In South Africa, differential performance in school mathematics with respect to social class remains an enduring concern as reflected in national and international large-scale assessments. The article examines the implications of evaluation for orientations to mathematics in a school populated by learners from upper-middle-class or elite backgrounds and a school populated by learners from working-class backgrounds. The particular focus is on mathematics problems featured in tests used by two Grade 10 teachers in each school and teachers’ marking of learners’ test scripts. A distinction between single-topic and multi-topic mathematics problem types is refracted through an analysis that draws on the adaptation by Davis of Lotman’s distinction between content orientation and expression orientation with respect to the reproduction of texts.

The analysis reveals a preponderance of single-topic mathematics problems and the absence of multi-topic mathematics problems employed in the school populated by learners from working-class backgrounds and the presence of both single-topic and multi-topic problem types in the school populated by learners from middle-class or elite backgrounds. Differences in the types of mathematical problems suggest differences in mathematical demand expected of learners and differences in their preparation for examinations in the two social class contexts.

The selection of test problems and the marking of test scripts as instances of evaluation construct an orientation to mathematics that is expression oriented in the working-class context whereas both expression and content orientations are evident in the middle-class or elite context. The analysis provides a potential explanation for the persistent disparity in mathematics performance along social class lines in South African secondary schools.

Narrowing the achievement gap in school mathematics in South Africa with respect to social class remains a persistent social justice issue in spite of extensive curriculum reforms. Difference in school mathematics performance between learners from middle-class families and learners from working-class families in South Africa has to a great extent been documented in large-scale quantitative studies (e.g., Reddy, Van der Berg, Lebani, & Berkowitz,

We have to bear in mind though that social class and ‘race’ remain intertwined in South Africa. Despite ‘racial’ desegregation of more affluent schools, poorer schools remain populated largely by ‘Black’ learners (Spaull,

Some smaller qualitative studies are concerned with the nature of the relation between mathematics performance and social class in an attempt to understand the underlying factors impacting on learner performance in school mathematics. Most research in mathematics education in South Africa (e.g. Carnoy et al.,

Despite diversity with respect to methodology, there is convergence in the findings of local and international comparative studies concerned with school mathematics in relation to the differential distribution of knowledge across different social class settings (Anyon,

The orientation to mathematics in schools populated by learners from working-class families is often described in this literature as procedural or weakly bounded from the ‘everyday’. In contrast, orientation to mathematics in schools populated by learners from middle-class families is commonly described in this literature as conceptual as opposed to procedural or as strongly bounded from the ‘everyday’. The literature thus partitions descriptions of mathematics realised in pedagogic contexts differentiated with respect to social class in terms of the academic-everyday distinction and procedural-conceptual opposition and remains relatively silent on similarities in the mathematics realised in those pedagogic contexts.

This article examines the orientations to mathematics in two schools differentiated with respect to learners’ social class membership by examining how evaluation structures orientation to mathematics. In particular, the research question pursued in this article is: How does evaluation function in relation to Grade 10 mathematics in two schools that differ with respect to the social class membership of their learner populations and what are the implications for learners’ orientations to mathematics?

The general methodology underpinning the analysis of data in this article is informed by Bernstein’s (

Bernstein (

Pedagogy entails a relationship between two or more notional pedagogic subjects, the teacher and the learner, with the reproduction of knowledge being the knot that ties the two together, referred to as a

For Bernstein, the centrality of evaluation is emphasised in his discussion of the

We can see that key to pedagogic practice is continuous evaluation. […] This is what the device is about. Evaluation condenses the meaning of the whole device. We are now in a position where we can derive the whole purpose of the device. The purpose of the device is to provide a symbolic ruler of consciousness. (Bernstein,

The pedagogic device, which is Bernstein’s attempt to relate social structure to individual consciousness, entails three hierarchically related ‘rules’ – the distributive, recontextualising and evaluative rules (Bernstein,

The evaluative rule is key in the pedagogic reproduction of knowledge. Bernstein argues that pedagogic practice is characterised by the ever-present evaluative activity where evaluation distinguishes legitimate from non-legitimate knowledge statements for learners (Bernstein, ^{2} = 4 where

Texts produced by teachers and learners make explicit the knowledge accepted as legitimate in a pedagogic situation and provide criteria that mark out legitimate knowledge statements from non-legitimate statements. The centrality of evaluation is underscored by Bernstein (

