Learning opportunity for Euclidean geometry in Further Education and Training mathematics textbooks

Objective: This study investigates the Euclidean geometry learning opportunities presented in Further Education and Training (FET) mathematics textbooks. Specifically, it examines the alignment of textbook content with the Curriculum and Assessment Policy Statement (CAPS) curriculum, levels of geometric thinking promoted, representational forms, contextual features, and expected responses.

Methodology: The research analyzed three FET mathematics textbook series to identify strengths and weaknesses in Euclidean geometry content. This study adopted the interpretivist paradigm. The study used a qualitative research approach and a case study research design. Purposive sampling techniques were used to select the textbooks currently used for teaching. This study used textbook analysis as the data collection method. Deductive content analysis was used as a data analysis strategy. In this study, interrater reliability was used to preserve the quality of data coding and reporting among coders as a percentage of agreement between three coders (Belur et al., 2021).

Data collection

This study employed various textbook analysis instruments that were specifically designed within its framework, including the content coverage instrument, mathematical activity instrument, geometric thinking levels instrument, representation forms instrument, contextual features instrument, and answer forms instrument. 1.1.1 Content coverage instrument

The study employed a content coverage instrument as a data collection tool, with a focus on textbook topics and subtopics. The content coverage instrument, in the form of a checklist, listed all the topics and subtopics of Euclidean geometry in the grade 10–12 curriculum and assessed whether each content was covered in the respective textbooks based on their corresponding grade levels. The aim was to provide a comprehensive assessment of the extensive range of content knowledge that students are required to acquire at each school level, specifically Grades 10-12, using a rubric. The rubric for assessment was designed to gather data and emphasised the extent of Euclidean geometry content coverage. The rubric focused on content coverage and provided a space to indicate if a subtopic was covered (by ticking) or not covered (-).

A checklist form was used to gather data from the textbook tasks by indicating the topics and subtopics covered in each textbook series. The checklist was developed from the CAPS guideline document for Grades 10–12. This instrument was used to examine the selected textbook content coverage to determine the extent to which the textbooks align with the CAPS Mathematics guideline document. This instrument divided the Euclidean geometry content into three categories: Grade 10, Grade 11, and Grade 12, as stipulated in the CAPS Mathematics guideline document for FET-level mathematics. To bolster results objectivity, all CAPS checklist items were quantified using dichotomous (yes/no) responses, summarised by scoring rubrics to justify different responses. A mathematical activity form tool was developed to collect data regarding the nature of mathematical activities in both worked examples and exercise tasks within each textbook. The form was designed in the format of a rubric based on Gracin’s (2018) mathematical activity framework: representation and modelling, calculation and operation, interpretation, and argumentation and reasoning. The rubric consists of five major sections, with the first section focusing on the nature of the mathematical activities required to successfully engage with geometry questions. A rubric was provided for the nature of mathematical activities for each geometry task, which was broken down into four categories to explore the nature of tasks more clearly. The categories of mathematical activities focused on representation and modelling, calculation and operation, interpretation, and argumentation and reasoning.

As this study intended to investigate the students’ OTL afforded by textbooks, an evaluation form was used to gather data. A form containing the four kinds of Euclidean geometry task types was included in the evaluation form used to examine the nature of each Euclidean geometry task. This form consisted of a list of the characteristics of each mathematical activity required to carry out the geometry tasks: Representation and modelling (R), Calculation and operation (C), Interpretation (I), and Argumentation and reasoning (A).” This form serves as a classification template, categorising tasks according to the competence the tasks demand of the students. Table 4.5 presents exemplary geometric tasks, categorised by skill, alongside corresponding evaluation indicators used to assess mathematical proficiency. A representation form instrument was utilised as a data collection instrument regarding the type type of representation used in presenting of the geometry ideas in each textbook sries (see section 3.3). A rubric was utilised to capture the type of representation, with a designated space for each task. This rubric provided a space for documenting the representation format for the tasks. To make the captured data clear, we divided the rubric into four distinct sections: pure mathematics, verbal, visual, and combined forms of problem presentation.

