As will be seen, the students (i.e. the teachers) engage in the task in a surprising way: their use of GeoGebra is very limited and they use various mathematical terms such as ‘asymptote’ and ‘undefined’ very loosely.

In order to understand what is happening and why, I situate my analysis of the students’ activities within Sfard’s (2007, 2008) theory of commognition. The term ‘commognition’ (a mixture of the terms ‘cognition’ and ‘communication’) was introduced by Sfard (2008) to refer to a theory, developed by her, in which cognition and communication are regarded as different expressions of the same phenomenon. Specifically, cognition is an intrapersonal expression whereas communication is an interpersonal expression of a phenomenon. The theory draws both on Vygotsky and Wittgenstein and assumes a view of mathematics and of mathematics learning which resonates strongly with my own experiences of teaching and learning mathematics. Moreover it provides tight analytic constructs within which to examine and interpret mathematical activity. I use these constructs to show how ‘words matter’ and how a change in the discourse (in particular the use of the term ‘vertical asymptote’) affects the mathematical activity of the students. I also use this theory to show how the use of computers as a tool in mathematical learning may require an explicit rewriting of the rules of what counts as mathematical activity.

In this article, I focus on:

• the ways in which two teachers engage in a task whose purposes is to enhance their knowledge of the distinction between removable discontinuities and vertical asymptotes.
• how the teachers deal with the affordances and limitations of graphing software, in this case, GeoGebra. This has important implications for their use of technology as a teaching tool.

These foci need to be contextualised within two constraints. The first is that all the students in the course were digital immigrants¹ and had not used any educational software (such as GeoGebra) prior to this course. Secondly, the intended purpose of GeoGebra in the particular task was as a generator of graphs. That is, it was intended as a tool of amplification rather than of cognitive reorganisation (Pea, 1993).

In the Southern African context, Pournara (2009) grapples with issues about what constitutes appropriate knowledge for pre-service mathematical teachers; Adler and Davis (2011) explore the constitution of mathematics for teaching in various South African education institutions. They show how different courses are differently constituted with different foci and they ask how the different types of courses they describe ‘relate to teachers’ learning from and experiences of mathematics for teaching and ultimately, the quality of their teaching’ (p. 20). The article I present here goes a little way to addressing the first part of this question.

The course was structured as a part reading, part activity course. The class met once a week for a three-hour session over 11 weeks. Students were expected to study a specific chapter from the prescribed pre-calculus textbook (Sullivan, 2008) prior to their weekly session (this included doing examples at home). During the three-hour session, the class and lecturer discussed the topic and examples they had studied at home for the first hour. They were then presented with tasks around the topic which they did on their own or in pairs. Some of the tasks required the use of GeoGebra, others did not. I designed the structure of the course; my colleague, Lynn Bowie, taught most of the course and designed, selected or adapted most tasks.

According to Sfard, mathematics is an autopoietic system; that is, it creates its own objects. Thus mathematical objects are necessarily discursive objects. Sfard’s notion of mathematical discourse derives directly from this ontological view of mathematics and the power of her framework lies in the connections it makes between this ontological perspective and the ways learners engage in the discourse. The foundational basis of Sfard’s mathematical discourse theory contrasts with other discourse theories which have developed from non-mathematical domains. Indeed I find the inner consistency and coherence of Sfard’s framework (Sfard, 2008) most appealing for understanding mathematical activity. (This is not to deny that other discourse theories have been successfully used in the mathematics domain, for example, Morgan’s, 2005, use of Fairclough’s critical discourse analytic approach; Shreyar, Zolkower & Perez’s, 2010, use of Halliday’s systemic functional linguistic analysis.)

Sfard (2008) and Ben-Yehuda, Lavy, Linchevski and Sfard (2005) have defined and illustrated the different components of mathematical discourse in the context of everyday and school mathematical activity. Viirman (2011), in Sweden, has used commognitive theory to examine the discourse of university mathematics lecturers teaching an undergraduate course on functions. However, the components of mathematical discourse have, as far as I know, not been elaborated to sites of teacher education. Since such contexts have their own peculiarities, I extend Sfard’s descriptions to include such contexts where necessary.

