1 Introduction

This paper develops a line of research on the incorporation of computer-based tools and resources into the mainstream practice of mathematics teaching in secondary schools. This line began by building a generic model of what teachers regard as successful practice (Ruthven & Hennessy, 2002, 2003), drawing on decontextualised accounts elicited through group interviews with mathematics departments. A subsequent project has sought out and examined professionally well-regarded mainstream practice in secondary mathematics and science teaching, drawing on more strongly contextualised accounts of specific instances of particular types of technology-supported practice, supported by examination of actual classroom events. This paper reports a study of teaching practice incorporating use of graphing software; another study has examined dynamic geometry (Ruthven, Hennessy & Deaney, 2005, 2008).

2 A practitioner model of the contribution of technology use to teaching mathematics

The earlier research addressed the broad question of how teachers—specifically mathematics teachers in English secondary schools—conceive the incorporation of computer-based tools and resources into their classroom lessons. Drawing on teachers’ accounts of successful practice, this work developed a ‘practitioner model’ (Ruthven & Hennessy, 2002) which was refined into a more compact form (Ruthven & Hennessy, 2003) in the light of cross-subject comparison (Ruthven, Hennessy & Brindley, 2004). The themes in this model highlight the contribution of digital tools and resources to:

  • Effecting working processes and improving production, notably by increasing the speed and efficiency of such processes, and improving the accuracy and presentation of results, so contributing to the pace and productivity of lessons;

  • Supporting processes of checking, trialling and refinement, notably with respect to checking and correcting elements of work, and testing and improving problem strategies and solutions;

  • Overcoming pupil difficulties and building assurance, notably by circumventing problems experienced by pupils when writing and drawing by hand, and easing correction of mistakes, so enhancing pupils’ sense of capability in their work;

  • Focusing on overarching issues and accentuating important features, notably by effecting subsidiary tasks to support attention to prime issues, and facilitating the clear organisation and vivid presentation of material;

  • Enhancing the variety and appeal of classroom activity, notably by varying the format of lessons and altering their ambience by introducing elements of play, fun and excitement and reducing the laboriousness of tasks;

  • Fostering pupil independence and peer exchange, notably by providing opportunities for pupils to exercise greater autonomy and responsibility, and to share expertise and provide mutual support.

Subsequent research has offered collateral support for the model. An independent Paris study (Caliskan-Dedeoglu, 2006; Lagrange & Caliskan-Dedeoglu, 2009) found that the original version of the model provided a useful template to describe teachers’ pedagogical rationales for the classroom use of dynamic geometry. However, when teachers were followed into the classroom, it became clear that these rationales sometimes proved difficult to realise in the lessons themselves. Teachers could be overly optimistic about the ease with which students would be able to use the software. Far from effecting working processes and overcoming pupil difficulties, computer mediation might actually impede student actions, with the teacher trying to retrieve the situation by acting primarily as a technical assistant. Likewise, students could encounter difficulties in relating the figure on the computer screen to its paper-and-pencil counterpart. Rather than accentuating important features, computer mediation might actually mask them. This serves to emphasise that while the model identifies the types of ‘normal desirable state’ which teachers associate with successful technology use, actually achieving such success depends on creating classroom conditions and pursuing teaching actions which establish and maintain these states (Brown & McIntyre, 1993). Certainly in a parallel Cambridge study of classroom practice in using dynamic geometry, teachers were found to have developed strategies to anticipate, avoid and overcome such potential obstacles (Ruthven et al., 2005). What these studies emphasise, then, is that the model represents a guiding ideal for teachers. To be able to realise this ideal in practice depends on teachers developing craft knowledge to underpin their desired classroom use of new technologies.

