# Mathematical Knowledge and Practices Resulting from Access to Digital Technologies

## Abstract

Through an extensive review o f the literature we indicate how technology has influenced the contexts for learning mathematics, and the emergence of a new learning ecology that results from the integration of technology into these learning contexts. Conversely, we argue that the mathematics on which the technologies are based influences their design, especially the affordances and constraints for learning of the specific design. The literature indicates that interactions among students, teachers, tasks, and technologies can bring about a shift in empowerment from teacher or external authority to the students as generators of mathematical knowledge and practices; and that feedback provided through the use of different technologies can contribute to students' learning. Recent developments in dynamic technologies have the potential to promote new mathematical practices in different contexts: for example, dynamic geometry, statistical education, robotics and digital games. We propose a transformation of the traditional didactic triangle into a didactic tetrahedron through the introduction of technology and conclude by restructuring this model so as to redefine the space in which new mathematical knowledge and practices can emerge.

### Keywords

Mathematical knowledge Mathematical practices Dynamic technologies Learning ecologies Didactic triangle Didactic tetrahedron## 8.1 Overview of the Chapter

We have structured this chapter into three major sections: (1) mathematical knowledge and learning that results from the use of technology, (2) mathematical knowledge on which the technologies are based, and (3) mathematical practices that are made possible through the use of technology. We preface these three major sections with a look back at the potential of digital technologies to transform the way mathematics could be taught and learned that emanated from the first ICMI study, and comment on the limited realization of that potential. We argue that the assimilation of the technologies to existing classroom practices rather than the technologies provoking an accommodation in those practices has limited that potential. In this preface we suggest a metaphor for how technology could transform the traditional didactic triangle (student, teacher, and mathematics) into a didactic tetrahedron, and use selected faces of that tetrahedron as the focus of subsequent major sections of the chapter.

The first major section begins with a discussion of what we mean by “mathematical knowledge” in a technological world and the different research perspectives on knowledge in a mathematical learning context. Following this discussion, we address the major question of the influence of technology on the nature of mathematical knowledge. We give particular attention to the operational and notational aspects of mathematical knowledge, and then look at how technology has influenced the contexts for learning mathematics, and the emergence of a new learning ecology that results from the integration of technology into these learning contexts. We conclude this section of the chapter with three different case studies that illustrate novel ways of learning mathematics within these different learning ecologies.

The second major section of this chapter discusses the mathematical knowledge that “resides” within the different technologies, or, rather the mathematics on which these technologies are based and that influences their design, especially the affordances and constraints for learning of the specific design. (Design issues are addressed more fully in Sect. 1of this volume.) This second section concludes with a discussion of how much of this mathematics the user should be aware or even understand.

In the third major section of the chapter we focus on new mathematical practices. We begin with a discussion of the link between knowledge and practice in mathematics learning and teaching. This is followed by a discussion of the interactions among students, teachers, tasks, and technologies, and the resulting shift in empowerment brought about by these interactions. We then look at the role of feedback that can be provided through the use of different technologies. We conclude this section with more detailed descriptions of technologies that have the potential to promote new mathematical practices in different contexts: dynamic geometry, statistical education, robotics and digital games. The chapter concludes with a summary in which we revisit our didactical tetrahedron, restructuring its vertices so as to redefine the space in which new mathematical knowledge and practices can emerge as a result of our review of the literature presented in this chapter.

### 8.1.1 Preface

As indicated in Chap. 7, the first ICMI study, held in Strasbourg, France in 1985 concerned the influence of computers and informatics on mathematics and its teaching. This first ICMI study reported (with considerable optimism) on the*potential* of these technologies to transform the way mathematics could be taught and learned (Howson and Kahane 1986). Initial predictions about the influences of technology built up in our minds an image that computing machines would replace arduous tasks, with computational giants freeing the human element (Pacey 1985). The uses of technology in education, however, have often simply replaced paper with computer screens without changing tasks; computers have been used to “simply transfer the traditional curriculum from print to computer screen” (Kaput 1992, p. 516) in ways that resemble traditional worksheets and structured learning environments, rather than working to transform learning (Tyack and Cuban 1995). This limitation, however, is less a limitation of the technology “than a result of limited human imagination and the constraints of old habits and social structures” (Kaput 1992, p. 515).

Piaget ( 1970) introduced the distinction between the assimilation and accommodation of concepts, contrasting adaptation of the environment to the organism with adaptation of the organism to its environment. In assimilation, learners would attempt to interpret a new idea into their current framework for conceptual understanding. This often meant remaking the concept to fit within their perspective, sometimes at the expense of its intent. Piaget argued that for understanding, it was necessary to sometimes adapt one's framework to take on and make the new concept viable within the environment (accommodation). In some ways, a similar revolution is taking place in mathematics classrooms - some are taking technology innovations and refitting them to retain the viability of the current classroom contexts. For example, many uses of technology take the form of creating electronic worksheets and structured lessons that more or less take the place of current classroom practices. Rather than have the technologies redefine classrooms, they are assimilated into current practice. Alternatively, technology can assist us in considering new forms of practice in profound ways, essentially accommodating new technologies rather than assimilating them. In this chapter we shall attempt to describe those situations in which the use of technology has brought about an accommodation in the ways people teach and learn mathematics and the new kinds of mathematical knowledge that results from such accommodations. Healy ( 2006), for example, describes the challenges of integrating technology into the Brazilian education system. Built on the assumption that the introduction of computer technology would act as a catalyst for change in classroom practice, researchers came to better understand the complexity of the educational system and in particular the critical role of the teacher in the process of learning, and the reciprocal relationship between technology and meaning-making.

## 8.2 Mathematical Knowledgein a Technological World

### 8.2.1 What Is Mathematical Knowledge?

*Children's mathematics:*The mathematics that children (or learners of any age) construct for themselves and is available to them as they engage in mathematical activity*Mathematics for children:*The mathematical activities that curriculum developers/writers and teachers design to engage students in meaningful mathematical activity*Adult mathematics:*The mathematics that adults have constructed through their years of schooling and experience in the world*Disciplinary mathematics:*The mathematics created and studied by professional mathematicians

Whether one takes a radical constructivist view of knowledge (Piaget and Szeminska 1965; von Glasersfeld 1995; Steffe 1992) or a social constructivist view (Vygotsky 1978), the question of who's mathematics we are focusing on is relevant when addressing our driving question of what new types of mathematical knowledge emerge as a result of access to digital technologies.

[H]ave long contrasted unconnected, disembodied, meaningless, context-bound, or mechanical procedures (what could be called a “weak scheme”) with well-connected, contextualized, integrated, meaningful, general, or strategic procedural knowledge (what could be called a “strong scheme” …). Analogously, well-connected conceptual knowledge has been contrasted with sparsely connected conceptual knowledge … For instance,

strong schemas- which involve generalizations broad in scope, high standards of internal (logical) consistency, principle-driven comprehension, and principled bases for a priori reasoning (i.e. predictions are derived logically) - have been proposed to underlie deep conceptual knowledge.Weak schemas- which entail generalizations local in scope, low standards of internal (logical) consistency, precedent-driven comprehension, and no logical basis for a priori reasoning (i.e. predictions are looked up) - have been posited to underlie superficial conceptual knowledge and to explain why younger children's concepts may be less deep and sophisticated (e.g., less general, logical, interconnected, or flexible) than older children's or adults'. (p. 117)

Steffe ( 1992, 2002, 2004) and Olive ( 1999; Olive and Steffe 2002; Olive and Vomvoridi 2006) would disagree with the association of younger children's mathematical concepts with weak schemas and superficial conceptual knowledge. They argue, instead, for a distinction between young children's mathematics and older children's mathematics (and adults' mathematics) based on the*nature and content* of their schemes and schemas (Olive and Steffe 2002). Within the constraints of their experiences, young children's schemas can be as strong as adult schemas (or as weak). What distinguishes both their schemes and schemas from those of adults are the mental constructs they have built based on their lived experiences.

Baroody et al. ( 2007) point out that “The construct of adaptive expertise, for one, unites the notions of deep conceptual knowledge, deep procedural knowledge, and flexibility” (p. 120). Hatano ( 2003) argues that flexibility and adaptability only seem possible when there are conceptual meanings to provide criteria for selecting among alternative procedures. That is, one needs conceptual knowledge to give meaning to processes. Gray and Tall ( 1994) put forward the notion of “procept” as a combining of process and concept. They argue that successful students can use symbols as procepts, whereas less successful students are limited to use of procedures.

In the examples that follow later in this chapter, we sh all attempt to distinguish among technologies that have the potential to help users combine procedural and conceptual knowledge (proceptual knowledge) and those technologies that enhance the learning of isolated (or disconnected) processes.

### 8.2.2 The Influence of Technology on the Nature of Mathematical Knowledge

The next step in our investigation of new knowledge comes out of the need to examine the way that learners view mathematics, both its nature and its utility. In keeping with the view that young children can (and do) construct*strong schemas*, Steinbring ( 2005) notes that “New mathematical knowledge is not merely still unfamiliar, added, finished knowledge, but new mathematical knowledge has ultimately to be understood as an extension of the old knowledge by means of new, extensive relations, which at the same time let the old knowledge shine in a new light and, even generalize the old knowledge” (p. 3). This requires quite a different conceptualization of the nature of mathematical knowledge, both by learners and by teachers. If one considers mathematics to be a fixed body of knowledge to be learned, then the role of technology in this process would be primarily that of an efficiency tool, i.e. helping the learner to do the mathematics more efficiently. However, if we consider the technological tools as providing access to new understandings of relations, processes, and purposes, then the role of technology relates to a conceptual construction kit. In this way, Steinbring argues that for learners, mathematical knowledge is always “on the way,” where knowledge bound in their concrete experiences emerges as new generalized understandings developed through ongoing interactions with ideas, relations, processes, structures and patterns viewed in new ways. Therefore, as we shall argue further in Sect. 8.4, knowledge is deeply embedded in practices and experiences.

