Chapter for book: Mathematics Curriculum Material and Teacher Development: from text to ‘lived’ resources icon

Chapter for book: Mathematics Curriculum Material and Teacher Development: from text to ‘lived’ resources


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Chapter for book: Mathematics Curriculum Material and Teacher Development: from text to ‘lived’ resources. Editors: G. Gueudet, B. Pepin & L. Trouche

Chapter 1: Knowledge resources in and for school mathematics teaching


Jill Adler: Wits – Marang; KCL


  1. Introduction


This book, and the range of chapters within it, takes as its starting point, the role of curriculum resources in mathematics teaching and its evolution. Teachers draw on a wide range of resources as they do their work, using and adapting these in various ways for the purposes of teaching and learning. At the same time, this documentary work (as it is referred to by Guedet & Trouche, Chapter 2) acts back on the teacher, and his or her professional knowledge. Documentary work is a function of the characteristics of the material resources, teaching activity, the teachers’ knowledge and beliefs, and the curriculum context. The chapters that follow explore and elaborate this complexity.


An underlying assumption, of course, is an increasing range of textual resources for teaching, and moreover, wide availability of digital resources. An expanding and diversifying range of resources for teaching is part of ‘normal’ practice. Such assumptions require pause and consideration in contexts of poverty. In other words, in contexts of minimal material and electronic resources, the relationship between the teacher, his or her knowledge and beliefs, and the curriculum and curriculum resources needs to be problematised. It is this kind of context that gave rise to a broad conceptualisation of resources in mathematics teaching, a conceptualisation that included the teacher and her professional knowledge i.e. epistemological resources, together with material and cultural resources, like language and time. In Adler (2000) I describe this broad conceptualisation, elaborating and theorising material and cultural resources in use in practice in mathematics teaching in South Africa. The discourse used is of a teacher ‘re-sourcing’ his or her practice - a discourse with strong resonances in documentary work.


This introductory chapter builds from and fills out that work by foregrounding and conceptualising knowledge as a resource in school mathematics teaching. The research I draw from is located in South Africa where, twenty years after the release of Nelson Mandela and the beginning of the dismantling of the deeply inequitable apartheid state, the availability of even basic resources like a single textbook per learner in all schools still cannot be taken for granted. Now, as then, resources in use in mathematics teaching, and particularly teachers’ professional knowledge, are a critical focus of attention.


I begin with a brief review of the conceptualisation of resources from my earlier research, and draw attention to the emerging difficulties we faced in describing and theorising the problem of knowledge as a resource in school mathematics teaching. I will then move on to the substance of this chapter - a discussion of the methodology we have developed in the QUANTUM1 research project to ‘grasp’ knowledge resources in use in mathematics teaching. This current research has as its major question, what and how mathematics comes to be constituted in pedagogic practice. The focus illuminates knowledges in use in practice, and how these shape what is made available for learning. I will illustrate the methodology we have developed through recent empirical work in secondary mathematics classrooms in South Africa. These illustrations add force to the argument for foregrounding knowledges in use in descriptions of classroom practice. Furthermore, while the context that gives rise to the methodological tools offered here, these, and the particular gaze that produces them, are, I propose, useful for studying the evolution of knowledge resources in use in teaching in any context.


  1. ^ Locating the study of knowledge resources


QUANTUM has is research roots in a study of teachers’ ‘take-up’from an upgrading in-service teacher education programme in mathematics, science and English language teaching in South Africa (Adler & Reed, 2002). By ‘take-up’ we mean what and how teachers appropriated various aspects of the programme, using these in and for their teaching. This discursive move is explained in Adler (2002, p. 10). We set out to examine teacher ‘change’. ‘Change’, however, produces a deficit discourse: teachers are typically found to be lacking. They have either not changed enough, or not changed in the right way. ‘Take-up’ enabled us to describe the diverse and unexpected ways teachers in the programme engaged more, or less, with selections from the courses offered, and how these selections were recontextualised in their own teaching.


Amongst other aspects of teaching, we were interested in resources in use. We problematised these specifically in school mathematics practice (Adler, 2000), where I argued for:

… a broader notion of resources in use that includes additional human resources like teachers’ knowledge base (as opposed to their mere formal qualifications), additional material resources like geoboards which have been specifically made for school mathematics, everyday resources like money, as well as social and cultural resources like language, collegiality and time.

(Adler, 2000, p. XXX)


I also argued for the verbalisation of resource as ‘re-source’. In line with ‘take-up’, I posited that this discursive move shifts attention off resources per se, and refocuses it on teachers working with resources; on teachers re-sourcing their practice. These arguments emerged in the changing socio-political context of South Africa in the late 1990s, where pervasive inequality in the educational system pushed the availability of resources into the foreground, back-grounding their use. A common and justified lament of teachers, particularly those in rural schools, was their “lack of resources”, and a singular plea for “more resources”. The complexity of resource availability and use as this emerged in the study led to our extended conceptualisation of resources.


