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Drijvers, P. (2013). Digital technology in mathematics education: why it works (or doesn’t).
PNA, 8(1), 1-20.
DIGITAL TECHNOLOGY IN MATHEMATICS
EDUCATION: WHY IT WORKS (OR DOESN'T)
Paul Drijvers
The integration of digital technology confronts teachers, educators and
researchers with many questions. What is the potential of ICT for learn-
ing and teaching, and which factors are decisive in making it work in the
mathematics classroom? To investigate these questions, six cases from
leading studies in the field are described, and decisive success factors
are identified. This leads to the conclusion that crucial factors for the
success of digital technology in mathematics education include the de-
sign of the digital tool and corresponding tasks exploiting the tool’s ped-
agogical potential, the role of the teacher and the educational context.
Keywords: Didactical function; Digital technology; Instrumentation
La tecnología digital en educación matemática: por qué funciona (o no)
La integración de la tecnología digital enfrenta a profesores, formado-
res de profesores e investigadores a muchas preguntas. ¿Cuál es el po-
tencial de las TIC en el aprendizaje y la enseñanza, y qué factores son
determinantes al trabajar en clase de matemáticas? Para investigar es-
tas cuestiones, se describen seis casos de estudio prominentes en el área,
y se identifican los factores decisivos para el éxito. Esto lleva a la con-
clusión de que los factores cruciales para el éxito de la tecnología digi-
tal en la educación matemática incluyen el diseño de la herramienta di-
gital y de las tareas apropiadas que exploren el potencial pedagógico de
la herramienta, el papel del profesor y el contexto educativo.
Términos clave: Función didáctica; Instrumentación; Tecnología digital
For over two decades, many stakeholders have highlighted the potential of digital
technologies for mathematics education. The National Council of Teachers of
Mathematics, for example, in its position statement claims that “Technology is
an essential tool for learning mathematics in the 21st century, and all schools
must ensure that all their students have access to technology” (National Council
of Teachers of Mathematics [NCTM], 2008). ICMI devoted two studies to the
2 P. Drijvers
PNA 8(1)
integration of ICT in mathematics education, the second one expressing that
“…adigital technologies were becoming ever more ubiquitous and their influence
touching most, if not all, education systems” (Hoyles & Lagrange, 2010, p. 2).
However, the integration of digital technology still confronts teachers, educa-
tors and researchers with many questions. What exactly is the potential of ICT
for learning and teaching, how to exploit this potential in mathematics education,
does digital technology really work, why does it work, which factors are decisive
in making it work or preventing it from working? What does a quarter of a centu-
ry of educational research and development have to offer here?
Of course, these questions are not clearly articulated. What do we mean by
“it works”? Does this mean that the use of digital technology improves student
learning, invites deeper learning, motivated learning, more efficient or more ef-
fective learning? Does it mean that ICT empowers teachers to better teach math-
ematics? And, concerning the effect of educational research, do studies on digital
technology work in the sense that they provide answers to these questions, or do
they just help the researcher to better understand the phenomenon, and as such
contribute only indirectly to improving mathematics education? My interpreta-
tion of “why it works” in the title of this contribution includes both learning and
teaching, and also refers to learning on the part of the researcher.
In this paper I will explore the question of “why digital technology works or
does not” by briefly revisiting a number of leading studies in the field, that are
paradigmatic for a theme, approach, method, or type of results. For each of these
studies, the focus is on what they offer on identifying decisive factors for learn-
ing, teaching and research progress. As such, this contribution reports on a con-
cise and somewhat personal journey throughor a helicopter flight overthe
landscape of research studies on technology in mathematics education.
FRAMEWORK FOR CASE DESCRIPTION
How to decide which studies to include in this retrospective and even somewhat
historical paper? Even if somewhat subjective and personal arguments cannot be
completely ignored, the case selection is based on a number of criteria. A first
criterion for including a study or a set of studies is that it really contributes to the
field, by providing a new perspective, a new direction or is paradigmatic for a
new approach to the topic. An indication for this is that the study is frequently
quoted and has led to follow-up studies. A second criterion for inclusion is that
the study under consideration contributes to theoretical development in the field
of integrating technology in mathematics education, and as such promotes
thought in this domain. A third and final criterion for the set of cases presented in
this paper as a whole, is variation. Variation does not only apply to theoretical
perspectives, but also to the mathematical topic addressed in the study, the type
of technological tools used, and the pedagogical functionality of the digital tech-
Digital Technology in Mathematics Education: Why it Works (or doesn’t) 3
PNA 8(1)
nology. Concerning this functionality, we use an adapted version of the model by
Drijvers, Boon, and Van Reeuwijk (2010) which distinguishes three main didac-
tical functionalities for digital technology: (a) the tool function for doing mathe-
matics, which refers to outsourcing work that could also be done by hand; (b) the
function of learning environment for practicing skills; and (c) the function of
learning environment for fostering the development of conceptual understanding
(see Figure 1). Even if these three functionalities are neither exhaustive nor mu-
tually exclusive, they may help to position the pedagogical type of use of the
technology involved. In general, the third function is the most challenging one to
exploit.