Within the field of mathematics, many changes have taken place as a result of technological advancements. According to the Australian Academy of Science ( 2006), however, professionals have rarely taken advantage of new developments in the mathematical sciences (e.g., genetic innovation research, optimization in imaging, stochastic modeling), most of which have emerged out of the intersection of mathematics and new technologies (MASCOS 2004; Australian Academy of Science 2006). Likewise, despite the strong influence of technology on new developments in the field, little has changed in the school mathematics curriculum (e.g., Sorto 2006). These resistances are likely due to conflicts in teachers' and curriculum publishers' beliefs about the nature of mathematics and the goals of school curriculum. The foundations of what and how mathematics should be taught are now being challenged with the infusing of technologies into mathematics education. Reconciling these conflicts requires a re-evaluation of our beliefs about the very nature of mathematics. Most research utilizing new technologies in mathematics education portray mathematics as experimental, challenging, and empowering (e.g. Buteau and Muller 2006; Kaput 1996; Noss and Hoyles 1996; Papert 1972, 1980). These images resonate with philosophical challenges to the nature of mathematics in the last century (Lakatos 1976, 1978).

There is a perception in the general population of mathematics as a field that is “hard, right or wrong, routinised and boring” (Noss and Hoyles 1996, p. 223); this perception is rampant in school mathematics and in the public domain. The divorce of mathematics from its epistemological roots has often created a by-product of perceptions by learners that mathematics is too difficult for ordinary people to grasp. Technological environments potentially reconnect the learner with contexts in which they regain the agency to create meaning. These situations can be authentic contexts supported by technological tools to control complexity or they can be imaginary worlds in which learners can try out ideas.

Borba and Villarreal ( 2006; drawing on work by Tikhomirov 1981) consider the epistemological role of computers in learning mathematics. They argue that conceptualizing knowledge as atomistic leads to a view that the role of computers in generating knowledge is one where technologies either substitute humans or supplement humans. In schools, this is often the conceptualization that is operationalized - technologies are used to substitute paper-and-pencil calculations or supplement graphing skills. However, they argue that this view is shortsighted. In conceptualizing technologies through a broader complexity framework, one begins to realize the challenges in separating technology's effect on the transformation of knowledge with the transformation of practice. Borba and Villarreal further contend that humans and technologies are often seen as disjoint, assuming that the “cognitive unit” is only the human being, not the humans-with-media perspective that they adopt. “The very idea of considering the human being as the unit that produces knowledge can underestimate the importance of technologies in this knowledge production” (p. 12). Noss and Hoyles ( 1996) argue that because technologies mediate knowledge construction, they not only alter this construction of knowledge, but the meaning of knowledge for individuals as well.

### 8.2.3 Mathematical Knowledge:Operational and Notational Aspects

*notational*aspects of the discipline. That is, the emphasis is on the symbolic and representational aspects of mathematics (Fey and Good 1985; Kaput 1987, 1998; Hitt 2002), particularly in the areas of algebra, geometry, and statistics. School mathematics has been primarily restricted to routine procedures that could be carried out by hand; time and effort focused on teaching students to calculate and perform procedures by hand. Therefore, the emphasis was on the written, notational, symbolic aspects of mathematics rather than its more operational aspects:

In the past, teachers and students were confined to a sequential approach to learning these procedures, with the mastery of each step in the procedure necessary before proceeding to the next step. Technology allows a different approach, with more complex procedures (or

macroprocedures) chunked into a series of simpler procedures (microprocedures) (Heid 2003). (Heid 2005, p. 347)

Technology has therefore allowed school mathematics to incorporate a more*operational* focus that adds another dimension to understanding. By an operational focus, we mean an emphasis on the practice and applications of mathematics through visualization, manipulation, modeling, and the use of mathematics in complex situations. With technology, students can use technology to solve an equation before they need to master factoring or the quadratic formula by hand, approximating solutions graphically (e.g., Fey and Heid 1995). In this way, there exist choices regarding which to do first (by hand or with technology) or at all. By operationalizing mathematics, mathematics is distributed between the student and the technology, with the student empowered to decide when and how to use the tool (Heid 2005; Geiger 2006). However, this requires the student to understand and make decisions about what mathematics might be useful and how it might be used. This is quite a new experience after conventional schooling has cued students into knowing which mathematics is needed for a problem up front (Boaler 1997).

Technology is likely to change not only the content of school mathematics but also the processes of school mathematics and the nature of mathematical understandings. Students in technologically rich classrooms are likely to develop multirepresentational views of mathematics. Some technologies will enable them to develop almost a kinematic understanding of functional relationships. (p. 357)

### 8.2.4 Contexts for Learning Mathematics

The focus of mathematics in school has been on teaching students the power that mathematics has to generalize and abstract from particular contexts. These abstractions have developed over centuries of work by thousands of mathematicians. Rather than require children to begin this road again, school mathematics allows them to benefit from the toil and uncertainty of previous generations of mathematicians, and instead work with mathematical tools that have already passed the test of viability. The drawback has been that in the process of abstracting from context, the purpose of mathematics as a tool for making meaning has sometimes been forgotten or put aside. Some of this putting aside has been because of efficiency - the contexts in which mathematical meanings can be derived are neither simple to design nor simple in themselves. This is no surprise given the elapse of time over which these ideas have arisen. Some content that students encounter in schools has developed only in the last century (e.g. fractal geometry and iterative or recursive functions) - in comparison to the millennia on which their foundations are built.

Rather than adopt new mathematics, many reform curricula have chosen to embed traditional mathematics into what they term “real-life contexts” (e.g. CMP, Core Plus, Mathematics in Context, in the US). Unlike problems found in the world, however, many of the problems found in these curricula remain “well-defined” in their attempts to simplify the complexity of the situation. These “pseudo-contexts” can actually make mathematics more difficult to learn (especially for lower socio-economic students), as they give children conflicting messages about whether unintended contextual and experiential factors should be ignored or drawn into play (Lubienski 2000). By oversimplifying the intellectual demands required to mathematize and interpret problems, and by trivializing the contribution of mathematics to solving real problems, the perception of mathematics as a subject with limited use outside of school is reinforced. Noss and Hoyles ( 1996) contend that technologies open the possibility for meaningful mathematics to be created within the context of school rather than simply brought in from the outside.

Contexts allow the learner to reflect on and control for the meaning and reasonableness of their developing ideas. This allows them to ensure that concepts are viable within the situation. Of course, a goal in mathematics is to abstract and generalize across contexts, but enough is not done to encourage meaning-making to begin with. These contexts come out of a diverse allowance of settings. Technological tools allow for one type of setting from which learners can play with ideas. Dynamic software packages can facilitate visualization (Presmeg 2006), connecting informal and formal mathematics (Mariotti 2006), and develop perceptions of mathematics as an instrument rather than an object (Rabardel 2002). In situating the student in a transformative position of agency, these technologies potentially redefine and expand the student's role as knower and creator of mathematical knowledge. and her colleagues (Laborde et al. 2006) remind us, however, that it is not only the interaction of the student and the machine that matters but also the design of tasks and learning environment (see Sect. 1 of this volume). They argue for an intrinsic link between mathematical knowledge and understanding of its use as a tool. Mackrell ( 2006) argues, for example, for the powerful influence that dynamic visualization software programs like Cabri 3D ( 2005) and Geometer's Sketchpad (Jackiw 2001) can have on students' understanding by enabling them to manipulate mathematical objects as tangible entities, observing and debating invariant relationships. Mathematics is, after all, primarily concerned with properties of*invariance* - the characteristics that describe attributes and relationships that remain constant under varying conditions. For example, in Euclidean geometry, the three angle bisectors of a triangle pass through the same point. Algebraic identities describe relationships that remain fixed as the values of the variables change. Coming to know these invariant properties through dynamically changing that which varies (rather than memorizing facts) can contribute to much more stable and powerful mathematical knowledge. Thus, dynamic technologies can become powerful contexts for learning mathematics.