The term ‘craft knowledge’ refers to the largely reflex system of situated expertise which teachers develop, tailored to their professional role and embedded in their classroom practice (Brown & McIntyre, 1993; Leinhardt, 1988). Compared to the more decontextualised and rationalistic approach to characterising a ‘professional knowledge base for teaching’ in which the ‘representations of [subject] knowledge in teaching’ associated with ‘pedagogical content knowledge’ are highlighted (Wilson, Shulman & Richert, 1987), the craft perspective focuses on the functional organisation of a broader range of teacher knowledge to accomplish concrete professional tasks. In particular, it recognises that much innovation calls for adaptation of craft knowledge with respect to key structuring features of classroom practice such as working environment, resource system, activity format, curriculum script and time economy (Ruthven, 2007, 2008). The use of new technologies often involves changes in the working environment of lessons in terms of room location, physical layout and class organisation (Jenson & Rose, 2006), requiring modification of the classroom routines which enable lessons to flow smoothly (Leinhardt, Weidman & Hammond, 1987). Equally, while new technologies broaden the range of tools and resources available to support school mathematics, they present the challenge of building a coherent resource system (Amarel, 1983) of compatible elements that function in a complementary manner and which participants are capable of using effectively. Likewise, innovation may call for adaptation of the established repertoire of activity formats that frame the action and interaction of participants during particular types of classroom episode (Burns & Anderson, 1987; Burns & Lash, 1986), and combine to create prototypical activity structures or cycles for particular styles of lesson (Monaghan, 2004). Moreover, incorporating new tools and resources into lessons requires teachers to develop their curriculum script for a mathematical topic. This ‘script’—in the psychological sense—is an event-structured organisation of variant expectancies for the lesson and of alternative courses of action, forming a loosely ordered model of goals, resources and actions for teaching the topic (Leinhardt, Putnam, Stein & Baxter, 1991); it interweaves mathematical ideas to be developed, appropriate topic-related tasks to be undertaken, suitable activity formats to be used and potential student difficulties to be anticipated, guiding the teacher in formulating a particular lesson agenda, and in enacting it in a flexible and responsive way. Finally, teachers operate within a time economy in which they seek to improve the ‘rate’ at which the physical time available for classroom activity is converted into a ‘didactic time’ measured in terms of the advance of knowledge (Assude, 2005).

3 Research on teaching practices incorporating use of graphing technology

Software for mathematical graphing on the coordinate plane was amongst the earliest computer applications developed for professional and educational use. From the start, mathematics educators have seen such software as having more than just a pragmatic, computation-effecting function in translating symbolic expressions (such as coordinates and equations) into graphic images (such as points, lines and curves); but as also having an epistemic, concept-building function, through supporting exploration of symbolic–graphic relationships. In an early survey, Fey (1989, p. 247) noted that “suggestions of ways to use such graphs [as pictures of algebra] have appeared in many places and the software tools available to facilitate graphing are really quite versatile and easy to use”. He indicated that, for teaching purposes, graphing software was typically first used to display series of graphs as a means of revealing the patterns associated with various rule types and the significance of parameters within each type. He reported that it was then common to ask students to generate rules to match a given graph or to fit given points; here Fey referred to the success of a piece of educational software which placed such tasks within a game format.

Over the course of the 1990s, several pioneering studies investigated aspects of teacher thinking and classroom practice in using graphic calculators at upper secondary level. The study with concerns most similar to those of this paper was conducted by Simmt (1997) who examined the beliefs and practices of teachers introducing the use of graphing calculators into a unit on quadratic functions. The machines were used primarily to generate patterns of graphic images; most commonly by students themselves; and in the majority of cases through a guided-discovery approach which some teachers reported as made possible only through use of the technology. The reasons that the teachers gave for using graphing technology related principally to saving lesson time and generating more examples, and to increasing instructional variety and enhancing student motivation, corresponding respectively to the Effecting working processes and improving production and Enhancing the variety and appeal of classroom activity themes of the practitioner model. In the majority of cases, teachers considered the calculator valuable as a tool for checking students’ sketches or manipulations, relating to the theme of Supporting processes of checking (but apparently with little trialling and refinement involved). One teacher mentioned aspects of Overcoming pupil difficulties and building assurance in reporting that using the calculator increased students’ confidence in the accuracy of their graphs, and that this enabled them to work with less dependence on the teacher, in line with Fostering pupil independence (but without reference to peer exchange). Finally, some teachers expressed concern about limitations of calculator graphing in respect of the frequent need to adjust the graphing window and to interpret approximate coordinate values, so qualifying the perceived contribution of this technology to an ideal of Focusing on overarching issues and accentuating important features.

