Transitions Toward Digital Resources: Change, Invariance, and Orchestration

Abstract

This chapter reports on the work of Working Group 4 and focuses on the integration of digital resources into mathematics teaching and learning practices. There are five central sections, focusing on, instrumental genesis, instrumental orchestration, the documentational approach to didactics, digital resources and teacher education, and the design of learning environments with the use of digital resources. A range of constructs and theoretical approaches are covered in these five sections, and the opening section comments on construct validity and issues in “networking” theoretical frameworks. The chapter can be viewed as a literature review which surveys past and present (at the time of writing) scholarship with an eye to possible future research. The chapter is extensive in several dimensions: a large range of digital resources and applications are considered; the subjects using digital
resources are not just teachers but also students, student teachers and student teacher sections, and the opening section comments on construct validity and issues in “networking”theoretical frameworks.

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389© Springer Nature Switzerland AG 2019
L. Trouche et al. (eds.), The ‘Resource’ Approach to Mathematics Education,
Advances in Mathematics Education,
https://doi.org/10.1007/978-3-030-20393-1_12
Chapter 12
Transitions Toward Digital Resources:
Change, Invariance, andOrchestration
PaulDrijvers, VerônicaGitirana, JohnMonaghan, SametOkumus,
SylvaineBesnier, CerenusPfeiffer, ChristianMercat, AmandaThomas,
DaniloChristo, FranckBellemain, EleonoraFaggiano, JoséOrozco-Santiago,
MdutshekelwaNdlovu, Mariannevan Dijke-Droogers, Rogérioda Silva
Ignácio, OsamaSwidan, PedroLealdinoFilho, RafaelMarinho de
Albuquerque, SaidHadjerrouit, TuğçeKozaklıÜlger, AndersStøleFidje,
ElisabeteCunha, FreddyYesid Villamizar Araque, GaelNongni, SoniaIgliori,
ElenaNaftaliev, GiorgosPsycharis, TiphaineCarton, CharlotteKrog Skott,
JorgeGaona, RosilângelaLucena, JoséVieira do NascimentoJúnior,
RicardoTibúrcio, andAndersonRodrigues
Abstract This chapter reports on the work of Working Group 4 and focuses on the
integration of digital resources into mathematics teaching and learning practices.
There are ve central sections, focusing on, instrumental genesis, instrumental
orchestration, the documentational approach to didactics, digital resources and
teacher education, and the design of learning environments with the use of digital
resources. A range of constructs and theoretical approaches are covered in these ve
P. Drijvers · M. van Dijke-Droogers
Freudenthal Institute, Utrecht University, Utrecht, The Netherlands
V. Gitirana (*) · R. Lucena
CAA- Núcleo de Formação Docente, Federal University of Pernambuco, Caruaru, PE, Brazil
e-mail: veronica.gitirana@gmail.com
J. Monaghan
Agder University, Kristiansand, Norway
University of Leeds, Leeds, UK
S. Okumus
Recep Tayyip Erdogan University, Rize, Turkey
S. Besnier
CREAD: Center of Research on Education, Learning and Didactic, University Rennes 2,
Rennes, France
C. Pfeiffer
Cerenus Pfeiffer, Stellenbosch University, Stellenbosch, South Africa
C. Mercat · P. LealdinoFilho
S2HEP (EA4148), IREM, Claude Bernard Lyon 1 University, Lyon, France

390
sections, and the opening section comments on construct validity and issues in “net-
working” theoretical frameworks. The chapter can be viewed as a literature review
which surveys past and present (at the time of writing) scholarship with an eye to
possible future research. The chapter is extensive in several dimensions: a large
range of digital resources and applications are considered; the subjects using digital
resources are not just teachers but also students, student teachers and student teacher
A. Thomas
University of Nebraska-Lincoln, Lincoln, NE, USA
D. Christo · S. Igliori
PUC/SP– Pontifíce Catholic University of São Paulo, São Paulo, Brazil
F. Bellemain · R. da Silva Ignácio · R. Marinho de Albuquerque · R. Tibúrcio · A. Rodrigues
Federal University of Pernambuco, Recife, Brazil
E. Faggiano
University of Bari Aldo Moro, Bari, Italy
J. Orozco-Santiago · F. Yesid Villamizar Araque
Cinvestav-IPN, Centre for Research and Advanced Studies, Mexico City, Mexico
M. Ndlovu
University of Johannesburg, Johannesburg, South Africa
O. Swidan
Ben-Gurion University of the Negev, Beer Sheva, Israel
S. Hadjerrouit · A. S. Fidje
Agder University, Kristiansand, Norway
T. KozaklıÜlger
Bursa Uludağ University, Bursa, Turkey
E. Cunha
Instituto Politécnico de Viana do Castelo, Viana do castelo, Portugal
G. Nongni
University of Laval, Quebec City, Canada
E. Naftaliev
Achva Academic College, Arugot, Israel
G. Psycharis
National and Kapodistrian University of Athens, Athens, Greece
T. Carton
Paris 8 University, Saint-Denis, France
C. Krog Skott
University College Copenhagen, Copenhagen, Denmark
J. Gaona
Universidad Academia Humanismo Cristiano, Santiago, Chile
J. Vieira do NascimentoJúnior
State University of Feira de Santana, Feira de Santana, Brazil
P. Drijvers et al.

391
educators. Issues raised in the sections include individual and collective use of
resources, the adaptation of these resources for specic learning goals and to pre-
pare (pre- and in-service) teachers for the use of digital resources.
Keywords Digital resources · instrumental genesis · instrumental orchestration ·
documentational approach to didactics · teacher education · design of learning
environments
12.1 Introduction
PaulDrijvers, VerônicaGitirana, JohnMonaghan and SametOkumus
This chapter reports on the work of Working Group 4 (WG4), which had the title of
this chapter. This introduction to the chapter describes the original remit of WG4,
outlines the range of papers accepted, describes and comments on the formation of
ve thematic subgroups formed during the conference, and comments on constructs
and theoretical frameworks referred to in these thematic sub-groups.
The remit of WG4:
In this working group, some of these issues will be addressed from theoretical
perspectives, including instrumental genesis, instrumental orchestration and docu-
mentational genesis.
WG4 was the only Working Group to focus on digital resources and the only one
to include a focus on students’ use of the digital resources; it is hardly surprising,
then, that it was the biggest Working Group– 25 papers and 2 posters. The papers
can be found in the conference proceedings. The titles, below, give a avor of the
issues discussed in WG4 at the conference:
Digital resources have become an important part of teachers’ and students’
resource systems. the integration of digital resources into teaching and learn-
ing practices, however, raises many questions to teachers and educators.
How to choose appropriate resources from the myriad of available options?
How to adapt these resources to the specic learning goals at stake?
How to orchestrate the students’ use of the digital resources?
What do student resource systems look like?
How to prepare pre- and in- service teachers for these challenging tasks?
Which role can digital resources play in assessment?
Which opportunities do they offer for new learning formats, such as blended
learning and ipped classrooms?
How do classroom experiences inform the (re)design of a digital resource?
What are the options for personalized learning in adaptive environments?
12 Transitions Toward Digital Resources: Change, Invariance, andOrchestration

392
– A proposal of instrumental orchestration to integrate the teaching of physics and
mathematics.
– Instrumental meta-orchestration for teacher education.
– Orchestrations at kindergarten: articulation between manipulatives and digital
resources.
– Orchestrating the use of student-produced videos in mathematics teaching.
– Pre-service mathematics teachers’ investigation of the constraints of mathemati-
cal tools.
– Transition from a paper–pencil to a technology-enriched teaching environment:
A teacher use of technology and resource selection.
– An examination of teacher-generated denitions of digital instructional materials
in mathematics.
– Teachers’ intervention to foster inquiry-based learning in a dynamic technologi-
cal environment.
– TPACK addressed by trainee teacher educators’ documentation work.
– The birth of the documentary system of mathematics pre-service teachers in a
supervised internship with the creation of a digital textbook chapter.
– Planning of the teaching of the standard deviation using digital documentary
resources.
– LEMATEC Studium: A support resource for teaching mathematics.
– Using an app to collect data on students’ use of resources for learning
mathematics.
– Analysis of the use of resources on internet by pre-service mathematics
teachers.
– From sample to population: A hypothetical learning trajectory for informal sta-
tistical inference.
– Teaching and learning of function transformations in a GeoGebra-focused learn-
ing environment.
– Creation of innovative teaching situation through instrumental genesis to maxi-
mize teaching specic content: Acid–base chemical balance.
– A proposal of instrumental orchestration to introduce eigenvalues and eigenvec-
tors in a rst course of linear algebra for engineering students.
– Teaching computational thinking in class: A case for unplugged scenario.
– A computational support for the documentational work mathematics teachers
documentational work in EFII.
– From digital “bricolage” to the start of collective work: What inuences do sec-
ondary teachers non-formal digital practices have on their documentation work?
– Digital resources: Origami folding instructions as lever to mobilize geometric
concepts to solve problems.
– Exploring teachers’ design processes with different curriculum programs.
– Prospective teachers’ interactions with interactive diagrams: Semiotic tools,
challenges and new paths.
– Instructors’ decision-making when designing resources: The case of online
assessments.
P. Drijvers et al.

393
The Working Groups met in three 2-hour sessions over the conference. At rst, it
was difcult to see themes through the diversity of approaches and foci, but ve
themes appeared: instrumental genesis, instrumental orchestration, the documenta-
tional approach to didactics, teacher education, and design. We (the WG4 organiz-
ers) suggested these themes to the WG4 members and a collective discussion
endorsed the themes as representative. Members were asked to pick their theme-
group by going to different areas of a large room– everyone went to an area without
fuss (a form of “embodied validity” for the ve themes). The theme-groups then
started discussing their theme: initially how their paper tted into the theme and
then structuring ideas and constructs around the theme. These ve theme groups
liaised after the conference and produced the next ve sections of this chapter. We
now move on to constructs and theoretical frameworks.
We rst comment on what we mean by “constructs” and “theoretical frame-
works.” We use the word “construct” for a mental image and name of a phenome-
non. “Instrumental genesis” and “instrumental orchestration” are examples of
constructs. Zbiek et al. (2007) use constructs “that have specic applications to
mathematics, that have an empirical basis, and that help one understand relation-
ships among tool, activity, students, teacher, a curriculum content” (p.1172). Also,
academics may use constructs to talk about general properties of “things” in the real
world. Academics should, of course, ensure that the constructs they use are clearly
tied to the real world and accurately describe the phenomenon under examination–
this is called “construct validity.” A “theoretical framework” (or “theory” or “theo-
retical approach”) is a perspective for interpreting reality that usually includes a
number of constructs specic to the theory. There are “grand” and “local” theoreti-
cal frameworks: Piaget’s (1955) genetic epistemology is a grand theory and radical
constructivism, and the theory of didactical situations includes local theories that
are aligned with Piaget’s grand theory (see Lester 2005). The documentational
approach to didactics is a local theory, but what, if any, is the grand theory to which
it is aligned? “Networking” theoretical frameworks (using a bit of one in another)
has occupied the attention of mathematics education academics for several decades
(see Kidron etal. 2018); the state of the art with networking theoretical frameworks
is that it is often possible (at some level) but must be done with careful attention to
detail. We now comment on constructs and theoretical frameworks referred to in
these thematic sub-groups.
The principal construct of Sect. 12.2 is instrumental genesis. It is aligned with
Rabardel’s instrumentation theory and constructs from Vergnaud’s (2011) Piagetian
approach (e.g., operational invariants). The authors utilize Gibson’s construct of
affordances in their discussion of instrumental genesis. The principal construct of
Sect. 12.3 is instrumental orchestration (IO) (which makes essential use of instru-
mental genesis). Central constructs of IO are “didactical congurations” and
“exploitation modes” and, in later formulations, “didactical performances.” There is
mention of possible networking with Koehler and Mishra’s (2009) Technological,
Pedagogical and Content Knowledge (TPACK) framework teachers’ professional
knowledge and Ruthven’s (2014) model of Structuring features of classroom prac-
tice. Besnier and Gueudet’s (2016) construct of “chaining orchestrations” (which
12 Transitions Toward Digital Resources: Change, Invariance, andOrchestration

394
itself arose from networking IO with the Anthropological theory of didactics) is also
used. The section ends by employing ideas from Lakoff and Núñez’s (2000) embod-
ied cognition perspective. The focus of Sect. 12.4 is the local theory documenta-
tional approach to didactics (DAD), which links, obviously, to IO. The section
explicitly discusses networking DAD to other theoretical frameworks, for example,
Activity Theory, the Joint action theory in didactics and TPACK (and a variant, the
model Mathematical pedagogical technological knowledge). A host of emerging
construct is considered, for example, “documentational trajectory” and “resource
system metamorphosis,” among others. Section 12.5 employs constructs introduced
in earlier sections but implicitly introduces new theoretical frameworks because this
section is essentially concerned with teacher education and how one views teachers,
and teaching depends very much on one’s theoretical perspective. There was no
room in that section to consider possible tensions in some of these perspectives, but
we take the opportunity here to mention that Teresa Assude’s approach is informed
by the Anthropological theory of didactics and Kathleen Heid’s by constructivism
and that networking these approaches is problematic. Section 12.6 is concerned
with the design of learning environments. As with other sections, it considers vari-
ous theoretical approaches and employs a number of specialized constructs, but an
added element of complexity is that the design of learning environments is not just
a meeting of approaches, it is a meeting place of disciplines– computer science and
didactics (with ideas and approaches from engineering).
We make these comments on constructs and theoretical frameworks partly as an
advanced warning to the reader but partly to remind ourselves to be aware of the
importance of construct validity and the difculty of networking theoretical
approaches.
12.2 Instrumental Genesis: ATheoretical Lens toStudy
Mathematical Activities withDigital Tools
CerenusPfeiffer, DaniloChristo, MdutshekelwaNdlovu, SaidHadjerrouit and
SoniaIgliori
This section focuses on instrumental genesis.1 For this, we will seek to investigate,
in a synthetic way, what instrumental genesis means. One answer to this question
was presented by Gueudet (Chap. 2) when she took up the foundation elements of
this theory, the distinction between an artifact (a digital artifact for the purpose of
this section), a product of human activity designed for human activity and directed
by objectives, and an instrument developed by a given subject (Rabardel 1995); the
notion of instrument as an artifact + utilization scheme; the notion of scheme with
1 This section also mentions “documentational genesis.” Sect. 12.4 below considers the
Documentational Approach to Didactics. The processes governing instrumental genesis and docu-
mentational genesis are similar, although the underlying artifacts these processes work on differ.
P. Drijvers et al.

395
its four components, the objective of the activity, rules of action, operational invari-
ants, and inferences (Vergnaud 1996); and highlighted two processes behind instru-
mental genesis– instrumentation and instrumentalization.
12.2.1 Theoretical Approaches toInstrumental Genesis
Drijvers and Trouche (2008) view instrumentalization as the process by which sub-
jects shape the instrument and its use, and instrumentation is the process by which
the artifact inuences the activity and the thinking of the subjects. Both aspects
inuence and are inuenced by the pedagogical design of the teachers, which gives
rise to this genesis. Ratnayake and Thomas (2018) argue that teachers have to adapt
digital resources and appropriate them to their practices by shaping and transform-
ing them (instrumentalization and instrumentation). Lagrange and Monaghan
(2009) argue that the availability of technology challenges the stability of teaching
practices; techniques that are used in “traditional” settings can no longer be applied
in a routine-like manner when technology is available. In order to help teachers to
benet from technological resources in everyday mathematics teaching, it is there-
fore important to have more knowledge about the new teaching techniques that
emerge in the technology-rich classroom and how these relate to teachers’ views on
mathematics education and the role of technology as a teaching resource therein
(Drijvers etal. 2010). Drijvers etal. (2013a) also contend that a deep understanding
of students’ learning processes is a core challenge of research in mathematics
education.
The theory of instrumental genesis (TIG) ascribes a major role to artifacts that
mediate human activity in carrying out a task (Drijvers etal. 2013a). When the arti-
fact is used to carry out a task, it becomes an instrument (Drijvers and Trouche
2008). Ndlovu etal. (2011) also view instrumentation as the process by which the
user of the artifact is mastered by his or her tools or by which the artifact inuences
the user by allowing him or her to develop activity or utilization schemes within
some boundaries. Such limits include constraints, which assist the user in one way
and impede in another; enablements, which effectively make the user able to do
something; and potentialities, which open up possibilities and affordances that favor
particular gestures or movement sequences (see also Noss and Hoyles 1996; Trouche
2004).
The notion of “affordance” is particularly important to the theory of instrumental
genesis. The notion was originally proposed by E & J Gibson in the 1950s. Gibson
(1977) is an authoritative account and refers to action possibilities, that is, what the
user can do with an object. Norman (1988) applied the notion of affordances to digi-
tal tools. In this context, affordances refer to the perceived and actual properties of
the tool, primarily those fundamental properties that determine just how the tool
could possibly be used. Kirchner etal. (2004) developed three levels of affordances
for digital tools. Firstly, technological affordances are properties of digital tools that
are linked to usability issues. Secondly, educational or pedagogical affordances are
12 Transitions Toward Digital Resources: Change, Invariance, andOrchestration

396
properties of tools that act as facilitators of teaching and learning, and, nally, social
affordances are properties of tools that act as social facilitators.
Given these considerations, we argue that, within the context of instrumental
genesis, the affordances of digital tools are actualized at the technological, didacti-
cal/pedagogical and social levels. Technological affordances provide opportunities
that facilitate the learning of mathematics, such as ease-of-use, ease-of navigation,
accurate and quick completion of mathematical activities, drawing of graphs and
functions, etc. Didactical/pedagogical affordances help in building and transform-
ing mathematical expressions that support conceptual understanding of mathemat-
ics, such as collecting real data and creating a mathematical model; using a slider to
vary a parameter or drag the vertex points of a triangle in geometry software; mov-
ing between symbolic, numerical, and graphical representations; simulating math-
ematical concepts; or exploring regularity, change, etc. Finally, social affordances
facilitate group work and discussion, collaborative learning, and students taking
greater control over their own learning (see Hadjerrouit 2017).
12.2.2 Papers Presented at theConference
Taranto etal. (2018) treat a Massive Open Online Course (MOOC) as an artifact,
that is, a static set of materials. They claim that when a MOOC module is activated,
it dynamically generates a complex structure that is called an ecosystem. The
researchers add that the process of transforming an artifact into an instrument is
replaced here by the evolution, artifact– ecosystem/instrument.
In a similar vein, Ratnayake and Thomas (2018) analyze the process of designing
tasks using the structure of documentational genesis and identify a series of items in
the set of resources employed by the research communities of teachers. These
include artifacts such as the criteria for designing rich tasks, the three-point frame-
work for lesson planning, delivery and review and an exemplary task, GeoGebra,
students’ worksheet, and an A-level syllabus. The tasks before and after an interven-
tion were evaluated using the Rich Task Framework, which comprises 12 factors
including the appropriateness of the tasks for the instrumental genesis of the stu-
dent. They claim that groups that freely shared ideas were more exible in their use
of digital technology than others, seeking and incorporating appropriate digital
technology techniques into the tasks to help students understand mathematical con-
cepts. This, they claim, allowed them to improve their personal instrumental genesis
by learning new techniques and follow-up schemes; this evidences a development
in professional instrumental genesis. Overall, this research suggests that there is
merit in encouraging teachers to design digital technology tasks by working col-
laboratively in small groups provided specic support is given to the professional
development of teachers to assist them. In turn, there may be benecial effects in the
broader documentary and instrumental geneses.
In Lucena etal. (2018), the notion of instrumental genesis appears in the scope
of IO, when metaphorically they say that an orchestra in general can be recognized
P. Drijvers et al.

