The Opportunity to Learn Euclidean Geometry in Two Mathematics Textbooks of Tenth Grade in South Africa

Textbooks play a central role in the teaching and learning of mathematics. In some schools, textbooks serve as the only resource material available to teachers and students. Nevertheless, little is known about the learning opportunities in mathematics textbooks in most countries. This study investigated the opportunities to learn Euclidean Geometry in two textbooks of tenth-grade mathematics in South Africa. It examined the content coverage, content organization, and the types of tasks used in teaching the textbooks. This study followed a case study research design and a qualitative approach. The Curriculum and Assessment Policy Statement's (CAPS) grade 10 Euclidean geometry curriculum and Gracin's mathematical activity types served as frameworks for the analyses. The data were analyzed following the deductive content analysis approach. The result shows that the contents of Euclidean geometry were well covered in both textbooks following the curriculum, and the contents were presented in logical and sequential order to enhance learning. In addition, it was found that the tasks in the textbooks were predominantly of argumentation and reasoning type. It was concluded that the textbooks offer sufficient opportunities for learning Euclidean geometry as specified in the curriculum for the grade level. However, the inclusion of a broader range of tasks in the future editions of the textbooks was recommended.


INTRODUCTION
Textbooks are vital resources in teaching and learning. They are used as familiar sources of information for the formal study of school subjects and instruments for teaching and learning (Graves, 2000;Sherman, Walkington & Howell, 2016). They are avenues for knowledge acquisition for teachers and students; hence they are commonly used in teaching and learning in most schools in different countries (Van Steenbrugge, Valcke & Desoete, 2013). As a fundamental resource, textbooks can shape the way a subject is taught and learned. They influence the teachers' instructional decisions because teachers often use textbooks to plan their lessons (Jones & Tarr, 2007;Remillard & Heck, 2014). Textbooks are expected to provide a framework for what is taught, how it may be taught, and in what sequence it may be taught (Lemmer, Edwards & Rapule, 2008); they "are designed to translate the abstractions of curriculum policy into operations that teachers and students can carry out" (Valverde et al., 2002, p.2).
Due to the central role of textbooks in teaching and learning, textbooks used for teaching and learning must provide optimal opportunities for students to learn the curriculum content. The textbooks must provide explicit, correct content and instructional support to teachers and students (Lemmer, Edwards & Rapule, 2008). The textbooks are essential regarding presenting important mathematics topics like Euclidean Geometry in textbooks used for teaching and learning.
Euclidean geometry studies planes and solid figures based on Euclid's axioms and theories (Artmann, 2020). It is a very significant branch of mathematics; hence, its study is included in the mathematics curricula of most countries, including South Africa (Fitriani, Suryadi & Darhim, 2018;Shongwe, 2019). It is one of the topics in mathematics in the South African School curriculum (Curriculum and Assessment Policy Statement [CAPS]) that reinforces problem-solving through critical thinking (Department of Basic Education [DBE], 2011). In the curriculum (CAPS), the teaching of Euclidean proof starts in Grade 10, where the students are expected to investigate, make and prove conjectures about the properties of plane geometric figures ([DBE, 2011).
Due to the importance of Euclidean geometry in mathematics, the difficulties encountered by teachers and students in its teaching and learning, and the poor achievement of students on the topic (DBE, 2019), several research studies have explored, among others, teachers and pre-service teachers' knowledge of Euclidean geometry or difficulties they have in teaching it (Alex, 2019; Tachie, 2020;Ubah & Bansilal, 2019), students' challenges in learning Euclidean geometry (Fitriani, Suryadi & Darhim, 2018;Matheou, & Panaoura, 2021;Ngirishi & Bansilal, 2019;Shongwe, 2019), and impact of different teaching strategies and technologies on students' learning of Euclidean geometry (Adeniji, Ameen, Dambatta & Orilonise, 2018;Bayaga, Mthethwa, BossÃ & Williams, 2020;Mukamba & Makamure, 2020). Nevertheless, Euclidean geometry learning opportunities in teaching and materials seem to be lacking in mathematics education research, especially in the African context. As the textbook is practical teaching and learning resource in South Africa, it was found necessary to investigate the opportunity to learn Euclidean Geometry in mathematics textbooks.
