Understanding proof by induction through a social constructivist lens using worked examples

• In Event: Social and justice issues in mathematics education

Proposal

Proof by induction is a fundamental mathematical concept that enables students to grasp mathematical structure and establish connections between different topics. It also fosters reasoning skills that are applicable across various mathematical domains. However, many students struggle with this topic. Previous research has primarily focused on post-secondary students who were already familiar with inductive proofs in a teacher-guided learning environment. This study aims to develop effective pedagogy tailored to secondary students who are just beginning to learn proofs through a student-led learning environment.

Recent global shifts in mathematics education support a social constructivist approach to learning (Cavalcante et al., 2022). While proof by induction is widely taught in mathematics curricula worldwide, including countries like Indonesia (Adinata et al., 2020), Australia (Hine, 2017), and Italy (Arzarello & Soldano, 2019), in Canada, it is primarily covered through the International Baccalaureate (IB) Analysis and Approaches (A&A) Higher Level (HL) mathematics course, which many schools have chosen to adopt. The IB A&A HL curriculum shares mathematical competencies, such as collaboration, metacognition, and inquiry, with the Ontario secondary mathematics curriculum (IBO, 2019; Ontario Ministry of Education, 2007; Ontario Ministry of Education, 2021). Therefore, one of the intervention's objectives was to encourage students to actively demonstrate these competencies within the classroom setting. The IB curriculum is renowned for its holistic approach to learning, and with over 5500 IB schools worldwide, the increasing emphasis on mathematical literacy in mathematics curricula worldwide (Cavalcante et al., 2022) creates a favorable environment for this intervention. Mathematical literacy, in this context, refers to the application of mathematics in real-world situations using problem-solving, analysis, and reasoning skills.

The proposed intervention draws from the social constructivist framework, emphasizing peer feedback, self-discovery, and the Thinking Classrooms approach (Liljedahl, 2021). The primary question driving this intervention was: How can working with peers to provide feedback on worked examples help IB students write better proofs? Although the participants in this study had worked with logic and argumentation previously, they have not had any exposure to formal proofs before this intervention. This intervention proceeded through four stages. The first stage consisted of students giving feedback to worked examples written by the teacher's previous class followed by the second stage where students created an IB style mark scheme based on patterns observed with the worked examples and feedback. Both stages occurred on the same day. For homework that night, students completed an inductive proof problem on their own to bring to class the next day. The third stage consisted of pairs of students who provided feedback to the previous night's homework and the last stage consisted of students self-assessing their own work. The four stages have been thoughtfully chosen to afford students abundant opportunities for learning to give feedback, making a seamless transition from outsiders to active members within the community, and ultimately to oneself.

Formative and summative assessments were administered the day after the intervention and two weeks later, respectively. The results demonstrated that students not only gained a deeper understanding of inductive proofs by comprehending the structure and steps involved, as well as avoiding common errors made by previous students, but also that this improvement was reflected in their assessment grades weeks later. With the increasing importance of mathematical literacy in the latest curriculum revisions, it becomes crucial to develop effective pedagogy applicable in curricula worldwide. Teachers should consider alternative methods to teacher-directed learning when teaching historically challenging mathematical concepts.

Author

• Xiao Feng, O.I.S.E. University of Toronto