Other studies throw more oblique light on the concerns of this paper. Farrell (1996) studied the classroom practice of teachers nearing the end of their first year of involvement in a development project in which they were working with teaching materials to which use of graphing technology was integral. The study concluded—with appropriate caution—that technology use helped to create conditions under which it became possible—but not inevitable—for formats for classroom activity to shift towards less teacher exposition and more student investigation and group work, providing greater scope for both teacher and students to take on explainer, consultant and co-investigator roles. Doerr and Zangor (2000) studied the practice of a teacher who was very experienced in graphing calculator use. The main findings relate to typical modes of classroom calculator use, as modelled and encouraged by the teacher, and grounded in her broader instructional approach: as computational tool, transformational tool, data collection and analysis tool, visualising tool and checking tool. Although this study has a strong pedagogical focus, the idea of graphing calculators as media for developing and deploying variant mathematical strategies also emerges as important: strategies differing markedly from those already recognised, and strongly dependent on distinctive affordances of the new tools, such as solving equations graphically or numerically rather than analytically. This idea is absent from, or marginal in, the other studies discussed here (for example, being generally rejected by the teachers participating in Simmt’s study).

Over the last few years, the use of graphing technology has become more widespread and commonplace, notably moving down from relatively advanced courses to earlier levels of secondary education. However, the teacher thinking and classroom practice associated with wider use appear to have received little attention from researchers. Moreover, few studies at any level have been conducted outside North America, and few have examined the use of graphing software as against graphic calculators. The study to be reported in this paper addresses all these aspects of underinvestigation in the field. It complements Godwin and Sutherland’s (2004) study which compared two lessons taught by teachers involved in an initiative to develop classroom use of new technologies. In one lesson, graphic calculators supported work on straight line graphs; in the other, graphing software provided a means of studying quadratic graphs. Describing the unfolding teaching sequences in each lesson, Godwin and Sutherland’s paper emphasises the way in which teachers played “a vital role in orchestrating and structuring classroom activities in such a way as to support students to focus attention on appropriate mathematical ideas” (p. 132) and the way in which this creates “an emergent and collective… community… in which knowledge construction converges to some acceptable ‘common knowledge’” (p. 150). In particular, they note how, in both lessons, teachers introduced critical attributes of the graph types through considering a succession of parameterised sub-families, with the graphing technologies used to give students the power to experiment with many members of each sub-family within a relatively short period of time. Our study complements this work by incorporating a stronger teacher perspective into the overall analysis of technology-supported classroom practice and by extending analysis of the specific handling of graph types to bring out the interaction between technological affordances, task designs and teaching actions.

4 Design and method for the study

In the first phase of the larger research project from which this study derives, recommendations from professional leaders and evaluations from school inspections were triangulated in order to identify subject departments in state-maintained schools that were regarded as successful both in terms of the general quality of their practice and their use of computer-based tools and resources within it. To ascertain what practitioners themselves regarded as successful practice, focus group interviews were then conducted with each of these subject departments (during the latter part of the 2002/2003 school year). In these interviews, teachers were invited to nominate and describe examples of successful practice involving use of computer-based tools and resources. Through this process, several practices were selected for more intensive investigation in the second phase of the research through conducting case studies (during the 2003/2004 school year).

From the focus group interviews with 11 mathematics departments, use of graphing technology was identified as a successful established practice, favourably mentioned in all the departments, and selected as a nominated example in seven, with graphing software generally preferred over graphic calculators. However, outside advanced courses, use of this technology still proved relatively infrequent, occurring on no more than a handful of occasions per school year with any particular class. The particular mathematical topics which were most frequently cited were linear equations (nominated in six departments and mentioned in four more) and quadratic equations (nominated in only one department but mentioned in four more, all of which nominated linear equations). The use of graphing technology to treat such equations also featured prominently in official curricular and pedagogical guidance (DfEE, 2001; DfES, 2003), and lessons of this type were cited favourably, both in reports of individual school inspections and in a national report on the impact on secondary mathematics of government ICT initiatives in schools (OfStEd, 2004; p. 7). A further index of the archetypicality of this practice was provided when one department (which had nominated it as an example) volunteered that for their forthcoming professional development session on ICT in mathematics teaching they had “specifically asked for it to be not drawing graphs and straight lines”!