Lotman’s distinction between

Cultures can be governed by a

A grammar-oriented culture [i.e. governed by a system of rules] depends on ‘Handbooks’, while a text-oriented culture [i.e. governed by a repetition of model texts] depends on ‘The Book’. A handbook is a code which permits further messages and texts, whereas a book is a text, generated by an as-yet-unknown rule which, once analyzed and reduced to a handbook-like form, can suggest new ways of producing further texts (Eco in Lotman,

A grammar-oriented culture which is governed by a system of rules is juxtaposed with a text-oriented culture which is governed by a repetition of model texts. Lotman’s categories, while useful as a heuristic for thinking about pedagogy, require adaptation for analysis of pedagogic situations. Pedagogic modalities in which learners are encouraged to reproduce texts, through repetition and rehearsal, that precisely conform with texts considered as legitimate in the pedagogic context, are suggestive of Lotman’s text-oriented cultures. The use of ‘model answers’ for classes of mathematics problems often used in pedagogic contexts resonates with Lotman’s text orientation.

His concept of grammar-oriented cultures, on the other hand, is comparable with pedagogic modalities that encourage ‘syntactical symbol manipulation and propositional descriptions of relations between mathematical objects’ (Davis,

Although combinatorial rules are embodied in model texts in text-oriented cultures, the repetition and rehearsal of texts seems to suggest that individuals can produce texts without the use of combinatorial rules (see Davis,

Lotman suggests that text-oriented societies are at the same time expression-oriented ones, while grammar-oriented societies are content-oriented. The reason for such a definition becomes clear when one considers the fact that a culture which has evolved a highly differentiated content-system has also provided expression-units corresponding to its content-units, and may therefore establish a so-called ‘grammatical’ system — this simply being a highly articulated code. On the contrary a culture which has not yet differentiated its content-units expresses (through macroscopic expressive grouping: the texts) a sort of

So, for Davis (^{2} = 9 which insists that the equation must be transformed into standard form (^{2} + ^{2} = 9 can be solved by using square roots, thus bypassing the need to transform (^{2} = 9 into standard form.

With a

Previous studies show that a large proportion of mathematics lessons is spent on solving mathematics problems (see, for example, US Department of Education, 2003). Mathematics problems are commonly used as the main vehicles for the elaboration of mathematics topics in schooling. Typically, teachers use mathematics problems as worked examples to illustrate particular solution procedures, as practice exercises to provide opportunities for learners to become proficient at executing those procedures and in assessment tasks such as tests and examinations to ascertain learners’ knowledge of mathematics. Mathematics problems, sometimes referred to as mathematics tasks in the literature (see eds. Shimizu, Kaur, Huang, & Clarke,

Mathematics tasks or problems are described in different ways in the literature. Boaler’s (

The categorisation of problem types used in this article references the topics indexed by the mathematics problems because of the larger project’s interest in the

I distinguish between

Problem 1 of test administered by Ivory College teachers.

Multi-topic problems encourage

However, the content realised through solving mathematics problems is not the issue under discussion in this article. Instead, mathematics problems serve as instances of pedagogic evaluation in that they are selected by teachers for the elaboration of mathematics topics and are used to assess learners’ acquisition of mathematics topics.

Social class serves as a background contextual variable in this study and not as an explanatory category. The intention here is not to set up causal relations between social class on the one hand and orientations to mathematics on the other hand. The selection of empirical sites on the basis of the social class membership of a school’s learner population was guided by the assumption that because social class continues to be aligned with differential mathematics achievement, such differences potentially point to contrasts in the way evaluation functions.

As indicated earlier ‘race’ and social class remain inextricably linked in South Africa. I use the term ‘race’ in quotation marks because ‘race’ has little biological validity (Yudell, Roberts, DeSalle, & Tishkoff,

Deregulation of ‘race’ as an admission requirement in all schools followed shortly after the demise of apartheid. Subsequently, post-apartheid South Africa has witnessed substantial transformation in the ‘racial’ demographics of school populations. ‘White’, ‘Coloured’ and ‘Indian’ schools have changed with respect to their ‘racial’ composition, but the learner populations of ex-Department of Education and Training schools have to a large extent remained exclusively ‘African’ (Sujee, 2004, as cited in Chisholm & Sujee,

Enrolment patterns across the South African schooling system are now determined largely by school fees. No-fee and low-fee-paying public and independent schools serve the majority of South African children – working-class and lower-working-class children who are predominantly ‘African’ and ‘Coloured’ (Franklin,

This article reports on a research study (Jaffer,

The study was designed as a comparative study of two schools that differ with respect to the social class membership of their student population. The selection of schools was based on learners’ social class membership. School fees were used as a proxy for learners’ social class membership which was confirmed through biographical information obtained from a learner questionnaire and school questionnaire completed by the principal. Arbor High is a no-fee school and Ivory College is a private school with school fees set at R85 000 per annum in 2012.

