Pre-service Primary Teachers’ Mathematical Content Knowledge: An Exploratory Study

Abstract

The Primary Teacher Education (PrimTEd) project was established in response to concerns about the pre-service preparation of primary teachers in South Africa. In order to inform the development of appropriate pre-service mathematics courses, an initial need in the PrimTEd project was to establish the nature of the mathematical knowledge of pre-service teachers both near the start and also near the end of their studies, through an online assessment. This paper describes the design of this PrimTEd online mathematics test and reports on the performance on the test of first-year and fourth-year primary mathematics student teachers, at three different universities. The overall performance of students together with the small differences between first and fourth year student performance indicate a need for student teachers to revisit primary school mathematics in a way that provides a deep understanding of key mathematical concepts in order to be better prepared for future teaching careers.

Keywords:

• Mathematical content knowledge
• pre-service teachers
• primary mathematics

Background

Primary teachers’ mathematical knowledge remains an issue of broad concern, not only in South Africa but also internationally. Most of the research on this issue in South Africa has been focused on in-service primary teachers, and particularly teachers at Intermediate Phase (grades 4–6) level (e.g. Carnoy & Chisholm, 2008; Taylor, 2011; Venkat & Spaull, 2015). A key convergence from the analyses in these studies has been the finding of substantial gaps in upper primary teachers’ mathematical knowledge. Additionally, teacher performance on assessment items requiring reasoning beyond the purely procedural consistently shows low results (Carnoy & Chisholm, 2008).

Predictably, in this context of concern, attention has turned towards pre-service teacher education and its role in supporting the growth of prospective teachers’ mathematical knowledge. This attention rests on two assumptions, both of which have backing in the research literature. The first is that improving teachers’ mathematical knowledge is important for their teaching to be more coherent and connected, and second, that pre-service teacher education is a key lever for changing the quality of mathematical knowledge for teaching in schools (Ma, 1999). There is, however, a gap in the research base, with limited data on the kinds of mathematical knowledge actually demonstrated by pre-service teachers or the sort of mathematics that they have the opportunity to learn. One study that did examine pre-service teacher education primary mathematics programmes, the Initial Teacher Education Research Project (ITERP) study, indicated substantial differences in the nature and level of the mathematics covered across five different pre-service primary teacher education programmes in South Africa, as well as differences in the time allocations for mathematical study across the programmes (Bowie & Reed, 2016). In that study, the authors noted that some institutions focused their courses on primary mathematics and early secondary mathematics-related content with a conceptual and pedagogic orientation, while other institutions focused more on upper secondary and tertiary mathematics content with procedurally oriented approaches. However, although it is likely that such differences will lead to graduates having widely varying knowledge, little information is available on outcomes of such courses specifically relating to the nature of pre-service teachers’ mathematical knowledge—the issue in focus in this paper, which we examine by drawing on initial findings from the Primary Teacher Education (PrimTEd) project.

The PrimTEd project was set up in response to the ITERP study findings. In PrimTEd those involved in the education of prospective primary teachers at different South African universities are, at the time of writing, collaborating on a research and development project aimed at improving pre-service teacher education. The work is aimed at establishing a shared understanding of appropriate curricula and standards across mathematical knowledge and classroom practice for pre-service primary teachers, together with developing quality support materials. One strand of PrimTEd has focused on pre-service primary teachers’ mathematical knowledge through exploratory assessments given close to the entry and exit points of pre-service programmes. In this paper, we outline the instrument used to assess pre-service teachers’ mathematical content knowledge and share outcomes from initial exploratory administrations of this instrument across three participating institutions. Driving our initial interest in this dataset are the following key questions:

• What can be said about the nature of first- and fourth-year BEd pre-service primary teachers’ mathematical content knowledge for each of the three institutions?

• What kinds of differences are seen between the first- and fourth-year pre-service teachers’ mathematical content knowledge in the three institutions?

Mathematical Content Knowledge for Teaching Primary School Mathematics

The kind of mathematics knowledge needed by primary school teachers has been the topic of a number of research studies. Many of these studies have developed and refined frameworks for assessing teacher knowledge from both mathematical content knowledge and pedagogical knowledge perspectives. Given that the strand of work with which we are involved focuses on the mathematical content knowledge of pre-service teachers, we focus our attention on two particular bodies of work: firstly, writing on the ways of theorising the mathematical knowledge of teachers; and second, the smaller subset of writing that has focused on the mathematical content knowledge of pre-service teachers. In both these foci, we draw on South African evidence relating to mathematics teacher knowledge as this evidence base provides the rationale for our selections from the literature.