Figure 1. Didactical functions of technology in mathematics education
CASE DESCRIPTIONS
In this section we will discuss the selected studies in a short frame. To do this in
a way that does justice to them and in the meanwhile serves the purpose of this
paper, we first present a global description of each case, including the mathemat-
ical topic, the digital tool and the type of tool use. Next, I will explain what is
crucial and new in the study, and why I decided to include it. Then the theoretical
perspective is addressed. Each case description is closed by a reflection on
whether digital technology worked well for the student, the teacher or the re-
searcher, and which factors may explain the success or failure.
Case 1: Concept-First Resequencing by Heid (1988)
The first case description concerns a study reported by Heid (1988), which is
considered as one of the first leading studies into the use of digital technology in
mathematics education. The study addresses the resequencing of a calculus
course for first-year university students in business, architecture and life sciences
using computer algebra, table tools and graphing tools that were used for concept
development (branch c, Figure 1). The digital technology allowed for a “concept-
first” approach, which means that calculus concepts were extensively taught,
whereas the computational skills were treated only briefly at the end of the
Didactical functions
of technology in
math education
Do mathematics
(a)
Learn mathematics
Practice skills
(b)
Develop concepts
(c)
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course. The results were remarkable in that the students in the experimental
group, who attended the resequenced, technology-intensive course, outperformed
the control group, who attended a traditional course, on conceptual tasks in the
final test, and also did nearly as well on the computational tasks that had to be
carried out by hand. The subjects in the experimental group reported that the use
of the computer took over the calculational work, made them feel confident
about their work and helped them to concentrate on the global problem-solving
process.
One of the reasons to discuss the Heid study here is that it is paradigmatic in
its approach in that its results form a first “proof of existence”: indeed, it seems
possible to use digital technology as a lever to reorganize a course and to suc-
cessfully apply a concept-first approach, using digital technology in the pedagog-
ical function of enhancing concept development, without a loss of student
achievement on by-hand skills.
From a theoretical perspective, Heid’s notion of resequencing seems closely
related to Pea’s distinction of ICT as amplifier and as (re-)organizer (Pea, 1987).
The former refers to the amplification of possibilities, for example by investigat-
ing many cases of similar situations at high speed. The latter refers to the ICT
tool functioning as organizer or reorganizer, thereby affecting the organization
and the character of the curriculum. In the light of that time’s thinking on the role
of digital tools to empower children to make their own constructions (Papert,
1980), the organizing function of digital technology was often considered more
interesting than the amplification.
So did technology work in this case? Yes, it did at the level of learning: The
final test results of the experimental group turned out to be very satisfying. And
yes, it also worked at a more theoretical level, as the notions of resequencing and
concept-first approach were operationalised and made concrete. Now why did it
work, which factors might explain these positive results? Even if nowadays we
would not consider the digital technology available in 1988 as very sophisticated,
I would guess that at the time the approach was new and motivating to the stu-
dents, and the representations offered by the technology did indeed invite con-
ceptual development. Decisive, however, I believe was the fact that the research-
er herself designed and delivered the resequenced course. I conjecture that she
was very aware of the opportunities and constraints of the digital technology, and
was skilled in carefully designing activities in which the opportunities were ex-
ploited, and in teaching the course in a way that benefitted from this. Whether the
course, if delivered by another teacher, would have been equally successful, is
something we will never know.
Case 2: Handheld Graphing Technology
The second case description concerns the rise of handheld graphing technology
in the 1990s. For several reasons, graphing calculators became quite popular
among students, teachers and educators at that time (for an overview, see
Digital Technology in Mathematics Education: Why it Works (or doesn’t) 5
PNA 8(1)
Trouche & Drijvers, 2010). Teaching materials were designed that made exten-
sive use of these devices and researchers investigated the benefits of this type of
technology-rich activities (Burrill et al., 2002). Very much in line with the work
by Heid (see Case 1), the focus of much of this work is on the pedagogical func-
tion of concept development. The main idea seems to be that students’ curiosity
and motivation can be stimulated by the confrontation with dynamic phenomena
that invite mathematical reasoning, in many cases concerning the relationships
between multiple representations of the same mathematical object. In many cases
this mathematical object is a function, but examples involving other topics, such
as statistics, can also be found.