Available project resources made it feasible to follow up only a limited number of cases of each practice. Given that there were relatively few differences in the forms of graphing practice that teachers described, and that graphing software (rather than graphic calculator) was clearly teachers’ preferred technology, we followed up two cases of this type which provided some degree of contrast in approach, located in schools in different regions of England. Other important considerations were that the teachers concerned had already provided quite full and thoughtful accounts during the focus group interviews, and that two lessons could be observed with each teacher, one on linear forms and the other on quadratic. Detailed observation records were made of each lesson, incorporating a transcript of the main episodes, integrated with further observational material including copies of other resources used and records of the graphs displayed. Post-lesson interviews were conducted with teachers after each observed session. These were organised around a standard sequence of printed cards asking teachers about their thoughts, first while preparing the lesson (what they wanted pupils to learn; how they expected use of the technology to help pupil learning), then looking back on the lesson (how well pupils learned; how well the technology helped pupil learning; important things that they were giving attention to and doing). At the end of the first interview, teachers were also asked about any pitfalls that they had experienced in using graphing technology. At the end of the second interview, they were invited to suggest any ways in which their approach differed between the two lessons. These interviews were audiotaped and transcribed.

The subsequent process of analysis was in three stages. First, the file for each lesson was analysed in the following ways, providing the basis for assembling a summary of the lesson, focusing specifically on the teaching practice and practitioner thinking associated with it. From the observation record, occasionally amplified by illuminating material from the interview transcript, a basic outline of the working environment, resource system, lesson agenda and activity structure of the lesson was compiled. From the interview transcript, occasionally amplified by illuminating material from the observation record, a narrative was constructed to set out the main lines of thinking that teachers reported as lying behind the lesson and emerging in response to it. Here, to provide a common core of organising constructs, the broad themes from the compact version of the earlier ‘practitioner model’ (Ruthven & Hennessy, 2003) were used as appropriate, but the narratives themselves drew directly on the source material, going beyond the earlier themes where necessary (Strauss & Corbin, 1994). These lesson summaries are presented in Boxes 1 to 4 and sketch the classroom practice observed in each lesson and the teacher thinking associated with it (with the indexing of segments used to facilitate later cross referencing from this main text). Second, an analysis was conducted across lessons and teachers, again organised in terms of the earlier practitioner themes and elaborating them where necessary, to produce a model of this technology-supported teaching practice, grounded in the teacher accounts and teacher actions captured in the evidence base. Third, sensitised by the earlier literature review and by wider theory in the field to aspects of the teaching practice which remained largely tacit in this model, some further key issues—concerned with instrumental induction, task design, teacher intervention and craft knowledge—were examined, drawing more reflectively on the case files to extend the earlier analysis.

5 A practitioner model of the contribution of graphing software to the teaching of algebraic forms

The themes from the earlier study provided a useful organising framework for synthesising the thinking reported by teachers in association with each lesson, making it possible to elaborate a practitioner model of the contribution of graphing software to the teaching of algebraic forms, grounded in the observation and interview data from this study.

Teacher accounts of all the lessons made reference to various aspects of the theme of Effecting working processes and improving production, suggesting, for example, that the software made it possible to produce graphs “extremely accurately and extremely quickly” [B1/7], making “doing the activity an awful lot easier and quicker and more efficient” [B2/8], so that—in terms of time economy—students could “move through everything at a much quicker pace” [A2/8], allowing a topic to be addressed in only a single lesson [A1/8; B1/6]. In the lower- and average-attaining classes, this also helped make tasks accessible to students who would have found “organisation and presentation challenging” [B1/8] and would “have really struggled” [A1/6], echoing aspects of Overcoming pupil difficulties and building assurance.

These factors also underpinned some aspects of the theme of Enhancing the variety and appeal of classroom activity, in terms of the use of graphing software making lessons less “laborious” [A1/6; B1/7; B2/7] and less dependent on pencil-and-paper work [A2/7; B1/8], and increasing the immediacy and interactivity of tasks [A1/5]. For the higher-attaining classes, the teachers also talked of the potential of using technology to make tasks more “challenging” [A2/5] or “demanding” [B2/7] in mathematical terms. With the lower-attaining class, technology was used to give students who “wanted to get the right thing in front of the class” [A1/7] the frisson of a very immediate and public test, by getting them to come out and check their proposal through using the software on the projected computer.

