Jocelyn Monteith and I decided to start this school year in Mathematics 7 focusing specifically on the development of Mathematical Habits of Mind. According to mathematicians and educational researchers Levasseur and Cuoco (2006), it’s the mathematical habits of mind, or modes of thought, that enable us to reason about the world from a quantitative and spatial perspective, and to reason about math content that empowers us to use our mathematical knowledge and skills to make sense of and solve problems. Mathematical habits of mind are also identified as key characteristics of mathematical problem solvers by the Galileo Educational Network and nRich Maths.
Our first few challenges have focused on developing students ability to “tinker” or to persist with “trial and error.” We recognize that many students can be reluctant to use a trial and error approach as they may feel they are only using it because they do not know the “right” way to solve the problem. In reality, trial and error involves trying something out and then using it to gain further insight into the context and to get a better idea of what to try next. This is often the start of working systematically.
Students responses to this problem were varied. We encouraged them to share their attempts, strategies, ideas, and frustrations. At first many were quick to assert that they had “finished” and to ask whether they were “right.” We challenged them to find a way of concluding unequivocally that they were right with questions like “how do you know for sure that it is not possible to hit all of the orange tiles within the given parameters?” Some students expressed frustration at their inability to use a quick formula to solve the problem. Some students appreciated that the emphasis was not on speed but on careful consideration following multiple attempts and an awareness of their thought process as they determined what adjustments to make. Here are some examples of student thinking:
A recurring question among some students as we worked through this problem was “how is this math?” It is a major misconception among young mathematicians that computational fluency is mathematics due to the fact that developing a high level of accuracy and automaticity in basic computational and procedural skills is so much of a focus in mathematics in schools. This introductory activity was a good reminder that higher-order skills such as problem posing, problem representation, solving complex problems, and transferring math knowledge and skills to problems in non-math disciplines are no less valuable. (See also: The Problem With Math Problems: We’re Solving Them Wrong)
Levasseur, Kenneth, and Al Cuoco. “Mathematics Habits of Mind.” Teaching Mathematics through Problem Solving: Grades 6-12. Ed. Harold L. Schoen. Reston, VA: National Council of Teachers of Mathematics, 2006. 27-37. Click to download