ZDM Mathematics Education
DOI 10.1007/s11858-010-0269-2
ORIGINAL ARTICLE
Handheld technology for mathematics education:
flashback into the future
Luc Trouche • Paul Drijvers
Accepted: 11 June 2010
FIZ Karlsruhe 2010
Abstract In the 1990s, handheld technology allowed
overcoming infrastructural limitations that had hindered
until then the integration of ICT in mathematics education.
In this paper, we reflect on this integration of handheld
technology from a personal perspective, as well as on the
lessons to be learnt from it. The main lesson in our opinion
concerns the growing awareness that students’ mathematical thinking is deeply affected by their work with technology in a complex and subtle way. Theories on
instrumentation and orchestration make explicit this subtlety and help to design and realise technology-rich mathematics education. As a conclusion, extrapolation of these
lessons to a future with mobile multi-functional handheld
technology leads to the issues of connectivity and in- and
out-of-school collaborative work as major issues for future
research.
Keywords Mathematics education Handheld technology Instrumentalisation Instrumentation Orchestration
L. Trouche (&)
INRP (National Institute for Pedagogical Research),
Lyon, France
e-mail: [email protected]
URL: http://educmath.inrp.fr/Educmath/recherche/educmath/
page_luc_trouche
P. Drijvers
Freudenthal Institute for Science and Mathematics Education,
Utrecht University, Utrecht, The Netherlands
1 Introduction
Since its origin, mankind has developed tools to assist in
labour. In ancient times, stones were shaped and used as
fist hammers for carrying out handicraft work. Later, the
abacus was used for arithmetic tasks in trade and bargaining affairs. In mathematics, a diversity of tools have
been in use, such as clay tablets, compasses, rulers, books,
paper, pencils, and, in present times, calculators and
computers (Maschietto & Trouche, 2010). Seen in a historical perspective, handheld tools have a long tradition of
being at the heart of mathematical and scientific practice.
In the 60s of the previous century, four-function calculators and scientific calculators became very popular types
of handheld devices for mathematicians who had to do
computations. Engineers used relatively big and complex
calculator tools, requiring reverse polish notation and
offering programming facilities to carry out high-precision
calculations. An era of tremendous technological developments was heralded with the emergence of new devices
for information and communication: the digital society.
Computers were big and expensive, but handheld calculators penetrated all sectors of society, including schools.
In the 1990s, HHT became popular in mathematics
education in some countries. Graphing and symbolic calculators in particular became affordable and widespread,
not in the least because of enthusiasm amongst teachers,
educators and researchers, who were interested in the
opportunities technology offers. The numbers of desktop
PCs in homes and schools also increased exponentially, but
access to computers still was a matter of concern, not to
speak of constraints on communication and technical
support.
Nowadays, PCs are widespread. The development of
laptop and notebook computers has moved PCs in the
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L. Trouche, P. Drijvers
Fig. 1 Graphing calculator
applications on an iPhone
handheld device
direction of HHT. Wireless networks allow for mobility
and Internet access from anywhere. PDAs and smartphones
are now in the pocket and offer applications for mathematics as well (see Fig. 1). It seems that HHT is entering a
new era.
On looking back at the recent role of HHT in mathematics education, one may wonder why such tools, and in
particular the graphing and symbolic calculators, became
so popular in the previous decades. What made them so
attractive to students and teachers? How did their implementation in mathematics education work? Were the
sometimes high expectations of the use of HHT, which was
supposed to improve mathematics learning and teaching,
met in practice? How do we reflect on these developments
in retrospect? It is time for a flashback.
We are also interested in lessons to be learnt for the
future. What can we learn from the integration of HHT into
mathematics education in the past decades? Which lessons
to learn, what guidelines to extrapolate for future, what
issues to take care of, and which directions to go? These
are the central questions of this contribution. In short, the
purpose of this flashback is to consider the implications for
the future.
In this paper, we try to address the above questions by
looking back from a personal perspective (see Fig. 2). We
try to trace our own work through these last two decades.
Of course, we do not mean to disregard the relevance of the
excellent work done by many teachers, educators and
researchers in the field (e.g. see Burril et al., 2002 for an
overview); rather, we try to backtrack our personal research
trajectories considering them as exemplary of the recent
history of research into the use of HHT in mathematics
education.
123
Our ambition is not to provide just a synthesis, but, by
looking for the main trends of recent research in this field,
to capture on the one hand the obstacles and the dead ends
and on the other the fruitful ideas, so as to provide some
hints for further research.
2 Hymn for HHT
As indicated in the previous section, in some countries the
use of HHT became widespread in the mid-1990s amongst
students, teachers, educators and researchers in the mathematics education community. How did this happen, and
why? What were the reasons for the successful dissemination of graphing calculators, in particular, and, to a lesser
extent, symbolic calculators? Why did these types of
technology become so widespread?