Theorising Teachers’ Mathematical Content Knowledge

Blomeke and Delaney (2012) provide an overview of research on assessment of mathematics teachers’ knowledge. They identify two key conceptual frameworks that have been used for distinguishing between different aspects of knowledge of mathematics for teaching. These are the frameworks emerging from the Teacher Education and Development Study in Mathematics (TEDS-M) and the Learning Mathematics for Teaching (LMT) studies.

In the TEDS-M framework, mathematics content knowledge (MCK) includes the substantive elements of mathematical knowledge, described by Grossman, Wilson, and Shulman (1989), which encompasses knowledge of procedures and concepts, and ways in which this knowledge is organised. Blomeke and Delaney (2012) describe this as:

not only basic factual knowledge of mathematics but also the conceptual knowledge of structuring and organizing principles of mathematics as a discipline: why a specific approach is important and where it is placed in the universe of approaches to mathematics (p. 225).

MCK, in their model, is divided into four content areas (number and operations, algebra and functions, geometry and measurement, and data and chance) and three cognitive subdomains (knowing, applying and reasoning) (Li, 2012). In the design of our study we found the idea of breaking up MCK into content areas, as well as a focus on cognitive demand, to be useful.

The LMT research group (e.g. Ball, Thames, & Phelps, 2008; Hill & Ball, 2004) worked on identifying mathematical knowledge for teaching by studying the mathematical work that expert teachers do while teaching. They originally categorised this knowledge into MCK and pedagogical content knowledge (PCK). We focused our attention particularly on MCK for two main reasons. Firstly, given the extensive South African evidence of gaps in primary teachers’ mathematical content knowledge (Taylor, 2011), and given also the evidence that PCK is built on MCK foundations (e.g. Beswick, Callingham, & Watson, 2012), improving pre-service teachers’ MCK is indicated to be a priority. Second, given that PCK relates to knowledge related centrally to the practices of teaching, it has been acknowledged as hard to assess in the online assessment format that we chose to use so that test administration could be standardised across institutions.

Of interest to our study were the distinctions Ball et al. (2008) made within the MCK category, dividing it into common content knowledge (CCK), specialised content knowledge (SCK) and horizon content knowledge (HCK). Common content knowledge is described as the kind of mathematical knowledge that would be held both by teachers and by those who are not teachers (e.g. how to multiply two two-digit numbers). Specialised content knowledge, in contrast, is described as the mathematics unique to the work of teaching, for instance, analysing different methods of multiplying large numbers and deciding which methods would hold generally, and which are specific to particular types of examples. Horizon content knowledge is described as involving an understanding of how mathematical topics are linked and progress, and is important for teachers so they have a sense of where what they are currently teaching might lead (e.g. which methods of multiplication extend beyond working with integers). In relation to the CCK/SCK/HCK distinction, Venkat and Spaull’s (2015) findings suggested gaps at the level of common content knowledge among Intermediate Phase teachers (grades 4–6) in South Africa, with particular difficulties associated with multiplicative reasoning-related topics. Therefore, there is the need to include items at the level of the primary grades within the assessment design, and emphasise multiplicative structures across topics like fractions, ratios and percentages.

Liping Ma’s seminal 1999 study in this area adds to these two models through her theorising about the differences between Chinese and American mathematics teachers’ knowledge. Ma described the knowledge that the Chinese teachers in her study displayed as involving a ‘profound understanding of fundamental mathematics’ (p. 120). This knowledge involved a depth and coherence that could be seen in their knowledge of connections between ideas, different forms of representation and the progression of ideas. Ma’s study found that such understanding was not as apparent in American teachers’ practices. Given that connections are viewed broadly in the mathematics education field as underpinning more conceptual working, and given, too, that connections are seen as central to higher-cognitive-demand tasks in the work of Stein, Smith, Henningsen, and Silver (2009), this feature directed attention to cognitive demand as a design feature in the assessment, with inclusion of items across lower and higher cognitive demand levels.