In focus were selected material (e.g. chalkboards) and cultural resources (language, time). With a theoretical orientation drawn from social practice theory, we proposed an elaborated categorisation of resources, supported by a description of some examples of their use in practice in terms of their transparency (Lave & Wenger, 1991; Adler 2000). These combined to illustrate that what matters for teaching and learning is not simply what resources are available and what teachers recruit, but more significantly how various resources can and need to be both visible (seen/available and so possible to use) and invisible (seen though to the mathematical object intended in a particular material or verbal representation), if their use is to enable access to mathematics. We were able to describe the diverse ways in which ‘new’ and ‘existing’ (and often maligned) resources (e.g. chalkboard) were used to support routine as well as new practices, and how they enabled or constrained possibilities for access to mathematics.


Out of focus in this work were human resources: teachers themselves, their professional knowledge base, and knowledges in use. The teachers in our study were studying courses in mathematics and mathematics education, courses designed for their learning. We were thus interested in their ‘take-up’ from these courses, in how they ‘re-sourced’ their practice in an epistemological sense. However, we had difficulty ‘grasping’2 teachers’ take-up with respect to subject and pedagogic content knowledge for teaching, and so their knowledge in use. We had anticipated clearer articulation of mathematical purposes by teachers over time. Many of the teachers in our study, and not only those with poor results in their courses, however, could not elaborate their mathematical purposes, despite probing in interviews. Our analysis of these interviews, together with observations in their classrooms over three years, nevertheless, suggested correlations between teachers’ articulation of the mathematical purposes of their teaching, and the ways in which they made substantive use of ‘new’ material and cultural resources (language in particular). These results are in line with a range of research that has shown how curriculum materials are mediated by the teacher (e.g. Cohen, Ball et al, 2003). Indeed, that the interaction between a teacher and the curriculum materials he or she uses is relational (Remillard, 2004) and thus co-constitutive, serves as a starting point for a number of chapters in this volume (Ruthven, Pepin). In addition, our analysis also pointed to unintentional deepening of inequality. The ‘new’ curriculum texts selected by teachers from their coursework and recontextualised in their classroom practice, appeared most problematic when teachers’ professional knowledge base was weak, and typically, this occurred in the poorest schools (Adler, Reed, Lelliott & Setati, 2002).

These claims are necessarily vague and tentative – our methodology did not enable us to probe teachers subject knowledge and pedagogic content knowledge and take-up with respect to these over time. Furthermore, we emerged far more appreciative of the non-trivial nature of the elaboration of the domains of mathematics and teaching in the construction of teacher education - a point emphasised recently by Chevellard (in Guidet & Trouche, 2010). In a context where contestation over selections from knowledge domains into mathematics teacher education continues (Parker, 2009), the importance of pursuing knowledge in use in teaching through systematic study was evident. Mathematical knowledge for and in teaching, what it is, and how it might be ‘grasped’, became the focus in the QUANTUM study that followed. Our object in QUANTUM is to be able to describe what comes to be constituted as mathematics in and across pedagogic contexts, including school classroom. This has resulted in a methodology that illuminates what comes to function as ground or criteria for mathematics, and so the domains of knowledge teachers call in as they go about their work. It is this conceptualisation that has enabled an elaboration of knowledge resources in use in mathematics teaching.


  1. ^ Conceptualising knowledge resources as ground


In Adler (2000), and as discussed above, I argued for a conceptualisation of ‘resource’ as both a noun and a verb. I argued for thinking about resource as “the verb ‘re-source’, to source again or differently where ‘source’ implies origin, that place from which a thing comes or is acquired”. In this chapter, as in the earlier paper, ‘resource’ is both noun and verb - ‘knowledge resources’ refers to domains of knowledge - the objects, processes and practices within these - that teachers call in as they go about the work of teaching. This conceptualisation of knowledge as resource coheres with the orientation to the notion of ‘lived resources’ that underpins this volume. While my focus is domains of knowledge (not curriculum material), I am similarly concerned with what is selected, transformed and used in practice, and what is produced as a result. Selecting from domains of knowledge and transforming these in use for teaching is simultaneously the work of teaching and its outcome, what comes to be legitimated and so constituted as mathematical knowledge in a particular practice. As will be elaborated below, knowledge resources in use in teaching are recruited (appealed to) as grounds for, and to ground, what counts as mathematics in a school classroom context.


As a study concerned with teachers’ knowledge in use, QUANTUM also has roots in Lee Shulman’s seminal work on the professional knowledge base of teaching. Shulman (1986) distinguished subject matter knowledge (SMK), pedagogical content knowledge (PCK) and curriculum knowledge (CK) as critical categories in the professional knowledge base of teaching. Over the past two decades, a range of studies have developed out of Shulman’s early work, a considerable number of which have been located in mathematical contexts (e.g. Ball, Thames and Phelps, 2007; Even, 1990; Ma, 1999; Thwaites, Huckstep and Rowland, 2005). One strand of these mathematical studies has interrogated and elaborated categories of knowledge for teaching mathematics (Ball et al., 2007; Even, 1990; Ma, 1999), and how such “specialised knowledge” can be measured (Hill, Sleep, Lewis & Ball, 2007). The categories SMK, PCK and CK continue to be widely used. Remillard (2004), for example, notes SMK and PCK as components of the teacher in her teacher-curriculum conceptualisation (p.235), and Pepin (Chapter 5) explores teachers learning mathematical knowledge for teaching (MKT) through their use of curriculum texts. Of interest to this volume as a whole is a concluding comment by Hill et al that “teachers’ mathematical knowledge for teaching ... appears amplified ... by the choices made around curriculum materials" (p.500). Teachers with weaker MKT made poor or inappropriate use of curriculum materials. This comment reinforces the observation in our earlier research of a correlation between teachers' articulated mathematical purposes (part of their knowledges in use) and their appropriate use of curriculum materials, and so our focus on knowledge in QUANTUM.