As an example, Figure 2 shows two graphing calculator screens which stu-
dents set up to explore the effect of changes in the formula of the linear functions
Y1 and Y2 on the graph of the product function Y3. This naturally leads to ques-
tions about properties of the product function and the relationship with properties
of the two components (Doorman, Drijvers, & Kindt, 1994).
Figure 2. Exploring the product of two linear functions
A paradigmatic study in this field is done by Doerr and Zangor (2000). The re-
searchers report on a small-scale qualitative study, in which 15-17 year old pre-
calculus students study the concept of function using a graphing calculator, with
a focus on the pedagogical tool functionality of concept development (branch c,
Figure 1). The authors identify five modes of tool use, namely computation,
transformation, data collection and analysis, visualisation, and checking. The re-
sults show that the teacher was crucial in establishing and reinforcing these
modes of tool use, for example by setting up whole-class discussions “around”
the projected screen of the graphing calculator, to develop shared meaning and
avoid a too individual development of tool use and mathematical insight. The re-
searchers stress that using digital technology in mathematics teaching is not in-
dependent from the educational context and the mathematical practices in the
classroom in particular.
The main reason to discuss the Doerr and Zangor study here is that it high-
lights the importance of the educational context in studies on the effect of digital
technology, and the crucial role of the teacher in particular. The relevance of the
educational context has later been elaborated in the notion of pedagogical map by
Pierce and Stacey (2010). Concerning the teacher, she establishes a culture of
6 P. Drijvers
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discussing graphing calculator output in a format that is close to what is called a
“Discuss-the-Screen orchestration” in Drijvers, Doorman, Boon, Reed, and
Gravemeijer (2010) and by these means contributes to the co-construction of a
shared repertoire of ways to use the graphing device.
From a theoretical perspective, Doerr and Zangor (2000) use frameworks on
learning as the co-construction of meaning, and that the “features of a tool are
not something in and of themselves, but rather are constituted by the actions and
activities of people” (p. 146). Even if this may sound somewhat trivial nowadays,
during the period of initial enthusiasm these were important insights with conse-
quences for the role of the teacher, who led the process of sharing and co-
construction, particularly in the case of personal, private technology.
So did technology work in this case? Doerr and Zangor did not assess learn-
ing outcomes, but it seems that the students developed a rich and meaningful
repertoire of ways to use the graphing calculator for their mathematical work.
Why did this work, which factors might explain these findings? My interpreta-
tion is that the use of digital tools for exploratory activities which target concep-
tual development is not self-evident, as it is hard for students, without the math-
ematical background that we as teachers have, to “see” the mathematics behind
the phenomena under consideration. It is here where the teacher comes in, and
where the study becomes very informative for both teachers and researchers. In
this case, I believe that the fact that the teacher herself was skilled in using the
graphing calculator, was aware of its limitations, and was willing to explicitly
pay attention to the co-construction of a shared and meaningful repertoire of tool
techniques explains the results. As in the Heid study described in Case 1, the role
of the teacher seems to be an important factor. The issue of how to deal with pri-
vate, handheld technology is very relevant nowadays, as many students have
smart phones with sophisticated mathematical applications, and again, teachers
are faced with the danger of too individually constructed techniques and insights.
Case 3: Instrumental Genesis
By the end of the previous century, French researchers who were working on the
integration of computer algebra and dynamic geometry in secondary mathemat-
ics education felt the need to go beyond the then current theoretical views. Even
if they still experimented with explorative tasks, such as finding the number of
zeros at the end of ! (Trouche & Drijvers, 2010), a theoretical perspective was
needed that would do justice to the complex interaction between techniques to
use the digital technology, conventional paper-and-pencil work and conceptual
understanding. This led to the development of the instrumental genesis frame-
work, or the instrumental approach to tool use (Artigue, 2002; Guin & Trouche,
1999; Lagrange, 2000). Even if there are different streams within instrumentation
theory (Monaghan, 2005), it is widely recognized that the core of this approach is
the idea that the co-emergence of mental schemes and tool techniques while
working with digital technology is essential for learning. This co-emergence is
Digital Technology in Mathematics Education: Why it Works (or doesn’t) 7
PNA 8(1)
the process of instrumental genesis. The tool techniques involved have both a
pragmatic meaning (they allow the student to use the tool for the intended task)
and an epistemic meaning, in that they contribute to the students’ understanding.
