Abstract: This paper reports from a case study with teachers at two schools in Norway participating in developmental projects aiming for inquiry communities in mathematics teaching and learning. In the reported case study, the teachers participated in one of the developmental projects focusing on implementation and use of computer software in mathematics teaching. I study teachers’ initial orchestration of dynamic geometry software (DGS) in mathematics teaching at lower secondary school. By utilising the notion of ‘instrumental orchestration’ from the theoretical perspective known as the ‘instrumental approach’ (Drijvers et al., in Educ Stud Math 75:213–234, 2010; Trouche, in Int J Comput Math Learn 9:281–307, 2004), I examine how teachers in their initial teaching with DGS empower students’ mathematics learning with the DGS software. According to this perspective, it involves teachers’ orchestration of two interrelated processes instrumentation and instrumentalisation. Analytical findings indicate that a difference in teachers’ empowerment is evident and consequences for students’ opportunities to engage with mathematics represented by the DGS are presented.

Abstract: In an attempt to study the emerging questions about design of mathematical tasks that could support the solving of challenging problems, we designed two settings of interactive diagrams that share an example represented as an animation of multi-process motion but differ in their organizational functions. The interactive settings, each comprising of one task, were designed to support investigation in which the embedded animated example was expected to offer representative and general views. We analyzed the inquiry processes of 13- and 14-year-old interviewees and hypothesized whether these cases could represent more general considerations for the design of animated examples within a setting of interactive diagrams. We argue that (a) self-controlling the number of representations, (b) self-controlling the appearance of simultaneous information, (c) flexibly using links between dynamic and static representations, and (d) establishing links or completing partial links between representations are valuable inquiry processes in solving a problem that includes animated examples to enable viewing the representativeness of a given example.

Abstract: Seeking to contribute to our understanding of the role of educational technology in mathematical learning, this paper takes a socio-genetic approach to tracing the ways technology becomes part of classroom mathematical activity. It illuminates the reflexive processes of inscription, translation and re-inscription as technologies evolve by examining the development and classroom use of Texas Instruments’ TI-Nspire™. To investigate the development and use of TI-Nspire, research from the field of Science and Technology Studies is drawn on that provides insights into the relationship between development, technology, and users while avoiding essentialist positions that obscure either technological or human aspects of the relationship. The findings show that rather than being a linear process where the technology is passed from developer to teacher to student, the development and use of TI-Nspire involves multiple feedback loops with constant reconfiguration. These loops occur at several levels as teachers and students integrate the technology into their mathematical activity and these reconfigurations feed into new versions of the technology.

Abstract: This paper presents the development process of a praxeology (theory-of-practice) for supporting the teaching of proofs in a CAS environment. The characteristics of the praxeology were elaborated within the frame of a professional development course for teaching analytic geometry with CAS. The theoretical framework draws on Chevallard’s anthropological approach to the didactics of mathematics and Duval’s analysis of transformations within and between registers of semiotic representations. The teachers (n = 43) were asked (a) to draw conjectures regarding unfamiliar behavior of tangents to hyperbola, before and after exploration using given slider bars; and (b) to prove their conjectures after being exposed to the algebraic expressions underlying the slider bars. The teachers were also asked twice, before and after (b), to rate the need to ask students for an algebraic proof in similar tasks. Three types of proofs are presented in an increasing order of the level of mathematical maturity exhibited in each proof. Based on results coming from the empirical study, we propose a praxeology for preparing teachers to teach proofs consisting of Task design, Techniques, and Didactical discourse.

Abstract: called the Golden Ratio.The goal of this article is to demonstrate how the integration of multiple software toolsinto a familiar educational context of Fibonacci numbers allows for the discovery of newmathematical knowledge. It is a snapshot of the authors’ recent work (Abramovich andLeonov 2008, 2009a, b) on the development of technology-motivated methods of teachingtopics in discrete mathematics at the university level, including programs for secondarymathematics teachers (referred to below as teachers). The snapshot highlights the potentialof the joint use of a spreadsheet and computer algebra systems for the discovery of cycles

