential; the notion ofpraxeology.

INTRODUCTION
In the early days of research on implementing CAS in mathematics education, the entrance of this
digital tool was met with applause by the community of researchers, and the focus was on the potential that
CAS could offer mathematics education (Dreyfus, 1994). The idea is, that, at the level of upper secondary
school, ‘low level’ work, such as solving an equation, can be outsourced (Bang, Grønbæk, & Larsen, 2017)
to the CAS. This would leave more time to focus on more complex issues such as the mathematical discourse.
This potential is denoted the lever potential (Winsløw, 2003).
The lever potential has been documented in several articles since the 1990s (which we will further
elaborate on in the background section). However, using a CAS on any problem in the subject of algebra
does not automatically lift the students’ learning to a higher level, quite the contrary. When solving a
traditional algebraic task, such as finding the solution for an equation (Kieran, 2007) with paper and pencil,
the mathematical discourse of the students includes the use of fundamental properties of algebraic structures,such as the distributive rule. When solving the equation with a CAS, the mathematical discourse becomesvery different, and most fundamental properties of algebraic structures are blackboxed (Carlsen, 2019).
In our study, we will consider the lever potential of CAS and explore the aspect, of the lever potential,of how CAS can be used to make the mathematical discourse explicit in the teaching of elementary algebra.
Furthermore, we will study how the four categories of task, technique, (technical) discourse and theoretical
discourse can be used to design appropriate activities in elementary algebra for the implementation of CAS.

Corresponding author: Louise Meier Carlsen

2Louise Meier Carlsen

BACKGROUND
In this section, we will introduce the lever potential of CAS for the teaching of school algebra, andoutline results from some of the studies depicting the lever potential of CAS. In addition, the role of CAS, inthe selected papers, is to perform part of the mathematical tasks such as time-consuming algebraic
manipulations, i.e., solve an equation or simplify an expression.
The lever potential of CAS was first introduced by Dreyfus (1994) and later further elaborated andnamed, as the lever potential, by Winsløw (2003). The idea is“for students to operate at a high conceptual level; in other words, they can concentrate on the operations
that are intended to be the focus of attention and leave the lower-level operations to the computer.” Dreyfus(1994) p 205.
That is, instead of focusing on applying a series of techniques e.g., to solve an equation, students caninstead progress to focus on more theoretical aspects such as technical discourse or theoretical discourse e.g.,
the number of solutions in relation to the type of equation.
One type of lever potential can be described as using CAS to swiftly consider multiple representationof a mathematical object. Then each different representation will accentuate different aspects of themathematical object (Dreyfus, 1994; Gjone, 2009; Kieran & Yerushalmy, 2004; Gyöngyösi, Solovej &Winsløw, 2011). Thus, supporting further development of the students’ concept of the mathematical object.
Another type of affordance of CAS is the ability to swiftly examine a series of comparable examples.
This enables students to consider similarities and differences between the examples and explicitly develop
and formulate their concept of the mathematical object in focus. Several papers have studied such possible
activities. For example, the doctoral thesis by Drijvers (2003) is a testimony to how carefully crafted CASbased activities can support students’ explicit development and formulation of their “definition” of parameters.
As an example, Heid et al. (2002) suggests working with the formula A(time) = deposit (1 + rate)time to havestudents develop and formulate their concept of parameters. Other suitable mathematical topics have also
been explored such as algebraic syntax (Artigue, 2002b; Guin & Delgoulet, 1997) and algebraic equivalence(Gjone, 2009; Lagrange, 2005) has also been conducted.
A different type of lever potential of CAS can be described as, despite outsourcing algebraic
manipulations to CAS such as expanding an expression. The pen-and-paper techniques can be used asarguments explicitly stated for validating the patterns produced and discovered with CAS. In a series of
papers, students work through a course of carefully constructed problems exploring the factorisation of thepolynomial xn‒1 using CAS and developing the telescoping technique (Kieran & Drijvers, 2006; Hitt &Kieran, 2009; Martínez, Kieran, & Guzmán, 2012).
The research mentioned, illustrating the different lever potentials, all have in common that the activities
are very carefully designed with very specific purposes in mind. In the last articles mentioned, the activities
are designed utilizing the dialectic between practice and reasoning (Artigue, 2002a). This dialectic is viewed
as a structural model using the three notions of task (the task to be solved), technique (that is applied to solve
the task), and theory (the reasoning that explains and validates the technique). The idea is taken from the
Anthropological Theory of Didactic (ATD). However, ATD suggests to further consider the theory part of thedialectic as consisting of two parts. One part is technology, which is the discourse that explains and justifies

Designing activities for CAS-based student work realising the lever potential

3

the technique. The second part is theory, which is the discourse that justifies and validates the technology.
This structural division can further benefit the design of CAS-based activities.
In our study, we want to show how the quartet-notion of praxeology can be used to design activities thatrealise the lever potential of CAS. The fist activity will focus on developing and having the students formulate
their definition of equations, while the second activity will include traditional algebraic technical discourses,
such as manipulation of equations, and link this to the theoretical discourse of having the solution stay thesame.

FRAMEWORK
In general, the ATD suggests considering human activity, such as cooking a soup, as an amalgam ofpractice and reasoning (Bosch & Gascón, 2014; Chevallard, 1998). To cook soup, one could start with frying
onions in a pot (the practice), with the reasoning being that onions give a nice flavour, and frying themsweetens and enhances their flavour. Alternatively, one might be frying onions because it is generally
perceived as a good way to start cooking a soup. Our knowledge is based on practice, and our practice isshaped by our knowledge.
The ATD proposes denoting the dialectic of practice and knowledge by praxeology. Further to structure
the practice as a twofold. One part the type of task, which is the type of task that is aimed at being solved.
For example, cooking a soup or solving an equation of the type s(x‒t) = n·s‒s·m·x, where s,t,m and n are
integers and s ≠ 0. The other part is the technique, which is the actions utilized in order to solve the type oftask, such as frying onions, or in the case of solving the equation above: rewriting the equation into x‒t =n‒m·x, and then another rewriting into the equation x = n+t . The knowledge related to the practice, which
1+m
is called logos from Greek meaning theory, can be split into two components. The technology, which is thediscourse that justifies and validates the technique, such as “we multiply with 1 on both sides of the equation”
sand “we can multiply with the same number on both sides of the equation”. The discourse can also containmotivation for the actions included in the technique, the effectiveness of the technique, etc. The theory
justifies and explains the technology, such as the solution set must remain the same. Again, the theory also
comprises of discourse such as the effectiveness of the technology and motivation for the technology.
However, the praxeology taught usually differs from the praxeology learned. Indeed, theory, or eventechnology, does not need to be “formal mathematics” but can, to individuals and even large institutions, also
include mere convictions and even falsities, such as “we must maintain the balance between the right-handside and the left-hand side of the equation” and “when we move a number from one side of the equation tothe other side, then the sign changes” or “it is correct because it is the same method our teacher illustrated”.
In general, technology is close to the actions of the technique and can also include motivation for the actionsof the technique, effectiveness of the technique etc. (Chaachoua, Bessot, Romo, & Castela, 2019). Whiletheory is the technology of the technology.
The following is an example of what a typical praxeology for a grade 8 student could look like whensolving the equation 3(x‒2) + 7 = 14 + 2x. The type of task is to solve the equation n(x‒m) + (n·m+1) = t +(n‒1)x, where n,m and t are positive integers. The technique involves rewriting n(x‒m) → nx‒nm (applying

4Louise Meier Carlsen

the left distributive rule from left to right), rewriting such that all terms containing the variable are on the left
side of the equation, etc. The prevailing technology would involve multiplying inside the parentheses, when
multiplying a positive number with a negative number, the result is a negative number; whenever weencounter a parenthesis, we “dissolve” it, etc. The theory: an equation must be kept in balance meaning thateach side of the equal sign must be equally heavy, the solution to the equation is a number such that each sideof the equal sign in the equation has the same value, parenthesis induce a different order of operations, etc.
The notion of praxeology is useful because the four components -type of task, technique, technology,
and theory- enable us to describe and analyse explicitly the elements in play in solving a type of task. This
tool lets us consider structures and relations between praxeologies within and across educational institutions
and mathematical themes, such as solving equations (of different types) or manipulating algebraic expressions.
A praxeology is determined by its type of task (and the institution in which it lives). An action or an ingredientof a technique can be applied in several different praxeologies or several times within the same praxeology.
For example, we can reduce an expression even when the task is not necessarily to solve an equation. One
technology can unify several actions and even whole techniques. For instance, multiplying both sides of theequation by a number does not necessarily reduce the equation. The theory, such as we must maintain thebalance between the right-hand side and the left-hand side of the equation, can join several technologies,
since it is also the theory for adding on both sides of the equation. Therefore, we can classify a type of leverpotential of CAS as to focus on (parts of) the technology or theory. By targeting the development of students’technology or theory, this will strengthen related techniques and the scope of those techniques.
As hinted above, the praxeology taught and the praxeology learned differ, particularly in the logos
block. This is because in a teaching and learning situation, there is not a direct transfer of knowledge andpractice. However, the praxeology is transposed, and this transposition is conditioned and regulated within
the institutional context. Therefore, we can refer to students’ personal praxeologies (Chaachoua, Bessot,
Romo, & Castela, 2019). Studying the structure of students’ praxeologies can help us identify some of the
difficulties they encounter in learning algebra. For instance, if we consider the two tasks of expanding thefollowing expressions: 3(x‒2) and (x‒2)3, students often find the second expression much harder to expand
than the first (the case of Danish upper secondary students (Poulsen, 2015)). For students at lower secondary
school, the two praxeologies generated by the tasks are not related, not even by theory. This is because thesestudents are not yet familiar with the formalised commutative property of the distributive rule, (a + b)c = c(a+ b), which would link the two praxeologies. This example emphasizes that if the technology and theory
block is not made explicit in mathematics education, such as putting it into words or making it the object ofstudy in a lesson, then many of the praxeologies are not linked, and thus, techniques will not be related toeach other. This makes the more than 90 elementary algebraic techniques at lower secondary school (Poulsen,2015) into a wilderness of rules.
If we consider the praxeologies of students at lower secondary school for solving equations, the use of
CAS will dramatically change part of the praxeology (Carlsen, 2019). Without CAS, it would make sense todistinguish between solving the equations (x‒3)2 = 5·2‒2x, and 12‒3x + 2 = ‒2, because they require different
techniques and technology to solve, such as multiplying inside the parenthesis. However, when solving witha CAS, the two equations do not require different techniques to solve, so the praxeologies would be the same.
Furthermore, since the technique has changed into entering the equation, entering the command solve, andDesigning activities for CAS-based student work realising the lever potential

5interpret the output, the technology has changed accordingly. However, a part of the theory block remains
unchanged, namely, that the solution to the equation is a number such that when substituted with x, both sidesof the equation yield the same number. By assigning technology and theory the same category and notion, as
in previous studies (Hitt & Kieran, 2009; Kieran & Drijvers, 2006; Kieran & Saldanha, 2008) we mightoverlook possible information or design opportunities for CAS-based activities.
If we consider the lever potential of CAS through the lens of praxeology and ATD, it can be viewed as
making explicit and formulating elements of the algebraic logos, including both technology and theory,
through carefully designed tasks that can only be reasonably approached with CAS-based techniques. As thiswill strengthen the theory and/or the technology, it will benefit a range of algebraic praxeologies.
Research questions●	How can CAS be used to engage students to work with elements of the theory block for praxeologies
in school algebra?
●	How can CAS be used to strengthen students’ technology related to standard techniques (such as
rewriting equations) in school algebra?
●	How can one design tasks that realise the use of CAS described in the previous research questions?

CONTEXT AND METHODOLOGYTo explore how the notion of praxeology can guide the design of tasks for teaching of school algebra
realising the potential of CAS a qualitative approach has been chosen. We have conditioned the design ofCAS-based activities to fit into two regular math lessons. It was decided that one lesson would focus on atheoretical aspect of solving equations, while the other lesson should focus on strengthening tradition pen
and paper techniques for solving equations. A series of tasks was designed, and two lesson plans were created
to describe the different didactical situations throughout the lessons. The two lessons were conducted in three
Danish grade 8 (14 - 15-year-old) classes by their regular mathematics teachers. Prior to carrying out thelesson plans, a meeting was held with each teacher to familiarize them with GeoGebra, a program forrecording of screen activities, and the learning objectives of the lessons. The first two participating classes
were already familiar with using GeoGebra, though they had never used the CAS application, while the thirdclass had no prior experience with GeoGebra, CAS tools, or other dynamic geometric environments.
To collect data for the study, seven recording devices were placed in the classroom to record thestudents’ and the teacher’s voices. Additionally, the students’ work in GeoGebra was recorded with
Screencast-O-Matic, software that records screen activities. Everything written on the blackboard wasphotographed, and field notes were written by the researcher during each lesson.
For each of the three lessons, one recording was chosen for a full transcript. For the remaining
recordings, only episodes including not yet transcribed work or discussions were transcribed. All thetranscriptions were analysed with the notion of praxeology, i.e., categorized by task, technique, technology,
and theory. The students’ written materials were also analysed using the notion of praxeology. The screencasts
and written materials were used to support the transcription and analysis of students praxeologies.

6Louise Meier Carlsen

THE LESSON PLANS AND THE RATIONAL OF THE DESIGNSIn this section, we will give a description of the lesson design based on the lesson plans written for theteachers, and part of the a priori analysis of the lesson. The description will be supplemented by the rationale
behind the design. It is assumed that the students already have some familiarity with working with simple
equations, such as manipulating them and describing simple relations using equations.
Lesson A: Describe what is an equation
The first lesson, we can, as in the Background section on the lever potential of CAS, describe thelearning objective as to further develop the students’ concept of a mathematical object, in this case: equations.
The focus of the lesson is to make explicit and formulate what makes an equation, such as the possible
solutions, and whether the equal sign or a variable can be omitted. It is not the purpose of the lesson that thestudents should develop a definition in par with scholars of mathematics, such as a statement that expresses
an equality between two expressions, but rather develop and formulate a definition that is appropriate for the
institution and tradition of lower secondary school.
The first task given to the students in the lesson is to write down an equation of their own and then try
to describe what an equation is. An important aspect of this activity is that the teacher does not provide thestudents with an example of an equation earlier in the lesson or interfere with the students’ work. The use of
CAS is not intended for this activity. The expectation is for the students to give a somewhat vague definition,
mentioning an unknown, and that “the unknown must be found”.
Next, the students are given a series of equations to solve in the CAS window of GeoGebra, while
recording their screen work with Screencast-O-Matic. For the series of equations containing different typesof equations, see Figure 1. For each equation that is solved with CAS, the students are asked to reconsider
their description of an equation.
Figure 1. A minimized version of the students’ worksheet for solving equations with CAS including the apriori analysis of added description.

Equation1)

14 a + 2 = 72

2)

14 y + 4 = 7 + (y‒11)34

3)
4)
5)
6)
7)
8)
9)

x‒2 = 0(x‒1)(x‒2) = 0(x‒1)(x‒2)(x‒3) = 0
2x + 4 = (x‒2)2 + 8
r‒23 = (3r‒4)2‒5r4(x‒436)‒326 = t‒6434
2(3‒4) + 7 = 5‒3(4‒2) + 18
22(3‒5)3(23‒11) = 15 +2
3t + 12 = 3(t+4) = 4(t‒2) + 20‒t12(3‒7) + 4 + y

10)
11)
12)

Addition to the description of what an equation is
An equation can have a fraction as a solution
Unknowns can be named other than x, and have a fraction as asolution
An equation can have two solutions
An equation can have three solutions
An equation can have infinitely many solutions
An equation can have no solution
An equation can have an unknown as part of a solution
Equations must contain an unknown
Equations must contain an unknown
An equation must only contain one equal sign
An equation must contain an equal sign

Designing activities for CAS-based student work realising the lever potential

7

The first equation is intended to help the students become familiar with entering and solving an equation
using CAS in GeoGebra. The second equation yields the outputy = 371. This equation was included20
because we have observed in other schools that many students implicitly expect an integer solution,
particularly if students are primarily working in a traditional paper-and-pencil environment. Therefore, thistype of equation is intended to serve as a gentle transition into questioning and further developing the logos
block for solving equations. The solutions for the fourth and the fifth equations are 1 and 2, and 1, 2, and 3
respectively, the output from CAS is {x=1, x=2} and {x=1, x=2, x=3} respectively. The equations areconstructed in a way that makes it easier for the students to test or find the solutions by trial and error byhand, in order to relate the output of CAS to the solutions for the equations. It is expected that the students
will add that it is possible for an equation to have multiple solutions to their description of an equation.
Equation six is true for all values of x, which CAS writes {x=x}, while equation seven does not have asolution, which CAS writes { }. It is expected that the students will spend a considerable amount of timeinterpreting the output of CAS but will ultimately conclude that equations can have an infinite number ofsolutions or no solution at all. Equation eight has the solution 1 t‒1091, which is to expose the students to4
a type of equation where the solution contains a parameter. However, we do not expect the students to know
the difference between a parameter and a variable since it is not an explicit part of the curriculum. Despite
this, the difference between parameters and variables could be a topic of study for a future CAS-based lesson.
We await that the students will conclude that a solution for an equation can contain another variable. It is
expected that the students will add that a solution to an equation can include an expression that contains an
unknown. Equations nine and ten are equations without an unknown, and CAS gives the output { } and{x=x} respectively. We expect that all students, in their first description of an equation, will mention
something about an unknown, and thus equation nine and ten are included to prompt a discussion of whether
an unknown is a requirement for an equation. It is also expected that the students will not regard theexpressions as equations. These examples are given as formulating a definition requires considering thelimitations of the object. Expression eleven and twelve contain two or no equal sign, respectively, and CAS
gives the outputs {t = t} and {y = 44}, respectively. The expressions have been included in the hope that thestudents will discuss the role of the equal sign in an equation. It is expected that, due to CAS being able togive a solution for both expressions, the students will question the role of the equal sign.
For the next section of the lesson, the teacher conducts the students’ sharing of their discoveries ofadditional descriptions and clarifications, followed by the teacher’s reformulation and clarification of thedescriptions. In the last section of the lesson, the teacher further challenges the students’ description, i.e.,asking if an equation can have four and a half solution. This is done in case any of the students have writtenthat equations can have any number of solutions, and that “number” must be specified as positive integers.
Another follow-up reflection questions is if it is fair that you can manipulate an equation (with an unknown)
into something that is not an equation (equation without an unknown). This question is asked to challenge
the perceived concept of equations in lower secondary school as having an unknown.
Lesson B: Make an equation where x = 2
For the second lesson, the learning objective is for the students to strengthen their praxeology forsolving equations, in particular, the relation between the techniques for manipulating an equation, such as

8Louise Meier Carlsen

adding a number to both sides of the equation, and the theory that the solution must remain the same.
The lesson starts with the teacher telling the students that today they will have a competition about
making “ugly” equations. He then proceeds to write his two contributions on the board 2(x+4) ‒ 7x+2 =3
4
3x + 2 ‒ 33x ‒ 18 and 3x‒4 = 4x‒2 . He further shows how to solve the two equations in CAS, and that both2
123
equations have the solution 2.
Then, the students are given five minutes to create their own “ugly” equation, but the solution must be
2. The students are asked to use CAS to experiment and to verify the solution. After the five minutes, one
student for each group presents their equation to the class, and the class votes to determine the “ugliest”
equation.
The next section of the lesson is introduced by the teacher. The students will continue working on
making the “ugliest” equations, but the focus is now, given an equation with solution two, to find, develop,
and describe techniques that complicates the equation further, but the solution is still two. The students are
given a series of equations that all have solution two, but they are free to start with any equation that has
solution two, see example in Figure 2. The students are asked to use CAS to experiment and verify that theequations have solution two.
Figure 2. An extract of the worksheet for developing methods for complicating equations.

Old equation with solution x=2
3x‒4 =New equation with solution x=2(4x‒2)
3

Method:The students get 15 minutes to develop their methods. It is expected that the students will develop
techniques such as adding a number to both sides of the equation, multiplying both sides of the equation witha non-zero number, adding an expression equal to zero on one side of the equation, multiplying one side ofthe equation with an expression equal to one, etc. The teacher then orchestrates the students’ sharing ofdifferent methods while taking notes on the board.
The last section of the lesson involves a new competition, in which the students have five minutes tomake the “ugliest” equation with solution 2. They are encouraged to try out the collection of shared methods.
The students are asked to use CAS to experiment and verify that the equations have solution two. After the
allotted time, a student from each group presents their “ugliest” equation, and the class choses the “ugliest”
equation them all.
The teacher ends the lesson by repeating the learning objective, particularly relating the traditional
algebraic technology for manipulating an equation, with the theory that the solution remains the same.

Designing activities for CAS-based student work realising the lever potential

9

EMPIRICAL FINDINGSIn this section, we will present the students’ praxeologies based on the data collected. In lesson A, ourfocus is on how and what theory is developed and clarified through the lesson. The discussions cited and thereplay of the lesson are based on voice recording transcripts, while the students’ work presented are cuttingsfrom their worksheets. In lesson B, we are interested in the techniques and the technology developed related
to making equations that has solution two. We mainly use the transcripts to give a replay of the lesson, andthe students’ production noted on their worksheets to illustrate the students’ technology related to thepraxeology of retaining the solution for the equations.
Lesson A: Describe what is an equation
Though the lesson was carried out in three different classes, the students’ praxeologies related to thelearning objective of the lesson were similar. The most noticeable difference between the classes was thedifference in time spent on introducing the use of CAS and Screencast-O-Matic. In our analysis, we will
focus on the theory that the students developed related to the praxeologies of solving equations.
Of interest for our study of the lesson A are the episodes around solving new types of equations in CAS
and revising the description of an equation. We will go through those parts of the lesson chronologically andrelay and present episodes from our data.
In this section of the lesson, the students work in groups. We will now consider the students’ work anddiscussions. Equation two, 14y + 4 = 7 + (y‒11)34, which has the solution 371 . For the first two classes who
20
regularly solve equations with a non-integer solution and use calculators to perform the calculations, the
outputx = 371 did not generate any discussion regarding its viability as a solution. Those groups who do
20 reflect on the output come to the conclusion that the fraction is a division that has not yet been performed,
and therefore, should be written as a decimal number instead. Many of these groups went on to perform thecalculation and write the first three to four decimal places of the number. In the third class, the solution 371
20
prompted the first discussion in the groups. Initially, the students were not accepting a fraction as a valid
solution for the equation. However, after either consulting the teacher or solving the equation by hand, thegroups eventually accept a fraction as a possible output for CAS. During the discussions, some groups
expressed that while CAS had successfully isolated x on the left-hand side of the equation, it had yet tocomplete the calculations on the right-hand side of the equation that would convert the fraction into a decimal
number. One group disappointedly remarked that CAS cannot do everything for them. Subsequently eachgroup added a statement like “The solution to an equation can be a fraction”, see Figure 3.

Figure 3. Addition of a fraction being a possible solution to an equation to the description of an equation,
cut out from worksheet.

10Louise Meier Carlsen

The fourth equation, (x‒1)(x‒2) = 0, with two solutions, sparked the most discussion among the groups.
After double-checking the input in CAS, it was generally accepted that CAS can solve the equation, and the(one) solution to the equation can be read from the output {x = 1, x = 2}. The disconnect between thestudents’ view of equations having exactly one solution and the solution offered by CAS sparks varied
discussions. One group proposes the hypothesis that you can choose between one or two, while anothergroup that one and two must be added together, resulting in the solution of three. Yet another group has thefollowing discussion about substitution and the role of the unknown in an equation:
Aza: But, it is x equal to one, and x equal to two!
Bab: Well, it [x] is the same [referring to the equation].
Aza: What do you mean? It is two different x’s [referring to the output]. [Small pause]
Cala: Are there no multiplication sign in between [the expression (x-1) and (x-2)]?
Aza: There is at least a multiplication sign there.
Cala: I think it makes sense. x minus one is one thing[equation], and x minus two is anotherthing[equation], and then they both must be solved.
Bab: But, both x’s [in the equation] must be the same.
Cala: Well, yes, it is like that.
Most of the groups eventually conclude that an equation can have two solutions by substituting (byhand) x with a value, and then discovering that either the term (x‒1) or the term (x‒2) must be zero, realising
thus that both 1 and 2 are solutions to the equation. As a result, the groups all wrote additions to theirdescription of an equation, see Figure 4 for example.

Figure 4. Addition of «can have several solutions» to the description of an equation, cut out from worksheet.

We have cited the above part of a discussion as it demonstrates how the students are beginning todevelop and articulate elements of theory for solving equations. Specifically, they are learning what thesyntax is for writing equations, and what the role of the unknown is if it appears more than one time in theequation. They are also learning how to approach equations with multiple solutions how to use substitution
in these cases.
In some groups, unforeseen inquiries into the theory continues. In one group, a student asks how one
knows if an equation has two solutions or only one solution. In another group, they discuss whether thenumber of solutions for an equation can be two and a half.
The sixth equation, 2x+4 = (x‒2)2+8, which has an infinite number of solutions, is not an example that allgroups reach. The groups that reach this equation will get the output {x = x}. In general, the students are notconfused about the CAS output, but instead, they are confused about what the meaning of the expression x = x.
Xia: x is equal to x. That makes sense [ironic].
Yao: Then the solution is x -that is just super [ironic]. [small pause] I do not understand x equal to x,
how can that [x=x] be?

Designing activities for CAS-based student work realising the lever potential

11

Xia: I don’t think we entered everything correctly.
Zhou: We entered correctly.
Yao: What does it mean: x is equal to x [directed towards the teacher]?
Teacher: That is a very good question.
Yao: We entered correctly.
Teacher: Yes, you did. But there is something very interesting here.
…
Xia: It [CAS] wrote that the x, that the unknown is equal to the unknown.
Teacher: So, what numbers can you substitute with x in order to solve the equation?
Xia: All.
Teacher: That is an interesting observation.
Yao: So, no matter if we substitute four or seven, then it would be correct.
Teacher: So, how many solutions does the equation have?
Zhou: Infinite.
Teacher: Okay Zhou, that is an interesting observation, which you might want to note on your paper.
Yao: And x is equal to x [writes aloud while she writes]
Zhou: Does it also apply for negative numbers?
Xia: I don’t think so. We can ask our teacher.
Zhou: Teacher, does it also apply for the negative numbers?
Teacher: Try. What happens if you substitute with a negative number?
…
We have chosen this extract from one of the groups’ transcript, as it illustrates not only the formulationof a new type of equation (containing an unknown) with an infinite many solutions, but also prompts students
to reflect on and formulate further elements of the theory block.
Equation seven, r‒23 = (3r‒4)2‒5r, is an equation with no solution, which GeoGebra prints as theoutput { }. The groups of students who have reached the equation conclude that a solution does not exist
since the curly parenthesis that beforehand have encompassed the solution(s) are now empty. In one group,
before reaching the conclusion that the equation has no solution, the students discuss, whether the unknown,r in this case, can be substituted by a curly parenthesis:
Sekai: What does it say?
Tabia: It just says the wrong [type of] parenthesis.
Sekai: Okay, we interpret it as if there should be a parenthesis on both.
Tabia: On both?
Sekai: Yes, because we must keep the balance [of the equation].
Tabia: No, that must be wrong, it [the solution] cannot be parentheses.
Sekai: If this is an equation, then there was a parenthesis here, and it is really heavy, then you also needa parenthesis here [on the other side of the equation], so it becomes equally heavy. Do you get it?
Ronja: No!
It is finally dismissed with the argument that a parenthesis is not a solution for an equation.
This bit of discussion between the students shows that questioning a part of the theory leads to further

12Louise Meier Carlsen

exposure of the theory, such as the view that an equation is a seesaw where the equilibrium must be maintained
and a clarification of what is possible and not possible to substitute with the unknown. Furthermore, thestudents implicitly raise the question of what do you substitute the unknown with if there is no solution to
the equation?
The eighth equation, 4(x‒436) ‒ 326 = t‒6434, is an equation with two different variables, and the
output of CAS is x = 1 t‒1091. The groups that reach this equation do not seem perplexed with now
4
having a solution for an equation that contains an unknown and relates the solution to previous obtained
solution type with infinitely many solutions. In one group, a student remarks, “x is equal to one fourth t [stops
mid-sentence]. It is that thing where x can have one or several unknown [as a solution]. Next!” Later, shereflects, “I had not considered that one could put an equation in relation to, err, an unknown in relation toanother unknown”.
The ninth equation, 2(3‒4) + 7 = 5 ‒ 3(4 ‒ 2) + 18, is an equation with no unknowns, and it is true. If
each side of the equation is simplified by calculations, it will yield 5=5. The output of CAS is {x = x}. The
groups of students who reach the equation immediately dismiss the expression as an equation since it does
not contain an unknown: “There is no x -that is not an equation!”.
The next episode of interest is the last episode of the lesson, where the students share their findings, andthe teacher reformulates the students’ findings and writes key phrases on the board. Following this, theteacher outsets further discussions and clarifications. In one of the lessons, the output {x = x} is further
discussed. During the students’ presentation of the implications of this type of output, i.e., an equation canhave an infinite number of solutions, it is implicitly understood that the solution type is considered a number.
The teacher relates this type of equation to equation eight, where the solution contains an expression with asecond unknown, and asks the students if, perhaps when x = x, the solution can also be an expression
containing an unknown.
The curly parentheses are also one of the objects that the teachers consider. After the students have had
their try at guessing the meaning of the curly parentheses, the teacher explains that the curly parentheses areusually used to denote a set. The students have not only encountered equations with more than one solution
but are also taught how to write such solutions.

