Catherine Martin
President, Colorado Council of Teachers of Mathematics
Our transition to the Colorado Academic Standards asks us to both raise the floor and lift the ceiling: to raise the bar of expectations for ALL students and to lift the ceiling to increase expectations for those students who aspire to enter STEM fields. At the same time, we are called to support all students in developing the varieties of expertise called for in the Standards for Mathematical Practice. The focus on these standards in tandem with the mathematics content standards provides us with a rich opportunity to rethink how we engage students in learning mathematics to ensure that we both raise the floor and lift the ceiling. I would suggest that we begin with the first standard: Make sense of problems and persevere in solving them. To do so requires that we consider both the mathematical task and the feedback on the task provided to students.
Selecting a task that is worth making sense of and persevering in solving is essential. But prior to that selection, we should first consider the purpose of using rich mathematics problems. One possible purpose is simply to teach students how to get answers. Juxtaposed with this purpose is a second purpose of using problems as a vehicle for students to learn new mathematics AND to learn how to learn mathematics. I believe this second purpose fits with the essence of making sense of problems and persevering in solving them. Consider the following phrases from this standard: “start by explaining to themselves the meaning of a problem and looking for entry points, . . . plan a solution pathway rather than simply jumping into a solution attempt, . . . monitor and evaluate their progress and change course if necessary, continually ask themselves, ‘Does this make sense?’”. Furthermore, using problems to help students both learn new content and learn how to learn mathematics will support students far beyond a single mathematics course. But this view may shift our thinking and the thinking of students who are well trained to get answers on problems that, by design, don’t necessarily require them to persevere in solving.