Opportunity to Learn (OTL) is a way of measuring whether students have access to the different ingredients (e.g., content domains, cognitive skills, qualified teachers) that make for quality learning (Akiba, LeTendre & Scribner, 2007;Hadar, 2017). OTL is used to determine whether students are provided with enough access and information to learn the curriculum for their age and grade level. According to McDonnell (1995), OTL was introduced as part of the First International Mathematics Survey in the early 1960s, but the concept was substantially refined in the Second International Mathematics Study (SIMS), conducted between 1976 and 1982. OTL includes the qualities and quantities of human and material resources available in schools, school conditions, and students' experience (Banicky, 2000). These issues are considered critical for students' learning and achievement (Mohale & Mafumbate, 2019;Stols, Kriek & Ogbonnaya, 2008). Hence, OTL includes students' access to qualified teachers, appropriate books and quality learning materials, quality teaching, and access to school conditions that provide them with enough opportunity to learn and achieve knowledge and skills. The OTL of a textbook measures what the textbook offers teachers and students for them to access the desired knowledge and skills in the intended curriculum (Hong et al., 2020;Otten et al., 2014). It includes the content covered in the textbook and the depth of coverage of the content (Charalambous, Delaney, Hsu & Mesa, 2010).
Some studies have explored student opportunities to learn using different measures of OTL.
For example, based on classroom instruction, Boston & Wilhelm (2017) and Weiss, Pasley, Smith, Banilower, & Heck (2003) used lesson observation to explore student opportunities to learn mathematics. Stols (2013) used students' workbooks to investigate the learning opportunities offered to Grade 12 mathematics students regarding content coverage, time on task, curriculum coherence, and cognitive demand of questions. Based on textbooks, Gracin (2018) explored the opportunity to learn mathematics available to students in grades 6-8 Croatian mathematics textbooks. Similarly, Hong and Choi (2018) studied the opportunity to learn linear functions available to students in Korean and American mathematics textbooks.
Textbooks are often considered expressions of the intended curriculum (what students are expected to learn) and are used to mediate the intended curriculum and the implemented curriculum (Hadar & Ruby, 2019;Polikoff, 2015). Hence, textbooks influence the implemented curriculum, indicating student opportunities to learn the curriculum (Houang & Schmidt, 2008). Student opportunity to learn in textbooks has been shown to relate to the students' mathematics performance (Wijaya, van den Heuvel-Panhuizen & Doorman, 2015 Similarly, Polikoff's (2015) analysis of textbooks' alignment to core curriculum standards in Florida, United States, showed that the three textbooks analyzed offered limited opportunities for students to learn problem-solving in that the textbooks "overemphasized procedures and memorization and underemphasize conceptual skills" (p.1206).
In their study on OTL available in textbooks, Van Zanten and van den Heuvel-Panhuizen (2018) explored OTL problem-solving in Dutch Grades 4 and 6 mathematics textbooks. The study showed that the textbooks offered minimal opportunities for students to learn problem-solving in that only 3-9% of tasks in the textbooks were found to be problem-solving tasks. Gracin (2018) analyzed the mathematical activities students need to engage in to do the tasks in the most used Grades 6-8 Croatian mathematics textbooks. Gracin found that the textbooks did not provide a full range of task types, and most of the tasks in the textbooks required computation. In a study with Grade 8 Arab community students in Israel, Hadar (2017) investigated the relationship between students' achievements in a national examination and the cognitive level opportunities provided in their mathematics textbooks. The study showed that the students' achievements at different cognitive levels correlated with the cognitive opportunities provided in their textbooks. In a similar study, Hadar and Ruby (2019) examined the cognitive demands of tasks in four different mathematics textbooks in Israel. They explored the level and complexity of understanding required to undertake the tasks successfully. The study found that the per cent of algorithmic tasks (requiring "ritual performances of algorithms and recall") in the textbooks ranged from 35.8% to 58.8%. Furthermore, the study found that the tasks in all the textbooks mainly were understanding tasks of levels one ('present knowledge') and two ('act with or on knowledge'). The textbooks had very few tasks on level three ('criticize or create knowledge').
Wijaya, van den Heuvel-Panhuizen and Doorman (2015) analyzed the opportunity to solve context-based tasks available in three Indonesian mathematics textbooks. Their study showed that the textbooks did not give enough opportunity for context-based problem-solving. Also, the study showed that only 2% of the context-based tasks were reflection tasks (highest cognitive level).  (Houang & Schmidt, 2008). It is the cognitive behaviours and attitudes expected of students after learning the content from the textbook.
To explore OTL in a textbook, Charalambous et al. (2010)  It is the arrangement of the content in logical and sequential order to make sense and enhance connections between concepts. Charalambous, Delaney, Hsu, & Mesa (2010)