### 8.2.5 A New Learning Ecology

It is not the technology nor the play themselves that evoke meaning, but rather careful interactions between the task, teacher support, technological environment, classroom and social culture, and mathematics (Noss and Hoyles 1996). We have learned much from the days of thinking the computer would solve the challenges of learning (Cuban 2001), but this does not mean that we should be tempted to err on thinking dichotomously that computers have failed to add value to learning. In the old way of thinking, computers were seen as human tutors and evoked a vision of the teacherless learning environment. New avenues for using technology take advantage of, rather than marginalize, teacher, task, and classroom cultures. For example, there is the question as to whether strong emphasis in school mathematics on developing expertise in symbolic manipulation should continue to be at the forefront of time in secondary mathematics given the accessibility of Computer Algebra Systems (CAS). Within geometry, traditional instruction utilizes a definition-theorem-proof (dtp) approach to teaching geometry, where students are first taught definitions and given theorems and proofs about geometric objects and relationships before having an opportunity to work with and investigate these relationships for themselves. Dynamic geometry environments (DGEs) are challenging this perspective, including the very nature of what counts as a proof, when one considers that students can test a conjectured relationship with thousands of cases to assess its viability. DGE objects and the assessment of their relationships, because they are based on student design in search of a question, depend in new ways on using argumentation and justification. Proof takes on new meaning in this context, and becomes a tool that learners can use to explain what they discover through their dynamic explorations and, thus enable them to convince their peers of their new conjectures - rather than mimicking a mathematician's proof to satisfy an unknown cultural construct (de Villiers 1999). In statistics, new visualization tools enable learners to interact with data through envisioning relationships informally before more formal tools are brought into play. For example, instead of being taught how to calculate a mean, learners might first examine distributions of heights of children of different age groups and look for viable ways to compare and talk about them. Research has found that in technological environments in which children can design their own tools to describe aspects of the data that they find useful, students can envision concepts of center and spread of data by talking about the “clump” in the data (Konold et al. 2002; Makar and Confrey 2005). In an environment supported by worthwhile tasks and a culture of inquiry, learners have an opportunity to operationalize mathematics and use it as a tool for a productive purpose, rather than apply pre-made mathematical concepts to a contrived situation. “The challenge is to focus on the learning ecology as a whole, considering the interactions between different dimensions - epistemological, technological (or perhaps instrumental), cognitive, and pedagogical - concomitantly” (Healy 2006, p. 3). In the following section we present some example cases of technologies that have been successfully used (in different ways) as an integral part of the learning ecology to bring about the construction of (new) mathematical knowledge (for the learners).

### 8.2.6 Example Cases of Effective Technologies

In this section we present case reports of the uses of three different technologies: Computer microworlds designed with what Zbiek et al. ( 2007) call high levels of “cognitive fidelity,” simulation software and curricula designed to introduce the Mathematics of Change and Variation (MCV) to middle school students, and dynamic geometry environments (DGE). The first case is an example of how young children, within the context of a constructivist teaching experiment, were able to construct powerful fractional schemes through the use of computer-based tools that enabled them to enact their mental operations. The second case reports results of a state-wide implementation of a curriculum unit that made use of specially designed simulation software to improve middle school students understanding of rates, ratios and proportions. The third case reports on the global use of dynamic geometry environments. Zbiek et al. ( 2007) would categorize the latter two cases as exhibiting high levels of “mathematical fidelity,” that is, they provide the users with mathematically accurate visualizations and feedback. We point out, however in Sect. 8.3 of this chapter, that mathematical fidelity cannot be taken as a given with several common technologies used in mathematics classrooms.

#### 8.2.6.1 The Fractions Project: Using Technology with High Levels of “Cognitive Fidelity”

Steffe and Olive ( 1990) at the University of Georgia (USA) designed and conducted a 3-year constructivist teaching experiment with 12 children (beginning in their third grade in school) in order to develop cognitive models of children's construction of fractions. Computer microworlds called Tools for Interactive Mathematical Activity (TIMA) (Biddlecomb 1994; Olive 2000b; Olive and Steffe 1994; Steffe and Olive 2002) were specifically designed for the teaching experiment and were revised during the teaching experiment based on the children's interactions within these environments. The TIMA provide children with possibilities for enacting their mathematical operations with whole numbers and fractions. They also provide the teacher/researcher with opportunities to provoke perturbations in children's mathematical schemes and observe children's mathematical thinking in action.

Being able to make a stick (in TIMA: Sticks) that is “9 times as long as the 1/7-stick” through repetitions of a 1/7-stick, provided Joe with an instantiation of his iterable unit fraction. He had made a modification in his whole-number multiplication scheme that enabled him to use a unit fraction in the same way that he could use units of one with his composite units. The TIMA software had provided Joe with the tools to build a bridge from whole numbers to fractions. (p. 360)

The TIMA software (and later adaptations) has been used by many researchers in different countries since the conclusion of the Fractions Project: Nabors ( 2003) used the TIMA: Bars software in her study of proportional reasoning; Norton ( 2005) used TIMA: Bars in his study of eliciting student conjectures; Hackenburg ( 2007) used a Java version of TIMA: Bars called JavaBars (Olive and Biddlecomb 2001) in her research on middle school students' rational number concepts; Chinnappan ( 2006) reported using JavaBars in his 2001 study with elementary children in Australia, in which “JavaBars mediated children's cognitive actions” (Chinnappan 2001, p. 102); and Kosheleva et al. ( 2006) also used JavaBars in their study on the effects of Tablet PC technology on mathematical content knowledge of pre-service teachers, where JavaBars was found to provide “a creative workspace to explore fractions”. (p. 298)

Cognitive fidelity is a particularly important consideration for researchers. By providing action choices to the learner that faithfully reflect potential cognitive choices, tools such as the TIMA technology can provide to researchers more powerful evidence of patterns in children's thinking. In turn, an improved understanding of children's thinking can better inform continuing development of the tools. (p. 1177)

For instance, one way that technology can enhance the learning of rational number concepts is through the use of computer tools that allow students to enact psychological operations that are difficult to perform with physical materials. In order to establish a relation between a part and a whole in a fractional situation, the child needs to mentally disembed the part from the whole. With physical materials it is not possible to remove a part from the whole without destroying the original whole. With static pictures the part is either embedded in the whole or is drawn separate from the whole. … Using a computer tool that provides the child with the ability to dynamically pull a part out of a partitioned whole while leaving the whole intact, the child can enact the disembedding operation that is necessary to make the part-to-whole comparison. (pp. 7–8)

In addition to enabling students to operationalize the part-to-whole relation, the disembedding action, combined with repeating the disembedded part, led to iterating operations that enabled students to construct meanings for fractions greater than one (improper fractions) (Tzur 1999). The ability to enact recursive partitioning led to reversible reasoning and splitting operations, essential for the construction of the “rational numbers of arithmetic” (Olive 1999; Olive and Steffe 2002).

#### 8.2.6.2 The SimCalc Project: Introducing the Mathematics of Change in Middle School - Technology with High Levels of “Mathematical Fidelity”

SimCalc software engages students in linking visual forms (graphs and simulated motions) to linguistic forms (algebraic symbols and narrative stories of motion) in a highly interactive, expressive context. SimCalc curriculum leverages the cognitive potential of the technology to develop multiple, interrelated mathematical fluencies, including both procedural skill and conceptual understanding. (p. 2)

In terms of the theoretical frameworks outlined in Chap. 7, the SimCalc software acts as a semiotic mediator, linking several different semiotic systems to develop both procedural skills and conceptual understandings. The results of this extensive implementation of the SimCalc MathWorlds curriculum do, indeed, indicate the cognitive potential of the technology, achieving what has been termed the “gold standard” for experimental research, both in design and effects.

The research project involved 120 grade 7 teachers recruited from 8 regions of Texas. A Treatment-Control experimental design was used, with teachers being randomly assigned (by school) to either group. Of the 120 teachers who originally attended the summer workshop, 95 returned complete data for the 2005–2006 school year. At the outset of the experiment the Treatment Group of 48 teachers and the Control Group of 47 teachers did not differ in any significant way. In the summer of 2005 both groups participated in a 2-day professional development workshop focused on rate and proportionality. The Treatment Group received “an integrated replacement unit incorporating SimCalc curriculum, software, and three additional days of teacher training.” The Control Group “used their existing curriculum but had the benefit of training and materials on the topic of rate and proportionality” (Roschelle et al. 2007, p. 3). Rate and proportionality are typically taught in a 2- to 3-week unit in grade 7 in Texas. The Treatment Group teachers were asked to replace this unit with the SimCalc unit, the Control Group teachers were asked to continue using their existing textbooks, enhanced with professional development support. “The main outcome variable was student learning of concepts of rate and proportionality, measured on identical tests administered before and after the 2- to 3-week rate and proportionality unit” (p. 4). These tests consisted of 30 items: 11 simple and 19 complex items. The simple items were based on items used on the Texas state test and typically asked students to find the missing term in a proportional relationship (e.g. “If 2/25 =*n*/500, what is the value of*n*?”). The complex items addressed understanding of a direct proportional relationship as a function*f*(*x*) =*kx*, and the concept of slope of a line graph as an indication of speed in a distance-time relationship. All 30 items went through rigorous validation processes, “including cognitive interviews with students, item-response theory analyses on field test data collected from a large sample of students, and expert panel reviews” (p. 5).

*p*< 0.0001), indicating that the students in the Treatment Group classrooms learned more than their counterparts in the Control Group classrooms. The difference was most pronounced across the 19 complex items, and held across SES, race and gender groups. Based on these results, the researchers claim the following:

- (a)
That the SimCalc approach was effective in a wide variety of Texas classrooms

- (b)
That teachers successfully used these materials with a modest investment in training

- (c)
That student learning gains were robust despite variation in gender, ethnicity, poverty, and prior achievement. (Roschelle et al. 2007, p. 6)

The researchers make the important point that the gains were accomplished by the Treatment students on the more complex items dealing with proportionality and rate, whereas all students made similar gains on the simpler items. For example, with respect to the comparison of two distance-time graphs on the same coordinate axes, the Treatment students were more likely to use the correct idea of “parallel slope as same speed,” whereas Control students were more likely to have the misconception “intersection as same speed.” (Roschelle et al. 2007, p. 7)

Other studies on students' conceptions of slope and rate (e.g. Lobato and Siebert 2002; Olive and Çağlayan 2008) have highlighted the difficulties students experience with these concepts; thus, the results obtained through the use of the SimCalc software and curriculum are seen as a breakthrough in this traditionally difficult and important mathematical topic. For the students in the Treatment Group, the use of the SimCalc technology promoted the construction of new mathematical knowledge.

