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A Causal-Comparative Study of South African Pre-Service Primary Mathematics Teachers' Spatial Visualization Ability: Does Common Content Knowledge Matter?

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Title: A Causal-Comparative Study of South African Pre-Service Primary Mathematics Teachers' Spatial Visualization Ability: Does Common Content Knowledge Matter?
Language: English
Authors: Shongwe, Benjamin (ORCID 0000-0001-8083-6462)
Source: International Journal of Mathematical Education in Science and Technology. 2022 53(9):2338-2363.
Availability: Taylor & Francis. Available from: Taylor & Francis, Ltd. 530 Walnut Street Suite 850, Philadelphia, PA 19106. Tel: 800-354-1420; Tel: 215-625-8900; Fax: 215-207-0050; Web site: http://www.tandf.co.uk/journals
Peer Reviewed: Y
Page Count: 26
Publication Date: 2022
Document Type: Journal Articles
Reports - Research
Tests/Questionnaires
Education Level: High Schools
Secondary Education
Higher Education
Postsecondary Education
Elementary Education
Descriptors: Mathematics Skills, Pedagogical Content Knowledge, Spatial Ability, Visualization, Mathematics Teachers, Teaching Methods, Comparative Analysis, Preservice Teachers, Teacher Education Programs, Elementary School Teachers, Misconceptions, Foreign Countries
Geographic Terms: South Africa
DOI: 10.1080/0020739X.2020.1869333
ISSN: 0020-739X
1464-5211
Abstract: This study used a modified mathematical knowledge for teaching (MKT) framework to compare the "common content knowledge" (CCK) of two groups of purposively selected pre-service primary mathematics teachers (n = 86) in relation to spatial visualization. Specifically, thirty-five pre-service teachers (n[subscript 1] = 35) with a pure school mathematics background and fifty-one pre-service teachers (n[subscript 2] = 51) with a high school mathematical literacy background worked on problems that involved spatial objects, their properties and relationships. The responses of pre-service teachers with a mathematical literacy content background to the "Surface Development Test" (SDT) were significantly different from their counterparts' which suggested that their CCK in relation to spatial visualization was thin. Analysis of the "Probes" questionnaire which inquired about the pre-service teachers' views on the concept of spatial visualization showed that misconceptions hindered their performance. Future research directions are discussed.
Abstractor: As Provided
Entry Date: 2023
Accession Number: EJ1367167
Database: ERIC
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  Value: <anid>AN0159297330;imt15aug.22;2022Sep27.08:20;v2.2.500</anid> <title id="AN0159297330-1">A causal-comparative study of South African pre-service primary mathematics teachers' spatial visualization ability: does common content knowledge matter? </title> <p>This study used a modified mathematical knowledge for teaching (MKT) framework to compare the common content knowledge (CCK) of two groups of purposively selected pre-service primary mathematics teachers (n = 86) in relation to spatial visualization. Specifically, thirty-five pre-service teachers (n<sub>1</sub> = 35) with a pure school mathematics background and fifty-one pre-service teachers (n<sub>2</sub> = 51) with a high school mathematical literacy background worked on problems that involved spatial objects, their properties and relationships. The responses of pre-service teachers with a mathematical literacy content background to the Surface Development Test (SDT) were significantly different from their counterparts' which suggested that their CCK in relation to spatial visualization was thin. Analysis of the Probes questionnaire which inquired about the pre-service teachers' views on the concept of spatial visualization showed that misconceptions hindered their performance. Future research directions are discussed.</p> <p>Keywords: Common content knowledge; school mathematics; mathematical literacy; spatial visualization ability</p> <hd id="AN0159297330-2">1. Introduction</hd> <p>It is widely acknowledged that teachers' mathematics content knowledge is an essential ingredient of high-quality instruction (for example, Ball et al., [<reflink idref="bib3" id="ref1">3</reflink>]). In particular, pre-service primary mathematics teachers need to have a profound, flexible, and adaptive knowledge of mathematics content (Ma, [<reflink idref="bib46" id="ref2">46</reflink>]). Why is this knowledge so important? It contributes significantly to students' achievement in mathematics (Bobis et al., [<reflink idref="bib11" id="ref3">11</reflink>]; Senk et al., [<reflink idref="bib60" id="ref4">60</reflink>]). In unpacking the idea of mathematics content knowledge, Ball et al. ([<reflink idref="bib2" id="ref5">2</reflink>]) identified <emph>common content knowledge</emph> (CCK) which is defined as the 'content knowledge that is used in the work of teaching in ways <emph>in common with</emph> how it is used in many other professions or occupations that also use mathematics' (Hill et al., [<reflink idref="bib39" id="ref6">39</reflink>], p. 436) and specialized content knowledge (SCK) which is the 'knowledge specific to the work of teaching' (Hill et al., [<reflink idref="bib39" id="ref7">39</reflink>], p. 82). It is the former knowledge that is of interest to the present study, in the context of spatial visualization ability (SVA). Van Nes and de Lange ([<reflink idref="bib70" id="ref8">70</reflink>]) point out that the application of SVA begins in childhood as we imagine, for example, 'where in the kitchen it is that they can find their snack before they walk into the kitchen to get it' (p. 215). That is, they define SVA as the capability to imagine the movements of objects and spatial forms.</p> <p>The importance of SVA is underscored by the plethora of research on teacher education conducted to gain a better understanding of the knowledge that pre-service teachers need in order to improve mathematical instruction (for example, Ball et al., [<reflink idref="bib4" id="ref9">4</reflink>]; Ma, [<reflink idref="bib46" id="ref10">46</reflink>]; Venkat & Spaull, [<reflink idref="bib71" id="ref11">71</reflink>]). However, pre-service primary teachers' mathematical content knowledge remains an international concern (Reeves & Muller, [<reflink idref="bib57" id="ref12">57</reflink>]; Ubah & Bansilal, [<reflink idref="bib69" id="ref13">69</reflink>]; Venkat & Spaull, [<reflink idref="bib71" id="ref14">71</reflink>]). Research reports not only show substantial gaps in the CCK of these teachers (Venkat & Spaull, [<reflink idref="bib71" id="ref15">71</reflink>]) but also that their performance in tests requiring conceptual rather than purely procedural show low achievements (Carnoy & Chisholm, [<reflink idref="bib16" id="ref16">16</reflink>]). In fact, Ball et al. ([<reflink idref="bib2" id="ref17">2</reflink>]) point out that the CCK of 'many teachers is dismayingly thin' (p. 14).</p> <p>Noteworthy is that building a teacher's pedagogical content knowledge (PCK)[<reflink idref="bib1" id="ref18">1</reflink>] is dependent on solid foundations of CCK (Beswick & Goos, [<reflink idref="bib7" id="ref19">7</reflink>]; Krauss et al., [<reflink idref="bib43" id="ref20">43</reflink>]). That is, knowing mathematics is a necessary component of knowing how to teach it. As Southwell and Penglase ([<reflink idref="bib62" id="ref21">62</reflink>]) point out, many pedagogical processes are of little benefit if sound mathematical knowledge is lacking. Hence, improving pre-service primary mathematics teachers' common content knowledge (CCK), which is the mathematical everyday knowledge that all educated adults should have and therefore requires no knowledge of teaching or students (Krauss et al., [<reflink idref="bib43" id="ref22">43</reflink>]), remains a priority. To this end, pre-service teacher education is a key lever for changing the quality of mathematical knowledge for teaching in schools (2010). However, a great deal of attention is currently being given to SCK, which is the knowledge that is exclusive to the profession of teaching (Welder & Simonsen, [<reflink idref="bib73" id="ref23">73</reflink>]). In addition, little is known about the CCK that pre-service teachers bring to their mathematics courses in the context of SVA.</p> <p>The purpose of this concurrent triangulation study was to gain insights into the differences (if any) in CCK, in the area of SVA, of two groups of pre-service primary mathematics teachers (those with a background in high school mathematics and those with a background in mathematical literacy[<reflink idref="bib2" id="ref24">2</reflink>]) – particularly their ability to visualize three-dimensional (3D) objects from two-dimensional (2D) drawings – and why the differences (if any) occur. To this end, Ball and her colleagues' (Ball et al., [<reflink idref="bib4" id="ref25">4</reflink>]) mathematical knowledge for teaching (MKT) theory with a special focus on describing and studying these teachers' CCK was the lens used to investigate CCK. In particular, the CCK part of the framework was used to answer the following questions: (<reflink idref="bib1" id="ref26">1</reflink>) What is the difference (if any) in common content knowledge relating to spatial visualization between pre-service primary mathematics teachers with grade 12 mathematics experiences and those with grade 12 mathematical literacy experiences? and (<reflink idref="bib2" id="ref27">2</reflink>) Why are these two groups of pre-service teachers' common content knowledge in relation to spatial visualization different, if at all?