Learning mathematical rules with reasoning

This research focused on students learning of mathematical rules with reasoning. A small group of students (age 11-12 yrs) was observed closely by the first author as she taught them fraction rules. The area of focus was fractions and activities were designed pertaining to the four rules of fractions: addition, subtraction, multiplication, and division. The study was undertaken from a constructivist perspective of learning according to which students' learn through active participation in the construction of knowledge (Glasersfeld, 1995). This was significant in the context of mathematics classrooms in Pakistan which usually subscribe to the objectivist epistemology i.e. knowledge of the ultimate reality is possible. An implication of adhering to this epistemology is the knowledge transmission view of teaching and learning (Halai, 2000). Key findings of the study were that there were two significant factors that enabled students to learn rules with reasoning were: Teachers questions and opportunities for students to explain thinking; and opportunity for students to engage with concrete and semi concrete materials. The study also provides some useful insights into the sequence of teaching the fraction rules and raises implications for mathematics teaching and teacher education.