Ann Kajander, Lakehead University, Thunder Bay, Canada
Jennifer Holm, Wilfrid Laurier University, Waterloo, Canada
Title of the talk:
How far can learning with representations and reasoning go? Examples with quadratic functions.
Mathematics learning via representations and reasoning has enjoyed some acceptance at the elementary school level, but in our country at least, is rarely extended very far into the secondary grades. In parallel, in Canada, there is new emphasis on visualizing and verbing (Lunney-Borden 2018) as important in supporting the mathematics learning of Indigenous students. In this talk we illustrate some of our recent work using representations and reasoning as an environment for secondary mathematics classroom exploration of quadratic functions. What is distinctive about this work is that the representations are used differently than to simply provide an 'explanation’ of a known (i.e. previously taught) procedure. Rather, the models we have explored have the potential for both student investigation and exploration to develop conceptual understanding, but also allow direct and intentional generalisation. The generalisations in turn are used to directly support the development of the formal procedures. The formal procedures, then, become necessary outcomes of the investigations, rather than something to be either formally 'taught’ first, or added later via teacher transmission.
Alik Palatnik, The Hebrew University of Jerusalem, The Seymour Fox School of Education, Israel
Title of the talk:
Embodied design for spatial geometry learning
Like Plato’s allegorical cave-dwellers, students of three-dimensional geometry seldom get to handle the real thing, working instead with two-dimensional silhouettes. Such historical sensory deprivation may partially explain students’ generally poor conceptual understanding of this core content and alienation from the field.
This talk suggests an alternative approach based on the conceptualization of constructing tangible models as embodied design for spatial geometry learning. I will present empirical results from a design-based research study inviting middle-school students to collaboratively construct and investigate voluminous objects.
In the first experimental activity, three students use a 3D pen to answer spatial geometry questions about a triangle inscribed in a cube. In the second experimental activity, four students construct a human-size model of a fractal tetrahedron and utilize available resources when exploring the volume and shape of an unfamiliar geometric body. In the second experimental activity, six students learned the properties of icosahedron by constructing and exploring physical models on different scales.
The analytical apparatus of Mason’s shifts of attention theory was used to investigate why and how using physical models can facilitate learning of (spatial) geometry. In all cases, students’ critical insights are characterized as shifts in perceptuomotor attention leading to the refinement of geometric argumentation. I argue that students’ realization of available 3D medium affordances catalyzed these shifts. A dynamic of student collaboration was influenced by the evolution of collective multimodal perception and physical actions of students with and through the models. The findings contribute to a socio-material elaboration of embodied learning for school geometry.