Ivory College is ‘racially’ mixed and serves learners from upper-middle-class or elite families. Arbor High is populated by ‘African’ learners from working-class families. At each school, two teachers and learners in one of their Grade 10 mathematics classes comprise the research participants of the study. Sara and Jada taught at Ivory College, Maya and Jono at Arbor High.

The data collected for the research study included video-recorded mathematics lessons, curriculum and teaching resources such as textbooks, worksheets, tests designed and administered by the teachers, test scripts from the four Grade 10 mathematics classes and interviews with selected learners in each class. This article focuses specifically on the mathematics problems used in a test administered by the teachers and the marking of test scripts as instances of pedagogic evaluation.

The announced topics in the observed lessons of the four teachers are all related to the CAPS curriculum topic Functions, which according to the CAPS pace setter is scheduled for teaching in the second term of the school year (Department of Basic Education,

Ivory College wrote a ‘common’ test, which was a test set by one of the Grade 10 teachers and written by all Grade 10 learners at the school. The test, referred to as

Ivory College learners were given considerable practice opportunities by their teachers, mostly independently of the teacher, since the tasks were either given as homework exercises or tutorials that were required to be submitted for marks. The test problems in the

In addition, Ivory College learners were exposed to variations in phrasing of mathematics problems. For example, problems on calculating the points of intersection of two functions were posed in different ways in the worksheet and the tutorial: (1) calculate the points of intersection of the functions ^{2} + 9 = 2

The selection of mathematics problems for the test, worksheet and tutorial suggests that evaluation functions in a way that attempts to move beyond recall and rehearsal of procedures for solving particular classes of mathematics problems. In other words, fostering content orientation to mathematics rather than an expression orientation. Furthermore, the test, like the worksheet and tutorial, encourages inter-topic connectivity in that mathematics problems focus on more than one topic simultaneously and so require learners to select appropriate computational resources. The evaluative activity instantiated in the selection of mathematics problems for the test suggests an orientation to mathematics that is content oriented rather than expression oriented.

The memorandum of

Sara mostly makes the evaluative criteria explicit to learners by correcting errors or providing evaluative commentary (see

A learner’s marked solution to Problem 1.1.

Sara awarded full marks for Problem 1.1 to two learners despite the fact that they generated the correct

Sara’s marking of Problem 1.6 (see ^{2} + 2

The statement 2

Sara’s marking of Learner 1’s and Learner 2’s solutions to Problem 1.6 suggests that the logical connectives and order relations are not explicitly required as computational resources and indicates that content that diverges both at the level of expression and at the level of content is accepted as correct. Furthermore, her marking indicates that the presence of the expressions /

Jada’s marking was consistent across learners’ test scripts and she made the evaluative criteria explicit to learners by correcting errors or providing evaluative commentary (see

Noa’s (P12) solution to Problem 1.1.

At Arbor High, each Grade 10 teacher set their own mathematics test. The tests administered by the two Grade 10 teachers at Arbor High differed in terms of announced topics assessed. Maya’s test covered the topics dealt with during the observed lessons and consisted of four test problems. Problems 1, 2 and 3 entailed finding the equation of a given function provided as a sketch (see

Extract of test on functions administered by Maya.

Problems 1–3 are of the type ‘calculate the equation of the function’ and Problem 4 of the type ‘sketch the graph of the function’, which were the problem types covered in class during the observed lessons. All the test problems are classified as single-topic problems. The test problem types are the same as the problems used in the observed lessons, but the examples differed.

The test, like the problems used during the observed lessons, directly named the procedure that learners were expected to carry out. Maya’s learners were not expected to analyse problems in order to select a particular procedure for solving a problem. Furthermore, learners in Maya’s pedagogic context were not exposed to variations in the phrasing of problem types. For example, the ‘calculate the equation of the linear function’ mathematics problems provided by Maya all entailed a sketch with given intercepts. Variants of the same mathematics problem could, for example, be achieved through changing the nature of the given points: two intercepts or a

The absence of problems that require analysis in order to select appropriate procedures for solving the problem and the lack of variation in problem statements are suggestive of an orientation to mathematics that attempts to elicit precise responses from learners through the rehearsal of particular procedures for solving particular problems. In other words, learners are encouraged to recognise problem types and then select the correct procedure.

