The ways teachers converse about their work in relation to information and communications technologies (ICTs) are worth studying. We analyse how a teacher converses about her local practices in relation to two spreadsheet algebra programs (SAPs) on variables. During the conversations we noticed that the teacher keeps different policy documents – boundary objects – firmly in view, in relation to the design of the two other boundary objects, namely the two SAPs. The policy documents provide details on the operative curricula which entail the intended, implemented and examined curricula. Of these curricula, the teacher regarded the examined curriculum and associated examinations as most important. Also, she conversed about how she intends to align the design features of the two SAPs with particular policy documents, especially in the context of the South African high-stakes National Senior Certificate examinations and the attendant examination pressure. Our results confirm current professional development (PD) literature suggestions that emphasise fostering coherence, for example between policy boundary objects details and what university-based PD providers do when they interact with school teachers.

The results provide guidelines for university-based PD providers to integrate SAPs or other ICTs related to algebra and variables by keeping teachers’ local practices in view. These providers should note that different policy-related boundary objects shape the ways teachers understand and converse about their local practices, namely their work at the classroom level.

The topic of this study is boundary-objects-related details. We unpack this topic by outlining the meanings of boundary objects. In this article the concept of boundary objects does not take on a singular meaning. First, we define boundary objects as ‘objects which inhabit several intersecting social worlds and satisfy the informational requirements of each of them’ (Star & Griesemer,

Another key notion in this article is teachers’ local practices. We define teachers’ local practices as ways teachers converse about their work within the schooling system and at the classroom level. Teachers respond to and interact with different stakeholders, for example parents, school principals and education department officials, to name a few. Also, teachers contend with operative curricula that include:

The intended curriculum: CAPS details per grade level.

The interpreted and implemented curricula: what they understand and do, that is, teach in their classrooms, informed by CAPS details.

The examined curriculum: the (mathematics) content present in examinations and assessments (Julie,

Operative curricula details are spelled out in policy documents, that is, the boundary objects we listed in the first paragraph. The operative curricula inform teachers’ ‘logic of practice’ and their ways of working (Bourdieu,

In terms of the professional development (PD) literature, we do not know much about ways teachers who work under conditions of high-stakes examinations in the greater Cape Town area, South Africa, converse on the work they do (Julie et al.,

Two sub-questions are:

What other studies set a foundation for this main research question?

How do the article’s findings relate to other studies?

We justify the main research question with its conversation focus as follows. First, in terms of working in the school and the education system, this study aims to bring to the fore ways teachers converse about boundary-objects-related details that impact on their work at the local, that is, classroom and school levels. Generally, teachers are stakeholders in mathematics education research (Krainer,

Here, the analysis can bring to the fore curricular details about what the two parties – UMEs and teachers – know relative to each other. Such details become helpful for ‘working with’ as opposed to a deficit view of ‘working on’ teachers (Setati,

In the operative curricula, we find boundaries, that is, separations, between products, factors, trinomials and parabolas. The design features of

Two sub-questions inform the literature review.

From the PD literature we find several studies that reference boundary-objects-related details. In South Africa, Julie et al. (

Boundary-objects-related details do not only refer to the high-stakes Grade 12 NSC examinations. These details also include references to school-based end-of-year summative assessments. In a recent survey on assessment in mathematics, Suurtamm et al. (

From the effective PD literature we also find references to boundary-objects-related details. Garet et al. (

The operative curricula become key to understanding ways teachers converse about the varying demands on their local practices, especially the examined curriculum. We define operative curricula as the intertwining intended, interpreted, implemented and examined curricula. In practical ways, in South African schools, and on a daily, weekly and monthly basis, the different policy documents, that is, boundary objects, seek to impose ‘order’ and provide details on the intended curriculum, namely the Curriculum and Assessment Policy Statements (CAPS) documents. Teachers interpret, that is, make sense of, and implement CAPS details in their classrooms. During the school year teachers also prepare their learners for the examined or assessed curriculum, in other words, for examinations. Examinations exert an ordering effect on teachers’ local practices because they occur during stipulated times and dates during the school year. Such examination details inform us that teachers will likely bring up issues of examinations and assessment. Examinations also operationalise significant components of the intended curriculum spelled out in policy documents (Julie,

Outline of the operative curricula showing interlocking interactions between curricular variations.