Common across these studies is agreement that a well-connected understanding of fundamental mathematical ideas is critical. These fundamental understandings include the ability to translate between representations, and to solve problems that require insight in devising solution pathways rather than simply applying known procedures. In this regard, Stein et al.’s (2009) description of the levels of cognitive demand of mathematical tasks provided a useful lens for considering items to include in our assessment. Stein et al.’s taxonomy consists of four categories of tasks: memorisation; procedures without connections; procedures with connections; and doing mathematics. Memorisation and procedures without connections tasks are considered to be low-level-demand tasks as they only involve reproducing previously learned facts or performing an algorithm, and the focus is simply on getting a correct answer without requiring the production of any explanation or justification. Procedures with connection and doing mathematics tasks are higher-level demand tasks. Procedures with connections refer broadly to ‘tasks that demand engagement with concepts and that stimulate students to make powerful connections to meaning or relevant mathematical ideas’ (Stein et al., 2009, p. 1). They also often require learners to use or interpret representations and to make connections between multiple representations. At a higher level still, tasks classified as doing mathematics require ‘complex and nonalgorithmic thinking’ (Stein et al., 2009, p. 1).

Mathematical Content Knowledge of Pre-service Teachers

A caveat in the literature base on primary teachers’ mathematical knowledge is that it is largely drawn from in-service, rather than pre-service teachers. A small number of key studies have, though, focused on the mathematical knowledge of primary teachers in the preparatory phase. At the cross-country level, Senk et al.’s (2012) summary of data from the TEDS-M international comparative study of future primary teachers found large differences between and within countries in terms of pre-service course offerings and outcomes related to mathematics. In their study, empirical data from teacher responses to MCK items were analysed to create two ‘anchor points’ of ‘lower’ and ‘higher’ mathematical knowledge. The lower level was marked by basic whole-number computations, straightforward procedures involving fractions, solving routine problems and visualising two-dimensional shapes. The higher level was marked by more complex fraction problem-solving, some awareness of the nature of argumentation and relationships among two-dimensional shapes.

Tirosh (2000) in Israel, Chinnappan and Forrester (2014) in Australia, van Steenbrugge, Lesage, Valcke, and Desoete (2014) in Belgium and Vula and Kingji-Kastrati (2018) in Kosovo all investigated pre-service teachers’ knowledge of fractions and all concluded that the student teachers tended to have an algorithmic or procedural understanding of fractions with weak conceptual knowledge and were thus often unable to explain the underlying rationale for the operations on fractions. These findings confirmed the usefulness of working with two broad levels of cognitive demand within our assessment design, across the broad topic dimensions.

Another important study from the perspective of the PrimTEd project is Beswick and Goos’s (2012) analysis of the MCK (alongside PCK and beliefs in their study) of a sample of 294 pre-service primary teachers from seven Australian higher education institutions. With developmental aims that mirror our own, this study concluded with a finding related to MCK that was of particular interest to us:

Overall the results for MCK suggest that although teacher education needs to attend to sophisticated concepts involving abstract thinking, the weak mathematics knowledge of entering pre-service primary teachers means that simple knowledge and skills, such as the correct use of a ruler, cannot be taken for granted. (p. 82)

When coupled with the teacher knowledge evidence and prior findings from the South African ITERP study showing wide variation between institutions in pre-service course offerings related to mathematics (Bowie & Reed, 2016), this body of work confirmed the need to include assessment items focused on lower-cognitive-demand skills in primary mathematics, and to work across topic areas.

The focus of the first set of studies here largely aimed at theorising and categorising the different types of mathematical knowledge that might be required, rather than the substantive content. The second set of studies provided some useful pointers that we could use to develop a more concrete framework for test design. The gap, from a South African perspective, was the absence of an evidence base of the kinds of mathematical knowledge pre-service BEd students have on entry. This left us with uncertainties over what could count as ‘previously learned facts’.

We also had pragmatic concerns arising from the need for an assessment that was practical for multiple institutions to use. This necessitated careful choices about the tasks to include in the assessment, and an online assessment format. In thinking about the assessment tasks we were also mindful of Venkat and Spaull’s (2015) evidence from their study of the SACMEQ data and backed up by other studies (e.g. Carnoy et al., 2011; Taylor, 2011) indicating substantial gaps in South African primary teachers’ mathematical knowledge at, or close to, the levels of their actual teaching. We also paid attention to Beswick and Goos’s (2012) caution over what should not be taken for granted. Thus, in designing tasks that would be likely to provide useful evidence, we chose to work primarily with items related to primary mathematics curriculum content, and tasks drawn from lower and higher-cognitive-demand levels for specific topic strands. Given also the emphasis in South Africa’s, and many other countries’, primary mathematics curricula on number topics, we paid particular attention to number in general, and multiplicative structure in particular, in our test development.