*Mathematics of Change and Variation*(MCV) to students who have traditionally been shut out by “the long set of algebraic prerequisites for some kind of formal Calculus, this despite the fact that the bulk of the core curriculum can be regarded as preparation for Calculus” (p. 7). Kaput goes on to state:

…we can see that while large amounts of curricular capital are invested in teaching numerical, geometric and algebraic ideas and computational techniques in order that the formal symbolic

techniquesof Calculus might be learned, the ways of thinking at the heart of Calculus, including and especially those associated with the Fundamental Theorem, donotrequire those formal algebraic techniques to be usefully learned. Indeed, by approaching the rates-totals connections first with constant and piecewise constant rates (and hence linear and piecewise linear totals), and then gradually building the kinds of variation, we have seen the underlying relations of the Fundamental Theorem become obvious to middle school students. (p. 7)

Thus, when we look at the strong results from the Texas implementation of the SimCalc curriculum in light of Kaput's major points concerning access to the important ideas of Calculus, we can look forward to a majority of students creating new kinds of mathematical knowledge concerning change and variation as a result of using such technologies within a well-conceived curriculum, implemented by enthusiastic and well trained teachers.

#### 8.2.6.3 Dynamic Geometry Environments

Dynamic Geometry Environments (DGEs) include any technological medium (both hand-held and desktop computing devices) that provides the user with tools for creating the basic elements of Euclidean geometry (points, lines, line segments, rays, and circles) through direct motion via a pointing device (mouse, touch pad, stylus or arrow keys), and the means to construct geometric relations among these objects. Once constructed, the objects are transformable simply by dragging any one of their constituent parts. Goldenberg and Cuoco ( 1998) provide an in-depth discussion on the nature of Dynamic Geometry. A common feature of dynamic geometry is that geometric figures can be constructed by connecting their components; thus a triangle can be constructed by connecting three line segments. This triangle, however, is not a single, static instance of a triangle that would be the result of drawing three line segments on paper; it is in essence a prototype for*all possible triangles*. By grasping a vertex of this triangle and moving it with the mouse, the length and orientation of the two sides of the triangle meeting at that vertex will change continuously.

A study by Olive ( 2000a) describes how a 7-year old child (Nathan) constructed for himself during just 5 min of exploration with the*Geometer's Sketchpad*® a fuller concept of “triangle” than most high-school students ever achieve. He had been shown how to construct a triangle with the segment tool and then experimented by dragging the vertices of this dynamic figure, all the time asking his father if the figure were still a triangle. His father threw the questions back to him and when Nathan responded that the figures were still triangles (fat triangles, skinny triangles, etc.) his father asked him why they were still triangles. Nathan responded “because they still had three sides.” But the real surprise came when he moved one vertex onto the opposite side of the triangle, creating the appearance of a single line segment. Nathan again asked his father if this was still a triangle. His father again threw the question back to him. Nathan thought for a while, then held out his hand with his palm facing outwards, vertically, and rotated it to a horizontal position with his palm facing down, while saying: “Yes. It's a triangle lying on its side!” This last comment and accompanying hand-motion indicates intuitions about plane figures that few adults ever acquire: That they have no thickness and that they may be oriented perpendicular to the viewing plane. [Had Nathan entered*Flatland* (Abbott 1884)?] Such intuitions are the result of what Goldenberg et al. ( 1998) refer to as “visual thinking.”

Nathan's use of the dynamic drag feature of this type of computer tool illustrates how such dynamic manipulations of geometric shapes can help young children abstract the essence of a shape from seeing what remains the same as they change the shape. In the case of the triangle, Nathan had abstracted the basic definition: a closed figure with three straight sides. Length and orientation of those sides was irrelevant as the shape remained a triangle no matter how he changed these aspects of the figure. Such dynamic manipulations help in the transition from the first to the second van Hiele level: from “looks like” to an awareness of the properties of a shape (Fuys et al. 1988). For Nathan, this was new mathematical knowledge.

Lehrer et al. ( 1998) found that children in early elementary school often used “mental morphing” as a justification of similarity between geometric figures. For instance a concave quadrilateral (“chevron”) was seen as similar to a triangle because “if you pull the bottom [of the chevron] down, you make it into this [the triangle]” (p. 142). That these researchers found such “natural” occurrences of mental transformations of figures by young children suggests that providing children with a medium in which they can actually carry out these dynamic transformations would be powerfully enabling (as it was for Nathan). It also suggests that young children naturally reason dynamically with spatial configurations as well as making static comparisons of similarity or congruence. The van Hiele ( 1986) research focused primarily on the static (“looks like”) comparisons of young children and did not take into account such dynamic transformations. The use of DGEs with school-age children brings about a need for research on dynamical theories of geometric knowledge.

At the secondary level dynamic geometry environments can (and should) completely transform the teaching and learning of mathematics. Dynamic geometry turns mathematics into a laboratory science rather than one dominated by computation and symbolic manipulation, as it has become in many of our secondary schools. As a laboratory science, mathematics becomes an investigation of interesting phenomena, and the role of the mathematics student becomes that of the scientist: observing, recording, manipulating, predicting, conjecturing and testing, and developing theory as explanations for the phenomena.

Laborde et al. ( 2006, citing Hoyles 1995), suggest that the process of decision-making and reflection in the interaction between manipulation and outcomes provide students “with hooks they need on which to hang their developing ideas” (p. 292). The software constrains students' actions in ways that require the teacher to conceptualize problems from a student's point of view and encourage students to conceptualize mathematics in new ways. As Balacheff and Sutherland ( 1994) point out, the teacher needs to understand the “domain of epistemological validity” of a dynamic geometry environment. This can be characterized by “the set of problems which can be posed in a reasonable way, the nature of the possible solutions it permits and the ones it excludes, the nature of its phenomenological interface and the related feedback, and the possible implication on the resulting students' conceptions” (p. 13).

*Geometry Turned On*(King and Schattschneider 1997) provides several examples of successful attempts by classroom teachers to integrate dynamic geometry software in their mathematics teaching in ways that generated new mathematics (for the students). Keyton ( 1997) provides an example that comes closest to that of learning mathematics as a laboratory science. In his Honors Geometry class (grade 9) he provided students with definitions of the eight basic quadrilaterals and some basic parts (e.g. diagonals and medians). He then gave them 3 weeks to explore these quadrilaterals using

*Sketchpad*. Students were encouraged to define new parts using their own terms and to develop theorems concerning these quadrilaterals and their parts. Keyton had used this activity with previous classes without the aid of dynamic geometry software. He states:

In previous years I had obtained an average of about four different theorems per student per day with about eight different theorems per class per day. At the end of the three-week period, students had produced about 125 theorems… In the first year with the use of

Sketchpad, the number of theorems increased to almost 20 per day for the class, with more than 300 theorems produced for the whole investigation. (p. 65)

Goldenberg and Cuoco ( 1998) offer a possible explanation for the phenomenal increase in theorems generated by Keyton's students when using*Sketchpad*. Dynamic geometry “allows the students to transgress their own tacit category boundaries without intending to do so, creating a kind of disequilibrium, which they must somehow resolve” (p. 357). They go on to reiterate a point made by de Villiers (1994 ,1998 cited in Goldenberg and Cuoco ), that “To learn the importance and purpose of careful definition, students must be afforded explicit opportunities to participate in definition-making themselves” (p. 357). Marrades and Gutiérrez ( 2000) found similar results in their studies of secondary school students using Cabri Géomètre in proof-oriented geometry classes. Hadas et al. ( 2000) also found that designing activities in dynamic geometry to cause surprise and uncertainty was effective in provoking proof on the part of their students.

Keyton's activity with quadrilaterals stays within the bounds of the traditional geometry curriculum, but affords students the opportunity to create their own mathematics within those bounds. Other educators have used dynamic geometry as a catalyst for reshaping the traditional curriculum and injecting “new” mathematics. Cuoco and Goldenberg ( 1997) see dynamic geometry as a bridge from Euclidean Geometry to Analysis. They advocate an approach to Euclidean geometry that relates back to the “Euclidean tradition of using proportional reasoning to think about real numbers in a way that developed intuitions about continuously changing phenomena” (p. 35). Such an approach involves locus problems, experiments with conic sections and mechanical devices (linkages, pin and string constructions) that give students experience with “moving points” and their paths.

Laborde et al. ( 2006) state that DGE “has provided access to mathematical ideas by allowing the bypassing of formal representation and access to dynamic graphing which is particularly important for the learning of geometry. … Just as digital technology provides means to by-pass formalism, it may also provide the means to transform the way formalism is put to use by students” (p. 284). DGEs allow for not only manipulations, but also macro-constructions, trace, and locus (Sträßer 2002). In DGE, geometric objects are constrained by their geometric properties (unlike paper-and-pencil sketches which can be distorted to fit expectations), similar to how physical objects are constrained by properties of physics when manipulated within the world. By observing properties of invariance simultaneously with manipulation of the object, there is potential to bridge the gap between experimental and theoretical mathematics as well as the transition from conjecturing to formalizing.