As a first explanation for this phenomenon, it should be
noted that in spite of the high expectations expressed by
many researchers and educators (e.g. Papert, 1980),
implementation of technology in the mathematics classroom was inhibited by infrastructural limitations. Hardware
was organised in computer laboratories that were difficult
to access, and the technology was operating far from
smoothly. Network technology was primitive, so installing
software in a computer laboratory was quite a job. Also,
most computer laboratories were not appropriate for
whole-class teaching or interactive teaching techniques.
Lessons in the computer laboratories needed a lot of
preparation by the teacher, and the use of technology
usually had to be teacher driven. Compared to this situation, HHT had some important trumps: it could be used in
any ordinary classroom, without additional infrastructural
Handheld technology for mathematics education
Fig. 2 Authors’ personal
backgrounds
Paul Drijvers
As a teacher trainer in the 90s, I was
fascinated by the phenomenon of computer
algebra, and excited that a machine was able
to carry out sophisticated procedures, such
as calculating limits and derivatives and
algebraic simplifications, techniques I
considered these procedures, we spent so
much time on teaching, to be at the heart of
mathematics.
Mathematics
trivialised?
Where is the heart of mathematics?
In the mid-90s, the availability of HHT
solved our infrastructural issues. Now
technology was really integrated! But what
do we want to teach? And why don’t
students see the mathematics in the
techniques the way we see them? How to
approach this difficulty?
requirements. There was no need to make computer laboratory reservations, or to have to use technology during the
complete lesson because the room did not allow for anything else. It was always available without dominating the
classroom. In short, HHT made it possible to bypass the
infrastructural limitations within schools.
In addition to this, a second explanation is that HHT also
made the teachers’ lives easier. Lesson preparation was no
longer that laborious, and the initiative and responsibility
of using technology could eventually be handed over to the
students, who could themselves decide on when and how to
use the device. Different teaching techniques, including
individual work, group work and whole-class work, could
be used and intertwined. Integrating technology into
assessment, an important concern if one wants the assessment to reflect the teaching, became feasible through the
use of HHT. In short, HHT offered new possibilities for the
teacher who wanted to make use of the opportunities
technology brought about. So finally, technology in the
mathematics classroom was no longer beyond reach, but
manageable!
The assessment argument also convinced authorities and
policy makers that HHT could, on the one hand, bring
technology into the classroom, but, on the other, leave
assessment formats unchanged, even if the content of the
test might be questioned. In several countries, this led to
HHT entering national examinations, though this was not
as straightforward as it might seem in many cases (Brown,
2010; Drijvers, 1998, 2009). In its turn, such national
measures left teachers and students who were less
favourable towards technology with no choice: once the
national policy was decided upon, nobody would want to
put their students at a disadvantage. A third explanation,
therefore, is that national policies made it difficult for ‘midadopting’ teachers to neglect HHT.
Luc Trouche
Beginning as a teacher (1975) facing the
introduction of scientific calculators,
following on as a teacher educator (1985)
facing the introduction of graphing
calculators, then as a researcher (1995),
facing the introduction of symbolic
calculators, my professional development
was, in some way, drawn by technological
development.
A threefold questioning rose: what are tools,
what is mathematics, what are learning
processes…? Finally, what was the most
important question? Difficult to say, but one
element seemed to have been decisive:
calculators were in the students’ hands and
in the classroom. If we wanted to teach
mathematics, we had to teach with such
technology.
A fourth factor in the popularity of HHT is the students’
appreciation. As is the case of other popular technological
devices such as television, at the heart of the HHT is a
screen with dynamic images (Trouche, 1994). Furthermore,
an advantage of the personal handheld device is that the
students’ familiarity and confidence with it develops
quickly because of its permanent availability (Lagrange,
1999). Also, HHT is personal technology. Students have a
sense of ownership and the means for personalisation and
customisation (e.g. through the installation of games or the
production of additional programs), which facilitate
appropriation. On the one hand, this personal and private
character of HHT, which makes students feel free to try
things and to make errors, contributes to its popularity
(Ruthven, 1990). On the other hand, this privacy might
hinder students from sharing their results, and questions,
with peers and their teacher (Doerr & Zangor, 2000).
Not only did teachers and students get excited about the
possibilities offered by HHT, but also researchers and
educators in the 1990s were optimistic about the positive
effect of its introduction into mathematics classrooms.
Many explorative studies were conducted and reports on
experiences with the use of HHT, such as graphing calculators, appeared in both scientific and professional forums (e.g. see Fig. 3). In most cases, these reports were
quite positive:
Use of the graphics calculator stimulates the posing
of new questions and the generalization of problems.
For the student, this involves […] a change of attitude
with respect to mathematics from one of ‘passiveperformance’ to that of ‘active-investigation’.
(Drijvers & Doorman, 1996, p. 429)
The interest and activity within the research community
contributed to the dissemination of ideas on the easy
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L. Trouche, P. Drijvers
Fig. 3 A dialogue making use
of graphical exploration options
offered by HHT (Drijvers &
Doorman, 1996)
Teacher: How can you shift the graph of y = x2
two places to the right?
Johan: y = x2 + 2
He enters this and presses GRAPH.
Johan: Oh no! y = x2 + 2x, GRAPH... that's not
right either.
