Have you ever had one of those days when things just weren’t clicking like they should? Then it dawns on you that you failed to carry out one of your daily habits or routines? Most likely you had some distraction, which resulted in you omitting just one simple act. But oh how that one simple omission can wreak havoc on your entire day!!!
Our lives are full of habits – routines, actions, or behaviors that are so well formed that they happen without any thought on our part. There’s just something about our habits and routines that adds organization and structure, helps us to make sense of our world, and overall helps our lives run smoothly.
Teachers are pros when it comes to modeling good organizational skills and habits. They’re also experts in demonstrating and modeling correct mathematical procedures. But as you and I both know, learning and doing math is more than knowing and understanding how to carry out math procedures.
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 (Levasseur & Cuoco, 2009). These habits, a composite of many skills, attitudes, and likings, enable us to behave intelligently when confronted with a problem to which the immediate answer is unknown. They’re the habits that empower us to use our mathematical knowledge and skills to make sense of and solve problems.
There are two types of problems solvers – experienced and inexperienced.
It’s been said that, “Inexperienced problem solvers don’t know what to do when they don’t know what to do. Experienced problem solvers do know what to do when they don’t know what to do.”
Experienced problem solvers know what to do when they don’t know what to do because they’ve developed the mathematical habits of mind that enable them to behave intelligently when confronted with a problem.
Without being reminded, experienced problem solvers automatically employ the steps in the Polya’s (1945) four-step problem solving process in an attempt to solve a problem presented to them.
Here are the four steps experienced problem solvers follow:
1. They begin by first making an attempt to understand the problem and ask clarifying questions that help them interpret and understand the conditions of the problem.
2. They make sense of the mathematical situation at hand and choose a strategy, tools, and/or models they see as relevant and applicable in solving the problem.
3. They use the strategy they’ve chosen and if that strategy doesn’t work, they try a different strategy – all part of perseverance in problem solving.
4. They take one last look at the original problem, the strategy and processes used, and determine if the solution is reasonable and viable.
But the mathematical habits of mind cannot simply be taught or learned by studying a list. They are best learned when students are immersed in classroom experiences that enable them to engage in the learning of mathematics concepts through problem solving, making and using abstractions, and developing and applying mathematical theories. Classrooms steeped in the Common Core Standards for Mathematical Practice (MP) provide ideal opportunities for students to learn and develop these habits as they:
When students have experiences in making sense of and solving problems and communicating and using precise and appropriate mathematics and mathematics language, they have opportunities to develop the overarching habits of a productive mathematical thinker. The productive thinker knows that it may take more than one attempt to find the solution to a problem and it may require trying more than one strategy.
Problems that can be approached from a variety of entry points are always more palatable to students.
When you present students relevant, interesting, and challenging problems that can be approached and solved using different strategies, you give them choices. They have choices in to using variety of tools and models for solving the problem. This in turn helps them to reason about, look for, and make use of structure in solving problems. All of this leads to increased opportunities for understanding important math concepts and skills which strengthens their ability to explain their thinking and reasoning as well as critique the reasoning of others.
You as a teacher play a vital role in the development of mathematical habits, in conveying a mindset that fosters productive struggle and emphasizing the value that wrong answers can have. You set the stage by embracing a mindset that fosters productive struggle and the importance and value in grappling with problems. The more you engage your students in learning and doing mathematics, the greater the chances of them developing the mathematical habits of mind of a productive mathematical thinker – an experienced problem solver who does know what to do when they don’t know what to do.
We value your feedback so we invite you to share your questions, comments, and suggestions about teaching and learning mathematics. Thanks!
Seen Cindy's free guide on Developing Mathematical Literacy?