### 8.2.7 Summary of Students' Mathematical Knowledgein a Technological World

In this first section of the chapter we have attempted to define, deconstruct, and illustrate what we mean by “mathematical knowledge.” In this endeavor, we have examined ways in which both the nature and construction of mathematical knowledge have been influenced by the integration of digital technologies in mathematics teaching and learning in ways that create a new learning ecology. We emphasized the different aspects of procedural and conceptual knowledge that have been discussed in the literature, and attempted to illustrate how certain dynamic technologies can enhance the development of “proceptual” knowledge and bring out the operational aspects of mathematics rather than focus on the notational aspects. In particular, we described the use of tools that exhibit “cognitive fidelity” (certain microworlds) and those that embody “mathematical fidelity” (Zbiek et al. 2007), such as SimCalc and DGEs. These kinds of technologies have been used successfully by researchers and classroom teachers in varying contexts to promote the learning of new mathematical knowledge (for the learners). As success stories, they illustrate the complex interactions among students, teachers and mathematics, mediated through technology as depicted by our didactic tetrahedron (Fig. 8.1).

We now turn to a discussion of the mathematical knowledge (if any) that is necessary to comprehend how certain technologies function so that we may use them both sensibly and sceptically.

## 8.3 Mathematical Knowledge“Within” Technologies

Within the general public, it is a common myth that the computer is always right - a perception of its “mathematical fidelity.” The main message of this section is that this notion is simply wrong (Sträßer 1992, 2001a). After an initial discussion of this issue, we suggest some pedagogical consequences of the “wrong-doing” of certain technologies.

### 8.3.1 Numbers and Arithmetic

As every computing machine has finite storage, it is obvious that it cannot correctly represent irrational numbers or rational numbers with more digits than the storage can hold. This is one reason why it is advisable to stay in “algebraic mode” in Computer Algebra Systems (e.g.,

*Derive*) as long as possible.Most computing machines internally work with “floating point arithmetic,” often not in a base-ten system but in a base-two or base-16-system. Consequently, the machine is simply unable to correctly represent fractions as simple as 1/3. To give an example widely used, the program

*Excel*does not have more than 15 digits plus three digits to represent the exponent, restricting the interval*Excel*can cover from −9,99999 99999 9999E307 to 9,99999 99999 9999E307. Near zero, the “Microsoft knowledge base” offers 1E−307 as the smallest positive number, and −1E−307 as the biggest negative number. An explanation for the maximal exponent “307” (307 being a prime number) could not be found.

From these two pieces of information, it is clear that even elementary arithmetic has its limitations when it is done on a computer. When it comes to very small and very big numbers or to limits and irrational numbers, computer arithmetic has its limitations and can become “wrong.” As a consequence, one can never prove the divergence of the harmonic series*a*(*n*) = Σ (1 + 1/2 + … + 1/*n*) by adding partial sums on a computer; it is necessary to go back to symbolic mathematics.

### 8.3.2 CASand Problem Spotting

To avoid these pitfalls and problems, it has been suggested that students use algebra as long as possible when doing complicated calculations, hoping that the algebra postpones rounding errors as long as possible (Sträßer 2001a). For many of the problems related to computer-based arithmetic, this works fine. In Computer Algebra Systems (CAS) the problems normally start as soon as calculations go beyond the simple development of a formula. Spotting problem situations like reducing (*x* − 1)^{2}/(*x*^{2} − 1) to (*x* − 1)/(*x* + 1) will be correctly handled by most modern CAS programs (recalling that the reduction is invalid for*x* = 1). Without greater understanding about the inner mechanisms of automatic algebraic calculations, the problem of this type of manipulation of equations and formulae serves as a warning to check the algebraic reductions provided by a particular CAS program.

### 8.3.3 Geometry with Linear Algebra

The algebraic and arithmetical problems seem far away when considering Dynamic Geometry Environments (DGEs) like Cabri, Cinderella, Geometer's Sketchpad or other comparable software. In fact, this is also a wrong assumption. To our best knowledge, all of the DGEs internally rely on a representation of Geometry by means of a multi-dimensional linear algebra (in most cases, on real numbers; Cinderella relies explicitly on complex numbers in order to avoid singularities). The “tangent monster” (Sträßer 2001b) illustrates the limitations of such a system: The geometric problem of identifying a tangent to a circle is condensed to solving a quadratic equation where the determinant is zero (converging the two solutions of the equation into one). With the problems of computer-based arithmetic in exactly handling zero, it does not come as a surprise that DGEs have difficulties with the number of intersections of a straight line and a circle when the Geometry indicates they must be tangent (implying just one solution of the quadratic equation).

Relying on an internal algebraic representation of Geometry has additional unwanted consequences. They emerge in situations as simple as constructing an angle bisector twice: Construct the angle bisector SB of an angle ASC and then construct the angle bisector of ASB. If one drags the point C, in most DGEs the second angle bisector will jump as soon as the original angle ASC gets bigger than 180°. This action produces the same drawing when you move the point C around to coincide with the original angle, as most DGEs are not “continuous,” but “deterministic” (i.e., when you reproduce an initial location of a point after some dragging around, you will get the initial drawing). Cinderella is proud of avoiding this “jump,” offering continuous dragging. The negative aspect of this is that when dragging C around S in a full circle, you will end in a different position of the second angle bisector, making Cinderella a non-deterministic DGE (for the example of this concept and more consequences of these design features of DGEs, see Gawlick ( 2001), who asserts that DGEs have to make a choice between continuity and determinism). There are other trivial examples where continuous and deterministic DGEs differ, but the point here is that there seems to be a need for a design decision in DGEs, which implies a choice that a geometer would like to avoid.

For statistical software like*Fathom*, developed for educational use, or professionally used software like SPSS, the consequences of the inner representation of statistical models are not as well researched like they are for DGEs. There is some interesting work on how “random” the random numbers are - some software packages even tell the user how the “random” numbers are generated. Without going into details, it is clear that these programs internally do not roll an ideal dice. It may be worthwhile to research the issue of consequences for the simulation of stochastic situations.

### 8.3.4 Who Has to Know What About the Underlying Mathematical Assumptions and Processes of Spreadsheets, DGEs, Statistical Packagesand CAS?

This question has not previously been discussed in depth. The position put forward here is a preliminary and tentative one. It seems obvious that typical end-users of these software programs cannot be aware of all the underlying mathematical assumptions and processes when using the software (some details are even impossible to explain to a prototypic end-user), even in considering only the representation of numbers inside a computer. To really “understand” what is going on, one has to be aware of rational and irrational numbers, representation systems by different bases and a developed concept of limits. Only a thorough scientific analysis of phenomena may lead to a fuller understanding, research that is being undertaken by mathematicians in the field of numerical analysis. As a consequence, complete understanding by every end-user is simply not viable, nor desirable. The computer will not be a transparent machine, but will remain a black box (or at best a grey box, cf. Buchberger 1989).

What can be reasonably hoped for is a basic understanding of the inner representation of mathematics (e.g., numbers, equations, stochastics, graphical representations, and geometric figures) within a computer and a global awareness of problems related to the difference between conceptual and computational mathematics. At the very least, teachers of mathematics and computer science should know about these assumptions and processes and be able to react in an appropriate way when the phenomena discussed above occur. The difficulties of such a position should not be underestimated and are open to misuse by being taken as an excuse for not knowing what the machine actually has produced. We suggest such a position because one can prove that the computer is NOT always correct, but in fact makes “mistakes” if compared to theoretical mathematics. In other words, there is a limit to the level of “mathematical fidelity” (Zbiek et al. 2007) for any digital technology.

## 8.4 New Mathematical Practices

### 8.4.1 Link Between Knowledge and Practice

*mathematical practices*(p. 24):

Noting that expertise in mathematics, as in any field, involves more than knowledge, we propose an explicit focus on mathematical know-how - what mathematicians and mathematics users

do. We refer to these things they do asmathematical practices. Being able to justify claims, using symbolic notation efficiently, and making generalizations are examples of mathematical practices. Such practices are important in both learning and doing mathematics. Their absence can hamper mathematics learning.

In many cases, the use of technologies in schools has encouraged a closer relationship between mathematical knowledge and mathematical practice, providing learners with opportunities to experiment, visualize, and test emerging mathematical understandings. From the use of digital technologies, a new model of interaction between the student, the mathematical knowledge and the instrument emerges. Briefly, technological tools can be experimental instruments whereby ideas can be explored and relationships discovered (Duke and Pollard 2004; Hoyos 2006; Hoyos and Capponi 2000), developing greater flexibility in analysis of complex situations. Realistic data, too complex to be used previously, can be brought into the classroom to make mathematics learning more interesting, challenging and practical (Kor and Lim 2004, 2006). Kosheleva et al. ( 2006) demonstrate how use of state-of-the-art technology (Tablet PC) allowed future teachers in their study to significantly improve learning of mathematical content knowledge through exploration and utilization of technology within their practice teaching.

There is a huge overlap between what is activated in a brain by thinking about an activity and what is activated when you actually perform that activity. And so I think that for example imagining that a cube rotates in space is deeply rooted in the physical act of rotating cubes with your hand.