Alex: y = x2 + 4x + 4, no, that's not right, the
graph shifted two places to the left. So then it is
y = x2 – 4x + 4.” That's right.
Teacher: And three places to the right?
Alex tries this on the graphics calculator.
Alex: y = x2 – 6x + 9.
Rewriting the formulas gives y = (x – 2)2 and y =
(x – 3)2.
Erik: And if you have y = x2 + 2x and you shift
that three places to the right?
Teacher: What do you think?
Erik: First complete the square, that'll give you
y = (x + 1)2 – 1, and then three places to the right,
so -3 in parentheses, that's y = (x – 2)2 – 1.
appropriation of HHT, opportunities for explorations by
students, and means to focus on mathematical thinking
rather than on procedural skills. In fact, HHT technology
was considered to be a means to implement reform agendas
for mathematics education, as a lever for educational
change that was ‘in the air’. Of course, there were also
critical comments, for example on the costs of personal
HHT, on its role in assessment and on the effects of its use
on basic skills and on conceptualisation of mathematical
objects (Trouche, 2000).
Altogether, HHT in the 1990s in the eyes of many
stakeholders seemed to be the right thing at the right
moment, allowing for a real integration of technology in
mathematics teaching and learning, and giving way to
contemporary, challenging and motivating mathematics
education. Meanwhile, things turned out to be more complicated and a growing need emerged for theoretically
based research, which would go beyond the somewhat
naı̈ve idea that the simple integration of HHT would work
out well.
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3 Students’ instruments, improvisation
or interpretation
3.1 First ideas on opportunities offered by HHT
As indicated above, initial ideas on the use of HHT for
mathematics education, as shared amongst researchers and
educators in the 1990s, focused on the opportunities that
technology in general, and HHT in particular, would offer
for students’ learning. For example, is was claimed that
graphs, in traditional mathematics education at that time
the final result of a long and routine process of function
analysis, would now be the starting point for interesting
function investigations rather than the end point (Kindt,
1992a, 1992b). Also, the graphical output of HHT could
constitute for the students an occasion for explorations
leading to algebraic thinking. Figure 4, for example,
shows that students can explore the effect, on their
product, of changing linear functions Y1 and/or Y2. This
naturally leads to questions about properties of the product
Handheld technology for mathematics education
Fig. 4 Exploring the product of
two linear functions (Doorman,
Drijvers, & Kindt, 1994)
Fig. 5 Procedure for
calculating the number of zeros
at the end of n! (Drijvers, 1999;
Trouche, 1998; Weigand, 1989)
Fig. 6 Student copying
inappropriate TI-81 graphical
representation of (x2 ? x - 1)/
(x - 1) on paper (Drijvers,
1995)
function and their relation to properties of the two
‘building blocks’.
As a second example of ideas for capitalising on
opportunities that HHT offers, Fig. 5 shows how students
captured a procedure for calculating the number of zeros at
the end of n! Like programming, setting up such a procedure after some paper-and-pencil explorations can be
considered as a means of condensing and reifying the
process. This suggests that the use of HHT can promote
object thinking and encapsulation of mathematical processes, which is seen as an important mathematical
achievement (e.g. see Sfard, 1991).
These examples show that HHT was initially used to
achieve high educational goals such as the ability to relate
graphic and algebraic properties, mathematical investigation and reification of processes into mathematical objects.
The introduction of HHT in mathematics education can be
characterised as a phase of improvisation, both by students,
who were confronted with new types of mathematical
activities, and teachers, educators and researchers, who
were searching for a means to exploit HHT’s potential for
the learning of mathematics.
3.2 Growing awareness of changing epistemologies
In spite of the inspiring teaching activities using HHT that
were available, teachers sometimes were confronted with
disappointing student results and with unexpected difficulties. Dealing with graphs, for example, was not as easy
for students as it might seem at a first glance. To understand the empty screens that in some cases appear as a
result of pressing a graph button, students need to become
aware of the idea that a viewing window represents a
rectangular view on just a limited part of the theoretically
infinite plane, which may or may not ‘hit’ the graph.
Strange graphical representations appeared on the screen,
due to inappropriate window settings or pixel effects, the
latter particularly for the first-generation graphing calculator, and were copied on paper inappropriately (see Fig. 6)
or interpreted incorrectly (see Fig. 7).
These difficulties and sometimes misconceptions
attracted attention. As a result, a growing awareness
emerged amongst educators and researchers, an awareness
of students’ concept images (Tall & Vinner, 1981), their
conceptual knowledge and the meaning they attach to these
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L. Trouche, P. Drijvers
Fig. 7 A ‘‘convincing image’’:
the limit of x ? ln x ? 10 sin x
should not exist (Guin &
Trouche, 1999)
In a graphing calculator environment,
most of students infer, from the
oscillation of the curve, that the limit of
this function, when x tends towards
infinity, does not exist.
Without access to a graphing calculator,
students used to say: the behaviour of
such a function depends on ln x, which
is “much bigger” than sin x.