Equation nine and ten, 2(3‒4) + 7 = 5‒3(4‒2) + 18 and 3(23‒11) = 1 + 22(3‒5) , are also brought to52
the attention of the students since they do not include an unknown. The students all agree that they do not
qualify as equations, even though CAS can solve them with an output of {x = x} or { }. In one of the lessons,the teacher suggests that they can be called statements, and these statements can either be true or false, justas the so-called equations can either be true or false.
In the last episode of the lesson, the students formulate elements of their theory. The teacher thenfurther questions the theory, and together, the students and the teacher develop and formulate theory regarding
equations, as well as other related subjects.
Overall, the students developed and clarified a wide range of theory related to solving equations,
including rational or decimal numbers as solution, equations having several or infinitely many solutions, andthe limitation of parenthesis as a solution. They also established syntax for equations i.e., whether there canexist 2-in-1 equations and syntax relating to parenthesis. In addition, the students determined the rules for
substitution in equations with multiple solutions. At the end of the class, the teacher institutionalised theory

Designing activities for CAS-based student work realising the lever potential

13

related to what can a solution for an equation be/ not be.
Lesson B: Make an equation where x = 2
We will now study the empirical findings for lesson B. In our analysis, we will focus on the techniques
and the technology that the students developed related to the tasks of making and further developing equations
with a solution of two. As the problems are open, there are many possible ways of attacking the task.
Although the lesson was carried out three times, the students’ praxeologies developed in the lesson weresimilar. We will present our findings in chronological order, i.e., first the task of making an equation with asolution of two, and then the task of further developing an equation with a solution of two while maintaining
the same solution.
The first section of interest is the students’ first try to create an equation with a solution equal to two.
One group of students suggest three main techniques. The first main technique:
Irene: Can’t we just make a random equation and then compensate?
Leo: What! Oh, you mean like just add minus [a number] at the end.
…
Leo: We just copy the one up there [from the board] but we make it longer.
…
Irene: I have got the greatest idea ever! Can we just do like last lesson, where x is equal to x. Infinitely
many answers.
Joules: How do you get x equal to x?
Leo: Then you have to make both sides [of the equation] such that when reduced they are the same.
The group of students has formulated the three main techniques for this section of the lesson. We willrefer to them as π1, which is the main technique for making a “random” equation, checking the solution with
CAS, and then compensating until the solution is two. π2, which starts with an equation with a solution of
two, complicating the equation further, and checking the solution with CAS. π3, which is making an equationthat is always true by making the right-hand side equivalent to the left-hand side.
The group cited above ends up trying out π1. After entering a “random” equation, they get the output
from CAS {x = 4}. This prompts the discussion on how to compensate such that the solution is two.
Leo: What if we divide it [the equation] with two?
Irene: Then it would be one-half x equal to two, that would still give the result x equal to four. If wewant two less on the right side perhaps, we should either add or subtract two up here.
The work and discussion of the group show the variety of techniques the students hold, their formulationof their technique, and the choice of technique. Even more specific technique of how to manipulate a simple
equation with solution of four such that the solution becomes two.
All the groups work with similar or variations of the above mentioned three main techniques.
Another group that works with π3 uses the technique of making two expressions that are equal to five, andthen place the expressions on opposite sides of the equal sign so that the equation has infinitely many
4x + x ‒ 1 x
2 + x = 3+x.
solutions. The result is 1
x+ 2 x14

Louise Meier Carlsen

Another group employing π3 uses the technique of adding and subtracting the same on both sides of theequation.
Gerð: We use a lot of difficult numbers, and then we subtract them in the end.
Fróði: Minus one thousand-seven-hundred-and-sixty-one minus something plus x minus sevenhundred-and-eleven. Then x is equal to x!
Gerð: I think we should use pi.
…
The discussion among the students and the results of their work show that the students have the theory
that equations with infinitely many solutions exist and how such equations can be made with the technology
of making an equation where the right-hand side of the equation is equivalent with the left-hand side.
Although some groups have discussed π2, further developing an equation with solution two, none ofthese groups choose to work with this strategy.
In the next section of the lesson of interest, the classes are tasked with π2. They are given a list ofequations, all of which have a solution of two, to make more complex. Our main interest is not the ugly
equations produced, but rather the techniques and technology that the students develop and employ. We will
first recount the main techniques and technologies developed and formulated by the students, and thenrecount some of the theory that was discussed and clarified.
We will present two of the techniques developed and formulated by the students. The first technique,which we will denote by τ1, is to divide a part of the equation by an expression that is equal to one when x is
equal to two, i.e., 3‒x. In the following transcript, the technique is developed, and the expression 3x‒4 is
3x‒4
rewritten into (x3‒ 2x)‒3 .
Margrethe: Perhaps we can make a fraction here, and then make some weird expression [as thedenominator] that is just equal to one … Then we say four minus two, all we need is for it to equal one.
x and we have two, we need [pause]. Here we have two x, thus we need four, minus x squared. Here
we have got. No. x to the power of three, then we have minus four, then we need to subtract with three
on the other side somehow [(x3‒2x) ‒3]…That is a very overcomplicated way of writing one.
As a method the group writes “overcomplicating by dividing with one”.
The other technique, which we will denote as τ2, was developed, and clarified by the students. It is to
add the same number on both sides of the equation. A group has the following discussion:
Emmy: We must make this one ugly.
Elliot: You can change this to three and then minus thee here [referring to the other side of the equation
sign in the equation].
Emmy: Plus six, plus two.
Elliot: [interrupts] You are doing it wrong. You cannot put plus on each side [of the equal sign] because
you have to put a minus. [Directed to the teacher] Isn’t it true that if you put a plus on one side, then on
the other side is has to be minus?
Teacher: Hmm, you have to keep the balance of the equation. If I have an equation and I make one of
the sides heavier, then in order of keeping the balance, the other sides must also get heavier. I.e. if I add
something on one side then I need to add the same thing on the other side.
Elliot: Does this make sense? [shows the teacher her equation]

Designing activities for CAS-based student work realising the lever potential

15

Teacher: Test it [with CAS]…
Elliot: Okay, we add plus three and twenty on both sides. And minus nineteen on both sides. And plus
four on both sides.
...
The extracts show the two techniques, τ1 and τ2, developed, formulated, and employed by students. τ1,
dividing a term of the equation with an expression that is equivalent to one, and τ2, adding the same number
on both sides of the equation. The two techniques developed and formulated play a fundamental role inschool algebra, as they are often the key to simplifying algebraic expressions or equations. In addition, they
are pivotal for the application and handling of school algebra, such as simple formulas, in other subjects andfurther education.
Although the focus of the activity was on developing techniques and technology, discussions andclarification of elements of the theory also emerged. The main theory present was that the solution must staythe same, as it was the condition for rewriting the equations. During the lesson, the students often referred toanother main theory, which is that an equation can be seen as a seesaw, and that the seesaw must be kept inbalance, i.e., if you add a number to one side of the equation, you must add a number or an expression thatis of equal weight on the other side of the equation.
We will now present some more of the technology and theory that emerged in this section of the lesson.
In one group of students that were employing τ1, a student wanted to divide by zero. In another group, also
employing τ1, a student asked what happens if you divided by x. As time was of the essence, the respective
groups did not engage in theoretical discussion but refocused on the task at hand and developing techniques
for which the technology and theory were already part of the students’ inventory.
The activity of handling equations all with solution two seemed to generate, in several groups, aquestioning about whether the equations were all the same. After seeing the teachers’ two examples ofequations with solution equal to two, a student asks
Uliuk: How can they all give the same result? Such a long equation can give the same result as that one.
Vaaltimaat: That is the question!
Teacher: I think that is a very good question!
Uliuk: Then you can just write that [the first equation] instead of all that [the second equation]. That[the second] one takes a lot longer to reduce.
In the second section of the lesson, other students commented that it did not feel like they had madenew equations, as it was a rewriting of the first equation. This happened in particularly if the students haddone simpler manipulations, such as rewriting 4x into 3x + x or 2x into x + x.
The extracts show that elements of theory are discussed and clarified, but they also inspire further
questioning of the theory, which could lead to entire new types of praxeologies such as when are two
equations the same? The students are becoming aware of equations as an object of study.
In the next part of the lesson, the students share their methods while the teacher reformulates and writes
them on the board. The techniques shared are: multiplying a term of the equation with an expression that isequal to one when x is equal to two; dividing a part of the equation with an expression that is equal to one
when x is equal to two; adding an expression that is equal to zero when x is equal to two, to one side of the16Louise Meier Carlsen

equation; adding the same number or equivalent expressions on both sides of the equation; and multiplying
with the same number or expression on both sides of the equation. In two out of the three classes involved in
the project, the teacher ends the sharing of methods by emphasises the connection to the theory that thesolution of the equation stays the same. In the third class, the teacher ends the sharing of methods by reflectingthat many of the methods developed for uglifying an equation, are the same methods one could use tosimplify an equation. Many of the discussions on techniques, technology, and questioning of theory is notshared, and neither is a relation to Lesson A made in this lesson.
The students’ sharing of their developed methods shows the variety of the different technologies present
in the lesson and the similarity to the technology for solving equations. Thus, embedding the theory elementthat the solution is maintained into the praxeology for solving equations.
In the last attempt at making an ugly equation with solution two led to the emergence of another pieceof theory concerning the equal sign. In one group the necessity of the equal sign in an equation is questioned.
However, after wondering how to solve such an equation, or how to check if the solution is two, they agree
that the equal sign is a requirement for an expression to be an equation.
The recount of the group’s discussion shows that the students are once again considering equations as
an object of study, and though the question of whether or not the equal sign is a necessity for an equation was
raised in the first lesson, the discussion in this account includes technology.

CONCLUSION
In this section, we will try an answer our research questions regarding the potential of CAS and thenotion of praxeology as a tool for designing tasks.
The use of CAS in the activities allowed for the students to approach new tasks (more exotic types ofequations) that would otherwise have required time-consuming algebraic work such as solving equations.
The new tasks involved themes within school algebra that strengthened, developed, and clarified bothtechnology and theory. In lesson A, the students worked with describing what constitutes an equation. The
students developed, formulated, and clarified a wide range of technology and theory involving equations andrelated topics such as substitution, thus strengthening an entire series of related praxeologies. In lesson B, thestudents worked on establishing a stronger connection between the theory of maintaining the solution of an
equation with the technology and techniques used for manipulating equations. Thus, sustaining a grander
logos for school algebra. However, lesson B did not solve all problems related to the praxeologies of solving
equations. During the lesson, both students and teachers mentioned the metaphor of an equation as a seesaw,
where the balance must be kept, as a valid argument for justifying techniques. There were instances where
this magic trick could have been questioned, such as when a student wanted to divide by zero or when one
student wanted to take the square root of both sides of the equation (without following using the absolutevalue), which could possibly change the solution space. But it did not happen.
By outsourcing time-consuming algebraic work, CAS can be used to introduce new types of equationi.e., equations with more than one solution, making it possible to study equations as an object and not justusing equations as a tool. This in turn developed both elements of the students’ theory and technology such

Designing activities for CAS-based student work realising the lever potential

17

as in lesson A. In lesson B, the use of CAS provided students with a technique for verifying the solution of
their equations, allowing students to experiment with creating and developing the equations. This led thestudents to develop and formulate not only an abundance of technology, but also formulate and clarify
elements of theory.
The students’ work with the logos further prompted them to question, formulate and discuss elementsof theory, such as rules and the role of substitution (from lesson A) or what happens if you divide by the
unknown, thus further enhancing the activity.
As both lesson A and lesson B contained many questions inquiring further into elements of technology
and theory, we conjecture that the activities could be further enriched by not being constrained by a lesson
plan. This would allow the students to direct the path of inquiry and development of technology and theory.
In addition, the object of lesson B, which is to further develop and explicitly state the relation between thetechniques and technology (of manipulating equations) with the theory (of the solution staying the same), theinstitutionalisation must be further emphasised.
The use of the notion praxeology, particularly the four notions of task, technique, technology, andtheory, allowed us to analyse the prevailing paper-and-pencil praxeology for solving an equation, and thus
select the themes “the students’ definition of an equation”, and “the relation between algebraic manipulationsof an equation, and maintaining the same solution”. Both themes are within school algebra, thus the designed
activities would strengthen the students’ logos by further developing, formulating, and clarifying it.
The notion of praxeology lets us further study the lever potential of CAS, enabling the design of future
activities more easily. In lesson A, the lever potential can be described as the development and formulationof the students’ concept of equation, which is an element of the theory block. This characterisation can also
be used to describe the potential of CAS studied in the thesis by Drijvers (2003). In lesson B, the leverpotential is the strengthening of the relation between the theory of maintaining the same solution, with thetechniques and technology for traditional algebraic manipulations. In the article by Hitt and Kieran (2009),
the lever potential can be described as the telescoping technique (using paper and pencil) being used not only
as a technique but also as technology for developing, justifying, and validating conjectures. The use of thenotion praxeology enables us to describe the students’ learning explicit and justifies the lever potential ofCAS.
As a final note, we would like to reflect on how these more theoretical CAS-based activities can upset
the traditional epistemological model for school algebra and thus necessitate new design tools for activities.
Traditionally, second degree equations with two, one or no solutions are introduced by their graphicalrepresentation at the end of lower secondary school. However, a more inquiry-based teaching approachcombined with CAS will naturally lead to question what an equation is, and following an examination of aseries of examples. These examples would feasibly include second- and higher-degree equations as thestrength of CAS is to handle more exotic examples than is possible in a pen-and-paper-only environment. Achange in the traditional curriculum when implementing CAS is not unheard of (Kendal, Stacey, & Pierce,2005) and perhaps even necessary. This would, however, only stress the need for design tools to support
teachers in analysing and crafting CAS-based activities.

18

Louise Meier Carlsen

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Gyöngyösi, E., Solovej, J. P., & Winsløw, C. (2011). Using CAS based work to ease the transition from calculus toreal analysis. In Proceedings of the 7th conference of European researchers in mathematics education (pp.
2002-2011).
Heid, M. K., Blume, G. W., Hollebrands, K., & Piez, C. (2002). Computer algebra systems in mathematics instruction:
Implications from research. The Mathematics Teacher, 95(8), 586-591.
Hitt, F., & Kieran, C. (2009). Constructing Knowledge Via a Peer Interaction in a CAS Environment with Tasks
Designed from a Task–Technique–Theory Perspective. International Journal of Computers for Mathematical
Learning, 14(2), 121-152. doi:10.1007/s10758-009-9151-0
Kieran, C. (2007). Learning and teaching algebra at the middle school through college levels In F. Lester (ed.).
Second Handbook of Research on Mathematics Teaching and Learning: A project of the National Council of
Teachers of Mathematics, (vol. 2, pp. 669-705). National Council of Teachers of Mathematics.
Kieran, C., & Drijvers, P. (2006). The co-emergence of machine techniques, paper-and-pencil techniques, andDesigning activities for CAS-based student work realising the lever potential

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theoretical reflection: A study of CAS use in secondary school algebra. International Journal of Computers forMathematical Learning, 11(2), 205-263. https://doi.org/10.1007/s10758-006-0006-7
Kieran, C., & Saldanha, L. (2008). Designing tasks for the co-development of conceptual and technical knowledge
in CAS activity: An example from factoring. In G. W. Blume & M. K. Heid (Eds.) Research on technology andthe teaching and learning of mathematics, (vol. 2, pp. 393-414). Information Age Publishing.
Kieran, C., & Yerushalmy, M. (2004). Research on the role of technological environments in algebra learning andteaching. In K. Stacey, H. Chick, & M. Kendal, (Eds.), The Future of the Teaching and Learning of Algebra The
12th ICMI Study (vol. 8, pp. 97-152). Springer, Dordrecht. https://doi.org/10.1007/1-4020-8131-6_6
Lagrange, J.-B. (2005). Using symbolic calculators to study mathematics -the case of task and technique. In D. Guin,
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Martínez, C., Kieran, C., & Guzmán, J. (2012). The Use of CAS in the Simplifi cation of Rational Expressions andEmerging Paper-and-Pencil Techniques. In L. R. Van Zoest, J.-J. Lo & J. L. Kratky (Eds.), Proceedings of the
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Louse Meier Carlsen
IT-University of Copenhagen, Copenhagen, Denmark
Email: oca@itu.dk
https://orcid.org/0000-0003-3987-9350

Special Issue:
Rethinking the Diversity of Theories inMathematics Education. Contributions Related to the
Topic Study Group 57 of ICME 14

HIROSHIMA JOURNAL OF MATHEMATICS EDUCATION 16: 21-26, 2023

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Guest Editorial
RETHINKING THE DIVERSITY OF THEORIES IN MATHEMATICSEDUCATION. CONTRIBUTIONS RELATED TO THE TOPIC STUDYGROUP 57 OF ICME 14
Angelika Bikner-Ahsbahs
Bremen University (Germany) and Western Norway University of Applied Sciences (Norway)

Ivy Kidron
Jerusalem College of Technology (Israel)

Yusuke Shinno
School of Education, Hiroshima University (Japan)

Takeshi Miyakawa
School of Education, Waseda University (Japan)
This Special Issue emerged from Topic Study Group 57 (TSG57) on the “Diversity of Theories inMathematics Education” conducted at ICME 14 in July 2021 (Bikner-Ahsbahs et al., in print). Based on thepresentations and discussions in the TSG57 at the conference, five papers have been extended and elaborated
for the special issue. To introduce this special issue, we would like to look back briefly at what we havediscussed in the TSG. Main topic was the Networking of Theories in mathematics education research andthis also resonates with this special issue.
In recent years, the theory networking enterprise has advanced (Kidron et al. 2018). Networking ofTheories means to relate different theories, to make use of them in research and scientifically reflect thisprocess. Key assumption of this approach consists of respecting theoretical identities that are often based on
cultural backgrounds and at the same time requiring from researchers to keep a critical stance on theoreticalcompatibility in the theoretical networking practice. To keep this balance, Prediger et al. (2008) havedeveloped pairs of networking strategies that express increasing degrees of integration; these areunderstanding other and making understandable own theories, comparing and contrasting theories,
combining and coordinating theories, and local integration and synthesizing of theories. Many authors inthis issue recount how they used these strategies in their research.
The TSG57 addressed three subthemes in three sessions: Reasons for the diversity of theories;
methodological approaches in design research related to the diversity of theories; and reconsidering specificcommitments in the diversity of theories. In the first session, the TSG articulated a double aim, to respect
theory development coming from various educational cultures and research areas in the field and at the sametime to overcome theoretical fragmentation in the field. This requires a balance of keeping the diversity
needed but simultaneously bridging theories to make communication across the theory cultures possible.
Two themes were discussed, first, networking of theories can be a source of (as well as it may unpack)‘tensions’ between theoretical discourses, depending on basic assumptions and, second, networking oftheories can offer ‘flexibility’ when relating basic assumptions and thus creating a new discourse. From thesecond session we learned that each theory is built on the three commitments, ontology, axiology, and22epistemology, in specific ways that underlie how teaching and learning are understood, in research as well asin practice. Main postulation raised in the discussion was to increase scientific sensitivity for thesecommitments, specifically in terms of theoretical choices in transformative research. From the third session
we have learned to make these commitments visible in research and to look for methodological ways topractice this sensitivity, as for instance by a vertical analysis grounded in historical or philosophicalconsiderations. A crucial part of the discussion resulted in the insight that each theory on teaching andlearning entails a certain understanding of ethics grounded in this theory.
Finally, two questions were raised: To what extent is it necessary or even mandatory to take ethicalissues of the teaching/learning into consideration for theorizing and what are the relevant concepts of ethics
for that? More generally, what are the pitfalls when such commitments go unnoticed?
This Special Issue does not address all the aspects and questions mentioned in the TSG57, but somerelevant aspects are captured by the publications. All the five papers focus on and ‘rethink’ the diversity oftheories from different perspectives. In the first contribution “Facing the Challenge of Theoretical Diversity:
The Digital Case” Michèle Artigue explores the advancement in the field of technology in mathematics
education in terms of research on two concepts conducted by networking of theory approaches. In the second
article “The Role of a Priori Analysis in Theories”, Ivy Kidron explores the notions of a priori analyses
related to two theories showing how the epistemological grounding of theories determines them. In the third
paper “Ethics in the Mathematics Classroom”, Luis Radford elaborates the concept of communitarian ethics
for the Theory of Objectification making explicit how theory and ethics are entwined. The fourth contribution“Networking Praxeologies and Theoretical Grain Sizes in Mathematics Education: Cultural Issues Illustratedby Three Examples from the Japanese Research Context” by Yusuke Shinno and Tatsuya Mizoguchi is acomparison of three Japanese networking examples that shows clearly the relevance of culture when
networking theories. The final contribution by Angelika Bikner-Ahsbahs et al. is a networking case on design
research leading to local integration of two theoretical approaches that allows to explain how learners
transformed their behavior when acting with a digital artifact.
In the following, we are going to summarize each article and thereby extract their main contributions
to the field.

SUMMARIES OF THE ARTICLES IN THE SPECIAL ISSUE
The theoretical diversity offered by different theoretical cultures is a real challenge in mathematics
education research. The analysis in the paper by Artigue (2023) provides an important overview of thetheoretical landscape and its main tendencies in theorizing the use of technology in mathematics education.
Artigue aims to contribute to the reflection on the question: “To what extent, are we now better equipped tomeet the challenge of theoretical diversity in relation to technology-based mathematics teaching andlearning?” To this end, two conceptual tools are introduced in the paper. Both tools have proven theireffectiveness in addressing issues of theoretical diversity: The scale of networking strategies and the conceptof research praxeology (introduced in Artigue et al., 2011). Two case studies are described: The Instrumental
Approach (IA) and the Documentational Approach to Didactics (DAD). The emergence and development of

HIROSHIMA JOURNAL OF MATHEMATICS EDUCATION 16: 23-26, 2023

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these approaches illustrate well the global dynamic of the field towards increasing theoretical diversity, thequestions raised by this dynamic and the insightful efforts made to deal with it. The evolution of teachers’
documentational work was induced by technological advances. One important contribution of Artigue’s
paper is her analysis that demonstrates how knowledge has advanced both, in terms of understanding thediversity challenge and in terms of developing strategies to address it.
Kidron (2023) investigates the role of a priori analysis in different theories in the context of networking
theories. A priori analysis here means an analysis, which is often carried out prior to the experiment or datacollection. While the a priori analysis was initially proposed as a part of the methodology of didactic
engineering developed within the Theory of Didactical Situations (TDS) (Artigue, 2020), it is today sharedin different theories. As the methodology is often considered a part of the theory (Radford, 2008), thecomparison of the methodologies reveal the nature of each theory as well as the relationship between them.
In the previous work of networking theories, several case studies have been carried out by analyzing thecommon empirical data from different theoretical perspectives to shed light on their differences,commonalities, and complementarities (see Prediger & Bikner-Ahsbahs, 2014), that is to say, these studies
consist of the comparative analyses of the processes and results of a posteriori analyses. Kidron’s paper is
unique in that it focuses on a priori analysis instead of a posteriori analysis. It shows by comparing the apriori analyses of TDS and the theory of Abstraction in Context (AiC), each theory has a different focus,
while both theories place importance on the epistemological perspective: on the one hand, AiC focuses on
the learner’s construction of knowledge with a question “What is the epistemic process of the student?”; on
the other hand, TDS focuses on the didactic system with a question “How this [epistemic] process ispossible?” One of the important contributions of Kidron’s paper is that it shows the relevance of analyzing apriori analyses in the context of networking theories. As the focus of the theory affects the data collection,
the study of a priori analyses allows us to better understand the choice of data.
Radford (2023) begins his elaboration of educational ethics by an example, sensitizing readers forseeing teaching and learning as a deeply ethical form of activity. Although always present, ethics seem tohave lost sight in mathematics education. Radford sees this being grounded in the conception of mathematics
education that has been reduced “to a matter of acquiring knowledge, making the question of being andbecoming peripheral aspects of teaching and learning” (p. 60). He traces two fundamentally distinguishable
forms of ethical action back to Hobbes (1841) and Kant (2006). With reference to Hobbes, the notion of
ethics consists of contractual obligations that regulate acting in society. Under the maxim of universal reason,
Kant anchors moral action, hence, ethical acting, in the individual. According to Radford, Hobbes’ conceptionof ethics informs the transmissive idea of teaching and learning, while ethics according to Kant rather inform
constructivist approaches. Thus, ethics “appear framed by the way in which we understand teaching andlearning” (Radford, 2023, p. 60), hence, by theories of teaching and learning. Radford elaborates the concept
of communitarian ethics for the Theory of Objectification as an example. His contribution is unique as hemakes explicit how the Theory of Objectification and communitarian ethics are interrelated by elaborating
the theory’s relational conception of the social and how this hands over to ethics. He defines communitarian
ethics through three dimensions: responsibility, commitment, and care. Making use of Lévinas (1982)’
understanding of responsibility he regards responsibility as “living and acting with and for others” (p. 69).
The key function of communitarian commitment is “to participate in the creation of the classroom common

24work” (p. 69) of joint labor. A core ingredient of educational ethics is care, meaning here that “the importanceof caring for the Other is to go beyond ourselves and to be dragged powerfully into the world, to position
ourselves there, with-the-Other” (p. 70). Communitarian ethics are thus relational and, consistent with the
Theory of Objectification, permanently materializing themselves within the classroom.
Networking of theories is a way to bring different theoretical perspectives into a dialogue often used to
unpack implicit theoretical assumptions. However, there is a lack of meta-theoretical knowledge for
conducting a networking theories study. With their article, Yusuke Shinno and Tatsuya Mizoguchi (2023)
want to contribute to filling this gap by answering the question How can we characterise researchers’
theoretical work on networking endeavours in terms of praxeology? Leaning on the work of Artigue andBosch (2014), they use the concept of praxeology introduced by the Anthropological Theory of the Didactic(Chevallard, 2019) and elaborate the concept of grain size of theories to characterize the researchers’
theoretical work in three Japanese research and development examples that follow networking theories
approaches. Their networking technique, one of the elements of praxeology, consists of networking strategies.
Shinno and Mizoguchi use the strategy of comparing and contrasting theories to explore a design approach(lesson study or didactical engineering), with the strategy of combining and coordinating theories they
investigate an empirical classroom study, and they use local integration for the case of a curriculum
development study where the reference epistemological model (Bosch & Gascón, 2006) plays a key role.
Shinno and Mizoguchi identify and compare the three networking praxeologies of the examples to describe
the theoretical discourses involved. Their contribution is unique as it shows that these discourses are
intrinsically related to culture, i.e., classroom culture, research culture, and curriculum culture. As reflectedby the authors, these forms of culture are specific in the Japanese culture of mathematics education in thattheory for research and theory for practice (p. 7) are distinguished.
Bikner-Ahsbahs et al. (2023) address theoretical networking between Activity Theory (Leontyev, 2009)
and Instrumental Approach (Rabardel, 2002) to investigate the role of digital feedback in supporting theteaching and learning of negative numbers with a multimodal algebra learning (MAL) system. Within
Activity Theory, tools such as the MAL system play an essential role as cultural objects. From the perspectiveof Instrumental Approach, the process of how the MAL system becomes an individual instrument can beconceived as instrumental genesis, shaped by two interrelated processes: instrumentation andinstrumentalization. Bikner-Ahsbahs et al. (2023) examine the teaching and learning of negative numberswith the MAL system using Activity Theory and a model about feedback loops. They characterize the roleof digital feedback as mediating between different types of action expressed by a four-layered model. One of
the important contributions of Bikner-Ahsbahs et al.’s paper is their theorizing of an empirical phenomenon
at the boundary of the two theories leading to ‘local integration’, a relatively challenging strategy of theory
networking that allows to go beyond the mere understanding of a special empirical phenomenon (Prediger etal., 2008). In their networking work, after coordinating the two theories with the help of the layered model,
they clearly demonstrate how local integration in terms of Radford’s (2008) theory concept is achieved.
Four of the five articles are about the networking of theories, and two papers address philosophicalcommitments underlying educational theories. While Artigue elaborates the state of the art of research andits advancement in technology in mathematics education realized by theory networking approaches, theother three networking papers contribute directly to advance its meta-theoretical sphere. Shinno andHIROSHIMA JOURNAL OF MATHEMATICS EDUCATION 16: 25-26, 2023

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Mizoguchi elaborate that cultural backgrounds affect notions of theory and thus also notions of theory
networking. Bikner-Ahsbahs et al. explore the role of digital feedback at the boundary of two theories as a
new phenomenon and show the advancement made by the take-up of the networking strategy of local
integration. Kidron, through theory networking, contributes to our understanding of epistemologicalcommitments of theories and their methodological and conceptual consequences. Grounded in philosophicalconsiderations, Radford improves our knowledge on ethical commitments, underlying theories on teaching/learning in general. All the five papers advance the field by meta-theoretical considerations included into
research resulting in deepening our understanding of what can make a difference in the diversity of theories
in mathematics education.
Acknowledgement
We would like to thank all the reviewers who contributed to this special issue. As part of the editorial
process for this special issue, we have invited at least one external reviewer for each paper who was not
involved as an author of a paper or as a guest editor of the issue, but who had expertise in the research areaof theories in mathematics education. The contributions of all the reviewers were crucial to the elaborationof each paper and to the publication of this special issue.
References
Artigue, M. (2023). Facing the challenge of theoretical diversity: The digital case. Hiroshima Journal of Mathematics
Education, 16. 27–43.
Artigue, M. (2020). Didactical engineering. In S. Lerman (Ed.), Encyclopedia of mathematics education. Second
Edition (pp. 202–206). New York: Springer. https://doi.org/10.1007/978-3-030-15789-0_44
Artigue, M., & Bosch, M. (2014). Reflection on networking through the praxeological lens. In A. Bikner-Ahsbahs,& S. Prediger (Eds.), Networking of theories as a research practice in mathematics education (pp. 249–265).
Springer International Publishing. https://doi.org/10.1007/978-3-319-05389-9_15
Artigue, M., Bosch, M., & Gascón, J. (2011). La TAD face au problème de l’interaction entre cadres théoriques en
didactique des mathématiques. In M. Bosch et al. (Eds.), Un panorama de la TAD. Actes du troisième congrès
de la TAD (pp. 33–56). Centre de Recerca Matemàtica, University of Barcelona.
Bikner-Ahsbahs, A., Kidron, I., Bullock, E., Shinno, Y., & Zhang, Q. (in print). Topic Study Group 57: Diversity oftheories in mathematics education. In Jianpan Wang (Ed.) Proceedings of the 14th International Congress on
Mathematical Education. World Scientific Publishing House.
Bikner-Ahsbahs, A., Vallejo-Vargas, E., Rohde, S., Janßen, T., Reid, D., Alexandrovsky, D., Reinschluessel, A.,
Döring, T., & Malaka, R. (2023). The role of feedback when learning with a digital artifact: A theory networking
case on multimodal algebra learning. Hiroshima Journal of Mathematics Education, 16. 95–116.
Bosch, M., & Gascón, J. (2006). Twenty-five years of the didactic transposition [ICMI bulletin], (Vol. 58) (pp. 51–
65).
Chevallard, Y. (2019). Introducing the anthropological theory of the didactic: An attempt at a principled approach.
Hiroshima Journal of Mathematics Education, 12, 71–114. https://doi.org/10.24529/hjme.1205
Hobbes, T. (1841). The English works of Thomas Hobbes of Malmesbury: Vol. 2 (W. Molesworth, Ed.). John Bohn.
Kant, I. (2006). Groundwork of the metaphysic of morals (Mary Gregor, Trans.). Cambridge University Press.
Kidron, I. (2023). The role of a priori analysis in theories. Hiroshima Journal of Mathematics Education, 16. 45–56.
Kidron, I, Bosch, M., Monaghan, J., & Palmer, H. (2018). Theoretical perspectives and approaches in mathematics

26
education research. In T. Dreyfus, M. Artigue, D. Potari, S. Prediger & K. Ruthven (Eds.), Developing researchin mathematics education. Twenty years of communication, cooperation and collaboration in Europe (pp. 254–
268). Routledge. https://doi.org/10.4324/9781315113562
Leontyev, A. N. (2009). Activity and consciousness. Marxists Internet Archive https://www.marxists.org/archive/
leontev/works/1978/index.htm
Lévinas, E. (1982). Éthique et infini. [Ethics and infinity]. Fayard.
Prediger, S., & Bikner-Ahsbahs, A. (2014). Introduction to networking: Networking strategies and their background.
In A. Bikner-Ahsbahs & S. Prediger (Eds.) Networking of theories as a research practice in mathematics
education. (pp. 117–125). Springer. https://doi.org/10.1007/978-3-319-05389-9_8
Prediger, S., Bikner-Ahsbahs, A., & Arzarello, F. (2008). Networking strategies and methods for connecting
theoretical approaches: First steps towards a conceptual framework. ZDM – Mathematics Education, 40(2),165–178. https://doi.org/10.1007/s11858-008-0086-z
Rabardel, P. (2002). People and technology: A cognitive approach to contemporary instruments. Université paris 8.
https://hal.archives-ouvertes.fr/hal-01020705
Radford, L. (2008). Connecting theories in mathematics education: challenges and possibilities. ZDM - The
International Journal on Mathematics Education, 40(2), 317–327. https://doi.org/10.1007/s11858-008-0090-3
Radford, L. (2023). Ethics in the mathematics classroom. Hiroshima Journal of Mathematics Education, 16. 57–75.
Shinno, Y. & Mizoguchi, T. (2023). Networking praxeologies and theoretical grain sizes in mathematics education:
cultural issues illustrated by three examples from the Japanese research context. Hiroshima Journal of
Mathematics Education, 16. 77–94.