Rather than exploration, the reconciliation of digital tool use, paper-and-pencil
use, and conceptual understanding is stressed (Kieran & Drijvers, 2006).
A paradigmatic study in this field is the one by Drijvers (2003) on the use of
handheld computer algebra for the learning of the concept of parameter. Four
classes of 14-15 year old students worked on activities using a handheld comput-
er algebra device both in its role of mathematical tool and for conceptual devel-
opment (branches a & c, Figure 1) to develop the notion of parameter as a “su-
per-variable” that defines classes of functions and that can, depending on the
situation, play the different roles that “ordinary” variables play as well. The re-
sults of the study include detailed analyses and descriptions of techniques that
students use, and the corresponding expected mental scheme development. Fig-
ure 3 provides a schematic summary of such an analysis for the case of solving
parametric equations in a computer algebra environment (Drijvers, Godino, Font,
& Trouche, 2012).
Figure 3. Conceptual elements related to the application of the solve command
The main reason to discuss this study here is that by providing elaborated exam-
ples it contributes to a concrete and operationalised view on the schemes and
techniques that are at the heart of the instrumental approach. The study shows
that the instrumental approach is a fruitful perspective that can provide tangible
guidelines for both the design of student materials and the analysis of student be-
haviour.
From a theoretical perspective, apart from the concretisation of the notions of
schemes and techniques, the author integrated this with a more general view on
mathematics education, namely the theory of realistic mathematics education
(Freudenthal, 1991). The two perspectives seemed to be complementary and both
provided relevant guidelines for design and analysis.
An equation should contain an = sign
Indicate the unknown to solve
Notice the scope of
the square root sign
“solve with respect to x” = “express x in terms of b”
Word /letter vars?
A solution can be
an expression
8 P. Drijvers
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So did technology work in this case? No and yes. The conclusions on the
learning effects of the intervention are not very clear. Even if the students learned
much about the concept of parameter, their work still showed weaknesses both in
the use of the tool and in the understanding of the mathematics. This suggests an
incomplete instrumental genesis. Factors that may explain these findings are: (a)
the difficulty of the mathematical subject for students of this age, (b) the com-
plexity of the computer algebra tool, and (c) the efforts and skills needed by the
teachers to not only go through their personal process of instrumental genesis,
but also to facilitate the students’ instrumental genesis by their way of teaching.
The latter aspect was addressed more explicitly later in the notion of instrumental
orchestration (Drijvers & Trouche, 2008; Trouche, 2004). The study did work in
the sense that it contributed to the researchers’ understanding of the complexity
of integrating sophisticated digital technologies in teaching relatively young stu-
dents. The close intertwinement of the students’ cognitive schemes and the tech-
niques for using the digital technology is identified as a decisive factor in the
learning outcomes of technology-rich mathematics education.
Case 4: Online Applications
With the growing availability and bandwidth of internet, researchers became in-
terested in the potential of online interactive applications or applets for mathe-
matics education. The advantages of online content include access without local
software installation, ease of distribution and updating for developers, and per-
manent availability for users as long as the internet is accessible.
Many studies investigate this potential. For example, Boon (2009) explores
the opportunities for teaching 3D geometry using online applets. Doorman,
Drijvers, Gravemeijer, Boon, and Reed (2012) describe a teaching experiment in
grade 8 focusing on the concept of function using an applet called AlgebraAr-
rows1 for building chains of operations. Apart from an instrumental perspective
(see Case 3), the theoretical framework includes domain-specific theories on rei-
fication, realistic mathematics education and emergent modelling. The applet is
used for concept development (branch c, Figure 1). A third example is the study
by Bokhove, who focuses on acquiring, practicing and assessing algebraic skills
(Bokhove, 2011; Bokhove & Drijvers, 2012). His teaching experiments took
place in grade 12 classes and made use of applets that offer means to manipulate
algebraic expressions and equations2. The theoretical framework in this case in-
cluded notions from algebra pedagogy such as symbol sense, which is expected
to support skill mastery, but also elements from educational science on assess-
ment and on feedback. In contrast to the studies described so far, the role of the
digital tool in Bokhove’s work includes the environment to practice skills
1 See http://www.fisme.science.uu.nl/tooluse/en/
2 See http://www.algebrametinzicht.nl/
Digital Technology in Mathematics Education: Why it Works (or doesn’t) 9
PNA 8(1)
(branch b, Figure 1), which might be the easiest role, even if the design of appro-
priate feedback is an issue to tackle.