METHODS
This study followed a case study research design and a qualitative approach. It examined the Euclidean geometry content in the two textbooks of tenth-grade mathematics in South Africa. It also examined the types of tasks (explanation and exercises tasks) used to present the content in the textbooks. The qualitative approach was necessary to produce rich descriptive data concerning Euclidean geometry coverage in the textbooks (Yin, 2018). The two textbooks were Classroom Mathematics (Pike et al., 2011), identified here as Book1, and Platinum Mathematics (Campbell & McPetrie, 2012), identified as Book2. The textbooks were purposively sampled for the study. They are approved for use in schools, and they seem to be the most popular and widely used mathematics textbooks in secondary schools in the country.
The Curriculum and Assessment Policy Statement's (CAPS) grade 10 Euclidean geometry content and organization were used as the yardstick for examining the content coverage and organization of Euclidean geometry in the textbooks. Deductive content analysis (Krippendorff, 2018) was used for the data analysis. In contrast, Gracin (2018) four categories of mathematical tasks analytical framework (Table 1) were used to analyze task types in the textbooks.

Representations and modelling
Representation concerns the translation of the given mathematical data into another mathematical representation. Modelling involves recognizing relevant mathematical relationships from the given situation and representing the same problem in a mathematical model (symbolic, graphical, and so on).
Example: Represent the following statement with a sketch (drawing): PQRS is a parallelogram with diagonals PR and QS intersecting at O. PQ = 8cm and PS = 5cm.

Calculation and operation
Operation is the concrete, sensible and efficient conducting of computational or constructional steps. The calculation is concerned with conducting elementary computation operations with concrete or generalized numbers. It also refers to transforming measure units, transforming mathematical expressions, solving equations, estimating results, approximations and conducting elementary geometrical constructions".
Example: Calculate the values of x and y.

Interpretation
Interpretation "concerns recognizing relations and relevant data given in the mathematical representations (graphical, symbolical and tabular) and their interpretations in the given context".
Example: Which pairs of lines are parallel? Give reasons. Argumentation "refers to the description of mathematical aspects that speak pro or contra a particular decision. It requires concrete and appropriate implementation of mathematical relations and characteristics, mathematical rules, as well as the correct usage of mathematical language. Reasoning concerns the sequence of true arguments that lead to a conclusion".

EDUMATIKA :
Example: ABCD is a parallelogram, AC and BD are equal, prove that ABCD is a rectangle.
All the explanation tasks (that is, solved examples tasks used in explaining the concepts) and exercise tasks on Euclidean geometry in the textbooks were analyzed concerning the mathematical activities, skills, or competencies required to do the tasks. There were no agreements on the analysts' classifications on the tasks. They discussed the tasks and arrived at a consensus.

RESULTS AND DISCUSSION
Euclidean geometry content is presented in chapters 9 and 14 (pages 195 -224 and 327 -342 respectively) in Book 1, and topics 6 and 10 (pages 161 -181 and 245 -252 respectively) in Book 2. The distribution of the Euclidean geometry tasks in the textbooks is shown in Table 2.

Content Coverage
According to the mathematics curriculum (DBE, 2011), the contents expected to be covered in Grade 10 under Euclidean are: (i) Revision of preliminary results: lines, angles, triangles, similarity, and congruence.
(ii) Investigation of line segments joining the midpoints of two sides of a triangle,

Content Organization
The content of Euclidean geometry was presented in two chapters in each of the textbooks. Figure   1 shows the order of presentation of the contents of Euclidean geometry in the textbooks. The organizations of the contents were similar in both textbooks. It was found that the organization of contents in the textbooks, from the revision of properties of lines, angles, and polygons to geometric proofs, was in a sequential and logical order to enhance the making of connections between the concepts. This sequencing shows coherence as the proceeding contents build on the knowledge from the preceding contents.

