By Chepina Rumsey and Ian Whitacre
(Newly published article here: https://www.tandfonline.com/doi/full/10.1080/07370008.2018.1491580)
As educators who prepare prospective elementary teachers (PTs) to teach math, it’s easy to get stuck in the idea that the PTs are lacking critical knowledge and mathematical understanding. At the same time, we tell our PTs to be wary of labels and of making assumptions about what their future students know. We ask them to look beyond “low” versus “high-achieving” groups and beyond labels like “smart” and “weak at math.” Rather than focusing on what children do not understand, we encourage PTs to start with what the children do know and use that as a starting point, scaffolding to bring the children to the mathematical understanding. Recognizing that every child brings knowledge that we can build on is an important lesson we want to impart to PTs.
Just as we advise PTs to reflect on the knowledge children bring to the classrooms and use that as a starting point, we want to use the knowledge that PTs bring to our university classroom as we teach mathematics content courses. Some of the areas commonly included in mathematics content courses for PTs are number sense and whole number operations, with a goal that the PTs develop a conceptual understanding with flexibility and efficiency in different strategies. Like other mathematics teacher educators, we see PTs each semester who can perform standard algorithms, yet do not have a deep understanding of what the steps represent and why they are doing them. When we ask students to mentally compute, we often see at least some students trying to imagine the numerals aligned vertically, while motioning steps in the air with their fingers. Observations about PTs’ knowledge are documented in literature as well, with many researchers recognizing that PTs often reason inflexibly about numbers and operations. How can PTs’ hidden flexibility be revealed and how do those latent strategies and ways of reasoning influence their learning during a content course? Are there sociomathematical norms that we can specifically emphasize to help promote PTs’ development of more complete strategy ranges?
In our article, we unpack these questions as we focus on one PT, Brandy, in a Number and Operations course. We document how her reasoning and flexibility changed throughout the course and how her prior knowledge shaped her learning trajectory. We also document the sociomathematical norms in the class, especially those involved in discussions of mental-computation strategies. Finally, we identify some connections between Brandy’s flexibility development and those norms. Rather than seeing PTs who are dependent on the procedures of the standard algorithms as deficient, we hope readers see them as possessing something potentially powerful and worthy of elaboration. Their hidden strategies may light the way in the direction of flexibility development.