In relation to Fostering pupil independence and peer exchange, both teachers reported or were observed allotting short periods for playful exploration of the software by students, and for consequent sharing of discoveries [A1/15; A2/14; B1/12; B2/1]. Equally, for their higher-attaining classes, the teachers talked of using the technology to support “exploration” [A2/6] of more “open ended” tasks [B2/5], in which students “have their own control over the situation” [B2/6] and are “investigating, exploring, almost on their own” [A2/4]. Comparing her two lessons, Teacher A commented that she gave her higher-attaining class “more open-ended questions” to which they responded through “a text box on [their] graph to explain what differences [they had] seen”, whilst her lower-attaining class were asked “much more particular questions” and “had a sheet to put very definite answers on… to focus them in”. However, in the lower-attaining class [A1/14] as well as the higher-attaining classes [A2/12&13], when students came up with mathematical ideas going beyond the lesson agenda, she supported them in “go[ing] off at a tangent” [A2/12]. The contrast that Teacher B drew between the framing of tasks for her average- and higher-attaining classes was less strong: the former “had a very specific task”, whereas the latter “had to do a slightly more open-ended task”. She too was observed supporting students in going beyond the lesson agenda [B2/12] in a way which, as she pointed out, was only possible because of the availability of the graphing software.

Teachers’ encouragement of informal exploration of the graphing software, and their assistance to students using it to engage in mathematical speculation and experimentation beyond the lesson agenda, also evidences how they saw this technology as a means of Supporting processes of checking, trialling and refinement. In both her lessons (suggesting that this had become part of her curriculum script), Teacher A posed the same speculative question about lines sloping in an opposite way, leading to a similar trialling episode being inserted into a conventional investigation [A1/13] and an introductory review [A2/12]. Likewise, the ‘target practice’ tasks in both of Teacher B’s lessons (with the explicit linking of them [B2/3], indicating both a developing curriculum script, and an emerging activity format tailored to this type of topic) were conceived more broadly as examples of ‘trial and improvement’ [B1/13], dependent on feedback from the graphing software. With the younger class, this required the teacher to renegotiate norms with students who were hesitant about the legitimacy of trialling [B1/13]; with the older class, the socio-mathematical agenda had moved on to developing habits of prediction and reflection to scaffold trialling processes [B2/13].

Finally, in terms of the theme of Focusing on overarching issues and accentuating important features, the teachers talked of how use of graphing software helped students to “get to grips with” [B2/10], “get an idea of” [B1/9] or “see straight away” [A1/9] the effect of altering a coefficient in the equation on the properties of its graph. Likewise, the teachers highlighted particular software devices which facilitated apprehension of equation/graph matching [A1/10; B1/10], comparison of gradients [A1/10] and examination of limiting trends [A2/9]. Nevertheless, teacher management and guidance also played an important part in helping students to gain such insights. In relation to one of the ‘target practice’ tasks, for example, key actions of Teacher B included constraining the type of expression to be graphed [B1/11], drawing attention to the equivalence of expressions [B1/11] and pressing students to seek further equations so as to generate graphs which were “steeper or shallower or sloping in the other direction” [B1/4]. Likewise, in both her lessons, Teacher A reported actively checking, and if necessary developing, students’ understanding of the relationship between the equation of a graph and the coordinates of points lying on it [A1/11; A2/10], and prompting students to attend to the key mathematical properties which investigations aimed to establish [A1/12; A2/11].

6 The significance of instrumental induction, task design and teacher intervention

The discussion in the previous section has highlighted the crucial part played by teacher prestructuring of technology-based tasks and by teacher shaping of technology-and-task-mediated activity in realising the ideals of the practitioner model. This section develops these points to examine the ways in which teachers, and the wider sources that they drew on, contributed to the conduct of well functioning lessons.

Analysis of ‘institutional’ and ‘instrumental’ aspects of tool use (Artigue, 2002; Guin, Ruthven & Trouche, 2005; Ruthven, 2002) has developed in response to difficulties encountered with the educational use of sophisticated technologies designed for use by professional mathematicians. It provides a conceptual framework for analysing the process through which students (and indeed teachers) progressively appropriate a material artefact to create a mathematical instrument. Essentially, in the school context, this calls for development of an institutionalised order of tool-mediated activity and the induction of users into this order. Graphing software, however, has been explicitly designed for educational use. The teachers described the packages they were using as “instinctive” [B] and “user-friendly” [A&B]. They identified several aspects of the user interface which made the software readily accessible and interpretable by students (contrasting the software favourably with graphic calculators in many of these respects): the clearly labelled scales [A&B] and the gridlines in the background to assist comparison of gradients [A]; the ‘hand’ tool for dragging the image to view sections of the graph outside the original display [A]; the colour coding which associated particular equations with their graphs when several were displayed simultaneously [A&B]; the acceptability of expressions defined in the form x= as well as y=, and defined implicitly as well as explicitly [A]. Nevertheless, however much these graphing packages had been “designed to do things easily” [B], the teachers still played an important role in inducting and supporting students in use of the software for mathematical purposes. It was on this foundation that classroom realisation of the teachers’ conceptions of successful technology use depended.

