Current curriculum initiatives (e.g., National Governors Association Center for Best Practices and Council of Chief State School Officers 2010) advocate that models be used in the mathematics classroom. However, despite their apparent promise, there comes a point when models break, a point in the mathematical problem space where the model cannot, or arguable should not, be used. In this work, we explore the breaking point of the chip model for integer subtraction and the area model for fraction addition. Breaking is inevitable-either because no one model is robust enough to be applicable in a very large problem space and/or because the modifications required to keep the model functioning weaken or even eliminate its benefits. While models are often intended to serve as visual illustrations or embodiments of a concept, adaptions at the model breaking point can turn model use into nothing more than executing the graphical analogue to a not-well-understood procedure. The act of identifying model breaking points illuminates the affordances and constraints of the model. This provides students a unique opportunity to discriminate across mathematical models to develop a meta-level understanding of the relationship between models and the mathematics those models are intended to support.
- Integer operations