</p> <p>Prior to providing a background to the study, a clarification of the term 'spatial visualization ability' is necessary. There are differing perpsectives to consider when searching for an adequate and workable definition of SVA. According to Nagy-Kondor ([<reflink idref="bib53" id="ref28">53</reflink>]), SVA refers to the disposition to depict 'situations when the components are moving compared to each other' (p. 266). The <emph>Standards</emph> (National Council of Teachers of Mathematics [NCTM], [<reflink idref="bib54" id="ref29">54</reflink>]) define SVA as the ability to build and manipulate 'mental representations of two- and three-dimensional objects and perceiving an object from different perspectives' (p. 41). Clements and Battista ([<reflink idref="bib18" id="ref30">18</reflink>]) defines SVA as understanding and performing imagined movements of two- and three-dimensional objects. Still, Clements and Battista ([<reflink idref="bib18" id="ref31">18</reflink>]) see SVA as capability to 'perform imagined movements of objects in two-dimensional and three-dimensional space' (p. 444). For McGee ([<reflink idref="bib49" id="ref32">49</reflink>]) conceptualizes SVA as the disposition 'to mentally rotate, manipulate, and twist two- and three-dimensional stimulus' (p. 896). Hegarty and Waller ([<reflink idref="bib37" id="ref33">37</reflink>]) consider spatial visualization as 'the ability to mentally manipulate, rotate, twist, or invert objects without reference to one's self' (p. 127).</p> <p>The multiplicity of meanings attached to SVA notwithstanding, I adopt Miller and Bertoline's ([<reflink idref="bib50" id="ref34">50</reflink>]) most compelling and comprehensive characterization of SVA as the capability 'to imagine the rotation of a depicted object, the folding and unfolding of flat patterns, and the relative changes of positions of objects in space' (p. 9). This definition is appropriate for this study given that the task is based on the tasks in the <emph>surface development test</emph> (SDT) used to assess pre-service teachers' SVA (Odell, [<reflink idref="bib56" id="ref35">56</reflink>]). Briefly, the task involves manipulation in which there is movement among the internal parts of a complex configurations that are spatial (not verbal or mathematical). The thesis of this paper is that there is a significant difference in the SVA of pre-service teachers who experienced high school mathematics and the SVA of those that took mathematical literacy. Establishing the veracity of this hypothesis is important in mounting a challenge to the disproportionate attention accorded to proficiency in numeracy over SVA in the mathematical literacy curriculum.</p> <p>The rest of the paper is organized as follows. situates this study within existing knowledge on SVA literature; mainly the literature on CCK but also on spatial visualization and mathematics, and the effect of technology on SVA. Then, the specific framework guiding the study is outlined. Next, the methodology (that is, methods and rationale behind them) highlighting the mixed methods design used to address the research questions underpinning this study (including an outline of modified version of the SDT which taps into pre-service teachers' CCK in relation to spatial visualization), is presented. This is followed by a presentation and discussion of results. The paper concludes by setting a course for the future research, including that which incorporates technology into spatial visualization instructional practices.</p> <hd id="AN0159297330-3">2. Background to the study</hd> <p></p> <hd id="AN0159297330-4">2.1. Pre-service teachers' common content knowledge</hd> <p>An important aspect of a teacher educator's knowledge of pedagogy is to have an awareness of the knowledge and expectations of the students (Harel & Sowder, [<reflink idref="bib35" id="ref36">35</reflink>]). Several studies over the past three decades have attempted to identify and categorize the different elements of knowledge required for effective teaching, derived chiefly from analyses of primary school teachers' work and their practices (for example, Ball et al., [<reflink idref="bib3" id="ref37">3</reflink>]; Delaney et al., [<reflink idref="bib25" id="ref38">25</reflink>]; Shulman, [<reflink idref="bib61" id="ref39">61</reflink>]). As a consequence, mathematics content knowledge which includes both CCK and SCK, has received significant attention in mathematics education research (Chapman, [<reflink idref="bib17" id="ref40">17</reflink>]). The relationship between CCK and SCK can be defined in terms of seeing SCK as the knowledge of selecting and designing instructional tasks, making representations to explain CCK (Iserbyt et al., [<reflink idref="bib42" id="ref41">42</reflink>]). Clearly, CCK is the foundation on which SCK builds. These forms of knowledge are additional to PCK which is knowledge of how to make mathematical ideas understandable to students, knowledge of students' difficulties in mathematics as well as their typical perceptions and misconceptions (Delaney et al., [<reflink idref="bib25" id="ref42">25</reflink>]).</p> <p>Common content knowledge (CCK) is the type of knowledge that is not exclusive to teachers; any adult may have well developed CCK but most likely will lack the knowledge to transform it into teaching. More precisely, CCK is 'knowledge that is used in the work of teaching in ways in common with how it is used in any other professions or occupations that also use mathematics' (Hill et al., [<reflink idref="bib38" id="ref43">38</reflink>], p. 377). According to Thames and Ball ([<reflink idref="bib66" id="ref44">66</reflink>]), CCK allows a person to successfully solve mathematical problems in non-classroom contexts, including 'being able to do particular calculations, knowing the definition of a concept, or making a simple representation' (p. 223). Further, CCK only requires that a person possesses the skills and procedures necessary for solving (not explaining or representing) mathematical problems (Welder & Simonsen, [<reflink idref="bib73" id="ref45">73</reflink>]).</p> <p>This scarcity implies that the extent to which pre-service teachers' CCK enables them to readily recognize wrong answers provided by their students and identify inaccurate definitions or faulty information in textbooks (Brijlall, [<reflink idref="bib13" id="ref46">13</reflink>]) is not clearly known. However, four studies are important to the research reported in this study. These studies are important to consider in order to show not only how the present study addressed the gap in literature on pre-service teachers' SVA but also, most importantly, they formed the foundation for the theoretical framework underpinning this study and helped in examining the methodological approaches in recent research studies. Considering a methodological approach in the review of each of these four works helped to prevent duplication of effort and provided an African perspective on the notion of CCK in pre-service teachers' SVA.</p> <p>In a large study, Ball ([<reflink idref="bib1" id="ref47">1</reflink>]) examined the CCK of 217 pre-service primary mathematics teachers as they entered formal teacher education in five sites of the Michigan State University in the United Sates (US). The focus of the investigation was division of fractions. Questionnaires and interviews with a smaller sample of pre-service teachers revealed that the CCK that both groups of teachers brought with them to teacher education from their pre-university and university mathematics experiences tended to be rule-bound, compartmentalized, and thin. Welder and Simonsen ([<reflink idref="bib73" id="ref48">73</reflink>]) investigated 48 mostly female pre-service teachers enrolled at a public, mid-sized university in the western US. They were interested in measuring the effects of an undergraduate mathematics content course for pre-service primary teachers' CCK and SCK using a one-group pretest-posttest design. The content covered in their course's curriculum was congruent to that covered in primary mathematics classrooms. Matched pairs <emph>t</emph>-tests showed significant gains in CCK in two areas of prerequisite algebra concepts (numbers and equations/functions). As they contend, these results provided evidence of pre-service teachers developing mathematical understanding beyond CCK.</p> <p>In another study, Yurt and Tünkler ([<reflink idref="bib74" id="ref49">74</reflink>]) investigated 234 prospective pre-service teachers' SVA at two universities in central and Southern Anatolia using an explanatory sequential design. They collected quantitative CCK data using 'Mental Rotation Test' and SDT, and qualitative data using 'Opinion Form for Spatial Ability Tests'. The results showed that the pre-service teachers' SVA was low and that pre-service teachers with higher academic averages had better SVA.</p> <p>Turgut and Nagy-Kondor ([<reflink idref="bib67" id="ref50">67</reflink>]) reported on an investigation of spatial visualization skills of two groups of pre-service primary mathematics teachers in Hungary (<emph>n</emph> = 78) and Turkey (<emph>n</emph> = 81). They used a reduced version of 'Heinrich Spatial Visualization Test' (HSVT), which is a paper-pencil test consisting 25 items of spatial visualization problems related to synthesis and decomposition of pieces. A correlational analysis revealed that there was a significant difference between Hungarian and Turkish pre-service primary mathematics teachers' SVA in favour of the Hungarian sample.</p> <p>To sum up this section, a common thread weaving through these four studies is that none of them investigated the CCK of pre-service primary mathematics teachers' SVA from the perspectives of the knowledge with which they enter university in the context of SVA. As already mentioned, the knowledge with which these teachers enter university is important in their learning of space and shape. The study reported here sought to redress that literature lacuna. Capturing primary school pre-service teachers' conceptions of spatial visualization is not only important in that it helps to minimize the perpetuation of misconceptions in the students that they will eventually teach, but also is useful in predicting their performance in mathematics modules. A grasp of the concept of spatial visualization is necessary for success in different domains in science, technology, engineering, and mathematics (STEM) education (Brus et al., [<reflink idref="bib15" id="ref51">15</reflink>]).