Fig. 8 Solutions of an equation
considered as intersection points
of graphs
concepts, in short their epistemologies being affected by
their work with HHT. Tool constraints needed to be dealt
with, and techniques for using the tools gave rise to new
meanings. For example, the technique of solving an
equation by drawing graphs of the ‘left-hand side’ and the
‘right-hand side’ and then approximating the coordinates of
the intersection points leads to a graphical, mental image of
an equation rather than an algebraic one (see Fig. 8).
A second example of new epistemologies concerns the
profound change in the relationships between a drawing
and a figure from a paper-and-pencil environment to a
dynamic geometry environment.1 Parzysz (1988) distinguishes a drawing (a picture object of seeing) from a figure
(integrating properties, object of knowing). In a paper-andpencil environment, a drawing has to be marked (for
example, to indicate that two sides are equal) to show its
properties. In a dynamic geometry environment, the technique of dragging reveals the construction’s properties,
which might remain hidden in the drawing (Falcade,
Laborde & Mariotti, 2007; Laborde & Capponi, 1994).
The growing awareness of changing epistemologies,
sometimes for better but sometimes for worse, clearly
evoked the need for theoretical reflection and
1
These are also now available on handheld devices.
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reconsideration, as exploiting the benefits from the integration of HHT in mathematics education turned out to be
not as simple as expected.
3.3 Theoretical advancements
As a result of the experiences described above, there was a
need for new theoretical approaches, which would do justice to the observation that using tools is not just a matter of
transforming mathematical thinking into commands for the
tool, but that the relation between user and tool is a bidirectional one: the user shapes the techniques for using the
tool, but the tool shapes the user’s thinking as well:
Tools matter: they stand between the user and the
phenomenon to be modelled, and shape activity
structures. (Hoyles & Noss, 2003, p. 341)
The relationship between techniques for using a tool and
mathematical thinking is a subtle and delicate one, which
requires theoretical frames of equal subtlety. For example,
the notion of situated abstraction (Noss & Hoyles, 1996)
refers to the mathematical knowledge, which emerges
within the frame of using technological tools in a particular
situation, and which to a certain extent remains ‘‘attached’’
to these technological experiences. Theories on semiotic
Handheld technology for mathematics education
mediation stress the meditational function of technological
tools in the process of the user making sense of a task (e.g.
see Falcade, Laborde & Mariotti, 2007). Gravemeijer
(1999) develops the notion of emergent modelling to point
out how mathematical meaning co-emerges with the
development of students’ symbolisations made with
whatever kind of tool. Though these theoretical approaches
are quite different, they share an interest in meaning
making, symbolising and in the relation between these
processes and the techniques with which technological
tools are used.
Another theoretical framework, which emerged in the
context of integrating HHT into mathematics education,
concerns instrumental approaches to using tools. As we
consider them to be highly relevant, we address them in
more detail. An essential starting point in instrumentation
theory is the distinction between artefact and instrument
(Rabardel, 2002). An artefact is the, often but not necessarily physical, object that is used as a tool. We speak about
an instrument if a meaningful relationship exists between
the artefact and the user for a specific type of task. Besides
the artefact, the instrument also involves the techniques
and mental schemes that the user develops and applies
whilst using the artefact. Put in the form of a somewhat
simplified ‘formula’: instrument = artefact ? schemes.
The process of an artefact becoming part of an instrument
in the hands of a user, in our case the student, is called
instrumental genesis (Artigue, 2002). During instrumental
genesis, a bilateral relationship between the artefact and the
user is established: whilst the student’s knowledge guides
the way the tool is used and in a sense shapes the tool (this
is called instrumentalisation), the affordances and constraints of the tool influence the student’s problem-solving
strategies and the corresponding emergent conceptions
(this is called instrumentation). As an anecdotal example of
instrumentalisation, we recall an incident involving some
students who programmed a RESET screen on a graphing
calculator, so as to simulate a system reset before the
written test in which this was a requirement.
Now, what are schemes and techniques? A scheme is a
more or less stable way to deal with specific situations or
tasks. As we see a scheme here as part of an instrument,
we speak of an instrumentation scheme. Within instrumentation schemes, schemes of instrumented action and
utilisation schemes are distinguished. Utilisation schemes
are directly related to the artefact and are building blocks
for more integrated schemes of instrumented action, which
are more global schemes directed towards an activity with
the object (Trouche, 2004). In these mental schemes,
technical and conceptual aspects are intertwined and codevelop. We cannot look into the heads of our students
[even if neuroscientists are advancing (Thomas et al.,
2008)!], so schemes cannot be observed directly but have
to be inferred from what can be observed, the instrumented techniques. Instrumented techniques are more or
less stable sequences of interactions between the user and
the artefact with a particular goal. In this interpretation,
the technique can be seen as the observable counterpart of
the invisible mental scheme. Finally, a scheme is constituted by instrumented techniques and knowledge guiding
these techniques. Vergnaud (1996) identifies concepts in
action and theorems in action as the heart of this knowledge, built through and for students’ instrumented activity
(which is not always what the teacher expects to be
built!). The students’ techniques can be seen as actual
interpretations of their schemes, which reflect their personal thinking. Techniques, of course, depend on the tools,
and reconciling paper-and-pencil techniques with HHT
techniques may be a challenge for students (Kieran &
Drijvers, 2006).