Boon ( 2006) provides descriptions of a variety of virtual activities that lead to new ideas on visualization in learning mathematics. Chiappini and Bottino ( 1999) allege that visualization allows students to access mathematical knowledge by integrating the symbolic re-constructive approach (the traditional teaching strategy) with the motor-perceptive approach that involves actions and perceptions and produces learning based on doing, touching, moving and seeing. However, in Kosheleva and Giron ( 2006) it was shown that students using virtual manipulatives often formed math ideas and approaches that were unexpected or unwanted by the teachers and the designers of these virtual manipulatives. For example, in the “Algebra Scales” activity from the National Library of Virtual Manipulatives (Utah State University 2007) children preferred the approaches that represented “shortcuts,” requiring less time to get the correct answer, thus circumventing the equation solving process for which the applet was designed.

It must be made clear, however, that it is not the technology itself that facilitates new knowledge and practice, but technology's affordances for development of tasks and processes that forge new pathways. Just as cases of innovative uses of technology have emerged, there are valid concerns and shortcomings in the ways that technology has been used. Tall ( 1989) expressed his concern that in the process of using technology, the “authority of the machine” might be an impediment to learning, especially in the early stages. He pointed out that students would lose some autonomy in the problem solving process if they ignored their common sense and followed the lead of the machine.

On the other hand, technology has been used “to motivate students to take on, more and more, the responsibility of mediator in their own mathematics learning” (Buteau and Muller 2006, p. 77). This responsibility can lead to engagement with different kinds of mathematical learning practices. For example, Drijvers and Doorman ( 1996) observed that the use of graphing calculators appeared to stimulate students' interest in participating in explorative activities. Farrell ( 1990) reported that students who used graphing calculators to learn mathematics were more active in the classroom. More group work, investigations, explorations, and problem solving were also observed among students using graphing calculators. Others, for instance, Dick and Shaughnessy ( 1988), noticed that there was a shift by teachers to less lecturing and more investigations being conducted by students. Dana-Picard and Kidron ( 2006), in their study of a computer algebra system (CAS) as an instigator to learn more mathematics, stated “… the implemented Mathematics has to be understood. In order to afford a real understanding of the process, the user has to learn new Mathematics. We called this occurrence a*motivating constraint* of the software” (p. 130). Although these examples are encouraging, it is important to look beyond using technology as a motivational tool and move towards using technology to deepen and extend mathematical learning. This is challenging when developing proficiency with the technology takes time before the technology can become an instrument. This working relationship (instrumental genesis, discussed in Chap. 2) develops through operationalizing the mathematics with the technology. Confrey (in Heid 2005) emphasized that an obstacle in implementing these kinds of mathematical practices lay in the difficulty that (1) the type of reasoning needed to grapple with complexity is not taught in mathematics courses; and (2) students typically do not “own” the problems they work on. Overcoming these difficulties requires attention to the complex interactions among students, teachers, tasks, and technologies.

### 8.4.2 Interactions Among Students, Teachers, Tasks, and Technologies: Shifts in Empowerment

A key change in students' mathematical practices spurred by technologies is the locus of control in a task. Technologies can often promote student engagement and command in key decision-making junctures during exploration. Capabilities are distributed between the student and the tool, with the user in charge of making decisions about when and how to use the tool (Heid 2005, p. 348). Control shifts more to the student in making decisions about how to utilize the technology in problems that do not “tell” which mathematics is needed up front (Heid 2005). Technologies allow students to check the validity of their answers and assess their own hypotheses. While engaging in explorative activities with the technological tools, students might encounter unexpected strategies that lead them further to ask new questions when working toward a solution (Makar and Confrey 2006).

Drijvers and Doorman ( 1996) assert that when students alternate between experimentation and reflection, mathematical concepts are strengthened. Artigue ( 2002) agrees with this assertion in her genesis of a reflection about instrumentation and the dialectics between technical and conceptual work. Olivero ( 2006) demonstrates that students with considerable dynamic geometry (DGE) experience and average mathematical background seem to use the software with more interactions and explorations, whereas students with stronger mathematical background and little experience with DGEs typically do not fully exploit the possibilities offered by DGEs.

*Technology as master*, where the student is subservient to the technology and the relationship is one of dependence. An example is where the technology is used as a timesaving device, but the student does not evaluate whether the output is accurate or useful.*Technology as servant*, where the technology is subservient to the student, typically used to replace pen and paper computation and used as a faster means to the output.*Technology as partner*, where the technology is used creatively to boost student empowerment, treating the technology almost as a surrogate human partner.*Technology as extension of self*, where users draw on their technological expertise as an integral part of their mathematical thinking.

An example of*technology as partner* can be found in the use of scientific probes and sensors (CBLs) to investigate problems. Probes and sensors connected to a computer or a graphics calculator offer opportunities for teachers to focus mathematics teaching on inquiring, understanding and reasoning instead of the drill and practice of routine problems typical in conventional instruction. Probeware open avenues for students to investigate and explore science in a mathematical setting. The data collected in a science experiment using the probeware can be stored and analyzed via the technological tools while conducting the experiment in real-time. The use of electronic probeware as a technological partner is capable of transforming mathematics into an interdisciplinary, authentic and participatory subject (Lyublinskaya 2004, 2006).

Sinclair ( 2003) noted the importance of the nature of the task in either promoting or discouraging students' engagement with exploration activities and geometric thinking in DGE. Laborde ( 2001) further distinguished four types of tasks used by teachers in DGE: (1) tasks for which the technology facilitates but does not change the task (e.g., measuring and producing figures); (2) tasks for which the technology facilitates exploration and analysis (e.g., identifying relationships through dragging); (3) tasks that can be done with paper-and-pencil, but in which new approaches can be taken using technology (e.g., a vector or transformational approach); and (4) tasks that cannot be posed without technology (e.g., reconstruct a given dynamic diagram by experimenting with it to identify its properties - the meaning of the task comes through dragging). For the first two types, the task is facilitated by the technology; for the second two, the task is changed by technology. The nature of the task turns into a modeling activity, where deductions are drawn from observations, and solution paths may differ from the mathematics the teacher intends. Kieran and Drijvers ( 2006), in their study of learning about equivalence, equality and equation in a CAS environment, report on the “intertwining of technique and theory in algebra learning in a CAS environment.” (p. 278). Their analysis revealed that the relation between students' theoretical thinking and the techniques they use for solving the tasks, and the confrontation of the CAS output with students' expectations were the two main issues. While this confrontation became one of the most powerful occasions for learning in the classroom, they assert that appropriate management of these complications by the teacher is a necessary precondition to foster learning.

The findings concerning appropriate management by the teacher in Kieran and Drijvers study suggest that tasks with technology should not be studied without careful attention to the classroom environment created by the teacher. Ruthven et al. ( 2005) found that teachers often restricted students' explorations in order to avoid meeting situations that did not align with the planned learning outcomes. For example, the teacher may structure students' experiences in Cabri to exploit the mathematical fidelity of the Cabri construction with respect to classic Euclidean Geometry (Mariotti 2006).

The focus initially was on the learner and his/her interactions with technology, giving rise to theoretical reflections about learning processes in mathematics by means of technology. The focus moved to the design of adequate tasks in order to meet some learning aims and then to the role of the teacher. The integration of technology into the everyday teacher practice became the object of investigation. Finally, the role of the features of software and technology design were also questioned and investigated in order to better understand how the appropriation of the technological environment by students could interfere with the learning of mathematics and how the teacher organizes students' work for managing this interaction between appropriation of the tool and learning. (p. 296)

Thus, the interactions among students, teachers, tasks, and technologies have now become the focus of research in the field. Laborde and her colleagues, however, argue that more research is needed to better understand links between students' instrumentation processes and their growth of mathematical knowledge.

In this section we have seen the emergence of another component that we need to encompass in our didactic tetrahedron: the nature of mathematical tasks. We shall address this additional aspect of the didactic situation in our concluding remarks for this chapter. In the next section we discuss the critical role of feedback that technologies can play in the development of new mathematical practices.

### 8.4.3 Role of Feedbackin Practice

Laborde et al. ( 2006) indicate that feedback through technology offers a great deal of opportunity for new ways of understanding mathematics; for example, feedback through DGEs and microworlds, generated by manipulating the environment and generalizing/abstracting through reflection on outcomes to actions. Such feedback creates a need to search for another solution if the feedback gives evidence that a solution is inadequate; it can also help students refine their thinking iteratively as they design (rather than at the end of the design process). “The software incorporating knowledge and reacting in a way consistent with theory impacts on the student's learning trajectory in the solving process” (p. 294). The effect of technology on students' learning trajectories is the focus of Chap. 9.

As described in Sect. 8.2, the use of DGEs encourages students to make conjectures and the feedback they get from both measuring and dragging elements of their constructions allows students to rapidly test these conjectures, and, thus, refine them in a recursive cycle of conjecture-test-new conjecture. Hollebrands ( 2007) distinguishes two different types of strategies employed in students' activities with the Geometer's Sketchpad: reactive and proactive. The critical difference between the two strategies is whether the choice of action is in response to what the computer produced (reactive) or in advance of what the student anticipates the computer is supposed to do (proactive). In either strategy, the feedback provided by the DGE is critical for the students' subsequent actions. Makar and Confrey ( 2006) found similar distinctions in ways that learners use and respond to feedback in dynamic statistical software (see Sect. 8.4.4.2).

*a*would be reflections of graphs associated with positive values of

*a*.