Angelika Bikner-Ahsbahs,
Bremen University, Germany & Western Norway University of Applied Sciences, Norway
bikner@math.uni-bremen.dehttps://orcid.org/0000-0002-0407-4897
Ivy Kidron

Yusuke Shinno

Jerusalem College of Technology, Israel

Hiroshima University, Japanivy@jct.ac.il

shinno@hiroshima-u.ac.jp

https://orcid.org/0000-0002-8986-0167
Takeshi Miyakawa
Waseda University, Japantmiyakawa@waseda.jp
https://orcid.org/0000-0002-1076-3592

https://orcid.org/0000-0001-6999-5075

HIROSHIMA JOURNAL OF MATHEMATICS EDUCATION 16: 27-43, 2023

27

FACING THE CHALLENGE OF THEORETICAL DIVERSITY:
THE DIGITAL CASE
Michèle Artigue
Université Paris Cité, France

Abstract
The presentation of TSG57 at ICME-14 outlined that mathematics education is a scientificfield in which many theoretical cultures coexist, and that “this diversity can be regarded as richness
but it also challenges research as well as communication and cooperation in the field”. It wasadded that “how the scientific community can cope with this diversity with scientific integrity
remains an open question”. In this paper, we propose to contribute to the reflection on thischallenging issue by considering research on technology-based teaching and learning. First, we
present a brief overview of the theoretical landscape in this area of research, highlighting its
diversity. We then introduce two conceptual tools that have proven their effectiveness in addressing
issues of theoretical diversity: the scale of networking strategies and the concept of researchpraxeology, before focusing on two case studies. These regard the instrumental approach and the
documentational approach to didactics, the emergence and development of which illustrate well
the global dynamic of the field towards increasing theoretical diversity, the questions raised bythis dynamic and the insightful efforts made to deal with it.
Keywords: mathematics education, digital technology, research praxeology, scale of networking
strategies, instrumental approach, documentational approach to didactics

1. INTRODUCTION
Increasing theoretical diversity is a general phenomenon in mathematics education, and research
focused on technological issues is no exception, as we already showed it in our plenary lecture at CERME5
fifteen years ago (Artigue, 2007). In fact, in that lecture we pointed out that, for decades, this area of research
has reflected the general trends and major developments in the field but has also been a source of inspiration
for it, as shown for instance, by the contribution of research in this area to constructivist and situatedperspectives as well as to semiotic and embodied approaches. We also argued that such a situation made thisarea of research an interesting window through which to look at theoretical diversity and at its practical
implications. This is all the more so as the development and the use of digital technologies, since theiremergence, have been given a transformative aim in mathematics education.
A similar vision is proposed in the volume resulting from ICMI Study 17, the second ICMI Study
dedicated to the teaching and learning of mathematics with digital technologies (Hoyles & Lagrange, 2010).
In fact, in this volume, Chapter 7 (Drijvers et al., 2010) is especially devoted to theoretical perspectives. The
Corresponding author: Michèle Artigue

28

Michèle Artigue

authors point to the influence on this area of research of the global evolution of the field from constructivist
to socio-cultural perspectives, and they illustrate this with three examples: the webbing and situatedabstraction frame in reference to (Noss & Hoyles, 1996), the theory of didactical situations and the related
notion of “antagonist milieu” (Brousseau, 1998) and the perceptuo-motor activity frame developed by
Nemirovski for use in research on modelling environments involving physical apparatus, such as the wellknown water wheel (Rasmussen & Nemirovski, 2003). They also highlight the theoretical creativity of thisarea of research. Two “current developments” are then presented with more details: the instrumental approachon the one hand, and mediation and semiotic mediation on the other hand. In the first case, the authors rightly
point out the fact that the theoretical combination at stake between cognitive ergonomics and theanthropological theory of the didactic has led to different variations reflecting the particular weight given tothese two perspectives, the first one cognitive, the second one institutional (see Section 3 for more details),
although all variations share fundamental distinctions and concepts, such as the distinction between artefact
and instrument, and the concept of instrumental genesis. For the second, the researchers involved share theclaim that the epistemological nature of mathematical objects makes them accessible only through themediation of representations, and that technological devices substantially alter these mediation possibilities.
However, once again, as the authors point out, these premises are dealt with through different theoreticalconstructions, for instance the concept of human-with-media introduced by Borba and Villareal (2005),
which emphasizes the unity between humans and tools, or the theory of semiotic mediation (Bartolini Bussi& Mariotti, 2008), which emphasizes the essential role of the teacher in the transition from personal meanings
rooted in the context of the artefact to mathematical meanings. Despite the fact that the chapter begins withthe assertion that, due to technological evolution, “communication has become a more integrated part oftechnology use”(p. 98), the theoretical influence of this evolution does not seem very strong at the time,rather a line for future research. Looking to the future, the authors write that the key word should be“connectivity”, because connectivity is a key word for approaching a technological evolution that isfundamentally changing our modes of communication and learning opportunities in and out-of-school. In
fact, this issue is mainly addressed in Chapter 11 (Beatty & Geiger, 2010), where it is emphasized thattechnological evolution calls for theoretical frameworks that allow researchers to approach mathematicallearning as a collaborative endeavor in technologically enhanced communities of practice. Various examples
are presented where the influence of theoretical constructs associated with communities of practice (Wenger,
1998) is clear. However, there is no doubt that at the time of the ICMI Study 17 research on these newer
technological affordances was not so much developed.
The chapter on technology in the book published for the 20th anniversary of ERME (Dreyfus et al.,2018) confirms the theoretical diversity observed in the ICMI Study volume (Hoyles & Lagrange, 2010), andorganizes it in relation to a didactic tetrahedron with knowledge, technology, students and teacher as edges(Trgalová et al., 2018). However, it does not show an obvious movement of research towards the latest
technologies. In fact, the evolution observed in the last decade, at least until recently, seems to be mainly due
to the increase of research on teachers, their knowledge and their practices in digital environments, their
preparation and professional development, their documentational work. For example, the documentational
approach to didactics began to develop about fifteen years ago, motivated both by this attention to teachers
and by the evolution of their documentational work induced by technological advances, as well explained in

Theoretical diversity: the digital case29

(Gueudet & Trouche, 2010). This increased attention is especially addressed in the book The Mathematics
Teacher in the Digital Era (Clark-Wilson et al., 2014) that makes clear the variety of theoretical constructs
developed to address these issues. These include the adaptation of Ball’s mathematical knowledge forteaching model ((MKT) to the technological pedagogical and content knowledge model (TPACK) by Mishra
and Koehler (2006), Ruthven’s structures features of classroom practice framework (Ruthven, 2007), and theextension of the instrumental approach to the teacher, leading to specific constructs such as instrumental
orchestration (Trouche, 2005), double instrumental genesis (Haspekian, 2011) and genesis of use (Abboud& Vandebrouck, 2013). They also include, of course, the specific constructs associated with the
documentational approach, such as the concepts of documentational genesis, resource system anddocumentational trajectory (for an updated vision, see Trouche, Gueudet & Pepin, 2019). Of particularinterest for a reflection about networking is the chapter written by Ruthven, which explores the similarities,
complementarities and contrasts between instrumental orchestration, TPACK and his own framework(Ruthven, 2014). Another point worth highlighting is the place given to the instrumental approach and its
extensions in these various syntheses. For instance, in (Clark-Wilson et al., 2014) an entire section with six
chapters is dedicated to related research.
In recent years, however, the context has been transformed by major technological changes: themassification of mobile multi-touch technologies, the development of virtual or augmented reality devices,and the growing influence of social networks on practices. The context has also been transformed by the
disruptions caused by the pandemic situation, with the abrupt shift to online or hybrid forms of teaching at
all levels of education, and the efforts made to limit the growth of educational inequalities by investing
massively in the most widely accessible technologies, such as mobile phones. Actions and research are
multiplying in response to the new challenges encountered, which will undoubtedly add to the existing
theoretical diversity.
In the aforementioned CERME lecture (Artigue, 2007), we argued for the collective development of
networking activities in order to limit the growing risk of fragmentation of the field. This need is also stressedin the ICMI Study 17 volume where it is written that the key word for the future should be “connectivity”,
not only for reasons of technological evolution, but also because efforts should be made to better connect theexisting diversity of theoretical perspectives. This is even more necessary today. Since then, however, thesituation has changed with the undeniable advances of the networking enterprise (Bikner-Ahsbahs &Prediger, 2014; Kidron et al., 2018). These are both conceptual, methodological and practical. To what
extent, then, are we now better equipped to meet the challenge of theoretical diversity in relation totechnology-based mathematics teaching and learning? Our aim in this paper is to contribute to the reflection
on this question. To this end, we introduce two conceptual tools in the next section before exploiting them totwo case studies. Over the last decade, we have indeed experienced the strong potential offered by the
combined use of these two conceptual tools to address the challenges posed by theoretical and linguistic
diversity (Artigue, 2019, 2021; Artigue & Bosch, 2014; Mesiti et al., 2021).

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Michèle Artigue

2. INTRODUCING TWO CONCEPTUAL TOOLS
2.1. The scale of networking strategies
As explained in (Bikner-Ahsbahs & Prediger, 2010), this construct aims to show the variety of strategies
that can be used to create connections between theories. The scale distinguishes eight strategies, paired andordered between two extreme positions: “ignoring other theories” and “unifying globally”, both of which
considered not desirable. However, as the authors themselves point out, this idea of a linear order must be
taken flexibly, as the degree of integration depends on the precise work done, not just on the strategy.

Figure 1: Scale of networking strategies (Bikner-Ahsbahs, 2016, p. 34, adapted from (Bikner-Ahsbahs &Prediger, 2010, p. 492) (CC BY 4.0))

The authors provide precise definitions for these different strategies. For example, on page 495, they
explain that coordinating means that “a conceptual framework is built by well-fitting elements from different
theories” and that this strategy therefore presupposes the complementarity of the theoretical approaches
involved, whereas combining means that “the theoretical approaches are only juxtaposed according to a
specific aspect”. The combining strategy can thus involve theories with some conflicting basic assumptions.
On page 496, we are told that integrating locally and synthesizing label a pair of strategies that focus “on thedevelopment of theories by putting together a small number of theories or theoretical approaches into a newframework”, and the authors differentiate between the two strategies by considering the dissymmetry/symmetry of the theories involved in terms of scope and degree of development. As explained in (Artigue2019), we find it more appropriate to differentiate between these two strategies according to the dissymmetrical/
symmetrical contribution of the theories involved in the resulting construction, and to consider that any time
there is a significant dissymmetry in the connection, even for theories having a similar state of development,
there is local integration. This is the case, for example, when a broad theory is enriched by theoreticalconstructs from another well-established theory in order to build an integrated theoretical framework thattakes into account research findings in a particular area produced in a different theoretical culture.
2.2. The concept of research praxeology
The notion of research praxeology introduced in (Artigue et al., 2011) extends to research practices theconcept of praxeology at the heart of the anthropological theory of the didactic (hereafter ATD) (Chevallard,

Theoretical diversity: the digital case312019). Indeed, a basic principle of ATD is that all human practices can be modelled in terms of praxeologies.
These consist of a praxis block and a theoretical block in dialectical interaction. A priori, thus, this is also thecase for research practices. By definition, the praxis block of research praxeologies includes the different
types of tasks that the research activity requires, and the associated techniques of study. Their theoretical
block consists of the discourse used to describe, justify and interpret research techniques (technologicaldiscourse in ATD language), and a theory consisting of “statements of a more general and abstract character,with a generally strong justifying and generating power.” (Bosch & Chevallard, 2020). At the simplest levelof point praxeologies, dealing with a single type of task, a typical research praxeology in mathematics
education is associated with a research question and a technique of study for this question. These form its
praxis block. This block is dialectically linked to a theoretical block consisting of technological andtheoretical discourses. The theoretical block includes at least some methodological discourse that explains
and justifies the technical choices; some research background, namely existing knowledge regarding thequestion at hand or related questions, and associated constructs that are considered of interest, plus at least
one theory that underlies the whole construction.
Of course, this is a very simple case. Most research questions cannot be solved by using a single
technique, and the theoretical block of research praxeologies, even point praxeologies, often, does notmobilize a single theory but some theoretical combination. Most often, research praxeologies emerge fromquestions about teaching and learning processes, or more globally about the functioning of didactic systems
or institutions, that is, through their praxis block. But the questions and their formulation are influenced by
the “theoretical” already present in the researcher’s or the research team’s environment. The study techniques
are guided by those that are implemented in close research praxeologies, or that are familiar to the researcher.
In this way, the praxis and the theoretical blocks interact from the outset. Moreover, the study of a researchquestion generally requires more than a single point praxeology. It is necessary to develop a coherent set ofresearch praxeologies that share, at least in part, the same theoretical block. Local and regional praxeologies
thus emerge. New questions also arise from research results; new elements enter the technological discoursein terms of distinctions, categorizations, didactic phenomena. If they become reasonably shared, they will
enrich the theory itself, thus contributing to the dynamic of research praxeologies.
We claim that approaching the relationships between theories through this praxeological lens can help
researchers to take better account of the functional role of theories in the research activity. The interest in
adopting such a functional stance towards theories became clear to us while working in the European project
TELMA whose aim was to capitalize European knowledge about technology enhanced learning inmathematics (Artigue, 2009). At that time, however, the concept of research praxeology had not yet been
introduced. In the context of research on the educational use of digital technologies, its first use took place
in a retrospective analysis of the theoretical activities carried out in the ReMath European project that had
emerged from TELMA (Artigue & Mariotti, 2014).
In the following two sections, we use these tools to present and discuss two case studies. These regard
the instrumental approach and the documentational approach to didactics, respectively. Both approaches
have obvious connections and both have generated a multiplicity of theoretical interactions, given thediversity of researchers who have contributed to their development or simply used them. They provide alimited, but sufficiently rich, ground to support the reflection aimed at in this article.

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Michèle Artigue

3. A FIRST CASE STUDY: THE INSTRUMENTAL APPROACH
3.1. Overview of the instrumental approach
The instrumental approach (IA in the following) emerged in France in the mid-1990s in a specifictechnological context, that of CAS technology (Artigue, 2002), and with two theoretical pillars: the ergonomic
perspective on contemporary tools developed by Rabardel and Vérillon (Rabardel, 1995; Vérillon &Rabardel, 1995) and ATD1. In fact, this approach quickly spread beyond the small community in which it wasborn. The rapid publication of books, both in French and in English, and of many articles certainly favoredthis dissemination. All over the world, researchers began to incorporate IA constructs into their theoreticalframeworks. This phenomenon was already evident at the time of the ICMI Study 17 mentioned above, andin the last decade the theoretical diversity has even increased. All constructions share the reference to the
Rabardel’s and Vérillon’s ergonomic perspective, in particular:●

the distinction between artefact and instrument;●

the concept of instrumental genesis with its two movements dialectically connected: instrumentalizationfrom the user to the artefact and instrumentation from the artefact to the user;●

and the conceptualization of instrumental genesis in terms of the elaboration or appropriation ofschemes.
However, these constructs are combined with or integrated into different theories, which leads tosignificantly different research praxeologies. This phenomenon was already visible in the first two doctoral
theses based on IA, those of Defouad (2000) and Trouche (1997), the latter having the theory of conceptual
fields (Vergnaud, 1991, 2009) as its main theoretical component. The impact on the different components of
their respective research praxeologies is clear, with obvious consequences on the results obtained. In
Defouad’s thesis, in line with the ATD conceptualization of human activities in terms of praxeologies, moreemphasis is placed on the instrumented techniques developed, the material signs (ostensives in ATD
language) used in them, and the discourse explaining and justifying the techniques (the technologicaldiscourse in the praxeological model), rather than on the schemes underlying them. Furthermore, special
attention is paid to the relationships between the students’ instrumental geneses and the classroom instrumental
genesis, and their management by the teacher. The results are mainly expressed in terms of regularities
identified in the students’ instrumental geneses and of didactic phenomena that show links between thecharacteristics of the students’ instrumental geneses and the institutional conditions and constraints thatshape the ecology of instrumented techniques in the classroom. In Trouche’s thesis, the results are mainly
expressed in terms of schemes of instrumented action and of their evolution in the transition from graphic tosymbolic calculators.
A few years later, in the ReMath project (Kynigos & Lagrange, 2014), thanks to the development ofspecific methodological tools, we were able to analyze the effect on research praxeologies of the integrationof instrumental perspectives in theories born outside the French didactic tradition such as the theory ofsemiotic mediation (Bartolini Bussi & Mariotti, 2008) or constructionism (Papert, 1980). New local
integrations have also resulted from the extension of IA to the teacher, such as its integration in the dual
1 More detailed information about the instrumental approach, its emergence and development, can be found in the Michèle
Artigue’s Unit of the ICMI AMOR project: https://www.mathunion.org/icmi/awards/amor/michele-artigue-unit

Theoretical diversity: the digital case33

ergonomic and didactic approach of teachers’ practices (Robert and Rogalski, 2002), well illustrated by the
GUPTEN project in France (Lagrange, 2013).
3.2. Networking challenges: the scheme/technique case
There is no doubt that all these connections have contributed to the wealth of knowledge and results
produced under the IA umbrella, but they have also raised serious challenges. We will illustrate thesechallenges with an example, the scheme/technique case, which has been the source of intense debate foralmost a decade (see the synthesis offered in (Monaghan, 2007)). IA emerged as a synthesis between
Rabardel’s and Vérillon’s instrumental perspective and ATD. In such a process, ATD categories anddiscourses were intertwined with those proposed by cognitive ergonomics. Instrumental geneses were
thought in terms of the development of praxeologies, and thus approached in terms of instrumented techniques
and associated technological discourses, rather than only in terms of schemes, as was the case in ergonomic
publications. This duality was discussed extensively in the first decade of IA. The collective work on thisissue helped to clarify the different points of view and, in particular, to reject simplistic assimilations such as
those that reduced techniques to gestures or to the observable part of schemes. In fact, thinking in terms ofpraxeologies means that techniques cannot be isolated from the technological discourse that describes,
explains and justifies them. In a sense, reducing techniques to gestures is akin to reducing schemes to theirobservable characteristics without considering the essential component of schemes that the operational
invariants underlying the observed regularities are. Indeed, the many contributions to the scheme/techniquedebate have made it clear that schemes and techniques correspond to two different and complementary waysof approaching instrumental issues, both insightful but irreducible to each other.
However, a new element deserves to be taken into account in today‘s reflection on this topic. In fact,
while maintaining its institutional anchorage, ATD has progressively incorporated the concept of personal(mathematics) praxeology, following research by Crozet and Chaachoua (2016)2. Like institutional
praxeologies, personal praxeologies are modelled in terms of quadruplets that dialectically articulate a praxis
block consisting of types of tasks and techniques and a theoretical discursive block. However, this personal
dimension, the very nature of the students‘ verbalizations used to identify the theoretical block of personal
praxeologies, should allow for more productive connections between the language of schemes and thepraxeological language.
3.3.The ReMath contribution
The European ReMath project also had important spin-offs in terms of networking involving IA, as itsoon became clear that, despite the diversity of their theoretical cultures, five of the six teams involved weresensitive to Rabardel’s and Vérillon’s conceptualizations and had incorporated them in some way into their
theoretical background. Thanks to a carefully designed methodology, we were able to document and analyze
the effects of these incorporations both in the design and development of digital dynamic artefacts(DDAs) – six DDAs were actually created or improved as part of this project – and in the use of these DDAs
by two different teams, the team in charge of the DDA design and another team from another country with a2 As explained in Module 9 of the Artigue’s unit of the AMOR project, the idea of personal praxeology had been alreadyintroduced in Defouad’s thesis, but it did not disseminate at the time and was soon forgotten.

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Michèle Artigue

different theoretical background. In this paper, we focus on the case of constructionism, and a comparison
involving the Greek team in the Educational Technology Lab (ETL), for which constructionism was the main
theoretical support, and the French team in DIDIREM, now the Laboratory of Didactics André Revuz(LDAR), whose theoretical background combined IA with the theory of didactic situations (Brousseau,
1998) and Duval’s semiotic theory (Duval, 1995), both very influential in French didactic research on digitaltechnologies (Artigue & Trouche, 2021). As explained in the special issue (Kynigos & Lagrange, 2014), in
ReMath ETL was in charge of two DDAs, Cruislet and MaLT, and DIDIREM was in charge of the DDA
Casyopée. Their constructionist culture made the Greek researchers particularly sensitive to the expressive
power of digital artefacts, which they had theorized in terms of “half-baked microworld”, that is to say“pieces of software explicitly designed so that their users would want to build on them, change them or decompose parts of them in order to construct an artifact for themselves or one designed for instrumentation byothers” (Kynigos, 2007, p. 336). This obviously influenced their (local) integration of Rabardel’s andVérillon’s ergonomic perspective into constructionism, leading them to emphasize the instrumentalization
dimension of the process of instrumental genesis. Indeed, instrumentalization captures the vision that
artefacts can be transformed by their users in ways anticipated by the designers, but also in ways that are notanticipated (catachresis phenomena). Instrumental genesis results from the dialectic interaction between
instrumentalization and instrumentation processes, and a constructionist perspective leads to emphasize therole of instrumentalization in this dialectic interaction. In the design of Cruislet and MALT, and in the tasks
they proposed to students, the ETL researchers were particularly sensitive to the need to support this
instrumentalization dimension of instrumental geneses and the students’ creativity associated with it. In linewith the constructionist culture, programmable simulations using the Logo language, open and challenging
tasks with strong potential for the students’ expressiveness were essential tools for this. The contrast with the
DDA Casyopée was clear. This DDA was very innovative in terms of dynamic connections between
representations – for example, it offered advanced tools for the functional modelling of geometric situations
involving covariation of lengths and areas –, but the focus was on institutional graphic and symbolic
representations, and the support offered by the DDA for their treatment within a given semiotic register orconversion from one register to another one, in line with Duval’s semiotics. In terms of design priorities,
therefore, more emphasis was placed on supporting instrumentation processes than on supporting
instrumentalization processes, and the learning potential offered by the emergence and progressive
transformation of alternative representations to the institutional ones was not considered. Such comparisons
sharpened our vision of instrumentalization, of the dialectical games between instrumentalization andinstrumentation, and made clear the influence on our vision of the theories with which we had networked
ergonomic perspectives.
The cross-case study regarding the DDA Cruislet, involving ETL and DIDIREM, was also particularly
insightful. It highlighted the profound differences between scenarios based on the theory of didactic situations
and constructionism, although both theories share socio-constructivist principles. The concept of fundamental
situation, the aim of optimizing the adidactic potential of situations through the selection and managementof didactic variables, and the attention paid to the organization of the dual processes of devolution andinstitutionalization, are at the very heart of the theory of didactic situations and of didactic engineering
designs based on this theory. Such constructs are alien to constructionist scenarios. In these scenarios, theTheoretical diversity: the digital case35

aim of ensuring that the tasks designed and their technological and social environment offer strong potential
for supporting the students’ expressiveness is more important than the aim of achieving a precise learning
goal and optimizing the possible trajectories. The Cruislet software, which offers various possibilities for
simulating airplane flights over a map of Greece, was perfectly adapted to a constructionist approach tolearning. For the French teachers and researchers, experimenting with it was a real challenge, and ultimately
only possible within a specific project device, less constrained than the usual lessons. However, thisexperience was also a source of questioning and enrichment for us, as shown in (Le Feuvre, Meyrier &Lagrange 2010). Today, the evolution of the concept of didactic engineering carried out by ATD through theconcept of study and research path and the related methodological tools such as question-answer graphs(Barquero & Bosch, 2015), undoubtedly provides a more appropriate framework for exploiting the potentialof such constructionist DDAs in the French didactic culture.
Beyond this particular case, the ReMath cross-experiments showed that the productive networking oftheories does not obey the same constraints when theories are networked, a posteriori, to analyze and interpretcollected data, and when they are networked to build a conceptual framework to support didactic design, be
it the design of digital artefacts or the design of tasks and learning situations involving them. The
experimentation of Cruislet mentioned above showed that the logics underlying constructionist design and
TDS design were so far apart that it was very difficult, if not impossible, to coordinate them in a conceptual
framework. Significant distance was also observed in the conceptual frameworks supporting the design ofsituations involving the DDA Casyopée by the French DIDIREM team and the Italian UNISI team from the
University of Siena. The UNISI design was guided by the theory of semiotic mediation (Bartolini Bussi &Mariotti, 2008), while the DIDIREM design, once again, was guided by TDS and Duval’s semiotics. In linewith their theoretical background, the Italian researchers conceived Casyopée as an instrument of semiotic
mediation, and, in their design, they carefully organized the transition from semiotic signs attached to theartefact to mathematical signs detached from the artefact. Much time was devoted to this transition, incollective discussions led by the teacher without access to Casyopée. For the French researchers, in line with
the TDS, Casyopée was a central element of the a-didactic milieu. Their design, while paying attention to theconnection to be made between paper-pencil and instrumented techniques, did not conceive this connectionin terms of transition. This was clear in the institutionalization phase where both instrumented and noninstrumented forms of knowledge were combined. Conversely, the various cross-case studies carried out in
ReMath have shown the potential offered by the combination of constructionism, TDS and the theory ofsemiotic mediation in the a posteriori analysis of the experiments (see, for example, Maracci et al., 2013). In
summary, the a posteriori combination of complementary and even conflicting theoretical perspectives in theanalysis and interpretation of data can be very productive, while the combination of theoretical perspectives
that are too far apart can make a coherent design impossible. More compatibility is needed. This is consistent
with the distinction made between the combining and coordinating strategies in the scale of networking
strategies (see Section 2) and confirms the pertinence of this distinction. Design based on theoretical
networking requires coordination, and not just combination, of theories or theoretical constructs.

36Michèle Artigue

4. A SECOND CASE STUDY: THE DOCUMENTATIONAL APPROACH TO DIDACTICS
The documentational approach to didactics (hereafter DAD) was born in a course given at the French
Summer school of didactics of mathematics in 2007 (Gueudet & Trouche, 2009). It was motivated by thesignificant changes in the documentational work of teachers brought about by technological evolution, theincreasing number of digital and online resources available to and used by teachers, and the growing role ofprofessional and social networks in their professional activity. Digital technologies are therefore at the heartof DAD. As Gueudet (2009) explains in her analysis of the development of DAD, its source was her andTrouche’s earlier work, which was mainly concerned with digital technologies. For Trouche, it was thecontribution to the development of IA and to the SFODEM project, “a project of in-service teacher training,
mainly at distance, aiming to support the integration of ICT in the teachers’ practices by a rich offer ofresources on a platform” (Gueudet, 2019, p. 19). For Gueudet, it was her research on learning processes with
online exercises, first at university level and then extended at all levels of schooling, in particular inconnection with the association of teachers Sésamath, an association that has produced very much usedonline resources, since its creation more than 20 years ago (https://www.sesamath.net).
The development of DAD in just fifteen years is impressive, as is impressive the number of researchers
from different cultures and backgrounds who have contributed to it, as shown in (Trouche, Gueudet & Pepin,2019). Another interesting element for our reflection is that, from the beginning, DAD has combined different
sources of inspiration: IA and it can be seen as an extension of IA to the documentational work of teachers,
but also many others. For instance, Adler’s conception that “resources for school mathematics extend beyond
basic material and human resources to include a range of other human and material resources, as well as
mathematical, cultural and social-resources” (Adler, 2000, p. 210) is explicitly used in the extended definitionof resources proposed in the DAD. Remillard’s participative approach to the use of curriculum resources(Remillard, 2005), or Wenger’s community of practice approach (Wenger, 1998), to mention just but a few,
have also been very influential.
Over the last decade, we can observe a progressive evolution of the connections, both on the side of thefounders of DAD and on the side of its contributors and users. Analyzing the first book published on thisapproach (Gueudet & Trouche, 2010), we have shown in (Artigue, 2019) that Gueudet and Trouche mention
many authors in the two chapters they co-author, but that few theoretical connections are really elaborated
beyond the foundational one with IA. For example, we observe only one example of local integration beyond
the already mentioned one with Adler’s conception of resources. It occurs when the authors link teachers’
systems of documents and systems of activities by introducing a categorization of teachers’ activities
explicitly inspired by the study moments of ATD (Chevallard 2002). The foundational connection with IA
leads to the distinction between resource and document, analogous to the distinction between artefact andinstrument. Indeed, a document is defined as a hybrid entity consisting of updated/recombined resources andof a scheme of utilization of these. This connection also leads to the concept of documentational genesis withthe dual processes of instrumentation and instrumentalization, and the development of schemes.
An interesting point is the awareness of researchers, since this emerging state of DAD, of the need forspecific methodologies that allow them to approach the teachers’ documentational activity in-class and outof-class, with a long-term follow-up, and a close involvement of the teacher in the collection of data leading

Theoretical diversity: the digital case37

to a reflective stance. Very soon, this methodology was given a name: “reflective investigation” and a specific
tool was associated with it: the SRRS (schematic representation of resource system), and both became
emblematic of the research praxeologies based on DAD. In the chapters of the same book written by otherresearchers, one can observe many cross-references but, once again, rather few advanced forms of networking,which is not surprising in this nascent state of DAD. In most chapters, the main purpose seems to be to makeanother approach understandable, or to contrast it with DAD. Only one chapter really goes further and opens
the way to local integration (Trgalová, 2010). As explained in (Artigue, 2019, p. 100), “the author connects
DAD with the model of teacher activity developed by Margolinas (2002) and Balacheff’s cK¢ model ofconceptions (Balacheff 1995), two constructions already connected by these two authors to analyze teachers’
didactic decisions in (Balacheff and Margolinas, 2005)”. It is interesting to note that, in this particular case,the data used comes from a doctoral thesis whose methodology has stimulated some form of documentational
work, which certainly makes the networking more practicable.
As shown in (Artigue, 2019), the theoretical core of DAD research praxeologies stabilized quickly,
which contributed to the coherent structuring of these praxeologies, together with the privileged role given
to the methodology of reflective investigation and the SRRS tool. This did not prevent the diversification ofresearch problématiques, to which the increasing cultural diversity of the researchers contributing to DAD
certainly contributed. When preparing our plenary lecture at the Re(s)sources 2018 conference, we asked
Luc Trouche for a selection of references to get an updated vision of the theoretical connections in DAD, andwe received a list of theoretical crossings with one or two references for each. The list mentioned thirteentheories and theoretical approaches. Some of them were already present in the first book such as informationand communication sciences, Remillard’s approach or communities of practice, but also were new such as
cultural historical activity theory, the theory of social creativity, constructionism or the meta-didacticaltransposition approach. Even if it remains partial, the analysis of the connections carried out in (Artigue,2019) shows a clear progression in their quality. The collaborative work that has been developed, evidenced
by the fact that most publications are co-authored, has obviously made this progress possible. There is no
doubt that DAD researchers have been able to create communication devices and forms of collaborative
work that help to meet the challenges raised by the increasing diversity of theoretical connections involvedin the development of the theory, and also by the linguistic diversity of their community as shown by the
DAD-MULTILINGUAL project (https://hal.science/DAD-MULTILINGUAL/).
In terms of research praxeologies, DAD today unifies, at a regional level, a diversity of researchpraxeologies aimed at better understanding of the transformations of teachers’ activity induced by the use ofdigital technologies, and at developing innovative uses of the increasing diversity of these technologies. As
already mentioned, DAD research has created a methodological tool, the methodology of reflective
investigation. This methodology, which provides a study technique well adapted to the research questions
addressed by DAD, has become emblematic of its research praxeologies, and has been progressively enrichedas shown in (Trouche, Gueudet & Pepin, 2019). The theoretical block of DAD research praxeologies has also
been progressively enriched in dialectic interaction with the enrichment of their praxis block. For example,
the study of documentational genesis over longer periods of time than was initially the case has led to theconcept of documentational trajectory, which in turn has led to new methodological and representational
tools (see, for example, Rocha, 2018).

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Michèle Artigue

In this second case study, the combined use of the networking and praxeological lenses helps us tounderstand a dynamic particularly rich in theoretical connections that has developed over the last fifteen
years, to understand why we observe such a theoretical diversity in it, and how this diversity has gradually
been better managed through more advanced forms of networking, and also thanks to the unifying
characteristics of the DAD research praxeologies.

5. CONCLUSION
In this paper, we have shared some elements of the knowledge we have gained through our experienceof networking between theories, focusing on research on teaching and learning in digital environments andthe ways in which these influence teaching and learning processes in mathematics. If we return to the questionthat has guided the reflection developed in this paper: “To what extent are we now better equipped to face thechallenge of theoretical diversity in what regards technology-based mathematics teaching and learning?”,
our reflection makes clear that conceptual and methodological tools have come a long way since the issues
raised by theoretical diversity were seriously put on the agenda of the international mathematics education
community at the beginning of this century. As a result, as we have tried to show, knowledge has advanced
both in terms of understanding the diversity challenge and in terms of developing strategies to address it. All
the forms of networking identified in the scale of networking strategies have been used productively,
involving a number of theories with different state of development, aims and scope. The connections
established and the knowledge built now form a solid background for supporting research in this area.
Networking research has also shown the interest of developing specific research praxeologies. These allow
us to examine our respective research practices, their theoretical frameworks and their results without
denaturing the theories and theoretical constructs involved, in order to build significant connections, andmake clear their potential and limitations, as has been done, for example, in ReMath and in (Bikner-Ahsbahs& Prediger, 2014). More globally, our reflection confirms the interest of adopting a praxeological lens toapproach the diversity challenge.
The theoretical landscape of research on technology-based teaching and learning is a dynamic one. On
the one hand, the development of multi-touch mobile technologies, of virtual and augmented reality devices,
is the source of new interesting theoretical connections, such as the connection between IA and radical
embodied cognitive science approaches presented in (Shvarts et al., 2021). On the other hand, the massive
move towards online and hybrid mathematics teaching provoked by the pandemic situation has abruptly
placed technology-based teaching and learning at the center of the educational agenda. This situation, andthe resulting increase in educational inequalities, is also a source of new theoretical needs and connections,
as illustrated by Borba (2021), for example. In this contribution to the special call on mathematics education
research in pandemic times launched by the journal Educational Studies in Mathematics, Borba indeed
reworks the theoretical construct of humans-with-media that he introduced years ago to emphasize that the
production of knowledge results from the collective agency of humans and media (Borba & Villareal, 2005)
to take into account the impact and agency of non-living things such as COVID-19. He also suggests
combining this theoretical perspective with that offered by critical mathematics education (Skovsmose,

Theoretical diversity: the digital case391994) to address the crucial issue of educational inequalities exacerbated by the pandemic. These are only
two examples but there is no doubt that networking needs will continue to grow in this area of research aswell as in mathematics education more generally.

