For as long as mathematics education has been considered to be a serious scientific domain, researchers, educators, and teachers have been theorizing about the learning and teaching of mathematics. This has led to an overwhelmingly broad spectrum of theoretical approaches, ranging from the philosophical to the practical, from the global to the local, some focusing on learning in general and others very much based in mathematical knowledge. This body of theoretical knowledge is still growing.

Now that the issue of integrating technological tools into the teaching and learning of mathematics has become urgent, one can wonder what the existing theoretical perspectives have to offer. Can they be applied to this new context? Or do we need specific theories appropriate for the specific situation of using tools for doing – and learning – mathematics? If yes, what is so specific about the integration of technology that justifies the need for such new paradigms? Are these paradigms to be considered as part of the body of knowledge of mathematics education, or do they rather belong to theories about humans’ interactions with technology? What kinds of problems do we want the theoretical frameworks, old or new, to solve for us? It is our conviction that theoretical frameworks are needed in order to guide the design of teaching, to understand learning, and to improve mathematics education.

From the time of the development of the mainframe computer in 1942, the first four-function calculator in 1967, the microcomputer in 1978, and the graphing calculator in 1985 (Kelly 2003), both mathematicians and mathematics educators have been intrigued by the possibilities offered by technology. However, it was not until the late 1960s when, according to Fey (1984), mathematicians and mathematics educators began to feel that computing could have significant effects on the content and emphases of school-level and university-level mathematics.

Among the earliest applications of the new technology to mathematical learning in schools was Computer Assisted Instruction – the design of individualized student – paced modules that were said to promote a more active form of student learning. Perhaps the most well known is the PLATO project (Dugdale and Kibbey 1980; Dugdale 2007). The next wave in technology-based approaches to mathematics learning involved programming, in particular, in Logo and BASIC. The development of the Logo programming language by (Feurzeig and Papert 1968; Papert 1980) was instrumental in this regard. Papert, a mathematician who was influenced by the theories of Piaget, was interested in the learning activities of young children and how the computer could enhance those activities (see, e.g., Papert 1970, for descriptions of children and junior high school students learning to program the M.I.T. “turtle” computer). In his 1972 article, entitled Teaching children to be mathematicians versus teaching about mathematics, Papert promoted “putting children in a better position to do mathematics rather than merely learn about it (Papert 1972).” At the time, programming in BASIC was also considered a means for enhancing students’ mathematical problem-solving abilities (Hatfield and Kieren 1972), even for students as young as first graders (Shumway 1984). The arrival of the microcomputer in the late 1970s not only increased the interest in programming activity, but also led to the development of more specialized pieces of software. Some of these specialized software tools were created specifically for mathematics learning (e.g., CABRI Geometry developed by Laborde 1990, and Function Probe developed by Confrey 1991), while others were adapted for use in the mathematics classroom (e.g., spreadsheets and computer algebra systems). The microcomputer and the graphing calculator also fed the growth of functional approaches in algebra and interest in multiple representations of mathematical objects (Fey and Good 1985; Heid 1988; Schwartz et al. 1991). However, by the 1990s, technological tools were still not widespread in mathematics classrooms, nor was there an abundance of qualitatively good software available (Kaput 1992). Against the above technological scene in mathematics education during the years from the 1960s to the early 1990s, we now examine the question of the theoretical frames that were used in the technology-related research in mathematics education during the same period. We begin with the Proceedings of the 1985 ICMI Study on technology.

**Tutor, Tool, Tutee**

With the arrival of the microcomputer and its increasing proliferation, a new framework was developed, which classified educational computing activity according to three modes or roles of the computer: tutor, tool, and tutee (Taylor 1980). To function as a tutor: “The computer presents some subject material, the student responds, the computer evaluates the response, and, from the results of the evaluation, determines what to present next” (p. 3). To function as a tool, the computer requires, according to Taylor, much less in the way of expert programming than is required for the computer as tutor and can be used in a variety of ways (e.g., as a calculator in math, a map-making tool in geography,…). The third mode of educational computing activity, that of tutee, was described by Taylor as follows: “To use the computer as tutee is to tutor the computer; for that the student or teacher doing the tutoring must learn to program, to talk to the computer in a language it understands” (p. 4). The rationale behind this mode of computing activity was that the human tutor would learn what s/he was trying to teach the computer and, thus, that learners would gain new insights into their thinking through learning to program.

**White Box – Black Box**

A theoretical idea that focused on the interaction between the knowledge of the learner and the characteristics of the technological tool was the White Box/Black Box (WBBB) notion put forward by Buchberger (1990). According to Buchberger, the technology is being used as a white box when students are aware of the mathematics they are asking the technology to carry out; otherwise the technology is being used as a black box. He argued that the use of symbolic manipulation software (i.e., CAS) as a black box can be “disastrous” (p. 13) for students when they are initially learning some new area of mathematics – a usage that is akin to the Tool mode within the Tutor-Tool-Tutee framework. However, other researchers (e.g., Heid 1988; Berry et al. 1994) have shown that students can develop conceptual understanding in CAS environments before mastering by-hand manipulation techniques. While the WBBB idea is pitched in terms of two extreme positions, others (e.g., Cedillo and Kieran 2003) have taken this notion and adapted it in their development of “gray-box” teaching approaches.

