Comparisons of Two Major Theories on Teaching and Learning Mathematics Through Variation

• In Event: 29.082 - Roundtable Session 9
  In Roundtable Session: 29.082-4 - Teaching and Learning Mathematics Through Variations: An International Perspective

Abstract

Both Variation Theory of Learning (Marton, 2015, VT hereafter) and Bianshi Teaching (Gu, Huang, & Marton, 2004, BT hereafter) presume that the pattern of variation and invariance constructed in classroom interaction has great influence on what students might learn. VT is intended for learning and teaching powerful ways of seeing in different subject domains (Marton & Pang, 2006; Marton, 2015). BT is intended to capture and make explicit good practices in mathematics teaching in China (Gu et al., 2004).
To discern and focus upon the critical aspects of a phenomenon, the learner must experience variation in these aspects against a background of invariance (Marton, 2015). Four patterns of variation and invariance including ‘contrast,’ ‘separation,’ ‘fusion,’ and ‘generalisation’ are suggested for organising learning instances. When one critical aspect varies while the other aspects are kept invariant, the pattern is called ‘separation.’ In contrast, when two or more critical aspects of the phenomenon vary simultaneously and are thus brought to the learner’s awareness at the same time, this pattern is called ‘fusion.’ For instance, to grasp the concept of a triangle, one must ‘contrast’ it against something that is not a triangle. Furthermore, to ‘generalise’ the idea of a triangle, one must see the sameness (three sides linking end-to-end) across the various contexts (regardless of their positions and sizes). After grasping what is a triangle, then the learner could classify triangles as acute, right, and obtuse triangles by focusing the sizes of angles while side lengths remain invariant, or as equilateral, isosceles, and scalene triangles by focusing on side lengths (generalization). The most important implication of VT to teaching is that there are certain necessary conditions without which learning cannot take place. It is important that “when learners need to discern more than two or more critical features, the most powerful strategy is to let the learners discern them one at a time, before they encounter simultaneous variation of the features” (Lo & Marton, 2012, p. 11).
According to Gu et al. (2004), the goals of learning include building substantial connection between new knowledge and existing knowledge, and developing problem-solving ability and an interconnected knowledge structure. The BT framework proposes two types of variation: conceptual and procedural variation. The conceptual variation refers to using multiple examples purposefully– from concrete to abstract, contrast with non-prototypical and non-concept images – highlighting the essential features and clarifying the connotation of concepts. The procedural variation focuses on understanding how the concepts have evolved and how problems could be solved using various strategies. The idea of procedural variation is to provide hierarchical-progressive mathematical tasks to scaffold students’ learning by considering core connections between concepts and learning progressions (Huang, Gong, & Han, 2016).
In conclusion, the BT theory is more elaborate and explicit regarding the design of learning tasks and is concerned with mathematics. VT, a framework for learning in general, can help to explain how teaching can be designed to bring about learning and thus the formulation of new meanings and relationships, and their transfer.

Author

• Rongjin Huang, Middle Tennessee State University