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Theoretical diversity: the digital casedoi.org/10.1017/CBO9780511803932

Michèle Artigue
Université Paris Cité, Paris, France
Email: michele.artigue@univ-paris-diderot.fr
https://orcid.org/0000-0002-8176-8243

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HIROSHIMA JOURNAL OF MATHEMATICS EDUCATION 16: 45-56, 2023

45

THE ROLE OF A PRIORI ANALYSIS IN THEORIES
Ivy Kidron
Jerusalem College of Technology, Israel

Abstract
The paper offers a theoretical reflection on the role of a priori analysis in theory. For eachtheory the a priori analysis has a role of reference while comparing with the a posteriori analysis.
Different theories offer different a priori analyses. The different a priori analyses result from thedifferent priorities of the theories with regard to the focus of analysis. In the a priori analyses thesepriorities and their differences are made explicit. This study demonstrates the important role of thea priori analysis in the different theories and the benefit of comparing a priori analyses for
networking the theories.
A priori analysis, a posteriori analysis, networking theories, epistemological
Keywords: perspective.

INTRODUCTION
This paper offers a theoretical reflection on the role of a priori analysis in different theories. In thewords “the role of a priori analysis in different theories” I mean both, the role of a priori analysis as amethodology within a given perspective and also the role of a priori analysis in the research of networking
theories as a methodological tool to better understand the specificities of different theories. In the following,
I explain why I consider both roles of a priori analyses, in a given theory and in networking theories.
In Kidron et al. (2018), the authors deal first with the diversity of theoretical perspectives and networking
theories and only then questions concerning the role of theory are discussed with regard to the diversity oftheoretical perspectives and with networking in the background. The authors explain how by means of
networking theories researchers understand better what a theory is:
In all the work related to networking theoretical approaches, deepening into the notion of theory
appears as a crucial issue. (Kidron et al., p.260)
The present paper deals with the role of a priori analysis. Trying to give a general description of theterm a priori analysis, we may describe an a priori analysis as an analysis, which is often carried out prior to
the experiment or data collection as a part of methodology proposed within some theoretical perspective.
Therefore, the answer to the question “What is an a priori analysis?” might be different for different theories.
In the present paper, I investigate the differences in the a priori analyses of some specific theories aswell as the influence of the different a priori analyses on the networking of these theories.
Dealing with a priori analyses, the focus of the paper is on methodology. We will also observe how the
Corresponding author: Ivy Kidron

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Ivy Kidron

notion of a priori analysis is closely related to the phase of design.
The main theories discussed in this paper are the Theory of Didactical Situations (TDS) and the theoryof Abstraction in Context (AiC). There are three main reasons for the choice of these theories. The first one
is that TDS as a well-established theory that belongs to the French school of didactics benefits from a strong
a priori analysis. The second reason of this choice is that this paper is based on my networking experience
with these theories (Kidron et al., 2008; Kidron et al., 2014). The third reason is that the a priori analyses take
into account the mathematical epistemology of the given domain and for both theories the epistemologicalperspective is of importance.
Then, other examples of the differences in the a priori analyses of couple of theories (TDS and APC(Action, Production, Communication); CWS (Connected Working Spaces) and AiC) will be presented
shortly as well as the influence of the different a priori analyses on the networking of these theories.
Only then, we will be in a better situation to understand the important role of the a priori analysis in the
different theories and the benefit of comparing a priori analyses for networking theories.

A PRIORI ANALYSES IN DIFFERENT THEORIES:THEORY OF DIDACTICAL SITUATIONS (TDS) and
THEORY OF ABSTRACTION in CONTEXT (AiC)
For both theories, one main focus of the a priori analysis is the epistemological perspective. We willobserve how the two theories consider the epistemological dimension in different ways. TDS combines
epistemological, cognitive, and didactical perspectives. TDS focuses on the epistemological potential of
didactical situations. AiC analysis is essentially cognitive and focuses on the students’ reasoning; mathematicalmeaning resides in the verticality of the knowledge constructing process and the added depth of the resulting
constructs. An epistemological stance is underlying this idea of vertical reorganization (I will explain thisterm in more details in a next subsection). For both theories, the epistemological dimension has a significant
role. In the literature, the important role of epistemology is discussed in detail in Artigue (1990, 1995).
Kidron (2016) analyses the connection between epistemology and networking of theories.
In the next two subsections, I will present shortly each theory and deal with the question what is an apriori analysis for each theory. Then, I will deal with the differences of the a priori analyses and demonstrate
that, in their effort of networking theories, researchers of both theories benefit of comparing their a priori
analyses.
TDS a priori analysis
A short introduction to TDS is offered in Artigue et al. (2014). As mentioned earlier, this well-establishedtheory belongs to the French school. It began to develop in the 1960s in France, initiated by Guy Brousseau(1997). In Artigue (2020, p. 203) we read “how the idea of didactical engineering which emerged in French
didactics in the early 1980s contributed to firmly establish the place of design in mathematics education
research”. We also read that the Didactical Engineering (DE) is structured in four different phases. Design
and a priori analysis is one of these four phases. Artigue (2020, p. 204) wrote: “The goal of the a priori

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47

analysis is… to build a reference with which classroom realizations will be contrasted in the a posteriori
analysis”. In Artigue et al. (2014, p. 48) we learn about the three characteristics of the way TDS considers
the teaching and learning of mathematics. The first characteristic, the systemic perspective of TDS isexpressed by the central object of the theory, the idea of situation, which is itself a system. In the present
paper, a special attention is given to the second characteristic, the epistemology of mathematical knowledge,
while discussing TDS a priori analysis. The third characteristic, the vision of learning as a combination ofadaptation and acculturation, relates to the cognitive dimension. These characteristics determine the questions
that TDS raises, as well as its methodology.
The theory is structured around the notions of a-didactical situation (in a-didactical situation there is no
explicit didactical intentions: students are working as if there is no didactical intention and the teacher
refrains from interfering) and didactical situations and includes concepts relevant for teaching and learningin mathematics classrooms. The social dimension also has a significant role in TDS. In essence, the central
object of the theory, the situation, incorporates the idea of social interaction.
It is this systemic view that led to the concept of DE (Artigue, 1989, 2020) that we mentioned earlier
in connection to TDS a priori analysis. Artigue et al. (2014, p.50) explains DE methodology:
It is a methodology which is structured around a phase of preliminary analysis combining
epistemological, cognitive, and didactical perspectives, and aiming at the understanding of theconditions and constraints to which the didactical system considered is submitted, a phase of design
and a priori analysis of situations reflecting its optimization ambition; and, after the implementation, aphase of a posteriori analysis and validation.
The notion of “milieu” is an important construct in TDS. In Brousseau (1997, p.9) we read:
Within a situation of action, everything that acts on the student or that she acts on is called the “milieu”.
A-didactical situations are well explained in Brousseau (1997, pp, 54-72). The a-didactic milieu wasinitially defined by Brousseau as the system with which the student interacts in the a-didactic game. In
Brousseau (1997, p.57) we read:
The analysis of the didactical relationships implies the definition or the recognition of these“fundamental” and adidactical games, bringing together a milieu and a player, these games being suchthat knowledge – a given precise knowledge – will appear as the means of producing winning
strategies”.
In the design of learning situations, there is a special attention to the constituents of the milieu organized
for the learner.
In her chapter “Perspectives on design research: the case of didactical engineering”, Artigue (2015)
presents the evolution of Didactical Engineering (DE) in the last three decades and explains its links with
TDS. She also presents its characteristics as a research methodology. In this chapter we read that design has
always played a fundamental role in the French school. We also read how design is connected to the a priori
analysis.
In Artigue et al. (2014, pp. 54–60), we have a detailed example that explains the components of the
TDS a priori analysis and the requests of the a priori analysis for developing the systemic analysis typical for
TDS. For example, the need of information of the mathematical knowledge of the students, of the particularsituation at stake, of the teacher’s expectations regarding this situation. The methodology for analysis is

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Ivy Kidron

described in the following sentence:
We developed thus our analysis using the usual techniques of TDS, that is to say, preparing an a priori
analysis focusing on the determination of the cognitive potential of an a-didactic interaction with themilieu, for a generic and epistemic student, that is, a student accepting the a-didactical game and able
to invest in it the mathematical and instrumental knowledge supposed by the teacher. (Artigue et al.,2014, p. 63)
Therefore, in TDS a priori analysis:–	The researchers make assumptions about the supposed mathematical knowledge of the students
which is required for a productive interaction with the “milieu”.
–	There is a need of information of the particular situation at stake, of the teacher’s expectations
regarding this situation.
–	There is a focus on the determination of the cognitive potential of an a-didactic interaction with themilieu.
–	The researchers make assumptions about the role of the teacher and how she extends the results of
the a-didactical situation.
The a priori analysis must then play its role of reference as well as its role of revealing the didactic
phenomena. Then the a posteriori analysis is compared to the a priori analysis and sometimes the hypotheses
which were done in the a priori analysis are not in accord with the a posteriori analysis of the collected data.
This comparison of the a priori analysis and the a posteriori analysis will allow the TDS researchers to deeply
understand the functioning of the “situation”.
AiC a priori analysis
Dreyfus & Kidron (2014) offers a short introduction to AiC. The theory is explained in more details in(Schwarz et al., 2009). AiC has been developed over the past 20 years with the purpose of providing atheoretical and methodological approach for researching, at the micro-level, learning processes in whichlearners construct deep structural mathematical knowledge. Methodologically (and this is the focus of thepresent study), the AiC researchers are offered tools that allow them to observe and analyze students’ thinking
processes. A detailed treatment of the methodology is offered in Dreyfus et al. (2015). AiC view of abstraction
is grounded in the works of Davydov (1990) and Freudenthal (1991). I wrote earlier that AiC focuses on thestudents’ reasoning and that mathematical meaning resides in the verticality of the knowledge constructing
process and the added depth of the resulting constructs. Freudenthal ideas led his collaborators to the idea of“vertical mathematization”. This idea is explained in Dreyfus et al. (2015, p. 186–187):
Vertical mathematization points to a process that typically consists of the reorganization of previous
mathematical constructs within mathematics and by mathematical means by which students construct
a new abstract construct. As researchers in mathematics education, we preferred the expression “verticalreorganization” to the expression “vertical mathematization” to discern between what is intended by
the teacher - the mathematization, and what often happens - a reorganization…. In vertical reorganization,
previous constructs serve as building blocks in the process of constructing.
Thus, AiC defines abstraction as a process of vertically reorganizing some of the learner’s previous
mathematical constructs within mathematics and by mathematical means in order to lead to a new construct

A priori analysis

49(for the learner). The expression used in the previous sentence within mathematics and by mathematicalmeans demonstrates the importance of the epistemological dimension for AiC. For the convenience of thereaders, I will report some more details about AiC. The process of abstraction has three stages: the need fora new construct, the emergence of the new construct and the consolidation of this new construct. The second
stage, the emergence of the new construct is analyzed by means of three observable epistemic actions:
Recognizing, Building-With and Constructing. Recognizing takes place when the learner recognizes that a
specific knowledge construct is relevant to the problem she or he is dealing with. Building-with is an action
comprising the combination of recognized knowledge elements, in order to achieve a localized goal, such as
the actualization of a strategy, or a justification, or the solution of a problem. Constructing consists ofintegrating previous constructs by vertical mathematization to produce a new construct.
In view of AiC essential cognitive perspective, the focus is on the students’ processes of constructionof knowledge. In the AiC approach, contextual aspects are considered to be integral factors of the learning
process. Context is regarded in a wide sense, comprising historical, physical and social context. Historicalcontext includes students’ prior learning history, physical context includes artefacts such as computers andsoftware, and social context refers to interaction with peers, teachers and others.
Design is important for AiC. This is in accord with the epistemological stance which is underlying theidea of vertical reorganization. The design is accompanied by its epistemological aspects. As a part of theAiC methodology, an effort is made to foresee students’ expected processes of construction of knowledge
and an a priori analysis of the activities is conducted.
The AiC a priori analysis consists first of assumptions about the previous mathematical knowledge ofthe students, in particular, previous constructs which have been constructed in the past and that may be
helpful in the present task. Then, the AiC a priori analysis consists of intended constructs that are required in
the given task. For each intended construct, the AiC researchers give in the a priori analysis an operational
definition. The operational definition will help the researchers in their decision if the student did express theintended construct. It will offer a criterium for evidence if the intended construct has been constructed.
Different researchers in the team perform separately their a priori analyses. Then the a priori analyses are
discussed until there is agreement between the researchers.
Like for the TDS researchers, the a priori analysis serves as a system of reference for the AiC researchers.
Comparing the a priori analysis and the a posteriori analysis, the AiC researchers note that sometimes thestudents achieve new constructs which were not expected in the AiC a priori analyses. This fact is an
important and interesting stage in the research. Sometimes, students only achieve constructs that partiallymatch a corresponding intended construct in the a priori analysis (Ron et al., 2010).
As explained in Dreyfus et al. (2015), the AiC a priori analysis is not only a list of intended constructs.
It is more a structure of intended constructs with some interactions between the different constructs. For
example, some constructs are contained in others. Some intended constructs might be a prerequisite forothers. Sometimes, possible paths of thinking are taken into account. This is relevant, for example, for apriori analyses of justification tasks. Justification is a specific case of construction of knowledge. Eachitinerary of thinking towards the justification might be in itself a kind of construction of knowledge anddifferent itineraries of thinking, each with a structure of intended constructs might appear in the a priori
analysis.

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Comparing the a priori analyses of TDS and AiC and the benefit for networking
The two theories share the same aim: to understand the epistemological nature of the episode but
different questions are asked:
AiC: What is the epistemic process of the student?
TDS: How this process is possible?
For both theories, TDS and AiC, the epistemological perspective is of importance but their a priori
analyses have a different focus. In AiC the focus of the a priori analysis is on the learner’s construction ofknowledge. The a priori analysis reveals hypotheses about constructs that might be observed during theconstruction process. For AiC, processes of abstraction are inseparable from the context in which they occur.
The notion of context is very wide in AiC. The context has many components. For example, as mentioned
earlier, the task, the computer, the teacher, the social interaction between students are considered as part ofthe context. The AiC a priori analysis with its focus on the learner’s processes of construction of knowledge
cannot take explicitly into account all the contextual factors. In a later phase, the researchers will analyze theinfluence of the context on the construction processes that were observed in the analysis of data. For example,
Kidron & Dreyfus (2010) analyzed the influence of the computer on the construction processes observed in
the analysis of data. They describe how instrumentation led to cognitive constructions and how the roles ofthe learner and the CAS intertwine. But there is an essential difference if the researcher analyzes her datataking explicitly into account in advance the contextual factors or if she first analyzes the data and theprocesses of construction of knowledge and only then she analyzes the influence of the contextual factors on
the construction processes.
For TDS, the situation is different. The focus is on didactical systems. TDS observes the entire situationand not only the student and the mathematical activity. For example, TDS is interested in relations between
systems and the teacher is an element of the system. Consequently, TDS considers already in the a priori
analysis the role of the teacher and how he extends the results of the a-didactical situation. Because of thedifferent foci between TDS and AiC, context is not theorized and treated in the same way in the different
theories. This fact has an important consequence on the differences of the a priori analyses.
AiC a posteriori analysis might be influenced by the fact that some contextual factors are not taken
explicitly into account in the a priori analysis. As a consequence, some excerpts which might add direct
knowledge in the analysis of the cognitive processes might be missed if one focuses first on the cognitive
processes and only then analyzes the influence of other parts of the context.
Kidron et al. (2014) refer to a networking case that links three theories. The issue of context is comparedand contrasted in the three theories. The analyses from the different perspectives refer to a set of data from a
video recording that show a session from the group-work of two students, during a teaching experiment on
the exponential function in secondary school.
In Kidron et al. (2014, p. 175), the authors noted that
An interesting, and also revealing, point is the fact that, in the analysis, AiC researchers focus on theautonomous work of the students, while TDS researchers pay more attention to the episode where thestudents interact with the teacher...
The a posteriori analyses of the two theories are influenced by the differences in the a priori analyses
and their different priorities in their focus of analysis. Different units of analysis are considered and as a

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consequence of the focus of analysis, as demonstrated in the a priori analysis, each theory shapes the kind ofdata that is appropriate to this focus. As pointed by Radford (2008):...it is through a methodological design that data is ﬁrst produced; then the methodology helps theresearcher to “select” some data among the data that was produced but also helps the researcher to“forget” or to leave some other data unattended.
As a consequence, the different a posteriori analyses conducted within the two theories complement
each other. Each analysis highlights a specific view which reflects the focus of research of the given theory.
AiC analysis, with its specific tools, offers a fine-grained analysis of the students constructing processes.
TDS, with its different focus, analyses the entire situation and, in particular, the interaction between theteacher and students. For example, in Kidron et al. (2014, p. 172) we read how TDS analyses the role of theteacher:
TDS complements the AiC analysis in analyzing how the teacher extends the outcomes of the
a-didactical interaction. The TDS analysis seems to start where the AiC analysis stops.
The different a priori analyses result from the different priorities of the theories with regard to the focusof analysis. Investigating these differences in the a priori analyses might lead to a better understanding of thedifferent a posteriori analyses and to the insights offered by one theory to the other one in the networking
process. I will extend this comment in the concluding remarks of the paper.

A PRIORI ANALYSES IN DIFFERENT THEORIES:CONNECTED WORKING SPACES (CWS) and ABSTRACTION in CONTEXT (AiC)
Psycharis et al. (2021) describes a research study in which students experience functions in a pluralityof settings: physical context, geometry, measures, algebra. Two frameworks are used: Connected Working
Spaces (CWS) (Minh & Lagrange, 2016) and AiC. CWS builds on the idea of “Mathematical Working
Spaces” (MWS). The MWS theory is well described in Kuzniak et al. (2016) and more recently in Kuzniak
et al. (2022).
Psycharis et al. (2021) wrote
Work in a MWS is organised around three dimensions: semiotic (symbol use, graphics, concrete objects
understood as signs); instrumental (construction using artefacts, such as geometric figure, graphs etc.)
and discursive (justification and proof using a theoretical frame of reference). CWS builds on this ideaof MWS by considering that in activities involving mathematics and other settings, students have towork in several working spaces and to coordinate the semiotic, instrumental and discursive dimensionsof these spaces.
In this specific research study, the students had to work in several working spaces and to coordinate thesemiotic, instrumental and discursive dimensions of these spaces. This is an example of networking thatbegins already in the networking of the a priori analyses. The a priori analyses are different in the sense thatthey complement each other. CWS a priori analysis identifies the different working spaces and the three
dimensions (semiotic, instrumental, discursive) in each of the working spaces as well as the opportunity forthe students to make connections between the working spaces. AiC a priori analysis relies on the structure

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Ivy Kidron

offered by CWS. CWS alone could not offer, in its a priori analysis, maximal assumptions on learning. AiC
a priori analysis is an effort in this direction, especially which constructs might be observed during theprocess of knowledge construction. In Psycharis et al. (2021) we read:CWS alone allows merely minimal assumptions regarding learning, but it provides a reliable structure
for more precise hypotheses by AiC.
In this example of networking between CWS and AiC, we observe how comparing a priori analyses
might enable to support the communication between the two different theoretical approaches. The explicit
differences between the two a priori analyses demonstrate how CWS combines well with AiC and how
connections between working spaces contribute to conceptualization.

A PRIORI ANALYSES IN DIFFERENT THEORIES:THEORY of DIDACTICAL SITUATIONS (TDS) andAPC (ACTION, PRODUCTION, and COMMUNICATION)
Artigue (2021) describes how the two theoretical frames TDS and Action, Production, andCommunication (APC) (Arzarello, 2008) were used in the thesis of Michela Maschietto (Maschietto, 2002).
Arzarello and Sabena (2014) offer a short introduction to the APC theory, especially the fact that APC
provides a frame for investigating semiotic resources in the classroom. The authors describe the importanceof gestures for communication and thinking. They underline that the main components of the APC-space are
the body, the physical world, and the cultural environment and cite Arzarello (2008, p. 162):
The APC-space is built up in the classroom as a dynamic single system, where the different components
are integrated with each other into a whole unit. The integration is a product of the interactions among
pupils, the mediation of the teacher and possibly the interactions with artifacts. The three letters A, P,
C illustrate its dynamic features, namely the fact that three main components characterize learning
mathematics: students’ actions and interactions, their productions and communication aspects.
We also read that “space” should not be considered as a physical entity, but rather in an abstract way.
Artigue describes the tensions that appear comparing the different a priori analyses offered by TDS and
APC. The theories belong to different cultures but, as in the case of TDS and AiC, for both the epistemologicaldimension is important. The main problem was how to create a DE that suits both theories, TDS and APC. I
translate freely how Artigue (2021, p. 37) points to the main source of tension between the two a priori
analyses:
In a didactic engineering consistent with APC theory, the gestures of the students expressing cognitive
constructions, the way in which these gestures will be taken up and exploited by the teacher throughsemiotic games, are essential ingredients of the cognitive dynamics of the class. They escape thepotential for anticipation and control of the trajectories of a priori analysis.
The interesting part is that, in this specific context, to overcome this source of tensions and in order tobe consistent with the Italian culture, the notion of a priori analysis was revisited, especially for what concerns
the role and interaction of the teacher in semiotic games with the students. After performing these revisions,the DE that resulted was perfectly successful for this research.

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53

CONCLUDING REMARKS
Considering each specific theory alone, we observe the role of reference of the a priori analysis while
comparing with the a posteriori analysis. Comparing the different a priori analyses offered in different
theories, we better understand the role of a priori analysis. In the present paper, the focus on comparing apriori analyses highlights the important role of a priori analyses in the networking of theories.
Kidron et al. (2018) wrote that in the last fifteen years:
Different aims in the efforts to network theories were differentiated. In some cases, the researchers wereinterested in the complementary insights that are offered when given data or an empirical phenomenon
is analysed with different theories. In other cases, the interest in the rich diversity of theories was toexplore the insights offered by one theory to the other.
This idea appeared already in Arzarello et al. (2007, pp. 1625-1626) in which examples of different
profiles of networking were presented.
Comparing a priori analyses, the aim is to explore the insights offered by each theoretical lense to theother from the very beginning. The networking of theories begins already in the stage of a priori analysis andnot only for comparing a posteriori analyses.
We wrote that the different a priori analyses result from the different priorities of the theories withregard to the focus of analysis. In the a priori analyses these priorities and their differences are made explicit.
Reflecting on the role of a priori analysis in both theories, TDS and AiC, we realize its importance andwhy it is necessary towards a better understanding of the a posteriori analysis of the collected data. For boththeories, the a priori analysis plays a role of reference while comparing the a priori and a posteriori analysis.
For TDS, it plays the role of revealing the didactic phenomena and helps to deeply understand the functioningof the “situation”. For AiC, it offers a structure of intended constructs that are required in a given task as wellas possible paths of thinking. We also realize the importance that each theory keeps the specific characteristics
of its a priori analysis. I wrote in a previous section that there is an essential difference if the researcher
analyzes her data taking into account in advance the contextual factors or if she first analyzes the data andthe processes of construction of knowledge and only then she analyzes the influence of the contextual factors
on the construction processes. This essential difference is tightly connected to the specific characteristics ofthe different a priori analyses for AiC and TDS. In Kidron et al. (2008, p. 262), we read that:In networking, we want to retain the specificity of each theoretical framework with its basic assumptions,and at the same time profit from combining the different theoretical lenses. What we aim at is todevelop meta-theoretical tools able to support the communication between different theoreticallanguages, which enable researchers to benefit from their complementarities.
Comparing a priori analyses might enable to support the communication between different theoretical
approaches:
Realizing some common points in the a priori analyses enables the beginning of a dialogue between thetheories. In the case of networking between TDS and AiC, for example, the common points in the
epistemological dimension help towards the beginning of the dialogue (as demonstrated in Kidron et al.,2014). This idea could be used for other theories and other cases of networking: Some other common points
in the a priori analyses, for example, the social dimension, might help towards the beginning of the dialogue.

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Realizing the differences in TDS and AiC a priori analyses, we better understand the choices of data (aswell as the “data which was left unattended”) that researchers of each theory select for a posteriori analyses.
Sometimes, the data which was left unattended by one theory might add direct knowledge in the analysis of
this theory and, as a consequence, some cognitive processes, for example, might be missed. The complementary
insights which are missing might be offered by means of networking theories. This situation might happen
in different cases of networking theories.
Also, for TDS and APC, the epistemological perspective, as a common point in the a priori analyses,
enables the beginning of a dialogue. In spite of the differences and tensions between the two cultures a DE
was created that suits both theories and was successful for the research study. In the case of CWS and AiC
the explicit differences in the a priori analyses led to the understanding that the a priori analysis of AiC relieson the structure offered by the a priori analysis of CWS. The two theories complement well each other, andthe networking experience begun at the stage of comparing a priori analyses.
As we noted in AiC and APC, “a priori analysis” exists outside the French context but with different
meanings in different theories.
Moreover, there are theories with which the researchers do not carry a priori analysis. Even so, eachtheoretical approach, as a research methodology, is used with task sequences that have been designed withwell-defined conceptual learning objectives in mind. Sometimes, the data required in order to do theappropriate analysis from the point of view of a specific theory might be different than the data required inanother theory. Reflecting on these differences might allow a beginning of dialogue between theories.
Acknowledgment
This research was supported by the Israel Science Foundation (grant number 1815/16).

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Ivy Kidron
Jerusalem College of Technology, Israel
Email: ivy@jct.ac.il
https://orcid.org/0000-0002-8986-0167

HIROSHIMA JOURNAL OF MATHEMATICS EDUCATION 16: 57-75, 2023

57

ETHICS IN THE MATHEMATICS CLASSROOM
Luis Radford
Laurentian University, Ontario, Canada

Abstract
Despite some recent progress, ethics remains a peripheral topic in mathematics education
research. My intention in the first part of the article is to show that ethics is omnipresent in themathematics classroom. However, ethics does not always operate in the same way. To better grasp
the nature of ethics, in the second part of the article, I discuss two ethical systems that have been
influential in Western thought (Hobbes’s and Kant’s). These systems have largely informed theunderstanding of the mathematics classroom, even if, more often than not, they remain implicit.
Then, I move to a short discussion of ethics in postmodern times and try to pinpoint what is atstake in ethics. The previous theoretical considerations pave the way to approach, in the last partof the article, ethics from an educational viewpoint. I argue that, in educational contexts, such as
the school, ethics appear framed by the way in which we understand teaching and learning. I end
the article with an outline of the communitarian oriented relational ethics articulated in the theoryof objectification—a communitarian ethics whose practice features responsibility, commitment,
and care.
Keywords: ethics, theory of objectification, Vygotsky, Spinoza, Kant, Hobbes

1. INTRODUCTION
The first question that might arise is the following: What does ethics have to do with mathematics education?
Let me present a short twofold answer.
First answer
Teaching-learning mathematics cannot avoid facing the question of the legitimation of particular forms ofknowledge and knowing that arise in the classroom. Classroom discussions usually lead to conflicting views
about what counts as mathematically valid and authentic.
Here is an example. In a Grade 5 class (10-11-year-old children) of a French-speaking school in
Sudbury, Canada, the students were invited to write a text for a student from another class explaining how tosolve linear equations. To write and solve simple equations, the students had been using an iconic semiotic
system (ISS) that comprises four signs:
⎧
⎨
⎩

,

⎫
, = , ↑⎬.
⎭In the ISS, the small rectangles represent cards; the envelope represents the unknown (as each envelope
Corresponding author: Luis Radford

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Luis Radford

contained the same unknown number of cards). The problems with which the students had been dealing
involved two individuals (e.g., Claudine and Sylvain) who each had a known number of cards and one ormore envelopes. The individuals’ total number of cards was the same.1
Some students suggested a text based on a concrete example: they used the equation=

These students divided the two sides of the equation by a vertical line (see Figure 1, left). But other studentssuggested a text without any concrete example (see Figure 1, right; a translation of the texts can be found in
the Appendix).
Which text is better? And if you were the teacher, what would you say to the students?

Figure 1. Two mathematical texts. Which one is better?

Since there are plenty of ways in which to think mathematically, taking sides or suggesting something else
involves a question of power, and since there is a question of power, there is also a question of subjecting
people to a particular image of mathematics. This is an ethical matter.
Second answer
My second answer is not about positions taken on questions of mathematical legitimacy but about relations
between people.
Teaching-learning mathematics in the school involves interaction between people. Teaching-learning
is based on relations with others, and these relations involve necessarily an ethical dimension: in classroom
interaction we have, for instance,●

relations of power and subjection,●

relations of authority and obedience, and●relations of solidarity and inclusiveness.

Here is an example. In a Grade 4 classroom (9-10-year-old children) of a French-speaking school in Ottawa,
Canada, the students were working in small groups of three or four trying to solve geometric problems. The
1 A paradigmatic example is the following: “Sylvain and Chantal have some hockey cards. Chantal has three cards andSylvain has two cards. Their mother puts some cards in three envelopes and makes sure to put the same number of cards ineach envelope. She gives one envelope to Chantal and two to Sylvain. Now the two children have the same number ofhockey cards. How many hockey cards are inside each envelope?” (Radford, 2017, p. 18). See also, Radford, Demers, andMiranda (2009).

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Ethics in the Mathematics Classroom

first problem revolved around the classical question of whether squares are rectangles. The pictures in Figure
2 provide a sample of body positions of the interaction of a group of four students. In Picture 1, Laura istalking to Sandra, the girl in front of her: “Yes, but they all have four sides.” In Picture 2, Mirna tries tocontribute to the group and says: “The squares have same umm … the same edges …” Laura turns to look at
Mirna for a short period of time; then, turns back to look at Sandra to continue their discussion. In Picture 3,
Híria (front left) tries unsuccessfully to get Sandra’s and Laura’s attention and says: “The squares haveparallel faces … because there is … Look!” In Picture 4, after recurring attempts to be heard, Mirna expresses
her frustration and utters an anguished “Ahhhhh!!!”

1

2

3

4

Figure 2. A group of Grade 4 students dealing with a geometry problem.

We are here in the presence of an ethics of exclusion. Híria and Mirna try hard to enter the conversation,
to contribute to the discussion, but they are not heard. The practice of an ethics of exclusion impedes a
genuine collective engagement in mathematics and raises an invisible, yet important, wall between us andthem.
The previous examples illustrate one of the main ideas of this article: teaching-learning is unavoidably
an ethical event—and this is so regardless of the pedagogical model that underpins it.
I would like to go a step further and contend that ethics is not only omnipresent in mathematics teaching
and learning but is also a crucial component of it for at least two reasons.
First, ethics shapes the manners in which teachers and students engage and assume (or not) certain
responsibilities in the mathematics classroom. Consequently, ethics shapes how teachers and students cometo understand mathematics and conceive of themselves as practicians of mathematics.
Second, ethics shapes the students’ and teachers’ relationships with others—for instance, in the various
manners by which the students voice (or not) their values and understandings, and how their voice is heard(or not). In this sense, ethics affects how teachers and students assert themselves as mathematical subjectivities(Radford, 2020).
Perhaps what is surprising in the claim that I am making, namely that mathematics teaching andlearning is unavoidably an ethical event, is not that ethics is omnipresent in teaching and learning but the fact
that, some exceptions put aside (e.g., Atweh, 2014; Atweh & Brady, 2009; Boylan, 2016; Dubbs, 2020;
Ernest, 2009; Maheux & Proulx, 2017; Neyland, 2004; Maheux & Roth, 2014; Roth, 2013; Silva D’Ambrosio

60Luis Radford& Espasandin Lopes, 2015; Walshaw, 2013), we have not been able to notice this presence before—or atleast not with the importance it deserves. I would like to suggest that the root of that omission is entrenchedin the long-standing conception of mathematics education as a technical process, a conception that reduces
mathematics education to a matter of acquiring knowledge, making the question of being and becoming
peripheral aspects of teaching and learning. With this same technicist stroke, we erase from view the fact thatthe mathematics classroom is not only a site of knowledge production but also a site where subjectivities areproduced every day.
The importance of ethics in an encompassing account of learning leads us to ask the question of thekind of ethics that we, implicitly or explicitly, nurture in classrooms and research. This question seems to beimportant in discussions on the diversity of theories in mathematics education; considering theories from the
ethical standpoint they convey (even if they do so only obliquely) may shed new light on the differences,similarities, and complementarities between theories.
In an attempt to better grasp the nature of ethics, in Section 2, I discuss two ethical systems that havebeen influential in Western thought (Hobbes’s and Kant’s ethical systems). These systems have largely
informed the understanding of the mathematics classroom, even if, more often than not, they have remained
implied. Then, in Sections 3 and 4, I discuss what is at stake in ethics. Intended always as a means to bettergrasp the nature of ethics, Section 5 revolves around some ideas of ethics in postmodern times. The previous
theoretical considerations pave the way to approach, in Section 6, ethics from an educational viewpoint. I
argue that, in educational contexts, such as the school, ethics appear framed by the way in which weunderstand teaching and learning. I end the article with an outline of the communitarian oriented relational
ethics articulated in the theory of objectification (Radford, 2019, 2021a)—a communitarian ethics whose
practice features responsibility, commitment, and care.