The overlapping circles in

First, the article’s findings relate to studies that mention teachers’ awareness of examinations or assessment issues. Pong and Chow’s (

On a similar issue, Gregory and Clarke (

Second, the article’s findings relate to policy documents and studies in algebra and variables. As noted above, diagnostic reports in South Africa note that learners struggle with the concept of a variable. Unsurprisingly, we also read about ways of ‘making algebra work’ in schools, which includes focusing on meanings of variables based on instructional strategies that deepen student understanding, within and between algebraic representations (Star & Rittle-Johnson,

This study followed a qualitative research design approach in which we adopted a case study. For the case study we examined the particularity and complexity of the case, namely the topic of boundary-objects-related details (Tomaszewski et al.

The idea for this study comes from teachers who participated in a small-scale PD initiative, on a voluntary basis. Activities with the teachers included discussions that focused on the design of different SAPs as instances of the application of ICTs (Leung,

For the study, we sampled conversation excerpts from one teacher because they reflected the particularity and complexity of the case. During the conversations, this teacher referenced and compared the boundary objects, namely the POA, SAG and CAPS, with the design features of the two SAPs. In addition, she conversed about other important boundary-objects-related details such as preparing for high-stakes NSC Mathematics examinations.

We collected data in the form of audio-taped conversations that focused on boundary objects, namely the two SAPs,

The design of ^{2} +

Screenshot of

Furthermore, the instructions in the upper left corner become key to understanding the variables or parameters in the case of ^{2} +

As before, key to understanding the design of ^{2} − 4^{2} +

In addition, the script starting with ‘consider the standard form of a quadratic function’ enables interpretive flexibility with respect to the cell-variables. This script addresses policy concerns about learners’ challenges with interpreting the discriminant, for example (see

Screenshot of

Based on the case study, we used a ‘constant comparative method’ to analyse the data excerpts, namely the transcriptions of audio-taped recordings (Tomaszewski et al.,

The case study calls for a main and embedded unit of analysis (Yin,

We used what Niss (

Research framework outlining the two data incidents.

Empirical situation | University-based mathematics educator (UME) meets with the teacher, with the design of the two spreadsheet algebra programs serving as a focus of conversation. |
---|---|

Analytical layer 1: |
During these conversations the teacher provides boundary-object-related details about the operative curricula with associated boundary objects that influence and structure her local practices; these include references to the intended, implemented and examined or operative curricula as well as ways that variables feature in the operative curricula. These meetings and conversations also instantiate a boundary encounter between two discursive practices: university-based mathematics education and school mathematics teaching. |

Analytical layer 2: |
Also, during conversation exchanges the teacher directly or indirectly compares and contrasts the operative curricula with the design of the two spreadsheet algebra programs with respect to variables. |

In each case, there is a table with three columns labelled: turns, speaker (T1 for ‘teacher’ and UME for ‘university-based mathematics educator’) and utterance. We use the acronym UME to emphasise the distance and boundary encounter between the university and the school.

Excerpt 1 (see

Excerpt 1.

Turn | Speaker | Utterance |
---|---|---|

1 | UME | ‘You were saying …’ |

2 | T1 | ‘The way we are basically guided by curriculum, and when the advisors come, we go to meetings etc. They will tell us, there’s the paper, I want the paper in March, this is the stuff you have to teach and then we set up our POA, call the parents to the office …’ |

3 | UME | ‘What is a POA?’ |

4 | T1 | ‘Programme of assessment. We set it up according to those lines; at the start of the year we will sit down, teachers will have a meeting and we’ll say okay we have two assessments for the quarter and a March examination. What do we teach? Now we open the SAG and the SAG will state, no graphs, it is multiplication, products, it is factorisation. Then we do the factorisation, that is ^{2} + 10 |

5 | UME | ‘You said something about what I was doing was reinterpreting the SAG, what does that mean?’ |

6 | T1 | ‘The SAG has certain guidelines that state what we have to do in a specific way etc. Very seldom do they say that we like, you saying connecting or taking three things, your variables, your numerical values and your sketch and doing it as a one completed lesson. What they have is almost like an apart session only for multiplication. That’s a concept that they need to understand. They will say, the learner must be able to, the learner must be able to … that’s what the SAG states. The learner must be able to …, the learner must be able to … They have been taught that way, but what I see now is, what I can do, is, you can give your products then you must be able to give you your whatever …’ |

7 | UME | ‘Factorise.’ |

8 | T1 | ‘… and connect it to the graphs all in one etc. and connect it inductively; they will see, hopefully they will see the picture emerging between numerical values and the things over there [ |

9 | UME | ‘Okay.’ |

10 | T1 | ‘That graph will go well with the projector. I am not really |

T1, teacher; UME, university-based mathematics educator.