397
as an instrumental grouping comprising a conductor and instrumentalists, their
instruments and scores, all well arranged in a space for the purpose of performing a
piece of music. This concept of IO aims to model the practice of the teacher to sus-
tain the instrumental genesis of students in rich mathematical learning. They cite
Rabardel (1995) to argue that instrumental genesis is a transformation of an artifact
by the action of someone, transforming it into an instrument while the subject goes
through the process of instrumentation integrating it into their practice. The trans-
formation of the artifact into an instrument is not characteristic of the structure of
the tool but of the schemes that the subject develops to integrate it. They go on to
say that, from their perspective of students’ instrumental genesis, two concepts are
fundamental for the orchestrating teacher: the concept of scheme and the concept of
situation (that does not assume here the meaning of didactic situation but the mean-
ing of task). The idea here is that any complex situation can be analyzed as a com-
bination of tasks, each with its own nature, and difculties are important to know.
Orozco etal. (2018) inform us that the integration and the use of new technolo-
gies in mathematics education have had an impact, but in many cases, this impact is
anarchic; the digital age induces change to the access of information and construc-
tion of knowledge, among other actions by human beings. It is a fact that these new
and sophisticated tools do not immediately become efcient instruments of teach-
ing-learning. The instrumental approach (Guin and Trouche 1999) is a structure that
allows one to take into account the role of technology in learning and teaching
mathematics, in which the role of the teacher in this structure is fundamental, since
s/he is responsible for the instrumental genesis of students, carried out by means of
orchestrations (Drijvers etal. 2010).
In the paper by Igliori and Almeida (2018), instrumental genesis is implicit, since
it is present in the production of the teacher’s documentation. The paper presents a
web tool, built for the purpose of providing digital resources, which favor the instru-
mentalization of the user teacher. The construction steps, from the digital objects to
the teaching of mathematics at the elementary school level, can still be used as sup-
port in the work of the instrumentalization of its students. The process of instrumen-
talization is the rst step of instrumental genesis.
Pfeiffer and Ndlovu (2018) describe their research carried out with students in a
bridging program at a South African university participating in a qualitative study
with TIG as a theoretical framework. This exploratory study investigates which
instrumentation processes are dominant in a GeoGebra-enhanced mathematics
learning environment to support students to develop an understanding of concepts
in function transformations and circle geometry. The instrumentation process in this
study was thus how GeoGebra shaped the thinking of the students and how it helped
them to understand concepts. The instrumentalization process, in turn, was how the
students used GeoGebra on their own as a tool, for example, to validate their answers
and test their conjectures. During in-depth and focus-group interviews in Pfeiffer’s
(2017) study, students were asked if GeoGebra had helped them with certain con-
cepts. Most of them afrmed that GeoGebra use had indeed helped them to better
understand function transformation and circle geometry. The following responses
concern perceived affordances of GeoGebra.
12 Transitions Toward Digital Resources: Change, Invariance, andOrchestration

398
– “I knew from last year that if you reected a graph about the y-axis, the x-values
change sign (from positive to negative or negative to positive). We were just
doing it mechanically, but with GeoGebra, this year, I could see what is going on
(visual affordance) and it made sense.”
– “I could see the signs. I understand now better why the sign changes if g(x) is
reected in x-axis ….. then I know the negative sign has to stand in front of g(x)
and it also meant that the new graph is h(x).”
– “With all the different circle geometry theorems, you could see which angles are
equal to each other or different segments. I could see them.”
– “It helped. Specially to see them visually (visual affordances). Like the chords,
the angle subtended by the same chord to show that they are equal.”
These responses suggest that the students acquired “physical and logico-mathe-
matical” knowledge of function transformations. GeoGebra use also afforded the
students an opportunity to link visual graphic representations to the algebraic repre-
sentations of the same concepts.
The visual affordances of GeoGebra identied by students are as follows:
– GeoGebra acted as a tool to visualize the transformations, the utilization scheme
of changing sliders, and gave them an enactive sense of what the parameter in the
equations mean: that the change in the equation transforms the original function;
the instrumented action scheme of typing the transformation notation gave them
visual understanding of how the horizontal translation, reection in x- and y-axis,
occurs; and that the reection in the y = x means the inverse graph of a
function.
– The utilization scheme of changing the colors of the different resultant graphs
helped students compare them to the original graph, resulting in better under-
standing the nature of the “shifts.”
– GeoGebra use gave students a better understanding of sketching the inverse
graph of an exponential function because they came to know it as a mirror image
of the exponential graph. It enabled them to use the instrumented action scheme
of sketching the graph by using critical points.
– GeoGebra use also helped with the understanding (instrumental genesis) of theo-
rems in circle geometry – for example, showing which angles are equal and
which angles are subtended by the same chord or arcs. Responses showed how
GeoGebra shaped the thinking of the students and how it helped them to under-
stand and visualize theorems.
With regard to the instrumentalization process, responses of the students showed
how GeoGebra was independently utilized as a tool to validate their answers and
test their conjectures. Observations showed how the students discussed and ana-
lyzed the properties of a GeoGebra applet and conjectured what the transformation
of the function should be. They tested and validated their conjectures by dragging
sliders in the applet. The students, therefore, had an opportunity to make and vali-
date conclusions about the type of transformation on the basis of intuition or experi-
P. Drijvers et al.

399
ence obtained through GeoGebra. Observations also showed how students acquired
or discovered physical knowledge of function transformations and circle geometry.
We now turn to visualization, the ability to use and reect upon pictures, graphs,
animations, images, and diagrams on paper or with digital tools with the purpose of
communicating information, thinking about and advancing understandings (Arcavi
2003). Visualization tools are becoming important in mathematics education.
Two papers emphasize the role of visualization tools for teaching and learning
mathematics. Barbosa and Vale (2018) highlight the potential of visual solutions
and strategies to promote mathematical learning. Even though the term “instrumen-
tal genesis” is not explicitly mentioned in the article, there are clear indications of
instrumentation and instrumentalization processes. The authors present two exam-
ples of tasks and their visual solutions. The rst one is related to the area of the area
of rhombus using a visual gure with colors (a square) as an artifact with four mid-
points of each side of the square. In the process of instrumentation, the students
shaped the artifact using their own mathematical knowledge to nd a visual solution
to the problem, while the artifact enabled the students to produce the solutions
within its constraints (instrumentalization). The second example involved the
manipulation of rational numbers, equations, and proportionality. The students pro-
duced many solutions including a solution obtained by visualization. Similar to the
rhombus task, instrumentation and instrumentalization processes were at work in
this case, too. The paper points to the affordances of visualizations to achieve more
efcient solutions, since these provide additional strategies. The social affordances
of the tool are also emphasized, since it allowed students to discuss their
strategies.
The second paper on visualization tools, Martinez etal. (2018), describes a case
of instrumental genesis at a Mexican university, where teachers use digital technol-
ogy to give feedback on their teaching practice in order to move from instrumenta-
tion to instrumentalization and orchestration processes. The intention is to establish
the importance of the digital tools as supports for didactical activities and mediators
of mathematical knowledge in classrooms. This can be characterized as the teach-
er’s instrumental genesis, where the processes of instrumentation and instrumental-
ization are intertwined, involving the planning of the class session, selection,
distribution, and management of the artifacts with their affordances and constraints;
giving rise to a scheme of use by identifying the features of the artifacts, the subjects
of the activity, and their knowledge; and thus providing a form of IO.
The nal paper we consider in this section demonstrating the usefulness of the
theory of instrumental genesis is the paper on teaching computational thinking (CT)
in classroom environments (Lealdino Filho and Mercat 2018). Even though the pro-
cess of instrumental genesis is not explicitly mentioned in this article, there are clear
indications of instrumentation and instrumentalization processes, affordances, and
constraints of the artifacts as well. As an example of CT, the article presents a binary
magic trick using ve cards with numbers. The task consists of asking the student to
choose a secret number between 1 and 31, showing her/him each card one after
another to decide whether the card contains the secret number. In terms of instru-
mentation, the work consists of understanding the binary magic trick and writing an
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400
algorithm which performs it. The algorithm is the artifact that is shaped by the stu-
dent using his/her knowledge of CT and mathematics. The algorithm itself has its
own rules with affordances and constraints that must be followed by the students in
order to return the same result independently of who performs the steps. This is the
instrumentalization process. The article shows the possibility of using CT to design
an algorithm without the use of digital tools. Implementing the algorithm on a com-
puter follows basically the same logic, but it requires understanding the program-
ming language in order to create an algorithm for the computer to yield the solution
to be achieved. In addition, instrumental geneses that use programming languages
as artifacts that mediate between the student and the task have an element of creativ-
ity in order to solve the problem. Programming is an iterative process where it is
common for a program not to work as expected and thus cannot yield the correct
answer the rst time it is performed, in contrast to conventional digital tools such as
GeoGebra, for example. The search for a better or more efcient solution can be
achieved in various ways such as testing the program with different data and strate-
gies, discussions with fellow students and the teacher, or conducting a search on the
Web for alternative solutions, etc. The programming process provides affordances
and constraints at the technological, pedagogical/didactical, and social level and
creates interactions that facilitate the emergence of varied utilization schemes for
the students. The combination of technological, pedagogical, and social elements,
in addition to the creativity element of the programming process has huge impact on
students’ instrumental genesis and the schemes they develop when using CT and
mathematics.
We conclude this section by noting that although much has been built on the
notion of instrumental genesis (e.g., instrumental orchestration and documenta-
tional genesis), there is still much to learn about instrumental genesis itself.
12.3 Revisiting Instrumental Orchestration: PastFindings
andFuture Perspectives
PaulDrijvers, SylvaineBesnier, JoséOrozco-Santiago, TuğçeKozaklıÜlger
and FreddyYesid Villamizar Araque
Soon after instrumental genesis was recognized as a key process in exploiting the
potential of digital technology in mathematics education, it was acknowledged that
teachers play a crucial role in enhancing this process. Instrumental orchestration
arose an answer to the question of how to foster students’ instrumental genesis.
Even though the focus may have shifted toward teachers practices in terms of the
DAD since then, this chapter revisits IO and identies ve future perspectives of
this notion, to further extend its value for mathematics educations, and for teacher
training in particular: (1) a shift toward student-centered orchestrations, (2) extend-
ing the repertoire of orchestrations, (3) chaining orchestrations, (4) didactical per-
formance, and (5) teachers’ and students’ gestures.
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401
12.3.1 Past Findings
As shown in Sect. 12.2, the notion of instrumental genesis was an important step
ahead in research on the use of digital tools in mathematics education. It acknowl-
edged the subtlety and the complexity of turning artifacts into (parts of) instruments
through the joint development of techniques for using a particular tool for a particu-
lar task, and the corresponding insights to understand the mathematics involved.
Soon, the crucial role of teachers in this process was recognized. The question was
what teachers can do to foster this co-emergence of techniques and schemes, i.e., to
create appropriate environments to make instrumental genesis happen. This is where
the notion of IO came into play.
An instrumental orchestration was dened by Trouche (2004) as the teacher’s
intentional and systematic organization and use of the various artifacts available in a
learning environment in a given mathematical task situation to guide students’
instrumental genesis. An IO consists of two layers, a didactical conguration and an
exploitation mode. A didactical conguration is an arrangement of artifacts in the
environment or, in other words, a conguration of the teaching setting and the arti-
facts involved in it. Through the didactical conguration, the teacher “sets the scene”
for instrumental genesis. An exploitation mode is the way the teacher wants to
exploit a didactical conguration for the benet of the didactical intentions. It is the
expected way in which the didactical conguration can be exploited for the targeted
instrumental genesis. As a paradigmatic example of an IO, Trouche (2004) presented
the “Sherpa orchestration,” in which a student uses an artifact in front of the class,
thus allowing the teacher to guide the use, the students to react to that and the Sherpa
student (and, through her/him, the class) to get feedback on the techniques in use.
This notion of IO soon received attention. Assude (2007) introduced the notion
of instrumental integration, including initiation, exploration, reinforcement, and
symbiosis (see also Hollebrands and Okumus 2018). Also, it was pointed out that,
in spite of the somewhat formal word “orchestration,” the teacher in this model
should not be considered a conductor of a symphony orchestra but, rather, a jazz
band leader who prepares a global partition but also is open to improvisation and
interpretation (Drijvers and Trouche 2008; Trouche and Drijvers 2010).
To do justice to the multiple ad hoc decisions that teachers take in split seconds
while teaching, the IO model was expanded with a third layer called didactical per-
formance (Drijvers etal. 2010). The didactical performance refers to all (bounded)
choices made on the y with respect to how to actually perform in the chosen didac-
tical conguration and exploitation mode: what question to pose now, how to do
justice to (or to set aside) any particular student input, how to deal with an unex-
pected aspect of the mathematical task or the technological tool, or other emerging
goals. Figure12.1 depicts the three IO layers.
Since its early years, the notion of IO has widened its scope. Its relationships with
other models for teacher behavior and teacher knowledge have been investigated. For
example, Tabach (2011, 2013) and Drijvers etal. (2013b) combined and contrasted
the IO approach with the TPACK model on teachers’ professional knowledge. The
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402
two lenses showed to be complementary and together provided a richer view on teach-
ers’ practices in ICT-rich classrooms. Also, the relationships with Ruthven’s model of
Structuring features of classroom practice (SFCP) framework have been explored
(Bozkurt and Ruthven 2017; Ruthven 2014). In particular, the instrumental orchestra-
tion shows resemblance with the Activity Structure notion in the SFCP framework. To
explore another connection, Trouche and Drijvers (2014) investigated the relation
between instrumental orchestration and the notion of webbing. Whereas webbing
focuses on the construction of a web of connected mathematical ideas, instrumental
orchestration stresses the situation that invites this process. A further focus on teach-
ers’ practices with respect to designing, using and arranging resources has been devel-
oped under the name of the documentational approach to didactics, which is elaborated
in Sect. 12.4. As far as student level and age are concerned, the work on instrumental
orchestration originally focused on the upper secondary level, but since then, it has
been widened, as far as kindergarten level (Besnier 2018; Carlsen etal. 2016).
If we look back at these developments, how well did IO do over the previous
15years? It did lead to the acknowledgement that the way in which teachers foster
instrumental genesis is a key issue. In addition to this, some orchestration types
have been identied. In spite of the widening scope described above, however, we
wonder if IO really had the impact that it might have had. Our view is that its poten-
tial has not yet been fully exploited, if we take into account the limited number of
publications on this topic on the one hand, and the increasing role of digital tools in
mathematics education on the other. The agenda for this section, therefore, is to
revitalize the notion of IO.To do so, we outline ve future perspectives that we
consider promising and address below: (1) a shift toward student-centered orches-
trations, (2) extending the repertoire of orchestrations, (3) chaining orchestrations,
(4) didactical performance, and (5) teachers’ and students’ gestures.
12.3.2 Future Perspectives
12.3.2.1 A Shift Toward Student-Centered Orchestrations
When digital technology became more common in mathematics education, it was
hoped that it would offer opportunities for students’ ownership of their learning and
that it would provide a “context where the learner is consciously engaged in con-
structing a public entity, whether it’s a sand castle on the beach or a theory of the
Fig. 12.1 The three-layer
model of an IO
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403
universe” (Papert and Harel 1991, p.1). In line with this view, one might be tempted
to expect new types of student-centered orchestrations to emerge, which invite stu-
dents engage in mathematics through creating mathematical objects.
Findings so far, however, seem to show a dominance of teacher-centered orches-
trations. Drijvers etal. (2010) quote teachers privileging teacher-centered orchestra-
tions such as Technical demo because they feel more in control of the situation,
compared to student-centered orchestrations. This reminds of the experiences in the
UK, where the large-scale introduction of interactive whiteboards in the UK led to
traditional teacher-centered teaching practices: “the mere introduction of such tech-
nologies is insufcient to promote greater interactivity in the classroom, and indeed,
that use may have had detrimental effects” (Rudd 2007, p.2).
As another example of teachers preferring teacher-centered orchestrations,
Kozaklı Ülger and Tapan Broutin (2018) described a study on one mathematics
teacher’s integration of technology in her course. Compared to her lessons, which
usually were traditional, new orchestrations were observed in her technology-
enriched lessons, and she implemented various orchestration types in the teaching
process: Explain-the-screen, Discuss-the-screen, Link-screen-board and Not-use-
tech (Drijvers etal. 2010). However, this did not prevent the teaching process from
being teacher-centered. In spite of tablets with GeoGebra being available, the
teacher hardly used them and stuck to whole-class teaching. This preference for
teacher-centered orchestrations may have different reasons. The rst reason is that
students lack the skills of using software, in this study GeoGebra and that the teacher
does not want to spend precious teaching time to make them more experienced. The
second reason is the lack of technological-pedagogical knowledge and experience
by the teacher. Consequently, she might feel losing control if much is left to the
students’ initiative. For example, students might come up with solutions, strategies
and questions that are beyond the teacher’s knowledge and experience.
To make students take full benet of the potential digital technology offers, it
might be good to use more student-oriented orchestrations. To be capable of doing
so, teachers should feel the condence on their own technical skills, trust their stu-
dent learning capacities with respect to using digital tools, and dare to be out of
control and to deal with unexpected situations. How pre- and in-service mathemat-
ics teachers can acquire these skills and how they can make a shift toward student-
centered orchestrations is a research question that deserves more attention.
12.3.2.2 Extending theRepertoire ofOrchestrations
In the literature, a small number of orchestrations have been identied. After
Trouche’s (2004) paradigmatic Sherpa orchestration, the collection of IOs remained
very limited until the publications by Drijvers etal. (2010, 2013b). This resulted in
the identication of classes of whole-class and individual orchestrations, ranging
from being more teacher-centered to more student-centered (see Fig.12.2). Since
then, other researchers used this typology as a point of departure to identify addi-
tional IOs or describe variations (Tabach 2011, 2013).
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404
The question is, however, how context specic this limited repertoire is, how
general are the orchestration types, and how do they depend on the digital tools in
use, the mathematical topic, the teachers’ views on teaching, and other possible fac-
tors? Also, we expect the repertoire to need further extension, for example, in the
light of the increasing diversity of digital tools that came into play, such as MOOCs,
ipped classroom tools, etc. For example, Orozco etal. (2018) study instrumental
orchestrations in the case of university-level courses in linear algebra and the topic
of eigenvalues and eigenvectors in particular. Digital tools include computer algebra
systems and dynamic geometry software, and the results might shed light on pos-
sibly new orchestrations in this context.
In short, many questions on the repertoire of IOs are waiting to be answered.
How general is the set of IOs identied so far? Do we need a more comprehensive
taxonomy of orchestrations? How exactly is the relationship between the IO and the
targeted instrumentation schemes? These questions are high on the future research
agenda in our eld.
12.3.2.3 Chaining Orchestrations
So far, the focus within IO research has been on isolated orchestrations. Hardly any
attention is paid to integrating them into instructional sequences. How can teachers
sequence orchestrations into productive chains? Are there specic chains that form
natural sequences, like IO trajectories? Even if this idea was present in the early
years of instrumental orchestration (e.g., see Trouche 2004), it has not been further
elaborated so far.
In addressing these questions, an interesting approach could be to rst identify
the teachers’ goals while setting up a classroom organization. To characterize such
Fig. 12.2 Whole-class and individual orchestrations. (From Drijvers etal. 2013b, p.998)
P. Drijvers et al.