In summary, on the lower-cognitive-demand side we included items which tested prospective teachers’ mastery of the facts and procedures they would be required to teach. On the higher-cognitive-demand side, we selected or developed items that explored whether the student teachers had insight into that mathematical content and whether their responses indicated a connected and coherent understanding of it. We drew, where possible, from tasks in existing research-based assessments, nationally and internationally, adapting these to fit with the South African context and with each item categorised on the basis of lower/higher cognitive demand, its content focus (whole numbers/rational numbers/patterns, functions and algebra/geometry/measurement) and the schooling phase (Foundation Phase, i.e. grades R–3, or Intermediate Phase i.e. Grades 4–6) that the central idea in the item could be connected to in terms of the South African Curriculum and Assessment Policy Statement (DBE, 2011a, 2011b).

Methodology

It is beyond the scope of this paper to detail the actual process of the development of the test, based on the above design principles. The first version of the test consisted of 50 multiple-choice or short-answer items administered in an online environment. Forty-five of these items were focused on MCK and five on PCK. This assessment was administered in the second half of 2017 to fourth year BEd students at four institutions and, on the basis of the analysis of this data, minor amendments were made to the test to replace/adapt two MCK and three PCK items in order to provide a better distribution across the content strands and cognitive demand categories. The revised version was administered in the first half of 2018 to 1177 first year BEd students across seven institutions.

Across the two administrations of the test, there were 43 items testing MCK that remained unchanged from 2017 to 2018. Table 1 indicates the number of items in each of the categories for these 43 MCK items. In the marking, each item was equally weighted, so the distribution of mark allocations across the categories reflects the distribution of items. In order to maintain test integrity in the ongoing project, actual items are not released; we do, however, in the findings below provide illustrative equivalent items.

Table 1. Distribution of marks per category for 43 common items considered in this paper.

	Total	Lower cognitive demand	Higher cognitive demand	Foundation Phase-related content	Intermediate Phase-related content
Whole numbers and operations on whole numbers	1 1	5	6	10	1
Rational numbers	15	8	7	2	13
Patterns, functions and algebra	6	2	4	1	5
Geometry	4	4	0	3	1
Measurement	7	5	2	1	6
Totals	43	24	19	17	26

In this paper we report on the results on these 43 items from the three institutions (universities A, B and C) in which the fourth year cohort was tested in 2017 (near the end of their BEd studies) and the first year cohort in 2018 (near the beginning of their BEd studies).¹ The results from these students (n = 488 first year students; n = 282 fourth year students) are considered to provide a quasi-longitudinal picture of standards, at the institutional level, at entry to and exit from the degree programmes.

Findings

Mean scores for the test overall and for each content strand across the first and fourth year cohorts in each of the three institutions are presented as percentages. We first present the average results across the five content strands for each of the groups of students (Table 2) and then across the cognitive demand type and Phase (Table 3). Subsequntly, we look at performance in each content area in more detail, using illustrative items similar to those in the assessment, to gain insight into underlying areas of difficulty for pre-service teachers that can inform the mathematical content foci in BEd programmes.

Table 2. Performance of first (2018) and fourth (2017) year student teachers by percentage mean score (with standard deviation in brackets) overall and for each mathematics content area.

	Number of students	Overall (43 items)	Whole numbers, operations (11 items)	Rational numbers (15 items)	Patterns, functions and algebra (6 items)	Geometry (4 items)	Measurement (7 items)
University A
First years	181	58 (20)	66 (20)	47 (23)	63 (27)	74 (23)	55 (28)
Fourth years	90	64 (16)	68 (18)	58 (19)	62 (26)	79 (21)	65 (22)
University B
First years	237	51 (15)	59 (18)	39 (20)	56 (22)	72 (23)	48 (23)
Fourth years	147	50 (16)	56 (19)	42 (20)	47 (25)	69 (23)	52 (23)
University C
First years	70	41 (13)	49 (16)	32 (14)	42 (23)	60 (25)	33 (23)
Fourth years	45	50 (17)	52 (19)	46 (20)	45 (25)	63 (19)	49 (25)

Table 3. Mean scores (with standard deviations in brackets) of first and fourth year student teachers for items grouped according to cognitive demand and Phase.