Computer-generated routines themselves require further scrutiny. That is, certain algorithmic routines are easily executed by the software; however, the computer’s routines do not always obey the rules of mathematical discourse. For example, GeoGebra (and most other graphical software) generates a continuous graph even if there are removable discontinuities in that function. Consequently the user needs to be aware of how to interpret the computer output so that it is compatible with endorsed mathematical narratives.

Sfard (2008) usefully distinguishes between three different types of routines: explorations, rituals and deeds, each of which is distinguished by its own set of goals. Explorations are routines whose purpose is to produce or verify an endorsable or endorsed narrative. For example, routines of solving equations, of proving a mathematical result, or of generating and investigating a mathematical conjecture are explorations. In the task below, students were expected to embark on a set of explorations in order to substantiate the definition of a ‘vertical asymptote’ and to determine the existence or not of a vertical asymptote at a particular point. A ritual is a routine whose goal is social approval. This is usually through the participant aligning her mathematical activity with other people’s routines. Rituals involve imitation of others’ routines; as such they may be a very important part of mathematical learning. Echoing Vygotsky (1978), Sfard (2008) argues that a ritual is the form that routines take in the zone of proximal development (p. 253). Arguably we can identify ritualised routines in the episode below. Finally, deeds are routines whose purpose is a change in objects, not just in narratives (as in the case of explorations). So for example, a child may be able to divide an actual set of six cookies amongst 12 children (a deed) although she may not be able to formulate or execute 12 ÷ 6. In the GeoGebra environment, a task which involves the use of a slider to change the value of a parameter and hence a graph may constitute a deed.

Eva has a BSc degree with a postgraduate diploma in education; Tom has a degree in education. In an earlier survey Eva claims to use the computer often and is very confident in its use; in contrast, Tom claims that he does not use a computer often, but he is confident, as opposed to ‘very confident’, in its use. Neither Tom nor Eva has used a computer in the learning or teaching of mathematics. Eva does very well in the course; she obtains 87% in mid-semester test. Tom is one of the weakest students in the class and he obtains 34% in the mid-semester mathematics test.

In this vignette, the students were audio taped and their work was screen recorded as they worked on a task that was given to the entire class to work on in pairs. The researcher sat just outside the room in which the students were working, walking in periodically to observe what was happening. The pairs of students were told that they must treat this as a normal classroom session and that they could ask the researcher any questions as they would in a classroom setting.

FIGURE 1: The task.

Consequent to this, we expected them to write something like:

We expected them to work similarly for y₂ (a second degree equation). Given the cumbersome nature of drawing a cubic or quartic equation by hand, we expected students to use GeoGebra to sketch y₃ and y₄ . Whether the graphs were hand drawn or computer generated, it was hoped that students would notice that each function was not defined at x = 1 but that by cancelling the factor (x − 1) in the numerator and denominator, the problematic point, x = 1, could be removed. Graphically this would be depicted by a hole at x = 1 (they could draw this on a printout of the graph). Alternatively, they could use a theorem (the endorsed narrative) about the location of a vertical asymptote as given in their pre-calculus textbook (Sullivan, 2008, p. 188) to decide that none of the given functions had a vertical asymptote at x = 1, or elsewhere.

The theorem reads:

A rational function , in lowest terms, will have a vertical asymptote x = r if r is a real zero of the denominator q(x). That is, if x – r is a factor of the denominator q(x) of a rational function , in lowest terms, R(x) will have a vertical asymptote x = r.

Note that that since this was a pre-calculus course we did not expect students to know the language of discontinuities. Rather we expected them to speak of holes or functions not existing at a point.

FIGURE 2: Hand plots of y1, y2, y3 and y4 by Eva and Tom.