2. HOBBES AND KANTAll ethical perspectives are directly tied to conceptions of self and the social world. Try to think of what self
is and you will quickly realize that it is entangled with how we perceive ourselves and perceive others. This
perceiving of the other involves social relations, some to keep, some to discard. Our concepts of self areindeed inseparable from the landscape of ethical issues, “from how one ought to be” (Taylor, 1989, p. 112).
These ethical issues change from culture to culture and within a same culture through spans of time. Thus,
from the end of the Middle Ages and the dawn of Renaissance, when European feudal structures were being
shaken by the rise of mercantile capitalism and the ascending power of the concomitant new economic andcommercial relations, the social world appeared more and more as an aggregate of monadic individuals(Elias, 1991). On the social plane, life became increasingly regulated by agreements or contracts between
people (Le Goff, 1956; Jeannin, 1957). In the 17th century, in his Philosophical Rudiments Concerning
Government and Society, Thomas Hobbes conceives of these contracts as means to regulate human conduct
and to avoid aggression against each other. In his account, these contracts bring individuals out of the stateof nature, which “is nothing else but a mere war of all against all” (Hobbes, 1841, p. xvii). These contracts

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elevate the individuals to civil society; they embody personal aspirations and interests while trying tosafeguard each other’s security. Ethics appears here as a contractual mechanism to preserve the social orderwhile affirming the monads’ interests in a way that reflects the underpinnings of the emergent bourgeois
ideology.
One century later, when Kant articulated his ethical system, capitalism had expanded and become more
systematic. Kant’s society, of course, is still perceived as an ensemble of monadic individuals, but with anincreased complexity out of which the modern concept of self finds a more precise shape.
Kant assumed that individuals are by nature provided with a same reason—a legislative Reason. He
also assumed that the individuals’ reason moves in a world of universal laws that can define the social andmoral orders. From those assumptions Kant formulated his famous categorical imperative: “act only inaccordance with that maxim through which you can at the same time will that it become a universal law”
(Kant, 2006, p. 31, original emphasis).
As we can see, the categorical imperative answers the question: What should I do? Through it, moral
content becomes specified not in terms of social ends that the rule would help to attain or achieve. Moralcontent can only be derived from the form of the rule as expressed and appraised by the individual.
MacIntyre (1966, p. 209) notes that “The eighteenth-century individualist sees the good as theexpression of his feelings or the mandate of his individual reason.” The categorical imperative is an attempt“by the individual to supply his own morality, and at one and the same time, to claim for it a genuine
universality” (p. 208). As I argued in (Radford, 2021a), the categorical imperative makes the Other appear asa reflection of the self in a mirror through which the self can see how the actions of the Other might affectitself. It assumes a universal human subject and a universal reason that needs to be understood as a response
to “the emergence of a formal law in Europe at that time, the state homogenization that was being imposed[on society], [and] the development of an instrumental rationality” (Bohy-Bunel, 2022, pp. 23-24). This
universal human reason is nonetheless an exclusive ethnocentric reason in its most intimate nature. As BohyBunel remarks, the Enlightened modern bourgeois individual “arrogates to himself all possible humanity”(p. 24). This individual is
the Westerner whose social, classist, patriarchal and colonial project of domination becomes clearer inthe modern age . . . Its “science”, which ends up fitting perfectly into this project, postulates that it
relies on some universal “human reason,” precisely in order to better assign to non-humanity those
whose activity and being must be submitted to a total managerial control. (Bohy-Bunel, 2022, p. 25;
emphasis in the original)
It is hence under the assumption of one and only one universal reason, the Enlightened managerial
reason, that Kant can think that he solves the problem of the tension between the individual and the socialworld and the tension of human action in that world.

3. WHAT IS AT STAKE
What is hence at stake in conceptions about ethics? The previous discussion provides us with a hint: what is
at stake is the social meaning of choices about human conduct. This social meaning refracts, on our dealings

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with others, the forms of life found in the social world. In the case of Hobbes and Kant, the forms of life were
determined to a large extent by the capitalism of their time, which became the “dominant reality in thehistorical life of European societies” (Foufas, 2020, p. 24), that is capitalism in its commodity form.
Confronted with a social understood as an assemblage of monads, Hobbes found in the contractual
mechanisms of capitalists the prototype of a juridical form where individuals could negotiate and combine
their own interests. Through contractual mechanisms individuals became subject to a new entity, an assembly,
through which the multitude becomes a state or civitas. Pressured by the needs of a more sophisticated socialand economic organization and the emergence of a formal law, Kant did not need to have recourse tosomething extraneous to the individuals, as Hobbes’s did through the idea of assembly, to which theindividuals relinquished the control of their own actions. Kant did not need to do something like that, as heassumed, as we just saw in the previous section, that there was a universal reason legislating choices abouthuman behaviour.
It might not come as a surprise that the growth and implementation of ethics during the 20th century
unfolded torn by two opposite forces that we find already in play in Kant’s categorical imperative: on the one
hand, there was an emphasis on the individual (we saw that the categorical imperative comes from and isenunciated by the individual). This is the view of the existentialist ethics (Sartre’s ethics, for example) where,
standing in the debris of WWII, authentic existence was “to be found only in a self-conscious awareness ofan absolute freedom of choice” (MacIntyre, 1966, p. 269). On the other hand, there was an emphasis on theuniversal validity of the law, without assuming, however, that the law emanates from the individual itself.
The ethical self should be educated through a series of prescriptions. This is what has been called“prescriptionist” ethics; that is, the ethics practiced in particular by the legislators facing the problem ofgoverning the masses. We find modern prescriptionist ethics at work in post-revolutionary France. For
example, in 1833, French Minister of Education François Guizot explains that
The great problem of modern societies is the government of minds… It is not only for the commune
and in a purely local interest that the law wants all the French to acquire, if it is possible, the knowledge
indispensable to the social life, and without which the intelligence languishes and sometimes becomes
stultified... This is because freedom is only assured and regular among people enlightened enough tolisten in all circumstances to the voice of reason. (Guizot, cited in Mayeur, 2004, p. 335)
Thus “modern legislators and modern thinkers alike felt that morality, rather than being a ‘natural trait’of human life, is something that needs to be designed and injected into human conduct” (Bauman, 1993, p.
6). The result was that, in the footsteps of Kant, modern ethics was mainly understood as deontology; that is,
as duty and obligation driven by general rational laws to be applied regardless of the context. At the practicallevel, and with the push of modernity, this deontology was supplemented with a utilitarian outlook of theworld. In the contemporary educational context, Neyland pictures the problem as follows:
Recent modernist reforms . . . have resulted in teachers increasingly being represented as objects rather
than as subjects in policy discourse; professionalism has been replaced by accountability, andcollegiality by competition and surveillance; initiative, creativity and teacher-led innovation have been
constrained; teaching has become technicized, and learning experiences impoverished. (Neyland,
2004, p. 62)
Here, as Neyland suggests, ethics amounts to proceduralism; that is, the implementation of the dictatesEthics in the Mathematics Classroom

63of the educational apparatus as how teachers and students should behave and what they have to accomplish.
Postmodernist thinkers have quarrelled with this proceduralist conception of ethics. What bothers
many of them is not the subjective aspect involved in ethical contexts. They retain the Kantian idea that
morality does not come from outside the subject but from within. Bauman, for example, argues that “Themoral call is thoroughly personal; it appeals to my responsibility” (1993, p. 60). What bothers these
postmodernists is the deontological dimension. They have reacted suggesting a non-deontological ethics, anethics that is context-sensitive and hence impossible to frame through a priori laws or principles. As Baumanputs it, “Moral phenomena are inherently ‘non-rational’… They are not regular, repetitive, monotonous andpredictable in a way that would allow them to be represented as rule-guided (1993, p. 11; emphasis in theoriginal) . . . morality is endemically and irredeemably non-rational” (1993, p. 60; emphasis in the original).
Now, how can such an ethics possibly work if postmodernist thinkers themselves have been very
successful in showing that the individuals are subtly shaped by a multitude of social mechanisms ofgovernance that make them think and act in specified ways? How can we make sure that the ideas with which
we ponder a delicate ethical situation are not those prescribed by the system even if they do not appear in alaw form? In other words, how can we make sure that it is me and not the system through its webs of power
and knowledge that is talking through me?
Foucault, the last modern Kantian and one of the first postmodern thinkers, raises the question as
follows: How can the subject be the locus of meaning, feeling, intentionality, and ethical decisions if thissubject must talk, feel, and intend through thoughts and words that are not its own?
Can I, in fact, say that I am this language I speak, into which my thought insinuates itself to the pointof finding in it the system of all its own possibilities, yet which exists only in the weight of sedimentations[that] my thought will never be capable of actualizing altogether? (Foucault, 1966, p. 335)
Rothenberg articulates this problem asking how the individuals produced within the social modes ofpower, “subsisting as nothing other than the intersection of various discursive determinants” (2010, p. 26),
could be able to act independently of ideological operations. These individuals, Rothenberg argues, “have totranscend their ideological determinations somehow and lift the veil of misrecognition, even though they
themselves are nothing other than the expression of ideology and thoroughly blinded by it” (Rothenberg,
2010, p. 25).
What is at stake, I argued before, is the social meaning of choices about human conduct. We now see
that, going a step further, what is at stake at a deeper level, is our conceptions of human nature. Are humans
capable of transcending their ideological determinations or are they trapped in them forever?

4. SELF AS LACK AND SELF AS EXCESSHistorically speaking, feminist, multiculturalist, psychoanalytic, and dialectical materialist thinkers, andscholars in related fields have contended that human nature is such that ideological determinations can beovercome. We are not trapped. The “trapped forever” position, they argue, is based on a conception of theself as lack. It features a self that lacks wholeness or plenitude, subjected forever to the limits of the given
structures and confining discourses. They counter this conception of self with a conception of self as excess.

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In feminist theory the subject is conceived of as having the power “to reflect on the social discourse andchallenge its determinations” (Alcoff, 1988, p. 417). Following a Hegelian thread, for Butler (1999), thesubversion of the subject is possible because all acts of signification not only restrict the subject’s actions but
are, at the same time, in their enactment, always located within the possibility of a variation in the “alternative
domains of cultural intelligibility” (p. 185).
Rothenberg (2010) approaches the problem from a psychoanalytic perspective and claims that meanings
have the peculiar property of being impossible to be totally controlled. Although individuals are socially
produced, the producing causes leave “a remainder or [meaning] indeterminacy, so that every subject bears
some unspecifiable excess within the social field. Every subject is an ‘excessive’ subject [and] the excess isineradicable” (2010, p. 10).
Drawing on Lacan, Žižek refers to a fundamental characteristic of the Symbolic: its “openness.”2
The cause of this irreducible “openness” of the Symbolic is not its excessive complexity (we neverknow in what decentered context our statement will be inscribed), but the much more refined, properly
dialectical impossibility of taking into account the way our own intervention will transform the field.
(Žižek, 2010a, pp. ix-x)
Through our interventions in the world, Žižek contends, we affect our reality, which means that wechange and affect our social determinations. These are not outside an undisturbed reality, nor is our free will.
When we feel thwarted in our freedom by the constraining pressure of external reality, there must be
something in us, some desire or striving, which is thus thwarted, but where should this striving come if
not from this same reality? Our “free will” does not then in some mysterious way “disturb the natural
course of things,” it is part and parcel of this course. (Žižek , 2010a, p. xii)
Dialectical materialist thinkers (e.g., Fischbach, 2014; Macherey, 2008) have drawn on Marx’s (1998)
work, particularly on the idea of transformative praxis quickly sketched in Thesis 3 of Theses on Feuerbach.
Following Marx’s ideas, in dialectical materialism the subject is featured as one that, while being produced
by its circumstances, has, inversely, the power to reflect on these circumstances and to transform them (foran enlightening analysis see Macherey (2008) and Fischbach (2015a)). Following Marx, what contemporary
dialectical materialists add to the agentic conceptions of self mentioned above (i.e., the self as excess) is thatsubversion is accomplished in praxis, with others (Freire, 1998). Subversive praxis is the arena of theemergence of a new form of social consciousness. For consciousness is not only a refraction of reality.
Consciousness, through its varying layers of depth, is a concrete relation that, given our biography andcultural background, propels us towards the world and leads us to act on/in it and transform it (Clot, 2015).
In short, what dialectical materialists add to the agentic conception of self mentioned before is that thesubversion of self is not the result of the deeds of a solitary being: it is a political and social project strictlytied to the transformation of the individuals’ consciousness. There is still hope, then, that in our Grade 4
example, Mirna and her twin sister, Híria, will be heard. However, for this to occur, there must be atransformation of circumstances. The classroom culture must be transformed. This transformation requires a
new praxis, a classroom praxis, out of which a new form of social consciousness can emerge.
To recap, in each one of these accounts, although for different reasons, there are always possibilities to2 The Symbolic is the sphere of the Law. Without submitting ourselves to the Symbolic, we would not be able to function insociety. The Symbolic includes grammatical rules, social norms, and unconscious prohibitions (see Žižek, 2007).

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interrupt the quotidian train of our actions and thinking. It is at this point that Scott’s (1990) The Question of
Ethics as a question of interruption of habits and values acquires its whole sense. For Scott, The Question of
Ethics indicates our capacity to bring forward “an interruption in which the definitive values that govern
thought and everyday action lose their power and authority” (p. 4). This interruption makes sense precisely
because of our cultural-historical agentic nature (i.e., our power to sublate and surpass the cultural andhistorical possibilities on which we draw when we re-act to, and re-enact, the world).

5. ETHICS IN POSTMODERN TIMES
5.1 Ambiguity
As stated before, one of the main features of postmodern ethics is to consider ethics as having an intrinsicallyambiguous nature. To be ambiguous means that ethics is neither something contractual (like teachers do this,students do that), nor is it something that works on the basis of rules and abstract principles (like do yourhomework!). Valero and Jørgensen (2021) give an example of the ambiguous nature of ethics in contemporary
ethical thought. A student submits an essay that, according to a plagiarism software, shows 21.5% ofcompatibility between a student’s text and a referred text. However, the institutional maximum ofcompatibility is 20%. For the institution, plagiarism is
unacceptable and grounds for expulsion. The teacher knows the student and knows that there is a caseof terminal cancer in the immediate family that has affected the student’s academic ability … For theteacher, ensuring access to education, particularly for poor people, is a fundamental part of his
understanding of fairness in education. It is part of his social responsibility and commitment as ateacher. What does the teacher decide to do: follow institutional norms or his personal norms? (Valero
and Jørgensen, 2021, p. 271)
As the authors remark, “The issue is to enter into a deep reflection to generate an ethical rule that is
applicable in this situation” (Valero and Jørgensen, 2021, p. 272; my emphasis).
So, to say that ethics is intrinsically ambiguous means that our acts and relations to others do not haveone obvious meaning. They are context sensitive. Ethics is seen here as a context-sensitive dynamic andopen-ended relational stance that is continuously materialized and assessed as teachers and students engage
with each other.
5.2 Ethics as thoroughly personal
Another main feature of postmodern ethics, or at least the one we find articulated in Bauman’s work, is, asmentioned before, that ethics is something “thoroughly personal,” something that “appeals to myresponsibility” (1993, p. 60).
Bauman continues here the Western modern tradition that posits the individual as the origin of
intentionality (Husserl, 1982) and ethical concerns (Kant, 2006). This feature of postmodernism is anything
but new. It is easily understood once one realizes that postmodernity “one may say, is modernity without
illusions (the obverse of which is that modernity is postmodernity refusing to accept its own truth)” (Bauman,
1993, p. 32). In other words, postmodernity is modernity disenchanted; modernity removed of the illusions

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Luis Radfordof the universal abstract reason of the Enlightenment and its overarching grand narratives. Of course,
postmodernism, even if it continues the modern tradition of the “new,” the “novel,” the “creative,” is more
complex than this (Macherey, 2006). Capital (understood as the general system of production of theindividuals’ lives, the system of production that characterized modernity) has not disappeared. It continues
to be the mechanism through which human life is shaped and quotidianly produced. But capital has beenrefined and expanded: it has become more complex, more nuanced, and much more oppressive than inmodern times. Bauman, hence, is right in seeing in postmodernism the story of an unending end; it is still amodernism, but one that seeks with relentless vigour to get rid of the encumbrances of the Symbolic (the“law” in Žizek’s sense) that has continuously threatened the autonomy of the modern subject. In this sense,
postmodernist ethics is still (or again) a subjective ethics, this time pushed to its radical limits—with all thecontradictions that such a push entails (Lipovetsky, 1989; Žizek, 2010b). If postmodern ethics recognizes therole of the Other in some qualified Levinas’s sense (as Bauman does), it does so within a monadic
understanding of the social featured by modernity, that is, as an arena of “competitive struggle” whose
pinnacle is “individual choice” (Bauman, 1993, p. 46).

6. ETHICS IN THE THEORY OF OBJECTIFICATION
6.1 Reconceptualizing the social
Ethics in the theory of objectification is neither modern nor postmodern. Although ethics is understood in a
non-deontological, context-sensitive manner, ethics is not considered as coming from within. It is not
something “thoroughly personal.” The reason is not because the individual is exempt of all moral
responsibility. The reason is that it understands the social not as an agglomeration of monads that, once
constituted as humans, enter the social realm. This monadic conception of the individual, where the social isseen as a derivative of the individual’s life, has been one of the chief marks of modernity and postmodernity,from where we have inherited the “acids of individualism” (MacIntyre, 1966, p. 266).
The conception of the social that underpins the theory of objectification is such that there is no
separation between the individual and the social. The social is always appearing within the individual andbetween individuals. One cannot exist without the other. The social is neither a container, nor a flat surface
where we live our life. The social is made up of relations (rapports) between individuals. These relations are
not merely connectors; they are susceptible of modifying the individuals they put into relations.
To the extent that social relations express and manifest in their enactment political, economic, culturaland social conceptions about the world, ourselves and others, they affirm certain forms of dealing withothers. This is what we see in our discussion of Kant’s and Hobbes’s ethical systems: each one of themsuggests a social form of self-other dealings (Kant’s ethics turned to a universal reason legislating thetransactions between individuals; Hobbes’s one turned to transactions of a contractual nature). Ethics isfound precisely in these forms, which are nothing else but forms of alterity. Ethics, then, appears as the formof Self-and-Other.
If the social is understood as made up of social relations in movement, what would then be the kind of
relations that are targeted in the theory of objectification?

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6.2 Ethics and learning
To answer this question, we need first to make explicit the educational project in which this theory is
subsumed. The theory of objectification inscribes itself in a Freirean and Vygotskian educational project that
posits the goal of Mathematics Education as a political, societal, historical, and cultural endeavour aimed at
the dialectical creation of reflexive and ethical subjects who critically position themselves in historically andculturally constituted mathematical discourses and practices, and who ponder new possibilities of action andthinking.
As in other educational theories and pedagogical models, ethics in the theory of objectification appears
in the relations between students and teachers. Ethics, understood as the form of Self-and-Other that manifests
itself in our dealings with other individuals, is directly linked to the understanding of how learning occurs.
Let us consider two examples.
Think of the theory of transmissive instruction. Learning is conceived of as the assimilation, throughpractice and repetition, of knowledge that the teacher possesses. This theory positions teachers as knowers
and the students as lacking knowledge. The ethics of the transmissive instruction, manifesting itself in theform of alterity that it promotes, reflects, but also operationalizes. It manifests itself concretely in thealienating relations of power and subjection it fosters and maintains, relations that are thematized along theobedient lines of superior/inferior, potent/impotent, knower/ignorant, authority/vassal (Radford & Lasprilla
Herrera, 2020).
Now think of constructivism. In contradistinction to the theory of transmissive instruction,
constructivism is based on an ethics that stresses the freedom of the student: since knowledge is conceivedof as what results from the autonomous deeds of the student, and learning is the very process of the student’s
subjective construction of knowledge, teachers and students are positioned otherwise: the student’s freedom
and autonomy configure the constructivist ethical space (Radford, 2012). It alienates teachers and students,
but in a way that differs from how transmissive instruction does (Radford, 2021a).
Learning in the theory of objectification is neither about transmitting knowledge (as in direct ortransmissive teaching), nor is it about the students’ constructing their own knowledge (as in constructivist
approaches). In the theory of objectification, learning is conceived of as a collective and truly social embodied
and material process through which students critically encounter culturally and historically constituted waysof thinking mathematically. This encounter happens in what the theory terms joint labour (Radford, 2021a,
2021b).
6.3 Joint labour
Joint labour is a sensuous, practical, material activity—activity understood as driven by collective concerns.
The German and Russian languages have a specific term for this type of activity: Tätigkeit and deyatel’nost’,
respectively. Activity in this sense is opposed to activity as being merely busy with something (as in watchingTV). Again, the German and Russian languages have a specific term for this other type of activity: Aktivität
or aktivnost’. Unfortunately, in the translation into English (and several other languages), the distinction islost and both types of activity are rendered as activity.
In the case of the theory of transmissive instruction, classroom activity is not oriented towards thesatisfaction of collective needs. This activity corresponds hence to Aktivität or aktivnost’. In joint labour, by68Luis Radford

contrast, students and teachers work hand in hand to produce something together, what Hegel termed “acommon work,” in our case, mathematics. It is this sense of labouring together (as opposed to simply
interacting or exchanging with others) that makes joint labour a truly social activity and learning a collective
process.
6.4 Transforming ethics
Classroom research has shown us, however, that for learning to become a truly collective process, radical
changes in the classroom culture might need to occur. Often, drawing on experiences shaped by transmissive
instruction, the students conceptualize the teacher as the possessor of knowledge and power, and conceptualize
themselves as submissive to the teacher and her knowledge, even when the teacher tries to conceptualize
herself differently and encourages the students to learn collectively and organizes the classroom into small
groups. Often, the students configure small, enclosed groups and erect aggressive or exclusive antagonistic
barriers between their group and other groups—they resort to what we may term a clique ethics or gang
ethics (Radford & Lasprilla Herrera, 2020), which is also what we see in the example from the Grade 4classroom briefly mentioned in the Introduction.
Of course, moving towards a more encompassing, democratic, and inclusive ethics cannot be achievedby imposing new social forms of conduct. “It is not obedience to someone or obedience to something, but
the free adoption of those patterns of behavior which will vouchsafe the consonance of all of behavior”(Vygotsky, 1997, p. 233). The educational problem around ethics becomes the problem of the creation ofclassroom conditions for new ethical relations (new forms of alterity) to emerge and to be collectively
pondered and discussed against the always contested background of culture and history. The process of
emergence of such an ethics has to overcome the dynamic power of capital, which constantly tries toappropriate the social “in order to put it at the service of its own development.” It is hence “a question of notletting ourselves be dispossessed: it is necessary to maintain and to develop [the spring of the production ofsocial life] against capital” (Fischbach, 2015b, p. 71). This can only be done by developing a form of
democracy that would “increase our power to generate ourselves our own social life” (p. 71).
6.5 A communitarian ethics
So, what are the new ethical relations that we strive to nurture in the theory of objectification? We focus ona classroom mathematics practice featuring a communitarian ethics which is consonant with the conceptual
historical-cultural bases of the theory and its conception of learning as a collective process.
Communitarian ethics is based on what we call three ethical vectors. They define a dynamic ethicalspace. Rather than be seen as fixed terms, we see them as repères; that is, reference points against which togauge our actions. These vectors work as “primary orientation of the ethical self” and should not be seen as“a modern uniform code”; they are “shared ethical ideas, priorities and principles that are subject to ongoingand ‘antagonistic’ negotiation’” (Neyland, 2004, p. 57). These vectors (or “virtues” to use MacIntyre’s term)
are responsibility, commitment, and care. It is here where we resort to the construct of “voice”—not voice ina linguistic sense exactly; rather we resort to voice as something that brings in the notion of difference andthe primacy of the political (Giroux, 2005). Finding one’s voice or having a voice is “moving from silence
into speech,” it is “a gesture of defiance that heals, that makes new life and new growth possible” (bell hooks,

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2015, p. 29), something that “assumes a primacy in talk, discourse, writing, and action” (p. 33).
Coming back to our Figure 1, when Mirna utters an anguished “Ahhhhh!!!” and moves her right hand
towards the two other girls who are not listening, she is moving into speech to express her frustration about
not being counted and heard. Her voice (which is much more than what she discursively says, as it also says
things in her body posture, facial expression, pitch, gesture) opens up new possibilities for action (for herself
and the other teammates). Mirna’s embodied utterance is, indeed, a call to the Other.
6.5.1 Responsibility
The call now must be responded to, and it is responded to within a certain node of social relations that tie thestudents together. Whatever path the teammates’ response takes, it is cast in a general ethical attribute that
Lévinas calls responsibility. For Lévinas, responsibility is “the essential, primary, and fundamental structureof subjectivity … [where] the very node of the subjective is knotted” (1982, p. 101). Since all educational
theories put into motion a certain ethics—for, as mentioned before, ethics is the substrate and the form thatis continuously materialized in our dealings with the Other—responsibility is a common denominator of allof them. Yet, the meaning of responsibility is not the same. In the theory of direct instruction, the students’
responsibility consists in assuming a submissive role vis-à-vis the teacher. The teacher’s responsibility
consists in subjecting the students through the power of knowledge. In the case of the theory of didacticalsituations (Brousseau, 2005), responsibility follows Hobbes’s contractual outlook and appears as a didacticalcontract; that is, a reciprocal distribution of duties. In the case of the theory of objectification, responsibility
means living and acting with and for others; it means to respond to the call of others as they are on their own
terms: in their “existence, in [their] being-for-other[s] . . . as free being[s]” (Hegel, 1978, p. 57).
6.5.2 Commitment
Commitment is both the promise and its realization of doing everything possible to work side by side withothers in the course of our joint labour (e.g., trying to understand the process being followed to solve a
problem, trying to contribute to the classroom common work).
The classical utterance “I do not understand” of classrooms operating within the transmissive instruction
model often conveys the stance of a de-responsabilization; it intimates that it is the teacher who has not done
his work properly. Through this utterance, the student invests the teacher with the role of patriarch ofknowledge and requests the teacher to deliver knowledge. However, if teachers and students understand thatknowledge is produced collectively in the classroom, the same utterance would mean something like: “I donot understand, but let’s work together so perhaps things will become clearer.” Commitment is this resolution
to engage in joint labour, to participate in the creation of the classroom common work (oeuvre commune),
which can be establishing collectively how equations can be solved or theorems proved, for example.
6.5.3 Care for others
Far from being an act of condescending or simply caring for someone, the care for others is a relational
involvement entailing the attention to, and recognition of, others and their material and spiritual needs.
Although caring for the Other opens up the possibility of seeing ourselves in the Other, of recognizing ourvulnerability in the vulnerability of the Other, the importance of caring for the Other is to go beyond ourselves

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Luis Radford

and to be dragged powerfully into the world, to position ourselves there, with-the-Other.
To understand Mirna’s “Ahhhhh!!!” within an ethical practice of responsibility, commitment, and care,
we need to broaden our conception of language and recognize (in its Hegelian sense) this painful expression
as voice; that is, as something where, as Lévinas suggests, the saying (that which I want so much to utter but
I will never be able to do) moves beyond the totalizing enclosure of the said (that which I manage to say) andbecomes rather the possibility of openness to the other (Radford, 2021a). In this conception of voice, power
does not disappear since power is not a thing, but something imbricated in our relations to others. What we
can expect in the transformative movement towards a communitarian ethics is that, through conscious,
reflective, and critical stance, power in the classroom goes beyond its own subjecting mechanisms of social
order and becomes rather something fluid, dynamic, to be exercised with responsibility for the Other.

7. CONCLUDING REMARKSIn the first part of this article, I argued that mathematics teaching and learning is unavoidably an ethicalevent. There are at least two reasons for this. The first reason has to do with the fact that teaching and learningalways involve interaction between teachers and students and this interaction is based on relations (relationsof power and subjection, relations of exclusion, or relations of solidarity, inclusivity, etc.). The second reason
has to do with what is to be learned, with knowledge. It has to do with the legitimation of particular forms ofknowledge and knowing. Teachers are confronted in their everyday practice with a variety of ways thatstudents bring about how to solve mathematical problems. Legitimation appears here as what counts as true,as mathematically right, sound, worth pursuing, etc. I presented an example in which students produced two
different mathematical texts explaining how to solve linear equations. The question revolved around theepistemological role of examples: should the text include an example? Should it be rather general? What text
is better? As soon as we move beyond Eurocentric views of mathematics (D’Ambrosio, 2006; Powell &Frankenstein, 1997), the question of legitimation acquires its full sense.
In the first part of the article, I complained that despite its omnipresent nature in the mathematics
classroom, ethics still remains largely unexplored (although it is gaining some traction, as we can see in theNo. 38 (December 2021) issue of the Philosophy of Mathematics Education Journal, edited by Paul Ernest).
To better understand conceptions of ethics, in the second part of the article, I dealt with two ethicalsystems—Hobbes’s and Kant’s—that have been influential in Western thought and have informed theunderstanding of the mathematics classroom, even if, more often than not, those ethical systems haveremained implicit (for instance, Kant’s subjectivist ethical stance has influenced constructivism; Hobbes’s
contractualism has influenced transmissive learning, and both Kant and Hobbes have influenced the theoryof didactical situations).
In the third part, I moved to a discussion of what is at stake in ethics. We saw that, on a first level, theanswer lies in the social meaning of choices about human conduct. On a second and deeper level, the answer
lies in our conception of human nature.
The discussion of ethics in postmodern times, intended always as a means to better grasp the nature of
ethics, allowed us to see that, on the one hand, the postmodern project breaks with the modern project, while,Ethics in the Mathematics Classroom

71

on the other hand, continues it. The postmodern project breaks with the modern one in refusing to resort to auniversal reason as the basis of the ethical self. At the same time, it continues the modern project informulating a subjective ethics; that is, an ethics that is at the end of the day “thoroughly personal” (Bauman,
1993, p.60). The postmodern understanding of ethics still rests on the modern conception of the social as anaggregate of individuals, monadic self-constituted agents.
The previous theoretical considerations paved the way to approach, in the last part of the article, ethics
from an educational viewpoint. I outlined the ethics in a Vygotskian theory of teaching and learning, thetheory of objectification (Radford, 2021a). To do so, drawing on the work of Fischbach (2015a, 2015b), I
started offering a reconceptualization of the social as made up of social relations, which are much more thansimple links between individuals. So, instead of being a mere background, the social appears to be a dynamic
totality in perpetual movement and transformation. The relations of which the social is made up express
conceptions about ourselves and others. They are always tinged with political and cultural valences. These
considerations led me to suggest that ethics can be considered as the form of alterity (the form of Self-andOther) that manifests in our dealings with other individuals. I argued that, in educational contexts, such as
the school, ethics appear framed by the way in which we understand teaching and learning.
Against this background, I outlined the communitarian oriented relational ethics that is articulated in
the theory of objectification—a communitarian ethics. This ethics follows Spinoza in stressing the body’s
power of acting in the world (agendi potentia; Spinoza, 1989, p. 210). The communitarian ethics also follows
Lévinas’s (1982) work which radicalizes previous ethical systems in acknowledging that our actions anddeeds are always modulated by the presence of the Other—a presence that comes to us in a sentient andfleshy manner: through the proximity of our bodies. In this proximity our conceptual epistemologicalcategories and mechanisms are put on hold, and we encounter the Other as is. “The Other is appreciatedprecisely as Other, in her radical alterity and irreducible singularity, only when thought renounces its
totalitarian hubris and learns to think of the Other on her own terms . . . ‘beyond essence’” (Min, 1998, pp.
573–574; emphasis in the original). The self appears here conceived not as the prerequisite of existence of
ethical relations. The self is rather seen as the result of those relations.
In conceiving of ethics as the form of alterity, the focus turns not to moral precepts but rather to thefluid and content-dependent relationships between subjects as they appear in the immediacy and banality of
everyday life. In this view, ethics is continuously materialized in praxis out of a myriad of possibilities, for,in this view, the individual appears as “full of unrealized possibilities every minute” (Vygotski, 2003, p. 76).
If we come back to Figure 2, we see that Mirna’s teammates could have opted for other actions. The
materialization of the students’ actions reflects their understanding of the context (consciousness) and themanner in which the context is lived through their unfolding collective affective experience (emotions), as
well as the relational stances (ethics) they (consciously or not) adopt towards each other. Through theprevious classroom examples, we see that embodied, emotional, contextual, cultural-historical action isrelated to ethical postures that students assume and show in practice.
What is specific to the communitarian ethics of the theory of objectification is that, braking with thesubjectivist stances of modern and postmodern ethics, its practice features responsibility, commitment, andcare. This communitarian ethics provides reference points for pedagogical actions in the classroom, where
teachers and students explore together new critical spaces that promote engagement, inclusiveness, debate,

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Luis Radford

and respect (for some classroom examples, see Radford, 2021, 2021b). Underneath the communitarian ethics
lies the recognition that our historical, cultural, and material origin embeds and refracts dynamic andantagonistic visions and conceptions of the world and of what a good life can mean. Although impossible toposit and describe a priori, the good life and the common good are understood here as ideas (idéalités), more
specifically, as “visées éthiques” (aims), generated in a joint unending project created itself within the
dynamic that produces the social relations, a project that will always be paved with tensions, contradictions,and difficulties. “Let us call ‘ethical aim’ (visée éthique) the aim of the ‘good life’ with and for others in justinstitutions” (Ricœur, 1990, p. 202). It is the vitality of contradictions that gives substance to social human
life, always changing, always challenged, and that makes ethics to always unfold in a context of struggle.