As an example of a scheme, Fig. 9 shows a screen of
symbolic calculator in which a solve technique is applied to
a parametric equation. The ovals around the screen sketch
some concepts in action, which are involved in this technique, and reflect the scheme. One of the main concepts in
action of the scheme is the notion of a solution of a parametric equation being an algebraic expression instead of
the usual numerical value.
Fig. 9 Schematic aspects of the
technique of solving a
parametric equation (Drijvers,
Kieran & Mariotti, 2009)
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L. Trouche, P. Drijvers
To summarise the theoretical proceedings outlined in
this section, we note that theoretical frameworks have been
developed to respond to the observed difficulties that students have whilst using HHT, as well as to its constraints
and opportunities. These theoretical approaches, with the
instrumental approach as one of the most promising, serve
to make explicit the subtle relationship between tool use
and the process of meaning making, as well as to study,
design and evaluate this process.
4 Educators’ instruments: leading a jazz band
or a symphonic orchestra
The theoretical developments described above focus on
HHT gradually becoming a personal instrument, integrated
by each student whilst interpreting her/his own scores. But
how about the teacher? How can s/he exploit the availability of HHT in the mathematics classroom? What
teaching techniques and working formats should be used?
As is the case for learning with technology, learning to
teach with technology is a subtle process, which has to be
tackled gradually through some crucial steps.
4.1 What about the musical scores?
Within the instrumental metaphor, tasks can be thought of
as musical scores. The awareness that problem solving is at
the heart of mathematics (Vergnaud, 1996) leads us to
think about the new mathematical tasks or problem situations that fit well to the new technological environments.
This is not specific for HHT, but, as will be pointed out in
Sect. 4.3, the students’ permanent access to this kind of
technology makes this issue more important.
After the first, somewhat too optimistic, illusions
(‘‘enjoy mathematics with HHT!’’), it appeared that it was
Fig. 10 Thinking on
calculators the required answers
(Trouche, 1998)
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not that easy to design new mathematical problem situations, which, on the one hand, took advantage of technology and, on the other, required mathematical thinking
about what was beyond a given HHT result or image.
Examples of such mathematical situations are quite rare in
textbooks and essentially can be found in research literature (for example, Artigue, in Guin et al. 2005). Figure 10
shows an example of a situation, which has a great
potential for exploring and learning mathematics. To find
the second expected root, a student has to exploit both the
functionalities of the calculator (looking for a ‘‘right’’
window) and his/her mathematical knowledge (transforming for example the equation, for the positive numbers, into
x = 20 ln(x) to get ‘‘reasonable’’ values). To be able to
design such tasks themselves, teachers need to master both
the functionalities of the artefact as well as the mathematical and didactical backgrounds of the mathematical
topic to be taught.
4.2 What about the tuning of instruments?
Once such a mathematical problem situation is designed
and presented, an essential question is how to ‘make it
work’ in the classroom, how to organise students’ work in
time and space, how to combine individual and collective
phases within problem solving, and how to integrate each
student’s instrument into the orchestra as a whole? For
answering these questions, the notion of instrumental
orchestration was introduced (Trouche, 2005). The strength
of the metaphor is that it stresses the need for whole-class
management, even when ‘individual’ technology is used.
Instrumental orchestration applied to a mathematical situation proposes didactical configurations for integration of
the available artefacts in the classroom activity and
exploitation modes for these configurations. Figure 11
sketches an example of an orchestration: the Sherpa
The teacher’s purpose is to
introduce the notion of infinite
limit of a function. More
precisely, she wants to
illustrate the theorem: “the
exponential function grows
faster than any power
function”, which means that,
from a certain point, the
exponential curve will be
above the graph of the power
function.
The answer provided by the
calculator is surprising: it
seems that, from the second
given root, the power curve
will “always” remain above
the exponential curve.
Handheld technology for mathematics education
Fig. 11 The Sherpa-student
configuration (Trouche, 2004)
The Sherpa-student configuration rests
on the exploitation of a particular role to
the so-called Sherpa-student, who is
using the technology in front of the class.
The teacher is thus responsible for
guiding, more or less, through this
student using the calculator, which in
fact is the whole classes calculator. The
teacher thus fulfils the function of an
orchestra conductor rather than a oneman’s band.
Several
exploitation
configuration
can
modes
be
of
this
considered:
sometimes work could be strictly guided
by
the
Sherpa-student
under
the
supervision of the teacher, sometimes
work could free; the role of Sherpa can
be
switched
to
different
students,
according on what they are doing at the
spot, or the Sherpa can be the same
student for the whole lesson, etc.
orchestration in which one of the students uses the technology in a way that all students can follow it, and the
teacher guides this student’s use.
in the context of a jazz band with a band leader rather than
the context of a symphonic orchestra with a classical
conductor…
4.3 An ambiguous metaphor
4.4 A metaphor in progress
The metaphor of orchestration is fruitful as it highlights
the importance of instruments for developing mathematical activity and stresses the teacher’s responsibility with
respect to these instruments. It reveals the need for not
only designing ‘‘good mathematical problems’’, but also
corresponding orchestrations that take into account the
technical aspects of the environment. It helps to identify
and design appropriate ways of teaching using technology.
