Document Type : Original Article
Authors
1 PhD Student, Department of Mathematics, SR.C., Islamic Azad University, Tehran, Iran.
2 Department of Mathematics, Ar.C., Islamic Azad University, Arak, Iran
3 Department of Mathematics, SR.C., Islamic Azad University, Tehran, Iran.
Abstract
Introduction: The intersection of content knowledge and pedagogical knowledge includes strong analogies, images, definitions, and—essentially—the manner in which subject matter is conveyed to students in an understandable way. Using various models to provide examples is a key method for making abstract mathematical ideas accessible and concepts more intuitive for learners. Alongside the use of examples in teaching—drawing on pedagogical content knowledge and its specific capabilities—providing concrete, tangible interpretations for abstract mathematical concepts is of particular importance. Since mathematical concepts require concrete, practical contexts for learning, one can also employ other interpretations rooted in pure mathematical concepts, one of which is the use of metaphor. Given the importance of teaching mathematics—particularly algebraic concepts, due to their abstract nature at the lower secondary level—and given their extensive use across other subjects in both earlier and later educational stages, and since these concepts pervade most lower secondary mathematics courses, mathematics teachers must possess adequate content and pedagogical knowledge to incorporate up-to-date and effective strategies into their teaching. The researcher finds the use of examples and analogies particularly noteworthy, as previous studies have shown them to play an effective role in mathematics teaching and curriculum. However, it appears that some lower secondary mathematics teachers face numerous challenges in these areas. Moreover, based on a review of previous studies, no scale has yet been concurrently proposed to assess teachers' knowledge across these three domains. Therefore, the main purpose of this research is to design and validate a questionnaire on the quality of teachers' pedagogical content knowledge and their use of examples and analogies in teaching algebra concepts in the first cycle of secondary education.
Methods and Data: The research method was descriptive-survey. Given that the present study aimed to validate an assessment scale, the statistical population comprised male and female mathematics teachers employed in the first cycle of secondary education across the northern, central, and southern districts of the Tehran Provincial Education Department. Using snowball sampling, 225 teachers were selected. Drawing on previous studies—specifically, Aksu et al. (2014) on pedagogical content knowledge, Alkan et al. (2017) on types of mathematical examples, and Hendryana and Ati Rohati (2017) on metaphor presentation—the researcher designed a 50-item questionnaire. The instrument included both open-ended and closed-ended sections, rated on a 5-point Likert scale. In the open-ended response section, several questions were developed for each of the three domains within the context of lower secondary algebra. The questionnaire was organized into four sections. Section One collected demographic information about the teachers. Section Two contained closed-ended items assessing pedagogical content knowledge. Section Three included closed- and open-ended items assessing the presentation of examples. Section Four included closed- and open-ended items assessing the presentation of metaphors. In the knowledge-of-examples section, the teaching stages were classified into three phases: the first phase comprised simple and suggested examples; the second phase comprised reference and model examples; and the third phase comprised counterexamples and non-examples. In the knowledge-of-metaphors section, three main categories were classified: (a) foundational, evaluative, and clarification metaphors; (b) relational, practical, efficiency, stability, and irrelevant metaphors; and (c) flexibility and innovation metaphors. The Content Validity Ratio (CVR) was used to assess content validity. After revising the wording of each item, the questionnaire was finalized. In the final stage, experts confirmed the instrument with a CVR of 75%, which met the minimum required threshold.
Findings: Based on the demographic questionnaire data, 65% of the participants were female teachers and 35% were male. Regarding teaching experience, 54% had more than 15 years of service, 24% had more than 10 years, and 22% had fewer than 6 years. In terms of educational level, 53% held a bachelor's degree, 31% held a master's degree, and 16% held a doctorate in mathematics. For data analysis, exploratory factor analysis (EFA) was conducted using SPSS version 26, and confirmatory factor analysis (CFA) was performed using SmartPLS version 3. Following the confirmatory factor analysis, the three domains—pedagogical content knowledge, example presentation knowledge, and metaphor presentation knowledge—were evaluated based on their goodness-of-fit indices and respective subcomponents. Finally, a model of the variables and sub-variables was proposed.
Discussion and Conclusion: This questionnaire was designed as a scale to assess three domains of knowledge among lower secondary mathematics teachers in the area of algebra, based on theoretical foundations and previous studies. Accordingly, after presenting instructional material on first-cycle secondary algebra—particularly for grades eight and nine—the teachers were asked to respond to the open-ended and closed-ended items on the questionnaire. The aim was to identify the main factors determining teachers' level of knowledge in teaching mathematical concepts, particularly algebraic ones. What was primarily observed in the teachers' responses was a lack of sufficient competence and knowledge in presenting examples and metaphors, as well as in integrative pedagogical content knowledge. A qualitative examination of pedagogical content knowledge in lower secondary mathematics classes is therefore important. Additionally, other types of content knowledge also play a significant role in mathematics instruction. Knowledge of presenting examples can provide valuable insight into the nature of mathematics—for instance, in complex tasks for demonstrating methods, in developing concepts to illustrate relationships, and in constructing proofs. Similarly, knowledge of using conceptual metaphors demonstrates that the focus in presenting a metaphor should be on the underlying concept rather than on the words themselves. This study, relying on the process of teaching mathematical concepts—particularly algebraic concepts—examined three pedagogical content knowledge areas: (a) pedagogical content knowledge, (b) example-presentation knowledge, and (c) metaphor-presentation knowledge. Accordingly, the researchers set out to design a scale to assess teachers' knowledge in these domains. After several rounds of revising and adjusting the questionnaire items during the stages of content validity, convergent validity, discriminant validity, and reliability reviews, eight factors were identified, indicating that the majority of mathematics teachers exhibited similar response patterns within each identified factor. Therefore, it is suggested that this scale be examined and validated in other geographical regions and content areas—such as algebra and geometry, or other lower secondary subjects. Furthermore, it is recommended that the open-ended responses to the items be graded and compared qualitatively. Finally, in-service training courses led by expert instructors should be organized to equip teachers with effective strategies for providing examples and analogies, thereby enhancing their essential knowledge, and ensuring the quality of instruction in these subject areas
Keywords
)