REASONING & PROBLEM SOLVING

REASONING & PROBLEM SOLVING

Reasoning & Problem Solving

Introduction

Principles and Standards for School Mathematics NCTM (2000) draw attention on developing of students' mathematical reasoning, as well as on the assessment of this competence. Teachers should encourage students to justify their assertions and statements, and search for new methods and means to develop students' mathematical reasoning. However, it is not easy to specify the type of arguments that should be expected by students and the kind of reasoning that should be taught to primary students. Research shows that not all students' statements and arguments in mathematical problem solving (MPS) are mathematically valid arguments (see e.g., Evens & Houssart, 2004). Students often reason according to their personal experiences, and teachers who seek to understand what is actually behind an argument should escape their "egocentricity" and think through a child's perspective (Tang & Ginsburg, 1999). Therefore, teachers' assessment of students' arguments is essential to developing of students' mathematical reasoning. However, no piece of research seems to have investigated how teachers appraise students' arguments.

Theoretical Background And Aims

Mathematical reasoning or justification is a type of "weak proof" for a mathematical assertion. Russel (1999, p.1) argues that reasoning refers to "what we use to think about the properties of these mathematical objects and develop generalizations that apply to whole classes of objects". Recent studies (e.g., Pehkonen, 2000) suggest that primary students have difficulty in mathematical reasoning. It is, however, important in Mathematics teaching to let students develop the habit to ask for reasons and provide arguments in their mathematical activities as a preparation for the ultimate goal, which is to produce formal proofs in high school.

In this study we adopt the categorization of students' arguments proposed by Evens and Houssart (2004), which refers to reasoning in MPS; they propose four types of responses: 1) wrong or irrelevant, 2) restatement or reinforcement, 3) providing numerical examples and 4) justification. The first type refers to responses that are irrelevant to the solution of the problem, either due to incorrect course of solution or to arguments that are not rationally connected to the problem. The second type refers to mere restatements of the data, most likely in ones' own words, without any substantial addition to already given information. The third type concerns arguments limited to direct or indirect use of examples, a type of justification that might be accepted for primary students, as non-well articulated inductive reasoning. The last ...