As a paradigmatic design research study in this field, let us now describe the
work done by Bakker in somewhat more detail (Bakker, 2004; Bakker & Grave-
meijer, 2006; Bakker & Hoffmann, 2005). Bakker investigated early statistical
reasoning of students in Grades 7 and 8. In one of the tasks, students investigate
data from life spans of two brands of batteries while using applets to design and
explore useful representations and symbolizations (see Figure 4). Clearly, the
digital tools’ pedagogical functionality is on concept development once more
(branch c, Figure 1). The design of the hypothetical learning trajectory and the
student materials was informed by the development of statistics in history. In his
analysis of student data, Bakker uses Peirce’s (1931-1935) notions of diagram-
matic reasoning and hypostatic abstraction to underpin his conclusion that the
teaching sequence, including the role of digital tools, invited students’ reasoning
about a frequency distribution as an object-like entity, as became manifest when
they started to speak about the “bump” to describe the drawings at Figure 4’s
right hand side.
Figure 4. Applets for investigating a small set of statistical data
The main reasons to discuss Bakker’s work here are not only the originality of
the dedicated digital tools which meet new ideas on statistical reasoning and sta-
tistics education, and which were designed in collaboration with others (Cobb,
McClain, & Gravemeijer, 2003), but also the rich relationships with the different
resources and approaches, such as the historical perspective, to inform the de-
sign.
From a theoretical perspective, it is interesting to notice that even if technol-
ogy plays an important role in Bakker’s study, the design and analysis are driven
by theoretical perspectives from outside the frame of research on the use of tech-
nology in mathematics education, but rather from the world of mathematics ped-
agogy and beyond. I believe that this is a meaningful and promising approach.
On the one hand, as researchers we should benefit from specific results and theo-
ries from studies on the use of digital tools in mathematics education. On the
10 P. Drijvers
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other hand, we should not forget to involve theories on mathematics education
and educational science in general.
So did technology work in Bakker’s case? Yes, it did in the sense that the au-
thor clearly reports on conceptual development by the students involved in the
study. Why did this work, which decisive factors might explain these findings? I
believe that an important lesson to be learnt from this study is that design re-
search on the use of digital technology in mathematics education should not limit
itself to the study of the tools alone, but should include the tasks, and their em-
bedding in teaching as a whole, in order to understand what works and why it
works. In this case, I would guess it is the combination of the digital tools, the
tasks and activities, but also the whole-class discussions, the paper-and-pencil
work, the established mathematical practices, in short the educational context as
a whole, that explains the result. A second lesson to learn for us as researchers is
that a theoretical framework which integrates different perspectives can be very
powerful for generating interesting and relevant research results.
Case 5: Mobile Mathematics
Research on the use of mobile technology in mathematics education is in its early
stages but its importance is rapidly growing. It is evident that mobile technology
and smart phones in particular are very popular among students and more and
more wide-spread. Wireless Internet access allows for the use of mobile applica-
tions (also called MIDLETS, Mobile Information Device applications), SMS and
email services offer communication and collaboration opportunities, GPS facili-
ties allow for geographical and geometrical activities and the tool’s mobile and
handheld characteristics invite out-of-school activities, for example the gathering
of real-life data that inform biology or chemistry lessons (Daher, 2010).
As a paradigmatic example, I now address the MobileMath pilot study car-
ried out by Wijers, Jonker, and Drijvers (2010). In this study, the tool consisted
of a mobile phone with GPS facilities and a “native” application, designed for the
purpose of this game, which generated the view on the game situation and ar-
ranged communication with other teams’ devices. The mathematical topic in-
volved is geometry: teams of Grades 7 and 8 students used the GPS and the ap-
plication to play an outdoor game on constructing parallelograms (including
rectangles and squares), and could eventually destroy other groups’ geometrical
shapes. This so-called MobileMath game aims at making students experience
properties of geometrical figures in a lively, embodied game context. While play-
ing the game, students look at the map to imagine where they want to make a
shape, walk to the location for the first vertex to enter this location in the mobile
device, which generates a dot on the map, walk to the location of the second ver-
tex of their imagined shape which provides a line on the screen connecting the
first vertex with the current (moving) location, etc. The map in Figure 5 shows
some student constructions. The deconstruction option brought extra challenge
and competition into the game. From the data the researchers conclude that Mo-
Digital Technology in Mathematics Education: Why it Works (or doesn’t) 11
PNA 8(1)
bileMath adds a geometrical dimension to the world, transforming it into a game
board. MobileMath also invites mathematical activity, such as the (re)discovery
and use of characteristics of squares, rectangles and parallelograms, and taking
notice of geometrical aspects of the world.