ACKNOWLEDGMENTS
This article is a result of a research program funded by the Social Sciences and Humanities Research Councilof Canada and Laurentian University Research Fund. The article was presented at the TSG 57 (Diversity oftheories in mathematics education) of ICME 14. A previous version of it appeared in Radford (2021c).

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Ethics in the Mathematics Classroom

APPENDIX
Translation of the texts shown in Figure 1.
First text:
Step 1: Count all cards and envelopes in the equation.
Step 2: Take the number with fewer envelopes and remove all the envelopes. Ex.

Step 3: Same thing for the cards. Ex.
Step 4: Give an answer ex.

Second text:
To write an equation you have to:
1. Read the problem.
2. Take the smaller amount of card[s] on one side and cross it out.
3. Cross out the same amount of cards on the other side.
4. Take the smaller amount of envelopes on one board and cross it out.
5. Cross out the same amount of envelopes on the other side.
6. Circle the remaining cards and envelopes.
7. Count the remaining cards and envelopes.
8. Divide the cards into each envelope if you have more than one envelope remaining.

Luis Radford
Laurentian University, Ontario, Canada
Email: lradford@laurentian.ca
https://orcid.org/0000-0001-6062-0605

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HIROSHIMA JOURNAL OF MATHEMATICS EDUCATION 16: 77-94, 2023

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NETWORKING PRAXEOLOGIES AND THEORETICAL GRAIN SIZESIN MATHEMATICS EDUCATION: CULTURAL ISSUES ILLUSTRATEDBY THREE EXAMPLES FROM THE JAPANESE RESEARCH CONTEXTYusuke Shinno1 and Tatsuya Mizoguchi2
1

Hiroshima University, Japan, 2Tottori University, Japan

Abstract
This study aims to identify the theoretical underpinnings of researchers’ work on networking
theories in terms of the Anthropological Theory of the Didactic. The study is based on the conceptof research praxeology, which allows us to characterise researchers’ practice and knowledge of
networking. Three research examples, conducted in the Japanese educational context, are used toillustrate a means to characterise the four praxeological elements (type of tasks, technique,technology, and theory). The results imply that the notion of theoretical grain sizes (grand,intermediate, and local levels) can be used to deepen our understanding of researchers’ work on
networking theories. Based on our results, different characteristics of networking theories are
discussed according to the cultural issues and specificities of the illustrative examples.
Keywords: 
Networking theories, networking strategies, research praxeology, logos block,theoretical grain sizes

INTRODUCTION
The growth and success of mathematics education as a scientific research field are marked by the existenceof diverse theories. However, such growth also presents a challenge. As Prediger et al. (2008) stated:[T]he more the number of theories grows, the more difficult it will be to get an overview of all of themand of small theoretical bricks using different languages with slightly different meanings. (p. 169)
This motivates networking between different theories wherein the researchers work towards creating a
dialogue between different theoretical approaches while respecting the identity of the different approaches inmathematics education (Prediger & Bikner-Ahsbahs, 2014). Several studies have worked within the context
of networking theories for over a decade, and some have reviewed and reflected researchers’ networking
endeavours (Bikner-Ahsbahs & Prediger, 2014; Kidron et al., 2018). It appears that the latter is a metatheoretical study of the former. For instance, Artigue et al. (2011) use the notion of research praxeology
based on the Anthropological Theory of the Didactic (ATD); and Radford (2008) conceives a dynamic
cultural semiotic space as a semiosphere introducing a triplet comprising a system of basic principles,
methodology, and a set of research questions. In this space, networking of theories takes place and hence
may also inform the dynamic development of the semiosphere. More recently, Tabach et al. (2020) elaborated
Corresponding author: Yusuke Shinno

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Y. Shinno and T. Mizoguchi

on an argumentative grammar for networking theories by adopting Toulmin’s model of argument.
While such meta-level studies clarify the nature of networking theories, attempting them can be difficult
because characterising researchers’ practices and knowledge in networking requires a meta-theoreticalframework; however, this is still under development. Research praxeology has been adopted as a metatheoretical framework in this paper, along with some research examples of networking theories that havetaken place in the Japanese educational context, to contribute to the development of a framework forresearchers’ activities. Many researchers in mathematics education in Japan have worked on different
theoretical approaches developed in Western countries for their studies in the local context. However, only asmall number has explicitly described their networking endeavours. In this study, we attempt a retrospective
analysis, which aims to make the implicit explicit in terms of praxeologies. Thus, the research question inthis study is as follows: How can we characterise researchers’ theoretical work on networking endeavours
in terms of praxeology? Through this research question we expect to formulate a method to characterise thepraxeologies of networking theories in different types of studies. Additionally, this paper discusses some
cultural issues related to the Japanese educational context.

RESEARCH PRAXEOLOGY AND NETWORKING THEORIES
Elements of research praxeology
The concept of praxeology is one of the main constructs of the ATD (Chevallard, 2019). Using this concept,
any human activity can be understood as consisting of two blocks: praxis and logos. Each of these blocks has
two elements: type of tasks (T) and technique (τ) for the praxis block, and technology (θ) and theory (Θ) forthe logos block. A task (T) indicates a problem of a given type; a technique (τ) is a way of performing tasksof this type; a technology (θ) is a way of explaining and justifying the technique; and theory (Θ) explains or
justifies the technology. Artigue et al. (2011) extended the concept of praxeology to researchers’ practices
and knowledge (see also Artigue & Bosch, 2014; Artigue & Mariotti, 2014; Artigue, 2019). Researchpraxeologies of researchers’ activities also comprise four elements (T/τ/θ/Θ) (Artigue & Bosch, 2014). The
type of tasks (T) of a research praxeology often refers to research questions and problems to be studied; thetechnique (τ) comprises the research method that addresses the research questions; the technology (θ)
corresponds to methodological discourse that justifies the choice of the method and explains the results; andthe theory (Θ) includes the main principle and notions of a given theoretical framework.
Networking praxeologies
Networking praxeologies are used to understand the research practices of networking theories (Artigue et al.,2011). The praxis and logos blocks of networking praxeologies can be identified through retrospective
analysis of studies (Artigue & Bosch, 2014; Artigue & Mariotti, 2014; Artigue, 2019). However, suchanalysis is sometimes difficult to practice, because:[T]he theoretical block of networking praxeologies is also in the process of emerging, perhaps at the
moment it is in a less-developed state. (Artigue & Mariotti, 2014, p. 351)
Our study attempts to make the logos block of networking praxeologies more visible, based on ourNetworking Praxeologies and Theoretical Grain Sizes in Mathematics Education

79

retrospective analysis of networking strategies and theoretical grain sizes. For this analysis, let us explain
here what we can locate as the four elements (T/τ/θ/Θ) of networking praxeologies.
The praxis block of a networking praxeology also corresponds to the networking questions or problems (T),and method (τ). Unlike research practice in general, the networking endeavour refers to two or moretheoretical approaches. The choice of theoretical frameworks and their usage are part of the type of tasks andthe techniques. The logos block of networking praxeologies is quite different from that of researchpraxeologies in general. The methodological discourse (θ) of networking theories describes, explains, andjustifies networking strategies. Prediger et al. (2008) propose a formulation in terms of: understanding andmaking understandable, comparing and contrasting, combining and coordinating, and integrating locallyand synthesising. However, the proposal of Artigue and Bosch (2014) includes other ingredients based on the
praxeological structure of research strategies. Finally, the theoretical discourse (Θ) explains and justifies the
networking strategies. Identifying the elements that qualify as a theory (Θ) of networking praxeologies isdifficult as “the technological and theoretical discourses are not fully articulated” (Artigue & Bosch, 2014,
p. 260) in the current state of networking praxeologies. This is probably because of the scientific challengeof sharing a common meaning of theory and the role of theory in mathematics education and why it isimportant – or even necessary for researchers – to network theories. Therefore, we attempted to identify
technological and theoretical discourses behind the networking endeavour. The theoretical grain sizes can beused to describe the praxis block of networking praxeologies as well as to interpret the implicit logos blockfor characterizing researchers’ discourses.
Theoretical grain sizes
The notion of grain sizes1 has been used to describe, review, and categorize different levels and magnitudesof theories in mathematics education (e.g., Kieran, 2019; Shinno & Mizoguchi, 2021; Silver & Herbst, 2007;
Watson & Ohtani, 2015). For examples, Silver and Herbst (2007) distinguished three theoretical levels:
grand, middle-range, and local theories. Kieran (2019) also distinguished between three theoretical levels in
the context of task design research: grand, intermediate, and domain-specific theories (or theoretical frames).
These categories allow us to understand and describe the nature and spectrum of a given theory in terms ofdifferent theoretical levels and magnitudes. However, it is not easy to determine the grain sizes of theories ina general way, because one could consider different categorizations from different perspectives.
For Silver and Herbst (2007), grand theories respond “to a need for broad schemes of thought that can help
us organise the field and relate our field to other fields” (p. 60); and middle-range theories grow “from theneed to inform a discrete variety of practices, including individual mathematical thinking, teaching, andlearning in classrooms, or mathematics teacher education” (p. 61). Local theories help mediate connections
between problems, research, and practices, in a particular study’s context. However, for Kieran (2019),
grand theories include the cognitive-psychological, the constructivist, the socio-constructivist, thesociocultural, and other general educational theories. Intermediate theories have a more specialised focus
1 Someone might interpret the meaning of the word grain sizes differently. In this paper, we used this term as a metonymy fordescribing and understanding different levels and magnitudes of theories in mathematics education. The use of this term issimilar to what Watson and Ohtani (2015) mentioned in the reviews of task design research: “Grain size descriptions areintended to be descriptive tools [emphasis is added] for thinking in a structural way about task design, rather than beingprescriptive” (p. 5)

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than grand theories and, as such, can contribute in a more refined way to the design of curricular areas (pp.
271–272). Domain-specific theories “deal with distinct mathematical concepts, procedures, or processes ofmathematical reasoning” (p. 272). The list below summarises Kieran’s (2019) categorisations with examplesof the theoretical grain sizes2.
●

Grand theoretical frames shape the background understanding of research in mathematics education(e.g., the cognitive-psychological, the constructivist, the socio-constructivist, the sociocultural)●

Intermediate level frames3 are located between the grand theories and the domain-specific frames (e.g.,
realistic mathematics education theory (RME), theory of didactical situations (TDS), ATD, lesson
study, variation theory, conceptual change theory)●

Domain-specific frames specify distinct mathematical concepts, reasoning processes, or tools (e.g., aframe for fostering mathematical argumentation within problem-solving, frame for proof problems withdiagrams, frame for learning algebra using technological tools)

Silver and Herbst’s (2007) categories are more general than Kieran’s (2019), despite both considering threelevels of grain size. This is because Kieran aimed to review and discuss different theoretical frameworks
which have been used in task design research, but Silver and Herbst argued for the role of theories inmathematics education research in general. More recently, Shinno and Mizoguchi (2021) distinguished threetheoretical grain sizes in the context of mathematics teacher education as follows:●

Grand theories are well-established frames for research inside and outside mathematics education thathave been developed in a broader context of human activity (e.g., ATD, CHAT)●Intermediate theories are frames that do not specify a feature, but rather a general aspect of teachers’
knowledge or activities (e.g., knowledge for teaching, professional growth, documentational approach)●

Local theories are frames that specify a particular feature of teachers’ knowledge or activities (e.g.,teacher noticing, teacher design).

In this way, the method of distinguishing the grain sizes of theories cannot be absolute but is rather relative
to the research area of the study being undertaken. For the analysis in this paper, we re-categorize Kieran’s(2019) distinctions of grain sizes into three levels (grand, intermediate, and local) by generalising the above
descriptions of the three levels of Shinno and Mizoguchi (2021). In Table 1, there are a few critical points
when re-categorizing the earlier version by Kieran (2019). Despite the local-level being almost identical todomain-specific frames in Kieran (2019), we reconceptualised the other two levels to distinguish the degreeof generality of a given theory in mathematics education. For instance, while both ATD and TDS werecategorised as intermediate-level frames by Kieran (2019), we acknowledge the differences between the twotheories in terms of grain size. Nonetheless, our categorisations cannot distinguish theories for research
inside and outside mathematics education (e.g., ATD and variation theory) as we do not consider theories
developed in other disciplines in this study. In addition, we excluded “the background understanding of2 In Kieran (2019), what she called the grand theoretical frame is based on Cobb (2007). Examples for the intermediate andthe domain-specific frames includes several frames used in the task design research. While Kieran (2019) includes references
to each frame or study, we omitted the references in this table.
3 Kieran (2019) provided an additional explanation of the intermediate-level frames referring to their roots as follows: “Inaddition, intermediate level frames can also be characterized according to whether their roots are primarily theoretical orwhether they are based to a large extent on deep craft knowledge. An example of the former is the Theory of Didactical
Situations and the latter, Lesson Study” (Kieran, 2019, p. 272).

Networking Praxeologies and Theoretical Grain Sizes in Mathematics Education81

research in mathematics education”, which might refer to a general philosophical standpoint, from thedescription of grand-level theories. It is because we think background theories (such as constructivism)
providing general principles that shape the universe of what can be a researchable object are relevant to alllevels of grain sizes 4.
Table 1. (Re)categorizations of different theoretical grain sizes

Grain sizes of theories
Descriptions
Grand-level
well-established frames for conducting
research inside and outside mathematics

ExamplesATD, commognitive framework,
variation theory, etc.

education, that have been developed in the
broader context of human activity
Intermediate-level

frames that do not specify any particular

TDS5, theory of realisticknowledge or activities but specify amathematics education (RME), etc.

context rather than a general aspect of theresearch object to be studied
Local-level

frames that specify a particular

A model of problem solving, vanmathematical or didactic knowledge orHiele levels, etc.

activity

METHOD
Seeking an interface between the logos and praxis blocks is an appropriate way to develop the researchers’
discourse of networking praxeologies (Artigue & Bosch, 2014). Three research examples, utilizing multiple
theoretical frameworks for different purposes, are used to illustrate the praxis and logos blocks of networking
praxeologies. The three case studies involved different types of research (comparative study, empirical study,
and curriculum development). While these studies can be considered as cases of networking theory research,
the networking strategies used are not explicitly mentioned in these papers. The theoretical grain sizes werealso implicit. Therefore, we must interpret these implicit aspects. Thus, we selected research examples fromstudies that have been conducted in the Japanese educational context (such as lesson study, classroom
teaching, and curriculum development). One study (Example 1) was selected because it involved a type ofstudy (lesson study) that is culturally situated in Japanese educational research. The other two studies(Examples 2 and 3) were conducted by the author(s), for which we can analyse their theoretical aspects4 For example, Shvarts and Bakker (2021) considered the grain sizes of levels in terms of the philosophical and historical rootsof theories under considerations. They distinguished six different levels by means of vertical analysis for networking
theories: such as epistemic presumption, ontological presumption, grand theory, local theory in mathematics education,
application for teaching, and application for educational design. From this vertical perspective, background theories refer tohigher levels of a certain level of theories.
5 Some researchers would also call TDS a grand-level theory as it can be applied to any situation wherein teaching andlearning take place. Therefore, it should be noted that the distinctions or categorisations of different theoretical grain sizes
are not reserved for the ones mentioned in Table 1.

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Y. Shinno and T. Mizoguchi

retrospectively. While the three research studies are analysed as illustrative examples, they cannot be
conceived as typical or representative, and the results of the analysis cannot be generalised. Our attention isdrawn to the different praxeological models of the research on networking theories. Thus, the primary aim of
this analysis is to make implicit discourses in the logos block explicit by focusing on the theoretical grain
sizes of each theoretical framework to be networked. It allows us to understand the interface between thelogos and praxis blocks of networking praxeologies.
Regarding the praxis block of networking praxeologies, we identify two elements (T/τ) by analysing theaims, questions, and methods of networking in the papers. For the types of networking tasks (T), we
especially focus on the theoretical frameworks and/or models being used for networking, rather than practicalmethods of procedures or analyses in implemented studies (such as data collection or data analysis). Then,the networking strategies used in each study are identified as techniques (τ). Concerning the logos block of
networking praxeologies, the technological discourses (θ) to explain and justify the networking strategies areinterpreted by referring to some relevant quotations from each paper. To better understand the theoretical
discourses (Θ) underlying the technologies in each study, we discuss the role of cultural issues in networking
praxeologies; for example, what is a theory in different educational research traditions? What is the role oftheory regarding educational practice? Why is it necessary to connect or reduce the number of theories? For
such praxeological analyses of research on networking theories, the theoretical grain sizes categorized inTable 1 may play a determinant role for both the praxis and logos blocks.

ILLUSTRATIVE RESEARCH EXAMPLES
Three research examples
Example 1 – Comparing and contrasting
Miyakawa and Winsløw (2009) compared two didactical designs to introduce primary school students toproportional reasoning. The designs were based on two different approaches: didactical engineering inFrance, and lesson study in Japan. The comparison is informative as both approaches are used for planninga lesson, but the method of designing is very different. In the former, the lesson was designed according to
the theoretical principles proposed in a specific theory; however, in the latter, the lesson was created according
to the teachers’ experiences in the classroom. Didactical engineering is a methodology for TDS introduced
by Brousseau (1997) to gain theoretical insight into the functioning of a didactic system. In contrast, lesson
study does not necessarily involve an explicit didactic theory, but often refers to a certain practical orpedagogical approach as a theoretical basis for teachers’ practice. Miyakawa and Winsløw (2009) referred toa Japanese teaching approach called open approach method by Nohda (2000), which is relatively known asa teaching method to enhance multiple ways of thinking in the process of problem-solving, and contrasted itwith the notion of fundamental situation6 in TDS through their analysis and observations. Below, someelements in the process of designing a lesson to introduce students to proportional reasoning are summarisedand contrasted according to theoretical basis, design formats, and realised lessons.
6 A fundamental situation for a concept is a mathematical situation for which the concept constitutes a priori an optimal
solution (Artigue et al., 2014, p. 49).

Networking Praxeologies and Theoretical Grain Sizes in Mathematics Education

●

●

83

A Japanese case○

Theoretical basis: Open approach (Nohda, 2000)○

Design formats: Lesson study○

Realised lesson: Hatsumon (questioning)7 to enquire about an open problem (Nohda, 2000)

A French case○

Theoretical basis: Fundamental situation (TDS) (Brousseau, 1997)○

Design formats: Didactical engineering○

Realised lesson: adidactical milieu8 in the puzzle situation (Brousseau, 1997)

According to Miyakawa and Winsløw (2009), a comparison of the two didactical designs shows crucial
similarities and differences in a lesson in the following ways.
Both of the designs emphasize the social interaction and independent thinking of students. Both formats
for design require quite similar kinds of analysis, including anticipating student strategies and revising
the design in an experimental cycle. […] In a fundamental situation, they should lead to thepersonalization and institutionalization of a target mathematical knowledge (savoir), consistent withthe ‘official’ mathematical knowledge. […] In the open approach, the aim is for students to apply andtest their mathematical knowledge, through two main processes: the process in which some conditions
and hypotheses from a ‘real world’ problem are formulated mathematically and the process ofgeneralization and systematization after solving a problem. (Miyakawa & Winsløw, 2009, p. 216)
Miyakawa and Winsløw (2009) also discussed how didactical engineering and lesson study differ at the levelof their objectives. Lesson study provides an opportunity for teachers to develop their teaching professions;the main objective is to improve lessons. In contrast, didactical engineering aims to establish scientificknowledge wherein a lesson confirms or rejects/questions the conditions for learning.
Example 2 – Combining and coordinating
Shinno (2018) aimed to characterise the development of mathematical discourses in a series of lessons interms of the semiotic chaining model (triadic nested model) proposed by Presmeg (2006) and the commognitive
framework introduced by Sfard (2008). There were two reasons for using two different theoretical frameworks
in this study. The first is related to the content-specific aspect of students’ difficulties of reification in thelearning of square roots. Shinno (2018) argued that the cognitive account of reification can be used to explain
this difficulty; however, conceptualising it differently from semiotic and discursive points of view may allow
us to arrive at a deeper understanding of the reification phenomenon. The second reason is related to thesetting of the mathematics classroom used in this study. Based on the observations of earlier studies on
Japanese classroom culture (e.g., Emori & Winsløw, 2006), Shinno (2018) mentioned that “the social
interaction between the teacher and the students in a Japanese mathematics classroom can be assumed toconstitute a culturally unique discursive community” (p. 279), and that the semiotic and discursive approaches
can be suitable for analysing the process through which students become familiar with the usage and meaningof square roots through classroom interactions. For example, Shinno (2018) analysed how a new expression
7 Hatsumon (in Japanese) is a general pedagogical term which refers to a teacher’s key questioning in a lesson.
 Within the TDS, milieu is a component of a didactical triangle (teacher, students, and milieu), which constitutes a didacticalsituation. In an adidactical situation, “students accept to take the mathematical responsibility of solving a given problem, andthe teacher refrains from interfering and suggesting the target mathematical knowledge” (Artigue et al., 2014, p. 51). The
adidactical is referring to “the situation has been temporally freed from its didactical intentionality” (ibid.)

8

84

Y. Shinno and T. Mizoguchi√2 ＋√3 becomes a mathematical object rather than a computational process through classroom interactions
to justify or reject a teacher’s question ‘is √2 ＋√3 ＝√2＋3 true?’As shown in Table 1, Shinno (2018) attempts to coordinate the commognitive terms9 with semiotic terms10
in order to gain a multidimensional insight of the reification phenomenon through the networking of the twotheoretical approaches. Shinno (2018) proposed two theoretical benefits of using two theoretical lenses in thestrategy of combining and coordinating:From the viewpoint of the commognitive framework, the implicit meta-discursive rule may become
explicitly identified as the third component of ‘interpretant’ by means of the triadic nested model. Fromthe viewpoint of the triadic nested model, the first and second components, the signifier and signified,can be characterised as distinct features, such as keywords, visual mediators, and endorsed narratives.
(Shinno, 2018, p. 302)
Table 2. Coordinating theoretical terms from the two different frameworks

Commognitive framework (Sfard, 2008)

Triadic nested model (Presmeg, 2006)

Word use (keywords)

Signifier or signified

Visual mediators

Signifier or signified

Endorsed narratives

Signifier or signified

Routines (meta-rules)

InterpretantNote. Adapted from “Reification in the learning of square roots in a ninth-grade classroom: Combining
semiotic and discursive approaches” by Y. Shinno, 2018, International Journal of Science and Mathematics
Education, 16(2), p. 302.

In this empirical study, the semiotic chaining model was used to identify the three components (signifier,signified, and interpretant) in the classroom episodes and then characterised by their discursive features froma commoginive point of view. For example, when a signifier √2＋√3 was reified (objectified), an interpretant‘treating √2 ＋√3 as one irrational number’ was also understood as a meta-discursive rule. In this way,
Shinno (2018) sought to use theoretical terms from the two different frameworks (Table 2) interchangeably
for the analysis. Thus, combining and coordinating the two theoretical frameworks allowed him to gain amulti-faceted insight into the reification phenomenon, meaning a change from process-oriented use to object-

9 The meanings of the commognitive terms are as follows: word use refers to mathematical vocabulary, syntax, and ordinary
words associated with mathematics; visual mediators include physical, diagrammatic, and symbolic mediators ofmathematical objects; endorsed narratives mean a set of mathematical statements, proofs, rules of calculation, which areaccepted within a given community; and routines refer to regularly employed and patterned repetitive activities (e.g.,
calculating, proving, generalising) as well as a set of meta-rules (e.g., how to calculate, how to prove, how to generalise).
10 The model of semiotic chaining by Presmeg (2006) based on Lacan’s inversion of Saussure’s dyadic model (signifier andsignified) as well as Peirce’s triadic model (object/signified, representation/signifier, and interpretant). For this model, therelationship between signifier and signified is considered to have a nested structure, and meaning-making in the twocomponents is understood by the third component interpretant.

Networking Praxeologies and Theoretical Grain Sizes in Mathematics Education

85

oriented use of a signifier11.
Example 3 – Locally integrating
Shinno et al. (2015, 2018) proposed a theoretical framework for curriculum development of proof insecondary schools in Japan. Unlike the other two research examples, the theoretical framework undertakenin their paper was aimed at being used for developing, designing, or improving a curriculum related tomathematical proof. For this aim, the theoretical framework was constructed by adapting and integrating
different theoretical models and concepts as follows:●

The Mathematical Theorem as a system, consisting of statement, proof, and theory (Mariotti et al.,1997)12●A model of mathematical proofs, comprising knowledge, formulation, and validation (Balacheff,1987)13●

The concept of local organisation and global organisation (Freudenthal, 1971, 1973)14●

The concepts of small theory and large theory (Hanna & Jahnke, 2002)15

On the one hand, the theoretical framework proposed by Shinno et al. (2015, 2018) was a framework for
curriculum development, describing and prescribing mathematical contents and sequences in the Japanese
educational context. On the other hand, it seems that the theoretical models and concepts to be integrated
have different theoretical natures and are not always considered the ones for educational development. For
example, the notion of Mathematical Theorem (Mariotti et al., 1997) is a model to understand mathematicalpractice wherein a proof is carried out, but originally this model does not prescribe curricular contents andactivities to be taught in schools. Therefore, when constructing the framework, Shinno et al. (2015, 2018)reconceptualized some concepts, adapting (or detaching from) their original senses, and integrated them into
one framework so that it is relevant to the Japanese curriculum for proof. Let us briefly explain how they
adapted and integrated different theoretical concepts.
In their framework, the three elements – statement, proof, and theory – were used as the foreground inshaping the framework. Balacheff’s (1987) categorisation of formulation and validation was used tounderstand levels of statement and proof. While the original notion of theory in Mathematical Theorem
11 Regarding the implications of using the two frameworks, Shinno (2018) concluded: “[This study] enables to characterizereification as the change in a set of meta-rules, which means the replacement of the process-oriented use with the objectoriented use of a new signifier. Although the model of semiotic chaining itself conceptualizes reification as the act of making
signifier-signified couple in each node, as far as the reification in the discursive process is concerned, it may be important todetect a set of interpretants (meta-rules)” (Shinno, 2018, p. 311).
12 A Mathematical Theorem consists of a system of relations between a statement, and its proof, and the theory in which theproofs make sense. “[The] existence of a reference theory as a system of shared principles and deduction rules is needed if
we are to speak of proof in a mathematical sense” (Mariotti et al., 1997, p. 182).
13 Balacheff (1987) proposed a theoretical framework composed of knowledge, formulation, and validation, which allow us
to analyse the complex nature of proof and proving. The four levels (naïve empiricism, crucial experiment, generic examples,
and thought experiment) are regarded as the levels of validation.
14 The local organisation by Freudenthal (1973) is a concept to explain short deduction chains consisting of a statement and its
proof in geometry, wherein some properties can be accepted as taken for granted, while the global organisation is a concept
to explain an axiomatized system (such as a system of Euclidean geometry).
15 The distinction between small theory and large theory is similar to that between local and global organization. Hanna andJahnke (2002) mention that “instead of building a large theory (namely, Euclidean geometry) in the course of the curriculum,
it seems to be more appropriate to work in several small theories” (p. 3). Thus, small theory and large theory are considered
concepts rather than part of an operational framework.

86

Y. Shinno and T. Mizoguchi(Mariotti et al., 1997) mainly refers to mathematical theory (such as arithmetic, algebraic, or geometricaltheories), Shinno et al. (2015, 2018) reconceptualised it as a component which includes different layers of
the system (such as local and axiomatic theory) to consider the wide range of contents and levels in the
curriculum. They adapted the notions of Freudenthal (1971, 1973) and Hanna and Jahnke (2002).
Their reconceptualization of the element (i.e., theory) of Mathematical Theorem was needed to account for
the axiomatic aspect of geometry curriculum in secondary schools in Japan, as the Japanese geometry
curriculum emphasizes Euclidean geometry (Miyakawa, 2017; Shinno et al., 2018). Miyakawa (2017)
characterised this nature of the system in secondary school mathematics as quasi-axiomatized geometry withrespect to some characteristics such as “the term axiom or postulate is not used; some properties are
introduced after observation; some are admitted implicitly and there is no long list of axioms as in Euclid’s
Elements” (p. 49). This is also related to what Hanna and Jahnke (2002) distinguished as small theory andlarge theory.
Shinno et al. (2018) called a theoretical framework for curriculum development, the reference epistemologicalmodel (REM) and leaned on the elaboration of RME by Bosch and Gascón (2006), which constitutes thebasic theoretical lens for researchers to analyse different types of mathematical knowledge among different
institutions. Thus, Shinno et al. (2018) attempted to integrate different theoretical constructs regarding proof
into one framework (REM), wherein an internal consistency within the framework is created to understandcontents and levels in a curriculum, summarised as follows:
Regarding the proposed REM, we conceptualized that the statements and proofs are interrelated
according to the nature of theory and illustrated how this model helps to describe the evolution of eachelement. This model may help us understand the gaps in the evolution of the formulation of a statement
and how different meanings of proof relate to the distinction between local and (quasi-) axiomaticlevels of theory. (Shinno et al., 2018, p. 30)
For example, there are three levels of the formulation of a statement, i) drawing, diagrams, manipulation, andgesture; ii) ordinary language and words; iii) mathematical words and symbols. Considering a universal
proposition (statement), a universal quantification can be formulated by ordinary words (e.g., all, any) inschools and by mathematical symbols (e.g., ∀) at advanced levels. Based on the proposed framework,
Shinno et al. (2018) implied that there is a gap in the levels of formulation in the Japanese curriculum(especially between the ordinary and mathematical language of a universal proposition), and there is a crucialtransition between the levels for development16.
From the perspective of networking praxeologies
Our retrospective analysis of the three case studies in terms of networking praxeologies and theoretical grain
sizes can shed light on the principal aspects of networking in each study (as summarised in Table 3), and thus
enable the comparison and discussion of the praxis (T/τ) and the logos (θ/Θ). The types of tasks (T) were
found in the choice of the theoretical frameworks in the papers. The techniques (τ) were identified byfocusing on the networking strategies that were most relevant to the studies (based on our interpretations andretrospections). In Example 1, the strategy of comparing and contrasting was chosen and the similarities and16 Regarding linguistic issues, we also discussed how Japanese language may affect the difficulty and gap in the levels of theformulation of quantifications (Shinno et al., 2019).