Excerpt 1 (see

The epistemic order in Turn 5 merits attention in terms of embedded unit of analysis. The teacher notes that the design of

From the two units of analysis, we should notice how the teacher converses about a coherence she sees between two different boundary objects, namely the POA and SAG on the one hand and

In Excerpt 2 (see

Excerpt 2.

Turn | Speaker | Utterance |
---|---|---|

1 | T1 | ‘In any textbook, if you go to any textbook, even an exam paper. I am busy moderating at the moment. In any exam paper, we have separate chapters in books, where they treat products separate from factors, factorisation. They would do products as a separate entity, factors as a separate entity. Then they’ll go over to equations, different types of equations, then they’ll do something on the function. There is never an integrated approach, where they do everything together. As teachers, we need to make that connection known to learners.’ |

2 | UME | Making algebra work: Instructional strategies that deepen ‘Why?’ |

3 | T1 | ‘Because ultimately, if you can understand that the graph is actually a visual of a function on a table or a visual of an algebraic, given in rubric form, they have to select their own input values. They can actually put everything together, because then they have the picture in terms of the graph.’ |

4 | UME | ‘Can I say something? You said that the learners will bring everything together. Now that can only happen, I would say, on the encouragement of teachers. Do you agree?’ |

5 | T1 | ‘Yes, or otherwise you must have a directed worksheet for them or a work programme for them like this that will lead them into that.’ |

6 | UME | ‘You said directed worksheets; are there particular worksheets that you have that you have designed in the past where you have such an integration, to use your words, or …’ |

7 | T1 | ‘You mean like the connection here?’ [ |

8 | UME | ‘Yes.’ |

9 | T1 | ‘We normally do this. We have the given function, |

10 | UME | ‘Right.’ |

11 | T1 | ‘Your parameters, it can quickly give you your coordinates and from there they quickly do the plotting. At the end, I will give them the shortcut.’ |

12 | UME | ‘Why?’ |

13 | T1 | ‘Because I know, the pressure of the exams doesn’t allow for them to more or less work out a table. Some of them are slow actually. Some of them don’t have the necessary capacity in terms of technology. They don’t have a calculator.’ |

T1, teacher; UME, university-based mathematics educator.

In Excerpt 2 (see

As in the case of Excerpt 1 (see

In Excerpt 3 (see

Excerpt 3.

Turn | Speaker | Utterance |
---|---|---|

1 | UME | ‘I want you to look at the screen here … and just tell me what you are looking at? Anyone. You have the heading there: ‘The discriminant … quadratic function …’ |

2 | T1 | ‘Basically, it’s information concerning the type of function being quadratic and then also the means of the method of finding the |

3 | UME | ‘Is this the new CAPS?’ |

4 | T1 | ‘Yes, this is the new CAPS. Next year when they go to Grade 11, this is the new question that they will need to know.’ |

5 | UME | ‘Now below on the screen there you’ve got a table there for |

6 | T1 | ‘Smart.’ |

7 | UME | ‘Why do you say it is smart?’ |

8 | T1 | ‘Because, again you can clearly see the bridge between the numerical values |

T1, teacher; UME, university-based mathematics educator.

In Excerpt 3 (see

As before, we need to note the teacher’s comments on how the design of

In Excerpt 4 (see

Excerpt 4.

Turn | Speaker | Utterance |
---|---|---|

1 | UME | ‘I think this is the second time I hear you mention lead questions. What is it you want to lead them to?’ |

2 | T1 | ‘You see I have certain objectives … I have certain objectives.’ |

3 | UME | ‘Which are?’ |

4 | T1 | ‘If I teach the parabola, right, without giving them any numerical values, right, and only give them the signs of |

5 | UME | ‘By that you mean?’ |

6 | T1 | ‘The positive or negative sign of |

7 | UME | ‘What about the zeros? I am thinking of the case where |

8 | T1 | ‘Yes, that can also be included as a lead.’ |

9 | UME | ‘Let me get to the instance if |

10 | T1 | ‘Because then we are moving away from the fact that it is a parabola, a parabolic function. It won’t have two roots, it won’t have the characteristics of a parabola.’ |

T1, teacher; UME, university-based mathematics educator.