Both teachers followed a dual approach to establishing a collective repertoire of computer graphing techniques. Prior to undertaking a classroom task involving graphing, unless the teachers had confidence that the core techniques required were already familiar to the class [B2/2], they introduced [A1/2; B1/2] or reviewed [A2/2] them. More serendipitously, they also allotted short periods to playful exploration of the software by students and to subsequent sharing of new possibilities [A1/15; A2/14; B1/12; B2/1]. Supporting and developing use of the software was also an important dimension of teacher interaction with students while they were working on tasks. In the observed lessons, teachers guided basic operation of the software, prompted strategic action with it and supported mathematical interpretation of its results. Teacher actions included: explaining how to enlarge a target point to make it more visible [B1], and how to enter x 2 in the equation editor [B2]; helping students to understand why the software had produced a horizontal line rather than the expected sloping one (as a result of entering \(y = 5 + 4\) rather than \(y = 5x + 4\)) [B1], or a straight line rather than the expected curve (as a result of entering \(y = x + 2^2 \) rather than \(y = \left( {x + 2} \right)^2 \)) [B2]; prompting students to drag the displayed image to expose more of a particular graph [A1], or to pursue the limiting trend of a graph [A2]; and prompting students to zoom out on the displayed image of \(0.00000009x^{2} + x + 1\) to test whether it was a straight line, then introducing the comparison with \(0x^{2} + x + 1\) [B2].

Likewise, the structuring of lesson tasks through prepared materials and teacher intervention was crucial in realising many of the benefits attributed to using the graphing software. The mathematical content of the prepared tasks used by teachers corresponded closely to examples suggested in official guidance, as illustrated by the extracts in Box 5. The prepared tasks used in both lessons by Teacher A followed the inductive format exemplified by official examples 1 and 2, as did the first prepared task used in the second lesson by Teacher B. Teacher B’s ‘target practice’ tasks can be viewed as a reframing of the format exemplified by official example 3, capitalising on the interactivity of the technology. Her reason for adopting this task format was to introduce a more exploratory style, breaking away from what she saw as the overly didactic style of the investigation genre [B2/5]. In the classroom, however, her intervention proved necessary to reframe the task in suitably didactic terms [B1/4]. This signals the crucial part that didactical structuring through task design and/or teacher intervention plays in Focusing on overarching issues and accentuating important features.

Essentially, whatever task format was adopted, the learning goal of these lessons was to induct students into an accepted mathematical organisation of the multimodal systems constituted by equations of the types \(y = mx + c\) or \(y = ax^2 + bx + c\) and their graphs. Achieving such an organisation depends on managing the double semiotic of the system through coordinating algebraic and geometric registers, while also managing its multi-dimensionality through isolating phenomena and controlling variables. The official example 4 (in Box 5) illustrates this (as do the example types and sets in Boxes 1–4). Each of the first three subtasks in this example isolates a geometric phenomenon and controls algebraic form accordingly, playing on a single parameter; only with the fourth subtask does some integration of these phenomena commence. Nevertheless, a breakdown in such management emerged in one quadratic investigation where students failed to formulate the intended property of the coefficient b from the family of forms selected to exemplify it [B2/11]. This example apart, however, the investigation tasks (and indeed the lesson expositions) employed by teachers were largely successful—through sequencing and patterning example types and sets—in providing a logical decomposition of the multimodal mathematical system under consideration. It is this didactical organisation of the topic which underpinned the use of graphing software to help students grasp the effect of altering a coefficient in the equation on the appearance of its graph. In the absence of such structuring, however, the ‘target practice’ tasks called for much higher levels of teacher mediation.