QUANTUM aligns with a practice-based notion mathematics in and for teaching. Our object, though, is somewhat different. Working with a social epistemology, we understand that comes to be constituted as mathematics in any pedagogical practice is dialectically structured by pedagogic discourse (Bernstein, 2000). In other words, there is a structuring of mathematics by the institutions of schooling and curriculum, and by the activity of teaching within these. Mathematics in and for teaching can thus only be grasped through a language that positions it as structured by, and structuring of, pedagogic discourse. In this sense, SMK, PCK and CK in use in practice need to be understood as structured by pedagogic discourse. Consequently, a methodology for ‘seeing’ knowledges in use in teaching requires a theory of pedagogic discourse.


An underlying assumption in QUANTUM, following Davis (2001) is that pedagogic discourse (in both teacher education and school) proceeds through the operation of pedagogic judgement. As teachers and learners interact, criteria will be transmitted of what counts as the object of learning (e.g. what an ‘equation’ is in mathematics) and how the solving of problems related to this object this is to be demonstrated (what are legitimate ways of knowing, working with and talking about equations). As teachers provide opportunities for learners to engage with the intended object, at every step they make judgements as to how to respond to learners, what to offer next, how long to pursue a particular activity. All pedagogic judgement, of necessity, will appeal to some or other ground for legitimation, and so transmit criteria for what counts as mathematics.


In QUANTUM we describe these moments of judgements as appeals, arguing that teachers’ appeals to some or other ground illuminate the knowledge resources they call in, and so what comes to count as valid knowledge3. An underlying assumption here is that the demands of teaching in general, and the particular demands following changes in the mathematics curriculum in South Africa bring a range of domains of knowledge outside of mathematics into use. Parker (2006), building on Graven (2002) describes the range of mathematical orientations embedded in the new South African National Curriculum as including: mathematics as a disciplinary practice, thus including activity such as conjecturing, defining, proof; mathematics as relevant and practical, hence a modelling and problem-solving tool; mathematics as an established body of knowledge and skills thus requiring mastery of conventions, skills and algorithms; and mathematics as preparation for critical democratic citizenship, and hence a use of mathematics in everyday activity. What mathematical and other knowledge resources teachers select and use, and how these are structured by pedagogical discourse is important to understand. In our case studies of school mathematics teaching we are studying what and how teachers call in mathematical and other knowledge resources in their classroom practice so as to be able to describe what comes to function as ground in their practice, how and why.

Five similar case studies of mathematics teaching in a secondary classroom have been completed, each focused on a particular topic and unit of work4. We pursued the following questions: What domains of knowledge and practice (knowledge resources) does the teacher call in as he/she teaches (topic and unit) in grade (7 – 12) (the what)? What tasks of teaching does the teacher employ as he/she goes about the work of teaching (the how)? How can these two questions and their inter-relation be explained? In this chapter, I focus on the first of these questions, and its elaboration in two of the five case studies, cognisant that as knowledge resources come into focus, so other resources, as well as details on tasks of teaching go out of focus.


  1. ^ Evaluative events, criteria at work and knowledge resources in use


As is described in more detail elsewhere (Adler & Davis, 2006; Davis, Parker & Adler, 2007; Adler & Huillet, 2008; Adler, 2009), our methodology is inspired by the theory of pedagogic discourse developed by Basil Bernstein, and its illumination of the “inner logic of pedagogic discourse and its practices” (Bernstein, 1996, p. 18). It is this inner logic that shapes mathematics for teaching in school mathematics practice. Bernstein sees knowledges in school, or any pedagogic context, as structured by pedagogic communication, and impacting on meaning potential. There is a set of rules/procedures via which knowledges are converted into pedagogic communication5 (2000). For Bernstein, evaluative rules constitute specific practices, regulating what counts as valid knowledge (p. 28). Any pedagogic practice, either implicitly or explicitly, “transmits criteria”; indeed this is its major purpose. What is constituted as mathematics in any practice will be reflected through evaluation, through what and how criteria come to work6. How then are these criteria to be ‘seen’?


Our unit of analysis is what we call an evaluative event, that is, an interactional sequence in a mathematics classroom aimed at the constitution of a particular mathematics object. The shift from one event to the next is marked by a change in the object of learning. We work with the proposition that in pedagogic practice, in order for something to be learned, to become ‘known’, it has to be represented. Initial orientation to the object, then, is in some (re)presented form. Pedagogic interaction then produces a field of possibilities for the object. Through related judgements made on what is and is not the object, possibilities (potential meanings) are generated (or not) for/with learners. All judgement, hence all evaluation, necessarily appeals to some or other locus of legitimation to ground itself, even if only implicitly. An examination of what is appealed to and how appeals are made (i.e. how ground is functioning) delivers up insights into knowledge resources in use in a particular pedagogic practice7.