Figure 5. Map of students’ parallelogram constructions using GPS
One reason to discuss this study here is that the digital toolthe modern smart
phone with GPS facilities rather than an “old school” computeracts in multiple
ways, and its use includes all branches of the diagram displayed in Figure 1. The
device enables the exploration of properties of quadrilaterals (branch c). It also
allows for practicing the construction of parallelograms, which meets branch b.
And finally, the tool also functions as an environment to outsource the mathe-
matical work, in this case the drawing of the shapes (branch a).
As seems to be the case in other studies on the integration of mobile technol-
ogy in mathematics education, the theoretical perspective used by Wijers, Jonker,
and Drijvers (2010) is different from the frameworks common in most research
on technology in mathematics education. It is closely associated with frame-
works from studies on serious gaming, and focuses (a) on student engagement
and (b) on task authenticity. Enhancing student engagement is seen as an im-
portant potential of educational games. In the MobileMath study, student en-
gagement is stimulated by the game’s hybrid reality character: On the mobile de-
vice’s screen, students see the map of the reality in which they are walking, as
well as the virtual geometrical shapes they are creating. Hybrid reality games are
seen as beneficial for student engagement. In addition to this, the authors refer to
Prensky (2001) for a model on heuristics for the design of engaging games,
which include clear rules and goals, outcome and feedback, conflict, challenge
and competition, and interaction. Concerning task authenticity, the authors claim
that the effectiveness of learning activities can be enhanced if the tasks are au-
thentic and realistic. In line with the framework of Realistic Mathematics Educa-
tion, realistic means that problem situations presented in learning activities
12 P. Drijvers
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should be “experientially” real to students and have meaningful, authentic prob-
lem situations as starting points, so that students experience the game’s activity
as making sense.
So did the digital technology work in this case? As far as engagement and
authenticity are concerned, the answer is “yes”. The researchers report that the
students were engaged in the game and experienced it as challenging. Apparent-
ly, the game factor, in combination with the possible attractiveness of the digital
device, works out well. A second factor might be the outdoor and physical char-
acter of the game, which students may experience as a welcome change from
regular classroom teaching. What is not clear yet, however, is whether these ef-
fects will persist if this type of activity were to become more common. Also, the
study presented here has a small-scale pilot character and would certainly need
further replication.
Case 6: Teachers’ Practices and Professional Development
If we recapitulate the previous cases, in all but the last one the teachers’ practices
and experiences were identified as an important factor explaining why digital
technology worked or why it did not. Therefore, this final case focuses on the
role of the teacher, teaching practices and teachers’ professional development.
One of the first studies focusing on teachers’ practices and professional de-
velopment was the one by Ruthven and Hennessy (2002). In this study and in
subsequent work (e.g., Ruthven, 2007) crucial factors are identified that affect
teachers integrating digital technology in their teaching. In relation to the instru-
mental genesis model, Trouche developed the notion of instrumental orchestra-
tion to stress the relevance of teaching practices (Trouche, 2004). Case studies
based on these models describe teachers’ practices in relation to their opinions
and beliefs (Drijvers, 2012; Drijvers, Doorman, Boon, Reed, & Gravemeijer,
2010; Pierce & Ball, 2009). Another model on teachers’ professional knowledge
is Technological Pedagogical Content Knowledge (TPACK), which became
widespread but is also criticized (Graham, 2011; Koehler, Mishra, & Yahya,
2007; Voogt, Fisser, ParejaRoblin, Tondeur, & Van Braak, 2013).
In addressing the questions of how to prepare teachers for technology-rich
teaching and how to enhance their professional development in this field, in line
with the work done by Wenger (1998) on communities of practice, it is suggested
that the participation in a community of teachers who co-design and use re-
sources for teaching, can contribute to this (e.g., see Fuglestad, 2007; Jaworski
2006). Digital technology in such an enterprise acts on two levels: first, the pro-
fessional development concerns its use in mathematics education, and second,
digital technology may support the community’s work by offering online and vir-
tual facilities for exchange. Digital technology is both the subject at stake and the
Digital Technology in Mathematics Education: Why it Works (or doesn’t) 13
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vehicle to address it. Efforts have been done to exploit digital technology’s po-
tential for teachers’ professional development by designing online courses3.