When they were not, she proceeded to unravel the mathematical situation, using supplemental lines and symbolic reasoning to explain why a sign change in the numerator of the fraction would not yield the mirror image for the graph. Her reaction to this technology-based surprise led to her deeper understanding of this function and to her enhanced ability to reason in other parameter explorations. (p. 41)

The difficulty faced by learners in this problem was to imagine the changes when the*x*-coordinate approached zero. Graphing calculators allow students to witness the oscillations around*x* = 0 as well as other properties of the graph such as symmetry. Changing the window parameters on the graphing calculator allows the student to capture different sizes and dimensions of the graphed functions. The “trace” command can be used to explore the functional values around*x* = 0 and enable students to make the deduction graphically that the limit is zero. These examples illustrate ways that feedback from students' interactions with technology can have a strong impact on their mathematical understandings and practices.

### 8.4.4 Example Technologies that Promote New Mathematical Practices

In this section we examine, in more detail, several examples of technologies that have promoted new mathematical practices on the part of students and/or teachers. We begin by revisiting dynamic geometry environments, with particular emphases on “dragging” and the new role of proof in DGEs. The second example examines the use of new technologies in the teaching and learning of statistics, with particular emphasis on the new dynamical statistics software programs that have recently made their way into pre-college classrooms. The third section looks at children's activities in robotics and digital game environments, and the potential of these activities to engage children in mathematical practices. We have already described (in previous sections) some of the new practices made possible by graphing and CAS-enabled calculators, and the use of probeware in conjunction with calculators and computers.

#### 8.4.4.1 New Mathematical Practicesin Dynamic GeometryEnvironments

(i)

wandering dragging, that is dragging (more or less) randomly to find some regularity or interesting configurations; (ii)lieu muet dragging, that means a certain locusCis built up empirically by dragging a (dragable) point P, in a way which preserves some regularity of certain figures. (p. 3)

They also describe a third modality:*dragging test* that is used to test a conjecture over all possible configurations. Their distinction between wandering dragging and the more focused lieu muet dragging and dragging tests are not unlike Hollebrands' (2007) description of reactive and proactive strategies in DGEs or Makar and Confrey's (2006)*wanderers* and*wonderers* in dynamic statistical software (below). These different types of dragging modalities can be thought of as new mathematical practices that have emerged in the context of dynamic geometry environments.

*Variational Dragging Scheme*that involves the following components:

- 1.
Create contrasting experiences by wandering dragging until a dimension of variation is identified.

- 2.
Fix a value (usually a position) for the chosen dimension of variation.

- 3.
Employ different dragging modalities/strategies to separate out critical feature(s) under the fixed value (i.e. a special case for the configuration).

- 4.
Simultaneously focusing, hence “reasoning,” on co-varying aspects during dragging. A preliminary conjecture is fused together.

- 5.
Attempt to generalize by a change to a different value for the chosen dimension of variation.

- 6.
Repeat steps 3 and 4 to find compromises or modifications (if necessary) to the conjecture proposed in step 4.

- 7.
Generalization by varying (via different dragging modalities) other dimensions of variation. (pp. 350–351).

Reaction alpha: ignoring the perturbation

Reaction beta: integrating the perturbation into the system by means of partial changes

Reaction gamma: the perturbation is overcome and loses its perturbing character. (p. 2)

It is only in the last stage (reaction gamma) that teachers make an adaptation in their teaching that truly integrates the technology, thus generating both new teaching and new learning practices. Hollebrands et al. ( 2007) reviewed approximately 200 research studies on the use of technology in secondary school geometry (about half of these studies involved DGEs). The following themes emerged from their review: the role of representation in the construction of geometrical knowledge and DGE diagrams, the design of tasks and the organization of the milieu, students' constructions within a computer environment, and the instrumental genesis and its relationship to construction of knowledge. (This latter theme was discussed in Chap. 2 of Theme C.) The importance of studies addressing the question of proof in a dynamic geometry environment was also a major topic in their review.

The primacy of “proof” as the ultimate mathematical practice has been accepted in the teaching and learning of geometry since the time of Euclid. The very nature of DGEs challenges this primacy of proof but also creates new roles for proof as a mathematical practice (de Villiers 1999, 2006). Hoyles and Healy ( 1999) indicate that explorations of geometrical concepts using DGEs help students to define and identify geometric properties, and the dependencies necessary for the development of a proof; however, they do not necessarily lead to the construction of a proof. Olive ( 2000a) provides an example of how the interplay between a dynamic geometric construction and the functional dependencies of the dynamic measurements obtained from that construction, did lead to the development of an algebraic proof. In a course for pre-service high school teachers, students found the dimensions of a rectangle with fixed perimeter that contained the largest possible area by constructing a dynamic rectangle in*Sketchpad*, the sum of whose sides was constrained by a fixed line segment. Their construction allowed them to change the base of the rectangle, which in turn, caused the sides to change length (in order to keep the perimeter fixed). By measuring the base of the rectangle and the resulting area, and then plotting a point in*Sketchpad* based on these dynamic measurements (base vs. area), they were able to construct the locus of the plotted point, which was a parabolic curve with a maximum. They discovered that their plotted point reached this maximum area when the rectangle appeared to be a square. However, the dynamic measures of base and height were not exactly the same when the plotted point appeared to be at its maximum. This discrepancy led to an interesting discussion, and a need to prove by algebraic means that the maximum area will be attained when the rectangle becomes a square. Thus, they made the transition from geometric conjecture to algebraic proof.

*before*a formal proof has been obtained. It is this conviction that propels mathematicians to seek a logical

*explanation*in the form of a formal proof. Having convinced themselves that something must be true through many examples and counter examples, they want to know

*why*it must be true. de Villiers ( 1999) suggests that it is this role of

*explanation*that can motivate students to generate a proof:

When students have already thoroughly investigated a geometric conjecture through continuous variation with dynamic software like Sketchpad, they have little need for further conviction. So verification serves as little or no motivation for doing a proof. However, I have found it relatively easy to solicit further curiosity by asking students

whythey think a particular result is true; that is, to challenge them to try andexplainit. (p. 8)

The group in Italy headed by Ferdinando Arzarello (Arzarello et al. 1998a) has conducted investigations of students' transitions from exploring to conjecturing and proving when working with Cabri. They applied a theoretical model that they had developed to analyse the transition to formal proofs in geometry (Arzarello et al. 1998b). They found that different modalities of dragging in Cabri (identified above) were crucial for determining a shift from exploration to a more formal approach. Their findings are consistent with the examples given in previous sections of this paper.

#### 8.4.4.2 Technologies that Encourage New Practices in Statistics

This judicious selection deprives students of the experience of data discovery. … What seems to us to be missing are data sets - especially large and highly multivariate data sets - that are ripe for exploration and conjecture driven by the students' intrigue, puzzlement, and desire for discovery. (p. 1).

The graphing calculator and statistical analysis packages designed specifically for*learning* statistics (e.g., Finzer 2007; Konold and Miller 2005; Hancock and Osterweil 2007) support the use of authentic and realistic data and therefore stimulate students towards exploratory activity. This allows for a shift in emphasis from a focus on graphs, calculations, and procedures taught in isolation for their own sake towards the active use of statistics as a tool to solve interesting problems. This move allows learners to focus on the context under study rather than on the statistical tools as the objects of study (Makar and Confrey 2007). Modeling software can support a better understanding of representation and form “a bond between the data and whatever mathematical model you are starting to make” (Finzer, in Heid 2005, p. 357).

Students can collect large sets of real life data for data analysis. They can download, sort, tabulate and store these data rapidly using CBLs or web-based technologies and thus avoid tedious compilations of data tables by hand. Students can manipulate the data with symbolic expressions, solve equations, analyze data, and graph functions to fit the data. They can switch between screens to observe the different representations of the data. They can make conjectures, test their hypotheses and check their estimations. These dynamic technologies allow learners to work flexibly as the rapid display of different graphical representations allows more time for students to explore larger data sets and make comparison between groups, thus making statistics more meaningful and interesting (Kor 2004, 2005). Students can experience and appreciate more the practices of statisticians when they run statistical tests on authentic data that they obtained. Bienkowski et al. ( 2005) allege that students who engage in investigative activities with data using technologies perform better than those who simply report data.

*Fathom*(Finzer 2007) can encourage users to simply wander through a data set aimlessly looking for an interesting pattern to jump out at them and then try to explain the outcome with anecdotal evidence - what Makar and Confrey ( 2006) call

*wanderers*; at the same time, the ease of creating graphs supports others to assess the validity of hunches generated from the “I wonder” type questions that the technology was meant to encourage (what Makar and Confrey 2006, term

*wonderers*). Rubin ( 2007) further worries that access to technology has not necessarily discouraged students from “over-believing” computers and accepting results calculated by software on blind faith.

This attitude can lead to students accepting implausible results. … [On the other hand], they may also have done everything right and be seeing a relatively unlikely result. In the end, there's a delicate balance to be struck here, given the uncertainty that is at the heart of statistical reasoning. We want students to question what the computer generates, but not to reject results simply because they are not within the expected probability. (pp. 29–30)

Overall, however, the newer technologies that focus on learning statistics discourage the “black box” mentality of previous statistical packages (Meletiou-Mavrotheris et al. 2007; Makar and Confrey 2004). If technologies can continue to encourage greater focus on the utility of statistics for solving problems over an emphasis on the statistics as an end in itself, there is potential to resolve the widely reported use of statistics in situations where they don't make sense (Pfannkuch et al. 2004).