405
an organization, Besnier (2016) developed a link between moments of study
(Chevallard 2002) and the notion of orchestration. Chevallard considered that
“whatever the concrete path of the study, certain types of situations are almost nec-
essarily present during the study” (Chevallard 2002, p.11).2 These types of situa-
tions are called moments of study. Chevallard identied four types of moments,
described by Besnier (2016) as follows: designing and implementing introduction
and discovery moments; designing and implementing learning and training
moments; designing and implementing synthesis moments; and designing and
implementing evaluation moments. While studying IO in Kindergarten, Besnier
(2018) observed an orchestration linked to the design and implementation of a
moment of synthesis, to support discussions between pupils about the procedures
they used for solving a mathematical task. This orchestration was called “the manip-
ulatives and software duo” and was considered a variant of the “link screen board”
orchestration already identied in secondary school (Drijvers etal. 2010). We con-
sider these two orchestrations as a part of a continuum, which starts with prior
orchestrations that give the students the opportunity to experience moments of
introduction and discovery and to experiment moments of learning and training.
Besnier and Gueudet (2016) identied specic chains of orchestrations within
the same lesson. Orchestrations took place successively but also simultaneously.
With regard to successive orchestrations, the authors observed, in a moment of
introduction and discovery, a chain of three types of orchestrations, “discuss the
screen,” “explain the screen,” and “Sherpa at work,” and note teachers combining
teacher-centered and student-centered orchestrations for the same goal. In this
chain, the teacher leaves more or less room for the students’ experience or actions.
When should students be given more control? When should the teacher take over?
In connection with these questions, this manipulation of orchestration chains, by the
teacher and for the benet of students’ learning, seems to require dexterity and
expertise from the teacher.
As for the simultaneous orchestrations, we observed orchestrations such as
“accompanied use” and “peer work” carried out simultaneously during learning and
training moments. The teacher’s expertise in choosing a particular orchestration
targeted at specic students and simultaneously managing several orchestrations
seemed crucial here, to do justice to the differences between students.
In spite of this example, much remains unknown about the ways in which IOs
may be chained and connected. This is an important topic to investigate in more
detail and to address in pre-service and in-service teacher training.
12.3.2.4 Didactical Performance
As shown in Fig.12.1, the IO model distinguishes three levels: a didactical congu-
ration (the setting), an exploitation mode (the way in which the teacher intends to
use this setting), and a didactical performance (the way in which the teacher actually
2 Our translation.
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406
carries out the teaching, including unforeseen events and follow-up decisions). So
far, research has mainly focused on the didactical congurations and exploitation
modes. It has hardly addressed the latter phase of didactical performance, which in
the end might be decisive in the IO’s effect. How do teachers take their decisions,
and how can they be empowered to do so in a fruitful way?
Villamizar etal. (2018) studied the teacher’s didactical performance in a high
school course that integrated mathematics, physics and digital technology using the
Cuvima model (Cuevas etal. 2017). The objective was to promote insight into both
sciences based on the modeling of a physical phenomenon. One of the didactical
congurations included printed guides, a projection room and tablets, with an app
for video analysis and dynamic geometry. In groups of three, the students investi-
gated the physical phenomenon of conservation of energy in the free fall of a ball.
The teacher’s exploitation mode was guided by the four phases of the Cuvima
model: experimentation of a physical phenomenon (use of guides and tablets), mod-
eling by digital device (use of apps in the tablets), and conceptual analysis in phys-
ics and mathematics (use of didactic guides, projector, and blackboard), in which
the teacher used Link-screen-board and Discuss-the-screen orchestrations (Drijvers
etal. 2010). The teacher’s didactical performance was evident during the discussion
of the results, in which the teacher pointed out that the experimental data were
imprecise. To improve data collection, a student proposed to add new artifacts as
pointers to the tablet (USB, On-The-Go and mouse); in response to this, the teacher
assigned this student the role of Sherpa-student (Trouche 2004). This decision
clearly illustrates the importance of the didactical performance.
To summarize, the “proof of the pudding” of an IO to an important extent depends
on the teacher’s didactical performance. Consequently, it is highly relevant to know
more about effective didactical performance and about the ways in which pre- and
in-service teachers can further develop their skills on this point.
12.3.2.5 Teachers’ andStudents’ Gestures
As part of the didactical performance, teachers use gestures while teaching. Students
gesture as well while using digital tools. What is the relationship between the type
of gestures and the techniques invited in the IO? Is there a relationship between the
gestures, seen from an embodied perspective, and the techniques in use?
Notions on embodiment (Lakoff and Núñez 2000) stress that cognition, even in
the domain of mathematics, is rooted in bodily experiences, which take place in
interaction with the world. Sensori-motor schemes, in this view, might form a foun-
dation for instrumentation schemes that are formed through instrumental genesis.
However, research on IO seems to have neglected the embodiment and gesture per-
spective, and, in fact, one might wonder how to incorporate this view in the integra-
tion of digital tools in mathematics education. For example, Kozaklı Ülger and
Tapan Broutin (2018) showed that even in technology-enriched lessons, teachers
may prefer typical teacher gestures, such as tracing out a curve in the air, to using
technological resources.
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407
In short, further research is needed to investigate how IOs can take into account
the bodily experiences in which mathematical experiences are rooted. How can we
use digital technology to overcome the limitation of just neglecting embodiment?
What is the relationship between mathematical concepts, body and the material
activity with instruments? Recent developments in this eld suggest promising rela-
tionships between the use of digital tools, gesture, and embodiment (e.g., see Ferrara
and Sinclair 2016), but much is to be explored in more detail in this eld.
12.3.3 Conclusion
This reection on the past and the future of the notion of IO, on the one hand, shows
its potential: it is widely acknowledged that teachers play a crucial role in enhancing
the process of instrumental genesis, and that appropriate support to students is a
subtle matter. The three-layer IO model may help teachers become aware of this
subtlety and to develop their skills in exploiting the affordances of digital technol-
ogy in their mathematics classes. For example, the notion of didactical performance
highlights the exibility that IOs need, to allow on-the-y adaptations by the teacher.
As such, the notion of IO is considered an answer to the question of how to foster
students’ instrumental genesis.
On the other hand, the increasing role of digital technology in mathematics edu-
cation and the wide variety of digital tools makes us feel the IO model has not yet
been fully exploited. We recommend further research in the ve directions outlined
above, to further develop IO as both a theoretical and a practical framework but also
to better align it with current trends in mathematics education, including foci on
student-centered learning and on the importance of gestures and embodiment as
foundations of mathematical knowledge.
12.4 Perspectives oftheDocumentational Approach
toDidactics withRegard toTransitions TowardDigital
Resources
SylvaineBesnier, VerônicaGitirana, Rogérioda Silva Ignácio, RafaelMarinho
de Albuquerque, GaelNongni, GiorgosPsycharis, CharlotteKrog Skott and
JoséVieira do NascimentoJúnior
The roots of the DAD (Gueudet and Trouche 2008) are interrelated with a transition
of research interest from resources used by teachers and/or teacher educators to
digital resources. The increasing development of the DAD (Trouche etal. 2018),
however, points to its potential to obtain deeper understanding of teachers’ practice
with resources, digital or not. A basic assumption of DAD is that the multiplicity of
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408
the digital resources (including applets and e-textbooks) offers increased opportuni-
ties for teachers to design their lessons and modify teaching approaches tradition-
ally adopted in the classroom. At the same time, new digital means such as
e-textbooks, offering new potential structures to the teacher and new interactions
with the users, inuences teachers’ work at the level of both design and professional
development. Also, the study of collective design work taking place in diverse con-
texts and communities raises the question of collective documentational genesis.
As in the evolution of other theoretical approaches, its use as a framework implies
the identication of gaps that lead to new developments within its own theoretical
construction. These advances are often strongly demarcated by characteristics of the
object or context analyzed. For example, Rocha (2018) introduces the notions of
“documentational experience” and “documentational trajectory” as theoretical and
methodological tools to analyze teachers’ documentation processes over long peri-
ods of time.
In this section, we discuss some perspectives of DAD appearing in research on
the transition toward digital resources. We address two main questions in this
chapter:
– What are the perspectives under which DAD has been used to study to support
teachers’/teacher educators’ effective transition toward digital resources?
– How is DAD inuenced by (and how does DAD inuence) these perspectives in
terms of networking, extensions, and new areas of research?
This section is structured in the six subsections: DAD, connections with IO, net-
working of DAD and other theories in the transition toward digital resources, indi-
vidual and collective documentation work in the transition toward digital resources,
the development of DAD in relation to pre-service teachers, using the reective
methodology of DAD to support teachers’ meta-cognitive reections on their prac-
tices, and DAD and the design of digital resources. The section closes with nal
remarks.
12.4.1 DAD, Connections withInstrumental Orchestration
As discussed in the previous section, the framework of IO (Trouche 2004) was cre-
ated to allow exploration of the ways by which teachers create systematic and inten-
tional arrangements of artifacts and persons at the classroom to facilitate learners’
instrumental genesis. Trouche (2005) and Drijvers and Trouche (2008) argue that it
is not enough to adapt classical mathematical situations, but teachers must design
new situations considering the affordances and constraints of the technologies.
Designing new situations partly based on digital resources requires the development
of specic skills and knowledge of the teacher.3 Several research studies that are
3 Concerning the question of knowledge and skills mobilized in a broader perspective of resource
use (not only with digital resources), reference can be made to the work of Working Group 3:
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409
based on DAD explore the complexity of teachers’ practices in relation to the use of
digital resources. To understand this complex work, these studies focus on the skills,
knowledge, and expertise of teachers in the context of the design and implementa-
tion of situations involving digital resources (e.g., Psycharis and Kalogeria 2018;
Ratnayake and Thomas 2018). This work also looks at the (bounded) choices made
by teachers and factors that may explain these (bounded) choices. To take these
issues into account, we note that explicit links between DAD and IO have been
developed as part of research that considers teachers’ practice toward the integration
of digital technology.
Kozaklı Ülger and Tapan Broutin (2018) use DAD to understand teachers’
(bounded) choices in classroom planning in rich digital environments, using DAD
and IO in a complementary way. To understand teachers’ practice in a technological
environment, it considers IO to analyze teachers’ actions in a technologically
enriched environment, and DAD to determine teachers’ (bounded) choices of
resource and changes in this process. This case study looks at the practices of a
teacher for whom designing and implementing instruction in a technological envi-
ronment is new. The study highlights an important aspect of this work as “weaving”
(Billington 2009) as a common occurrence in observations. They observed teach-
ers’ movement with the available tools. Three tools were used: board, computer +
screen, and body movements. During the lesson, digital tools were intentionally
used, while in spontaneous situations, teachers used wooden boards or gestures.
The complementarity of the two frameworks is also explored in Besnier (2018),
who uses DAD to study aspects of teachers’ documentation of their process of
orchestrating classroom lessons for teaching numbers at kindergarten (4- and
5-year-old pupils) with digital and analogic materials. The research focus is on the
teachers’ adaptations of resources as well as on their classroom orchestrations. For
Besnier, orchestration is considered as part of the document developed by the
teacher. Orchestration corresponds to the recombined resources, and the action rules
part of the document. In this context orchestrations are the emergent part of the
scheme. In this research, Besnier identied a variant of the “link- screen- board
orchestration” (Drijvers etal. 2010), called “the manipulatives and software duo”
orchestration. This orchestration and its implementation are linked in the case of the
teacher to professional knowledge related to importance of verbalization and peer
exchanges. To allow pupils to discover the procedures, they must experiment in the
technological environment and discuss this experimentation with each other. Besnier
argues that it is necessary for the teacher to create a new resource and to implement
an orchestration and to make a link between manipulatives and software. The
changes in orchestrations observable in classrooms are then considered as the mark
of changes in the teachers’ resource systems. They reect on changes in teachers’
knowledge.
Considering the importance of IO within teacher documentation, Lucena (2018)
and Lucena etal. (2018) propose the notion of “instrumental meta-orchestration” to
“Instrumentation, skills, design capacity, expertise”; see Chap. 4 of this book, “Documentation
Work, Design Capacity, and Teachers’ Expertise in Designing Instruction.”
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410
promote teacher reection about IO with regard to their documentational genesis in
integrating digital resources. They work within a composition of IO, sometimes
sequenced, sometimes overlapping, focusing on meta-situations, which allow teach-
ers to reect on the notion of IO.
12.4.2 Networking ofDAD andOther Theories
intheTransition TowardDigital Resources
The assumptions and challenges, reinforced with existing research work based on
DAD over the last decade, suggest advantages of using additional theoretical lens to
study phenomena related to teachers’ and/or teacher educators’ (TEs) documenta-
tion work (DW) including the use of digital resources. We now consider attempts to
connect DAD with other theoretical frameworks and constructs. One strand of this
research targets elaboration and renement of theoretical terms traditionally used to
describe mathematics teachers’ work inside and outside the classroom such as
“mathematics teacher design” and “mathematics teacher design capacity.”
Networking of DAD with Brown’s (2009) theory of “teachers as designers” is based
on the common perception of teacher interaction with curriculum resources by the
two theories as a participatory two-way process of mutual adaptation (Pepin etal.
2017). This research is anchored in the French Sésamath association for the design
of a grade 10 e-textbook and a European funded project targeting inquiry-based
learning in mathematics and science (PRIMAS). The study leads to a new denition
of “teacher design capacity” as comprising (1) an orientation or goal, (2) a set of
design principles (called robust principles) that are evidence-informed (e.g., from
own practice) and supported by justication for their (bounded) choices, and (3)
“Reection-in-action” type of implicit understanding developed in the course of
instruction (“design-in-use”). This denition is used to investigate design capacity
development stemming from teachers’ transformation of digital curriculum
resources to (re-)design instruction and work with/in collectives.
Another case of networking, between DAD and Cultural-Historical Activity
Theory (CHAT), was triggered by the need to investigate design processes in teacher
collectives working on the development of e-textbooks (Gueudet etal. 2016). The
authors study the activity system of a community of teachers working in the context
of a teacher association (Sésamath) for about 4years to design/redesign a chapter
(functions) of an e-textbook. At the micro-level, DAD allowed the researchers to
capture the evolution of resources and rules shared by the community. At the macro-
level, CHAT helps them to understand different types of collective geneses that
result from tensions in the system indicating a change of the object of the activity at
different moments: from designing a “toolkit” for mathematics teachers to interac-
tive exercises and, nally, to a more “classical e-textbook.” However, the authors do
not provide a theoretical explanation of the term collective geneses. Similarly,
Essonnier and Trgalová (2018) connect DAD with Engeström’s (1987) activity the-
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411
ory and Fischer’s (2001) concept of community-of-interest, as described later in this
chapter.
Another strand of studies concerns networking of DAD to theoretical frame-
works focusing on aspects of teachers’/TEs’ knowledge. Psycharis and Kalogeria
(2018) network DAD and the TPACK framework (Mishra and Koehler 2006) to
study trainee TEs’ DW in technology enhanced mathematics. They investigate
which TPACK forms of knowledge targeted by trainee TEs in their documents and
which operational invariants are related to these forms of knowledge. The analysis
reveals one type of documents emphasizing the T aspect of TPACK (instructive) and
two types of documents emphasizing the P aspect of TPACK (explanatory, facilita-
tive). Operational invariants underlying trainee TEs’ DW are directly linked to the
trainees’ teaching practice as well as to their epistemologies concerning the role of
technology in the teaching and learning of mathematics and the ways they conceive
trainee teachers (“as students”/“of students”). Ratnayake and Thomas (2018) con-
nect the DAD with the theoretical model of Mathematical Pedagogical Technology
Knowledge (MPTK) (Thomas and Hong 2005) to study what factors inuence sec-
ondary mathematics teachers’ development and implementation of digital technol-
ogy algebra tasks. Although knowledge is not explicitly considered as a resource in
DAD, MPTK includes an extension of the concept of resources to embrace aspects
of Schoenfeld’s (2010) decision-making theory which includes teacher’s knowl-
edge as a primary resource.
Another aspect of networking concerns connections between DAD and frame-
works used to study teachers’ DW in different subject elds. For instance, Messaoui
(2018) connects DAD and Personal Information Management (Jones 2007) to study
the operational invariants underlying the scheme of how a teacher classies a new
resource in her/his resource system. The analysis, based on the observation of teach-
ers’ classication of resources in using computers, reveals operational invariants
related to didactic knowledge (e.g., type of activity, teaching grade) as well as
knowledge linked to digital literacy (e.g., create a le, drag and drop a folder).
Another example is the study of Jameau and Le Hénaff (2018) who combine DAD
and the Joint action theory in didactics (Sensevy 2011) to explore how a science
teacher uses digital resources (e.g., videos) for her Content and Language Integrated
Learning lessons to support language and science learning.
12.4.3 Individual andCollective Documentation Work
intheTransition TowardDigital Resources
In their seminal article introducing DAD, Gueudet and Trouche (2009a) emphasize
teachers’ involvement in professional collectives as one out of three fundamental
factors of the theory. Despite this early emphasis on the collective dimension, they
do not theoretically detail it further. Rather, they describe teachers’ DW as highly
personal, as it results from their professional, social, and personal background. It is
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412
thus interesting to see how this distinction or interplay between collective and indi-
vidual DW is treated in the ongoing development of DAD.
The need for further development of this interplay is spurred by the evolution of
digital technologies offering both new opportunities for learning formats for teach-
ers, teacher educators, etc., and new forms of collaboration (e.g., e-mail communi-
cation, designing and sharing of resources on platforms, and noninstitutional digital
spaces). Such new formats and forms are the primary focus in our selection of
papers from both inside and outside the Re(s)source 2018 International Conference.
Gueudet and Trouche (2011) and Gueudet etal. (2012) investigate an innovative,
online teacher-training program in France (Pairform@nce) designed to sustain ICT
integration but from two different perspectives, teachers and online teacher educa-
tors. Both papers focus on the teachers’ collective DW and provide empirical evi-
dence of professional development in terms of documentational genesis. However,
in this early stage in the development of DAD, the conception of the interplay
between the individual and collective in DW is rather vague. In recognition of this,
Gueudet and Trouche (2011) suggest further developments of this interplay: “What
is the ‘common part’ of the individual documents generated by a collective work?
To what extent is it possible to speak of a common knowledge coming from a com-
munity documentation genesis?” (p.410).
More recently, Carton (2018a) and Essonnier and Trgalová (2018) investigate
entirely new digital forms of teacher collaboration. Carton (2018a) studies how
teachers use non-institutional digital spaces to enrich their DW using an early de-
nition of “the social” by Gueudet and Trouche (2008). The paper provides empirical
evidence that these spaces offer favorable settings for collective work. Essonnier
and Trgalová (2018) study the inuence of designers’ resource systems and knowl-
edge on their (bounded) choices when collaboratively designing a c-book (c for
creative) in the MC2-project (Mathematical Creativity Squared Project– http://mc2-
project.eu/) by supplementing DAD with activity theory (Engeström 1987) and the
concept of Community-of-interest (Fischer 2001). Networking DAD with other
approaches, the authors argue, provides a more coherent theoretical conceptualiza-
tion of the collaborative design, where they foreground the designers’ joint enter-
prise and social interactions by viewing the collaborative design “as a collective DG
(documentational genesis), starting from a resource or a set of resources contributed
to the joint enterprise by the designers and resulting in a c-book resource” (Essonnier
and Trgalová 2018, p.62).
The work of other professionals rather than teachers also inspired and promoted
developments of DAD. For example, Kieran etal. (2013) extend the framework to
the collective activity of design researchers (i.e., the authors), contributing convinc-
ing analytical interpretations of the interplay between individual and collective
documentational genesis. Focusing on “the team’s documentational genesis”
(p.1048), they analyze what they call a “taken-as-shared genesis” by using a dual
perspective. An individual perspective: “a document relates directly to the cognitive
structures of those who have been involved in its design” (p.1047), combined with
a social perspective: “Each round of the process (of genesis) encouraged the sharing
of individual IOs (Operational invariants) (and associated ARs (action rules)), so
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413
that eventually the nal version of the (…) document came to be based on a shared
set of IOs” (p.1049). However, the authors do not theoretically elaborate these con-
cepts and processes to provide a coherent taken-as-shared approach.
Hence, in recent developments of DAD, we see promising theoretical and ana-
lytical proposals of how to interpret the interplay between individual and collective
DW in the transition to digital resources, either by linking DAD to other theoretical
approaches or by extending the framework beyond primary and lower secondary
teachers’ work. Despite this, there is a need for further theoretical elaboration of the
interplay to provide more accurate answers to the requests mentioned by Gueudet
and Trouche (2011).
12.4.4 The Development ofDAD inRelation toPre-service
Teachers Exploiting Digital Resources
In the rst publications introducing DAD (Gueudet and Trouche 2008, 2009a, b,
2010), and even in more recent ones (Besnier 2016), the research considers teachers
in the middle of their careers. Prieur (2016) includes teachers at the beginning of
their careers in his investigations of teachers’ documentational genesis (the heart of
DAD). Nonetheless, further studies of teachers’ documentational genesis are
needed. Indeed, the elements of a scheme’s development often take place during
initial teacher training.
Nongni and DeBlois (2018) discuss documentational genesis in the transition of
pre-service teachers to becoming teachers and their “epistemological stances” when
planning lessons. Leroyer (2018) investigates the inuence of these stances (Bailleul
and Thémines 2013) on the interactions between teachers and their resources.
According to them, the teacher can adopt three epistemological stances: the ancient
pupil, the university student, and the teacher (DeBlois 2012). Nongni and DeBlois
(2018) observe the inuence of these epistemological stances on the documenta-
tional genesis, in part on the use patterns and arrangement variables. They also
observe the inuence of pre-service teacher’s documentational genesis on episte-
mological stances, in particular how documentational genesis allows the transition
of pre-service teachers to becoming teachers and their epistemological stances
when they are interested in students’ understanding. Nongni and DeBlois (2017,
2018) also orient the documentational genesis toward the arrangement variables,
artifacts and didactic variables, when studying how the pre-service teachers exploit
digital resources. They posit a reciprocal inuence among these variables that could
provide a framework for understanding the documentational genesis of pre-service
teachers with regard to digital resources, by observing these epistemological stances
(DeBlois 2012). The epistemological stances adopted by the pre-service teachers
can then be used to understand pre-service teachers’ development in their anticipa-
tion of activities while planning their teaching.
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414
Assis etal. (2018) also investigate pre-service teachers’ activities, within a docu-
mentational trajectory (Rocha 2018), and go toward the understanding of what they
call a resource metamorphosis, from the “resource to study” toward the “resource to
teach.” When studying an early-career teacher’s resource system, they analyze the
pre- service mathematics teacher training to consider how they structure their
resource systems. The concept of resource system metamorphosis helps them under-
stand the transition from a system of study-oriented resources to a teaching-oriented
resource system. Their study presents the activities of two teachers who transpose
between two classes of different situations: one structured to perform mathematical
tasks using Dynamic Geometry and another to create tasks for students to learn
mathematics using Dynamic Geometry. The results suggest that pre-service teach-
ers rely on their study-oriented resources, including textbooks to develop their
teaching-oriented resource system, which includes dynamic geometry tasks.
Ignácio etal. (2018) focus on a pre-service teacher who is developing a super-
vised internship project that involves two cycles of the production and use of a digi-
tal textbook chapter on the role teaching. The analysis of the production of this
material shows that, in addition to the visible adaptations of printed textbook parts
for the digital medium, the pre-service teacher mobilized a vast system of resources
previously developed. The analysis provides evidence that the pre-service teacher
has developed professional knowledge related to the development and use of digital
resources for the teaching of functions.
12.4.5 Using aReective Methodology ofDAD toSupport
Teachers’ Metacognitive Reections onTheir Practices
The term “reective investigation methodology” was introduced in the context of
DAD to study teachers’ DW (Pepin etal. 2013). Enlarging the term beyond methods
of data collection and analysis, Ignácio etal. (2018) use it to organize a teacher
education program for pre-service teachers involving design, use, reection and
validation of an e-textbook chapter.
Reecting about one’s own documentation process also appeared as an impor-
tant tool for action research. Nascimento Jr. etal. (2018) use DAD, networked to
other theories, to analyze and modify their own actions while designing and experi-
menting with innovative lessons integrating digital technology for university sci-
ence teaching and learning. Among other aspects, their own documentational
genesis was analyzed considering the analysis of students’ instrumentation.
Conventional and innovative digital resources interacted and played relevant roles
in the process. Drawing attention to their own experiences, they acknowledge how
little they can control the outcomes of such interactions.
DAD research which involves a self-reective methodology allows the subject to
focus on his/her own documentation, documentational genesis, systems of docu-
ment, and documentation work, in particular, on how one creates his/her own “indi-
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415
vidual schemes of use” (Gueudet and Trouche 2009a, p. 204). For example,
Nascimento Jr. et al. (2018) argue that an attempt to adapt multiple materials
(including traditional textbooks, e-books, online familiar and unfamiliar materials
making use of large databases) demands that teachers require not only design capac-
ity to prepare lessons, but also expertise and decision-making skills. DAD can help
teacher educators to be aware of these needs, as is discussed in (Males etal. 2018,
p.207) with regard to emerging methodology “What do teachers attend to in cur-
riculum materials?”
Thomas and Edson (2018) also focus on the need to consider meta-cognitive
processes when analyzing teachers’ documentation work. They examine teachers’
conceptions of digital instructional resource as a way to understand how digital
resources impact on teachers’ work. They contrast teachers’ denition of theoretical
terms. As regards DAD, they consider resource and document from the teachers’
perspective (i.e., who designs, selects, and implements resources). They show that
while dening the term, the teachers “tended not to distinguish between the resource
and the genesis through which it becomes a document” (p. 343). Thus, they argue,
teachers’ DW may also occur in the meta-cognitive process of dening “what
counts” as digital instructional materials in a more general sense.
12.4.6 DAD andtheDesign ofDigital Resources
DAD has been used to analyze not only teachers’ work but also the work of other
professionals (researchers, software designers, artists, etc.) involved in the process
of designing digital resources. It has also been used to interpret, mainly for classify-
ing, actions and principles related to the design of digital resources, as well as to
design curricular digital resource for their effective use by teachers.
Essonnier and Trgalová (2018) consider the DAD as a tool to identify the design-
er’s resource system and its inuence in choosing digital resources, which is conso-
nance with the motivation of Bellemain etal. (2018), for using the DAD to identify
and establish requirements for a web environment to support teachers’ DW and the
design of digital resources. Indeed, they develop the idea of a web document, based
on the DAD concept of a document: software composed of other software programs
or digital components, that is, a set of digital resources and utilization schemes
designed by a teacher for a specic teaching aim. They propose a classication for
such resources (static, dynamic and active) depending on the kind of content dis-
played and/or interaction made possible with these digital documents. For them, an
activity is interpreted as a web document activity, since teachers organize both the
activity to be done by students and students’ actions in the activity, generating a new
document. Design issues are considered further in Sect. 12.6.
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416
12.4.7 Final Remarks
DAD has its origins within the digital resources integration problem, and its evolu-
tion within digital scenarios brings into the approach new needs and new concepts
such as documentational expertise, documentational trajectory, and metamorphosis
of the resource system. These new concepts and tools now comprise part of the
framework. In its origin, the networking of theories, sometimes articulated and
sometimes contrasted, has led to new networks that bring new issues into DAD
discourse, especially within teachers’ transition toward digital resource systems.
The potential of networking between IO and DAD is especially important in
research. It sheds light on how to support teachers’ use of digital resources and, at
the same time, the effects on their documentation as well as the correlation between
teachers’ documentation and teachers’ choice of resources. This dialectic leads us
to consider both frameworks for understanding teachers’ effective use of digital
resources and also goes toward an extension of IO into an instrumental meta-orches-
tration framework to teachers’ education toward using IO as a support to design this
use. The use of DAD in teacher education goes even further, with extensions of
DAD examining the beginning of the documentation process within initial teacher
training, as well as extending the idea of resource systems to pre-service teachers; a
reective methodology is important in dealing with teachers’ initial education. This
elicitation of the characteristics of teachers’ documentation can also be used to
improve one’s own practice in action research. Characteristics of teachers’ work on
the web also lead to perspectives of analyzing collective documentation and indi-
vidual documentation within the collective work. The research considered in this
section suggests the need for more investigations and greater precision regarding
the collective documentation approach. The continuous evolution of research using
DAD to support teachers’ effective transition toward digital resource also leads to
the emergence of new concepts and research tools for improving DAD.
12.5 Digital Resources andTeacher Education
SametOkumus, AmandaThomas, EleonoraFaggiano, OsamaSwidan,
ElisabeteCunha, ElenaNaftaliev and RosilângelaLucena
Mathematics teachers are the principal actors who are responsible for planning and
enacting school mathematical activity; and their enactment of technology into the
mathematics classroom is inuenced by a wide range of factors (Assude etal. 2010;
Zbiek and Hollebrands 2008). The factors of technology integration are like a jig-
saw puzzle in that each component must be supported and merged into another for
a successful implementation of technology into practice. A missing, or weakly con-
nected, piece in this jigsaw puzzle may impede or impoverish the use of
technology.
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417
One factor that inuences the enactment of technology in mathematics class-
rooms is teachers’ knowledge. For instance, the TPACK framework (Koehler and
Mishra 2009) describes aspects of teacher knowledge that interact to inuence how
teachers integrate technology for the teaching of content (i.e., mathematics). Assude
etal. (2010) categorize some of the factors that inuence mathematics teachers’
utilization of technology into four components: “the social, political, economic, and
cultural level, the mathematical and epistemological level, the school and institu-
tional level, the classroom and didactical level” (p.406). Heid (2008) stresses the
importance of how teachers use technology and how their educational beliefs affect
educational settings and student learning with technology, since students are likely
to use technology in the way that their teacher designed curriculum. She draws from
research studies to highlight that teachers who have constructivist teaching beliefs
are more likely to integrate technology into curriculum and allow students to be
explorers through the use of technology.
12.5.1 Teaching Activity Prior toaMathematics Lesson
Artzt etal. (2015) characterize mathematics teaching activity in three stages: deci-
sions teachers make before, during, and after a lesson. Teachers’ activity prior to a
mathematics lesson includes lesson planning and considering the affordances and
constraints of tools and resources to be integrated in the lesson. During the lesson,
teachers’ work focuses on monitoring and regulating. Evaluating and revising are
the main mathematics teaching activities after a lesson.
12.5.1.1 Lesson Planning
Artigue (2002) identies four key dimensions in technology-enhanced mathematics
learning: the mathematics, the teacher, the learner, and the tool. These dimensions
also apply to the documentation work in which teachers engage prior to teaching a
lesson (Gueudet and Trouche 2009a). Along with the four key dimensions, Zbiek
and Hollebrands (2008) identify external factors that must be considered when
planning for technology-enhanced mathematics teaching, including ready access to
technology and support staff, technology training and professional development,
time constraints, logistical constraints, technology and device availability, and the
availability of curriculum materials that capitalize on technology. The relationship
between curriculum materials and technology resources is particularly salient to
consider as teachers engage in documentation work prior to teaching mathematics
lessons because the nature of this work requires teachers to select and integrate cur-
riculum materials with technology and other resources.
With respect to the epistemological stance of curriculum materials, Choppin
(2018) nds that teachers who engage in lesson design work with new types of cur-
riculum materials tend to exhibit practices aligned with the curriculum programs to
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418
which they are most accustomed. Thus, when teachers plan lessons that integrate
emerging technology resources with existing curriculum materials, prior practices
and habituation to curriculum resource should be considered. Teachers’ documenta-
tion work during the planning phase of lessons also includes the identication of
potential resources that could enhance the mathematical and didactical goals of the
lesson. In dening what constitutes digital instructional materials, ndings from
Thomas and Edson (2018) suggest that teachers consider not only the resource itself
but how, where, and by whom it might be used. That is, teachers consider resources
in relation to the context and learners, as well as their potential use for the lesson.
Zbiek and Hollebrands (2008) note that “teachers’ conceptions, beliefs, knowl-
edge, and use of technology seemed to inuence the activities they created for their
students who were using technology to learn mathematics” (p.310). According to
Farrell’s (1996) ndings, technology affects teachers’ selections of tasks and activ-
ity types in mathematics curriculum. Based on her research study, Farrell (1996)
observes that teachers are more likely to prefer using activities that require investi-
gation and group work when they use technology. In addition, with the use of tech-
nology, teachers adopt tasks requiring more problem solving and higher level
thinking (Farrell 1996). Such preferences could be considered as a result of shifts of
teachers’ and students’ roles when technology is integrated. When planning to teach
mathematics with technology, teachers should consider the resources available to
them, the affordances and constraints associated with those resources (Kennewell
2001) and, more importantly, how those resources relate to the lesson’s mathemati-
cal and didactical goals (Artigue 2002).
12.5.1.2 Affordances andConstraints ofTools
In recent years, digital tools and online resources have become increasingly acces-
sible for teachers. When teachers incorporate technologies into mathematics les-
sons, “they may also utilize activities and examples from curricula that use
technology. Finally, they include representations and strategies specic to technol-
ogy” (Hollebrands etal. 2016, p.273). However, as Dick and Burrill (2016) empha-
size, “realizing the unique benets of dynamic interactive mathematics technologies
to enhance students’ conceptual learning of mathematics depends heavily on teach-
ers having the skills and knowledge necessary to make sound judgments in choos-
ing and using these technologies in the classroom” (p.43).
Mathematical tools have different affordances and constraints for mathematical
learning. An affordance is considered by means of what an environment offers the
agent who uses the tool (Gibson 1977) and “a constraint of an environment is related
to affordance in as much as it species what the environment does not afford”
(Monaghan 2016, p.168). Okumus and Ipek (2018) emphasize that teachers should
be able to identify not only the affordances but also constraints of the tools. In
Okumus and Ipek’s (2018) study, pre-service mathematics teachers work out the
Triangle Inequality Theorem with hands-on manipulatives and digital tools, and
identify tools that do not “stay true to the mathematics” (Dick and Burrill 2016,
P. Drijvers et al.

