Abstract:

Innovating mathematics pedagogy depends on how educators implement learning tasks. Tasks are critical for mathematics education. Formal mathematics teaching and learning primarily occur through designing, implementing, and working on tasks. Mathematics education workers such as textbook writers, curriculum developers, and classroom teachers design tasks to enable students to hone their conceptual and procedural understandings of mathematics. In turn, students work on those tasks to uncover and practice essential mathematical ideas and forms of reasoning. Nevertheless, an essential and persistent unsettled question for mathematics educators is how to design tasks inviting students to use their innate learning powers to generate mathematical ideas and reasoning, individually or collaboratively.
To respond to that critical, unresolved question, we will exemplify the design of instructional tasks for promoting proportional reasoning and principles of a pedagogical approach coined the Subordination of Teaching to Learning. Specifically, we aim to use the tasks, appropriate for students from approximately 7 to 15 years old, to illustrate how the pedagogical approach enables students to use their learning powers to engage and educate their awareness and to create mathematical ideas. That is, implemented through the approach, the tasks involve teachers of mathematics inviting students to use their learning powers to educate their awareness so that, individually and collaboratively, students can discern and articulate patterns and relations among mathematical objects. We will also suggest how to implement the tasks and indicate mathematical ideas students working on it will likely surface.

Keywords:

Mathematical protagonist, Task Design, Creating Mathematical Ideas and Reasoning.