Types of Tasks
Both textbooks used mainly argumentation and calculation tasks in teaching the topic (See Figure   2 This study explored the opportunity to learn Euclidean geometry in two grade 10 South African textbooks, focusing on the content coverage, content organization, and the types of mathematical tasks in the textbooks. The content presented on a topic serves as an indicator of what students would learn if the teaching of the topic in their classes covered all the content (Mesa, 2004). Both textbooks presented just about the same Euclidian geometry content and were organized in the same order. Both books started with the revision of angles and lines from previous grade levels followed by polygons (triangles and quadrilateral in book 1) and quadrilateral (in Book 2); the following content presented was the Midpoint theorem (in both books) and finally, Geometric proofs (also in both books). Starting the content with the revision of some entire contents from previous grade levels was necessary to lay a foundation and transition to the grade 10 work. Except for the detailed presentations of the investigation and making conjectures about the properties of a trapezium in Book1 and prove of conjectures about the properties of square and trapezium in Book2, the presentations of quadrilateral, midpoint theorem, and Geometric proofs in both textbooks is an indication that both textbooks addressed the core content of grade 10 Euclidian geometry specified in the curriculum.
The organization of content in both books seems to respond to the order of the topic contents in the curriculum except for the swapping of the midpoint theorem and definition of quadrilaterals.
In the curriculum, the midpoint theorem comes before the definition of quadrilaterals, while the definition of quadrilaterals is presented before the textbook's midpoint theorem. The authors might have placed the midpoint theorem after the definition of quadrilaterals to allow for an immediate application. It might also be that the authors wanted to place the midpoint theorem directly before the geometric proofs as a precursor for exploring the geometric proofs. The sequencing of the content in a mathematics textbook must have some underlying mathematical and pedagogical philosophy that the author believes will help interpret the content and render it teachable and comprehensible by the student (Dewey, 1906in Charalambous et al., 2010. The sequencing of content in the textbooks shows the progression from simple to complex as espoused in the curriculum (DBE, 2011). This, from both mathematical and pedagogical perspectives, might make the content easy to teach and accessible to students.
Both textbooks offered substantial numbers of worked examples and exercise tasks on the topic. Regarding the mathematical activities, skills, or competencies required to do the tasks, the tasks (worked examples and exercises) in both textbooks were mainly of argumentation and calculation types. The textbooks did not provide a full range of task types to give the optimal opportunity for students to learn the topic. This finding parallels the findings of Gracin (2018)  The dominance of argumentation and calculation tasks (especially argumentation tasks) in both textbooks might, however, be in response to the Grade 10 Euclidean geometry curriculum emphasis on investigating, making, and proving conjectures and theorems. Hence, the finding could suggest the textbooks' authors' thorough interpretation of the curriculum and their belief that Euclidean geometry is mainly about calculating sides and angles, investigating, making, and proving conjectures and theorems. Nevertheless, using worked examples and exercises that draw from multiple types of tasks would afford students more opportunities to develop a more complex, connected, and robust understanding of the content (Charalambous et al., 2010).

CONCLUSIONS
This study explored the OTL Euclidean Geometry available in two textbooks of tenth-grade mathematics. Specifically, it explored the content coverage, organization, and mathematical tasks used to teach the topic. The study found similarities in the content coverage and organization of the content in the textbooks. As specified in the curriculum, the topic contents were well covered in both textbooks, and the contents were presented in logical and sequential order. Besides, it was found that the tasks in the textbooks were predominantly argumentation and calculation type. In all, the textbooks were found to offer sufficient opportunities for learning Euclidean geometry as specified in the curriculum for the grade level. However, the author recommends including a broader range of tasks in the future editions of the textbooks. This research will likely help the teachers to use a broader range of mathematical tasks in teaching the topic and consequently enhance the Euclidean geometry learning opportunities of the students.