Grasping the slippery slope : the construction of conceptual understanding of slope in a social constructivist mathematics classroom

This study investigates secondary mathematics students‚ÄövÑv¥ learning of slope concepts through a social constructivist teaching approach. It examines the design and implementation of learning experiences based on constructivist pedagogy and their effectiveness in producing conceptual understanding of the measurement of slope as a ratio. A two-week Teaching Intervention was conducted in a heterogeneous year 8 mathematics classroom. Measurement of the understanding developed was determined from a pre-test and a post-test, and later assessment indicated the retention of concepts learned. Assessments were also conducted with comparable and older groups of students for the purpose of generating a broader view of slope conceptualisation held by a range of high school mathematics learners. A local instruction theory was developed to explain the relationships observed among learning activities and learning outcomes. Variation was observed in the learning outcomes. Many participants gained an understanding of the measurement of slope as a ratio from the Teaching Intervention, but this understanding was not retained by all. As a result of their learning experiences, some participants appeared to have formed appropriate connections to applications of slope that were assessed but not included in the learning activities. Factors beyond the learning activities that were conjectured to have also influenced learning include the participants‚ÄövÑv¥ prior knowledge of ratio applications and notation, as well as their perception of themselves as mathematics learners, and their familiarity with mathematical reasoning. Participants who, in the pre-test and initial activity, perceived steepness as having two components, were most likely to develop a conceptual understanding of the measurement of steepness as a ratio. A description of conceptual understanding of slope emerged throughout this study that allowed for an elaboration of the network of conceptualisations currently perceived to be Grasping the Slippery Slope associated with this concept. Some students at all levels struggled to recognise the slope ratio when presented as a percentage. It is recommended that the network of conceptualisations of slope include reference to slope notations as fractions, decimals, and percentages. The findings also recommend that the conceptualisation of percentage notation as a valid representation of ratios be emphasised more generally by secondary mathematics teachers. The local instruction theory formed through this study provided evidence for the effective combination of discovery, problem-based learning, discussion, and direct instruction teaching strategies in secondary mathematics. Analysis of the learning situations and outcomes appeared to suggest that if the participants had greater familiarity with mathematical reasoning as a part of classroom discourse, the social constructivist teaching strategies used would have been more effective in developing conceptual understanding. In conclusion, learning activities for mathematics designed from a social constructivist perspective can be effective in developing conceptual understanding. To be effective, learners need to have the necessary prior knowledge and engage purposefully in the learning activities.