Of course, what teachers appeal to is an empirical question. Our analysis to date has revealed four broad domains of knowledge to which the teachers across all cases appealed (though in different ways and with different emphases) as they mediated mathematics in their classrooms: mathematical knowledge, everyday knowledge, professional knowledge8 and curriculum knowledge. These overlap but are not synonymous with Shulman’s categories of SMK, PCK and CK as key in the professional knowledge of teaching, and in a context of a hybrid curriculum, they do so in interesting ways.


Teachers, in interaction with learners, grounded discussion in the domain of mathematics itself, and more particularly school mathematics. We have described four categories of such mathematical knowledge/activity reflecting the multiple mathematical demands in the discourse of new curriculum texts: mathematical objects have properties, mathematical activity follows conventions (e.g. there are two diagonals in a square and these bisect each other; in an ordered pair we write the x-co-ordinate first); mathematical knowledge includes knowledge of procedures, mathematical activity is following rehearsed procedures (e.g. the first step to add two proper fractions is finding a common denominator); mathematical activity is empirical (e.g. testing whether a mathematical statement is true by examining an instance – substituting particular numbers or generating a particular visual display); mathematical activity involves generalising (e.g. examining whether a statement is always true).


The second domain of knowledge to which teachers appealed was non-mathematical, and is most aptly described as everyday knowledge and/or practice. Across the data teachers appealed to practical, sensible or experiential knowledge to legitimate or ground the object being attended to9. For example, the likelihood of events was discussed in relation to the state lottery, or obtaining a ‘6’ when throwing dice; collecting like terms was exemplified by grouping similar material objects; in a task that required students to cut up a fraction was containing a whole, halves, thirds, quarters, fifths etc. up to tenths, and then reorganise/mix the fraction pieces and make wholes from different unit fractions, some students pasted pieces that together formed more than a whole. The teacher’s explanation as to why this was inappropriate was grounded in the way bricks are cemented to form walls. Connecting or attempting to connect mathematical ideas to everyday knowledge and experience is a topic of considerable interest, indeed concern in mathematics education in South Africa, where the multiple mathematical goals of the curriculum have produced a prevalence of such discourse in many classrooms.


A third domain is teachers’ own professional knowledge and experience: what they have learned in and from practice. For example, all five teachers called on their knowledge from practice of the kinds of errors learners make, and built on these in their teaching. Knowing about student thinking and misconceptions is a central part of Shulman’s category of pedagogic content knowledge (PCK), and its centrality in teachers’ practice is well described in Margolinas (Chapter 13 – previous volume). We call this experiential knowledge to distinguish it from what we have elsewhere referred to as mathematics education knowledge (Adler, forthcoming), i.e. knowledge derived from research reported in the field.


There were additional criteria at work in the data where authority was located in what we have loosely called curriculum knowledge. In all our cases, and in some cases this was a significant resource for the teacher, the criteria transmitted were based either on what was stated in a textbook, or expected/required in an examination. In other words, what counted as legitimate was based on exemplification or description in a text or what would count for marks in an examination (e.g. let’s look at what the text book says; you get marks in the examination for labelling your axes). Of interest is whether and how this legitimation is integrated with or isolated from any mathematical rationale. Hence the use of curriculum knowledge here is not synonymous with Shulman’s category of CK, which, as part of content knowledge for teaching, is clearly so integrated. In the remainder of this chapter, I present two of the five cases to illustrate our methodology and to illuminate the knowledge resources in use in mathematics teaching.


  1. ^ Knowledge resources in use in school mathematics teaching


The five case studies noted above, have been described in detail elsewhere (Adler, 2009; Adler & Pillay, 2007; Kazima, Pillay and Adler, 2008). Briefly, selection of cases was based firstly on the teachers being viewed as competent, yet ‘ordinary’ or ‘somewhat typical’; and secondly on them being accessible and willing participants. We were interested in observing and interpreting the ordinariness of mathematics teaching, in contrast to a designed practice, and through this being able to engage with both presences and absences in teaching. Data collection was organized in each case around at least one week of teaching focused on a particular topic. In effect between four and eight consecutive lessons in one class (of 35+ students on average) were observed in each case. Each lesson was video recorded, transcribed and complemented by field notes taken during observation, together with copies of materials produced by both teacher and learners in the lessons. The teacher was interviewed before, during and after all the lessons were taught. All three interviews provided for conversation with the teacher on interpretations of what was observed in the lessons, as well as with the opportunity to probe why things were done in the way that they were.


The two cases discussed in this chapter are telling: they present different orientations to mathematical knowledge, and similar and different uses of knowledge resources. In so doing, and akin to material resources, they problematise notions of professional knowledge that are divorced from practice and context. They also open up challenging questions for mathematics teacher education.


