Investigating pedagogical content knowledge of high school teachers on mathematical proofs

Abstract: This study investigated the pedagogical content knowledge (PCK) of mathematics teachers regarding the teaching of mathematical proofs, focusing on three domains: knowledge of curriculum, knowledge of students, and knowledge of teaching strategies. Six mathematics teachers from Grades 7 to 10 and sixteen senior high school students from Tagana-an National High School in Surigao del Norte participated in the study. Employing a qualitative research design and grounded theory methodology, data were gathered through semi-structured interviews and a researcher-made proof test. Findings revealed that while teacher-respondents generally followed the sequence of lessons as presented in the learning materials and demonstrated familiarity with topics requiring proof, their implementation varied. Teachers identified common student difficulties such as lack of prior knowledge, comprehension issues, and challenges in reasoning and retention. To address these, demonstration and step-by-step explanation were the most commonly employed strategies. However, some teachers admitted to skipping proof-related lessons due to time constraints and difficulty in delivering the content effectively. Student respondents, on the other hand, expressed limited understanding of proofs, associating them mainly with justification of solutions. Results of the proof test indicated that students did not meet the expected proficiency in mathematical reasoning, coherence, and clarity. Despite students’ recognition of their teachers’ efforts, a significant gap remained in proof writing performance. The study underscores the need for enhanced teacher training on proof instruction, greater emphasis on student thinking, and sustained curriculum support to improve the teaching and learning of mathematical proofs.