On slopes of perpendicular lines: Teachers’ explanations and mathematical connections

In this paper, we introduce the construct of a mathematical-pedagogical task (MP task) and analyse the responses of prospective secondary mathematics teachers as they attempted to explain a well-known relationship between the slopes of perpendicular lines. We suggest that the notion of “solution spaces”, introduced by Leikin and Levav-Waynberg (2008) in discussion of multiple solutions to a mathematical problem, is applicable in considering an MP task. Findings illustrate a variety of possible approaches and mathematical ideas that can reveal different solution spaces. We argue that MP tasks are an appropriate avenue for strengthening teachers’ knowledge. • Investigates prospective teachers’ attempts to explain “why so” about a well-known fact from school mathematics. • Qualitatively analyses solution spaces related to the relationship between slopes of perpendicular lines. • Extends notion of multiple solution tasks to include MP Tasks , i.e., mathematical tasks embedded in pedagogical situations. • Extends the notion of solution spaces to include curricular, pedagogical, and horizon solution spaces.