Victor V. Cifarelli

Associations of Students’ Beliefs With Self-Regulated Problem Solving in College Algebra

We examined the associations between the expressed mathematical beliefs of students and their self-regulated actions in solving mathematics problems. We conducted surveys and interviews that focused on students’ self-regulated problem solving and identified students’ self-reported beliefs about mathematics. Our findings suggest that even though students may possess rigid instrumental views about mathematics, they may still be able to achieve success by incorporating some general heuristics into their problem solving if they have first broadened their definition of mathematics to legitimize such activity. Instructional activities that allow students opportunities to share and defend their ideas for solving particular problems prior to actually solving them help students develop self-advocacy and contribute to a proactive sense of agency. Students need support to develop as self-regulated problem solvers. This can be achieved through coaching and one-on-one tutoring; however, it is difficult to achieve in classroom practice. For students to broaden their view of mathematics and what their role as a mathematical problem solver can be, they must be provided with ample problem-solving opportunities. Encouraging students to reflect on their problem solving helps promote the monitoring and assessment necessary for self-regulated learning to occur.

The Importance of Abductive Reasoning in Mathematical Problem Solving

Charles Sanders Peirce (1839—1914) made a distinction between formal and informal reasoning, and argued that the formal reasoning processes of induction and deduction were not sufficient to explain those instances when individuals entertain new ideas to explain surprising facts. Peirce asserted the existence of another kind of reasoning, abduction, through which the individual generates a novel hypothesis to account for or explain surprising facts under consideration.

Problem Posing as Reformulation and Sense-Making Within Problem Solving

This chapter examines a type of problem posing that has received little attention in the mathematics education literature. Silver (For the Learning of Mathematics 14:19–28, 1994) defined within-solution problem posing as “problem formulation or reformulation [that] occurs within the process of problem solving” (p. 19). Our analysis documents and explains the role that within-solution problem posing plays during problem solving, focusing on episodes of students from two grade levels: (a) Two fourth-grade students solving a multiplication task, and (b) A mathematics education graduate student solving a number array task. Our research examines: (a) How problem posing evolves from the students’ ongoing interpretations of problematic situations, and (b) How these posed problems contribute to the students’ problem solving. The results provide an explanation of how problem posing and problem solving coevolve in the course of solution activity and thus indicate the beneficial role that problem posing can play in the solution of mathematics problems.