Furthermore, the test, like the problems used in the observed lessons, treated topics separately. Therefore, unlike the Ivory College test, the test set by Maya suggests a lack of inter-topic connectivity.

Only sketches of functions were provided in Problems 1, 2 and 3 of the test (see

The total mark for the test and mark allocation per problem were not provided to learners and the teacher’s memorandum did not show the mark allocation. From the marked scripts, it became apparent that the teacher allocated four marks per problem, bringing the total of the test to 16. In the marking of learners’ test scripts, Maya at times indicated that an error was produced and deducted marks to penalise the learner. On a number of occasions though, errors produced by learners were not highlighted by the teacher and were marked as correct. An example is illustrated in

Learner 1’s marked solution to Problem 1.

Learner 1 identifies the sketch as representing a parabola as indicated in her choice of general formula ^{2} +

Maya awards full marks to Learner 2 for her solution to Problem 1 despite a number of computational inconsistencies with respect to mathematics. The learner’s final equation ^{2} −4^{2} +

Maya’s marking of the learner’s solution suggests a strong expression orientation to mathematics since her assessment of learners’ work validates the production of the expected expressions despite divergence from the mathematics content associated with the topic. Mathematics constituted in this pedagogic context is primarily a form of mathematical knowledge which diverges from mathematics at the level of expression and at the level of the content associated with the topic. The teacher’s marking of learners’ solutions suggests that her evaluation cultivates an orientation to mathematics that is expression oriented.

Jono’s test (see

Extract of test on domain and range of functions administered by Jono.

Test problem A1 corresponds with worksheet problem A5, test problem A2 with worksheet problem A1, test problems B1–B3 with worksheet problems B1–B3, test problems C1–C3 with worksheet problems C1–C3, test problem D1 with worksheet problem D1 and test problem D2 with worksheet problem D5. Thus, all the problems from the test were selected from the worksheet, sometimes in the same order.

All the test problems are classified as single-topic mathematics problems. Jono’s learners, like Maya’s learners, were not expected to analyse problems in order to select a particular procedure for solving a problem. The function type, for example linear functions or quadratic functions, was identified for the learner, thus generating a test of low complexity because learners mostly needed to recall the propositions with respect to each function type established during the observed lessons. For example, learners were expected to recall that the domain of a linear function is{

Jono’s learners, like Maya’s learners, were not exposed to variations in problem types. For example, mathematics problems related to domain and range of functions could be set in graphical form. In other words, learners have to deduce the domain and range from the graph of a function. The absence of problems that required analysis in order to select appropriate procedures for solving them and the lack of variation in problem statements suggest that learners were expected to rehearse and repeat particular procedures for solving particular problems, typical of an expression orientation to mathematics.

Furthermore, the learners had seen the test problems and worked through the problems in class. It seems that the test assesses whether learners are able to repeat the texts produced in class under test conditions. In other words, the evaluation encourages learners to reproduce texts that precisely conform with texts that are considered as legitimate in the pedagogic context, through repetition and rehearsal. The test is therefore strongly suggestive of an orientation to mathematics that is expression oriented.

The test memorandum provides solutions to the test problems but how marks ought to be awarded is not made explicit. The marked tests show that two marks were allocated per test problem, half a mark each for the domain and range expressed in set builder notation and in interval notation. The memorandum provided the domain for Problem A1 as {

Solutions to Problem A1 of Jono’s learners selected to be interviewed.

Name | Code | Domain (set builder) | Domain (interval) | Range (set builder) | Range (interval) | Mark awarded |
---|---|---|---|---|---|---|

Tim | P01 | { |
(−∞; ∞) correct | { |
(−∞; ∞) correct | 2 |

Ali | P02 | { |
(−∞; ∞) correct | { |
(−∞; ∞) correct | 2 |

Ozi | P11 | { |
(∞; ∞) incorrect | { |
(−∞; ∞) correct | 1.5 |

Lea | P12 | { |
(0; ∞) incorrect | { |
(−∞; ∞) correct | 1 |

Ory | P17 | { |
(−∞; ∞) correct | { |
(−∞; ∞) correct | 1 |

Zoe | P18 | { |
(−∞; ∞) correct | { |
(−∞; ∞) correct | 1.5 |

Jono at times corrected errors and on other occasions he neglected to identify the errors. For example, inconsistencies are evident when we compare the marking of two learners’ (Ozi and Ali) solutions to Problem A1. The set builder notation for the domain and range in both learners’ solutions is incorrect as they violate order relations. Jono corrects Ozi’s domain in set builder notation by putting rings around the inequality signs to indicate that they are incorrect and writing the correct expression –∞

Jono’s evaluative activity as instantiated in the marking of learners’ test scripts validates content that differs from mathematics at the level of expression as well as the level of content. In addition, his evaluation of learners’ test scripts represents an extreme version of expression orientation in that as long as the expressions produced by learners resemble the correct expression according to the memorandum they are accepted as correct.