**Microworlds and Constructionism**

Papert and Harel (1991) encapsulated the theoretical ideas underlying the educational goals of microworlds (e.g., the Turtle environment in Logo) in the notion of constructionism, that is, “learning-by-making.” In terms of the Tutor-Tool-Tutee framework, we are now in Tutee mode. Papert and Harel (ibid.) have described constructionism as follows:

Constructionism – the N word as opposed to the V word – shares constructivism’s connotation of learning as “building knowledge structures” irrespective of the circumstances of the learning. It then adds the idea that this happens especially felicitously in a context where the learner is consciously engaged in constructing a public entity, whether it’s a sand castle on the beach or a theory of the universe. (p. 1)While admitting in 1991 that the concept itself was in evolution, Papert and Harel provided examples of studies that Papert himself was involved with during the 20 years previous and that fed the early evolution of the idea. Microworlds, such as turtle geometry, were a central component of the theory:

The Turtle World was a microworld, a “place,” a “province of Mathland,” where certain kinds of mathematical thinking could hatch and grow with particular ease (Papert 1980: p. 125) The Turtle defines a self-contained world in which certain questions are relevant and others are not … this idea can be developed by constructing many such “microworlds,” each with its own set of assumptions and constraints. Children get to know what it is like to explore the properties of a chosen microworld undisturbed by extraneous questions. In doing so they learn to transfer habits of exploration from their personal lives to the formal domain of scientific theory construction. (Papert 1980, p. 117)Some critics (e.g., Becker 1987) suggested that Papert’s theory needed further elaboration. Later development of the notion of “microworld” would not restrict the term to Logo-based environments or even to computer environments (Edwards 1998; Hoyles and Noss 2003). In a paper prepared for the ICMI Study 17, Ainley and Pratt (2006) describe how they drew on Constructionist ideas to develop a framework for task design that involved the linked constructs of purpose and utility. In bringing this example on Constructionism to a close, we would be remiss if we did not note that the potential of Logo learning environments grabbed the attention, and research activity, of hundreds of researchers in mathematics education during the 1980s (see, for example, the 4 years of Proceedings of the International Conference for Logo and Mathematics Education from 1985 to 1989).

**Amplifier – Reorganizer**

Pea (1987) re-elaborated the psychological notion of cognitive tools for the case of technology in education. Computers have the potential for both amplifying and reorganizing mathematical thinking. However, Pea argued that the one-way amplification perspective, whereby tools allow the user to be more efficient and to increase the speed of learning, misses the more profound two-way reorganizational possibilities afforded by the technology. By this he meant that not only do computers affect people, but also that people affect computers (both by the way they decide

on what are appropriate ways of using them and on how in refining educational goals they change the technology to provide a better fit with these goals). Meagher (2006), in his ICMI Study 17 contribution, has described how digital technology introduced into the classroom setting can bring to the fore two-way effects that are unanticipated and that can lead to unintentional subversion of the expressed aims of a given curriculum. He proposes an adaptation of the Rotman (1995) triangular model of mathematical reasoning as a tool for better understanding the complex interaction among student, technology, and mathematics.

Pea’s theoretical work also included the development of a taxonomy comprising two types of functions by which information technologies can promote the development of mathematical thinking skills: purpose functions and process functions. The purpose functions engage students to think mathematically; the process functions aid them once they do so. The purpose functions focus on constructs such as ownership, self-worth,

and the use of motivational “real-world” contexts and collaborative learning environments. The process functions include, according to Pea, five categories of examples: “tools for developing conceptual fluency, tools for mathematical exploration, tools for integrating different mathematical representations, tools for learning how to learn, and tools for learning problem-solving methods” (p. 106). Some of this work fed into the development of theories on distributed cognition (Pea 1989) and on situated cognition (e.g., Brown et al. 1989) – the latter construct being taken up in a later section of this chapter on situated abstraction.

**Theoretical Ideas Emanating from the Literature on Mathematical Learning**

Not only did local theorization concerning the use of new technologies in education begin to grow during these years; gradually, links with recently developed theory from the learning and teaching of mathematics were established. In that an exhaustive coverage is not possible, the following three examples of theoretical ideas emanating from the literature on mathematical learning illustrate some of the ways in which such theories and frameworks were used in research involving technological environments during the years leading up to the early 1990s.