</p> <hd id="AN0159297330-5">2.2. The role of spatial visualization ability in mathematics</hd> <p>There is a significant relationship between SVA and achievement in mathematics (Battista, [<reflink idref="bib5" id="ref52">5</reflink>]). Hegarty and Kozhevnikov ([<reflink idref="bib36" id="ref53">36</reflink>]) conducted a study to demonstrate this relationship, in the context of mathematical problem solving. The results were consistent with Battista ([<reflink idref="bib5" id="ref54">5</reflink>]) assertion. Also, a large-scale international Teacher Education and Development Study in Mathematics (TEDS-M) (Tatto et al., [<reflink idref="bib65" id="ref55">65</reflink>]) has addressed pre-service teachers' knowledge in geometry, a domain rich in spatial visualization, among other content subdomains. The results highlighted this relationship.</p> <p>Using the TEDS-M research dataset on the basis that it is (a) publicly available, (b) draw on a large and representative sample of participants and (c) include participants across 17 different countries, found the relationships between CCK and PCK may not be as strongly linked as previously thought (Murray et al., [<reflink idref="bib52" id="ref56">52</reflink>]). This result suggests that knowing mathematics does not necessarily help a pre-service teacher to become an effective mathematics teacher. This points to the conclusion that the findings on the relationship between CCK and PCK are inconsistent. However, a critique of this relationship is beyond the scope of this study.</p> <p>Returning to SVA, research studies have reported on the difficulties encountered by both students and undergraduates pertaining to geometry (de Villiers, [<reflink idref="bib23" id="ref57">23</reflink>]; Hanna, [<reflink idref="bib33" id="ref58">33</reflink>]; Harel & Sowder, [<reflink idref="bib34" id="ref59">34</reflink>]). The genesis of such difficulties has been attributed to various aspects: cognition (Harel & Sowder, [<reflink idref="bib34" id="ref60">34</reflink>]), school mathematics education system (Lockhart, [<reflink idref="bib44" id="ref61">44</reflink>]), spatial ability (Del Grande, [<reflink idref="bib24" id="ref62">24</reflink>]), and concept formation (Tall & Vinner, [<reflink idref="bib64" id="ref63">64</reflink>]). Del Grande ([<reflink idref="bib24" id="ref64">24</reflink>]) attributes this difficulty to the school mathematics curriculum which places emphasis on the deductive aspects of geometry education and neglects the underlying SVA. Recently, the significance of SVA has gained recognition for its influence on the development of mathematical content knowledge (Bobis, [<reflink idref="bib10" id="ref65">10</reflink>]).</p> <p>The <emph>Standards</emph> (National Council of Teachers of Mathematics [NCTM], [<reflink idref="bib54" id="ref66">54</reflink>]) recommends that the mathematics curriculum for grade 5–8 should include the study of the geometry of one, two, and three dimensions in a variety of situations, so that students can visualize and represent geometric figures with special attention to developing spatial sense. Thus, given the consensus on the importance of SVA, how to improve one's SVA has become one of the central research topics in mathematics education and has wide implications for instructional practices. Technology is one tool envisaged to improve SVA.</p> <hd id="AN0159297330-6">3. Theoretical perspective</hd> <p>The research reported in this study draws from the practice-based theory of <emph>mathematical knowledge for teaching</emph> (MKT) which was developed by researchers at the University of Michigan. This theory was inspired by Shulman's ([<reflink idref="bib61" id="ref67">61</reflink>]) idea of pedagogical content knowledge (PCK) and categorizes knowledge needed to perform the recurrent tasks of teaching mathematics into six domains (Ball et al., [<reflink idref="bib4" id="ref68">4</reflink>]). These domains are grouped into two broad categories each with three domains: on the one hand, mathematics content knowledge comprised common content knowledge (CCK), specialized content knowledge (SCK), and horizon content knowledge (HCK); on the other hand, PCK which comprised knowledge of content and students (KCS), knowledge of content and teaching (KCT), and knowledge of content and curriculum (KCC).</p> <p>For this study, the focus was on describing and studying pre-service teachers' common content knowledge (CCK) for teaching grades 1–7; that is, knowledge and skill used that others outside mathematics teaching profession commonly understand. In distinguishing between CCK and SCK, Suzuka et al. ([<reflink idref="bib63" id="ref69">63</reflink>]) point out that CCK involves the doing of mathematics for oneself as opposed to attending to others' thinking which is a distinguishing characteristic of SCK.</p> <p>Although the teacher's ability to trace students' thinking is a manifestation of specialized content knowledge (SCK) that those would not need, hence, may not know (Ball et al., [<reflink idref="bib4" id="ref70">4</reflink>]), is important, teachers' CCK in relation to SVA was a priority in this study. The reason for using the CCK lens to investigate pre-service teachers' SVA is not only that there are gaps at the level of CCK among primary mathematics in-service teachers (Venkat & Spaull, [<reflink idref="bib71" id="ref71">71</reflink>]) but also that this kind of knowledge is also accessible to pre-service teachers with no pure mathematics background; it is not specialized knowledge. Thus, the notion of CCK provided an orderly, efficient scheme for bringing together observations and facts from separate investigations and thus guide the understanding of pre-service teachers SVA – both the 'what' and the 'why' of their occurrence (Evans et al., [<reflink idref="bib29" id="ref72">29</reflink>]).</p> <hd id="AN0159297330-7">4. Significance of the study</hd> <p>First, the significance of this study was that it challenges the disproportionate attention accorded to proficiency in numeracy over spatial visualization in school curricula. For instance, Mudaly ([<reflink idref="bib51" id="ref73">51</reflink>]) analysed and investigated students' responses to questions requiring SVA and found that most students had difficulty in using visualization as a means to understand mathematical concepts and concluded that this area of mathematics has been neglected. According to Gutiérrez ([<reflink idref="bib32" id="ref74">32</reflink>]), this disproportion is a consequence of the myopic confinement of spatial visualization to geometry. Yet SVA is a helpful aid in supporting intuition and concept formation for many topics in mathematics (Dreyfus, [<reflink idref="bib27" id="ref75">27</reflink>]). Second, this study makes a contribution to the efforts to reconsider the adequacy of some pre-service teachers' SVA at primary mathematics level. This effort is manifested in the focus on pre-service primary mathematics teachers' CCK rather than SCK because the former underpins the decisions that teachers make in relation to students' learning.</p> <hd id="AN0159297330-8">5. Methods</hd> <p></p> <hd id="AN0159297330-9">5.1. Ethical considerations</hd> <p>In an attempt to comply with ethical research practices, these five issues were considered: institutional permission; informed consent; privacy and confidentiality; and, anonymity. Initially, an application to conduct the study was made to the University's Ethics Committee. Informed consent of pre-service teachers to participate was sought by detailing the purpose, duration, methods, and potential value of the research so that they did not feel coerced to participate in the study. Participants' identity and the location of the research sites were not disclosed. Instead, pseudonyms were used. Anonymity was ensured by assigning a number as a code rather than using pre-service teachers' names on each of the instrument administered to the participants.</p> <hd id="AN0159297330-10">5.2. Setting</hd> <p>The setting for the present study was a module in space and shape and statistics taught to pre-service primary school teachers who were in their final year of the Bachelor of Education (B. Ed) programme at a large South African public university. This is a 4-year programme for teaching mathematics (among other modules) to primary school students, including two weeks of field-based (practicum) teaching experience during each year of the programme. The programme comprises three core primary mathematics education modules spanning over three years. The institution has a 'Forum Period', which is a 45-minute time in between Thursday's lectures, reserved for interested students to engage in cultural activities.</p> <p>The university draws students from diverse socioeconomic backgrounds in which most schools were vastly under-resourced. Based on the literature, the ability in spatial visualization was consequently a logical prerequisite for success in the primary mathematics module. This module, known as Primary Mathematics Education 000, was designed to develop knowledge for teaching space and shape, and statistics. The design and content of the module were guided by Ball and her colleagues' mathematical knowledge for teaching (MKT) model, spatial visualization and statistical concepts aligned with the aims of the <emph>Curriculum Assessment Policy Statement</emph> (CAPS) (Department of Basic Education [DBE], [<reflink idref="bib26" id="ref76">26</reflink>]).</p> <p>Participants in both groups had completed mandatory mathematics in the first 10 years of schooling. The topic of space and shape in the CAPS document is covered in all stages of the primary and secondary school curriculum. Thus, in relation to the participants' knowledge of space and shape prior to their university studies, it can reasonably be assumed that participants have had experiences with space and shape in their primary and high school mathematics curriculum.</p> <p>Briefly, the CAPS document specifies the content area and its accompanying concepts and skills from grade 1 to 7. In addition, each content area is broken down into several topics: algebra, financial mathematics, trigonometry, probability and statistics, differential calculus, analytical and Euclidean geometries. Given the research literature provided in this study, it seemed reasonable to hypothesize that school mathematics (which included instruction in analytical geometry), rather than mathematical literacy, predicted success in the spatial visualization component of the module because, as Clements and Sarama ([<reflink idref="bib19" id="ref77">19</reflink>]) point out, mathematical ideas are essentially spatial. The reasonable conclusion to reach in relation to the SDT is that it requires a repertoire of strategies gained in social interaction or experiences to successfully execute.