419
p.29). Tools may give rise to misconceptions or obstacles for students, and teachers
have important roles in identifying them (Okumus and Ipek 2018).
Dick and Burrill (2016) use design principles that may assist teachers in choos-
ing digital tools. For example, according to the Sandbox Principle, “technology-
based environments should be constrained to minimize the change that students
inadvertently escape or get lost in irrelevant aspects of the technology” (p.29). A
constraint in the design of a digital tool may also give an opportunity for mathemati-
cal learning, rescuing students from irrelevant aspects of the technology (Dick and
Burrill 2016; Naftaliev and Yerushalmy 2017).
Naftaliev and Yerushalmy (2017) design interactive diagrams that are “relatively
small unit(s) of interactive text in e-textbooks or another materials” (p.154) with
built- in constraints. Students generate different representations using the tool and
attempt to overcome the built-in constraints of interactive diagrams by modifying
the given representations or constructing new ones. According to the researchers,
constraints of interactive diagrams play an important role in mathematical investi-
gation. In this sense, Naftaliev and Yerushalmy’s (2017) use of constraints of inter-
active diagrams seems to align with Kennewell’s (2001) perspective who asserted:
The affordances are the attributes of the setting which provide potential for action; the con-
straints are the conditions and relationships amongst attributes which provide structure and
guidance for the course of actions… Constraints are not the opposite of affordances; they
are complementary, and equally necessary for activity to take place (p.106).
Dick and Burrill (2016) claim that constraints of digital tools are helpful in directing
students to think mathematically and “serve to support student attention and focus
on the mathematical implications of the actions they take on the mathematical
objects in the environment” (p.30). Some researchers argue that constraints of digi-
tal tools can give students more opportunities for mathematical learning than hands-
on manipulatives (Dick and Burrill 2016; Kaput 1995). According to Kaput (1995),
“most physical actions on physical manipulatives do not leave a trace sufciently
complete to reconstitute the actions that produced them” (Kaput 1995, p.167). Dick
and Burrill (2016) claim that hands-on manipulatives do not have any constraints
and “can be arranged in ways that are mathematically nonsensical” (p.30). On the
other hand, digital tools can be constrained, which allow for removing irrelevant
aspects of the technology. The built-in constraints can assist students with focusing
on relevant aspects of technology that are linked to mathematics.
In recent years, several researchers have used duos of artifacts: pairs of hands-on
manipulatives and digital tools that support one another (Faggiano etal. 2016;
Maschietto and Soury-Lavergne 2013; Voltolini 2018). In Cunha’s (2018) study,
students follow written directions and fold papers during the origami activities.
Then, they are asked to reproduce the required construction steps using a dynamic
geometry program. The researcher states that the activity enables students to explore
the mathematical relationships between hands-on manipulatives and the digital tool
and stimulates students to produce representations to accomplish the origami task.
In Maschietto and Soury-Lavergne’s (2013) research study, primary school stu-
dents produce turning gestures using a gear train of ve wheels (Pascaline) to per-
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420
form arithmetic operations. Based on the feedback from the pascaline students used,
Maschietto and Soury-Lavergne design a new artifact (e-pascaline) that is the pas-
caline digital counterpart. The design of the e-pascaline is inuenced by student-
produced signs that emerge during the use of the pascaline, and the semiotic
potential of the e- pascaline is promoted by the continuity (similar usages) and dis-
continuity (different usages) between the two artifacts. Design decisions for the duo
of the artifacts are made with regard to didactic goals, so that using one artifact adds
value to the other. The researchers stressed “that the pascaline also has added value
compared with the e-pascaline, which explains why one cannot be substituted for
the other” (Maschietto and Soury-Lavergne 2013, p.969).
In this line, Voltolini (2018) proposes a duo, combining digital and pen-and-
paper environments, through triangle-construction tasks, taking into attention the
links between the two, highlighting the continuity and discontinuities of the duo of
artifacts to promote “the evolution of pupil knowledge” (p. 87). Also, Faggiano
etal. (2016) investigate the synergic use of manipulative and digital artifacts (pass-
ing from one to the other) to construct and conceptualize axial symmetry and its
properties, trying to understand how this synergic action is developed so that each
task improves the learning of the others.
12.5.2 Teaching Activity During aMathematics Lesson
According to Artzt etal. (2015), the main tasks of mathematics teachers during les-
son implementation are monitoring and regulating. When the teacher monitors stu-
dents, he or she “observes, listens to, and elicits participation of students on an
ongoing basis to assess student learning and disposition toward mathematics”
(p.87). Regulation refers to in-the- moment lesson adjustments, “teachers must be
exible and able to modify their lessons based on their formative assessment of the
students” (p.75).
12.5.2.1 Monitoring
Researchers have characterized teachers’ utilization of digital tools in technology-
enhanced mathematics learning with a focus on how they position technology with
regard to mathematics and students. For example, Drijvers etal. (2010) observe
three teachers’ dynamic algebra java applets integration into the mathematics class-
room with a focus on how they orchestrate the whole-class discussions. The results
indicate that each teacher’s focus differs. The rst teacher, who focuses on students’
learning using technology, utilizes student-centered orchestrations. On the other
hand, the second teacher, who focuses on conventional representations of mathe-
matics, associates technology with representations. The third teacher, whose focus
is technology, gives technology directions and utilizes teacher-centered
orchestrations.
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Swidan etal. (2018) identify the orchestration processes of teachers who aim to
promote inquiry-based learning in a classroom setting where students collaborate in
small groups and use digital resources. The researchers pay special attention to the
ways the teachers use the digital resources to boost inquiry-based learning. While
teachers are monitoring students, they stand beside students without intervening for
a while. This passive teacher action is noted as necessary but insufcient to boost
the inquiry processes of students. After a short passive intervention, observing what
students are doing, asking students about their exploration processes and requiring
them to provide a short summary of their reasoning are found to be helpful in focus-
ing the teachers’ attention on the learning objects.
Erfjord (2011) examines three mathematics teachers’ utilization of a dynamic
geometry program and how teachers organize conditions for instrumental genesis
(e.g., organization of students’ work, central focus of lessons, etc.). Classroom
activities include drawing, constructing geometric gures, and working on parallel
and perpendicular mathematical objects using technological and non-technological
tools. Two of the teachers focus on technical aspects of the technology and instru-
mentalization-related tasks (e.g., making constructions that did not mess up). On the
other hand, the teacher whose focus is instrumentation (e.g., have students discuss
different methods of constructions) utilizes student- centered orchestrations. Tabach
(2011) associates a mathematics teacher’s orchestration of digital tools with her
technological pedagogical knowledge. The researcher reports that the mathematics
teacher utilizes more student-centered utilizations over time as her technological
pedagogical content knowledge changes.
12.5.2.2 Regulating
Research indicates that teachers regulate their instruction by making ad hoc deci-
sions due to feedback from students and factors such as time shortages (Artzt etal.
2015; Cayton et al. 2017; Drijvers et al. 2010). Stockero and Van Zoest (2013)
emphasize pivotal teaching moments that may prompt teachers to regulate their les-
sons. They dene pivotal teaching moments as “instance(s) in a classroom lesson in
which an interruption in the ow of the lesson provides the teacher an opportunity
to modify instruction in order to extend or change the nature of students’ mathemat-
ical understanding” (Stockero and Van Zoest 2013, p.127). Cayton etal. (2017)
identify pivotal teaching moments in technology-rich geometry classrooms. They
nd that a teacher who utilizes student-centered approaches pursues students’ think-
ing and extends their mathematical thinking by asking follow-up questions in
response to pivotal teaching moments. Leung and Bolite-Frant (2015) emphasize
mathematics teachers’ regulating of instruction as opening a pedagogical space
when they use a digital tool with discrepancy potential. According to the
researchers:
The discrepancy potential of a tool is a pedagogical space generated by (i) feedback due to
the nature of the tool or design of the task that possibly deviates from the intended mathe-
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422
matical concept or (ii) uncertainty created due to the nature of the tool or design of the task
that requires the tool users to make decisions (p.212).
On the other hand, teachers may not manage to capitalize on the discrepancy
potential of tools. For example, Ruthven etal. (2008) nd that one mathematics
teacher conceals anomalous situations of dynamic geometry software and makes
changes to the lesson on the y. However, teachers’ mathematical knowledge and
familiarity with the technology is an important factor for such ad hoc decisions.
12.5.3 Teaching Activity After aLesson
According to Artzt etal. (2015), as a post-active stage of teaching, teachers should
be able to evaluate and revise their lessons using “information from evaluations of
student learning and instructional practices” (p.88) that should develop students’
mathematical thinking better than their earlier plans. Self-reectivity may contrib-
ute in capturing the important changes that digital resources bring to the teachers’
practice. In this line of thought, researchers emphasize that pre-service and in-ser-
vice teachers should be supported in building up reective competencies or in
becoming reective practitioners (Atkinson 2012; Jaworski 2014).
12.5.3.1 Evaluating
Self-reectivity is not usually a spontaneous practice and requires motivation. The
impact of teachers’ beliefs about the role of digital resources in teaching and learn-
ing of mathematics plays an important role in technology integration, and “the
greatest challenge for professional development aimed at effectively using dynamic
interactive mathematics technologies: moving the teachers’ tool perspective to one
supporting student investigation and exploration” (Dick and Burrill 2016, p.46).
Analysis and design of mathematics tasks, the exploration of overarching ideas
linked to mathematical contents and the analysis of videotaped classroom situations
may enhance teachers’ instruction (e.g., Scherrer and Stein 2013). However, as
Barth-Cohen etal. (2018) point out, videotaping of classroom discourse remains a
challenging and understudied tool.
Lucena etal. (2018) use IO to develop teacher capabilities in integrating digital
resource in classroom and propose a new framework, instrumental meta-orchestra-
tion, that embraces theory and practice. According to the researchers, “an instru-
mental meta-orchestration is a systematic and intentional design of artifacts and
human beings, in an environment of formation by an agent, to execute a meta-situ-
ation of formation which aims to guide teachers in their instrumental genesis about
the theoretical model of instrumental orchestration” (Lucena etal. 2018, p.300). A
sequence of orchestrations is integrated, sequenced and imbricated to enable theo-
retical reection on the practice of IO. Instrumental meta-orchestration requires
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423
active involvement in observation, analysis of discourse using a theoretical lens, and
also promotes reection on different aspects (e.g., content, theory, and practice),
particularly when a digital tool integration is utilized.
12.5.3.2 Revising
Several researchers have developed frameworks to assist teachers, teacher educators
or curriculum writers in evaluating, creating and rening tasks that support stu-
dents’ thinking (Naftaliev 2018; Scherrer and Stein 2013; Sherman etal. 2017;
Trocki and Hollebrands 2018). These frameworks bring theories into practices as
teachers revise and (re)create their tasks/lessons embracing a critical lens. For
example, Sherman etal. (2017) combine two ne-grained frameworks for pre-ser-
vice teachers: cognitive demands of tasks (high-level vs low-level) (Stein and Smith
1998) and the roles of technology in using these tasks (amplier vs. reorganizer)
(Pea 1985). The researchers nd that pre-service teachers most often create high-
level tasks that may support students’ thinking. Furthermore, they most often use
technology as a reorganizer in which “technology has the capability to transform
students’ activity, supporting a shift in students’ mathematical thinking to some-
thing that would be difcult or impossible to achieve without it” (Sherman and
Cayton 2015, p.307).
Naftaliev (2018) examines pre-service teachers’ interactions with interactive
curriculum materials. The study uses a semiotic framework for analyzing the peda-
gogical functionality of interactive materials (Naftaliev and Yerushalmy 2017).
Naftaliev’s (2018) study includes ve interaction stages. Pre-service teachers rst
develop intended curriculum with interactive materials, then analyze classroom sce-
narios where interactive materials are enacted. In the third stage, pre-service teach-
ers build upon their experiences to develop comic representations of scenarios about
classes engaged with the interactive materials. The comics are developed in
LessonSketch (Herbst etal. 2011), a media-rich environment that “allows creating
experiences around classroom scenarios performed with cartoon characters in the
form of a slide show” (Naftaliev 2018, p.305). During the fourth stage, teachers
engage in learning mathematics units with interactive materials and reect on their
own processes of learning. In the last stage, the pre-service teachers design their
own unit for mathematics teaching and learning with interactive materials and pre-
sented an episode of a classroom scenario in which the class is engaged with the
units. The semiotic framework for pedagogical functionality of interactive materials
and the ve-stage procedure enable facilitating the pre-service teachers’ design pro-
cesses, to share, to discuss, and to modify their decisions.
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12.5.4 Concluding Remarks
Adequately incorporating technology in the mathematics classroom may be a battle
many teachers encounter. Some may only use it sparingly, while others do not use it
at all. This, almost certainly, stems from the traditional nature in which teachers
have learned and subsequently teach. With the use of technology, teachers may nd
an increase in classroom discourse as a positive outcome. Applets, computer soft-
ware, calculators, and other forms of technology may allow students to think more
conceptually while offering multiple representations quickly. As a result, students
may be able to have more focused discussion about why or how something works,
rather than just accept one way of doing something. With technology use, question-
ing strategies may also change (Zbiek and Hollebrands 2008). The use of technol-
ogy can increase the questions that can be asked about a given situation and even
heighten the demand of questions. However, “technology itself is not a panacea that
will remedy students’ difculties as they learn mathematics. Rather, it is teachers’
decisions about how, when, and where to use technology that determine whether its
use will enhance or hinder students’ understandings of mathematics” (Hollebrands
and Zbiek 2004, p.259).
12.6 The Design ofLearning Environments withtheUse
ofDigital Resources
ChristianMercat, FranckBellemain, Mariannevan Dijke-Droogers,
PedroLealdinoFilho, AndersStøleFidje, TiphaineCarton, JorgeGaona,
RicardoTibúrcio and AndersonRodrigues
The use of digital resources in learning environments, designed and used in a wide
variety of ways, is growing. In this context, the discussion of the effectiveness of a
designed resource for stimulating learning is an important debate, requiring research
in this design process. In this chapter, we will discuss two approaches to gain more
information about how to design digital resources: (1) design for use and (2) design
in use. After explaining this difference, we describe how this distinction can shed
light on different approaches to digital resources design for learning. Digital
resources used in any given didactic situation may range over many different types
of resources, and encompassing this complexity in a single theoretical framework is
challenging. Hypothetical Learning Trajectories (HLT) (Simon 1995; Simon and
Tzur 2004) is a means that can help structure the context and use of a design. IO and
the DAD can function as pivotal theoretical constructs to observe teachers’ designs.
These two approaches can guide the two forms of designs, we have introduced,
involving their collective aspects in the life-cycle of a resource, going through
diverse disseminations, appropriations, uses and redesigns. These redesigns, ulti-
mately addressed to the students, happen in a variety of contexts ranging from
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425
horizontal (socially creative and collaborative group work) to more vertical situa-
tions in a one-to-many dissemination from a “guru” to her followers or informal
numerical spaces such as blogs or social media.
12.6.1 Design forandin Use
There are various approaches to design, with regard to the use and design of (digital)
resources. In this part of the text, we elaborate on “design for use” (1) and “design
in use” (2).
By design for use, we refer to studies where a theoretically based design includes
conjectures and hypotheses about the way (digital) resources can be used to pro-
mote learning in practice: we look at studies that focus on the design of learning
environments based on theories, sometimes in combination with teaching skills. In
this approach, we focus on the teacher’s system of resources and on ways to struc-
ture all elements involved in the implementation, such as the necessary and specic
educational software engineering, the role of various actors, and the role of instru-
mentation and instrumentalization.
By “design in use,” we refer to studies that focus on the way in which learning
environments with digital resources are used, particularly through the orchestration
of the use, although envisioned a priori by the designers but put into practice by
teachers and students. Investigating how this is actually done in practice is a rich
source about what works and how it works for further agile design loops, rapidly
taking into account actual use. Based on these two approaches, linked in a dialectic
way, we can enrich the knowledge about the design of stimulating learning environ-
ments with digital resources. Of course, the line between design for use and design
in use is not so clear because the learning environment design already anticipates
usage, and the actual use by teachers implies in return adaptations, additions, modi-
cations or in-depth changes of these environments. This ambiguity is related to the
teacher’s own work, which, for the orchestration of an environment rich in technol-
ogy, nally develops an activity close to that of an engineer and assistants rather
than the usual metaphor of the orchestral conductor where each musician should
master his/her own instrument. The difculty of clearly distinguishing the two
designs for use and in use is also relative to the vocabulary. The verbs we use when
talking about the actions of either a computer engineer or a teacher are more or less
the same: they both conceive, design, elaborate, develop, and create resources, but
the level of actions is usually different, leading to the design of technical resource
for the rst and pedagogical resource for the second, all addressed ultimately to the
nal user, the student. To use a concrete example in order to try to explicit the dif-
ference, let us take the case of the use of videos as pedagogical resource. It is a
commonly used type of resource, and for the needs of its didactic exploitation,
many adjustments can be useful: indexation, selection of extracts, insertion of sub-
titles and comments, incrustation, etc. The development of interfaces that allow
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426
such adjustments is typically a design for use issue and their use by the teacher is
design in use.
The dialectic between design for use and design in use translates into an articula-
tion between two engineering process with a certain tension between them, on the
one hand, the design, founded by theoretical principles, of resources and supports,
structures and bindings, for the teacher to use, and, on the other hand, the need for
support structuring his actual orchestration, offering some exibility and document-
ing the needed adjustments and revisions.
To better highlight this dialectic between design for use and design in use, and
the articulation that it assumes between production-engineering and use-engineer-
ing, we can use the elements of IO and associate at some point the design for use
with a didactical conguration, its elaboration by the engineer and its conguration
by the teacher and the design in use when the exploitation mode and didactical per-
formance are the primary concerns.
12.6.2 Tackling Complexity
Designing and developing resources and their supports for teachers and their pupils
to use are an extremely broad and complex problem; thus even if we limit the study
to digital resources, there are many kinds of resources and many ways to use them.
A rst step to reduce this complexity is suggested by Adler (2000) who proposes a
classication of resources as object and action. In other words, inspired by the IO
and the DAD, the classication of resources is based on their own characteristics
and their utilization schemes:
– Developed by the teacher for the instrumentation and instrumentalization of
these resources.
– Developed by learners when these resources are involved in activities.
Silva (2018) develops this theoretical framework to provide a basis for specica-
tions of a digital system that allows teachers to describe and store resources by
integrating them into his/her resource system according to their specic character-
istics and utilization schemes. We regard the creation of such systems, for the orga-
nization and articulation of existing resources, as having potential to enrich the
range of object-action-activity of the teacher.
12.6.3 Designing New Resources forUse
The engineering processes underlying the creation of these resources and supports
are various and depend on the kind of resource conceived. Indeed, Tchounikine
(2011) argues that we do not implement the same theoretical and methodological
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427
principles in, for example, designing a microworld4 or a supporting environment for
collaborative learning.
In the context of design for use of digital resources, an important line of research
and development is interested in the conception of artifacts that effectively enable
the teacher to offer mathematical activities to the learners in a computational envi-
ronment. Typically, microworlds, simulations or games are considered useful in
activities designed to foster mathematical thinking. Common software of choice in
school mathematics is dynamic geometry systems, but we are interested as well in
more general microworlds and simulations, offering tools which may be:
– Used by the teacher for the orchestration of didactical situations.
– Used by the learner for the exploration and the resolution of problems related to
specic mathematical content.
Many other parameters have to be taken into account: the context of design;
individual or group use (and, in the latter case, collaborative or cooperative); the
context of use; whether for use in the presence of the teacher, collectively in syn-
chronized distance learning or individually in asynchronous learning. We focus here
on the contexts of a few examples.
Designing a new resource for use may be approached from a multidisciplinary,
even transdisciplinary, perspective. The design for use of new resources can be ana-
lyzed with regard to the “transposition informatique” (Balacheff 1994) supported
by a prior analysis of the epistemological, cognitive, didactic, and informatic dimen-
sions. Didactical Informatic Engineering (Tiburcio and Bellemain 2018), a reread-
ing of the didactic engineering (Artigue 1990) considering the Information
Technology (IT) dimension, proposes a systematic, operational and anchored
approach in the didactics of the mathematics of the “transposition informatique.”
By integrating the IT dimension to the didactic engineering, it is a matter of care-
fully analyzing the actual contributions of IT to support the mathematical activity of
the learners. Thanks to the interfaces and operational capabilities of the computer,
Siqueira and Bellemain (2018) are particularly interested in the contribution of
dynamic representations and articulations between these representations. Such a
resource can create an interactive object that provides feedback on abstract notions
it represents. The theoretical and methodological principles used in didactical infor-
matic engineering and the specication (design) of these digital resources are rooted
in epistemology, the theory of semiotic registers of representation (Duval 1993), the
theory of didactical situations (Brousseau 1997) and the Anthropological Theory of
Didactics (Chevallard 2002).
As an example of an implementation of this specic didactic informatic engi-
neering model, we consider the LEMATEC project (www.lematec.net.br), in which
design of artifacts allows for the dynamic articulation of various representations of
mathematical objects. The mathematical contents addressed by these resources are
the notion of function (Function Studium, Bellemain etal. 2016), the conics (Conic
Studium, Siqueira and Bellemain 2018), and area and perimeter (Magnitude
4 See Hoyles & Noss (1992) for an explication of this term.
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428
Studium, Rodrigues etal. 2018). In the context of teaching, the “designing-a-new-
resource” open question can become the guiding thread of the teaching of various
disciplinary contents with varying focus depending on the specic content approach.
In the study of Lealdino Filho and Mercat (2018) on teaching computational think-
ing in classroom environments, unplugged resources can be used to promote com-
putational thinking, and this activity leads the students to design digital resources.
Material resources are therefore designed for use and digital resources are not the
initial teaching resource but the product of the activity. This study, in the Computer-
Science unplugged framework, elaborates computational thinking competences
through implementing a design without an a priori use of computers. An initial step
in the convergent thinking phase (Mercat etal. 2017) is describing impressions,
beliefs about what is experienced, here a magical trick but generative art and optical
illusions in other works. In order to express them, thinking and expressing takes
place, iteratively replacing an abstract and subjective construct, by a concrete,
objective, and meaningful method which makes any information-processing agent
return the expected result. Solving the task of writing an algorithm to perform the
magical trick and to solve this particular problem did not need the use of any digital
resource. Implementing it on a computer requires further work in order to translate
the phenomena into a programming language. Implementing the activity requires
versatility and exibility on the side of the teacher. The possibility exists, of course,
to restrict the tools made available to the students and conduct a thorough a priori
analysis of the possible implementations that might emerge.
12.6.4 From Resource forUse toResource inUse
In the perspective of “design in use,” these artifacts have to be increased with tools
that, when used by the teacher, allow their orchestration in her teaching, with min-
ute tweaking and documentation process (Gueudet and Trouche 2008). To continue
with the example of dynamic geometry, in addition to tools for editing and manipu-
lation of gures, we nd functions for the conguration of menus, the elaboration
of a statement, the sharing of a gure at a distance, etc.
The implementation of learning activities using (digital) resources requires a
system in which all elements involved in the design and implementation of learning
activities are structured and organized. On the designer side, to make this instru-
mentalization and organization possible requires developing interfaces, supports,
guides to instrumentation, as well as, on the teacher’s side, robust resources sys-
tems, in an IO or in a documentation process. Investigating the design for use by the
teacher of digital resources is the best way to gather information of how teachers
build and use resources and systems of resources.
Brown (2009) investigates how the teacher works as a designer and regulates his/
her Pedagogical Design Capacity (PDC). He proposed to draw a parallel between
design and teaching, showing that these two activities share common procedures:
“Teachers must perceive and interpret existing resources, evaluate the constraints of
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429
the classroom setting, balance tradeoffs and devise strategies– all in the pursuit of
their instructional goals.” Stating that we should consider “teaching as design,” he
developed the concept of PDC in order to describe how teachers would interpret and
use curriculum materials. Using the example of a middle school science teacher try-
ing to set up a science lab in her class, he dened PDC as a “skill in perceiving the
affordances of the materials and making decisions about how to use them to craft
instructional episodes that achieve her goals.” Pepin etal. (2017) also argued that
design could be considered, when applied to teachers, as “designing for teaching.”
We could therefore say that there is a strong link between teachers’ design activities
and their DW.
In a design for use, HLT (Simon 1995; Simon and Tzur 2004) can structure a
priori information on both sides, of the teacher and of the expected users, helping
the designers to shape the resources for use. During a teaching experiment, an HLT
is implemented and tested, gathering data on the use of the resources, leading to a
revision of the design. For example, in the study by van Dijke-Droogers et al.
(2018), a HLT was designed and a teaching experiment (in the Netherlands) was
conducted to evaluate and revise this HLT.The challenge was to invite ninth grade
students, inexperienced with sampling, to making informal statistical inferences
without the knowledge of the formal probability theory. As educational materials
that focus on the development of informal statistical inferences for grade 9in the
Netherlands hardly exist, the materials had to be designed. In the HLT, the students
were expected to proceed from a rst experience with sampling physical objects,
through an understanding of sampling variation and resampling, to reasoning with
the simulated empirical sampling distribution. Design guidelines were identied
through a literature review, and the possibilities of (digital) resources were explored.
The designed eight-step HLT included information about the theoretical back-
ground, the learning steps, teaching approach, lesson activities, tools and materials,
practical guidelines, expected student behavior, and data collection. For example, in
step 6, students investigated what happened if the sample size increased. The
hypothesis in this step was that students would understand that the characteristics
(e.g., the mean) and the shape of the distribution of a larger sample usually better
resemble the underlying population. To conceptualize this idea, students used
TinkerPlots (Konold and Miller 2005) to easily and quickly simulate samples of
different sizes. A learning activity based on growing samples (Bakker 2004) and the
use of TinkerPlots was expected to help students develop aspects of informal infer-
ence and argumentative reasoning (Ben-Zvi 2006). Next, the students were asked in
step 6 to compare similarities and differences between their simulated sample
results and during a whole-class session, to the underlying population. Embedding
students’ ndings in a classroom discussion was expected to enhance their statisti-
cal reasoning (Bakker 2004). This HLT was, as a next step, tested in a teaching
experiment. The teaching experiment comprised a ten 45-minute lesson series and
was piloted in one class with 20 students. The data analysis consisted of verifying
whether the designed hypotheses actually occurred. To this end, for each step of the
design, the formulated hypotheses were translated into visible student behavior.
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430
12.6.5 Design inUse
When designing in use, the investigation can focus on the instrumentation of spe-
cic resources to enrich and rene the schemes used by the teachers in this instru-
mentation. There are many ways to observe orchestration, that is, to say the way the
teacher appropriates these available resources and rely on them to conduct the activ-
ity of the students.
For example, Fidje (2018) in his study investigates the way teachers use student-
produced video in mathematics teaching. This research aimed to identify and char-
acterize different orchestrations used by a teacher in a mathematical discussion with
regard to student-produced videos. Open coding was used to propose a framework
adopted from Brown’s (2009) degrees of artifact appropriation: ofoading (use as
is), adapting the resource, and improvising (disregard the resource and enact without
specic guidance from the presentation). The ndings show that the teacher orches-
trated the use of videos in distinctly different ways, capitalizing on the affordances
and working around the constraints of the medium. The teacher applied what
appeared as a quite xed framework for every mathematical problem presented in
the discussion, rst, with a presentation of the problem, followed by an elaboration
through a back-and-forth discussion, and ending with a conclusion and connecting
the current problem with the succeeding problems. This xed framework was evi-
dent throughout the lesson; a new problem was never presented without a conclusion
to the former. Within this framework, a number of orchestrations related to the stu-
dent-produced videos emerged. Firstly, there were ofoading orchestrations where
the teacher used the videos as they were. The most notable examples of ofoading
were when the teacher used the videos as an introduction to a problem or as the
conclusion to the problem. Secondly, the teacher used adaptation orchestration, as
he chose to adapt most of the student-produced videos in some way or another. For
example, the teacher started the video, paused it, and directed a question to the pre-
senters in the video. Thirdly, the teacher used hybrid orchestrations, where students
were asked to present something from their video. The teacher used this orchestra-
tion to improve the video or to elaborate on the problem addressed. Fourthly, the
teacher gradually improved orchestrations. The improving orchestrations were all
prompted by the presentations in the videos, even though they were not used to pres-
ent or elaborate the questions. This study showed how the teacher identied per-
ceived affordances in the different use of the resource in his lesson design, while
planning the lesson, culminating in utilization schemes for the set of resources used.
12.6.6 Collaboration asaWay toOptimize Design
Gueudet etal. (2013) reected upon conditions which were necessary for collective
work to happen. They dened this collective work as “teachers working with ‘other
participants’, that is, teachers working with and in teams, communities and net-
works.” They proposed the following criteria: a common working room, “ofcial”
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431
working hours, and possibly the intervention of institutions linked to school. After
analyzing the DW of two mathematic teachers, their representations and practice of
collective work, they came to the conclusion that collective lesson or task prepara-
tion was very important for teachers’ DW.Nevertheless, they argued that the simple
fact of being colleagues– working with the same students or in the same schools
were not accurate sufcient criteria to guarantee satisfactory collective work.
According to them, collective work and design could develop owing to conditions
very similar to that of “communities of practice” (Wenger 1998)– groups of teach-
ers who share a “joint enterprise, a mutual commitment, and a resource repertoire”
(Pepin etal. 2013): a “mutual endeavor,” that is to say, agreeing to work on resources
according to similar objectives; “minding the system,” that is to say agreeing on
norms of participation and pedagogical actions; and “common forms of addressing
and making sense of resources,” in other words, allowing shared resources to
become collective resources appropriated by the group. Therefore, these conditions
are complex to gather as material settings (getting specic time and space, e.g., a
common room to work together) are not sufcient for satisfactory collective design
to happen. It requires both a sharing of values about teaching and teachers’ subject-
matter and a sharing of resources. It also requires a particular attention to boundary
crossing allowed by brokers, bringing new acceptable techniques and ideas into a
community. They enrich the community without disrupting it, allowing for social
creativity in the realm of technology enhanced learning, as Essonnier (2018) shows
in her PhD thesis.
Carton (2018a) showed that indirect collaboration on non-formal digital shared
spaces could foster teachers’ Pedagogical Design Capacity, but also that non-formal
digital common spaces could offer favorable settings for collective work even
though it might lead to individual design. The analysis she carried out showed that
networks and platforms that were not originally dedicated to education or linked to
school institutions could offer favorable settings for collective work, for instance,
small groups of teachers connected through apps (Google Drive, Dropbox,
WhatsApp), e-mail correspondence or social media (Facebook groups or pages
which are not institutional but linked to subject matters or groups dealing with
teachers’ professional identity and experiences). These groups appeared to be either
dened by precise circumstances (teachers who met during their internship year
during their teacher education and wanted to stay in touch), or by teachers who
already knew each other personally or professionally or who already met or built an
online relation because they shared afnities or a similar status.
Different degrees of collaboration seemed to happen in these non-formal digital
spaces: rst, each teacher interviewed admitted they consulted, were inspired by,
copied, printed, or used colleagues resources available on the Internet, through per-
sonal spaces like blogs, websites, social media like Facebook pages, or subject-
matter dedicated groups, in order to “see what others do” and to “inspire” oneself,
most of the time “without saying thank you.” This pedagogical monitoring activity
seemed to trigger a documentational genesis (Gueudet and Trouche 2008), starting
with a selection of the initial resources owing to a follow-up of the colleagues’
work, even if there was no communication between the teachers.
12 Transitions Toward Digital Resources: Change, Invariance, andOrchestration

