This post summarises the chapter The pedagogy of mathematical inquiry (Makar, 2012). What does the pedagogy of mathematical inquiry mean specifically for teachers and learners? This chapter provides an indepth overview of the key elements of mathematical inquiry pedagogy, what it looks like in a real year 6/7 classroom (ages 1013) and provides theoretical connections to highlight important implications for teaching and teacher education. The idea of exploring the question Which bubble gum is best? enthuses students, as they determine the qualities that they value in bubble gum. Benefits of posing mathematical problems through illstructured questions include opening up ways for students to enter the solution process, providing opportunities for students to negotiate with peers how to address such problems, and presenting a need for students to justify their conclusions using the process they use to answer the question. The inquiry teacher in this chapter mathematises the context of exploring bubble gum to determine which is best and students create mathematical strategies to compare different brands of bubble gum. Four phases provide a framework for teaching and learning mathematics through inquiry: Discover, Devise, Develop and Defend. These phases offer teachers and their students, purposeful ways to navigate mathematically through an inquiry. The chapter illustrates how one teacher, April Frizzle, conducts an inquiry in her multiage classroom to find the best bubble gum. Although she initially refers to the experience as "the disaster of the bubble gum experiment!" April reflects on the challenges she has in trying to balance taking control of the lesson, and giving her students the opportunity to develop ways to collect and interpret the data they collect. The author of this chapter is able to summarise the positive experiences that April identifies and how these experiences benefit the learners in April’s classroom the next time they encounter a mathematical inquiry. What is the best orange? is a similar inquiry task that the students tackle the following term. It becomes clearer how the struggles from the first inquiry are now beneficial to the teachers and learners and April articulates some transfer between the two units. In particular, April comments on how the inquiry experiences have materialised her students’ beliefs about the nature of mathematics and how that changed the way they were now willing to approach problems. The final sections of the chapter considers theoretically, how the pedagogy of mathematical inquiry aligns with research on teaching and learning mathematics. These links highlight the importance of engaging diverse learners in mathematics and the author refers particularly to a theoretical framework (Harel & Koichu, 2010) that analyses learning through inquiry in three ways: as struggle, as purposeful, and as providing students with repeated opportunities to reason in meaningful ways. Ambiguity in inquiry topics offers a space for students to construct understandings as they struggle to reach their mathematical conclusions. Teachers are required to balance letting students construct their understandings with the need to support and scaffold that learning. The relevance of mathematics becomes important to students when teachers successfully mathematise a problem situation. Meaningful contexts help students see a problem as important to solve and the openness of inquiry questions lets students see there can be more than one way to solve such problems. Over time, engaging with mathematical learning through inquiry presents repeated opportunities to apply and build understandings. Students and their teacher in the class described above transferred “metbefore” (McGowena & Tall, 2010) mathematical struggles with the bubblegum unit of understanding the need for measurement, to quantitatively measure the qualitative characteristics of oranges. Bubblegum and oranges surprisingly presented opportunities for Ms Frizzle to negotiate what a culture of inquiry meant in her classroom, with her students. This illustration may not reflect the habits and norms in your own classroom and this process requires guidance and explicit support. Consider some of the research on this site on developing a culture of inquiry in your classroom, to help you decide upon what you want your mathematical inquiry classroom to look like. Summary by Kym Fry Harel, G., & Koichu, B. (2010). An operational definition of learning. Journal of Mathematical Behavior, 29, 115–124. McGowena, M. A., Tall, D. O. (2010). Metaphor or metbefore? The effects of previous experience on practice and theory of learning mathematics. Journal of Mathematical Behavior, 29, 169179. 
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