D T M IGITAL ECHNOLOGY IN ATHEMATICS E : W W ( ' ) DUCATION HY IT ORKS 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 Drijvers, P. (2013). Digital technology in mathematics education: why it works (or doesn’t). PNA, 8(1), 1-20. 2 P. Drijvers 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 through(cid:127)or a helicopter flight over(cid:127)the 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- PNA 8(1) Digital Technology in Mathematics Education: Why it Works (or doesn’t) 3 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. Do mathematics (a) Didactical functions Practice skills of technology in (b) math education Learn mathematics Develop concepts (c) 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 PNA 8(1) 4 P. Drijvers 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 PNA 8(1) Digital Technology in Mathematics Education: Why it Works (or doesn’t) 5 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 PNA 8(1) 6 P. Drijvers 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 (cid:1866)(cid:488) (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 PNA 8(1) Digital Technology in Mathematics Education: Why it Works (or doesn’t) 7 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). Indicate the unknown to solve Notice the scope of An equation should contain an = sign the square root sign Word /letter vars? A solution can be “solve with respect to x” = “express x in terms of b” an expression 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. PNA 8(1) 8 P. Drijvers 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/ PNA 8(1) Digital Technology in Mathematics Education: Why it Works (or doesn’t) 9 (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 PNA 8(1) 10 P. Drijvers 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- PNA 8(1)