Academic journal article Asian Social Science

Assessing Students' Mathematical Problem-Solving and Problem-Posing Skills

Academic journal article Asian Social Science

Assessing Students' Mathematical Problem-Solving and Problem-Posing Skills

Article excerpt

Abstract

Problem-solving and problem-posing have become important cognitive activities in teaching and learning mathematics. Many researchers argued that the traditional way of assessment cannot truly reveal what the students learnt and knew. Authentic assessment was used as an alternative method in assessing the students' mathematical learning. A performance rubric is an appropriate tool in examining students' ability to solve and pose mathematical problems.

Keywords: problem-solving, problem-posing, authentic assessment

1. Introduction

The shiftin learning theory from behaviourism to constructivism has had an enormous impact on the teaching and learning of mathematics (Hatfield, Edwards, Bitter, & Morrow, 2003). According to von Glasersfeld (1989) students acquire knowledge by constructing and restructuring it over time which is similar to the experiential learning theory by Dewey (1938/1997). An individual learns by doing or experiencing, and teachers should facilitate students' learning for attaining knowledge developmentally (Dewey, 1938/1997). In order to make the learning of mathematics meaningful, teachers are responsible for choosing and posing tasks that engage students actively in building their understanding, mathematical thinking, and confidence (Kulm, 1994).

For many years, scholars have discussed the difficulty of assessing students' mathematical understanding using traditional assessment (Anderson, 1998). Kulm (1994) argued that traditional tests only focus on students' mathematical skills and procedures. Therefore, the use of authentic assessment tools to measure students' learning is critically needed in mathematics. In this paper, a discussion on appropriate tools is presented for assessing student performance in mathematical problem-solving and problem-posing. The paper begins with a brief summary of problem-solving and problem-posing tasks and activities that influence the need of authentic assessments. The focus will be on the use of performance rubrics for examining a student's ability to solve and pose mathematical problems. The advantages and disadvantages of using performance rubrics for mathematics assessment will be discussed, and then followed by some concluding remarks.

2. Mathematical Tasks

The National Council of Teacher of Mathematics' [NCTM] Principles and Standards for School Mathematics urges teachers to use authentic mathematical tasks in the classroom to facilitate knowledge construction (NCTM, 2000). The mathematical problems and activities should engage students with real-world contexts by using and applying mathematical content they have learned into their workplace (Lajoie, 1995). Teachers should consider choosing and creating tasks that utilize the application of mathematical procedures and concepts with a variety of solution approaches that can demonstrate students' understanding (Kulm, 1994). According to Cohen and Fowler (1998) and Kulm (1994), teachers can adapt some beneficial strategies to generate many new mathematical tasks by changing the way the tasks are presented. For example, "Calculate 2.4 x 5.3", teachers can reformulate this routine task into a variety of meaningful tasks as following:

(1) How is 2.4 x 5.3 = 12.72?

(2) Generate a story problem with 2.4 x 5.3.

(3) Display 2.4 x 5.3 using pictorial representation.

(4) How much difference is there between 2.4 x 5.5 and 2.4 x 5.3?

Tasks (1) through (4) are considered authentic that utilized the constructivist stance for stimulating students' mathematical learning (Silver, 1994). These types of task have become commonly used in schools of mathematics in addition to journals, portfolios, and class projects. The openness feature of the tasks allows students to communicate their thinking and reasoning mathematically through their writing (Berenson & Carter, 1995; Burns, 2007). When using this type of task, teachers can help students extend what they know for developing their mathematical fluency and engage them in higher-order thinking (NCTM, 2000). …

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