	Number of students	Overall	Lower cognitive demand	Higher cognitive demand	Foundation Phase content	Intermediate Phase content
University A
First years	181	58 (20)	65 (23)	49 (19)	67 (20)	52 (22)
Fourth years	90	64 (16)	74 (17)	52 (18)	72 (17)	59 (17)
University B
First years	237	51 (15)	60 (17)	40 (15)	58 (16)	46 (17)
Fourth years	147	50 (16)	60 (18)	38 (18)	55 (19)	47 (17)
University C
First years	70	41 (13)	46 (17)	34 (12)	49 (15)	36 (14)
Fourth years	45	50 (17)	58 (19)	39 (15)	53 (16)	47 (19)

Several features of the summary data in Table 2 are of interest. Firstly, overall performance across first and fourth years at all three institutions on an assessment based on primary mathematics content is relatively low—a 64% mean score at best in the fourth year cohort at Institution A. Second, the extent of the difference in overall mean score between the first and fourth year cohorts is relatively small—a 9% point difference at best in Institution C. At all three institutions, the first year students’ performance was lowest on rational numbers, followed by their performance on measurement. In these two content areas, the fourth year students at all three institutions did perform better than the first year students. In contrast to this, at both universities A and B, the fourth year students performed worse than the first year students in the area of patterns, functions and algebra. At all three institutions the performance in the geometry content area was the highest for both the first and fourth years. A caveat to note here is that while two of the geometry items in the initial test were amended for the 2018 version of the test, the four geometry items from 2017 that remained in the 2018 test (and thus included in this analysis) were all lower-cognitive-demand items. This is likely to be part of the explanation for the higher score for the geometry items.

The pattern of results varies across the three institutions. In University B, the fourth year students’ overall performance was worse than that of first year students, with the fourth years students performing worse than the first year students on three of the five strands: geometry, whole numbers and operations, and patterns, functions and algebra. While first year students’ performance at University C was lower than that of the first year students at universities A and B, the fourth year students at University C did perform better than the first year students at that university in all of the content areas.

The limited difference between first and fourth year mean scores suggests a need for time to be given to fundamental mathematical ideas in BEd programmes. The mean scores across all content areas suggest that this attention does need to be spread across all topics (including geometry, given the caveat mentioned above), but with particular attention given to rational number and measurement where notably low mean scores were seen. Given the broad agreement that multiplicative reasoning, on which rational number ideas rest, is a fundamental focus of middle grades’ mathematics, an emphasis on this would appear to be critical not only for Foundation Phase BEd students—who need to be attuned to the preparatory role in the move from additive to multiplicative situations—but also for Intermediate Phase BEd students, whose work centrally involves the development of multiplicative reasoning.

Our next analysis focuses on mean scores based on the cognitive demand and Phase-based categorisations of items—see Table 3. Table 3 provides a more nuanced insight into the differences between first and fourth year performance. Predictably, mean scores on the higher-cognitive-demand items are lower than those on the lower-cognitive-demand items. However, we restate that all of the items in the assessment were drawn from primary mathematics content, so the fact that the highest mean scores on lower-cognitive-demand items stood at 74% (fourth years in University A) points to gaps in what ought to be the application of routine procedures. Across the higher-cognitive-demand items, performance was similar for the first and fourth year cohorts. The fourth year candidates did, however, perform better on the lower-cognitive-demand items at universities A and C. These patterns of performance are mirrored in the mean scores seen on the basis of the Foundation Phase/Intermediate Phase item categorisation.

The variations between the first and fourth year students’ outcomes across the different institutions provide options for probing the quasi-longitudinal data in more detail. A further analysis based on the extent of difference in mean scores between fourth and first year cohorts was carried out for both the lower- and the higher-cognitive-demand items in each topic strand.

