Date of Original Version
When we open up the classroom to student thinking we run into discoveries and conjectures that adults would likely not see, blocked by the constraints and conventions we have accepted in our own thinking. It is this lack in established constraints and conventions that can enable young thinkers to explore the world of mathematics more unencumbered. The risk of content unfamiliarity in such learning environments is not for the students; it is for their teachers (Ball, Thames, & Phelps, 2008; Chazan, 1999; Liping Ma, 1999). We want to relay one such instance in which a third grade student saw a new connection that stimulated us to investigate the mathematics of the situation in greater depth. In this article we want to show how this novel idea can be explored by students at multiple grade levels in different ways. The tasks at hand are especially well suited to address several of the mathematical practices standards from the Common Core State Standards, such as reason abstractly and quantitatively, look for and make use of structure, and look for and make use of regularity in repeated reasoning.
de Groot, C. and Hersey, N. (2015). When products are sums. New York State Mathematics Teachers’ Journal. 65 (1): 22-28.