Learning ambitious teaching of multiplicative properties through a cycle of enactment and investigation

This study explores components of ambitious teaching practices for teaching multiplicative properties that can give teachers opportunities to learn through a quick images activity in a cycle of enactment and investigation. Throughout the cycle, it is found that the participants have opportunities to learn the following components of ambitious teaching: 1. the mathematical ideas (strategies and concepts) involved, 2. mathematical work on representations, 3. predicting student responses, 4. mathematical language, and 5. setting and working towards a focused mathematical learning goal for the students. The findings point out the complexities of learning to teach elementary mathematics, such as the associative property of multiplication. The findings also highlight the opportunities these complexities provide for teacher learning when it comes to distinguishing between and understanding commutative and associative computations.