Whole Number

Examples of lower-cognitive-demand whole number items

1. 700–292 =

2. Identify the approximate position of 706 on a number line marked from 700 to 800 in divisions of 10.

Examples of higher-cognitive-demand whole number items

1. Fill in the number to make the number sentence true 623–298 = 622–□

2. Container A weighs 23 kg more than container B. Container A weighs 87 kg. How much does container B weigh?

Given the centrality of whole-number work to all grades in the primary school, the performance of both cohorts of BEd students on these items is of particular concern (see Table 4). The average scores on the lower cognitive demand point to gaps in procedural fluency and/or recalled and derived facts related to whole numbers and number sense. The low performance across all three institutions on the higher-cognitive-demand items suggests that many student teachers at both first and fourth year university level experience difficulty working structurally with numbers, and with making sense of whole-number problem situations. As the difference column in the table indicates, the fourth year students’ performance on both lower and higher cognitive demand was either no different or showed very little difference from that of the first year students.

Table 4. Mean scores of first and fourth years and percentage point difference between these scores for whole-number lower- and higher-cognitive-demand items.

University	Lower cognitive demand, 5 items			Higher cognitive demand, 6 items
	First years	Fourth years	Difference	First years	Fourth years	Difference
A	75	78	+3	58	60	+2
B	67	66	−1	52	47	−5
C	60	61	+1	41	45	+4

Rational Number

Examples of lower-cognitive-demand rational number items

1. Is $3/7$ smaller than $1/2$?

2. Calculate 45,304 × 1000

Examples of higher-cognitive-demand rational number items

1. $57/150$ is closest to 0.004; 0.04; 0.4; 4.

2. Which is true: 5 ÷ $13/17$ is smaller than 5; 5 ÷ $13/17$ is equal to 5; 5 ÷ $13/17$ is greater than 5?

Two things are notable in Table 5. Firstly, at all institutions and at both first and fourth year level, the students’ performance on rational number was poor. Second, however, rational numbers was one of the topics areas in which the fourth year students outperformed the first year students at each institution and this was true for both lower- and higher-cognitive-demand items (although at institution B this difference was small).

Table 5. Mean scores of first and fourth years and percentage point difference between these scores for rational-number lower- and higher-cognitive-demand items.

University	Lower cognitive demand, 8 items			Higher cognitive demand, 7 items
	First years	Fourth years	Difference	First years	Fourth years	Difference
A	53	67	+14	40	48	+8
B	46	49	+3	32	34	+2
C	35	53	+18	30	39	+9

Measurement

Examples of lower-cognitive-demand measurement items

Calculate the perimeter and area of a given rectangle.

Example of higher-cognitive-demand measurement items

A rectangle has an area of 40 cm². If all the sides of the rectangle are doubled, what will the area of the new rectangle be?

As with rational number, we see in Table 6 that the fourth year students performed better than the first year students, particularly on the lower-cognitive-demand items. However, the performance of all students on the higher-cognitive-demand items was particularly low with average score mostly below 30%.

Table 6. Mean scores of first and fourth years and percentage point difference between these scores for measurement lower- and higher-cognitive-demand items.

University	Lower cognitive demand, 5 items			Higher cognitive demand, 2 items
	First years	Fourth years	Difference	First years	Fourth years	Difference
A	66	78	+12	29	33	+4
B	60	64	+4	17	20	+3
C	39	57	+18	18	28	+10

Patterns, Functions and Algebra

Example of lower-cognitive-demand patterns, functions and algebra items

A is a number and B is a rule (an operation and number) in the diagram below. What is A?

Example of higher-cognitive-demand patterns, functions and algebra items

Given that $3n^2+6=10$ what will the value of $3n^2+8$ be?

The trend in patterns, functions and algebra, particularly with regard to the higher-cognitive-demand items shown in Table 7, is that fourth year students did worse than first year students. It may be the case that the more recent algebra experience that first year students have from high school feeds into this result.

Table 7. Mean scores of first years and fourth years and percentage point difference between these scores for patterns, functions and algebra lower- and higher-cognitive-demand items.

University	Lower cognitive demand, 2 items			Higher cognitive demand, 4 items
	First years	Fourth years	Difference	First years	Fourth years	Difference
A	73	76	+3	58	55	−3
B	74	66	−8	47	38	−9
C	51	61	+10	37	37	0

Geometry

Example of lower-cognitive-demand geometry items

How many lines of symmetry does a square have?