432
Then, teachers seemed to value these non-formal groups because they felt they
could express themselves or ask questions about didactical practices or resources
without fear of judgment or assessment, which could be considered as an indirect
way of getting feedback about one’s resources. Expression of shared trust and good-
will seemed to be two essential criteria to reach the rst step of collective work: not
only getting teachers to upload their resources but also getting them to express their
own “voice” (Remillard 2010, p.206) or affordances about their resources or their
practices. Other connections between participants of these non-formal groups were
1) their desire to develop their resource system alongside their didactical practices
in order to avoid routines and to adapt to their students; 2) a feeling of loneliness
regarding these interrogations or due to their interest in digital resources among
their school team.
Although every teacher who engaged (either actively or indirectly) in digital
groups admitted these spaces were a melting pot which fostered their documentary
work, they almost never mentioned that actual collective design happened directly
within the space where they found the resources. The feedback around posted activ-
ities or documents seemed more frequent than the actual reposting of transformed
resources. Digital spaces, which constituted small groups bound by close ties
(WhatsApp, Dropbox, email correspondence, private mailbox on apps), seemed to
be more favorable for collective feedback on resources, as modied resources were
exchanged and commented upon, while in bigger groups, especially on social media
(Facebook and Twitter) dialogue and interactivity seemed to serve each participant’s
professional development more than collective work.
In the MC2 project (Essonnier 2018; Essonnier etal. 2018), a platform, named
CoICode, was designed for capturing some of the social interactions regarding the
path of an idea, documenting its diverse sources and inuences until the nal rst
cycle of a pedagogical resource. The analysis of the produced traces allowed for the
characterization of traits in a community that promote social creativity. Of course,
the TPACK of its members have to be compatible and complement each other. The
context and atmosphere have to be free and trustful enough to allow for a fruitful
divergent phase but professional enough to succeed in producing something usable
as the conclusion of a convergent phase.
Teachers’ collective work is also shaped by and for students, mainly through
non-formal interactions. Carton’s (2018b) analysis of 24 semi-structured interviews
around secondary teachers’ creativity showed that teachers described their DW as if
it was a kind of “addressed creativity,” in the rst place addressed to their students.
Participants of the study seemed to consider students both as an “audience” and as
feedback providers, offering the most direct and genuine assessment teachers could
get, which turned into a strong motivation for documentational genesis (Gueudet
and Trouche 2009b), or design.
Lastly, the analysis also revealed that most teachers felt that their PDC and skills
in crafting pedagogical episodes were mostly underestimated by school institutions.
Therefore, some of the interviewees chose to turn to companies (either publishing
houses or edtech players) that would “publish” their work– either through text-
books or instructional kits. They seemed to expect a symbolical, nancial and pro-
P. Drijvers et al.

