Classroom Instruction That Fosters Mathematical Thinking and Problem Solving: Connections Between Theory and Practice

QUOTES:

"Doing mathematics cannot be viewed as a mechanical performance or an activity that individuals engage in by solely following predetermined rules. Mathematical activity can be seen more as embodying the elements of an art of craft than as a purely technical discipline." (p. 293)

"...mathematics is a rational human creation. It is a vast collection of ideas derived as a consequence of searching for solutions to social problems. The abstractions and inventions help us make sense of our world and ourselves." (p. 297)

"The 'intended' curriculum can only include our best guesses about what will both interest students and lead all toward development of mathematical power. At the same time, the 'actual' curriculum depends on teacher choice, and the 'achieved' curriculum depends on each student's interest and prior knowledge." (p. 300)

"Nussbaum and Novick suggest a three-part instructional sequence designed to encourage students to make the desired conceptual changes. They propose the use of an exposing event, which encourages students to use and explore their own conceptions in an effort to understand the event. This is followed by a discrepant event, which serves as an anomaly and produces cognitive conflict. It is hoped that this will lead the students to a state of dissatisfaction with current conceptions. A period of resolution follows, in which the alternative conceptions are made plausible and intelligible to students, and in which students are encouraged to make the desired conceptual shift." (p. 298)

"Classrooms should be places where interesting problems are explored using important mathematical ideas…This vision sees students studying much of the same mathematics currently taught, but with quite a different emphasis." (p. 302)

Romberg, T. (1994). Classroom instruction that fosters mathematical thinking and problem solving: Connections between theory and practice. In A. Schoenfeld & A. Sloane (Eds.), Mathematical Thinking and Problem Solving (pp. 287-304). Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.