Abstract: This issue marks a new phase in the evolution of this journal. As you can see the journal has a new title, Technology, Knowledge and Learning: Learning mathematics, science and the arts in the context of digital technologies. Despite the dramatic change to the title, the story is one of continuity, rather than discontinuity. For the last several years, our executive editors have sought to broaden the mission of the journal; most centrally, we have tried to move beyond a focus on mathematics to include a wider range of disciplines, especially science and the arts. Although it was easy for us to change our own conception of the mission of the journal, it has been quite difficult to alter how the mission is perceived by the wider world. We thus decided that we needed to send a stronger signal, and that a change to the title of the journal was the best way this could be accomplished. Thus, in the minds of our editors, the core mission of our journal has not changed, just how we communicate this mission to the outside world. It is worth taking a moment to reiterate what that core is, and what makes TKL unique as a journal. Our new title mentions technology, knowledge, and learning. But TKL is not an ‘‘educational technology’’ journal in the usual sense. Part of the difference is that we are interested in uses of technology that push the envelope of what is possible. But, more than that, our core mission is grounded in the belief that novel technologies do not just provide a means of teaching the usual subject matter in a manner that is more effective, entertaining, or efficient. Rather, new technologies hold the potential to radically reshape what is taught and learned. That is why the words ‘‘technology,’’ ‘‘knowledge,’’ and ‘‘learning’’ together make up our title. It is not because the use of technology can lead to better knowledge and learning. It is because all three are tightly related and they change together. The bread and butter of TKL will always be articles that explore these tight interconnections. This issue also marks a change in leadership of the journal. Uri Wilensky is stepping down as Editor-in-Chief after 6 years of leading the journal, and I will be taking his place. In addition, Andrea diSessa is stepping down as an executive editor. But here again the story is one of continuity rather than discontinuity. Aside from Andrea diSessa, the other

Abstract: This installment of the computational diversions column introduces a new game (at least I think it’s new, and original–I haven’t seen it anywhere before). The game is called HullGrams, which is intended to suggest a blend of the classic mathematical pastime of tangrams with the geometric notion of a convex hull. By way of preface–before we get to the rules of HullGrams–let’s begin with its inspiration in the puzzle of tangrams. Many readers will be familiar with tangrams; but for those who have never seen the puzzle, books such as (Crawford, 2002) and (Read, 1965) are recommended. Martin Gardner, in his Scientific American column, discussed the pastime and researched its history (Gardner (1988), chapters 3–4); that history, by the way, is resolutely less romantic than the fable originally spun by the larger-than-life American ‘‘puzzle king’’ Sam Loyd, who did not invent tangrams but popularized it early in the twentieth century. Briefly, the basic idea of the tangram puzzle is that we are provided with a set of seven geometric pieces: five of these are isosceles right triangles (two large, two small, and one medium-sized), one is a square, and one a parallelogram, and all angles within the shapes are multiples of 45 degrees. By placing the seven shapes flat on a plane in different arrangements, we can create an astonishing range of composite shapes. The essential idea of the tangram puzzle is conveyed in Fig. 1, which shows at left a photograph of my own set of plastic pieces, arranged to form a square. A typical tangram puzzle will begin with a solid black silhouette (think of this as the ‘‘goal shape’’), and the job of the player is to arrange the pieces so that their overall silhouette matches the goal shape. In Fig. 1, then, the simple black square shown at right would thus be the goal shape for the arrangement at left. I could spend much more time on tangrams, but there is already an extensive literature; and my hope is that by now you’ve gotten the idea, even if this is your first encounter with the puzzle. For those who have spent time with tangrams, you will probably have observed that in fact most goal shapes are pretty easy to match. The truly tough puzzles are the ones with relatively simple silhouettes, like the square in Fig. 1; but most goal shapes are more challenging in their original composition than they are in their solution. That is, it’s hard to

Abstract: Puzzles are an odd form of literature. For the most part, puzzle-writers—even the best of them—simply aren’t concerned with narrative, plot, character development, or even simple plausibility. Take, for example, the ‘‘island of truth-tellers and liars’’ found in so many logic puzzles: what kind of a place would really and truly be populated with (exclusive) truth-tellers and liars, anyhow? Could such people exist? (Even my best friends aren’t exclusive truth-tellers, and even my least favorite political figures aren’t exclusive liars.) And squeezed together on an island, no less—wouldn’t that lead to civil war? Or the farmer crossing the river—what’s the man doing walking around with both a fox and a goose? Maybe he should just invest in a muzzle for the fox? I was thinking about these matters the other day when browsing a fantastic puzzle book by Peter Winkler, entitled Mathematical Puzzles: a Connoisseur’s Collection (Winkler 2004). The book is a gem, and (among the relatively simpler challenges) it includes the following:

Abstract: Fans of ‘‘mathematical games’’ (I’m one) often end up with a small library of books on the subject, collectively containing descriptions of a vast trove of games. There are board games, or informal pencil-and-paper games, with applications to (e.g.) graph theory, number theory, combinatorics, logic, and many other branches of mathematics. In fact, the range and variety of mathematical games is so wide—and so rich in content—that one could plausibly base a pretty good introductory mathematics curriculum on these pastimes. Perhaps a few of the more advanced subjects would be underrepresented in this exercise (real analysis? differential geometry?); but a typical library of mathematical games provides, overall, a broad portrait of mathematics in general. Now, coverage of mathematics is one thing; but does our library provide reasonable coverage of what we mean by games? Interestingly, there are certain important types of games that tend to be near-invisible in recreational mathematics. For one thing, almost all the examples in our library are either solitaire games (which by some definitions of the term would hardly qualify as ‘‘games’’ at all), or two-person games. Think of the standards: tic-tac-toe, Nim, Sim, Sprouts, chess, Go, and so forth. Occasionally there are threeor four-person variations of these games, but for the most part we’re talking about n players, where n is a small positive integer. (Perhaps, if we program a computer to play a mathematical game against itself, we could say that n = 0 in that case?) Not all games are like this. After all, baseball is a game, isn’t it? You need to round up 18 people at the very least to play a weekend-afternoon informal game of baseball. Even a pick-up basketball game will usually have at least six players; touch football would require a dozen or so. And how about children’s favorites, such as musical chairs? Starting a game of musical chairs with only two or three kids on hand seems hardly worth the effort. Other children’s classics—Tag, Duck-Duck-Goose, Capture the Flag—are best played with numbers ranging from a half-dozen (at the low end) to twenty or more. For older children and adults, there are games such as charades, or twenty questions, and these are typically played in moderate-sized groups (three would be a bit small for these games). In other