^ Case 1: Knowledge resources in use in teaching linear functions, Grade 10. 10


Nash11, is an experienced and qualified mathematics teacher. He teaches across grades 8 to 12 in a public school where learners come from a range of socio-economic backgrounds. He has access to and uses curriculum documents issued by the National Department of Education (DoE), a selection of mathematics textbooks, a chalkboard and an overhead projector. He collaborates with other mathematics teachers in the school, particularly for planning teaching and assessment. He is well respected and regarded as a successful teacher in his school, and in the district.


Nash’s approach to teaching can be typically described: he gave explanations from the chalkboard; learners were then required to complete an exercise sheet he prepared. He did not use a textbook nor did he refer his learners to any textbook during the lessons observed. A six page handout containing notes (e.g. parallel lines have equal gradients), methods (steps to follow in solving a problem) and questions (resembling that of a typical textbook) formed the support materials used. This handout was developed by Nash in collaboration with his Grade 10 teaching colleagues12.


In the eight lessons observed, Nash dealt with the notion of dependent and independent variables; the gradient and y-intercept method for sketching a line; the dual intercept method; parallel and perpendicular lines; determining equations of straight lines when information about the line is given in words and also in the form of a graph; solving linear simultaneous equations graphically. He completed the unit with a class test. The overall pass rate was 94%; class average was 65%; and 34% obtained over 80%. Of course, success is relative to the nature of the test and the pedagogy of which it forms part. The test questions were a replica of questions in the handout given to learners, and so a reproduction of what had been dealt with in class.


In the first two lessons, Nash dealt with drawing the graph of a linear equation first from a table of values, and then using the gradient and y-intercept method. In Lesson 3, he moved on to demonstrate how to draw the graph of the function: 3x – 2y = 6, using the dual intercept method. The extract below is from the discussion that followed. It illustrates an evaluative event, the operation of pedagogic judgement in this practice, and the kinds of knowledge resources Nash called in to ground the dual intercept method. The beginning of the event – the (re)presentation of the equation 3x – 2y = 6 is not included here. Extract 1 picks up from where Nash is demonstrating what to do. The appeals - moments of judgement - are underlined, and related grounds described.


Extract 1. Lesson 3, Case 1. (Lr = learner)

Knowledge resources in use

Nash: first make your x equal to zero that gives me my y-intercept. Then the y equal to zero gives me my x-intercept. Put down the two points we only need two points to draw the graph.

Lr 1: You dont need all the other parts?

Nash: You dont have to put down the other parts … its useless having -6 on the top there (points to the y axis) what does the -6 tell us about the graph? It doesn’t tell us much about the graph. Whats important features of this graph we can work out from here (points to the graph drawn) we can see what the gradient is … is this graph a positive or a negative?

Lrs: (chorus) positive.

Nash: it’s a positive gradient … we can see theres our y-intercept, theres our x-intercept (points to the points (0;-3) and (2;0) respectively)

Grounds: procedural. Steps to carry out, legitimated by assertion by Nash.


Grounds questioned by a learner (who could wish only to secure procedural understanding).


^ Grounds: empirical.

Important feature of a graph are what can be ‘seen’


Mathematics is procedural and justified empirically

(in next minute, Nash emphasises importance of labelling points in an exam, in response to a question)

^ Grounds curriculum knowledge. Mathematics is what is expected in the examination

Lr 2: Sir, is this the simplest method sir?

Lr 3: How do you identify which side must it go, whether it’s the right hand side (Nash interrupts)

^ Nash: (response to Lr 2) You just join the two dots.

Lr 2: That’s it?

Nash: Yeah the dots will automatically … if it was a positive gradient it will automatically … if this was (refers to the line just drawn) negative … that means this dot (points the x-intercept) will be on that side (points to the negative x axis) … because if the gradient was negative, how could it cut on that side? (points to the positive x axis).

Lr 2: ^ Is this the simplest method sir?

Further questioning of ground

Grounds: procedural assertion. Again further reflection – though meaning of ‘simplest’ not apparent.


Mathematics is procedural, and based on authority of teacher

^ Nash: The simplest method and the most accurate ...

Learner 4: Compared to which one?

Nash: Compared to that one (points to the calculation of the previous question where the gradient and y-intercept method was used) because here if you make an error trying to write it in y form that means it now affects your graph whereas here (points to the calculations he has just done on the dual intercept method) you can go and check again you can substitute if I substitute for 2 in there (points to the x in 3x 2y = 6) I should end up with 0.


Grounds: avoiding error.


Mathematics demands accuracy and is error free


Judgments in this extract emerge in the interactions between Nash and four learners who ask questions of clarification, thus requiring Nash to call in resources to ground and legitimate what counts as mathematical activity and so mathematical knowledge in this class. Learners’ questions were of clarification on what to do, with possibilities for why this was the case. The opportunities for mathematical engagement were not taken up, e.g. the explanation of why only two points are needed, and the direction of the graph are grounded in what can be ‘seen’. The simplicity of the method is that it avoids errors of calculation and so is accurate. In this event, Nash’ responses were about what to do. Legitimation was provided by steps to follow or what could be ‘seen’. Appeals were to procedural knowledge or to some empirical feature of the object being discussed, or to curriculum knowledge (what counts in the examination).