**Visual Thinking vs. Analytical Thinking**

The interplay between visual and analytical schemas in mathematical activity and students’ tendencies to favor one over the other (Eisenberg and Dreyfus 1986) was a theoretical notion that was adopted in some of the past research studies involving technology. For example, Hillel and Kieran (1987) distinguished between the two, within the context of 11- and 12-year-olds working in turtle geometry Logo environments, as follows:

*By a visual schema we refer to Logo constructions of geometric figures where the choice of commands and of inputs is made on visual cues, Rationale for choices is often expressed by, “It looks like…”. By an analytical schema we refer to solutions based on an attempt to look for exact mathematical and programming relations within the geometry of the figure. (p. 64)*

These researchers found that the students did not easily make links between their visualizations and their analytical thinking. While research in nontechnology learning situations had disclosed (older) students’ preferences for working with the symbolic mode rather than with the graphical, the advent of graphing technology provided the potential for a shift toward valuing graphical representations and visual thinking (e.g., Eisenberg and Dreyfus 1989). These issues would continue to be explored in the years to come.

**From Past to Present**

A very interesting inventory of the mid-1990s research on technology in mathematics education is the one carried out by Lagrange et al. (2003). In their review of the world-wide corpus of research and innovation publications in the field of Information–Communication–Technology integration, they point out that “the period from 1994 to 1998 appeared particularly worthy of study [662 published works], because during these years the classroom use of technology became more practical, and literature matured, often breaking with initial naïve approaches” (p. 238). In the entire corpus of papers that was reviewed, Lagrange et al. found that the only theoretical convergences were at a general level and touched upon issues related to visualization, connection of representations, and situated knowledge. The study shows that less than half of the publications surveyed appeared to go beyond descriptions of the environment or phenomena being observed – and this literature was intended to reflect a certain maturity in the field.

To summarize this section on the theoretical frames that were used in the technology-related research in mathematics education in the period from the 1960s to the 1990s, we notice an initial concern with the potentials of technology use rather than with theoretical foundations. Gradually, both local technology-driven theories emerged and recently developed theories from mathematics education research were adapted to

the case of learning with technology. This overall development can be extrapolated and applied to the current situation, which is the issue at stake in the next section.

**Learning Theories from Mathematical Didactics**

As mentioned above, the early 1990s witnessed the beginnings of a more mature literature with respect to the use of technology in mathematical learning. However, this maturing did not occur in a vacuum. Within the mathematics education community at large, not only had theorizing become a more widespread activity, but also the nature of the theories being utilized within research on the learning of mathematics was experiencing a shift.

While Constructivism and its Piagetian roots provided the underpinnings of the theoretical elaborations that emerged in research related to technology use during the previous decades – elaborations that focused primarily on cognitive aspects of learning – the theoretical writing of Vygotsky with its sociocultural emphasis began to percolate through the international mathematics education community in the 1980s (see Streefland 1985, for the first mention within PME1 Proceedings of Vygotsky’s work). The steady growth in the development of sociocultural perspectives by researchers during the ensuing years was reflected in, for example, the scientific program of the 1995 PME Conference where plenary and panel presentations were devoted to Vygotskian theory (Meira and Carraher 1995). One of the first aspects of Vygotskian theory to be appropriated was his zone of proximal development (ZPD). However, later work by mathematics education researchers focused on the role played by language and other mediational tools in the teaching and learning of mathematics (Lerman 1998; Bartolini et al. 1999; Kieran et al. 2001). Yet another theoretical direction appeared during these years with the notion that knowledge is situated and is a product of the activity, context, and culture in which it is developed and used (Brown et al 1989).

The sociocultural perspectives that became popular during the late 1980s and 1990s were adopted, and adapted, by researchers with an interest in the role of technological tools in mathematical learning. The first example of a present-day theory used in research on the teaching and learning of mathematics within technological environments that we present in this section deals with a frame that resulted from the adapting of aspects from both sociocultural and situated learning theories, as well as from classic ideas on abstraction: the Webbing and Situated Abstraction frame. A second example, which shares aspects of both Vygotskian and Piagetian theories but is quite distinct from either of these, features Brousseau’s Theory of Didactical Situations and illustrates the way in which one of its concepts has served to inform research involving the integration of technology into mathematical learning situations: the concept of milieu. The third and last example of this section presents an

emerging and still developing theoretical frame, one that was conceptualized for use in research on modeling environments involving physical apparatus: the Perceptuo-Motor Activity frame.

**Source:**

Drijvers, P., Kieran, C., Mariotti, M-A., Ainley, J., Andresen, M., Chan, Y. C., Dana-Picard, T., Gueudet, G., Kidron, I., Leung, A. dan Meagher M. *Integrating Technology into Mathematics Education: Theoretical Perspectives. *In Lagrange, J-B., Hoyles, C. (Eds.), Mathematics Education and Technology-Rethinking the Terrain. New ICMI Studies, Volume 13. New York: Springer.