</p> <hd id="AN0159297330-11">5.3. Design</hd> <p>The present study is underpinned by a pragmatic paradigm using concurrent transformative mixed methods to gather both numerical and verbal data in line with Creswell and Plano Clark's ([<reflink idref="bib22" id="ref78">22</reflink>]) guidelines. The study adopted a causal-comparative design given that the focus was on determining whether having taken mathematics rather than mathematical literacy in high school affected pre-service teachers' performance in a paper folding test meant to establish their SVA. A causal-comparative research is a design that is intended 'to find relationships between independent and dependant variables after an action or event has taken place' (Salkind, [<reflink idref="bib59" id="ref79">59</reflink>], p. 124). Further, it was necessary to seek insight into the qualitative segment of the study through probing the pre-service teachers' CCK on the concept of spatial visualization.</p> <p>The mixing of quantitative and qualitative techniques was helpful in capturing both teachers' knowledge and thinking in relation to spatial visualization. As Holm and Kajander ([<reflink idref="bib40" id="ref80">40</reflink>]) note, '[o]nly when the beliefs and knowledge of the teacher are both considered, can changes in mathematics teaching have real and lasting effects on future generations of students' (p. 13). To control for an extraneous variable that could be unrelated to the investigation, relatively homogenous groups on spatial visualization were created by recruiting mostly pre-service teachers who finished school from predominantly no-fee paying schools. Traditionally, these schools are characterized by low socioeconomic conditions and are situated in previously disadvantaged or neglected areas such as townships and rural areas.</p> <hd id="AN0159297330-12">5.4. Participants</hd> <p>Spatial visualization ability and the sources of their ability were respectively assessed and understood through a purposive sample of 86 pre-service teachers enrolled in a mathematics education module as described in the section above. As is typical in schools of education and in the teaching profession in general (Correa et al., [<reflink idref="bib21" id="ref81">21</reflink>]), the majority of the participants were women (68%). This sample of pre-service teachers was drawn from the target population of primary mathematics education students (<emph>N</emph> = 223) who were in their final year of a B. Ed degree programme and comprised two groups of pre-service teachers: those with a pure high school mathematics background</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mrow><msub><mi>n</mi><mn>1</mn></msub></mrow><mo>=</mo><mn>35</mn><mo stretchy="false">)</mo></math> </ephtml> and those with mathematical literacy background</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mrow><msub><mi>n</mi><mn>2</mn></msub></mrow><mo>=</mo><mn>51</mn><mo stretchy="false">)</mo></math> </ephtml> . The sample mean age was 21.7 and the actual ages ranged from 19 to 41 years. The rationale for purposively selecting these two groups of participants was not only that they were representative of the population of pre-service teachers registered for the module and easily accessible to the researcher, but primarily also that they were appropriate in capturing the major variations in their CCK which could not be obtained from other choices (Maxwell, [<reflink idref="bib48" id="ref82">48</reflink>]). This sampling strategy helped in addressing the two aims of the study in that the participants were only those pre-service teachers who took high school mathematics and those who took mathematical literacy.</p> <hd id="AN0159297330-13">5.5. Measures</hd> <p>The materials consisted of a paper-and-pencil test measuring spatial visualization ability and a questionnaire probing participants views on spatial visualization.</p> <hd id="AN0159297330-14">5.5.1. Surface development test</hd> <p>Pre-service teachers' spatial visualization ability was assessed by the administration of Surface Development Test (SDT). Spatial visualization tests measure individual's ability to the ability to integrate two components, to decompose images into parts and to perform spatial transformations (Lohman, [<reflink idref="bib45" id="ref83">45</reflink>]). The SDT consisted of 6 problems and included demographic choices relating to the pre-service teachers' age, the mark obtained in either mathematics or mathematical literacy in high school (Appendix A). An example for completing this test is shown in Figure 1; numbers 1 and 4 have already been matched with their corresponding letters H and C, respectively. Thus, respondents can then match the sides indicated by the numbers 2, 3, and 5 with the letters B, G, and H, respectively. Respondents needed to take note of the fact that two of the letters can be linked to the same number. The cross (X) must always be on the outside of the object when the paper is folded.</p> <p>Graph: Figure 1. Sample item from the SDT from the Kit of Factor-Referenced Cognitive Tests.</p> <p>The measurement of SVA is standardized by international tests: Mental Cutting Test (MCT), Heinrich Spatial Visualization Test (HSVT), Purdue Spatial Visualization Test (PSVT) and Purdue Spatial Visualization Test – Visualization of Rotation (PSVT-R). However, one of the paper-and-pencil tests selected to measure SVA of primary school pre-service mathematics teachers in this study was a reduced version of the Surface Development Test (SDT). The SDT was developed by Ekstrom et al. ([<reflink idref="bib28" id="ref84">28</reflink>]) to collect data for determining if a difference existed between two groups of pre-service teachers' SVA.</p> <p>Taking Voyer et al.'s ([<reflink idref="bib72" id="ref85">72</reflink>]) discussion of the efficacy of assessment tools, I found the SDT (Appendix A) to be at the right level for the target population, easy to analyze and less costly to access, and its psychometric properties have been assessed. The original test contains 12 spatial visualization problems. In this study, the test consisted of only six problems each requiring pre-service teachers to match the edges of a 2D shape with the edges of the shape when it is folded up along the dotted lines to form a 3D object, as shown in Figure 1. The tasks required the participants to identify lines of symmetry in order to match locations on a 2D shape (net) with the same location on the 3D shape. This type of item may require the rotation of the 3D shape to find lines of symmetry and project the location of identical areas from 2D shape to 3D shape.</p> <p>The decision to limit the investigation to only half the problems was threefold. One was that the concepts required to solve the problems in the Test the items tap into general mathematical knowledge of geometry nets which involves the building of 3D shapes from flattened out nets. Geometry nets allow the exploration of the more familiar 2D shapes that go into making 3D shapes. Second, although this activity involves information a teacher would use, it is not specific to the work of teaching; therefore, it requires no knowledge of students or teaching (Blömeke & Delaney, [<reflink idref="bib9" id="ref86">9</reflink>]). Third, the decision was purely economic; time constraints.</p> <p>Bofferding ([<reflink idref="bib12" id="ref87">12</reflink>]) suggests that the Test seems to measure mental visualization, folding and rotation of shapes, visual comparison or mapping ability, spatial relationships, and to a smaller extent logic. This suggestion is consistent with the definition of spatial visualization adopted in this study. The psychometric properties reported in the SDT suggest a satisfactory reliability coefficient, and conclusive information on the validity of the test battery is given. The internal consistency coefficient of the Test in this study was calculated as</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>=</mo><mn>.81</mn><mo stretchy="false">(</mo><mrow><mi>n</mi><mo>=</mo><mn>86</mn></mrow><mo stretchy="false">)</mo></math> </ephtml> .</p> <hd id="AN0159297330-15">5.5.2. The Probes questionnaire</hd> <p>To collect qualitative data, I designed a Probes questionnaire (Appendix B). In this questionnaire, cartoons depicting pre-service teachers engaged in exchange of views on the concept of spatial visualization were used. Ibrahim et al. ([<reflink idref="bib41" id="ref88">41</reflink>]) suggest that the use of real-life figures and names can lead to prejudice toward making of a decision. As a consequence, the cartoons were used not only because they did not refer to gender, race or culture but also to improve the trustworthiness of the results (Ibrahim et al., [<reflink idref="bib41" id="ref89">41</reflink>]). This questionnaire consisted of three Probes. A sample Probe (Figure 2) depicts a pre-service teacher's thinking on spatial visualization.</p> <p>PHOTO (COLOR): Figure 2. A sample item in the Probes questionnaire.</p> <hd id="AN0159297330-16">5.6. Procedure</hd> <p>Informed consent forms were distributed to the population of these primary school pre-service teachers. The Test and the Probes questionnaire were administered in the beginning of the second semester of 2018. Transformative triangulation of quantitative and qualitative data took place at the discussion stage of this mixed methods study. Central to the concept of triangulation is the notion that different methods leading to the same results give more confidence in the research findings (Rothbauer, [<reflink idref="bib58" id="ref90">58</reflink>]).