This event, and the operation of pedagogic judgement is typical of how Nash conducted his teaching. Table 1 below summarises the full set of 65 events across the eight lessons, and the knowledge resources Nash recruited. As indicated above and in the numbers in the table, more than one kind of knowledge resource could be called on within one event. Nash’s appeals to everyday knowledge and his professional experience were not evidenced in this event. Briefly, his calling in of everyday knowledge, which were to add meaning for learners, were often problematic from a mathematical point of view. For example, he attempted to explain independent and dependent variables by referring to a marriage, husband and wife and expressed amusement and concern when discussing this in his post lesson interview!


Table 1: Case 1, Linear functions, Grade 10

Total occurrences

%Occurred

Events

65

Appeals/knowledge resources







Mathematics

Empirical

24

37

Procedures/conventions

43

66

Experience

Professional

18

28

Everyday

14

22

Curriculum

Examinations/tests

6

9

Text book

7

11


In overview, mathematical ground in this class was procedural, empirical, error free and authoritative, and supported by professional and curriculum knowledge. That these latter are key in Nash’s practice were reflected in his post lesson interview. Nash talked at length about how he plans his teaching, key to which is a practice he calls ‘backwards chaining’.

First and foremost when you look[ing] at the topic / my preferred method is … backwards chaining. [which] means the end product. What type of questions do I see in the exam, how does this relate to the [Gr 12] exams, similar questions that relate to further exams and then work backwards from there … what leads up to completing a complicated question or solving a particular problem and then breaking it down till you come to the most elementary skills that are involved; and then you begin with these particular skills for a period of time till you come to a stage where you’re able to incorporate all these skills to solve a problem or the final goal that you had.

He also illuminated how his experience factors into his planning and teaching, and his attention to error free mathematics. Learners’ misconceptions and errors are a teaching device rather than a feature of what it means to be mathematical.


You see in a classroom situation … you actually learn more from misconceptions and errors … than by actually doing the right thing. If you put a sum on the board and everybody gets it right, you realise after a while the sum itself doesn’t have any meaning to it, but once they make errors and you make them aware of their errors or … misconceptions – you realise that your lessons progress much more effectively … correcting these deficiencies … these errors and misconceptions.

  1. ^ Case 2. Knowledge resources used teaching Geometric thinking in Grade 1013.


Ken14, is also an experienced and qualified mathematics teacher. He has a 4-year higher diploma in education majoring in mathematics, an Honours degree in Mathematics Education, and at the time of the data collection was studying for his Masters. He has thus had opportunity to learn from the field of mathematics education research. He has eleven years secondary teaching experience across grades 8 to 12. The conditions in his school are similar to those in Nash’s school, and grade level teachers similarly prepare support materials and assessments for units of work. Ken too is well respected and successful in his school.


Ken prepared and presented a week’s work focused on polygons, the relationship between its sides, vertices and diagonals, generalisation and proof. He described his plans for the lessons as a set of ‘different’ activities to ‘revise’ and enable learners to reflect more deeply on geometry. The five lessons were organised around two complex, extended tasks. The first involved the relationship between the number of sides of a polygon and its diagonals. The second was an applied problem requiring learners to interpret a situation and recognise the need for using knowledge of equal areas of parallelograms on the same base and with same height to solve the problem.


The extract below is from the first lesson and work on the first task. Learners were to find the number of diagonals in a 700-sided polygon, a sufficiently large number to require reasoning, and generalising activity. The extract captures an evaluative event, with the presentation of the task marking the beginning of the event. It continues for 14 minutes as the teacher and learners interact on what and how they could produce an orientation and solution to the problem. Some progress is made, as learners are pushed to reflect on specific empirical cases. As with extract 1, the underlined utterances illustrate the kinds of appeals and so knowledge resources Ken calls on in his practice. All judgements towards the object – a justified account of the relationship between the number of sides and diagonals in a polygon - emerge from utterances of either or both learners and the teacher.


Extract 2. Lesson 1, Case 2.

Knowledge resources in use

In the first seven minutes of the class, the Ken (standing in the front of the class), puts the following problem onto the Overhead Projector: How many diagonals are there in a 700-sided polygon? After seven minutes, Ken calls the class attention.

Ken: Ok! Guys, time’s up. Five minutes is over. Who of you thinks they solved the problem? ….

Lr 1: I just divided 700 by 2.

Ken: You just divided 700 by 2.

Lr 1: Sir, one of the side’s have, like a corner. Yes … (inaudible), because of the diagonals. Therefore two of the sides makes like a corner. So I just divided by two (Inaudible).

Ken: So you just divide the 700 by 2. And what do you base that on? So what do you base that on because theres 700 sides. So how many corners will there be if theres, 700 sides?

[…] there is some discussion on about 700 sides and corners, and whether there are 350 or 175 diagonals.



^ Grounds: empirical and procedural

Grounds: properties of mathematical object


Mathematics is procedural and justified empirically; mathematical objects have properties.

^ Ken: Let’s hear somebody else opinion.

Lr2: Sir what I’ve done sir is …First 700 is too many sides to draw. So if there is four sides how will I do that sir? Then I figure that the four sides must be divided by two. Four divided by two equals two diagonals. So take 700, divide by two will give you the answer. So thats the answer ...