The orchestration metaphor, however, could be questioned. As Hoyles (2003) pointed out, orchestrations might
neglect students’ creativity, particularly if prepared
beforehand and applied in a rigid way. These criticisms led
to a more balanced definition of orchestration (Drijvers &
Trouche, 2008). Teachers should consider the students’
instrumentalisation processes as potential enrichments of
the artefacts. Therefore, whilst orchestrating a mathematical
situation in the classroom, the teacher should combine the
guidance of instrumentation processes with the recognition
of new instrumented techniques proposed by students. In
this sense, the metaphor of orchestration is to be understood
On looking back at the short history of the notion of
orchestration, we realise that so far it has not been widely
adopted (except for our own area) in the community of
research, nor in the domain of teacher education. Two
reasons might explain this weak dissemination: the small
number of examples of practical orchestrations and the lack
of elaboration of the concept itself. Let us consider these
two shortcomings.
The main configuration that we have exploited is the
Sherpa-student one. In the context of isolated HHT (each
student has a personal HHT, with a very small screen, and
that does not communicate easily with the others), this
configuration had a great potential. Nowadays, the
increasing connectivity of HHT opens ways for a greater
diversity of configurations. For example, the TI-Navigator
environment, offering connectivity between the calculators
of groups of four students (students’ quartets, following the
metaphor), can be seen as a generalisation of the Sherpastudent configuration: it is up to the teacher, at any time, to
project one or several calculator screen(s) on the classroom
whiteboard. Many configurations are possible, reflecting
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L. Trouche, P. Drijvers
different didactical choices. We thus experienced (Fig. 12)
how small changes in the configuration of the classroom
can have important consequences for the communication
between the students, and between the teacher and the
students. These remarks are not restricted to HHT: similar
experiences with computer laboratories have been reported
(Drijvers et al., 2010, submitted data). Also, students’
access to laptop or notebook computers with wireless
Internet connection will increase. This offers new means
for orchestrating the learning.
In this era of technological expansion, the need for and
the design of rich orchestrations will develop. Drijvers
et al. (2010, submitted data) propose, for this purpose, a
repertoire of new configurations covering a diversity of
possible orchestrations for the teacher (e.g. see Fig. 13).
Certainly, to design orchestrations that fit to her didactical
objectives, it is necessary for the teacher to have at her
disposal both a repertoire of problems and a repertoire of
configurations.
A second weakness of the notion of orchestration is a
theoretical one. When introducing orchestration, we
defined it as an intentional and systematic management of
artefacts, aiming at the implementation of a given mathematical situation in a given classroom. This clearly suggests an a priori design before the implementation in a
classroom. However, according to the instrumental point of
view, with its focus on instrumental genesis, techniques,
schemes and orchestrations naturally emerge and evolve
through using tools during teaching. To address this issue,
we distinguished between the orchestration work (preparing the teaching) and the band leader’s work (facing the
students).
With this in mind, Drijvers et al. (ibidem) add to the
configurations and exploitation modes a third level, the
essential level of didactical performance:
‘‘A didactical performance involves the ad hoc
decisions taken while teaching on how to actually
perform in the chosen didactic configuration 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 unexpected aspect of the
mathematical task or the technological tool, or other
emerging goals’’ (Drijvers et al., 2010, submitted
data).
An orchestration can be seen in this frame as an artefact
for a teacher, evolving through successive phases of design
and implementation in classroom situations.
So far, we have considered the integration of HHT only
in classroom situations. The technological evolution,
mainly the development of connectivity, makes the issue of
integration even more complex for the teacher as well as
for the students, as learning and teaching will take place
both inside and outside of the classroom.
Fig. 12 A small change in the
configuration with important
consequences (Hoyles et al.,
2009)
Fig. 13 The spot-and-show
configuration (Drijvers et al.,
2010, submitted data)
A Spot-and-show configuration refers to
the situation in which the teacher has the
opportunity to spot the students’ work
while preparing the lesson and to decide
deliberately to show selected parts of it as
a starting point for whole-class discussion
during the lesson. In the sketch on the
right, the lesson preparation takes place
late in the evening….
123
Handheld technology for mathematics education
5 Future themes: all/one-man/woman band?
Reflecting on the history of HHT since the 1980s from a
certain distance, one is tempted to frame the developments
as a convergent process with calculators becoming more
sophisticated, PCs becoming increasingly smaller and
therefore PCs and HHT merging into one single, rather
complex tool. However, this somewhat simplistic view
needs additional considerations for at least three reasons.
First, with the development of connectivity between
HHT as well as PCs, the unit of analysis is not a single
HHT or PC, but a network composed of HHT and PCs.
