This commentary aims to discuss an overarching boundary crossing framework under which integrated STEM (Science, Technology, Engineering, Mathematics) pedagogy can be conceptualized. Four potential learning dialogical processes for boundary crossing are presented and used as the main theoretical construct for the discussion. A proposal of an interactive pedagogical framework is put forward accompanied by a provisional statement to relate the connective factors that are critical in the formation of a boundary crossing STEM pedagogy. These factors are situated learning, communities of practices, problem solving, learning dialogical processes, and boundary objects. A Hong Kong school STEM case is employed to illustrate the applicability of this framework. The commentary ends with a reflective remark on boundary crossing STEM pedagogy.

This chapter concerns designing teaching/learning tasks that involve the use of tools in the mathematics classroom and consequently how, under such design, tools can mediate the representations and discourses of mathematical knowledge. The authors focus particularly on digital tools and how task design can be used to transform an artifact into a pedagogical instrument. Design considerations and heuristics are discussed to explore possible research directions in tool-based task design. In particular, the chapter exemplifies heuristics or principles (theoretical or pragmatic) for tool-based task design that are conducive to teaching and learning of mathematics, including epistemological, mathematical, representational, and pedagogical considerations of tool-based task design.

This chapter discusses some strands of experimental mathematics from both an epistemological and a didactical point of view. We introduce some ancient and recent historical examples in Western and Eastern cultures in order to illustrate how the use of mathematical tools has driven the genesis of many abstract mathematical concepts. We show how the interaction between concrete tools and abstract ideas introduces an “experimental” dimension in mathematics and a dynamic tension between the empirical nature of the activities with the tools and the deductive nature of the discipline. We then discuss how the heavy use of the new technology in mathematics teaching gives new dynamism to this dialectic, specifically through students’ proving activities in digital electronic environments. Finally, we introduce some theoretical frameworks to examine and interpret students’ thoughts and actions whilst the students work in such environments to explore problematic situations, formulate conjectures and logically prove them. The chapter is followed by a response by Jonathan Borwein and Judy-anne Osborn.