This puts in perspective, on one side, Teacher A’s suggestion that students were “investigating, exploring, almost on their own”, and, on the other, Teacher B’s (partial) renunciation of the “Do this, what do you notice? Do this, what do you notice?” style of the investigation genre. We have seen that the teachers were certainly aware of structuring their introductory expositions and student investigations to differing degrees for their two classes. But whatever the class, realising the benefits of using the graphing software to enhance the generation and comparison of multiple examples was dependent to a large extent on the intellectual organisation provided by other prepared resources and further teacher contributions. In the investigation tasks and introductory expositions, such organisation was introduced through carefully structured example sets. But because the ‘target practice’ tasks incorporated no comparable structuring, given students’ limited experience and knowledge of the mathematical topics in question, the organising function then fell predominantly on teacher prompting and questioning. This emphasises that graphing software and lesson tasks form a resource system, in which the technology’s contribution to supporting learning is powerfully conditioned by task structuring.

7 The adaptation of teaching practices and the development of craft knowledge

Earlier sections have highlighted the crucial part played by teacher structuring and shaping of technology-mediated activity in realising the ideals of the practitioner model. These sections have evidenced the adaptation of teaching practices and development of craft knowledge associated with teachers appropriating graphing software as an instrument for teaching and learning mathematics.

In terms of working environment, many of the aspects observed were not specific to graphing software. In three of the four lessons [A1; A2; B1], for example, technical difficulties arose which required some modification of normal working procedures. Likewise, in the changed working environment of the computer suite, teachers had to modify classroom routines, notably those concerned with managing the start of lessons, to include getting students seated appropriately, and their computer workstations and resources opened for use. Equally, adaptation was required to routines for securing the attention of students during periods of independent work, so as to efficiently make important points to the class as a whole [B1]:

I did a little [whole-class intervention during] the main activity… because if you didn’t do that you’d have repetitive questions. Which is something I’d do in a lesson anyway, so it’s just translating that to computers, and getting them to turn their monitors off and face the board… That’s something I’d be quite keen to get them into the habit of doing… Tearing them away from their computers… I find that very difficult because you know they’re not listening to you and they are missing out on something.

Developing a functional resource system incorporating the use of graphing software required teachers to extend their practice in several ways. As reported earlier, they developed strategies both to familiarise students with (and later to review) core techniques for using the software and to allow students to explore (and then to share their discoveries of) a wider range of technical possibilities. Complementarily, the teachers had devised or appropriated suitable tasks and supporting textual materials to underpin classroom activity that employed computer graphing to investigate the topic of algebraic forms. They were also developing a repertoire of strategies to support students in tackling these tasks, concerned not just with guiding software operation but with prompting strategic action and supporting mathematical interpretation.

In terms of activity structures, teachers suggested that use of technology made investigative lessons more viable. Equally, it seems that the availability of projection facilities permitted all the investigative lessons observed in this study to be organised within an activity structure in which episodes of individual or paired student activity at workstations were interleaved with whole-class activity, concluding with plenary review. Moreover, in the practice of Teacher B, the emergent type of ‘target practice’ task was associated with a rather different activity format for individual or paired student work, capitalising on the interactivity of the software to centre investigative activity around a process of trial and improvement. Notable also was the similar way in which both teachers had adapted the whole-class exposition and questioning format to exploit the opportunity to use the software to provide immediate feedback on student predictions, for example by students ‘taking the stage’ to use the projected computer to test their suggestions.

These preceding elements of adaptation are all interwoven in the development of teachers’ curriculum scripts for the topic of algebraic forms, as evidenced in the lesson agendas they formulated and in the detail of their classroom action (including interaction) during the observed lessons. On the basis of explicit comment by teachers (such as Teacher B referring her older class back to their previous encounter with the ‘target practice’ genre) or recurrent patterns of teacher action (such as Teacher A posing to both her classes the same speculative question leading to similar trialling activity), some of these examples clearly represent mature developments in teachers’ curriculum scripts for the topic. Other examples provide more evidence of teachers extending their repertoire of approaches to supporting students and (re)directing them towards desired states, intended responses and resultant learning. This included teachers’ extension of their capacity for reactive teaching in response to new types of student initiative made possible by the graphing software (such as Teacher A responding to students’ observations of the overlap of graphs [A1/14; A2/12] by explaining these properties through symbolic reduction of the more complex equation; and Teacher B guiding students’ exploration of a ‘barely’ quadratic expression through zooming out on its graph and comparing it to the ‘null-ly’ quadratic expression [B2/12]).