Ken: So you say that, there’s too many sides to draw. If I can just hear you clearly; … that 700 sides are too many sides, too big a polygon to draw. Let me get it clear. So you took a smaller polygon of four sides and drew the diagonals in there. So how many diagonals you get?

^ Lr2: In a four sided shape sir, I got two.

Ken: Two. So you deduced from that one example that you should divide the 700 by two as well? So you only went as far as a 4 sided shape? You didnt test anything else.



Grounds: empirical, pragmatic and procedural.

Mathematical activity is deductive and inductive

Lr2: Yes, I don’t want to confuse myself.

Ken: So you dont want to confuse yourself. So youre happy with that solution, having tested only one polygon?

Lr2: Inaudible response.



Ken: What about you Lr4? You said you agree.

Lr4: He makes sense. (referring to Lr1)He proved it. He used a square.

Tr: He used a square? Are you convinced by using a square that he is right?

Lr5: But sir, here on my page I also did the same thing. I made a 6-sided shape and saw the same thing. Because a six thing has six corners and has three diagonals.

^ Lr1: So what about a 5 - sided shape? Then sir.

Ken: What about a 5 - sided shape? You think it would have 5 corners? How many diagonals?

Interaction continues. Ken intervenes as he hears some confusion between polygon and pentagon, and turns the class’ attention to definitions of various polygons having learners look up meanings in their mathematics dictionaries.


^ Ken challenges the empirical ground and single case.


Grounds: empirical


Challenge to the empirical ground and single case.


Learners confirm, then challenge with counterexamples where Ground remains empirical.


Mathematical activity involves reasoning; providing examples and counterexamples.


Mathematical objects have properties and are defined.


The discussion and clarification of different polygons continued for some time, after which Ken brought the focus back on to the problem of finding the number of diagonals in a 700-sided figure, and work on this continues through the rest of this lesson and the next two lessons. It is interesting to note that in all the discussion on the 700-sided figure and the empirical instances discussed, a polygon is assumed to be regular and convex. Properties discussed focus on the number of sides and related number of angles in a polygon (again regular and convex), and diagonal is defined as a line connecting two non-consecutive corners. One route to solving the problem – noticing a relationship between the number of corners and the number of diagonals from each corner – and so the possibility of a general formula, becomes dominant. Ken’s focus throughout the two lessons is on conjecture, justification, counterexample and proof as mathematical processes. The mathematical object itself – a polygon – through which these processes are to be learned and developed, is assumed.


Judgements in this extract flow in interaction between Learners 1, 2, 4, 5 and the teacher. The knowledge resources called in fit within the broad category of mathematics. In particular, the ground for the teacher is reflected in his insistence on mathematical justification. However, these grounds are distinctive. The first appeal (Lr1) is to the empirical, a particular case that can be ‘seen’ (two of the sides makes like a corner) and a related procedure (I just divided by 2), followed by Ken’s challenge through an appeal to properties of a 700-sided polygon. The appeal of Lr2, is also to the empirical, to a special case (four sides), and this is supported by Lr4, and then by L5 (who did ‘the same thing’ with six sides). It is interesting to reflect here on what possible notion of diagonal is being used by Lr5. While there has been discussion on diagonals as connecting non-consecutive corners, it is possible Lr5 is only considering those that pass through the centre of the polygon. Ken does not probe this response, rather picking up on Lr1’s suggestion of a counterexample (what about a 5-sided shape?), which is also an empirical case. The appeals by the teacher, as he interacts with, revoices and responds to learner suggestions, are to the empirical and through counterexample, suggesting, and so providing criteria, that the justifications provided are not yet adequate – they do not go beyond specific cases. The grounds that came to function over the five lessons are summarised below


Table 2. Case 2, Geometric thinking, Gr. 10

Total occurrences

%Occurred

Events

37

Appeals/knowledge resources







Mathematics

Empirical

23

64

General

14

36

Procedures/conventions

8

23

Experience

Professional

0

0

Everyday

2

5

Curriculum

Examinations/tests

11

32

Text book

0

0


In sum, a range of mathematical grounds (with empirical dominant, and including appeals to mathematics as generalising activity) overshadowed curriculum knowledge, with everyday knowledge barely present. In the pre-observation interview Ken explained that his intentions with the lessons he had planned was to focus on the understanding of proofs. He wanted them to see proof as “a way of doing maths, getting a deeper understanding and communicating that maths to others”. In the post lessons interview, interestingly, Ken explained that these lessons were not part of his normal teaching. He used the research project to do what he thought was important, but otherwise didn’t have time for. He nevertheless justified this inclusion in terms of the new curriculum, which had a strong emphasis on proof, on “how to prove and what makes a proof”. When probed as to why he did not do this kind of lesson in his ‘normal’ teaching, he explained that there was shared preparation for each grade, and “because of time constraints and assessments, you follow the prep and do it, even if you don’t agree”.