This raises orchestrational issues. Second, through the
development of the Internet, which obviously relies on
connectivity, the HHT is no longer a stand-alone device,
which needs to contain all the necessary resources. Additional resources can simply be downloaded or accessed
directly online. The third additional consideration to the
simplistic view of merging tools relates to the previous
one. As it is not practical to have a dedicated HHT for each
type of task or subject (1 HHT for mathematics, 1 HHT for
geography, etc.), HHT is becoming quite generic, not
specifically dedicated to learning, but used for everyday
life (see the smartphone example in Fig. 1), raising new
questions about learning and teaching.
5.1 Connectivity and new challenge for orchestration
Connectivity is identified as an important issue in future
development of technology in education:
‘‘Thinking about the evolution of ICT in education,
the key expression that comes to the fore is connectivity. The interest in personal communication
strongly drives the need for connectivity. Even more
than is the case nowadays, students and their teachers
will communicate in oral or written form through the
Internet, through electronic learning environments,
and through classroom connectivity facilities that
allow for gathering students’ results from handheld
devices and projecting them on an interactive
whiteboard. […]. Computer tools offer options for
file transfer between handheld and desktop devices,
and between different types of software applications
such as DGE and CAS’’ (Drijvers, Kieran, &
Mariotti, 2009, p. 121).
Through connectivity, new opportunities enter the world
of education, with a focus on collaboration:
Digital technologies are already changing the ways
we think about interacting with mathematical objects,
especially in terms of dynamic visualisations and the
multiple connections that can be made between different kinds of symbolic representation. At the same
time, we are seeing rapid developments in the ways
that it is possible for students to share resources and
ideas and to collaborate through technological devices both in the same physical space and at a distance
(Hoyles et al., 2009, p. 439).
The potential of connectivity is particularly important in
the case of HHT. Section 4.4 shows how the small screens
of HHT beg for the development of devices for communication, such as TI-Navigator (Fig. 14, left). What appears
(for us) as a necessity in the case of HHT is an opportunity
in the case of PC: an opportunity to open the space of
debate into the classroom. For example, the confrontation
on a common screen of ‘‘rapid scribbles’’ made on each
student’s laptop (see Fig. 14, right screen) opens the way
for building a shared notion of fraction. In line with this,
Patton et al. (2008) introduce the idea of rapid collaborative knowledge building (RCKB) based on six principles
developed by Scardamalia (2002):
(1)
(2)
(3)
(4)
(5)
Make everybody think, as individuals and in teams
Class accepts new ideas, and constantly improves
ideas
Explore many ideas, and from many different angles
Students take initiative for their own learning
Everybody participates actively and contributes
knowledge
Fig. 14 From a TI-Navigator
experience (Hoyles et al., 2009)
to a GroupScribbles experience
(Patton et al., 2008)
123
L. Trouche, P. Drijvers
Fig. 15 A duo of HHT–PC
inviting the development of a
system of instruments (Aldon
et al., 2008)
(6)
Students organise their ideas and are self-reflective.
This type of connectivity nowadays is wireless, which
certainly makes it technically easier to set up in a classroom.2 It is not didactically easier for the teacher, as these
configurations confront her with a complex process of
management. Whilst using a ‘‘quick poll’’, for TI-Navigator, or ‘‘rapid’’ collaborative knowledge building in the
case of GroupScribbles, multiple ad hoc decisions have to
be taken by the teacher, and have to be taken ‘just on time’.
The new didactical configurations lead to complex didactical performances, including the need to adjust orchestration under the ‘fire of action’.
5.2 Connectivity and new challenge for systems
of instruments
Connectivity concerns connecting students’ as well as
teachers’ different HHT and PC tools. The TI-Nspire
environment (Aldon et al., 2008, see Fig. 15), for example,
provides such connectivity. It offers similar and compatible
environments on both HHT and PC, and resources can
move between the two platforms. It is also possible to have
an active image of the HHT on the PC screen; in this way,
the HHT can be used ‘‘through the PC’’ (Fig. 15, right
part).
USB flash drives also offer this kind of connectivity: one
can easily bring her/his own resources and implement them
on another device. These keys certainly have potential for
disseminating and sharing new resources. For this reason,
the French Ministry of Education has decided to give each
new teacher a ‘‘key to start’’ (‘‘une clé pour démarrer’’, in
French), which provides access to a portal of resources
dedicated to teaching the subject matter in question. This
key does not make teacher training redundant, but provides
novice teachers with an easy access to a set of resources,
2
The wireless connection works between the students’ HHT and the
teacher’s PC, not between the students’ HHT, which is certainly a
result of institutional constraints: the students are not allowed to
communicate during the examinations.
123
considered by the Ministry as key resources to initiate
‘‘good practices’’.