433
fessional recognition of their expertise from these partnerships, even though they
admitted the deals did not often offer them satisfactory conditions, most of all from
a nancial point of view. Interestingly, some of the interviewees seemed to imple-
ment design habits born from their DW into paid projects, for instance, lessons
presented as sheets which were used as models for an instructional kit. A hypothesis
which needs further research would be to consider that teachers accept these kinds
of partnerships because they throw light upon their PDC and therefore serve their
professional development.
12.6.7 From Design inUse toDesign forUse
The discussion presented in this chapter shows the richness of the theoretical con-
structs such as IO and DAD to observe, analyze, systematize and anticipate the
activity, individual or collective, of the teacher using digital resources and systems,
and this from multiple insights. The rst contribution of the works presented is
obviously relative to the models by allowing their validation, renements and evolu-
tions. A second contribution is relative to the conception and development of digital
resources, interfaces, supports and systems, which scaffold the engineering-teach-
ing activity undertaken by the teacher.
The realization of resources and platforms founded on theoretical and theoretical
reections is useful for several reasons. The rst is that engineering questions theo-
ries because it requires tangible operational answers, which can be programed and
computed, and this in turn promotes the evolution of the theories. The second is that
produced artifacts and platforms provide ways of validating the answers provided
by theories. We can consider a theoretical validation by the evaluation of the ade-
quacy between the realization and its specications. The adequacy in a semi-theo-
retical setup in laboratory with technologically experienced teachers might differ
from practical experimentation by ordinary teachers. The third is that the designed
artifacts and platforms are products that enrich teachers’ resource systems, partici-
pating in their professional development, and infusing theoretical research into
society.
A rst focus of the research presented concerns on resources and their character-
ization by their own specicities, by the utilization scheme implemented by the
teachers, and by the instrumental geneses implemented by the students. From this
rst insight, Adler’s (2000) systematization of resources (object-action-activity)
helps us to better analyze the choice and use of teachers’ resources, and provides
theoretical and methodological principles to produce the specications for com-
puter supports for these choices and uses.
A second focus is on the activity of the teacher as an engineer observing and
analyzing his activity of preparing his teaching and producing material from digital
resources. In particular, the teacher’s PDC or HLT can be evaluated.
Although we focus on the conception and implementation of supports for didac-
tic material production for the teacher, many possible orchestrations and articula-
12 Transitions Toward Digital Resources: Change, Invariance, andOrchestration

434
tions of resources can be built since the produced didactic material can be a didactic
situation based on problem solving, a list of training exercises, a multimedia presen-
tation of a specic content, a digital textbook, etc. For each of these possible
resource orchestrations, specic supports can be provided. Generally, conceiving
and implementing supports for the “design in use” of resources needs a wide variety
of investigations, mostly built on the IO and DAD to understand the way teachers
and researchers are selecting, taking decisions, combining and articulating resources,
freely or with the support/constraint of platforms, individually or collectively. It has
the purpose of several works presented during the Re(s)sources 2018 international
conference.
12.7 Conclusion: What Has andHas Not Been Addressed
PaulDrijvers, VerônicaGitirana, JohnMonaghan and SametOkumus
The questions in the original remit of WG4 have been unevenly addressed in the
Working Group papers and, consequently, in this chapter. Neither the question
regarding opportunities for new learning formats, such as blended learning and
ipped classrooms, nor the question on what student resource systems look like
have been addressed. There has also been little consideration of the role that digital
resources play in assessment. The questions on how to choose appropriate resources
and how to adapt them to specic learning goals, as well as the question regarding
options for personalized learning, have been considered, among other things, in
Sects. 12.3 and 12.4. The question on how to prepare pre- and in- service teachers
has been addressed in Sect. 12.5.
New foci (or, at least, new takes on existing foci) have been introduced. The
relationship between instrumental genesis and affordances is considered in some
depth in Sect. 12.2 (and mentioned elsewhere). This is, we feel, an important focus
for further work and could link with issues in the design of resources for teaching
and for learning. Section 12.3 considers ve areas (student-centered orchestrations;
extending the repertoire of orchestrations; chaining orchestrations; didactical per-
formance; and gestures) where the model of IO is not fully exploited. This section
could/should be used as a springboard for further work in these areas. Section 12.4
raises and partially addresses a number of questions regarding networking the DAD
to other theoretical framework, but as we noted in our Introduction, further work
needs to be done here. Section 12.5 considers many conceptions of teaching (with
digital resources) and advances knowledge in doing so, but further advancement
requires networking these conceptions. Section 12.6 helps us appreciate that how
learning environments are designed and used in practice and what works (and how
it works). Further work in this area includes not just networking theoretical frame-
works but networking elds of study (designers and didacticians).
P. Drijvers et al.

435
References
Adler, J. (2000). Conceptualising resources as a theme for teacher education. Journal of
Mathematics Teacher Education, 3(3), 205–224.
Arcavi, A. (2003). The role of visual representations in the learning of mathematics. Educational
Studies in Mathematics, 52(3), 215–241.
Artigue, M. (1990). Ingénierie didactique. Recherches en didactique des mathématiques, 9(3),
281–307.
Artigue, M. (2002). Learning mathematics in a CAS environment: The genesis of a reection
about instrumentation and the dialectics between technical and conceptual work. International
Journal of Computers for Mathematical Learning, 7, 245–274.
Artzt, A.F., Armour-Thomas, E., Curcio, C., & Gurl, T.J. (2015). Becoming a reective mathemat-
ics teacher: A guide for observations and self-assessment (3rd ed.). NewYork: Routledge.
Assis, C., Gitirana, V., & Trouche, L. (2018). The metamorphosis of resource systems of prospec-
tive teacher: From studying to teaching. In V.Gitirana, T.Miyakawa, M.Rafalska, S.Soury-
Lavergne, & L.Trouche (Eds.), Proceedings of the re(s)sources 2018 international conference
(pp.39–42). Lyon: ENS de Lyon, Retrieved on November 8th, 2018, at https://hal.archives-
ouvertes.fr/hal-01764563
Assude, T. (2007). Teacher’s practices and degree of ICT integration. In D. Pitta-Pantazi, &
G.Philippou (Eds.), Proceedings of the Fifth Congress of the European Society for Research in
Mathematics Education (pp.1339–1348). Larnaca: University of Cyprus and ERME.
Assude, T., Buteau, C., & Forgasz, H. (2010). Factors inuencing implementation of technology-
rich mathematics curriculum and practices. In C.Hoyles & J.-B.Lagrange (Eds.), Mathematics
education and technology – Rethinking the terrain. The 17th ICMI study (pp. 405–419).
NewYork: Springer.
Atkinson, B. (2012). Rethinking reection: Teachers’ critiques. The Teacher Educator, 47,
175–194.
Bailleul, M., & Thémines, J.F. (2013). L’ingénierie de formation : formalisation d’expériences
en formation d’enseignants. In A. Vergnioux (Ed.),. Traité d’ingénierie de la formation
L’Harmattan (pp.85–112). Paris.
Bakker, A. (2004). Design research in statistics education: On symbolizing and computer tools.
Utrecht: CD Beta Press.
Balacheff, N. (1994). La transposition informatique. Note sur un nouveau problème pour la didac-
tique. In M.Artigue, R.Gras, C.Laborde, & P.Tavignot (Eds.), Vingt ans de Didactique des
Mathématiques en France (pp.364–370). Grenoble: La Pensée Sauvage.
Barbosa, A., & Vale, I. (2018). Math trails: A resource for teaching and learning. In V.Gitirana,
T.Miyakawa, M.Rafalska, S.Soury-Lavergne, & L.Trouche (Eds.), Proceedings of the re(s)
sources 2018 international conference (pp. 183–186). Lyon: ENS de Lyon. Retrieved on
November 8th, 2018, at https://hal.archives-ouvertes.fr/hal-01764563.
Barth-Cohen, L.A., Little, A.J., & Abrahamson, D. (2018). Building reective practices in a pre-
service math and science teacher education course that focuses on qualitative video analysis.
Journal of Science Teacher Education, 29(2), 83–101.
Bellemain, F., Tiburcio, R., Silva, C., & Gitirana, V. (2016). Function Studium software. Recife-PE:
LEMATEC Research Group.
Bellemain, F., Rodrigues, A., & Rodrigues, A.D. (2018). LEMATEC Studium: A support resource
for teaching mathematics. In V.Gitirana, T.Miyakawa, M.Rafalska, S.Soury-Lavergne, &
L.Trouche (Eds.), Proceedings of the re(s)sources 2018 international conference (pp.255–
258). Lyon: ENS de Lyon. Retrieved on November 8th, 2018, at https://hal.archives-ouvertes.
fr/hal-01764563.
Ben-Zvi, D. (2006). Scaffolding students’ informal inference and argumentation. In A.Rossman &
B.Chance (Eds.), Proceedings of the Seventh International Conference on Teaching Statistics
(CD-ROM). Voorburg: International Statistical Institute.
12 Transitions Toward Digital Resources: Change, Invariance, andOrchestration