</p> <p>In the Test, respondents needed to solve the problems by visualizing the open and closed forms of the shape. Specifically, they needed to match the letters on the 3D object with the numbers on the 2D object. The Test took participants 15 min to complete. The Probes questionnaire was administered after the Test and participants took 10 min complete. In this questionnaire, participants were requested to select only one of the three alternatives provided, which they deemed to be closest to their views. Methodological triangulation of data from both sources was done at the discussion stage of this paper to understand the extent to which qualitative data complemented and clarified the quantitative findings. The next section reports on the results by blending both quantitative and qualitative results from the two data collection tools jointly.</p> <hd id="AN0159297330-17">5.7. Data analyses</hd> <p>An independent-samples <emph>t</emph>-test was used to answer the first quantitative questions which examined the significant difference (if any) between two different groups of pre-service teachers: those with a pure school mathematics background and those with a mathematical literacy background. The analyses of qualitative data were guided by the CCK component of MKT; which activates individual pre-service teacher's prior knowledge and thinking. To this end, the analysis involved the thematic coding of pre-service teachers' written responses and frequency counts in the Probes questionnaire. An overview of the analyses follows.</p> <p>Although these data were derived from non-random samples, at least they were representative samples in that they reflected the key characteristics (gender, age, year of registration) of the larger population of primary mathematics pre-service teachers at the university (Clements & Sarama, [<reflink idref="bib19" id="ref91">19</reflink>]). There were no outliers or missing data. To test whether the data were normally distributed, the Shapiro–Wilk test for normality was conducted. Further analysis involved the testing for the homogeneity of group variances. For this purpose, the Levene's test for equality of variances was used to examine the equality of the variances (the spread of scores around the mean) between the two groups of data. Having established that the difference between these two groups was statistically significant, it was necessary to examine their practical significance by considering the 'So what question' (Clements & Sarama, [<reflink idref="bib19" id="ref92">19</reflink>], p. 20). As a consequence, the actual magnitude of the effect of this significance was assessed to gain insight into whether the observed difference was meaningful.</p> <p>In scoring the Test, 1 point was allocated for each correct answer but no points are given for incorrect answers. The maximum score that could be received on the test was 30 and the minimum score was 0. High scores indicated high spatial visualization ability. A low mean was classified as signalling under-achievement which suggested that participants experienced difficulty in seeing 3D properties in 2D diagrams.</p> <p>Participants' open-ended responses to prompts were categorized into either informed, naïve or not classifiable if the response was blank or irrelevant to the prompt. The coding process involved the categorization of the written responses into 'informed' if the explanation made references to visualizing, imagination, and mind work and 'naïve' if references to drawing, reasoning, or explaining were made. These responses were then transformed into percentages for clarity and comparison purposes. Although data were collected from each individual respondent, in reporting the results, the Probes were used as themes to organize group rather than individual preservice teacher's response. Included in this presentation are frequency counts for the relevant themes. Participants' choices and written descriptions were coded into two categories. One was 'informed' if they were consistent with the mathematical view.</p> <hd id="AN0159297330-18">5.8. Triangulating quantitative and qualitative data</hd> <p>This approach is in line with the notion of methodological triangulation which involves using multiple data collection methods. This method has been found beneficial for providing confirmation of findings, more comprehensive data, increased validity and enhanced understandings of phenomena (Bekhet & Zauszniewski, [<reflink idref="bib6" id="ref93">6</reflink>]). In this study, data from quantitative analyses were clarified and enriched by themes from participants' written responses on spatial visualization. The questions in Probes were structured to facilitate asking multiple participants the same questions so as to achieve data saturation and thus avoid data saturation being a constantly moving target. Guest et al. ([<reflink idref="bib31" id="ref94">31</reflink>]) define the concept of data saturation as the point at which no new information is observed in the data. Consistent with this approach, data obtained from both quantitative and qualitative analyses are presented jointly in the next two sections.</p> <hd id="AN0159297330-19">6. Results</hd> <p>Preliminary statistical results showed that there were no outliers or missing data and the Shapiro–Wilk test for normality showed that the distribution was close to normal with a <emph>p</emph>-values larger than.05, suggesting that the sample scores for both groups were normally distributed. Levene's test for equality of variances for the homogeneity of group variances results determined that there was significant variance</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mrow><mi>p</mi><mo>=</mo><mn>.50</mn></mrow><mo stretchy="false">)</mo></math> </ephtml> between the subgroups. These results, taken together, suggested that the normality assumption was met and therefore the <emph>t</emph>-test for examining the differences between two groups was a suitable statistical technique for the analysis of the data (Salkind, [<reflink idref="bib59" id="ref95">59</reflink>]). Descriptive statistics such as the means, standard deviations, and measures of skewness and kurtosis are presented in Table 1.</p> <p>Table 1. Examining the SDT descriptives and normality tests.</p> <p> <ephtml> <table><thead valign="bottom"><tr><td /><td>Gr 12 subject</td><td>Measure</td><td>Stat.</td><td>Std. Err</td></tr></thead><tbody><tr><td>SDT score</td><td>Mathematics</td><td>Mean</td><td char=".">16.29</td><td char=".">.76</td></tr><tr><td>Std. Deviation</td><td char=".">4.47</td><td char="." /></tr><tr><td>Skewness</td><td char=".">−.36</td><td char=".">.40</td></tr><tr><td>Kurtosis</td><td char=".">.27</td><td char=".">.78</td></tr><tr><td>Mathematical Literacy</td><td>Mean</td><td char=".">8.72</td><td char=".">.72</td></tr><tr><td>Std. Deviation</td><td char=".">5.13</td><td char="." /></tr><tr><td>Skewness</td><td char=".">.99</td><td char=".">.33</td></tr><tr><td>Kurtosis</td><td char=".">.97</td><td char=".">.66</td></tr></tbody></table> </ephtml> </p> <p>The results of an independent-samples <emph>t</emph>-test analysis conducted to compare the SVA in pre-service teachers who took mathematics and those who took mathematical literacy in their grade 12 class showed a statistically significantly difference in the scores for mathematics (</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>M</mi><mrow><mrow><mtext>Maths</mtext></mrow></mrow></msub></mrow></math> </ephtml> = 16.3,</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mrow><msub><mi>D</mi><mrow><mrow><mtext>Maths</mtext></mrow></mrow></msub></mrow></math> </ephtml> = 4.47) and mathematical literacy</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mrow><mrow><msub><mi>M</mi><mrow><mrow><mtext>Maths Lit</mtext></mrow></mrow></msub></mrow><mo>=</mo><mn>8.73</mn><mo>,</mo><mi>S</mi><mrow><msub><mi>D</mi><mrow><mrow><mtext>Maths Lit</mtext></mrow></mrow></msub></mrow><mo>=</mo><mn>5.13</mn></mrow><mo stretchy="false">)</mo></math> </ephtml> conditions;</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo stretchy="false">(</mo><mrow><mn>86</mn></mrow><mo stretchy="false">)</mo><mo>=</mo><mn>7.25</mn><mo>,</mo><mi>p</mi><mo><</mo><mn>.001</mn><mo>;</mo><mi>d</mi><mo>=</mo><mn>.16</mn></math> </ephtml> . Put another way, the results showed that the difference between the two means was not only statistically significant but also of practical significance.</p> <p>Information about the second research question, 'Why are these two groups of pre-service teachers' common content knowledge in relation to spatial visualization different, if at all?' came from response to both Probes questionnaire responses. Qualitative results obtained from Probe 1 are displayed in Table 2. The table shows distribution of 'informed' and 'naïve' categories based on written responses to Probe 1. Just over 30% of the participants opted to write their responses to this Probe rather than use the existing choices of A and B (Appendix A).</p> <p>Table 2. Categorization of participants' responses to Probe 1, 'Think about what spatial visualization is'.</p> <p> <ephtml> <table><thead valign="bottom"><tr><td /><td /><td /><td>Mathematics</td><td>Maths Literacy</td><td /></tr><tr><td>Probe</td><td>Choice</td><td>Category</td><td>Number</td><td>%</td><td>Number</td><td>%</td><td>Total</td></tr></thead><tbody><tr><td>1</td><td>A</td><td>Naïve</td><td>18</td><td>21</td><td>38</td><td>44</td><td>56</td></tr><tr><td>B</td><td>Informed</td><td>3</td><td>3</td><td>0</td><td>0</td><td>3</td></tr><tr><td>C</td><td>Naïve</td><td>8</td><td>9</td><td>9</td><td>10</td><td>17</td></tr><tr><td>Informed</td><td>6</td><td>7</td><td>4</td><td>5</td><td>10</td></tr></tbody></table> </ephtml> </p> <p>One participant with a mathematical background explained spatial visualization as '<emph>Imagining invisible objects</emph>'. Another participant, who identified herself as having passed high school mathematics, wrote that '<emph>Spatial visualization involves moving something in your mind</emph>'. One other participant wrote, '<emph>When we do spatial vision, we're like dreamin'. It is like a vision you create</emph>'. Further, one female participant with a mathematics background answered this way, '<emph>Probably the ability to see things that are too far for a normal person to see</emph>'.