Practically, both teachers as well as students have to
combine sets of artefacts: different HHT (calculators, USB
keys…), different computers (personal computers, shared
computers, in school for example), overhead projectors,
view screens, interactive whiteboards, etc. Clearly, there is
a need, for each individual, to constitute a system of
instruments (Trouche, 2005). This new level of instrumental genesis is certainly an important issue for further
research. Finally, it leads to considering each individual as
a ‘one-man band’, managing sets of artefacts, and, inside
the classroom, the teacher as the person who is in the
uncomfortable (and challenging) position of managing a
band of one-man bands…
5.3 A duo of HHT and Internet
The most revolutionary manifestation of connectivity is
certainly the access to the World Wide Web via Internet. It
gives access to pedagogical resources and to online artefacts for the exploitation of theses resources (Fig. 16, left
side). As a consequence, it is no longer necessary to have
these resources physically installed on one’s own PC/HHT.
The device can become a ‘‘light’’ machine, making use of
distant resources. As the Internet is not particularly dedicated to mathematics, there is a tendency to use generic
artefacts. For example, we observed with surprise students
in 6th grade using Google as their calculator (Fig. 16, right
side)!
This example gives rise to some crucial reflections.
First, the conceptualisation of mathematical objects and
processes will be deeply modified: using Google to multiply two numbers develops a new view of what is a
multiplication. The result lies somewhere (just as it does if
we ask ‘‘what is the main town in Transylvania?’’); we
have to search for the result and not imagine a constructive
way to build it by our own means. Second, as soon as each
student can download particular resources such as applications that ‘‘increase the power of the calculator’’,
Handheld technology for mathematics education
Fig. 16 A duo of HHT and
Internet, towards light HHT?
Fig. 17 Learning as a (serious)
game, in the thread of HHT
dynamics? (Habgood et al.,
2005)
dedicated functionalities for example for investigating
function behaviour, or games, the process of instrumentalisation is amplified. This increases the teachers’
responsibility to incorporate each instrument in her new
orchestrations. Third, the need for developing high-quality
resources for mathematics teaching as well as criteria for
this quality is clearly urgent; this is currently the purpose of
several research projects at the European level (Trgalova
et al., 2009).
5.4 What is a learning tool, and what is a learning
activity?
At the beginning of this HHT-in-school story, the issue was
how to benefit from HHT dedicated to mathematics education (mainly calculators). Nowadays, it seems that we
have to face another process: students, on a daily basis, use
HHT, which is not dedicated to mathematics learning. Is it
possible to also gain benefit inside the classroom from
these kinds of HHT? This question opens a wide field of
research on mobile learning (Roschelle, 2003). For example: how would we integrate smartphones for the purpose
of mathematical activities in and out of school? More
generally, this raises two major questions, not only about
artefacts, but also about activities: how would it be possible
to motivate students to engage in mathematical activity out
of school, and how do we exploit the results of this activity
in school? In line with this, serious games and its
opportunities deserve serious attention by mathematics
educators and researchers (Habgood et al., 2005, Fig. 17).
Modelling mathematics learning as a challenge, or a
game (Brousseau, 1997), where the learner builds his/her
knowledge by engaging in this game, is quite usual in our
research community. Usually, however, the teacher designs
the game, its rules and its management. In the case of
serious games, the game and its rules are designed outside
of schools. In the early times of calculators, educators had
to think about the integration of these HHT into classrooms; perhaps, nowadays we should seriously consider the
integration into school of serious games, which might be
considered as ‘‘HHA’’ (handheld activities), due to their
quite natural appropriation. A shift in focus from technologies towards activities requires a rethinking of forms of
instrumentalisation and orchestration.
6 Conclusion
What do we learn from this history of the use of HHT for
learning mathematics from 1980 until today? First, we have
learnt to be less naı̈ve about machines and mediation,
primarily involving the learner who uses the tools and, at a
second level, the teacher who learns to integrate HHT into
her teaching. Machines are not neutral, but deeply influence activity, conceptualisation and, more generally, students’ and teachers’ development.
123
L. Trouche, P. Drijvers
Second, HHT, like every other tool, is not ‘‘ready to do’’
computing, graphing, investigation, problem solving,
learning or teaching. Doing requires appropriating a given
tool, and appropriating in its turn presupposes ‘‘putting in
the machine something of oneself’’ or, to put it another
way, personalisation and customisation. This strong point
of view leads to the consideration of each user as an
essential partner in the process of designing artefacts,
mathematical situations, orchestrations and resources.
Third, we have learnt that teaching is a responsibility of
the teacher, of course, but also of the students. In this
perspective, the Sherpa-student configuration can be considered as a good metaphor of the essential contribution of
students to teaching.
Fourth, bearing in mind the wide scope of HHT devices
and of tools for mathematics learning, we wonder how
mathematics education might benefit from students’ out of
school activities such as engagement in serious games,
often incorporated in multipurpose mobile HHT.
Fifth and last, but not least, our flashback leads us
towards the future of research, thinking beyond HHT in a
double direction. The first challenge concerns extending
the notions of mathematical situations and their orchestrations to out-of-school learning environments. The second challenge concerns renewing, from a practical and
theoretical point of view, the notion of artefacts for
learning and teaching. To enter these fields, we need to be
aware that HHT is no longer an isolated artefact, but
integrated in and articulated with a network of resources,
particularly online resources (Gueudet & Trouche, 2009).
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