#### 8.4.4.3 Children's Mathematical PracticesUsing Roboticsand Digital Games

The use of robotics in schools is a fairly recent phenomenon. Although the Turtle Geometry of Logo (Papert 1970) was initially developed as a control language for a physical, dome-shaped robot (dubbed the “turtle”), the expense of the physical device and control mechanisms in the late 1970s and early 1980s made the physical robot turtle prohibitive as a classroom-based learning tool. Mass production of similar control systems with small robotic devices for the toy market, have now made the use of robotics a possibility again in K-12 classrooms. Programming robotic vehicles to travel around obstacle courses, or navigate a specific route, while providing a fun, game-like context, has the potential for rich mathematical learning.

Although the potential of digital games as rich learning tools is widely recognized (Sanford 2006), this potential in schools has not yet materialized (Wijekumar et al. 2005). From a practical standpoint the majority of games released are not the kind of games that educators will find value in using as part of their teaching, and while a recent report (MacFarlane and Kirriemuir 2005) describes some of the issues reported by teachers, it also points to a pressing need to establish a better understanding of the value of games in school environments and the difficulties faced by teachers when using them (Sanford 2006). According to Wijekumar et al. ( 2005), it is still necessary to work on moving students from a game affordance in a computer environment to a mathematical learning situation in which they may use that affordance.

- 1.
With respect to an understanding of the angle concept in primary school, Hunscheidt and Koop ( 2006) introduced in the classroom a small robot on wheels programmed to move in centimeters, turn in degrees and wait in tenths of a second (p. 229). This artifact (named Pip) enabled the students to estimate and check distances and angles.

- 2.
In their work with eighth graders, Fernandes et al. ( 2006) used robots in order for their students to learn functions. In the context of being given two pictures of distance-time graphs that represented a robot's possible travel from a given starting point, students were to design programs for the robot to travel the represented routes. To begin, students made hypotheses about the routes represented in the two graphs, then discussed the possibilities of these situations, realizing that one of the graphs was not feasible (as the robot would have to be in different places at the same time) and finally understood that the graphs were not pictures of the robot's route but a representation of the relationship between time and distance of the robot's travel.

- 3.
Using another kind of virtual scenario, Kahn et al. ( 2006) designed a Space Travel Games Construction Kit (STGCK) to build a variety of games similar to

*Lunar Lander*(p. 261). They tested these STGCK with two student classes (one aged 11–12, and the other one aged 12–13) and a small group of three students aged 12–14. The results were that students developed understandings of Newton's Laws, showing engaged activity and active experimentation with the materials. In particular, students solved the different tasks posed using iterative strategies and repeatedly refined their strategy. Kahn et al. ( 2006) evidenced collaboration, competition and motivation as the most prevalent student activities. In addition, the authors came to realize that students could analyze and use the relationships hidden in the programming code as an integral part of the game, when they gave students easy access to the programming code and provided situations where they would want to analyze and adjust that code. - 4.
The emergence of mathematical strategies and consecutive refining strategies were also some characteristics of the results obtained by the instrumentation of a computational board game named Domino (Raggi 2006) with two classes of seventh and eighth graders. In this context, symmetry was the underlying mathematical structure for the game. When playing against the computer, the winning strategy is to place your dominos symmetrically opposite the computer-opponent's placements. The computer game was introduced into the classroom as an exploratory material. Each student had to initially play against the computer (named Robi). The task asked of students was to find a way to beat Robi or, if Robi won, to try to explain why Robi was able to beat them. Two groups of seventh and eighth graders were involved in this experience in order to discover possible affordances of the computer game for helping students learn symmetry (Rodriguez 2007).

The purpose of the Domino game (Raggi 2006) is the search for winning strategies that allow the winner to activate the last two consecutive squares on the game board. The game is immersed with symmetry, yet this (mathematical) structure that the game is intended to foment defines a potential organization that the children concretize in different ways once they are engaged in the task (Saxe and Bermudez 1996). For example, a result of seventh graders playing the game against the computer was a rapid turn toward a different winning strategy, one which consisted of trying to leave blank spaces, counting out how many were necessary according to which turn they had. Concerning eighth grade students, it was observed that when they used a strategy that they believed to be a winner, they continued to use it and perfected it as long as it was functional. Moreover, an opponent who began to win was a cause for reflection and the reformulation or construction of a new winning strategy.

As stated at the beginning of this section, the major question to be answered with respect to new mathematical practices that might develop from either the use of robotics or digital games in the school context, is the movement from the game context to mathematical problem solving situations. To connect to another new or learned context, David Shaffer proposes the notion of epistemic frames as “ways of looking at the world associated with the ways of knowing of a particular community” (cited in Sanford 2006, p. 13). Shaffer's epistemic frames can be regarded as a tool for building accounts of students' use of experience that was gained in one context and applied within another different context. According to Sanford ( 2006), “building on this concept will contribute to an attempt to build an understanding of the ways in which knowledge may be transferred from the game to other domains” (p. 13). The heuristics of students performing in the competitive situations that the mentioned games created corresponded to general action patterns for solving mathematics or science problems (cf. Polya 1945). Nonetheless, the potential of this type of psychological instrument (epistemic frames, Shaffer 2006) to learning specific mathematical topics is still to be determined, for example how it is related with solving specific mathematical problems.

### 8.4.5 Summary of New Mathematical PracticesMade Possible with Technology

We opened this section with findings that suggest that the link between mathematical practices and mathematical knowledge is strengthened in didactical situations that involve effective uses of technology. A major affordance of technology is how it can be used to help students visualize abstract mathematical concepts. Students can model, experiment, and test their emerging mathematical understandings using dynamic visualization software in many mathematical domains. There is a risk, however, that students (and teachers) may relinquish their mathematical authority to the computing machine (see Sect. 8.3 for the inherent danger in relinquishing this authority to machines that are mathematically limited). We emphasized how the technology could be used to motivate students to mediate their own learning, and how it has brought about a shift in teaching practices from lecturing to student-centered investigations.

In Sect. 8.4.2 we focused on ways in which the interactions among students, teachers, tasks and technology have the potential to bring about a shift in empowerment in the didactic situation. We introduced the need to pay particular attention to the design of the mathematical tasks in order to avoid students perceiving the role of the technology as their master rather than their servant or partner. Ultimately, we would like to see students use technology as an extension of themselves (Galbraith et al. 2001; Geiger 2006). The focus here is on where the locus of control lies in a mathematical task. Technology can be used to shift that locus of control towards the students and, thus, empower the students to take more responsibility for their own learning.

Several researchers have focused on the importance of task design (e.g. Sinclair 2003; Laborde 2001) in technological environments. They argue for designing tasks that are transformed by the technology, leading to new mathematical practices (e.g. modeling real-life phenomena, making deductions based on observations), rather than tasks that could be just as easily completed without the technology. One possible outcome with such tasks, however, is that students may engage with mathematics that the teacher did not intend (and with which s/he may not feel competent). The role of the teacher becomes critical in managing these rich didactical situations involving technology. The teacher can attempt to constrain the situation so that students engage with the intended mathematics, or they can be more open and willing to go where the students' investigations lead them.

The nature of different software tools also has a constraining effect on the possible mathematical practices. When computers were first introduced into the mathematics classroom, their use was primarily for teaching programming. With the development of the mouse interface and dynamic visualization software, the advocacy for programming has diminished in favor of what Laborde et al. ( 2006) call “expressive tools.” While programming tools (such as Logo) support the link between students' actions and symbolic representations (programming code), expressive tools (such as DGEs) assist students in the move from action and visualization to conjectures and reasoning. This shift towards expressive tools has brought about a shift in the focus of research on the interactions among students, teachers, tasks and technology.

Research on the role of feedback provided by technological tools suggests that learning is most likely to occur when the feedback is unexpected. Feedback provided by computational tools (such as CAS) can shift the focus of the student from micro-procedures (that the tool performs) towards macro-procedures that involve higher-level cognitive processes. New solution methods are made possible by the graphical feedback provided by graphing calculators and graphing software. For example, the ZOOM feature on most calculators can provide students with visual solutions to the limits of functions at critical points.

In the last part of this section on mathematical practices we examined examples of several different technologies that have been used successfully to generate new mathematical practices. We revisited the research on DGEs from the perspective of new mathematical practices, emphasizing the important aspect of the different dragging modalities and the utilization schemes that students could develop through use of these different dragging modalities. The introduction of DGEs into the didactical situation often created perturbations for the teacher. When teachers overcame these perturbations (rather than ignoring them) they made adaptations in their teaching that more authentically integrated the technology. The use of DGE also brings about new approaches to proof in geometry and an increased emphasis on the role of proof as explanation rather than only verification. Likewise, the use of dynamical statistics software has made it possible for students to work with large, authentic data sets, which they can download or generate through their own experiments. The ease with which students can represent, explore and manipulate data with these tools has brought about a shift in focus from studying statistical processes for their own sake towards the active use of statistics as a tool to solve interesting problems.

We concluded this section with a look at the introduction of robotics and digital games as contexts for learning mathematics. While the potential of these contexts as rich learning situations has been recognized, this potential has not yet been realized in mathematics classrooms. Teachers do not yet see the value of digital games as learning tools. The problem for teachers is finding ways to move students from a game affordance to a mathematical learning situation in which they may use that affordance. Several researchers have suggested that the use of Shaffer's epistemic frames (Sanford 2006) as a theoretical tool could help teachers organize such movement.

## 8.5 Final Words: An Adaptation of Our Didactical Tetrahedron

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