As a paradigmatic design research study in this field, let us now describe the
work done by Sabra (2011) in somewhat more detail. In his PhD dissertation,
Sabra describes two case studies of teachers’ collaborative process of profession-
al development in detail. In the first case ten teachers in the same school collabo-
rate on the design of a final assessment training session and a mathematics inves-
tigation task while integrating the use of TI Nspire in their teaching. The second
case study concerns a project in which eleven teachers, all members of the Sesa-
math community from all over France, collaboratively design resources on the
concept of function that are part of the course manual. The analysis shows that
the two communities develop in quite different ways, but that in both develop-
ments some critical incidentscalled documentary incidents in the thesisare
decisive. The digital tools in this case include web facilities for collaborative
work, file exchange and communication; the role they play for the participating
teachers is best characterized by branch a in Figure 1, the role of a tool for doing
mathematics, or rather a tool for collaborating on the design of mathematical re-
sources.
The main reason to discuss this study here is that its rich data including in-
terviews, blogs and observations and its fine-grained data analysis provide a de-
tailed insight in how communities of teachers may work (or may not) and how
technology may support this.
From a theoretical perspective, Sabra uses the notion of documentational
genesis as a main concept. Figure 6, taken from Gueudet and Trouche (2009),
shows how this is analogue to the notion of instrumental genesis, but now ad-
dressing the level of teachers using and designing digital resources. The interest-
ing point here, in my opinion, is that a similar framework is applied to and elabo-
rated for different situations and different levels of technology integration.
3 E.g., see http://www.edumatics.eu/
14 P. Drijvers
PNA 8(1)
Figure 6. Schematic representation of a documentational genesis (Gueudet &
Trouche, 2009)
So did technology work in the Sabra study? Maybe the answer is different for the
two cases that are described. In the case of the team of teachers within the same
school, it seems that the digital technology does not have so much to offer, and
that the professional interest of the community members does not invite a real
engagement in an effective collaboration. As a result, one can wonder whether
the targeted professional development really took place, and whether the com-
munity really contributed to it. In the second case of the teachers all over France,
the analysis shows a very lively process of collaboration, which is clearly afford-
ed by the digital technology and would not have been possible without it. Similar
to the other cases described in this paper, it seems that decisive factors that ex-
plain the phenomena go beyond the straightforward point of the available tech-
nology. My impression is that for a school team of teachers, collaboration is far
from self-evident, whereas teachers who volunteer for a role in the Sesamath pro-
ject share a professional interest to engage in a virtual community and in a shared
process of distant collaboration. This, I would conjecture, might be the main ex-
planation for the different results in the two cases Sabra describes.
CONCLUSION
Theslightly provocativequestion raised in the title of the paper is why digital
technology in mathematics education works or does not. The underlying aim was
to identify factors that promote or hinder the successful integration of digital
technology in mathematics education. The analysis of the six cases described in
this paper show that the integration of technology in mathematics education is a
subtle question, and that success and failure occur at levels of learning, teaching
and research. In spite of this complexity, three factors emerge as decisive and
crucial: the design, the role of the teacher, and the educational context.
The first factor concerns design. Cases 1, 3, and 4 reveal the crucial role of
design. This concerns not only the design of the digital technology involved, but
Digital Technology in Mathematics Education: Why it Works (or doesn’t) 15
PNA 8(1)
also the design of corresponding tasks and activities, and the design of lessons
and teaching in general, three design levels that are of course interrelated. In
terms of the instrumental genesis model, the criterion for appropriate design is
that it enhances the co-emergence of technical mastery to use the digital technol-
ogy for solving mathematical tasks, and the genesis of mental schemes that in-
clude the conceptual understanding of the mathematics at stake. As a prerequi-
site, the pedagogical or didactical functionality in which the digital tool is
incorporated (see Figure 1) should match with the tool’s characteristics and af-
fordances. Finally, even if the digital technology´s affordances and constraints
are important design factors, the main guidelines and design heuristics should
come from pedagogical and didactical considerations rather than being guided by
the technology´s limitations or properties.