Data analysis

This study used a qualitative deductive content analysis (QDCA) approach to analyse the collected data. In a DCA, research findings are allowed to emerge from the textbooks examined (Pertiwi & Wahidin, 2020). A deductive approach was appropriate because the codes and categories were drawn from theoretical considerations, not from the text itself (Islam & Asadullah, 2018).The researcher created nine Excel files, each with a four-column table, as shown in the figure below. Every column represents the type of mathematical activity category: Representation (R), Calculation (C), Interpretation (I), and Argumentation (A). Based on the Gracin (2018) framework, the researcher and two scorers read every worked example task and exercise task in each textbook examined in this study, extracted the mathematical activity required to complete the task successfully, and recorded it in the corresponding Excel file. If the tasks required more than one activity, the researcher considered the one that was dominantly required by the task author. The figure below shows the Excel sheet used to score the mathematical tasks for this study. To examine the geometric thinking embedded in textbook tasks, a comprehensive analysis framework was employed. This involved utilising a rubric to categorise tasks according to their corresponding geometric thinking levels, spanning from Level 0 to Level 4. For instance, tasks requiring students to define properties of a geometric figure were classified as informal deduction, whereas tasks demanding formal proofs were coded as formal deduction.

The analysis process commenced with a meticulous review of worked examples and exercise tasks to identify the embedded level of geometric thinking. Subsequently, Excel tables were utilised to record the geometry levels present in Euclidean geometry tasks, and their frequencies were calculated. The results, which highlighted the predominant levels in the textbook series, were then subjected to in-depth analysis. This study classified each task based on the dimensions of Zhu and Fan's (2006) answer forms and subsequently coded the problem as depicted in Figure 4.13. In this study, the researcher conducted the process of classifying the tasks based on the answers to the question forms by reading the task questions and coding them as either open-ended or closed-ended problems.

The researcher examined the types of tasks within the Euclidean geometry content in terms of their representation form and contextual features. This study used Zhu and Fan's (2006) framework to classify and code Euclidean geometry tasks found in textbooks. This study analysed the following classification of tasks: "Pure mathematical (R1), verbal (R2), visual (R3), and combined form (R4), based on Zhu and Fan's (2006) theoretical framework. In particular, each task was analysed against these representation-type categories in each textbook. An Excel table, as shown in the figure above, recorded the analysis of the representation forms.

To investigate the contextual features of mathematical tasks, the researcher systematically collected tasks from each textbook and created an Excel sheet to score the type of context presented in each problem. Zhu and Fan's (2006) theoretical framework provided the foundation for categorising and coding tasks, enabling a comprehensive analysis. This study classified the tasks into two distinct categories: Zhu and Fan (2006) define application problems (C1) as tasks presented in real-life situations, illustrating practical applications of mathematical concepts. Non-application problems (C2) are tasks that lack context and solely concentrate on mathematical procedures and calculations. We coded tasks presented in situations mirroring real-life scenarios as application tasks and tasks lacking context as non-application tasks. The coded data was meticulously counted, and the frequencies were recorded in tables using Microsoft Excel, as depicted in Figure 4.13. This systematic analysis facilitated a nuanced understanding of the contextual features of mathematical tasks across the examined textbooks. This study used the CAPS Mathematics guidelines as the foundation for developing an OTL analytical tool to classify the mathematical content. The CAPS Mathematics analytical tool encompasses the content areas that students should master in all grades. Next, I outlined the OTL categories, offering comprehensive details on the interpretation and analysis of the data. To analyse the data, I used a rubric for each textbook series. The researchers conducted a thorough review of each textbook task, utilising the CAPS Mathematics document as a benchmark to score the content concepts covered within each textbook. The content coverage rubric listed the topics and subtopics of Euclidean geometry as stipulated in the CAPS document. Content coverage was assessed through three questions, each tailored to a specific grade level: Grade 10, Grade 11, and Grade 12. I have identified and described the content coverage in the Euclidean geometry content, as well as the progression within the content, as detailed in Anneture A. As an illustration, if the content was covered in the tasks, a tick mark ü was used, and if a certain concept was not included in a particular textbook, it was marked with an X.

Key Findings:

1. Misalignment with CAPS curriculum

2. Overemphasis on calculation and operational tasks

3. Dominance of formal deductive reasoning

4. Mixed forms of representation

5. Non-application contexts

6. Closed-answer formats

Implications: The study's insights provide valuable guidance for teachers, textbook developers, and policymakers to enhance the teaching and learning of Euclidean geometry.