Comparing the tests across the two schools reveals differences in the types of problems set in the two schools and differences in the preparation for tests and examinations. Ivory College learners’ preparation involved classwork and independent work on worksheets and tutorials that pose mathematics problems in different ways. The types of problems encountered in class and independent work as well the test included both single-topic problems as well as multi-topic problems that required learners to draw on a number of different topics.

In contrast, Arbor High learners were only exposed to single-topic mathematics problems in their tests which were restricted to mathematics problems encountered during the observed lessons with no variation in the way problems are phrased. The Arbor High tests appear to encourage an expression orientation given the similarity of the mathematics problems used in the tests to those used during the observed lessons, with Jono’s test representing an extreme case of expression orientation because the test problems were extracted from the worksheet used during the observed lessons. The tests suggest that rehearsal of model texts in the form of set solution procedures for set problem types is the primary mode of mathematics reproduction in the Arbor High pedagogic contexts.

The absence of multi-topic mathematics problems in the Arbor High tests corresponds with the absence of multi-topic mathematics problems in the observed lessons, which indicates that topics are treated in isolation by Arbor High teachers thus resulting in a lack of inter-topic connectivity. As such, Arbor High learners are left to make connections between topics independently of the teacher. It could be argued that synthesis of school mathematics topics into a coherent whole is made much harder for the learners from working-class backgrounds than the learners from upper-middle-class or elite backgrounds.

In addition, Ivory College learners’ exposure in class and in tests to multi-topic problems which resemble examination type problems means that they appear to be better prepared with the support of their teachers for more mathematically demanding problems than their counterparts at Arbor High. It appears that in the working-class contexts, learners are left to make connections to topics and tackle more complex problems on their own, that is, without instruction and support from teachers, which perhaps provides insight into why mathematics performance for the majority of secondary learners in South Africa is so poor.

Comparing the marking of the test also reveals differences across the four pedagogic contexts. Maya and Jono’s marking included instances where mathematical violations were not corrected by the teacher and were accepted as correct. Furthermore, their marking is inconsistent and at times learners’ solutions marked as correct did not match their memoranda. Sara and Jada corrected learner errors, thus making learners’ errors explicit to them. Sara, however, at times made marking errors and did not deduct marks even though the solutions contained errors and on one occasion marked an incorrect solution as correct. Thus, Jono’s and Maya’s marking encourages an expression orientation to mathematics. Jono’s and Maya’s marking confirms the expression orientation observed in the observed lessons (see Jaffer,

We, therefore, see differences as well as similarities in the pedagogic practices across social class contexts, disrupting entrenched narratives in the literature about stark differences between schools populated by learners from working-class backgrounds and schools populated by learners from upper-middle-class or elite backgrounds. The crucial difference, however, does appear to hinge on the absence of content orientation in the school populated by learners from working-class backgrounds and the extensive preparation and type of problems that learners in the upper-middle-class or elite context are exposed to, which perhaps provides a possible explanation for the persistent differential performance in mathematics along the lines of social class.

This article derives from my doctoral research project. I hereby thank my supervisors, Associate professor Zain Davis and Professor Emeritus Paula Ensor, for their guidance and support during the supervision of my PhD study. I further wish to thank both supervisors for their valuable comments on a draft version of this article.

I declare that I have no financial or personal relationships that may have inappropriately influenced me in writing this article.

I declare that I am the sole author of this article.

Ethical clearance for this research was obtained from the Research Ethics Committee of the University of Cape Town in the Western Cape (ethical clearance number EDNREC202000909). Ethical issues including anonymity, confidentiality and voluntary participation were discussed with participants in the study and written consent for participation in the research was obtained from participants.

This work is based on research supported in part by the National Research Foundation of South Africa under grant number 92639. The research also benefited from funding from the university’s research capacity initiative. Any opinion, findings, conclusions and recommendations expressed here are those of the author and are not necessarily attributable to either of these organisations.

Data sharing is not applicable to this article.

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