As explained, two geometry items that were in the original 2017 version of the test were removed and replaced with other items in the 2018 test. For this reason there is only a small set of (four) lower-cognitive-demand geometry items on both tests presented in Table 8.

Table 8. Mean scores of first and fourth years and percentage point difference between these scores on items on geometry lower-cognitive-demand items.

University	Lower cognitive demand, 4 items
	First years	Fourth years	Difference
A	74	79	+5
B	72	69	−3
C	60	63	+3

Discussion and Conclusion

The poor performance of BEd student teachers, and particularly of the fourth year students who were close to finishing their degree, is a cause for concern. This study adds to our awareness of primary pre-service teachers’ mathematical content knowledge at the pre-service level, showing that their performance mirrors that of the studies highlighted earlier pointing to concerns about South African in-service primary teachers’ mathematics content knowledge (Carnoy & Chisholm, 2008; Taylor, 2011; Venkat & Spaull, 2015), also being applicable to the hitherto under-researched knowledge of pre-service teachers in South Africa.

Our interest in this paper and in the broader PrimTEd project relates to the lessons we can learn for mathematics teacher education in South Africa. While this dataset is quasi-longitudinal rather than longitudinal, there is cause for concern over the limited differences between first and fourth year BEd students’ mathematical knowledge across a range of topic areas, and particularly on higher-cognitive-demand items.

The fact that the trend in student performance in the patterns, functions and algebra topic is for first year students to do better than fourth year students is clearly counter to what we would want. Although the direct teaching of algebra is not a focus in primary school, the foundations of algebraic thinking are laid throughout primary school (Cai & Knuth, 2011).

Whilst the fact that fourth year students perform better than first year students on rational number and measurement items suggests that some learning is taking place over the course of the BEd, this does not mean that we can assume that this is taking places during the courses. It could equally well be the case that having to teach these topics during teaching practice in schools is what provides the student teachers with this knowledge. This would echo Brodie and Sanni’s (2014) finding that teachers ‘come to know well what they teach regularly’ (p. 188). Furthermore, we note that the improvement is largely in the area of lower-cognitive-demand items and that the differences between first and fourth year students on the higher-cognitive-demand items is small. There are two implication resulting from this. Firstly, we note that our first year data suggest that incoming student teacher knowledge in these areas is relatively weak (even on the procedural level) and thus pre-service teacher education cannot take it for granted that student teachers will arrive with a command of the mathematics they will be expected to teach. Second, we need to examine the mathematics courses in the BEd to understand what we are offering student teachers in the way of conceptual understanding of these topics and why we are seeing such small improvements in fourth year student performance.

Not taking for granted simple knowledge and skills is highlighted by the student performance in the area of whole numbers. This is a core area of work in primary mathematics and we would want all teachers to have a deep, connected and flexible understanding of it. The indication from this study is that there are a number of students who need work on the routine procedures connected to whole numbers and that most of the students need considerable work to achieve the profound understanding of fundamental mathematics that Ma (1999) suggests is important for teaching.

As uncomfortable as it may seem to include attention to primary school mathematics in a university course, the results of this assessment suggest that it is necessary to revisit much of the primary level content in pre-service teacher education. Furthermore, for the most part, fourth year student teachers’ performance on higher-cognitive-demand items across all topic areas did not rise above the 50% mark. Given the characterisation of the work of teaching as requiring a deep, connected knowledge of mathematics, this is something that teacher education in South Africa needs to pay attention to. These three institutions, which we see as typical, seem to be making little headway in this regard. We therefore argue that careful work needs to be done on looking at various offerings in pre-service mathematics education to try to tease out the key elements of courses that are able to produce improvements in conceptual understanding.

Disclosure Statement

No potential conflict of interest was reported by the authors.

ORCID

Lynn Bowie http://orcid.org/0000-0002-0054-373X

Mike Askew http://orcid.org/0000-0003-0826-461X

Additional information

Funding

This paper has been developed through the Teaching and Learning Development Capacity Improvement Project which is being implemented through a partnership between the Department of Higher Education and Training and the European Union [grant number PrimTed].

Notes

1 We have, intentionally, not provided contextual details about the three institutions as our focus—in this paper and in the broader project—is not related to institutional comparison, but instead to understanding and supporting the development of mathematical knowledge for teaching in the national landscape. The absence of contextual information also ensures anonymity of the participating institutions.