Eva and Tom now use GeoGebra to generate a graph of y₃; however they ignore this GeoGebra-generated graph and draw the graph by hand. Eva factorises the expression to get x³ + x² + x + 1 and she and Tom expend much effort hand plotting the resultant cubic using calculus. As before, Tom hand draws the graph correctly with a gap at x = 1 (and also with vertical line at x = 1; see Figure 2).

One possible reason for Eva and Tom’s limited use of GeoGebra is an entrenched cultural attitude: mathematics is done by hand. Technology is there as a tool for confirmation of hand-done mathematics, but not for doing mathematics. Ontologically speaking, mathematical outputs produced by a computer are not part of the official mathematical narrative. Indeed Eva, who took the leading role (Sfard, 2007) in the discourse, declared right at the beginning of the task: ‘We will do it sketching first and then we’ll check on GeoGebra’ (line 3). This attitude surprised me. This was the sixth week of this course and my colleague and I had stressed the value of using technology in doing mathematics: its narratives were endorsable, although some care had to be taken when interpreting its outputs.

Another possible reason for the privileging of hand-drawn graphs is in the students’ reading of the caveat (‘be careful. GeoGebra isn’t perfect here’). This statement was intended to alert the students to the fact that GeoGebra did not reveal removable discontinuities in its graphs. But the statement may have reinforced the belief (discussed previously) that narratives of computer-generated mathematics are not consistent with the official mathematical narrative. Indeed Eva implicitly justifies the hand sketching of all graphs by invoking this cautionary statement at the beginning of the task (line 3).

It is important to note that the existence of the cautionary statement cannot be taken as the sole or even main reason why students privilege hand-drawn over computer-generated graphs. After all, the students use GeoGebra graphs to confirm their hand sketches and they are aware that the GeoGebra graph looks like their hand sketches, other than the lack of visible discontinuities: ‘But you see GeoGebra doesn’t do that. Can you see that it’s like a continuous graph, né?’ (line 39). Indeed I strongly propose that more technologically enculturated students would have printed GeoGebra graphs and then hand drawn holes or asymptotes.

The question is: Why do the students prefer to use hand-drawn graphs rather than computer-generated graphs as visual mediators? Several reasons, some of which are discussed above, are feasible.

For example, it may be that they do not accept GeoGebra graphs as compatible with the endorsed mathematical narrative (as previously discussed), or it may be that they need to represent the value at which the function was not defined iconically. In this task, computer-generated graphs were not iconic and students lacked the experience of how to turn them into iconic mediators (print the GeoGebra graphs and draw a hole on the graph). So they preferred to use hand-drawn graphs in which a ‘hole’ could be directly and iconically represented.

With regard to the symbolic manipulation (an activity which is visually mediated), Eva is able to simplify the expressions y₁, y₂, y₃ and y₄ into polynomials which are easier to hand sketch (although this hand drawing is still very time-consuming and effortful). Further discussion is provided under the heading ‘Routines.’

However, the researcher’s modest intervention (lines 323–351)², in which the difference between a hole (usually represented by an open circle) and a vertical asymptote is explained, triggers a distinct change in Eva’s discourse. Indeed, post intervention Eva ‘unsames’ the terms ‘not defined at a point’ and ‘vertical asymptote’ (lines 423–426) and she unsames the term ‘open circle’ (used to represent a point through which there is a vertical asymptote) with ‘hole’ (which she uses to represent a removable discontinuity) (lines 406, 435, 439). She moves on to characterise the conditions which lead to a ‘hole’ (numerator and denominator cancel at zero, line 444). Mathematically speaking, neither ‘hole’ nor ‘open circle’ is a well-defined mathematical term. Nonetheless, by no longer speaking of an ‘open circle’ but rather speaking of a ‘hole’ Eva shifts her discourse from one in which ‘not defined at a point’, ‘open circle’ and ‘asymptote’ are samed to one in which these are unsamed; indeed, in the later discourse (lines 423–444) Eva produces an endorsed narrative in which she reserves the word ’hole’ to refer to a removable discontinuity (line 444). In contrast there is little change in Tom’s discourse; he still refers to the discontinuity as an ‘open circle’ (lines 407, 441) and he is uncertain as to whether he should erase the vertical asymptote at x = 1 or not (lines 345, 348). Just to reiterate: although the term ‘open circle’ is as acceptable as the term ‘hole’ in pre-calculus discourse, ‘open circle’ was used, before the researcher’s intervention, to signify a point at which there was an asymptote. Tom does not change his usage of this term and he does not sever this connection.