Networking Praxeologies and Theoretical Grain Sizes in Mathematics Education87

differences between the French and Japanese approaches (TDS and open approach) to didactical designs are
examined. In Example 2, the strategy of combining and coordinating is used to analyse classroom teachingwith a commognitive framework, and with the semiotic chaining model. In Example 3, local integration isused to construct a theoretical framework for the curriculum development of proof in secondary schools inJapan. The strategy in this case also offers a coordination of theoretical concepts (such as the relationshipbetween Freudenthal’s local/global organizations and Hanna & Jahnke’s small/large theories). We think thisis natural since coordinating is a strategy that “can be a starting point for a process of theorizing” (Prediger
et al., 2008, p. 173).
Table 3. Networking praxeologies and theoretical grain sizes in the three research examples

Example 1
To compare TDS and open

Example 2
To combine the model of

Example 3
To construct a theoretical

approach method

semiotic chaining and theframework for curriculum

commognitive framework

development, using theT

Mathematical Theoremτ

A strategy of comparing andA strategy of combining andA strategy of ‘coordinating

contrasting in terms of thecoordinating in terms of theand integrating locally’ intheoretical basis, design

theoretical terminologies

terms of statement, proof,

format, realized lesson

between two frameworks

and theory(see Table 2 (Shinno, 2018,p. 302))

θDiscourses on similarities

Discourses on the

Discourses on how to

and differences on thecompatibilities andintegrate different theoretical

didactic designs based on theinterchangeabilities between

elements, levels, andtwo approaches (see thethe semiotic and theconcepts into a common

quotation above (Miyakawa

commognitive terms

framework

Intermediate-level (TDS) andGrand-level (commognition)

Local-level (all theoretical

local-level (open approach)and intermediate-level

constructs)& Winsløw, 2009, p. 216))

GSa(semiotic chaining)
GS; grain sizesa

To determine the elements of the logos block, we needed to interpret the descriptions explicitly or implicitly
written in the papers. We considered the technological discourses (θ) by focusing on the descriptions to
explain and justify how two or more theories were networked by a certain strategy and why the strategy wasmost relevant to the study undertaken. In the previous subsection, we tried to show such descriptions by thequotations or elaborations from the papers. Table 3 included summarised technological discourses. However,
it is more difficult to determine the theoretical discourses (Θ) because they are mostly implicit in the papers.

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Y. Shinno and T. Mizoguchi

We considered such theoretical underpinnings by focusing on cultural elements included in each study; forexample, what is regarded as a theory in the Japanese educational and research context? What is the role oftheories regarding educational practice? What is the influence of the educational system in attempting toanswer these questions? We develop the discussion related to the cultural issues in the section titled‘Discussion and conclusion’.
From the perspective of theoretical grain sizes
To better understand the networking praxeologies in each study (Table 3), different theoretical grain sizes
were considered. Based on our categorizations of grain sizes mentioned earlier, TDS from Example 1 isconsidered an intermediate-level theory, since it is often used to understand and analyse the epistemologicalprocesses in mathematical classrooms from a systemic perspective. TDS is not a domain-specific theory
which specifies any mathematical knowledge or activity. Rather, it is primarily developed to model thefunctions and mechanisms between teachers, students, and the didactical milieu in a didactical/adidacticalsituation (in the classroom). In contrast, the open approach is seen as a local-level framework or instructional
principle which does not specify any mathematical content but does specify teachers’ teaching methods. For
instance, using an open problem is a characteristic aspect of this approach which is comparable with thenotion of fundamental situation from TDS (Miyakawa & Winsløw, 2009). However, TDS and the open
approach are not fully comparable because the latter does not provide a tool to analyse teaching and learningof mathematics at a general level (e.g., the approach cannot be applied to a lesson with a closed problem).
Therefore, the two frameworks are comparable only if one can focus on a specific notion of one approachthat has a counterpart in the other (such as a fundamental situation and an open problem). Thus, the discourses
on two didactical designs, as cited above from Miyakawa and Winsløw (2009, p. 216), have focused on somecomparable constructs from each framework.
In Example 2, the commognitive framework is viewed as a grand-level theory which is a comprehensive
theory of learning from a discursive point of view. The model of semiotic chaining originates from Peirce’s
semiotics. However, it was introduced by Presmeg (2006) into mathematics education for analysing therelations between signifier, signified, and interpretant in the teaching and learning of mathematics. Thismodel was used as an intermediate-level framework in Example 2, although one could also consider it as acollection of theoretical concepts. Shinno (2018) combined commognitive and semiotic approaches toprovide different accounts of reification in mathematical learning. Since the commognitive frameworkcomprises a much broader range of theoretical constructs, coordination with semiotic chaining is related toonly some semiotic parts of this grand theory. This implies that the strategy of coordinating between different
grain sizes requires an explicit discourse on complementarity; for example, which part of one theory can becoordinated with another theory.
Networking in Example 3 offers local integration, but the connectivity of different theoretical concepts israther complex. All theoretical models or concepts used in the study are regarded as local-level frames whichhave been developed in the context of research on proof and proving. However, this does not mean that all
the theoretical constructs have the same theoretical grain size. For example, the triplet (statement, proof,
theory) has been used as a foreground frame to integrate other theoretical concepts. There have been many
discourses on argumentative connectivity about how to integrate different models or concepts since Shinno

Networking Praxeologies and Theoretical Grain Sizes in Mathematics Education89

et al.’s (2015, 2018) attempt to propose a theoretical framework. A characteristic aspect of such a discussionis that it changes or extends the original meaning of the theoretical terms (statement, proof, theory) to thosewith a broader sense; this allows the inclusion of different contents and levels in the curriculum in secondary
schools in Japan. Additionally, they created new terminologies (for the three different layers of the system;
the logic of the real world, local theory, and quasi-axiomatic theory) through the adaptation of existing
notions (such as small/large theory)17.

DISCUSSION AND CONCLUSION
A deeper understanding of researchers’ work on networking theories
The praxis block corresponds to how researchers work when they choose different theories and network
them. The researchers may consider the grain sizes of theories according to the types of networking tasks (T)for the study undertaken. We then considered networking strategies as a technique (τ) and explored thetechnological discourse (θ) to explain and justify the strategy through a retrospective analysis of threeillustrative research studies as examples. Artigue and Bosch (2014) argue that the characterisation of thelogos block (θ/Θ) of a networking praxeology has not yet been established. In our study, we offered thenotion of theoretical grain sizes to deepen our understanding of the networking praxeologies. As mentioned
in the previous section, the networking techniques are affected by the grain sizes of the theories involved.
While we have only described the grain sizes (e.g., grand, intermediate, or local levels) of the theoreticalframeworks used in the three research examples shown in Table 3, attention was paid to the discourses on
networking different theoretical frameworks, which may have different sizes or are of the same grain sizes.
The characteristics of such networking praxeologies may differ according to the study type, such as empiricalstudies, design studies, or theory development. For example, Bikner-Ahsbahs and Prediger (2014) discussed
the notion of perspective triangulation (Denzin, 1970), meaning a research practice to increase the validityof empirical analysis. Although this notion does not refer to theories, increasing the validity of empiricalstudies can bring forth a benefit from such networking. Combining theories in/for empirical studies often
provides deeper insights into complex phenomena (Prediger et al., 2008), wherein the results of the empiricalanalysis can respond to the original theories. Bikner-Ahsbahs and Prediger (2010, 2014) also mentioned
other perspectives (explicitness, empirical scope, stability, and connectivity) to discuss the empirical andtheoretical benefits of networking theories for different types of studies. Some of these perspectives are
related to what we identified as technological discourse (a justification of the strategies involved in each casestudy). For instance, perspective triangulation or connectivity can support the strategy of combining andcoordinating in Example 2; connectivity can explicate the strategy for locally integrating in Example 3; andexplicitness can be the strategy for comparing and contrasting in Example 1, as well as the strategy for17 For example, a geometric statement “the sum of the interior angles of a triangle is 180°” can be explained by a physicalapproach (such as a measurement or experiment) which is acceptable within the logic of the real world. The same statement
can be proved by previously accepted properties of parallel lines, based on local theory which is often regarding one
particular proof. The quasi-axiomatic theory can be relevant when considering a system of statements and proofs in schoolgeometry (even though the term axiom or postulate is not introduced). More descriptions of the three layers are mentionedin Shinno et al. (2018).

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Y. Shinno and T. Mizoguchi

locally integrating in Example 3. In addition, it is also worth considering how theoretical terminologies maychange or preserve their meaning before and after networking attempts when examining linguistic
connectivity (Shinno, 2017). As shown in Example 2, by coordinating, the theoretical terms in one framework
are interchangeable with those in another framework. In Example 3, some original concepts are locallyelaborated and integrated into new terminologies in the new framework for curriculum development;
however, such new terms do not preserve their meanings in the original sense.
Cultural issues to be considered
Networking praxeologies can provide a model for understanding researchers’ practices and knowledge of
networking theories. This implies that it is important to consider the diversity of research activities related totheories of mathematics education. One of the reasons for this diversity is cultural issues, which may affectthe researchers’ work. This is also the case in the three research examples involving Japanese educational
contexts. If the theory (Θ) in the logos block of networking praxeologies is implicit, its description can only
be based on the researchers’ conception of a theory in research and practice.
For instance, Example 1 indicated the differences between TDS and the open approach; where the open
approach was a practice-oriented theory or a prescriptive theory, which is often used to improve teachers’classroom teaching and facilitate students’ problem-solving activities. In fact, both the descriptive and theprescriptive nature of a theory in mathematics education are often discussed in Japan (e.g., Koyama, 2006;
Yamada, 2011). This is related to what has been discussed in the previous Topic Study Group in ICME-13(Dreyfus et al., 2017); for example, “[i]n Japan, ‘theory’ is always a theory of some practice. A practicedevelops, somebody notices it, reflects upon it, and constructs a theory of practice” (p. 617). This implies two
distinct aspects. One is that a theory emerges and develops from/for some practice. The other is that a theory
is very close to practice and the theory often seems to be a pragmatic one18. However, in the international
context, some researchers might not or cannot accept such a pragmatic theory as a theory in mathematics
education (cf. Miyakawa & Shinno, 2022). Thus, there is miscommunication among international researchers
regarding what is called theory in Japanese educational research (e.g., Zazkis & Zazkis, 2013). This may
open a discussion that reveals the cultural specificity of Japanese (or East Asian) educational research, which
contrasts with the French (or European) tradition of didactics (Blum et al., 2019). Here, we do not discuss
the nature of theory in different research cultures (although such discussion is promising, it lies beyond thescope of the paper), but merely wish to point out that there is a cultural issue that may shape researchers’
knowledge, even in the context of networking theories.
In addition, in Example 2, Shinno (2018) considered the cultural specificity of the Japanese mathematics
classroom to be related to semiotic activities (Emori & Winsløw, 2006); however, this specificity of classroom
is generally attributed to the Japanese teaching pattern (Stigler & Hiebert, 1999). Such a cultural factor may
influence the researcher’s choice of theory or his/her perspective of using a theory for classroom study.
Recently, Funahashi and Hino (2014) have also investigated the interactive classroom pattern, called a
guided focusing pattern, which is often observed in Japanese classroom lessons, in terms of a discursive orcommognitive perspective for their theoretical and methodological framework. Although other theoretical18 Here we use the term pragmatic as an ordinary word without any theoretical underpinning.

Networking Praxeologies and Theoretical Grain Sizes in Mathematics Education

91

approaches can be used to analyse a classroom culture, what we would imply is that the researchers’
perspective of using a theory (or multiple ones) can be affected by the local educational context and by the
researchers’ conceptions of how the theory can be useful for investigating such a cultural specificity.
Furthermore, in Example 3, the specificity of the Japanese mathematics (geometry) curriculum in secondary
schools was considered when adapting some theoretical concepts that were integrated into a framework for
curriculum development. This also implies that the educational contexts (particularly the curriculum) in acountry may affect researchers’ work on theories. For example, given that the Japanese geometry curriculum
is characterised as quasi-axiomatized geometry (Miyakawa, 2017), we could adopt or adapt the related
concepts (such as, local/global organization, small/large theory) in their framework. In addition, developmental
research, which is often aiming at improving educational practices (including task design, curriculum
development, or lesson study) is a dominant type of research and/or study in Japan. Although a theoreticalframework is often constructed for such developmental work, what is important for the framework is to be
useful for development, rather than for scientific research (Miyakawa & Shinno, 2022). This is one reason
why the proposed framework in Example 3 referred to different constructs (models, concepts, frameworks)
to be integrated. For this reason, the researchers see all constructs as theoretical underpinnings of theframework for development, even though some constructs are not usually brought forth as theory for
research. This may have the implication that one can distinguish between theory for research and theory forpractice.
Finally, we attempted to translocate the cultural issues (such as research culture, classroom culture, andcurriculum culture) into the theoretical discourses (Θ), as shown in Table 4, which are implicit in the
networking praxeologies (Table 3). Furthermore, the logos block of networking praxeologies is also related
to the researchers’ conception of the field of mathematics education and the nature of mathematics education
as a scientific discipline. This raises at least two questions: How can we make such implicit discourses more
explicit in networking theories? Why is such a meta-level and a self-referential study important for
researchers? While we believe that the cultural issues in different educational research traditions play a
determinant role, further research is needed to address these questions and issues.
Table 4. Theoretical discourses (Θ) regarding cultural issues in the three research examples

Θ

Example 1
Discourses on the nature of

Example 2
Discourses on a culturally

Example 3
Discourses on the necessity of atheories in different research

unique discursive community in theoretical framework for

traditions (research culture)a Japanese mathematics

curriculum development inclassroom

Japanese mathematics education

ACKNOWLEDGEMENTS
This work was supported by JSPS KAKENHI, Grant Numbers JP20K02913 and JP18H01015. An earlier
version of this paper was presented at the Topic Study Group 57 related to the diversity of theories in ICME14 in 2021.The authors would like to thank the reviewers for their helpful comments to improve the paper.

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Y. Shinno and T. Mizoguchi

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Yusuke Shinno

Tatsuya Mizoguchi

School of Education, Hiroshima University, Japan

Faculty of Regional Sciences, Tottori University, Japan

Email: shinno@hiroshima-u.ac.jp

Email: mizoguci@tottori-u.ac.jp

https://orcid.org/0000-0001-6999-5075

https://orcid.org/0000-0002-4399-8988

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THE ROLE OF FEEDBACK WHEN LEARNING WITH A
DIGITAL ARTIFACT: A THEORY NETWORKING CASE ON
MULTIMODAL ALGEBRA LEARNING
Angelika Bikner-Ahsbahs1, Estela Vallejo-Vargas2, Steffen Rohde2, Thomas Janßen2,David Reid3, Dmitry Alexandrovsky4, Anke Reinschluessel4, Tanja Döring4, Rainer Malaka41

Bremen University & Western Norway University of Applied Sciences
2

Bremen University,

3

University of Agder,4

Bremen University, TZI, Digital Media Lab

Abstract
How digital feedback supports teaching/learning with a digital tool is not yet well understood.
In a networking of theories approach, Activity Theory and the Instrumental Approach are
combined to investigate the role of digital feedback for the teaching/learning of integers with the
MAL-system, a multimodal algebra learning system. The MAL-system is designed as a multitouch tangible user interface with feedback functions allowing students to mathematically operatewith (negative) numbers represented as virtual tiles. We explore the role of digital feedback by amulti-case study at the grade five level, in which we conducted experimental task-based interviewsof four student pairs, each supported by a tutor. Findings show that (digital) feedback mediates theteaching/learning activity in a supportive way. Reflection on the way the two theories are combined
reveals that they can be regarded as locally integrated into a layered model enriching the describingof the transformation of teaching/learning mediated by digital feedback.
Keywords: 
Networking of theories, activity theory, instrumental genesis, digital feedback,negative numbers, transformation

INTRODUCTION
Hattie and Timperley (2007) emphasized feedback to be an essential part of teaching and learning. Whenlearning is supported by digital tools, digital feedback is highly relevant (Fyfe, 2016; van der Kleij et al.,2015). Both immediate feedback as well as feedback variation have been shown to be effective for learning(Bockhove & Drijvers, 2012). A recent study on algebra learning shows that any digital feedback is better
than no feedback and the most important digital feedback takes an explanatory form (Fyfe, 2016). However,“… the computer can serve as an effective problem-solving tool only if accompanied by more traditional
forms of discourse between pupils and teacher” (Hillel et al., 1993, p. 38). But it is not so clear how feedback
may foster processes of teaching/learning, and how teacher feedback and digital feedback differ and interact
Corresponding author: Angelika Bikner-Ahsbahs

A. Bikner-Ahsbahs, E. Vallejo-Vargas, S. Rohde, T. Janßen,
D. Reid, D. Alexandrovsky, A. Reinschluessel, T. Döring, and R. Malaka

96

in the use of an artifact to support in-depth learning. Answering these questions depends on the backgroundtheory on learning (Evans, 2013). We use a networking of theories approach to investigate this problem
through a multi-case study in which students learn about negative numbers with a digital tool specificallydesigned for this purpose. This paper aims at answering the following research question: What is the functionof digital feedback in the support of the teaching/learning of negative numbers with this specific digital
artifact? To answer this question we will conceptualize the main terms in our theoretical framework, using
Activity Theory and the Instrumental Approach within a theory networking approach.

THEORETICAL FRAMEWORKOur approach aims at networking two theoretical approaches in mathematics education (Bikner-Ahsbahs &Prediger, 2014), Activity Theory (Leontjew, 1987)1 is used as a background theory (Mason & Waywood,
1996) for teaching/learning coordinated with the Instrumental Approach (Artigue, 2002; Trouche, 2020). We
will conceptualize the teaching/learning of negative numbers using activity theory. The instrumental
approach will allow us to embed the use of the artifact by the learners into the activity.
The networking of these two theoretical approaches builds on Radford’s elaboration of the notion of theory(2008) and how theories can be related. Radford defines a theory as:
a way of producing understandings and ways of action based on:●	A system, P, of basic principles, which includes implicit views and explicit statements that delineate
the frontier of what will be the universe of discourse and the adopted research perspective.
●	A methodology, M, which includes techniques of data collection and data-interpretation as supportedby P.
●	A set, Q, of paradigmatic research questions (templates or schemas that generate specific questions
as new interpretations arise or as the principles are deepened, expanded or modified) (2008, p. 320)
He abbreviates this definition by the triplet (P, M, Q). Radford (2012) proposes later to expand this notion of

Figure 1: Networking Strategies(CC BY 4.0, Bikner-Ahsbahs et al., 2016, p. 34, adapted from Prediger et al., 2008)

1 The author’s name Лео́нтьев is transliterated differently into Roman characters in different articles. Leontjew is the spellingin German literature and Leontjev, Leont’ev, Leontyev and Leontiev are spellings found in English literature.

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theory by including results R as a relevant constituent because research results rebound on the theories usedin research, thus pointing to the purpose of theories as tools for research. He represents this expanded
understanding of theory as a dynamic structure by [(P, M, Q), R].
The networking of theories means to relate theories with respect to their constituents. Four pairs of networking
strategies positioned on a scale of increasing integration (Figure 1) have shown to be useful to identify
relationships. The two pairs of strategies that are relevant in this study are coordinating and local integration.
While combining is a kind of triangulation, i.e., viewing an empirical phenomenon from different theoreticalperspectives, coordinating can be considered, when “a conceptual framework is built by fitting together
elements from different theory elements for making sense of an empirical phenomenon” (Prediger & BiknerAhsbahs, 2014, p. 120). The latter becomes relevant when we complement an already existing way oftheoretical understanding with an additional theory element to achieve a more comprehensive understandingof an empirical situation. Local integration means theorizing a phenomenon at the boundary of two or moretheories so that the theories themselves are expanded in their local understanding of a specific phenomenon.
Activity Theory
The concept of activity in activity theory is transformative as it is “the specifically human form of activity,of interacting with the world in which man changes it and himself at the same time” (Giest & Lompscher,
2006, p. 27, our translation). According to Leontjew (1987), activity is driven by its motive, i.e., the object
that initiates the activity on which it is built, in our research this is the activity of teaching/learning ofnegative numbers. An activity is embedded in the cultural-historical situation of society, its developmentchanges the society as well as society changes the activity, and by “interacting with the world” (Giest &Lompscher, 2006, p. 27, our translation) learners change it and themselves at the same time. Activity (Figure2) is made and renewed by actions directed towards goals, which are related to the motive, here for example
to add positive and negative numbers. Each goal is pursued by a task, e.g., to manipulate one side of theequality 2 – 3 = 2 – 3 preserving the equality relation. This task could result in sub-goals with respective

Figure 2: Activity system

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A. Bikner-Ahsbahs, E. Vallejo-Vargas, S. Rohde, T. Janßen,
D. Reid, D. Alexandrovsky, A. Reinschluessel, T. Döring, and R. Malaka

sub-actions, e.g., to manipulate the numbers on the right side without changing its result (e.g., 2 – 3 = (2+1)– (3+1)).
The manner in which these actions are performed is determined by the conditions of the setting and theconstraints the goals meet in the specific situation, e.g., by available artifacts (e.g., digital or non-digital
tangibles), which allow specific operations to be performed but constrain others. For example, symbols usedwith paper and pencil differ from using a symbolic calculator or our MAL-system in which numbers are
represented by virtual tiles, which can be placed, removed or re-arranged (Figure 3), as we will see. Activity
can only be observed through its actions on the object that is the motive of the activity. Leontjew (1987)
accentuates that activity is in constant transformation, it is (re)constituted by variations of actions developing
over time. It even may change into an action, and when routinized it may be transformed into an operation(p. 109). While “[a]ctions becoming ever richer, outgrow the circle of activity that they realize, and enter into
a contradiction with motives that engender them” (Leontyev, 2009, p. 175), then new motives may enter thescene, restructure actions and establish a new activity, for example solving algebraic equations including
negative numbers after the latter is routinized.
Tools play an essential role in the transformative nature of an activity as they are cultural objects throughwhich people act and interact with the world around them and so change it and themselves (see Leontjev,
2009). Tools mediate an activity and thus, its transformation. However, when new tools like the MALsystem are developed, it is not so clear how they contribute to these transformations as their use in culture is
not yet established. This has yet to be explored.
In this paper, we focus on teaching/learning as an irreducible activity (see Shvarts & Abrahamson, 2019)with negative numbers shaping its motive. How students transform themselves and the teaching/learning, inwhich they interact with others and the tool, is mediated by the MAL-system, a digital artifact designed tosupport Multimodal Algebra Learning (MAL). For this setting, the MAL-system is elaborated as a digital
tool for teaching/learning algebra. Here we address negative numbers (Figure 3) with a specific focus on therole of digital feedback. When the students use the MAL-system, they may transform the artifact into an
instrument for a specific situation e.g., for calculating with negative numbers. To grasp this process, we adopt
the Instrumental Approach (Artigue, 2002) as a theoretical frame that complements Activity Theory throughconceptualizing the use of the tool. As it is rooted in activity theory (Vérillon & Rabardel, 1995) we expect

Figure 3: Teaching/learning setting (left figure), the MAL-system (right figure) with a task (1, text on top),
the subtraction zone (2, frame in red), zero-pair (3, red & blue tile), unit-tile (4, blue tile), minus-one-tile (5,red tile), grouping (6, tiles surrounded by a yellow frame), symbolic expression (7) of the tiles equation2 – 3 = 1 + (-1) – 1 (see Figure 5)

The Role of Feedback When Learning with a Digital Artifact:
A Theory Networking Case on Multimodal Algebra Learning99

to be able to coordinate the two theories empirically towards a theoretical local integration.
The Instrumental Approach

Figure 4: Instrumental genesis (© Springer Nature, permission received, see Abb. 2.12 in BiknerAhsbahs, 2022, p. 25; adapted from Fig. 3.1 in Trouche, 2020, p. 395)

In line with activity theory, we consider tools as mediating the dynamic of an activity in the way people use
the artifact to achieve a goal. According to this view, instrumental genesis is the process of how a (digital)
artifact becomes an individual instrument (Artigue, 2002), where an instrument is the hybrid collection of an
artifact with individually and/or socially developed schemes of using the artifact in a specific situation(Trouche, 2020). Instrumental genesis is shaped by two interrelated dialectic processes (Figure 4); by
instrumentation (“directed towards the subject”) and by instrumentalization (“directed towards the artifact”)(Trouche, 2020, p. 408). These are
two intrinsically intertwined processes constituting each instrumental genesis, leading a subject todevelop, from the artifact, an instrument for performing a particular task; the instrumentation process
is the tracer of the artifact on the subject’s activity, while the instrumentalization process is the tracerof the subject’s activity on the task (Trouche, 2020, p. 409).
Rabardel (2002) emphasizes that “[t]he two processes jointly contribute to the emergence and evolution of
instruments, even though, depending on the situations, one of them may be more developed, dominant oreven the only one implemented.” (p. 103) He further explains that artifact functions “are a characteristic
property of the instrumental entity, and because in our perspective this entity is born of both subject andartifact, functions are also mixed in nature. They are rooted in both the artifact and scheme components of
the instrument.” (p. 104) There are two levels of the evolution of instrumentalization by the attribution offunctions to an artifact enriching it with new extrinsic properties, that are acquired momentarily (local level)or durably (global level) (p. 106). This research only addresses the local level of instrumentalization. Figure
3 shows our teaching/learning setting in which the MAL-system is used as the artifact, with which students
learn about negative number.
The MAL-system designed for learning about negative numbers
Activity Theory supported the design of tasks for the MAL-system (Reinschluessel et al., 2018), which
provides virtual multi-touch tiles that can be arranged on a virtual “mat” to represent key structures of the

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D. Reid, D. Alexandrovsky, A. Reinschluessel, T. Döring, and R. Malakaarithmetic symbol system. In addition to the tiles, there are various features included in the system. An
unlimited stock of tiles (Figure 5, 1) and a waste basket (2) on the left and the right side of the mat to allow
two learners to work at the same time. The white area on the top (3) is for task descriptions. Pressing anarrow button to the top right (4) either moves on to the next task or restarts the current task. The pen-buttonon the top left (5) switches the interaction mode between either direct interaction with the tiles or ‘painting’mode that allows, drawing or deleting a ’Subtraction Zone’ (SZ, the red frame in Figure 5). The general area
to place the tiles is an Addition Zone’ (AZ).

Figure 5: Functionalities for interacting with the MAL-system artifact.

The main representational part of the MAL-system is the mat with two sides (left and right), representing anequation, and an equal or unequal sign in the middle giving feedback on the correctness of the equation
represented by the tile arrangements on both sides. Numbers are represented by square unit-tiles. Two
different colors indicate the sign of the numbers (red for minus, blue for plus) and mathematical operations

Figure 6: Key features of the MAL-system – defining (-1) as 0 – 1 by 2 – 3 = 0 – 1 = -1 + 1 – 1 = -1

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with numbers are represented by practical actions with the tiles to achieve specific goals. The design starts
from three fundamental actions: placing, removing, and re-arranging (Reid & Vallejo-Vargas, 2019), which
students spontaneously conduct. Placing tiles in the ‘Addition Zone’ (AZ) (Figure 6) means adding them,removing means subtracting them, and re-arranging produces visual patterns without changing the results
but expressed in the symbolic feedback e.g., if the tiles are pushed together in certain ways. A red “blob”
drawn by a finger on the mat signals to subtract; it is called a ‘Subtraction Zone’ (SZ). A SZ shows what is
to be subtracted from the AZ that contains the SZ. It is also possible to insert a SZ within a SZ. When tiles
are pushed together they are grouped and this is shown by a yellow surround (Figures 5, 6), meaning that thetiles are considered to represent a single number by the system.
The basic action rule ‘perform a calculation by acting on tiles on one side of the equation while keeping theequal sign’ allows for legitimate action schemes e.g., removing two unit-tiles, one from the AZ and one fromthe SZ (Figure 6), or placing or removing zero-pairs made of a blue and a red tile. Through preserving theequality in this setting, the MAL-system emphasizes the relational notion of the equal sign rather than theoperational notion.
Five kinds of digital feedback functions are implemented into the system (Figure 3). The equal or unequal
sign produces “balance feedback” showing if the equation is correct or not. In the “symbolic feedback” thesystem automatically converts the tiles collections into a symbolic arithmetic expression above the mat.
When tiles are grouped, this is shown by a yellow frame. If a zero-pair is grouped it vanishes from the mat
and if moves are illegal the tiles wiggle or return to their original place.
A layered model and feedback
Learning with an artifact that models operations on negative numbers encompasses both learning to use it
and letting it fade away from awareness. We assume that the activity of teaching/learning is layered (seeSwidan et al., 2020) and conceptualize this layered process by focusing on four interrelated layers of acting
and operating, as Leontjew’s conceptualizing of an activity system considers (Figure 2). Layer 1 consists oflearning to operate and handle the MAL-system as an artifact, Layer 2 is learning with the MAL-system e.g.,
how to handle equations based on operations with tiles. Layer 3 addresses learning to link MAL-expressions
with their respective arithmetic expressions, and Layer 4 addresses the emancipation from the MAL-system.
The third and fourth layers may capture transforming processes of the students as indicated by Leontjew. In
Layer 3, the students’ learning of negative numbers requires converting and expanding MAL-expressions tomore general mathematical expressions by the use of the symbolic system of arithmetic. In Layer 4, thestudents emancipate themselves from the artifact in that the students’ actions with negative numbers do notdepend on the artifact anymore but rather are used for other activities. In the latter case, the students becomeaware of the instrument’s limits although they also may use the tool on purpose if necessary. When theactivity of learning negative numbers is enriched, handling negative numbers may be transformed from a
motive for an activity to an action towards further learning goals, for example for learning to solve algebraic
equations with negative numbers. As Figure 2 indicates, the four layers of handling the MAL-system are notseparated but rather interrelated with Layer 2 building on Layer 1, and Layer 3 building on both. We
conjecture that the layered model can be empirically coordinated with the process of instrumental genesis of
the MAL-system when the students pursue goals provided by tasks or by the tutor.

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Feedback is an essential part of our digital tool. It is defined as a response that comes back to an eliciting
action of an initial situation (Bikner-Ahsbahs et al., 2020; Reid et al., 2022), either by feedback of the MALsystem or in speech (Figure 7). Using the MAL-system, digital feedback is always present as the MALsystem shows the result of the student’s moves on the mat. It becomes relevant only if someone in the group,
tutor or learner, reacts to it, bringing the feedback into the group’s discussion e.g., by identifying a mistake.
Only then it may initiate further reactions and contribute to learning. In this way the digital feedback becomespart of a feedback loop, which starts with an initial situation from which a feedback eliciting action emerges
e.g., making a mistake on one side of the mat shown by an unequal sign. The reaction to feedback initiates
retroactive actions directed to the initial situation, for example reflecting on a mistake and correcting it(Bikner-Ahsbahs et al., 2020). In the case of tutor feedback, the tutor orients his/her feedback to a goal tohelp the students solve a task, thus, tutor feedback is an action, provoking retroactive actions directly, without
intermediate reactions.

Figure 7: Feedback loop (© Springer Nature, permission received, see Abb. 2.14, Bikner-Ahsbahs, 2022,
p. 30; see Bikner-Ahsbahs et al., 2020).

Fyfe (2016) distinguishes between four kinds of feedback, which in our case are not all digital. The MALsystem may offer verification feedback (e.g., through balance or symbolic feedback), (indirect) correct
answer feedback (translating MAL-expressions into arithmetic expressions) and the tutor or the students
may offer explanatory feedback and try-again feedback, and we add giving-approval feedback that appeared
repeatedly. All these kinds of human feedback are directed to goals in the teaching/learning activity.

