The top face of the cube represents the Habits of Mind that students should develop in both scientific and mathematical thinking. A habit is any activity that is so well established that it occurs without thought on the part of the individual. The Habits of Mind are those well-established thinking behaviors and patterns characteristic of methodical and logical thinking. Specifically: problem solving, communicating, reasoning, and making connections.
Problem Solving
"To solve a problem is to find a way where no way is known off-hand, to find a way out of a difficulty, to find a way around an obstacle, to attain a desired end that is not immediately attainable, by appropriate means."
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Problem situations for younger children arise from everyday experiences. ("Do you have enough money to buy your lunch and to pay for your yearbook?"). As students become more mathematically sophisticated, problem situations also spring from within mathematics itself ("Can the product of two prime numbers be a prime number?"). Problem solving is much more than applying specific techniques to the solution of a class of problems. It is a process by which the fabric of mathematics is both constructed and reinforced.
Communicating
All students need extensive experience communicating mathematics, by listening to, reading about, writing about, speaking about, reflecting on, and demonstrating mathematical ideas. By emphasizing the communication of mathematics, teachers can direct instruction away from a focus on the recall of terminology and routine manipulation of symbols and procedures, toward a deeper conceptual understanding of mathematics ("Janet, can you explain to Celia why you think the round pizzas are a better buy?").
Reasoning
Reasoning is fundamental to the knowing and doing of mathematics. Conjecturing and demonstrating the logical validity of conjectures are the essence of the creative act of doing mathematics. Starting from the intuitive and concrete thinking of younger children, reasoning and the development of logic become more formal as children develop higher levels of reasoning ability. The role of reasoning should be emphasized in all areas of mathematics ("If the square pizzas are larger, how can the round pizzas be a better buy?").
Making Connections
Students need to make connections between mathematics and other disciplines. When making connections within mathematics, one might ask, for instance, to geometrically represent slope on a graphing calculator and describe the linear relationship with an algebraic equation. When connecting mathematics with other disciplines, one might begin with data collected on a bouncing ball to develop the linear model.
The Habits of Mind are not specific mathematical or scientific topics, but are ways of thinking that influence the way students approach their world. These Habits of Mind will be encouraged and encountered whenever discussions of other ideas or concepts occur.
Academic expectations for students in the four areas of problem solving, communicating, reasoning, and making connections are summarized across the grade levels in Table 1: Mathematical Habits of Mind.