</p> <p>Results of Probe 2, 'Think about the purpose of spatial visualization', were relatively similar to this Probe 1 in character. It is for this reason than only Probe 1 results are presented in this study. Worth reporting is that all written responses to Probe 3, 'Think about which object has edges', were misconceptions (naïve views). Overall, in Probe 1, slightly above two thirds of the pre-service teachers (65%) held naïve views about the meaning of spatial visualization. Another result was that the majority (73%) of pre-service teachers held misconceptions about the purpose of spatial visualization in mathematics.</p> <p>These explanations were useful in providing insight into participants' thinking on spatial visualization because each pre-service teacher's thinking reflected their own unique perspective and experiences of spatial visualization. It was through these explanations that conclusions were drawn about why the two groups of pre-service teachers' common content knowledge in relation to spatial visualization was different.</p> <p>The discussion that follows is organized around the two research questions: (<reflink idref="bib1" id="ref96">1</reflink>) What is the difference (if any) in common content knowledge relating to spatial visualization ability between the primary school pre-service teachers with grade 12 mathematics experiences and those with grade 12 mathematical literacy experiences? and (<reflink idref="bib2" id="ref97">2</reflink>) Why are these two groups of pre-service teachers' common content knowledge in relation to spatial visualization different, if at all?</p> <hd id="AN0159297330-20">7. Discussion</hd> <p>The aims of this study were to (<reflink idref="bib1" id="ref98">1</reflink>) describe the difference (if any) in CCK relating to spatial visualization ability between the primary school pre-service teachers with grade 12 mathematics experiences and those with grade 12 mathematical literacy experiences, and (<reflink idref="bib2" id="ref99">2</reflink>) explain why these two groups of pre-service teachers' CCK in relation to spatial visualization was different, if at all. It is through these aims that this paper adds a new perspective on pre-service teachers' SVA, particularly so from a different context, the African continent. In keeping with the concurrent transformative method adopted in this study, quantitative and qualitative data were methodologically triangulated. Hence, the discussion here focused on simultaneously using the results obtained from the two datasets to describe where and why one dataset provided aspects of pre-service teachers' CCK that were different from or similar to those generated by another.</p> <hd id="AN0159297330-21">7.1. The difference in CCK relating to SVA between two groups of preservice teachers</hd> <p>Preliminary statistical results suggested that meaningful conclusions about that population of pre-service teachers with grade 12 mathematics and those with grade 12 mathematical literacy could be drawn. The data obtained from the Surface Development Test shed some light on the pre-service teachers' SVA. In particular, preliminary statistical results determined that there was significant variance between the two subgroups. This statistically significant result of an independent-samples <emph>t</emph>-test represented a real rather than a chance difference between the achievement scores of pre-service teachers with grade 12 mathematics and those with grade 12 mathematical literacy.</p> <p>Analysis of the data not only showed that there was a statistically significant difference between these two groups of preservice teachers but also that the difference was of practical significance. Although the study relied on the responses of 86 participants who attended a very specific module, the generalizability of the results to the population of fourth-year primary mathematics education pre-service teachers enrolled at the university where the study was conducted is supported by the use of an inference technique (a <emph>t</emph>-test for independent means).</p> <hd id="AN0159297330-22">7.2. Factors influencing these two groups of pre-service teachers' CCK in relation to spatial...</hd> <p>The results of the Probes questionnaire on pre-service teachers' views on the concept of spatial visualization were considered to be a reflection of pre-service teachers' CCK. As a consequence, they shared light on the efforts that these teachers may need to make in relation to the development of their PCK, particularly those with a mathematical literacy background. Qualitative analysis not only pointed to the detrimental effects of misconceptions on the development of preservice teachers' spatial visualization ability but also confirmed quantitative findings on the differences in CCK in relation to spatial visualization of the two groups.</p> <p>One interesting result was that most pre-service teachers with a mathematical literacy background harboured naïve views on spatial visualization. In particular, when prompted to 'Think about what spatial visualization is', more than twice as many pre-service teachers with a mathematical literacy as preservice teacher with a mathematics background believed that spatial visualization involves learning about triangle, squares, circles, lines, and so on. Similar results were found in a study of Turkish pre-service teachers conducted by Yurt and Tünkler ([<reflink idref="bib74" id="ref100">74</reflink>]). Specifically, they found that misconceptions were one of the primary causes of the pre-service teachers' low level of achievement in spatial visualization.</p> <p>The <emph>Principles and Standards for School Mathematics</emph> (National Council of Teachers of Mathematics [NCTM], [<reflink idref="bib54" id="ref101">54</reflink>]) states that 'one aspect of spatial visualization involves moving between 2-and-3D shapes and their representations' (p. 43). In light of this statement, another interesting result was that, in Probe 3, approximately the same proportion showed weak conception of the term 'edge'; in both cases, the majority of pre-service teachers conceived of an edge as a 2D mathematical object. Yurt and Tünkler ([<reflink idref="bib74" id="ref102">74</reflink>]) found similarly. In fact, they point out that pre-service teachers' knowledge of the difference between 2D representations of 3D structures and mentally manipulate their edges and faces, was generally weak. Overall, the results seem to be consistent with those of Ball ([<reflink idref="bib1" id="ref103">1</reflink>]) who found that pre-service teacher bring with them into their modules thin common content knowledge.</p> <p>As hypothesized, pre-service teachers with grade 12 mathematics experience scored significantly higher on the SDT given the differences in the mathematics content between the two high school subjects. Having taken this into consideration, it is worth noting that the proportion of participants who were able to provide meaningful explanations to the choices they made came from those participants with a pure mathematics background. These findings were consistent with those of Turgut and Yilmaz ([<reflink idref="bib68" id="ref104">68</reflink>]) who found that mathematics pre-service teachers' spatial visualization abilities were at a low level. Further, rather than seeing an edge as a segment where two faces intersect, almost all the pre-service teachers chose one of the first two alternative (which I deliberately presented as incorrect) options in the questionnaire; they most probably saw it as one of the 'easiest'. Yurt and Tünkler ([<reflink idref="bib74" id="ref105">74</reflink>]) attribute these results to the scarcity of opportunities for developing spatial abilities beginning from preschool education to higher education. Another interpretation of the results could be that a mathematics background is a predictor of rather than it has an effect on spatial visualization.</p> <p>Thus, instruction can be one of the artefacts that can enable them to organize the new information to perform spatial visualization tasks, and in that way help the pre-service teachers to successfully perform such tasks independently. Considerable attention is accorded to developing proficiency with numeracy in high school mathematical literacy curriculum and, not surprisingly pre-service teachers with grade 12 mathematical literacy performed poorly in the SDT. The plausible reason for the disproportionate attention accorded to numeracy over SVA in school curricula is the attitude that students require explicit instruction to become proficient in numeracy; they have an intuitive understanding of SVA and therefore require little instruction.</p> <p>According to Gutiérrez ([<reflink idref="bib32" id="ref106">32</reflink>]), this disproportion is a reasonable consequence of the associating spatial visualization with geometry. However, based on the results, this assumption is challenged because it is not premised on principle. In fact, spatial visualization ability is a helpful aid in supporting intuition and concept formation for many topics in mathematics learning (Glass et al., [<reflink idref="bib30" id="ref107">30</reflink>]). Perhaps, there is some grain of truth in Bishop's ([<reflink idref="bib8" id="ref108">8</reflink>]) suggestion that students' previous experience with manipulative apparatus might contribute to their performance in activities requiring spatial visualization ability. Given the <emph>Mathematical Education of Teachers'</emph> (Conference Board of Mathematical Sciences [CBMS], [<reflink idref="bib20" id="ref109">20</reflink>]) suggestion that 'the key to turning even poorly prepared prospective elementary teachers into mathematical thinkers is to work from what they do know' (p. 71), the findings in this study are useful in informing teacher education programmes of the need to focus attention on CCK of pre-service teachers who enter university without a pure mathematics background.</p> <p>Overall, the results in this study were consistent with the expectation that pre-service teachers who have high school mathematics experience were to perform better in the SDT partly due to higher aspirations, motivations, and engagement with mathematics (Martin & Ruble, [<reflink idref="bib47" id="ref110">47</reflink>]). In addition, these preservice teachers were expected to provide more informed explanations on the concept of spatial visualization because of their experience with problem solving frequently encountered in mathematics learning and teaching. The next section concludes the study.