METHODOLOGICAL CONSIDERATIONS
We undertook an empirical study to investigate the function of digital feedback in the support of the teaching/learning of negative numbers with the MAL-system. The study consisted of four cases of experimental taskbased interviews of a task sequence (Maher & Sigley, 2020) of about 110 minutes each. A tutor acting as ateacher conducted the interview that was shaped as a teaching/learning arrangement with two students(Figure 3, left). The student pairs worked collaboratively with one iPad, but distributed their common
responsibility: each student was responsible for one side of the equation on mat while collaborating. The

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interview followed a sequence of 25 tasks (see appendix) that aimed at building knowledge for handlingnegative numbers. The first five tasks introduced the main functionalities of the MAL-system including the
SZ through well-known calculation tasks with natural numbers. Tasks 6–9 introduced negative numbers
based on zero-pairs and the SZ. Tasks 10–23 addressed adding and subtracting with negative numbers andreflecting and comparing on these more complex tasks. Tasks 24 and 25 were oral tasks to get an impressionof the students’ imagery of handling negative numbers.
When a tile is moved from the SZ to the AZ (respectively from AZ to SZ) the sign of the tile changes. Thiscan be modeled by an exchange action (placing a zero-pair in the AZ (SZ) and then removing matching tiles
from the AZ and the SZ) or an automatic color change shortcut (when the tile passes the boundary of the SZ).
Such a color change happens also to a tile that changes from one side of the equation to the other (notintroduced here). Two interviews were conducted with automatic color change and two without it toinvestigate if shortcuts foster or hinder handling negative numbers. In this paper, we will focus mainly on thetwo cases without color change. Color change is not the focus here but turned out to be an interesting
feedback function in itself.
We conducted all the interviews in one Grade 5 class of a German gymnasium, a school in the two-tier school
system with the higher achievement requirements. At this time of the school year, the students were familiar
with natural numbers, they had met fractions but negative numbers were not systematically taught before thestudy, but rather a next step in the mathematics curriculum.
The interviews were video recorded, then paraphrased and images from the videos were extracted. These
preliminary analyses showed that one case stood out as it provided specifically rich data because the students
worked on all the tasks, addressed all layers with minimal guidance of the tutor and, thus, maximal activationof the students. This case served as a focus case, with which we compared the other cases. For the focus case,
we transcribed the video recording until Task 19 verbatim and analyzed this case for each task episode. Tasks
20–25 were analyzed video based. The other cases were included by theoretical sampling i.e., if the analysisof the focus case yielded a conjecture, this was checked by analyzing respective episodes in the other cases
for evidence.
In our analyses of the focus case, we investigated turn-by-turn the teaching/learning of negative numbers byreconstructing the reactions and retroactions to digital feedback, identifying the goals and related actions andoperations. The findings of the analysis of the focus group are presented. Then they are used to coordinate
the Activity Theory with a focus on the activity system with the Instrumental Approach towards local
integration.

FINDINGS
Feedback loops of digital and tutor feedback
We first illustrate the loop of digital feedback for task 6.2 (Place a blue and a red tile on the side where you
are performing the calculation.). Before, the students have experienced what is represented in Figure 6,
upper row, where the symbolic feedback says: 2 – 3 = 0 – 1.
1 Timo: (places a blue tile in the AZ, right side) keeps the same I think (it appears ≠, he adds a red tile,

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then = appears, see Figure 3, right) yes!
2 Tutor: (translates symbolic feedback into words) mhm yeah, together a red and a blue tile equals zero
3 Timo: because a red is always negative, so -1
4 Simon: that is (refers to the blue tile) 1 and this then (refers to the red one) I think -1
Timo solves the task (#1), where he first conjectures that equality will be kept. His eliciting actions are
placing the two tiles one by one. Digital feedback shows an unequal sign first and then an equal sign. His
reaction is “yes!” directly taking this digital feedback into the group. A second reaction comes from the tutor
who refers to the symbolic feedback. Retroactive actions follow in two steps by Timo and Simon, whotogether explain why a red and a blue tile add up to 0 (#3, #4). (Figure 3)
Directly after this scene we can observe a loop for tutor feedback in task 6.3 (Could you now further calculate
as you just have done?). Timo misunderstands the goal and puts into the same AZ an additional zero-pair (thefeedback = confirms). This serves as an eliciting action for a tutor feedback, pointing to the symbolic
feedback area and saying that just one number is expected to appear on the right side. The tutor says, “Nowat the moment you have 2 + (–2) – 1” (Figure 6, lower row), providing explanatory feedback to clarify thegoal to yield one number.
5 Timo: Then I would push away (refers to the zero-pairs) the reds, and the other tiles, then we would
have only –1, hence only one number.
6 Simon: (pushes the blue and red tiles together, symbolic feedback is 0 – 1) this would be two
numbers
7 Tutor: Before you have always tried this tile (taps on the blue tile in the SZ, right side) could you doit here, too (puts one zero-pair back)8 Timo: I have an idea (takes a blue tile from the AZ and the SZ away, = appears), this is still equal,
my idea then is to do so (takes the zero-pair away) and then one can do this (makes the SZ disappear),then we only have one number (refers to the red tile)
Tutor feedback indicates what is expected, the goal (one number) initiating directly retroactive actions (#5,
#6) by Timo and Simon. Timo does not take 0 as a number, but Simon corrects him, pointing to the two
numbers in 0 – 1. The tutor takes #6 as an eliciting action (Figure 6) for the next tutor feedback (#7) and now
re-directs the task goal. Timo uses a previously developed instrumented actions to solve the task by showingthat taking a blue tile away from AZ as well as from SZ and then pushing a zero-pair together to delete itleaves one red tile only. After deleting the SZ tapping three times on it, (–1) is the only number left (#8).
(Figure 6).
What we observe here is a feedback chain of two feedback loops (Figure 7) without reaction. Such a reaction
is not necessary because a tutor feedback is already part of the social interaction in the group as tutor
feedback is normally directed towards the goal of the task to ensure the task will be solved.
The role of feedback in teaching/learning
By using the MAL-system, Timo and Simon went through different learning experiences. Learning the
MAL-system itself (Layer 1, see Figure 9) means acquiring instrumented action schemes to meet theaffordances of the system (e.g., pushing together tiles to group them standing for adding them). The firstphase of acting addressed handling well-known natural numbers, which helped the students to learn to use

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the MAL-system for performing operations, for example how to place five unit-tiles on the left and the rightside and to decompose them to learn the grouping function. Here knowledge about natural numbers is used
to explain what grouping means, that a yellow frame around the tiles indicates grouping and that thetranslation into the symbols serves as symbolic feedback. While the beginning of teaching/learning mainly
addresses the instrumentation of acting with the features of the artifact, instrumentalization is locally present,
too, through enriching the artifact with attributing to it external knowledge about natural numbers and theirdecomposition as additional properties.
Layer 1 is also addressed when going beyond natural numbers, for instance, when solving task 6.1 (Set theexpression 2 – 3 on both sides and try to find the result of the calculation. How far do you get with yourapproach so far?), and task 6.2 (Place a blue and a red tile on the addition zone on the side where you are
performing the calculation). The (balance and symbolic) digital feedback supports defining a zero-pair as‘the red and the blue tile making together 0’. Likewise, task 8.1 (Place together a unit-tile and a minus-onetile and see what happens) allows the students to learn how the MAL-system “behaves” after these tiles arepushed together; as Simon reacts with “they disappear”, he points to the action of ‘making a zero-pairdisappear’, which is being instrumented by the feedback from the MAL-system.
The students also learned with the MAL-system (Layer 2) e.g., Timo explains how this “vanishing” happens:“the result is also zero and therefore they are only additional tiles which are not needed”. Timo’s explanation
is based on how “the result” is experienced on Layer 1 whereas previously it was associated with the idea of
having only one number (the result) represented by symbolic feedback. His claim, “the result is also zero”,
points at least to two different situations, first, it can be read that 0 = 0, and second, after the blue and the redtiles were pushed together on the left side the result is also 0. In this case, Timo’s retroaction to the feedback
plays the role of an explanation initiated by the tutor. Thus, the fact that the zero-pairs vanished, initiatedlearning on Layer 1 (when we push a zero-pair together both tiles vanish) and mediated the transition toLayer 2, which resulted in an explanation (because they make together zero they are not needed) and wasfurther used as an idea for treating tiles arrangements shown next.
Making generalizations based on examples are also supported by the MAL-system’s feedback and as such
constitutes learning with the MAL-system. For example, when solving task 8.2 (Place additional tiles on theleft side only so that the equal sign is still displayed, where the other side shows zero). Unexpectedly, Timo
offers an idea saying “well, simply place zero-pairs” while Simon places zero-pairs on the left side of the mat
as eliciting actions. As the equal sign is kept, Timo explains “it always works with as many zero-pairs as wewant”, and the tutor reinforces him (“when you have the same number of minus-one-tiles and one-tiles thenthe result is always zero, right?”), which serves as a giving-approval tutor feedback. So balance feedback
mediated teaching/learning on Layer 2. In addition, the students’ expressing the rule “it always works withas many zero-pairs as we want” shows that they have developed an instrument for deleting zero-pairs tosolve MAL-tasks. In this instrumental genesis, the following dialectic of instrumentation andinstrumentalization is at play: the instrumented action of pushing pairs of tiles together enriched by knowledge
about zero-pairs and simultaneously it is Timo’s independent choice to place zero-pairs on the mat, check animplicit conjecture and finally infer a rule, thus, expressing an enriched understanding of the task situation
by instrumentalization.
Learning to link MAL-expressions and arithmetic expressions (Layer 3) is also supported by the MAL-

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system’s digital feedback. For instance, Timo solves Task 7 (Place a minus-one-tile on the left and no tiles
on the right. Represent a math task by placing tiles on the right side that has the result –1. You may start with
red tiles, but in the end you should only see blue tiles on the right side) by representing 1 – 2 with tiles,
placing a unit-tile in the AZ and two unit-tiles in the SZ on the right side. When doing so, balance feedback
shows that this is correct and Timo elaborates symbolically saying “because one minus two is exactly thesame as zero minus one”. Additionally, Simon suggests that it is always possible to add two blue tiles,
placing one on the AZ and one on the SZ to keep the equal sign. Thus, Timo and Simon learned to consider
the number –1 as the result of a mathematical operation e.g., 1 – 2, as well as a set of equivalent subtractions(–1 = 1 – 2 = 2 – 3 = 0 – 1 = …). This generalization of treatments of –1 represented in the arithmetic symbol
system (see Duval, 2008) is a consequence of the learning with the MAL-system and its feedback on Layer
2 that mediates the transition to Layer 3 in Timo’s explanation.

Figure 8: Making sense of the negative number -3.

In Task 8.3 (Place minus-one-tiles together. How does this affect the symbolic representation above?), theyplace three blue and three red tiles, representing 0, on the left and none on the right side. The symbolic
feedback shows 3 + (–1) + (–1) + (–1) and Simon reads it loudly (Figure 8). This symbolic feedback makes
Timo act on Layer 3 by identifying “So there appears a double bracket because otherwise it would calculate
plus minus three and that would not make sense and therefore there appears the bracket because it is anothercalculation” (Figure 8). Simon asks “will they disappear if they all are placed together?” His conjecture inthis question comes from the perception of how single zero-pairs vanished, an instrumented action learned
before, which he now generalizes. Simon tried it out but nothing happened.
As the MAL-system is not programmed to fulfill this conjecture, the expected feedback is not shown resulting
in a feedback by non-feedback, which points to Layer 1. However, pushing single zero-pairs together issuccessful. They disappear as Simon and Timo had experienced earlier. Simon’s expectation indicates a step
towards emancipating from the MAL-system (Layer 4) considering its limits and overcoming them byanother way of acting. This situation shows that, as the students proceed in their process of instrumental
genesis their flexibility in working at the three layers increases, interconnecting the layers, continuously
supported by feedback.
Reflective summary
Taken together, digital feedback mediates teaching/learning between the four layers (Figure 9). We haveshown shifts from Layer 1 to Layer 2 and from Layer 2 to Layer 3, but also to Layer 4. However, it may

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Figure 9: A layered model of teaching/learning mediated by the MAL-system and its feedback (doublearrow), to be coordinated with the process of its instrumental genesis (green rectangle)

happen that symbolic feedback or symbolic tasks initiate translating the task into a MAL-expression moving
tiles on the mat (shift from Layer 3 to Layer 2) and then coming back to the symbolic expression (Layer 3),
thus mediating acting forth and back between Layer 3 and Layer 2. Balance feedback may keep the students
acting on Layer 2, for example when they try to solve a problem with the tiles. An example is Task 9(Represent the number –5 on the left side. Then, on the right side, find different possibilities for representing
the same number.) In this task, the students discover the exchange-strategy i.e., Timo places a red tile on the
AZ and removes a blue tile from the SZ. A bit later, Simon varies this strategy, he removes a blue tile of the
SZ first and then places a red tile on the AZ instead and finally Simon removes a red tile from the AZ andplaces a blue tile on the SZ. In each case, balance feedback confirms the correctness of these exchange
strategies. The students provide several explanations addressing Layer 2 by referring to their experience withtiles, such as Timo: “Yes that would work too because it results in zero, and then we can exchange them.”
However, finally Timo converts tile expressions into symbolic language (see Duval, 2008) to explain: “Thatis because a blue one in the subtraction zone is also minus 1 and the red one is already minus 1 and that is
why one can simply exchange them because they are both the same.“ Here, first the balance feedback
mediates the students’ actions on Layer 2, but then an explanation is produced with symbolic terms linking
Layer 2 and Layer 3 without support by symbolic feedback. Although the students cannot manipulate symbols
as these only appear as symbolic feedback, they expand their symbolic knowledge.
Working through the 25 tasks (see appendix) the students become more and more fluent in changing andbridging layers, translating between them, and using them in entwined ways. Thus, their actions becomericher and richer, but they also struggled when subtracting a negative number, for example in the task 2 –(–5). The students in the end choose tile expressions and symbolic expressions depending on their own
needs, adding verbal explanations when necessary. Symbolic language was more often used for actions ofconjecturing and explaining whereas operations with tiles of the MAL-system were often used for actions ofchecking, showing, testing and exploring. Within this single session, we only found one indication for
emancipating from the MAL-system by going beyond its limits. However, the fact that the students became
able to decide, which representation system they wanted to use as an instrument for a specific situation showsthat the students have passed through a transformation process as described by Leontjew (1987).

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THE TEACHING/LEARNING ACTIVITY PASSING THROUGH THE LAYERS
For teaching/learning negative numbers the MAL-system is used as an artifact with digital feedback. Task 6(Set the expression 2 – 3 on both sides and try to find the result of the calculation) addresses acting with theartifact practically on Layer 1. For this, students must draw a blob to get a SZ placing tiles in the AZ and SZ(Layer 1). How far do you get with your approach so far? is the subsequent task aiming at reflecting andexplaining, hence, learning with the MAL-system in Layer 2 about the current limit of calculating 0 – 1,
since the result is not just one number as in other cases where the difference is positive. This results indefining a new number, (–1), by 0 – 1, which means practically placing a red tile in the AZ (Layer 1). (–1)
then substitutes 0 – 1, practically and symbolically (Layers 2, 3). Hence, in this process learning on Layer 1
and Layer 2 are intermingled by instrumental genesis. Specifically, when the students proceed in working
with negative numbers, more and more features are attributed to the MAL-system on Layer 1, addressing
learning with the system on Layer 2 that even goes over into Layer 3. This makes clear that Layer 1 andLayer 2 are neither separated nor can they be regarded as ordered stages during the teaching/learning.
Through the dialectic duality of instrumentation and instrumentalization they become entwined impacting
the students’ further personal development.
Task 7 (Place a minus-one-tile on the left and no tiles on the right. Represent a math task by placing tiles on
the right side that has the result –1. You may start with red tiles, but in the end you should only see blue tiles
on the right side) addresses a goal on Layer 3, to substitute a whole class of tile pairs, resulting in 2 – 3 = 1– 2 = 0 – 1 …, by (–1) or practically by placing a red tile, but also by other suitable tile pairs (Layer 1). Thus,
operations on Layer 1 realize actions on Layer 2, on which epistemic insight on Layer 3 is built-up andexpressed by generalizing, supported by the symbolic feedback and transforming the learners’ abilities
towards the use of symbolic expressions.
Feedback assists in bridging between and within the layers. When the students proceed to act in an additional
layer, the activity of teaching/learning is transformed in that it changes its nature. If Layer 2 adds to Layer 1,then the students use the MAL-system to explore a situation in their own way. However, this already happenson Layer 1 when the students begin to make sense of the actions they conduct and use the tiles in new ways
not expected by the researchers. Layer 3 is already at stake in the symbolic feedback, when tile representations
are converted into symbolic expressions by the system. In this way, the students may enlarge their capacity
to act by using the symbolic language to explain, whereas on Layer 2 they instead can show how a goal canbe achieved or explain – based on tiles – why the goal is achieved or not achieved. When Layer 4 is added,
the MAL-system does not constrain acting any longer; the MAL-system can even be left aside.

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LINKING TWO THEORIES TOWARDS LOCAL INTEGRATION

Figure 10: Local integration of activity theory and instrumental approach into a phenomenon of entwining
the layers and thus transforming the activity (see Bikner-Ahsbahs, 2022, p. 25)

Activity Theory was empirically coordinated with the Instrumental Approach to explore the support functionof digital feedback in the activity of teaching/learning negative numbers with the MAL-system (Figure 10).
We assumed that this activity is layered and we elaborated theoretically a four-layer model related to thestructure of the activity system. Our findings brought about the new phenomenon that these four layers
became continuously more entwined by digital as well as tutor feedback and the instrumental genesis of the
MAL-system in the course of the teaching/learning process while the activity develops and through that thestudents develop.
We could empirically confirm four layers of teaching/learning with the MAL-system (Layer 1: Learning the
MAL-system, Layer 2: Learning with the MAL-system, Layer 3: Linking MAL-expression with arithmeticalgebraic symbols, Layer 4: Emancipating from the MAL-system) and thus confirming our assumption of afour-layer model. We showed that digital feedback and tutor feedback foster the accumulation of students’learning passing through these layers and thereby becoming more flexible while finally indicating
emancipation from the MAL-system.
The MAL-system provides the condition for operations that shape the actions with negative numbers. These
conditions are given but the students had to learn how to handle the MAL-system as well, while transforming
this artifact into an instrument in a process of instrumental genesis. Analyzing the use of the artifact from theperspective of the Instrumental Approach made us identify instrumentation processes leading to various(sub-) instruments. These included instruments for:

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keeping the equality by adding zero-pairs, by simultaneously taking the same tiles out of the AZ andSZ on the same side or by exchanging a blue tile by a red tile (respectively a red tile by a blue tile)e.g., when the tile is shifted from the AZ to the SZ or vice versa,●

transforming the action of taking a tile away into representing this by inserting the tile into a SZ andvice versa●

expressing –1 by n blue and (n+1) red tiles (while n is a natural number) or placing n blue tiles in
the AZ and (n+1) blue tiles in the SZ.

These processes involved Layers 1 and 2, but also Layer 3 when symbolic expressions were used to explainthese processes.
After coordinating the two theories, Activity Theory and the Instrumental Approach, we will now justify thatthey can be considered being locally integrated by the new phenomenon in that we base our argumentationon Radford’s theory concept [(P, M, Q), R] (Figure 11).

Figure 11: [(P, M, Q), R] for local integration

We have added the principle P of assuming the four-layer model for the activity of teaching/learning. We
have elaborated that feedback mediates between the layers and that the instrumental genesis of the MALsystem entwines these layers. This is empirically shown by the methodology M of the dual-theory approachfor experimental task-based interviews, conducted in the students’ school. The goals of the tasks were exactly
those that could come next according to the curriculum. Thereby, we could study an experimental situationof using the artifact in the natural setting for which the MAL-system was developed. The paradigmatic
research question Q we have explored was: What is the function of digital feedback in the support of theactivity of teaching/learning of negative numbers with the MAL-system? This question links the activity withthe digital artifact that was not provided by the natural cultural class environment but that is a new tool. Our
main results are that the digital feedback of this artifact mediates between the layers, thereby supporting theprocess of instrumental genesis that – as shown empirically – entwines the layers and assists the students’
passing through the four layers in their transformation to emancipate from the MAL-system.
Figure 10 summarizes the local integration we have achieved by adding a principle P, a methodology M anda research question Q (see Figure 11). With the results a new linking principle on the role of feedback
becomes apparent, that feedback entwines the layers by instrumental genesis. Thus, the results R rebound on
the principles by adding to Activity Theory a better understanding of Leontjew’s observation that activity isin constant transformations becoming “richer and richer” (1987, p. 200, our translation) thus pointing to theThe Role of Feedback When Learning with a Digital Artifact:
A Theory Networking Case on Multimodal Algebra Learning111

accumulative nature of teaching/learning.

DISCUSSION
As previously investigated in another setting (Bikner-Ahsbahs et al., 2020) also in this study we foundevidence that digital feedback is embedded in a feedback loop shaped by the tutor and the students. Feedback
does not speak by itself, the effect of feedback is determined by the way students retroact to the feedback andthis may happen in a pragmatic way or in an epistemic way (Artigue, 2002; Reid et al., 2022). Feedback is
considered to play the pragmatic role if the student reacts by (re)performing a practical action, for example
drawing the blob on Layer 1 to subtract or tipping three times with two fingers on the SZ to make it disappear.
It is epistemic when students feel invited to express or approach understanding e.g., with an explanation ora generalization expressed with symbols on Layer 3. The MAL-system’s feedback normally does not show
any preferred function. The function arises through the students’ reactions and retroactive actions, when they
just act practically, reflect about their actions or try to explain the feedback. Tutor feedback however is anaction, directed to a teaching goal. It may show a pragmatic or epistemic function related to the goal thatindicates what is expected from the students.
Digital feedback functions, like the wiggling of tiles in cases of illegal moves, are artifacts themselves. They
need to be instrumented by the students. It is interesting to note that the students also used digital feedback
for their own purposes. For example, some students used the balance feedback to explore legal moves onboth sides of the mat as for an algebraic equation. Others provoked feedback to test conjectures and pushed
three red and three blue tiles together to see if they vanish. As this did not happen, they learned to take thismissing feedback as a new kind of feedback for what the system is not programmed to do. These and otherexplorations the students undertook are clear indicators for instrumentalization processes.
Applying Leontjew’s (1987) view on the transformative nature of activity to the teaching/learning with the
MAL-system the students’ personal transformation happened within this activity through feedback thatmediates between layers of learning. Instrumental genesis of the MAL-system is the basic process nested in
the more comprehensive learning process in which tiles of the artifact are transformed into arithmetic
symbols to be used as instruments in future. This transformation in learning advances the students’ humanability to act. More generally, Leontjev (2009) explains:
For man a tool is not only a certain object with some external shape and certain mechanical properties;
he sees it as an object embodying socially developed ways of acting with it … An adequate relation
between man and tool is therefore primarily expressed in his appropriating … the operations fixed init, by developing his own human abilities to be developed and used as instruments. (p. 266).

CONCLUDING REMARKSAs the Instrumental Approach is rooted in Activity Theory, our local integration is not a surprise. What is
new, however, is the fact that the data provide evidence that the transformations of the activity can be

112

A. Bikner-Ahsbahs, E. Vallejo-Vargas, S. Rohde, T. Janßen,
D. Reid, D. Alexandrovsky, A. Reinschluessel, T. Döring, and R. Malaka

considered as layered and that the digital feedback of the artifact impacts on this transformation by mediating
between the layers in the teaching/learning where instrumental genesis plays a key role. The networking of
the two theories allowed us to observe the process of intertwining of the layers in the instrumental genesis
and how the students developed in this process. A similar result – but not based on a layered model – wasattained in our study on dividing algebraic equations with the MAL-system when students transcended
integers (Janßen et al., 2020, 5.1.3).
Leontjew (1987) has emphasized that transformations may entail actions that are transformed into operations
and activities that are transformed into actions. We could not identify these kinds of transformations. The
reason is probably that the transformations we have observed within an activity consist of incremental
transformative steps whereas we can expect that a transformation going beyond an activity consists of astronger shift of view, for example, the shift that is necessary from arithmetic equations to algebra equations.
Such transformations go beyond just one session, they must be considered as long-term developments.
A glance into the layered model draws our attention to a third theory: Duval’s cognitive theory ofrepresentations (2008) warrants the relevance of transitions, within the same register (treatments) as within
the MAL-system and transitions between different registers (conversion) here between the registers of thearithmetic symbol system and the MAL-system. Our results indicate that students use these registers fordifferent things. For example they explore how to generalize with the MAL-system within Layer 2, but they
explain perceiving generalizations with tiles using arithmetic symbols in Layer 3. The problem we are facedwith in the networking of theories when we consider a cognitive approach is its incompatibility with thecultural-historical approach. Due to the contrasting principles and the lack of appropriate data, Duval’s
cognitive theory of representations could not be integrated into our theorizing process. But the ReMath
project has shown that networking of it with an activity theory perspective could be conducted in the design
studies (Artigue & Mariotti, 2014).
As transitions between registers were essential in our study, our layered model calls for further research ontransitions between registers and the role of digital feedback there, in the MAL-system and with otherartifacts. Our results raise the question whether such transitions may also lead to an emancipation from theartifacts by becoming more flexible in its use and how it is possible to simultaneously improve proficiency
in the use of symbolic systems when going beyond the artifact.
Acknowledgements This research is funded by the German Federal Ministry of Education and Research(BMBF) in the grant program “Erfahrbares Lernen” (experienceable learning) (grant no. 16SV7550K).
Note This article is an extended version of the paper “Role of feedback when learning with an artifact”
presented by Angelika Bikner-Ahsbahs, Estela Vallejo-Vargas and Steffen Rohde in the Topic Study Group
57on the Diversity of Theories of the 14th International Congress on Mathematics Instruction in Shanghai(2020, postponed to 2021). It encompasses more information about the digital resource (the MAL-learning
system) and more in-depth elaboration from the perspective of the networking of theories.

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113

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APPENDIX
Table: The list of tasks

Task 1
Task 2

Task 3

Task 4

Task 5
Task 6

Task 7

Task 8

Task 9
Task 10
Task 11
Task 12
Task 13
Task 14
Task 15
Task 16
Task 17
Task 18
Task 19

1.1
1.2
2.1.
2.2.

Move individual tiles to one side of the mat and observe what appears at the top.
What does each tile stand for? What happens when you push tiles together?
Represent the number 5 on the left side of the mat.
So far, you see an unequal sign in the middle. Move tiles from the stock to the right side andobserve what happens. What do you have to do to make the unequal sign an equal sign?
2.3. Find as many different representations as possible for the number 5 on the right.
3.1. So far, everything on the mat has been added up. If we want to subtract, we can draw a circle
with our fingers.
3.2. A subtraction zone is created. Everything that is placed here is subtracted from what is
outside. Try it!
4.1. Set the expression 3 – 2 on both sides.
4.2. Make a guess and justify it: What will happen if you remove a tile from each zone from one
side of the mat?
4.3. Try it: Remove a tile from each of the two zones on one of the two sides of the mat.
4.4. Use the procedure to get the result of the task.
Determine in the same way the result of the arithmetic task 5 – 3. Explain your approach.
6.1. Set the expression 2 – 3 on both sides and try to find the result of the calculation. How far
do you get with your approach?
6.2. Place a blue and a red tile on the addition zone on the side where you are performing the
calculation. When added together, they give 0, so an equal sign is still displayed.
6.3. Find out how the red tile is represented in the symbolic equation.
6.4. Now you can complete the task.
7.1. Place a minus-one tile on the left, no tiles on the right.
7.2. Represent a math task by placing tiles on the right side that has the result -1. You may startwith red tiles, but in the end you should only see blue tiles on the right side.
8.1. Place a unit-tile and a minus-one tile together and see what happens.
8.2. Place additional tiles on the left side only so that the equal sign is still displayed.
8.3. Place minus-one tiles together. How does this affect the symbolic representation above?
Represent the number -5 on the left side. Then, on the right side, find different possibilities forrepresenting the same number.
Calculate the result of 2 + 5 with the tiles.
Calculate the result of –2 + (–5) with the tiles
Calculate the result of 3 + (–2) with the tiles
Calculate the result of –2 + 3 with the tiles
Calculate the result of 3 + (–5) with the tiles
Calculate the result of –5 + 3 with the tiles
Calculate the result of 3 – 2 with the tiles
Calculate the result of 3 – 5 with the tiles
Calculate the result of –2 – 5 with the tiles
Calculate the result of 2 – (–5) with the tiles

A. Bikner-Ahsbahs, E. Vallejo-Vargas, S. Rohde, T. Janßen,
D. Reid, D. Alexandrovsky, A. Reinschluessel, T. Döring, and R. Malaka

116

Task 20 Link all tasks with the same solution
Task 21 How would you calculate 3 – (2 – 1) without tiles?
Task 22 22.1. Represent the solution of 3 – (2 – 1) with tiles and calculate the result.
22.2. Carlos believes that you can just calculate 3 – 2 – 1 = 0. Explain by comparing with yoursolution why this is wrong.
Task 23 Four students have solved the task 2 – (–3 – 1) in the subsequent way:a. Alex: 2 – 3 – 1 = –2
b. Kim: 2 + 3 – 1 = 4
c. Karina: 2 – (–3 – 1) = 2 – (–4) = 2 + 4 = 6
d. John: 2– (–4) = 2 – 4 = –2
23.1. Explain with tiles the correct solution path.
23.2. Explain in each step what you have done.
Task 24 24.1. What does –5 mean to you?
24.2. How would you explain the number –5 to your classmates?
24.3. How would you explain –5 to a classmate, who only knows about positive numbers (such as
the number 5)?
Task 25 25.1. Do you have an idea of what –4 – 2 means and how you can calculate it?
25.2. Do you have an idea of what –4 + (–2) means and how you can calculate it?

Angelika Bikner-Ahsbahs, bikner@math.uni-bremen.deBremen University, Germany & Western Norway University of Applied Sciences, Norway
https://orcid.org/0000-0002-0407-4897
Estela Vallejo-Vargas, e.vallejo@pucp.pe

Steffen Rohde, strohde@uni-bremen.deBremen University, Germany

Bremen University, Germany

https://orcid.org/0000-0003-0431-9708

https://orcid.org/0000-0001-5578-893X

Thomas Janßen, janssent@uni-bremen.deDavid Reid, david.reid@uia.no

Bremen University, Germany

University of Agder, Norway

https://orcid.org/0000-0001-7031-2251

https://orcid.org/0000-0001-5363-5948

Dmitry Alexandrovsky, dimi@tzi.de

Anke Reinschluessel, anke.reinschluessel@uni-bremen.deBremen University, TZI, Digital Media Lab, Germany

Bremen University, TZI, Digital Media Lab, Germany

https://orcid.org/0000-0001-9551-719X

https://orcid.org/0000-0002-6389-6436

Tanja Döring

Rainer Malaka

Bremen University, TZI, Digital Media Lab, Germany

Bremen University, TZI, Digital Media Lab, Germany

https://orcid.org/0000-0001-8648-340X

https://orcid.org/0000-0001-6463-4828

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The theory distinguished sequential levels of geometrical thought (Freudenthal, 1973; Hoffer, 1983;
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Radford, L. (2000). Signs and meanings in students’ emergent algebraic thinking: A semiotic analysis.
Educational Studies in Mathematics, 42 (3), 237-268.
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Fischbein, E. (1987). Intuition in science and mathematics: An educational approach. Dordrecht:
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Sierpinska, A. & Lerman, S. (1996). Epistemologies of mathematics and of mathematics education. In
A. J. Bishop et al. (Eds.), International handbook of mathematics education (pp. 827-876).
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HIROSHIMA JOURNAL
OF
MATHEMATICS EDUCATION
Volume 16 

2023

Tatsuya Mizoguchi
Editorial for Volume 16
Research Articles
Louise Meier Carlsen
Designing Activities for CAS-based Student
Work Realising the Lever Potential

1

Special Issue: Rethinking the Diversity of Theories in Mathematics
Education. Contributions Related to the Topic Study Group 57 of ICME 14
Angelika Bikner-Ahsbahs, Ivy Kidron,
Yusuke Shinno, and Takeshi Miyakawa
Guest Editorial

21

Research Articles
Michèle Artigue
Facing the Challenge of Theoretical Diversity: The Digital Case

27

Ivy Kidron
The Role of a Priori Analysis in Theories

45

Luis Radford
Ethics in the Mathematics Classroom

57

Yusuke Shinno and Tatsuya Mizoguchi
Networking Praxeologies and Theoretical Grain Sizes inMathematics Education: Cultural Issues Illustrated byThree Examples from the Japanese Research Context

77

Angelika Bikner-Ahsbahs, Estela Vallejo-Vargas, Steffen Rohde,
Thomas Janßen, David Reid, Dmitry Alexandrovsky,
Anke Reinschluessel, Tanja Döring, and Rainer Malaka
The Role of Feedback When Learning with a Digital Artifact:
A Theory Networking Case on Multimodal Algebra Learning95