  1. ^ The significance of knowledge resources in use in practice


In the introductory sections of this chapter, I argued that the knowledges teachers call in in their practice matters. Earlier research, beyond my own project, suggested that teachers’ professional knowledge was a significant factor in the relationship between teachers and curriculum materials, and particularly so in contexts of poverty. Where curriculum resources are minimal, the insertion of new texts critically depends on what and how teachers are able to use mathematics and other knowledge domains appropriately for their teaching. By implication, a study of curriculum text as ‘lived’ needs to foreground knowledge resources in use. This chapter has offered a methodology – structured by evaluative events and criteria in use to ground objects of learning and teaching – for illuminating knowledges in use.


The methodology was put to work in two classrooms, enabling a description of the knowledge resources two different teachers called in to ground the mathematics they were teaching. Nash drew on extra-mathematical domains of knowledge, particularly curriculum knowledge and everyday knowledge, together with procedural knowledge of mathematics. Ken drew largely from the mathematical domain. The knowledge resources that sourced the work of these two teachers were substantively different, and so too was the mathematics that came to be legitimated in these classrooms. Nash backward chained from valued school knowledge reflected in national examinations, and built in teaching strategies to elicit errors from learners that he could then correct; and he did this by focusing on procedural knowledge and what is empirically verifiable. This practice produces student ‘success’, though, in Ruthven’s terms, he could be described as following a mathematically constrained script and activity format (Chapter XXX). Ken on the other hand, uses mathematics in extended ways to engage learners in reasoning practices like conjecturing leading to proof. However, he does this outside of his normal teaching. In Ruthven’s terms, he is not able to integrate new mathematical teaching practices into the well oiled activity format, curriculum script, and time economy that structures teaching practices in his school.


The object of QUANTUM’s research is not on what a particular teacher does or does not do, in some decontextualised sense, but rather on what comes to be used, and thus how mathematics is constituted in specific practices. Through the cases in this chapter, we see that observing teachers in practice is a window into the varying knowledge resources in use within a particular curriculum practice and set of institutional constraints. These insights were ‘revealed’ through the notion of ‘ground’ as that which is called on to legitimate what counts as mathematics in teaching. The methodological tools developed in the QUANTUM project probe beneath surface features of pedagogic practice to reveal substantive differences in the way teachers recruit and ground knowledge objects as they go about their mathematical work, and so into how knowledges become ‘lived’ resources. These, in turn, open up space for engaging with what is and is not included in teacher education.


ACKNOWLEDGEMENTS


This paper forms part of the QUANTUM research project on Mathematics for Teaching, directed by Jill Adler, at the University of the Witwatersrand. Dr Zain Davis from the University of Cape Town is a co-investigator and central to the theoretical and methodological work in QUANTUM. The methodological innovation described here has its roots in our joint work in mathematics teacher education. The elaboration into classroom teaching was enabled by the work of Masters students at the University of the Witwatersrand. This material is based upon work supported by the National Research Foundation under Grant number FA2006031800003. Any opinion, findings and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Research Foundation.


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1 QUANTUM is an R & D project on mathematical education for teachers in South Africa. Its development arm focused on qualifications for teachers underqualified in mathematics (hence the name) and completed its tasks in 2003. QUANTUM continues as a collaborative research project.

2 I use ‘grasp’ here in a technical sense to convey the message that knowledge in use in practice is not unproblematically ‘visible’, but is made so through the deployment of specific methodological tools and analytic resources.

3 For Bernstein (2000, pp. 32-33), what comes to count is never neutral. Pedagogic discourse necessarily delocates and relocates knowledges and discourses, and recontextualisation (transformation) creates a gap wherein ideology is always at play. What teachers call on is no simple reflection of what they know. In this chapter I do not explore the ideological and so political in the constitution of mathematics in and for teaching. We have done this elsewhere, particularly in our reporting of the constitution of mathematics for teaching in teacher education.

4 Studies in school classrooms have been undertaken by Masters students and a post doctoral fellow at the University of the Witwatersrand, working in QUANTUM. I acknowledge here the significant contribution of Mercy Kazima, Vasen Pillay, Talasi Tatolo, Shiela Naidoo and Sharon Govender and their studies to the overall work in QUANTUM, and specifically to this paper.

5 As a sociologist, Bernstein was ultimately concerned with the social distribution of knowledge, with how the internal logic of pedagogic discourse came to specialised consciousness and in inequitable ways. This social project is not in focus in this chapter.

6 It is important to note this specific use of ‘evaluation’ in Bernstein’s work. It does not refer to assessment, nor to an everyday use of judgement. Rather it is a concept for capturing the workings of criteria for legitimation of knowledge and knowing in pedagogical practice.

7 This set of propositions is elaborated in Davis (2005), and Davis et al (2003), and the result of the collaborative work in QUANTUM, and Davis’ study.

8 In Adler (2009), everyday knowledge and professional knowledge are collapsed, both viewed as knowledge from practical experience. The separation comes from the development of this chapter.

9 In our description of ground, we are not concerned with their mathematical correctness or whether they are appropriate. Our task is to describe what teachers call in, whatever this is.

10 For a detailed account of this study see Pillay (2006) and Adler & Pillay (2007)

11 This is a pseudonym.

12 This documentation practice, unfortunately in the light of this book, was not in focus in our research.

13 For detailed account of this study see Naidoo (2008)

14 This is a pseudonym.





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