We interpret ‘signs’ to mean different integer values, which are needed to compute the discriminant and to decide on the nature of the roots. She keeps the content of the examined curriculum in mind or in view (see Excerpt 1). The UME asks her to be specific about her lead questions (see the epistemic order in Turns 3 and 5). The UME continues to ask her to consider boundary instances, that is, where the parameters as placeholder variables assume the values of zero (see the epistemic order in Turns 7 and 9). She makes clear that as for her local practices, the resulting changes of the parameters will not display the ‘characteristics of a parabola’ or a ‘parabolic function’ (see Turn 10). As we can see, in the ecology of studying and teaching the mathematical object – a parabolic function, in this case – she keeps the content of the examined curriculum in view (see Turn 4 in Excerpt 3 [see

Excerpt 5 (see

Excerpt 5.

Turn | Speaker | Utterance |
---|---|---|

1 | UME | ‘I want to get back to your use of drilling. Drilling has been used in particular ways and sometimes it has negative meanings. Is that how you …’ |

2 | T1 | ‘Obviously drilling has a positive meaning.’ |

3 | UME | ‘Say a little bit more about that. |

4 | T1 | ‘Ask them whether they understand, they will say yes, and when it comes to an exam. But then ask them the next day, then are not that sure.’ |

5 | UME | ‘So, is that how you use drilling? Namely?’ |

6 | T1 | ‘To consolidate, to consolidate certain topics, especially in your teaching. You see our whole education system is based on that, it’s based on preparing learners for examinations.’ |

T1, teacher; UME, university-based mathematics educator.

Answers to the main research question – What boundary-objects-related details about the teacher’s local practices emerge during conversations that focus on the design of SAPs based on research related to variables? – offer ways for UMEs involved in PD to understand their work better. Concerning the main or primary unit of analysis, we gain insights into ways the teacher speaks about her local practices in a challenging socioeconomic school environment. In this regard, she keeps in view different boundary objects integral to the school’s operations, namely the different policy documents that outline the intended, implemented, interpreted and examined curricula. These entangled curricula form what we called the operative curricula. More importantly, she works in a high-stakes examinations environment where examination pressure, for example examination contents, counts. Hence, she makes it clear that she keeps the examinations firmly in mind, for instance the contents of the examination papers in the high-stakes NSC (matric) Mathematics examinations. The reality of the high-stakes examinations reflects the teacher’s experiential world, which has parallels, in the case of Hong Kong and the United Kingdom, for example. This examination pressure reality, in turn, should help UMEs to better understand what it takes to cross boundaries to school when they do PD work. In this regard, we ask: Can UMEs afford a reduced analytical representation of teachers? In schools, teachers contend with boundary objects.

In the case of the embedded unit of analysis, UMEs need to note the following. The teacher mentions how the features of the

In conclusion, from these topic boundary-objects-related details, we should note that the teacher aims at coordinating and aligning different boundary objects such as policy document details, which impact on her local practices. She works in a school located in a low-income socioeconomic environment, combined with the reality of the high-stakes NSC Mathematics examinations and associated examination pressure. As noted before, high-stakes examinations are not an uncontested area but have become a permanent and vital part of education. Such entangled and circumstantial conditions should signal to UMEs what is at stake for teachers as stakeholders in the education system. Boundary encounters between SAPs – or other types of ICTs, for that matter – and teachers, provide UMEs as PD providers and knowledge brokers with opportunities to improve their work with teachers’ functioning milieu of their schools and classrooms with their varying demands.

Alwyn Olivier is the designer of the spreadsheet algebra programs.

The authors declare that they have no financial or personal relationships that may have inappropriately influenced them in writing this article.

F.G. is the principal investigator, W.D. and A.A are collaborators. F.G. was responsible for data collection and analysis. The two collaborators contributed to the writing process.

Stellenbosch University Research Ethics Committee provided approval on 2 September 2011 (524/2011).

Data sharing is not applicable to this article as no new data were created or analysed in this study.

The views and opinions expressed in this article are those of the authors and do not necessarily reflect the official policy or position of any affiliated agency of the authors.