Abstract: Anyone knowledgeable about cubics knows that they are symmetrical by rotation through 180 about their inflection point. Slightly less well known is that the tangent to a cubic at the midpoint of two of the roots, passes through the third root (see Horwitz undated). Indeed, Aude (1940) used this property in reverse to locate the real midpoint of a pair of complex roots of a cubic. Arne Amdal (private communication) reported the observation as arising spontaneously from a student in his high-school in Norway using dynamic geometry to explore cubics. Kaye Stacey (private communication April 2002) working with colleagues also found some of the results to be recorded here.

Abstract: In this paper, we describe the design and implementation of computer programming activities aimed at introducing young students (9–13 years old) to the idea of infinity, and in particular, to the cardinality of infinite sets. This research was part of the WebLabs project where students from several European countries explored topics in mathematics and science by building computational models and programs, which they shared and discussed. We focus on a subset of the work in which students explored concepts of cardinality of infinite sets by interpreting and constructing computer programs in ToonTalk, a programming language and environment that is especially well-suited for young students. Our hypothesis is that via carefully designed computational explorations within an appropriately constructed medium, infinity can be approached in a learnable way that does not sacrifice the rigour necessary for mathematical understanding of the concept, and at the same time contributes to introducing the real spirit of mathematics to the school classroom.

Abstract: Integrating technology in school mathematics has become more and more common. The teacher is a key person in integrating technology into everyday practice. To understand teacher practice in a technological environment, this study proposes using two theoretical perspectives: the theory of technological pedagogical content knowledge to analyze teachers’ knowledge, and instrumental orchestration to analyze teachers’ actions. Applying this dual perspective to one teacher’s practice can shed light on the complexities faced by a teacher who integrates technology in her practice.

Abstract: Radical constructivists advocate discovery-based pedagogical regimes that enable students to incrementally and continuously adapt their cognitive structures to the instrumented cultural environment. Some sociocultural theorists, however, maintain that learning implies discontinuity in conceptual development, because novices must appropriate expert analyses that are schematically incommensurate with their naive views. Adopting a conciliatory, dialectical perspective, we concur that naive and analytic schemes are operationally distinct and that cultural–historical artifacts are instrumental in schematic reconfiguration yet argue that students can be steered to bootstrap this reconfiguration in situ; moreover, students can do so without any direct modeling from persons fluent in the situated use of the artifacts. To support the plausibility of this mediated-discovery hypothesis, we present and analyze vignettes selected from empirical data gathered in a conjecture-driven design-based research study investigating the microgenesis of proportional reasoning through guided engagement in technology-based embodied interaction. 22 Grade 4–6 students participated in individual or paired semi-structured tutorial clinical interviews, in which they were tasked to remote-control the location of virtual objects on a computer display monitor so as to elicit a target feedback of making the screen green. The screen would be green only when the objects were manipulated on the screen in accord with a “mystery” rule. Once the participants had developed and articulated a successful manipulation strategy, we interpolated various symbolic artifacts onto the problem space, such as a Cartesian grid. Participants appropriated the artifacts as strategic or discursive means of accomplishing their goals. Yet, so doing, they found themselves attending to and engaging certain other embedded affordances in these artifacts that they had not initially noticed yet were supporting performance subgoals. Consequently, their operation schemas were surreptitiously modulated or reconfigured—they saw the situation anew and, moreover, acknowledged their emergent strategies as enabling advantageous interaction. We propose to characterize this two-step guided re-invention process as: (a) hooking—engaging an artifact as an enabling, enactive, enhancing, evaluative, or explanatory means of effecting and elaborating a current strategy; and (b) shifting—tacitly reconfiguring current strategy in response to the hooked artifact’s emergent affordances that are disclosed only through actively engaging the artifact. Looking closely at two cases and surveying others, we delineate mediated interaction factors enabling or impeding hook-and-shift learning. The apparent cognitive–pedagogical utility of these behaviors suggests that this ontological innovation could inform the development of a heuristic design principle for deliberately fostering similar learning experiences.