The second factor concerns the role of the teacher, which is crucial in Cases
1, 2, and 6. The integration of technology in mathematics education is not a pan-
acea that reduces the importance of the teacher. Rather, the teacher has to orches-
trate learning, for example by synthesizing the results of technology-rich activi-
ties, highlighting fruitful tool techniques, and relating the experiences within the
technological environment to paper-and-pencil skills or to other mathematical
activities. To be able to do so, a process of professional development is required,
which includes the teacher’s own instrumental genesis, or, in terms of the
TPACK model, the development of his technological and pedagogical content
knowledge. Case 6 suggests that technology can help the teacher to advance on
this, together with colleagues in technology-supported collaboration. What seems
to be an open question is how the role of the teacher changes if we consider the
use of technology in out-of-school learning, gaming, and other forms of informal
education (see Case 5).
The third and final factor concerns the educational context, which includes
mathematical practices and the elements of the pedagogical map designed by
Pierce and Stacey (2010). Case 2 reveals how important it is that the use of digi-
tal technology is embedded in an educational context that is coherent and in
which the work with technology is integrated in a natural way. Case 5, the Mo-
bileMath example, shows that taking into account the educational context in-
cludes attention for important aspects such as student motivation and engage-
ment. Another factor that is not so much elaborated in the case descriptions but is
important to mention here, is assessment, which should be in line with the stu-
dents’ activities with technology; not doing so would suggest that in the end the
use of digital technology is not important. Finally, the use of digital technology
may lead to an extension of the educational context towards out-of-school set-
tings, as exemplified in Case 5.
The three factors identified above may seem very trivial, and to a certain ex-
tend they are quite straightforward indeed; however, their importance, I believe,
can hardly be overestimated and to really take them into account in educational
practice is far from trivial.
16 P. Drijvers
PNA 8(1)
DISCUSSION
Let me first acknowledge that the study presented here clearly has its limitations.
The discussion of the studies addressed cannot be but somewhat superficial in the
frame of this paper. Also, the number of studies is small, and the choice of the
studies included is not neutral. This being said, I do believe the article provides
avery roughmap of the landscape of research studies on technology in math-
ematics education and reveals some trends in the domain over the previous dec-
ades.
So what trends can be seen in retrospective? Globally speaking, a first trend
to identify is that from optimism on student learning in the early studies towards
a more realistic and nuanced view, the latter acknowledging the subtlety of the
relationships between the use of digital technology, the student’s thinking, and
his paper-and-pencil work. A second trend is the focus not only on learning but
also on teaching. The importance of the teacher is widely recognized and models
such as TPACK, instrumental orchestration and the pedagogical map help to un-
derstand what is different in teaching with technology and to investigate how
teachers can engage in a process of professional development. The third and final
trend I would like to mention here concerns theoretical development. Whereas
many early studies mainly use theoretical views that are specific for and dedicat-
ed to the use of digital technology (e.g., Pea’s notions of amplifier and reorganiz-
er in the Heid study), recent studies often include more general theories on math-
ematics education or learning in general, and also combine different theoretical
perspectives (e.g., see the work by Bakker, using Pierce, RME, and other theoret-
ical views).
To close off this discussion, I would like to express my strong belief that
these theoretical developments are crucial for the advancements in the field. The
studies addressed in this paper show strong relationships between the theoretical
frameworks, the digital tools and the mathematical topics (Kieran & Drijvers,
2012). We now have a myriad of theoretical approaches available in our work,
including very specific theories on the use of technology in mathematics educa-
tion, domain-specific instruction theories, and very general views on teaching
and learning. One of the challenges in our work, therefore, is to combine and
contrast the lenses each of these approaches offer (Drijvers, Godino, Font, &
Trouche, 2012). The notion of networking theories (Bikner-Ahsbahs & Prediger,
2010) provides a good starting point that may help to better understand the role
of digital technology in mathematics education and, as a consequence, to con-
tribute to the learning and teaching of the topic.
ACKNOWLEDGEMENT
I thank Arthur Bakker, Vincent Jonker, Carolyn Kieran, Hussein Sabra and Luc
Trouche for their helpful comments on the draft version of this paper.
Digital Technology in Mathematics Education: Why it Works (or doesn’t) 17
PNA 8(1)
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This document was originally published as Drijvers, P. (2012). Digital technolo-
gy in mathematics education: Why it works (or doesn’t). In Proceedings of the
12th International Congress on Mathematics Education (pp. 485-501). Seoul,
Korea.
Paul Drijvers
Freudenthal Institute for Science and Mathematics Education
Utrecht University
p.drijvers@uu.nl
Received: December 2012. Accepted: January 2013.