Finally, change in the time economy is evident in teachers’ comments on the contribution of graphing software to Effecting working processes and improving production. Equally, teachers reported that use of the software improved rate of learning return from classroom time by virtue of other contributions identified in the practitioner model, notably Overcoming pupil difficulties and building assurance and Focusing on overarching issues and accentuating important features. On the other hand, teachers had to manage the development of a double instrumentation of graphing both by hand and by machine. Doing this efficiently was assisted by development of a coherent resource system as noted earlier and by the availability of classroom projection permitting collective instrumented activity involving the class as a whole.

8 Conclusion

This study of teacher thinking and classroom practice can be read at two levels: at a more specific level, it seeks to provide a holistic understanding of how English secondary-school mathematics teachers use graphing software to teach about algebraic forms; at a more generic level, it aims to provide an example that illustrates the potential of an evolving framework for understanding technology use in school mathematics teaching.

Viewed from a strictly mathematical perspective, the archetypical classroom usage of graphing software that has been identified and examined in this study focuses on those same ideas identified in Fey’s (1989) discussion of pioneering work some 20 years ago: namely relations between symbolic expressions and coordinate graphs, notably the connection between particular coefficients or parameters in an expression and particular features of the corresponding graph. Likewise, the same types of mathematical task and attendant reasoning are prominent: namely induction of relationships between expression and graph through identification of pattern, and matching of expression to graph through trial and improvement. What this study shows, however, is that these features are simply the mathematical face of a larger pedagogical adaptation. Here indeed, both in official guidance and in classroom practice, graphing software was treated essentially as a pedagogical aid (replicating the predominant trend evidenced in the earlier work reviewed).

Teachers were particularly drawn to use graphing technology to support classroom activity that they variously described as involving investigation, exploration or discovery (as reported also in earlier studies by Farrell, 1996; Simmt, 1997). More specifically, in terms of the general themes of the practitioner model developed in our earlier work, teachers saw graphing software as contributing to:

  • Effecting working processes and improving production through making it easier to produce graphs accurately and rapidly, so increasing the efficiency and pace with which related topics can be covered;

  • Overcoming pupil difficulties and building assurance, through making graphing tasks more accessible to students who have difficulties with organisation and presentation;

  • Supporting processes of checking, trialling and refinement, through enabling lesson tasks based on trial and improvement, and supporting mathematical speculation and experimentation within and beyond the lesson agenda;

  • Focusing on overarching issues and accentuating important features, through helping to bring out the effects of altering particular coefficients or parameters in an equation on the properties of its graph and through facilitating comparison of gradients and examination of limiting trends;

  • Enhancing the variety and appeal of classroom activity, through reducing ‘laborious’ written work, increasing the immediacy and interactivity of classroom tasks and helping to create new forms of playful challenge within lessons;

  • Fostering pupil independence and peer exchange, through providing support for exploration by students and consequent sharing of discoveries, including software techniques and mathematical ideas within and beyond the lesson agenda.

At the same time, this study has highlighted the crucial part played by teacher structuring and shaping of technology-and-task-mediated student activity (as also shown in earlier studies by Doerr & Zangor, 2000; Farrell, 1996; Godwin & Sutherland, 2004) in realising the ideals of the practitioner model. Although teachers consider graphing software very accessible, successful classroom use still depends on their inducting students into using computer graphing for mathematical purposes, providing suitably prestructured lesson tasks, prompting strategic use of the software by students, and supporting mathematical interpretation of the results. Equally (as noted also by Godwin & Sutherland, 2004), the example of teaching about algebraic forms illustrates how, by providing prestructured lesson tasks and intervening to shape student work on them, teacher contributions play a fundamental part in managing the underlying semiotic system to make mathematical relations apprehensible to students, through coordinating algebraic and geometric registers, isolating phenomena and controlling variables.

Finally, this study has illustrated ways in which teachers, in the course of appropriating graphing software, adapt their classroom practice and develop their craft knowledge. Specifically, it has highlighted how teachers:

  • establish a coherent resource system incorporating software-mediated lesson tasks aligned with teaching goals, and supported by a common repertoire of suitable graphing techniques;

  • adapt formats for classroom activity to capitalise on the interactivity of the software;

  • extend curriculum scripts to encompass these features and to provide for proactive structuring and responsive shaping of student activity on software-mediated lesson tasks;

  • rework lesson agendas, both to include induction to computer graphing and to take advantage of the resulting time economy.