</p> <hd id="AN0159297330-23">8. Conclusions</hd> <p>Discussions about teacher ability bring into focus the CCK base that drives teachers' ability in spatial visualization. To investigate this, a concurrent transformative mixed methods study was conducted. In particular, a survey was conducted to describe the differences between pre-service teachers with a high school mathematics background and those with a mathematical literacy background. The research in this article was motivated by a concern that pre-service primary mathematics teachers bring into their training programmes weak background knowledge of spatial visualization which contributes to the poor CCK which in turn may affect students' performance in primary school mathematics. To test this hypothesis, a Test was administered and Probes were conducted. Using the CCK as a theoretical foundation, it was found that pre-service teachers with a mathematics background performed significantly better that those with mathematical literacy experiences in high school. Qualitative results reflected poor conceptual understanding of spatial visualization; pre-service teachers' CCK was punctuated by misconceptions. These results underscore the importance of high school mathematics as a vehicle for visualizing spatially.</p> <hd id="AN0159297330-24">8.1. Implications</hd> <p>The implication thereof was that there is a need to reaffirm the importance of providing pre-service primary mathematics teachers who took grade 12 mathematical literacy with support to mitigate the discrepancy in their SVA when compared with those with grade 12 mathematics. These results also inform future curriculum planning and development for the pre-service teacher training programmes; unless steps are taken to improve spatial visualization content knowledge of pre-service teachers entering university studies, it means that there are areas of the curriculum in which some pre-service teachers would not be competent. This discrepancy in common content knowledge needs to be addressed because it could have implications on these teachers' and their students' future. The practical significance of this results was that they can inform teacher education practice in relation to the unequal skill with which some pre-service teachers begin their studies and therefore there is a need to offer the requisite support to such teachers.</p> <hd id="AN0159297330-25">8.2. Limitations</hd> <p>Although methodological triangulation was used to increase the trustworthiness and validity of the findings, and thus mitigate fundamental biases arising from the use of a single method, several factors might affect the internal and external validity of this study. The limitation of this study lies in the fact that the results could be affected by the existence of other plausible rival hypotheses. Also, a true cause-and-effect relationship could not be definitively stated given the design adopted in this study. Again, the findings need to be read with caution since it might be possible that some of the pre-service teachers might have taken mathematics at grade 12 and changed to mathematical literacy after unsuccessful attempt(s) at mathematics. Put differently, the generalizability of the findings beyond the context of the study is questionable; in fact, the findings are of contextual relevance. These limitations notwithstanding, the use of multiple sources of data and perspectives insured that the results of this study demonstrated trustworthiness through data saturation. As a consequence, two recommendations emanated from these results.</p> <p>First is that explicit instruction and training need to focus on providing high school mathematical literacy students with opportunities to work and train in tasks that involved the visualization of translation of manipulatives into 3D shapes presented in various orientations. Specifically, DGS can be employed to create a continuum of images of 3D objects in different orientations. The integration of this technology into instructional practices is supported by the fact that spatial visualization ability is a malleable characteristic and thus amenable to intervention (Brus et al., [<reflink idref="bib15" id="ref111">15</reflink>]). In addition, pre-service primary mathematics teachers can utilize TPACK strategic thinking as they design plans to guide their students in exploring content topics with technologies (Niess, [<reflink idref="bib55" id="ref112">55</reflink>]).</p> <p>Second is that given that this group of pre-service teachers (with mathematical literacy background) has had limited exposure to spatial visualization tasks in high school, teacher education programmes need to provide opportunities to upskill themselves in spatial thinking and provide exemplars of rich activities. In addition, the finding that pre-service primary mathematics teachers' conception of the term 'edge' was thin as they conceived on an edge as a 2D mathematical object suggests the need for an investigation of pre-service primary mathematics teachers' CCK from the Tall and Vinner's ([<reflink idref="bib64" id="ref113">64</reflink>]) framework of concept formation. Given the variability of cultural backgrounds of pre-service teachers and the difficulty in catering for these varieties, nonstandard assessment measures were needed. Thus, the development and design of such measures may be the concern for future research.</p> <hd id="AN0159297330-26">Disclosure statement</hd> <p>No potential conflict of interest was reported by the author.</p> <hd id="AN0159297330-27">Appendix A</hd> <hd1 id="AN0159297330-28">Surface Development Test</hd1> <p>[Adapted from Ekstrom et al. ([<reflink idref="bib28" id="ref114">28</reflink>])]</p> <hd1 id="AN0159297330-29">Part A (Test Information)</hd1> <p>In this test you are to try to visualize how a plane figure can be folded to form some kind of 3D object. Consider the two drawing below. The drawing on the left is a piece of paper which can be folded on the dotted lines to form the 3D object drawn on the right. You are to imagine the folding and are to figure out which of the lettered edges on the object are the same as the numbered edges on the piece of paper at the left.</p> <hd1 id="AN0159297330-30">Part B (Demographic Information)</hd1> <p> <emph>Code: ------------</emph> </p> <p> <emph>Please, circle/tick one answer for each of the following.</emph> </p> <p></p> <p> <ephtml> <table><tbody><tr><td><italic>Personal particulars</italic></td></tr><tr><td><italic>Gender</italic></td><td><italic>Female</italic></td><td><italic>Male</italic></td><td><italic>Age (in years)</italic></td><td><italic /></td></tr><tr><td><italic>Grade 12 subject (Tick)</italic></td><td><italic>Mathematics</italic></td><td><italic>Mathematical Literacy</italic></td><td><italic>% mark obtained</italic></td><td><italic /></td></tr><tr><td><italic>Mathematics in Grade 11?</italic></td><td><italic>Yes</italic></td><td><italic>No</italic></td><td><italic>Home Language</italic></td><td><italic /></td></tr><tr><td><italic>Attended no-fee paying school</italic></td><td><italic>Yes</italic></td><td><italic>No</italic></td><td><italic>Rural school</italic></td><td><italic>Yes</italic></td><td><italic>No</italic></td></tr></tbody></table> </ephtml> </p> <hd1 id="AN0159297330-31">Part C (Instruction)</hd1> <p></p> <ulist> <item> Write the letters of the answers in the numbered spaces in the Table at the far right.</item> <p></p> <item> The side of the plane piece marked the X will always be the same as the side of the object marked with the X. therefore, the paper must always be folded so that the X will be on the outside of the 3D object.</item> <p></p> <item> The test consists of six (<reflink idref="bib6" id="ref115">6</reflink>) problems each with 5 marks. A correct solution is worth 1 mark thus bringing the total marks to 30. Your score on this test will be the number of correct letters.</item> <p></p> <item> Notice that two of the answers can be the same.</item> </ulist> <p>Now try to practice the problem below.</p> <p>Graph</p> <p>The solution to this problem is as follows: 1 is H; 2 is B; 3 is G; 4 is C; and 5 is H.</p> <hd1 id="AN0159297330-32">Part D (Surface Development Test)</hd1> <p>You will have 12 minutes to solve ALL these problems.</p> <p>Graph</p> <p>Graph</p> <p>Graph</p> <p>*End of Test. Please, turn over the page to complete the Probes Questionnaire.*</p> <hd id="AN0159297330-33">Appendix B</hd> <hd1 id="AN0159297330-34">Probes Questionnaire</hd1> <p>The statements below are intended to understand your views of spatial visualization. Just indicate what you believe in or think.</p> <p></p> <ulist> <item> <bold> Please, read each Probe carefully, and then _B_circle</bold> (A, B, or C) the statement with which you agree most and <bold>explain y</bold>our choice.</item> <p></p> <item> <bold> Please, do not spend a long time on any one question – _B_your first thoughts are usually your best</bold>. There is no right or wrong answer; your answers will <bold>NOT</bold> affect your marks.</item> <p></p> <item> This questionnaire will take you 10 minutes to complete.</item> </ulist> <hd1 id="AN0159297330-35">Probe 1</hd1> <p>Think about what spatial visualization is.</p> <p>Graph</p> <hd1 id="AN0159297330-36">Probe 2</hd1> <p>Think about the purpose of spatial visualization.</p> <p>Graph</p> <hd1 id="AN0159297330-37">Probe 3</hd1> <p>Think about which object has edges.</p> <p>Graph</p> <p>*End of Probes Questionnaire. 