436
Besnier, S. (2016). Le travail documentaire des professeurs à l’épreuve des ressources tech-
nologiques : Le cas de l’enseignement du nombre à l’école maternelle. PhD.Brest: Université
de Bretagne Occidentale, https://tel.archives-ouvertes.fr/tel-01397586/document
Besnier, S. (2018). Orchestrations at kindergarten: Articulation between manipulatives and digital
resources. In V.Gitirana, T.Miyakawa, M.Rafalska, S.Soury-Lavergne, & L.Trouche (Eds.),
Proceedings of the re(s)sources 2018 international conference (pp.259–262). Lyon: ENS de
Lyon. Retrieved on November 8th, 2018, at https://hal.archives-ouvertes.fr/hal-01764563.
Besnier, S., & Gueudet, G. (2016). Usages de ressources numériques pour l’enseignement des
mathématiques en maternelle : orchestrations et documents. Perspectivas em Educação
Matemática, 9(21), 978–1003.
Billington, M. (2009). Establishing didactical praxeologies: teachers using digital tools in upper
secondary mathematics classrooms. In V.Durand-Guerrier, S.Soury-Lavergne & F.Arzarello
(Eds.), Proceedings of the Sixth Congress of European Research in Mathematics Education
(pp.1330–1339). Lyon, France: ENS de Lyon.
Bozkurt, G., & Ruthven, K. (2017). Classroom-based professional expertise: A mathematics teach-
er’s practice with technology. Educational Studies in Mathematics, 94(3), 309–328.
Brousseau, G. (1997). Theory of didactical situations in mathematics. Didactique des mathé-
matiques, 1970–1990 (edited and translated by N.Balacheff, M.Cooper, R.Sutherland, &
V.Wareld). Dordrecht, NL. Kluwer Academic Publishers.
Brown, M.W. (2009). The teacher-tool relationship: Theorizing the design and use of curriculum
materials. In J.T. Remillard, B.A. Herbel-Eisenmann, & G.M. Lloyd (Eds.), Mathematics
teachers at work: Connecting curriculum materials and classroom instruction (pp.17–36).
NewYork: Routledge.
Carlsen, M., Erfjord, I., Hundeland, P.S., & Monaghan, J.(2016). Kindergarten teachers’ orches-
tration of mathematical activities afforded by technology: Agency and mediation. Educational
Studies in Mathematics, 93(1), 1–17.
Carton, T. (2018a). From digital “bricolage” to the start of collective work– What inuences
do secondary teachers non-formal digital practices have on their documentation work? In
V.Gitirana, T.Miyakawa, M.Rafalska, S.Soury-Lavergne, & L.Trouche (Eds.), Proceedings
of the re(s)sources 2018 international conference (pp.263–266). Lyon: ENS de Lyon. Retrieved
on November 8th, 2018, at https://hal.archives-ouvertes.fr/hal-01764563.
Carton, T. (2018b). Case study: How teachers’ everyday creativity can be associated with an
edtech player’s strategies. Présenté au XXIe Congrès de la SFSIC – Création, créativité et
médiations, Saint-Denis, 13-15 juin 2018.
Cayton, C., Hollebrands, K., Okumuş, S., & Boehm, E. (2017). Pivotal teaching moments in tech-
nology-intensive secondary geometry classrooms. Journal of Mathematics Teacher Education,
20(1), 75–100.
Chevallard, Y. (2002). Organiser l’étude. 1. Structures & fonctions. Actes de la XIe école d’été de
didactique des mathématiques (pp.3–32). La Pensée Sauvage : Grenoble. http://yves.cheval-
lard.free.fr/spip/spip/IMG/pdf/Organiser_l_etude_1.pdf
Choppin, J.(2018). Exploring teachers’ design processes with different curriculum programs. In
V.Gitirana, T.Miyakawa, M.Rafalska, S.Soury-Lavergne, & L.Trouche (Eds.), Proceeedings
of the re(s)sources 2018 international conference (pp.267–270). Lyon: ENS de Lyon. Retrieved
on November 8th, 2018, at https://hal.archives-ouvertes.fr/hal-01764563.
Cuevas, C.A., Villamizar, F.Y., & Martínez, A. (2017). Aplicaciones de la tecnología digital para
activiDADes didácticas que promuevan una mejor comprensión del tono como cualiDAD del
sonido para cursos tradicionales de física en el nivel básico. Enseñanza de las Ciencias, 35(3),
129–150.
Cunha, E. (2018). Digital resources: Origami folding instructions as lever to mobilize geometric
concepts to solve problems. In V.Gitirana, T.Miyakawa, M.Rafalska, S.Soury-Lavergne, &
L.Trouche (Ed.), Proceeedings of the Re(s)sources 2018 International Conference (pp.271–
274). Lyon: ENS de Lyon. Retrieved on November 8th, 2018, at https://hal.archives-ouvertes.
fr/hal-01764563
P. Drijvers et al.

437
Deblois, L. (2012). De l’ancien élève à l’enseignant. Quel parecours. In J.Proulx, C.Corriveau, &
H.Squalli (Eds.), Formation mathématique pour l’enseignement des mathématiques (pp.313–
320). Québec: Presses de l’Université du Québec.
Dick, T., & Burrill, G. (2016). Design and implementation principles for dynamic interactive math-
ematics technologies. In M.Niess, S.Driskell, & K.Hollebrands (Eds.), Handbook of research
on transforming mathematics teacher education in the digital age (pp.23–52). Hershey: IGI
Global Publishers.
Drijvers, P., & Trouche, L. (2008). From artifacts to instruments: A theoretical framework behind
the orchestra metaphor. In G.W. Blume & M.K. Heid (Eds.), Research on technology and
the teaching and learning of mathematics. Cases and perspectives (Vol. 2, pp. 363–392).
Charlotte: Information Age.
Drijvers, P., Doorman, M., Boon, P., Reed, H., & Gravemeijer, K.P. (2010). The teacher and the
tool: Instrumental orchestrations in the technology-rich mathematics classroom. Educational
Studies in Mathematics, 75(2), 213–234.
Drijvers, P., Godino, J.D., Font, V., & Trouche, L. (2013a). One episode, two lenses. Educational
Studies in Mathematics, 82(1), 23–49.
Drijvers, P., Tacoma, S., Besamusca, A., Doorman, M., & Boon, P. (2013b). Digital resources
inviting changes in mid-adopting teachers’ practices and orchestrations. ZDM– Mathematics
Education, 45(7), 987–1001.
Duval, R. (1993). Registres de représentation sémiotique et fonctionnement cognitif de la pensée.
Annales de didactique et de sciences cognitives, 5(1), 37–65.
Engeström, Y. (1987). Learning by expanding: An activity-theoretical approach to developmental
research. Helsinki: Orienta-Konsultit.
Erfjord, I. (2011). Teachers’ initial orchestration of students’ dynamic geometry software use:
Consequences for students’ opportunities to learn mathematics. Technology, Knowledge and
Learning, 16(1), 35–54.
Essonnier, N.K. (2018). Étude de la conception collaborative de ressources numériques mathé-
matiques au sein d’une communauté d’intérêt. PhD.Lyon: Université de Lyon, https://tel.
archives-ouvertes.fr/tel-01868226/document
Essonnier, N., & Trgalová, J.(2018). Collaborative design of digital resources: Role of design-
ers’ resource systems and professional knowledge. In V.Gitirana, T.Miyakawa, M.Rafalska,
S.Soury- Lavergne, & L.Trouche (Eds.), Proceeedings of the re(s)sources 2018 international
conference (pp.61–64). Lyon: ENS de Lyon. Retrieved on November 8th, 2018, at https://hal.
archives-ouvertes.fr/hal-01764563.
Essonnier, N., Kynigos, C., Trgalová, J., & Daskolia, M. (2018). Role of context in social creativity
for the design of digital resources. In L.Fan, L.Trouche, S.Rezat, C.Qi, & J.Visnovska (Eds.),
Research on mathematics textbooks and Teachers’ resources: Advances and issues (pp.215–
233). Cham: Springer.
Faggiano, E., Montone, A., & Mariotti, M.A. (2016). Creating a synergy between manipula-
tives and virtual artefacts to conceptualize axial symmetry at Primary School. In C.Csíkos,
A. Rausch, & J. Szitányi (Eds.), Proceedigns of the 40th conference of the international
group for the Psychology of Mathematics Education: PME 40 (Vol. 2, pp.235–242). Szeged:
International Group for the Psychology of Mathematics Education.
Farrell, A. (1996). Roles and behaviors in technology-integrated precalculus classrooms. Journal
of Mathematical Behavior, 15(1), 35–53.
Ferrara, F., & Sinclair, N. (2016). An early algebra approach to pattern generalisation: Actualising
the virtual through words, gestures and toilet paper. Educational Studies in Mathematics,
92(1), 1–19.
Fidje, A.S. (2018). Orchestrating the use of student-produced videos in mathematics teaching. In
V.Gitirana, T.Miyakawa, M.Rafalska, S.Soury-Lavergne, & L.Trouche (Eds.), Proceedings
of the re(s)sources 2018 international conference (pp. 275–278). Lyon: ENS de Lyon. In
Retrieved on November 8th, 2018, at https://hal.archives-ouvertes.fr/hal-01764563.
12 Transitions Toward Digital Resources: Change, Invariance, andOrchestration

438
Fischer, G. (2001). Communities of interest: Learning through the interaction of multiple knowl-
edge systems. In S.Bjørnestad, R.Moe, A.Mørch, & A.Opdahl (Eds.), Proceedings of the 24th
Information Systems Research Seminar in Scandinavia (pp.1–14). Bergen: The University of
Bergen
Gibson, J.J. (1977). The ecological approach to visual perception. Boston: Houghton Mifin.
Gueudet, G., & Trouche, L. (2008). Du travail documentaire des enseignants?: Genèses, collectifs,
communautés. Le cas des mathématiques. Education & Didactique, 2(3), 7–33.
Gueudet, G., & Trouche, L. (2009a). Towards new documentational systems for mathematics
teachers? Educational Studies in Mathematics, 71(3), 199–218.
Gueudet, G., & Trouche, L. (2009b). Vers de nouveaux systèmes documentaires des professeurs
de mathématiques?). In I.Bloch & F.Conne (Eds.), Nouvelles perspectives en didactique des
mathématiques. Cours de la XIVe école d’été de didactique des mathématiques (pp.109–133).
Paris: La Pensée Sauvage.
Gueudet, G., & Trouche, L. (2010). Des ressources aux documents, travail du professeur et genèses
documentaires. In G.Gueudet & L.Trouche (Eds). Ressources vives: le travail documentaire
des professeurs en mathématiques (pp.57–74). Paideia, Rennes: Presses Universitaires de
Rennes & INRP.
Gueudet, G., & Trouche, L. (2011). Mathematics teacher education advanced methods: An exam-
ple in dynamic geometry. ZDM– Mathematics Education, 43(3), 399–411.
Gueudet, G., Sacristan, A., Soury-Lavergne, S., & Trouche, L. (2012). Online paths in mathemat-
ics teacher training: New resources and new skills for teacher educators. ZDM– Mathematics
Education, 44(6), 717–731.
Gueudet, G., Pepin, B., & Trouche, L. (2013). Collective work with resources: An essential dimen-
sion for teacher documentation. ZDM– Mathematics Education, 45(7), 1003–1016.
Gueudet, G., Pepin, B., Sabra, H., & Trouche, L. (2016). Collective design of an e-textbook:
Teachers’ collective documentation. Journal of Mathematics Teacher Education, 19(2),
187–203.
Guin, D., & Trouche, L. (1999). The complex process of converting tools into mathematical
instruments: The case of calculators. International Journal of Computers for Mathematical
Learning, 3(3), 195–227.
Hadjerrouit, S. (2017). Assessing the affordances of SimReal+ and their applicability to support
the learning of mathematics in teacher education. Issues in Informing Science and Information
Technology Education, 14, 121–138.
Heid, M., K. (2008). Calculator and computer technology in the K-12 curriculum some observa-
tions from a US perspective. In Z.Ususkin, & E.Willmore (Eds.), Mathematics curriculum
in Pacic rim countries-China, Japan, Korea, and Singapore: Proceedings of a conference
(pp.293–304). Charlotte, Information Age Publishing.
Herbst, P., Chazan, D., Chen, C., Chieu, V.M., & Weiss, M. (2011). Using comics-based repre-
sentations of teaching, and technology, to bring practice to teacher education courses. ZDM–
Mathematics Education, 43(1), 91–103.
Hollebrands, K., & Okumus, S. (2018). Secondary mathematics teachers’ instrumental integra-
tion in technology-rich geometry classrooms. Journal of Mathematical Behavior, 49(1), 82–94.
Hollebrands, K., & Zbiek, R. (2004). Teaching mathematics with technology: An evidence-based
road map for the Journey. In R.Rubenstein & G.Bright (Eds.), Perspectives on the teaching
of mathematics: Sixty-sixth yearbook (pp.259–270). Reston: National Council of Teachers of
Mathematics.
Hollebrands, K., McCulloch, A.W., & Lee, H.S. (2016). Prospective teachers; incorporation of
technology in mathematics lesson plans. In M.Niess, S.Driskell, & K.Hollebrands (Eds.),
Handbook of research on transforming mathematics teacher education in the digital age
(pp.272–292). Hershey: IGI Global.
Hoyles, C., & Noss, R. (1992). A pedagogy for mathematical microworlds. Educational Studies in
Mathematics, 23(1), 31–57.
P. Drijvers et al.

439
Igliori, S.B. C., & Almeida, M.V. (2018). Un support numérique pour le travail de documentation
des enseignants de mathématiques de l’EFII (Collège, en France). In V.Gitirana, T.Miyakawa,
M.Rafalska, S.Soury-Lavergne, & L.Trouche (Eds.), Proceeedings of the re(s)sources 2018
international conference (pp.288–291). Lyon: ENS de Lyon. Retrieved on November 8th,
2018, at https://hal.archives-ouvertes.fr/hal-01764563.
Ignácio, R., Lima, R., & Gitirana, V. (2018). The birth of the documentary system of mathematics
pre-service teachers in a supervised internship with the creation of a digital textbook chapter. In
V.Gitirana, T.Miyakawa, M.Rafalska, S.Soury-Lavergne, & L.Trouche (Eds.), Proceeedings
of the re(s)sources 2018 international conference (pp.292–295). Lyon: ENS de Lyon. retrieved
on November 8th 2018 at https://hal.archives-ouvertes.fr/hal-01764563.
Jameau, A., & Le Hénaff, C. (2018). Resources for science teaching in a foreign language. In
V.Gitirana, T.Miyakawa, M.Rafalska, S.Soury-Lavergne, & L.Trouche (Eds.), Proceeedings
of the re(s)sources 2018 international conference (pp.79–82). Lyon: ENS de Lyon. retrieved
on November 8th 2018 at https://hal.archives-ouvertes.fr/hal-01764563.
Jaworski, B. (2014). Reective practicioner in mathematics education. In S. Lerman (Ed.),
Encyclopedia in mathematics education (pp.529–532). Dordrecht: Springer.
Jones, W. (2007). Personal information management. Annual Review of Information Science and
Technology, 41(1), 453–504.
Kaput, J.J. (1995). Overcoming physicality and the eternal present: Cybernetic manipulatives. In
R.Sutherland & J.Mason (Eds.), Exploiting mental imagery with computers in mathematics
education (pp.161–177). Berlin: Springer.
Kennewell, S. (2001). Using affordances and constraints to evaluate the use of information and
communications technology in teaching and learning. Journal of Information Technology for
Teacher Education, 10(1–2), 101–116.
Kidron, I., Bosch, M., Monaghan, J., & Palmér, H. (2018). Theoretical perspectives and approaches
in mathematics education research. In T. Dreyfus, M. Artigue, D. Potari, S. Prediger, &
K.Ruthven, (Eds.), Developing Research in Mathematics Education: Twenty years of com-
munication, cooperation and collaboration in Europe. London: Routledge.
Kieran, C., Boileau, A., Tanguay, D., & Drijvers, P. (2013). Design researchers’ documentational
genesis in a study on equivalence of algebraic expressions. ZDM– Mathematics Education,
45(7), 1045–1056.
Kirchner, P., Strijbos, J.-W., Kreijns, K., & Beers, B.J. (2004). Designing electronic collabora-
tive learning environments. Educational Technology Research and Development, 52(3), 47–66.
Koehler, M.J., & Mishra, P. (2009). What is technological pedagogical content knowledge?
Contemporary Issues in Technology and Teacher education, 9(1), 60–70.
Konold, C., & Miller, C.D. (2005). TinkerPlots: Dynamic data exploration (computer software,
Version 1.0). Emeryville: Key Curriculum Press.
Kozaklı Ülger, T., & Tapan Broutin, M.S. (2018). Transition from a paper-pencil to a technology
enriched environment: A teacher’s use of technology and resource selection. In V. Gitirana,
T.Miyakawa, M.Rafalska, S.Soury-Lavergne, & L.Trouche (Eds.), Proceedings of the re(s)
sources 2018 international conference (pp. 344–347). Lyon: ENS de Lyon. Retrieved on
November 8th, 2018, at https://hal.archives-ouvertes.fr/hal-01764563.
Lagrange, J.B., & Monaghan, J.(2009). On the adoption of a model to interpret teachers’ use of
technology in mathematics lessons. In V.Durand-Guerrier, S.Soury-Lavergne & F.Arzarello
(Eds.). Proceedings of the Sixth Congress of European Research in Mathematics Education
(pp.1605–1614). Lyon: ENS de Lyon.
Lakoff, G., & Núñez, R.E. (2000). Where mathematics comes from: How the embodied mind
brings mathematics into being. NewYork: Basic Books.
Lealdino Filho, P., & Mercat, C. (2018). Teaching computational thinking in classroom environ-
ments: A case for unplugged scenario. In V. Gitirana, T.Miyakawa, M.Rafalska, S.Soury-
Lavergne, & L.Trouche (Eds.), Proceedings of the re(s)sources 2018 international conference
(pp.296–299). Lyon: ENS de Lyon. Retrieved on November 8th, 2018, at https://hal.archives-
ouvertes.fr/hal-01764563.
12 Transitions Toward Digital Resources: Change, Invariance, andOrchestration