Furthermore, even though Tom and Eva no longer talk of an asymptote, Tom argues that: ‘For x equals to 1 it’s undefined because of division by zero’ (line 434). This is not correct: at x = 1 the function is not defined because we have division of zero by zero. So unlike Eva, Tom does not change his discourse; nor does he produce an endorsed narrative around the notion of a removable discontinuity.

Implicit in the post-intervention discourse is a commognitive conflict. A commognitive conflict is a struggle, often implicit, which is the result of the concurrent use of two incompatible discourses. Eva uses the term ‘undefined’ to refer to a point at which the function is not defined (because of a zero in the numerator and denominator) and at which there is a removable discontinuity − a ‘hole’ (lines 406, 435, 439). In contrast, Tom uses the term ‘undefined’ to refer to a point at which a function is undefined because of division by zero (line 434). He calls such a point an ‘open circle’. For learning to take place, the commognitive conflict needs to be acknowledged and resolved. Arguably it is the role of the teacher to support the resolution of such a conflict, rather than to bypass it, as in this vignette.

Also, when simplifying the expressions for y₁, y₂, y₃ and y₄, the students do not explicitly state where the function is indeterminate. For this reason, the routines that they execute with regard to simplification contradict endorsed mathematical narratives. For example, Eva writes:

She does not indicate that x ≠ 1 although she and Tom acknowledge that x ≠ 1 several times, for example, lines 7−8, 36, 310, 407.

A further (non-endorsable) routine, extensively discussed above, involves the students’ sketch of the vertical line x = 1 to indicate a vertical asymptote at x = 1 (see Figure 2). This drawing of an asymptote at x = 1 reflects and reinforces the saming of the notion ‘not defined at a point’ and ‘asymptote’ as discussed above.

Arguably, Tom’s discourse after the researcher’s intervention (lines 421 onwards) has the quality of a ritual. That is, he is prepared to go along with Eva’s explanation of what is happening at x = 1 (lines 422–444), even though he does not seem convinced that there is no vertical asymptote at x = 1 (evidenced by his checking whether he can erase the vertical asymptote from the sketch in lines 345, 348). Sfard (2008) argues that ritual is a necessary stage in routine development. Through the use of thoughtful imitation in routines the learner gains knowledge of the how of a routine. ‘Imitation … is the obvious, indeed, the only imaginable way to enter new discourse’ (p. 250). In contrast, Eva’s routines are primarily explorations. Her goal is to produce an endorsed narrative (which she does).

Another very important aspect of the mathematical activities, central to Sfard’s theoretical framework and evidenced through empirical data here, is the crucial role that words play in mathematical discourse. That is, it really does matter how we use words when talking about mathematical phenomena. This is starkly shown by the students’ unsaming of ‘vertical asymptote’ and ‘point at which the function is undefined’. It is further demonstrated by Eva’s embracing of a new word, ‘hole’, rather than ‘open circle’ (which was previously samed with ‘vertical asymptote’) to describe a removable discontinuity. Likewise, Tom’s continued use of the term ‘open circle’ to describe a point at which the function is undefined both reflects and is constitutive of his non-distinction between different types of undefined points. This points to the importance of a teacher or researcher carefully listening to what students actually say and using their discourse as a way into their understandings.

Finally, considering learning as a change or extension of discourse, we can say that some learning has taken place: Eva has changed her discourse and Eva is able to distinguish between a vertical asymptote and a point at which a function is undefined. For Tom, the learning is less evident. Indeed I suggest that, for Tom, learning in this vignette is in the form of ritual and imitation.

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2. Most of this transcript is not reproduced here.