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Educational Sciences: Theory and Practice, 16 (3), 965 – 986. https://doi.org/10.12738/estp.2016.3.0324</bibtext> </blist> </ref> <aug> <p>By Benjamin Shongwe</p> <p>Reported by Author</p> </aug> <nolink nlid="nl1" bibid="bib46" firstref="ref2"></nolink> <nolink nlid="nl2" bibid="bib11" firstref="ref3"></nolink> <nolink nlid="nl3" bibid="bib60" firstref="ref4"></nolink> <nolink nlid="nl4" bibid="bib39" firstref="ref6"></nolink> <nolink nlid="nl5" bibid="bib70" firstref="ref8"></nolink> <nolink nlid="nl6" bibid="bib71" firstref="ref11"></nolink> <nolink nlid="nl7" bibid="bib57" firstref="ref12"></nolink> <nolink nlid="nl8" bibid="bib69" firstref="ref13"></nolink> <nolink nlid="nl9" bibid="bib16" firstref="ref16"></nolink> <nolink nlid="nl10" bibid="bib43" firstref="ref20"></nolink> <nolink nlid="nl11" bibid="bib62" firstref="ref21"></nolink> <nolink nlid="nl12" bibid="bib73" firstref="ref23"></nolink> <nolink nlid="nl13" bibid="bib53" firstref="ref28"></nolink> <nolink nlid="nl14" bibid="bib54" firstref="ref29"></nolink> <nolink nlid="nl15" bibid="bib18" firstref="ref30"></nolink> <nolink nlid="nl16" bibid="bib49" firstref="ref32"></nolink> <nolink nlid="nl17" bibid="bib37" firstref="ref33"></nolink> <nolink nlid="nl18" bibid="bib50" firstref="ref34"></nolink> <nolink nlid="nl19" bibid="bib56" firstref="ref35"></nolink> <nolink nlid="nl20" bibid="bib35" firstref="ref36"></nolink> <nolink nlid="nl21" bibid="bib25" firstref="ref38"></nolink> <nolink nlid="nl22" bibid="bib61" firstref="ref39"></nolink> <nolink nlid="nl23" bibid="bib17" firstref="ref40"></nolink> <nolink nlid="nl24" bibid="bib42" firstref="ref41"></nolink> <nolink nlid="nl25" bibid="bib38" firstref="ref43"></nolink> <nolink nlid="nl26" bibid="bib66" firstref="ref44"></nolink> <nolink nlid="nl27" bibid="bib13" firstref="ref46"></nolink> <nolink nlid="nl28" bibid="bib74" firstref="ref49"></nolink> <nolink nlid="nl29" bibid="bib67" firstref="ref50"></nolink> <nolink nlid="nl30" bibid="bib15" firstref="ref51"></nolink> <nolink nlid="nl31" bibid="bib36" firstref="ref53"></nolink> <nolink nlid="nl32" bibid="bib65" firstref="ref55"></nolink> <nolink nlid="nl33" bibid="bib52" firstref="ref56"></nolink> <nolink nlid="nl34" bibid="bib23" firstref="ref57"></nolink> <nolink nlid="nl35" bibid="bib33" firstref="ref58"></nolink> <nolink nlid="nl36" bibid="bib34" firstref="ref59"></nolink> <nolink nlid="nl37" bibid="bib44" firstref="ref61"></nolink> <nolink nlid="nl38" bibid="bib24" firstref="ref62"></nolink> <nolink nlid="nl39" bibid="bib64" firstref="ref63"></nolink> <nolink nlid="nl40" bibid="bib10" firstref="ref65"></nolink> <nolink nlid="nl41" bibid="bib63" firstref="ref69"></nolink> <nolink nlid="nl42" bibid="bib29" firstref="ref72"></nolink> <nolink nlid="nl43" bibid="bib51" firstref="ref73"></nolink> <nolink nlid="nl44" bibid="bib32" firstref="ref74"></nolink> <nolink nlid="nl45" bibid="bib27" firstref="ref75"></nolink> <nolink nlid="nl46" bibid="bib26" firstref="ref76"></nolink> <nolink nlid="nl47" bibid="bib19" firstref="ref77"></nolink> <nolink nlid="nl48" bibid="bib22" firstref="ref78"></nolink> <nolink nlid="nl49" bibid="bib59" firstref="ref79"></nolink> <nolink nlid="nl50" bibid="bib40" firstref="ref80"></nolink> <nolink nlid="nl51" bibid="bib21" firstref="ref81"></nolink> <nolink nlid="nl52" bibid="bib48" firstref="ref82"></nolink> <nolink nlid="nl53" bibid="bib45" firstref="ref83"></nolink> <nolink nlid="nl54" bibid="bib28" firstref="ref84"></nolink> <nolink nlid="nl55" bibid="bib72" firstref="ref85"></nolink> <nolink nlid="nl56" bibid="bib12" firstref="ref87"></nolink> <nolink nlid="nl57" bibid="bib41" firstref="ref88"></nolink> <nolink nlid="nl58" bibid="bib58" firstref="ref90"></nolink> <nolink nlid="nl59" bibid="bib31" firstref="ref94"></nolink> <nolink nlid="nl60" bibid="bib68" firstref="ref104"></nolink> <nolink nlid="nl61" bibid="bib30" firstref="ref107"></nolink> <nolink nlid="nl62" bibid="bib20" firstref="ref109"></nolink> <nolink nlid="nl63" bibid="bib47" firstref="ref110"></nolink> <nolink nlid="nl64" bibid="bib55" firstref="ref112"></nolink>
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Items – Name: Title
  Label: Title
  Group: Ti
  Data: A Causal-Comparative Study of South African Pre-Service Primary Mathematics Teachers' Spatial Visualization Ability: Does Common Content Knowledge Matter?
– Name: Language
  Label: Language
  Group: Lang
  Data: English
– Name: Author
  Label: Authors
  Group: Au
  Data: <searchLink fieldCode="AR" term="%22Shongwe%2C+Benjamin%22">Shongwe, Benjamin</searchLink> (ORCID <externalLink term="http://orcid.org/0000-0001-8083-6462">0000-0001-8083-6462</externalLink>)
– Name: TitleSource
  Label: Source
  Group: Src
  Data: <searchLink fieldCode="SO" term="%22International+Journal+of+Mathematical+Education+in+Science+and+Technology%22"><i>International Journal of Mathematical Education in Science and Technology</i></searchLink>. 2022 53(9):2338-2363.
– Name: Avail
  Label: Availability
  Group: Avail
  Data: Taylor & Francis. Available from: Taylor & Francis, Ltd. 530 Walnut Street Suite 850, Philadelphia, PA 19106. Tel: 800-354-1420; Tel: 215-625-8900; Fax: 215-207-0050; Web site: http://www.tandf.co.uk/journals
– Name: PeerReviewed
  Label: Peer Reviewed
  Group: SrcInfo
  Data: Y
– Name: Pages
  Label: Page Count
  Group: Src
  Data: 26
– Name: DatePubCY
  Label: Publication Date
  Group: Date
  Data: 2022
– Name: TypeDocument
  Label: Document Type
  Group: TypDoc
  Data: Journal Articles<br />Reports - Research<br />Tests/Questionnaires
– Name: Audience
  Label: Education Level
  Group: Audnce
  Data: <searchLink fieldCode="EL" term="%22High+Schools%22">High Schools</searchLink><br /><searchLink fieldCode="EL" term="%22Secondary+Education%22">Secondary Education</searchLink><br /><searchLink fieldCode="EL" term="%22Higher+Education%22">Higher Education</searchLink><br /><searchLink fieldCode="EL" term="%22Postsecondary+Education%22">Postsecondary Education</searchLink><br /><searchLink fieldCode="EL" term="%22Elementary+Education%22">Elementary Education</searchLink>
– Name: Subject
  Label: Descriptors
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22Mathematics+Skills%22">Mathematics Skills</searchLink><br /><searchLink fieldCode="DE" term="%22Pedagogical+Content+Knowledge%22">Pedagogical Content Knowledge</searchLink><br /><searchLink fieldCode="DE" term="%22Spatial+Ability%22">Spatial Ability</searchLink><br /><searchLink fieldCode="DE" term="%22Visualization%22">Visualization</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Teachers%22">Mathematics Teachers</searchLink><br /><searchLink fieldCode="DE" term="%22Teaching+Methods%22">Teaching Methods</searchLink><br /><searchLink fieldCode="DE" term="%22Comparative+Analysis%22">Comparative Analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Preservice+Teachers%22">Preservice Teachers</searchLink><br /><searchLink fieldCode="DE" term="%22Teacher+Education+Programs%22">Teacher Education Programs</searchLink><br /><searchLink fieldCode="DE" term="%22Elementary+School+Teachers%22">Elementary School Teachers</searchLink><br /><searchLink fieldCode="DE" term="%22Misconceptions%22">Misconceptions</searchLink><br /><searchLink fieldCode="DE" term="%22Foreign+Countries%22">Foreign Countries</searchLink>
– Name: Subject
  Label: Geographic Terms
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22South+Africa%22">South Africa</searchLink>
– Name: DOI
  Label: DOI
  Group: ID
  Data: 10.1080/0020739X.2020.1869333
– Name: ISSN
  Label: ISSN
  Group: ISSN
  Data: 0020-739X<br />1464-5211
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: This study used a modified mathematical knowledge for teaching (MKT) framework to compare the "common content knowledge" (CCK) of two groups of purposively selected pre-service primary mathematics teachers (n = 86) in relation to spatial visualization. Specifically, thirty-five pre-service teachers (n[subscript 1] = 35) with a pure school mathematics background and fifty-one pre-service teachers (n[subscript 2] = 51) with a high school mathematical literacy background worked on problems that involved spatial objects, their properties and relationships. The responses of pre-service teachers with a mathematical literacy content background to the "Surface Development Test" (SDT) were significantly different from their counterparts' which suggested that their CCK in relation to spatial visualization was thin. Analysis of the "Probes" questionnaire which inquired about the pre-service teachers' views on the concept of spatial visualization showed that misconceptions hindered their performance. Future research directions are discussed.
– Name: AbstractInfo
  Label: Abstractor
  Group: Ab
  Data: As Provided
– Name: DateEntry
  Label: Entry Date
  Group: Date
  Data: 2023
– Name: AN
  Label: Accession Number
  Group: ID
  Data: EJ1367167
PLink https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=eric&AN=EJ1367167
RecordInfo BibRecord:
  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1080/0020739X.2020.1869333
    Languages:
      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 26
        StartPage: 2338
    Subjects:
      – SubjectFull: Mathematics Skills
        Type: general
      – SubjectFull: Pedagogical Content Knowledge
        Type: general
      – SubjectFull: Spatial Ability
        Type: general
      – SubjectFull: Visualization
        Type: general
      – SubjectFull: Mathematics Teachers
        Type: general
      – SubjectFull: Teaching Methods
        Type: general
      – SubjectFull: Comparative Analysis
        Type: general
      – SubjectFull: Preservice Teachers
        Type: general
      – SubjectFull: Teacher Education Programs
        Type: general
      – SubjectFull: Elementary School Teachers
        Type: general
      – SubjectFull: Misconceptions
        Type: general
      – SubjectFull: Foreign Countries
        Type: general
      – SubjectFull: South Africa
        Type: general
    Titles:
      – TitleFull: A Causal-Comparative Study of South African Pre-Service Primary Mathematics Teachers' Spatial Visualization Ability: Does Common Content Knowledge Matter?
        Type: main
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            NameFull: Shongwe, Benjamin
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            – D: 01
              M: 01
              Type: published
              Y: 2022
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            – Type: issn-print
              Value: 0020-739X
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              Value: 1464-5211
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              Value: 53
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              Value: 9
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            – TitleFull: International Journal of Mathematical Education in Science and Technology
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