440
Leroyer, L. (2018). The capacity to think of transmission of knowledge from learning supports:
A proposition of a conceptual model. In V. Gitirana, T.Miyakawa, M. Rafalska, S. Soury-
Lavergne, & L.Trouche (Eds.), Proceeedings of the re(s)sources 2018 international conference
(pp.203–206). Lyon: ENS de Lyon. Retrieved on November 8th, 2018, at https://hal.archives-
ouvertes.fr/hal-01764563.
Lester, F.K. (2005). On the theoretical, conceptual, and philosophical foundations for research in
mathematics education. ZDM– Mathematics Education, 37(6), 457–467.
Leung, A., & Bolite-Frant, J. (2015). Designing mathematics tasks: The role of tools. In A.Watson
& M.Ohtani (Eds.), Task design in mathematics education:. The 22nd ICMI study (pp.191–
225). Cham: Springer.
Lucena, R. (2018). Metaorquestração Instrumental: um modelo para repensar a formação
teórico-prática de professores de matemática. Doctoral thesis. Mathematics and Technological
Education Pos- graduation Program. Recife-Brazil: UFPE.
Lucena, R., Gitirana, V., & Trouche, L. (2018). Instrumental meta-orchestration for teacher edu-
cation. In V.Gitirana, T.Miyakawa, M.Rafalska, S.Soury-Lavergne, & L.Trouche (Eds.),
Proceedings of the re(s)sources 2018 international conference (pp.300–303). Lyon: ENS de
Lyon. Retrieved on November 8th, 2018, at https://hal.archives-ouvertes.fr/hal-01764563.
Males, L., Setniker, A., & Dietiker, L. (2018). What do teachers attend to in curriculum mate-
rials? In V. Gitirana, T.Miyakawa, M. Rafalska, S. Soury-Lavergne, & L. Trouche (Eds.),
Proceedings of the re(s)sources 2018 international conference (pp.207–210). Lyon: ENS de
Lyon. Retrieved on November 8th, 2018, at https://hal.archives-ouvertes.fr/hal-01764563.
Martinez, M., Cruz, R., & Soberanes, A. (2018). The mathematical teacher: A case study of instru-
mental genesis in the UAEM.In V.Gitirana, T.Miyakawa, M.Rafalska, S.Soury-Lavergne,
& L.Trouche (Eds.), Proceedings of the re(s)sources 2018 international conference (pp.211–
214). Lyon: ENS de Lyon. retrieved on November 8th, 2018, at https://hal.archives-ouvertes.
fr/hal-01764563.
Maschietto, M., & Soury-Lavergne, S. (2013). Designing a duo of material and digital artifacts:
The pascaline and Cabri Elem e-books in primary school mathematics. ZDM– Mathematics
Education, 45(7), 959–971.
Mercat, C., Lealdino Filho, P., & El-Demerdash, M. (2017). Creativity and technology in math-
ematics : From story telling to algorithmic with Op’Art. Acta Didactica Napocensia, 10(1),
63–70.
Messaoui, A. (2018). The complex process of classifying resources, an essential component of
documentation expertise. In V. Gitirana, T.Miyakawa, M.Rafalska, S. Soury-Lavergne, &
L.Trouche (Eds.), Proceedings of the res(s)ource 2018 international conference (pp.83–87).
Lyon: ENS de Lyon. Retrieved on November 8th, 2018, at https://hal.archives-ouvertes.fr/
hal-01764563.
Mishra, P., & Koehler, M.J. (2006). Technological pedagogical content knowledge: A framework
for teacher knowledge. Teachers College Record, 108(6), 1017–1054.
Monaghan, J.(2016). Developments relevant to the use of tools in mathematics. In J.Monaghan,
L. Trouche, & J. M. Borwein (Eds.), Tools and mathematics: Instruments for learning
(pp.163–180). NewYork: Springer.
Naftaliev, E. (2018). Prospective teachers’ interactions with interactive diagrams: Semiotic tools,
challenges and new paths. In V.Gitirana, T.Miyakawa, M. Rafalska, S.Soury-Lavergne, &
L.Trouche (Eds.), Proceedings of the res(s)ource 2018 international conference (pp. 304–
307). Lyon: ENS de Lyon. Retrieved on November 8th, 2018, at https://hal.archives-ouvertes.
fr/hal-01764563.
Naftaliev, E., & Yerushalmy, M. (2017). Design digital tasks: Interactive diagrams as resource and
constraint. In A.Leung & A.Baccaglini-Frank (Eds.), The role and potential of using digital
technologies in designing mathematics education tasks (pp.153–173). Cham: Springer.
Nascimento, J., Jr., Carvalho, E., & Farias, L.M. (2018). Creation of innovative teaching situa-
tion through instrumental genesis to maximize teaching specic content: Acid-base chemical
balance. In V.Gitirana, T.Miyakawa, M.Rafalska, S.Soury-Lavergne, & L.Trouche (Eds.),
P. Drijvers et al.

441
Proceedings of the res(s)ource 2018 international conference (pp.308–311). Lyon: ENS de
Lyon. Retrieved on November 8th, 2018, at https://hal.archives-ouvertes.fr/hal-01764563.
Ndlovu, M., Wessels, D., & De Villiers, M. (2011). An instrumental approach to modelling the
derivative in sketchpad. Pythagoras, 32(2), 1–15.
Nongni, G, & DeBlois, L. (2017). Planication de l’enseignement de l’écart-type en utilisant les
ressources documentaires. In A.Adihou, J.Giroux, A.Savard, & K.M.Huy (Eds.). Données,
variabilité et tendance vers le futur. Acte du Colloque du GDM (pp. 205–2012). Canada,
Québec: Université McGill.
Nongni, G., & DeBlois. (2018). Planning of the teaching of the standard deviation using digi-
tal documentary resources. In V.Gitirana, T.Miyakawa, M.Rafalska, S.Soury-Lavergne, &
L.Trouche (Eds.), Proceedings of the re(s)sources 2018 international conference (pp.312–
315). Lyon: ENS de Lyon. Retrieved on November 8th, 2018, at https://hal.archives-ouvertes.
fr/hal-01764563
Norman, D.A. (1988). The psychology of everyday things. NewYork: Basic Books.
Noss, R., & Hoyles, C. (1996). Windows on mathematical meanings: Learning cultures and com-
puters. Dordrecht: Kluwer Academic Publishers.
Okumus, S., & Ipek, A.S. (2018). Pre-service mathematics teachers’ investigation of the con-
straints of mathematical tools. In V.Gitirana, T.Miyakawa, M.Rafalska, S.Soury-Lavergne,
& L.Trouche (Eds.), Proceedings of the re(s)source 2018 international conference (pp.316–
319). Lyon: ENS de Lyon, retrieved on November 8th, 2018, at https://hal.archives-ouvertes.
fr/hal-01764563
Orozco, J., Cuevas, A., Madrid, H., & Trouche, L. (2018). A proposal of instrumental orchestra-
tion to introduce eigenvalues and eigenvectors in a rst course of linear algebra for engineering
students. In V.Gitirana, T.Miyakawa, M.Rafalska, S.Soury-Lavergne, & L.Trouche (Eds.),
Proceedings of the re(s)sources 2018 international conference (pp.320–323). Lyon: ENS de
Lyon. Retrieved on November 8th, 2018, at https://hal.archives-ouvertes.fr/hal-01764563.
Papert, S., & Harel, I. (1991). Preface, situating constructionism. In I.Harel & S.Papert (Eds.),
Constructionism, research reports and essays, 1985–1990 (p.1). Norwood: Ablex.
Pea, R.D. (1985). Beyond amplication: Using the computer to reorganize mental functioning.
Educational Psychologist, 20(4), 167–182.
Pepin, B., Gueudet, G., & Trouche, L. (2013). Re-sourcing teachers’ work and interactions: A col-
lective perspective on resources, their use and transformation. ZDM– Mathematics Education,
45(7), 929–943.
Pepin, B., Gueudet, G., & Trouche, L. (2017). Rening teacher design capacity: Mathematics
teachers’ interactions with digital curriculum resources. ZDM– Mathematics Education, 49(5),
799–812.
Pfeiffer, C.R. (2017). A study of the development of mathematical knowledge in a GeoGebra-
focused learning environment. Unpublished PhD dissertation. Stellenbosch: Stellenbosch
University.
Pfeiffer, C.R., & Ndlovu, M. (2018). Teaching and learning of function transformations in a
GeoGebra-focused learning environment. In V.Gitirana, T.Miyakawa, M.Rafalska, S.Soury-
Lavergne, & L.Trouche (Eds.), Proceedings of the re(s)sources 2018 international conference
(pp.324–327). Lyon: ENS de Lyon. Retrieved on November 8th, 2018, at https://hal.archives-
ouvertes.fr/hal-01764563.
Piaget, J. (1955). The construction of reality in the child. London: Routledge & Kegan Paul
Limited.
Prieur, M. (2016). La conception codisciplinaire de métaressources comme appui à l’évolution des
connaissances des professeurs de sciences. Les connaissances qui guident un travail de prépa-
ration pour engager les élèves dans l’élaboration d’hypothèses ou de conjectures. PhD.Lyon:
Université de Lyon, https://hal.archives-ouvertes.fr/tel-01364778v2/document
Psycharis, G., & Kalogeria, E. (2018). TPACK addressed by trainee teacher educators’ documenta-
tion work. In V.Gitirana, T.Miyakawa, M.Rafalska, S.Soury-Lavergne, & L.Trouche (Eds.),
12 Transitions Toward Digital Resources: Change, Invariance, andOrchestration

442
Proceedings of the re(s)source 2018 international conference (pp.328–331). Lyon: ENS de
Lyon. Retrieved on November 8th, 2018, at https://hal.archives-ouvertes.fr/hal-01764563.
Rabardel, P. (1995). Les hommes et les technologies: Approche cognitive des instruments contem-
porains. Paris: Armand Colin.
Ratnayake, I., & Thomas, M. (2018). Documentational genesis during teacher collaborative devel-
opment of tasks incorporating digital technology. In V.Gitirana, T.Miyakawa, M.Rafalska,
S.Soury-Lavergne, & L.Trouche (Eds.), Proceedings of the re(s)source 2018 international
conference (pp.219–222). Lyon: ENS de Lyon. Retrieved on November 8th, 2018, at https://
hal.archives-ouvertes.fr/hal-01764563.
Remillard, J.(2010). Modes d’engagement : comprendre les transactions des professeurs avec les
ressources curriculaires en mathématiques. In G.Gueudet, & L. Trouche (Eds.), Ressources
vives: le travail documentaire des professeurs en mathématiques (pp.201–216). Rennes/Lyon:
INRP/PUR.
Rocha, K. (2018). Uses of online resources and documentational trajectories: The case of Sésamath.
In L.Fan, L.Trouche, C.Qi, S.Rezat, J.& Visnovska (Eds.), Research on mathematics text-
books and teachers’ resources: Advances and issues. ICME-13 monograph (pp.235–258).
Cham: Springer.
Rodrigues, A., Baltar, P., & Bellemain, F. (2018). Analysis of a Task in three Environments: paper
and pencils, manipulative materials and Apprenti Géomètre 2∗. In V.Gitirana, T.Miyakawa,
M.Rafalska, S.Soury–Lavergne, & L.Trouche (Eds.), Proceedings of the Re(s)source 2018
International Conference (pp.223–226). Lyon: ENS de Lyon. Retrieved on November 8th,
2018, at https://hal.archives-ouvertes.fr/hal-01764563
Rudd, T. (2007). Interactive whiteboards in the classroom. Bristol: Futurelab Report– IWBs.
Ruthven, K. (2014). Frameworks for analysing the expertise that underpins successful integra-
tion of digital technologies into everyday teaching practice. In A.Clark-Wilson, O.Robutti,
& N.Sinclair (Eds.), The mathematics teacher in the digital era (pp.373–393). Dordrecht:
Springer.
Ruthven, K., Hennessy, S., & Deaney, R. (2008). Constructions of dynamic geometry: A study
of the interpretative exibility of educational software in classroom practice. Computers and
Education, 51(1), 297–317.
Scherrer, J., & Stein, M.K. (2013). Effects of a coding intervention on what teachers learn to notice
during whole–group discussion. Journal of Mathematics Teacher Education, 16(2), 105–124.
Schoenfeld, A.H. (2010). How we think: A theory of goal–oriented decision making and its edu-
cational applications. NewYork: Routledge.
Sensevy, G. (2011). Le Sens du Savoir. Éléments pour une théorie de l’action conjointe en didac-
tique. Bruxelles: De Boeck.
Sherman, M. F., & Cayton, C. (2015). Using appropriate tools strategically for instruction.
Mathematics Teacher, 109(4), 306–310.
Sherman, M.F., Cayton, C., & Chandler, K. (2017). Supporting PSTs in using appropriate tools
strategically: A learning sequence for developing technology tasks that support students’ math-
ematical thinking. Mathematics Teacher Educator, 5(2), 122–157.
Silva, A. (2018). Concepção de um suporte para a elaboração de webdocumentos destinados ao
ensino da geometria: o caso das curvas cônicas. Dissertação do mestrado. Programa de Pós-
graduação em Educação Matemática e Tecnológica. UFPE, Recife.
Simon, M.A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective.
Journal for Research in Mathematics Education, 26(2), 114–145.
Simon, M.A., & Tzur, R. (2004). Explicating the role of mathematical tasks in conceptual learning:
An elaboration of the hypothetical learning trajectory. Mathematical Thinking and Learning,
6(2), 91–104.
Siqueira, J.E. M., & Bellemain, F. (2018). A dynamic multirepresentational resource for conics. In
V.Gitirana, T.Miyakawa, M.Rafalska, S.Soury-Lavergne, & L.Trouche (Eds.), Proceedings
of the re(s)source 2018 international conference (pp.359–361). Lyon: ENS de Lyon. Retrieved
on November 8th, 2018, at https://hal.archives-ouvertes.fr/hal-01764563.
P. Drijvers et al.

443
Stein, M.K., & Smith, M. S. (1998). Mathematical tasks as a framework for reection: From
research to practice. Mathematics Teaching in the Middle School, 3(4), 268–275.
Stockero, S.L., & Van Zoest, L.R. (2013). Characterizing pivotal teaching moments in beginning
mathematics teachers’ practice. Journal of Mathematics Teacher Education, 16(2), 125–147.
Swidan, O., Arzarello, F., & Sabena, C. (2018). Teachers’ interventions to foster inquiry-based
learning in a dynamic technological environment. In V. Gitirana, T.Miyakawa, M.Rafalska,
S.Soury- Lavergne, & L.Trouche (Eds.), Proceedings of the re(s)source 2018 international
conference (pp.332–335). Lyon: ENS de Lyon. Retrieved on November 8th, 2018, at https://
hal.archives-ouvertes.fr/hal-01764563.
Tabach, M. (2011). A mathematics teacher’s practice in a technological environment: A case study
analysis using two complementary theories. Technology, Knowledge and Learning, 16(3),
247–265.
Tabach, M. (2013). Developing a general framework for instrumental orchestration. In B.Ubuz, Ç.
Haser, & M.A. Mariotti (Eds.), Proceedings of the Eighth Congress of the European Society
for Research in Mathematics Education (pp. 2744–2753). Ankara: Middle East Technical
University and ERME.
Taranto, E., Arzarello, F., & Robutti, O. (2018). MOOC as a resource for teachers’ collabora-
tion in educational program. In V.Gitirana, T.Miyakawa, M.Rafalska, S.Soury-Lavergne,
& L.Trouche (Eds.), Proceedings of the re(s)source 2018 international conference (pp.167–
170). Lyon: ENS de Lyon. Retrieved on November 8th, 2018, at https://hal.archives-ouvertes.
fr/hal-01764563.
Tchounikine, P. (2011). Computer science and educational software design—A resource for multi-
disciplinary work in technology enhanced learning. NewYork: Springer.
Thomas, A., & Edson, A. J. (2018). An examination of teacher-generated denitions of digital
instructional materials in mathematics. In V.Gitirana, T.Miyakawa, M.Rafalska, S. Soury-
Lavergne, & L.Trouche (Eds.), Proceedings of the re(s)source 2018 international conference
(pp.340–343). Lyon: ENS de Lyon. Retrieved on November 8th, 2018, at https://hal.archives-
ouvertes.fr/hal-01764563.
Thomas, M.O. J., & Hong, Y. Y. (2005). Teacher factors in integration of graphic calculators
into mathematics learning. In H.L. Chick & J. L. Vincent (Eds.), Proceedings of the 29th
conference of the International Group for the Psychology of mathematics education (Vol. 4,
pp.257–264). Melbourne: University of Melbourne.
Tiburcio, R., & Bellemain, F. (2018). Computational engineering, didactical, educational soft-
ware, software engineering. In V.Gitirana, T.Miyakawa, M.Rafalska, S.Soury-Lavergne, &
L.Trouche (Eds.), Proceedings of the re(s)source 2018 international conference (pp. 262–
264). Lyon: ENS de Lyon. Retrieved on November 8th, 2018, at https://hal.archives-ouvertes.
fr/hal-01764563.
Trocki, A., & Hollebrands, K. (2018). The development of a framework for assessing dynamic
geometry task quality. Digital Experiences in Mathematics Education, 4(2–3), 110–138.
Trouche, L. (2004). Managing the complexity of human/machine interactions in computerized
learning environment: Guiding students’ command process through instrumental orchestra-
tions. International Journal of Computers for Mathematics Learning, 9(3), 281–307.
Trouche, L. (2005). Construction et conduite des instruments dans les apprentissages mathé-
matiques : Nécessité des orchestrations. Recherches en Didactique des Mathématiques, 25,
91–138.
Trouche, L., & Drijvers, P. (2010). Handheld technology: Flashback into the future. ZDM –
Mathematics Education, 42(7), 667–681.
Trouche, L., & Drijvers, P. (2014). Webbing and orchestration; two interrelated views on digital
tools in mathematics education. Teaching Mathematics and its Applications, 33(3), 193–209.
Trouche, L., Gueudet, G., & Pepin, B. (2018, Online First). Documentational approach to didactics.
In S.Lerman (Ed.), Encyclopedia of mathematics education. NewYork: Springer. doi:https://
doi.org/10.1007/978-3-319-77487-9_100011-1.
12 Transitions Toward Digital Resources: Change, Invariance, andOrchestration

444
van Dijke-Droogers, M., Drijvers, P., & Bakker, A. (2018). From sample to population: A hypo-
thetical learning trajectory for informal statistical inference. In V. Gitirana, T.Miyakawa,
M.Rafalska, S.Soury-Lavergne, & L.Trouche (Eds.), Proceedings of the re(s)source 2018
international conference (pp.348–351). Lyon: ENS de Lyon. Retrieved on November 8th,
2018, at https://hal.archives-ouvertes.fr/hal-01764563.
Vergnaud, G. (1996). The theory of conceptual elds. In. L.P.Steffe, P.Nesher, P. cobb, G.a. Goldin,
& B. Greer (Eds.), Theories of Mathematical Learning (pp. 210–239). Mahwah: Laurence
Erilbaum.
Vergnaud, G. (2011). Au fond de l’action, la conceptualisation. In J.M. Barbier (Ed.), Savoirs
théoriques et savoirs d’action (pp.275–292). Paris: Presses Universitaires de France.
Villamizar, F., Cuevas, C., & Martinez, M. (2018). A proposal of instrumental orchestration to
integrate the teaching of physics and mathematics. In V.Gitirana, T.Miyakawa, M.Rafalska,
S.Soury- Lavergne, & L.Trouche (Eds.), Proceedings of the re(s)source 2018 international
conference (pp.352–355). Lyon: ENS de Lyon. Retrieved on November 8th, 2018, at https://
hal.archives-ouvertes.fr/hal-01764563.
Voltolini, A. (2018). Duo of digital and material artefacts dedicated to the learning of geometry at
primary school. In L.Ball, P.Drijvers, S.Ladel, H.Siller, M.Tabach, & C.Vale (Eds.), Uses
of technology in primary and secondary mathematics education (pp.83–99). Cham: Springer.
Wenger, E. (1998). Communities of practice: Learning, meaning, and identity. Cambridge:
Cambridge University Press.
Zbiek, R.M., & Hollebrands, K. (2008). A research-informed view of the process of incorporat-
ing mathematics technology into classroom practice by inservice and prospective teachers. In
M.K. Heid & G.Blume (Eds.), Research on technology in the learning and teaching of math-
ematics: Syntheses and perspectives. Charlotte: Information Age Publishers.
Zbiek, R.M., Heid, M.K., Blume, G.W., & Dick, T.P. (2007). Research on technology in math-
ematics education: A perspective of constructs. In F.K. Lester (Ed.), Second handbook of
research on mathematics teaching and learning (Vol. 2, pp.1169–1207). Charlotte: Information
Age.
P. Drijvers et al.