The reformed curriculum suggested that every instructional activity is an assessment opportunity for teachers and a learning opportunity for students. The movement emphasized classroom assessment in gathering information on which teachers can inform their further instruction. Assessment integral to instruction contributes significantly to all students’ mathematics learning.
The new vision of assessment suggested that knowing how these assessment processes take place should become a focus of teacher education programs. Problem-posing task referred to in the study was that the task teachers designed requires students to generate one or more word problems. The professional standards suggested that teachers could use task selection and analysis as foci for thinking about instruction and assessment. According to De Lange (1995), a task that is open for students’ process and solution is a way of stimulating students’ high quality thinking. Training teachers in designing and using assessment tasks has also been proposed as a means of improving the quality of assessments (Clarke, 1996).
However, the design of open-ended tasks is a complex and challenging work for the teachers who are used to the traditional test. Thus, the tasks involving in the study were considered as an informal way of assessing what and how individual student learned from everyday lesson. Thus, the preparation of the tasks involving in this study was not prior to instruction; rather, teachers generated them from the activities in which students engaged in everyday lesson. The mathematics contents covered in the textbooks were a dimension of the assessment framework of the study.
The reformed curriculum calls for an increased emphasis on teachers’ responsibility for the quality of the tasks in which students engaged. The high quality of tasks should help students clarifying thinking and developing deeper understanding through the process of formulating problem, communicating, and reasoning (MET, 2000). Thus, these cognitive processes were the other dimension of the assessment framework. The tasks teachers designed in the study were to assess students’ problem posing, communicating, and reasoning. Due to the limitation of space, this paper is primarily concerned with the problem-posing tasks. Problem-posing is recognized as an important component in the nature of mathematical thinking (Kilpatric, 1987). More recently, there is an increased emphasis on giving students opportunities with problem posing in mathematics classroom (English & Hoalford, 1995; Stoyanova, 1998). These research has shown that instructional activities as having students generate problems as a means of improving ability of problem solving and their attitude toward mathematics (Winograd, 1991). Nevertheless, such reform requires first a commitment to creating an environment in which problem posing is a natural process of mathematics learning.
Second, it requires teachers figure out the strategies for helping students posing meaningful and enticing problems. Thus, there is a need to support teachers with a collaborative team whose students engage in problem-posing activities. This can only be achieved by establishing an assessment team who support mutually by providing them with dialogues on critical assessment issues related to instruction.
Problem-posing involves generating new problems and reformatting a given problems (Silver, 1994). Generating new problems is not on the solution but on creating a new problem. The quality of problems in which students generated depends on the given tasks (Leung & Silver, 1997). Research on problem posing has increased attention to the effect of problem posing on students’ mathematical ability and the effect of task formats on problem posing (Leung & Silver, 1997). Such problem-posing tasks that situations were presented in a story form were created by researcher rather than by classroom teachers. Moreover, there is a little research on teachers’ responsibility for the variety and the quality of the problem-posing tasks. The way in which teachers explored to create tasks for students generating problems from a contrived situation was investigated in the present study.
For teachers, the problem-posing tasks allowed them to gain insight into the way students constructing mathematical understanding and served to be a useful assessment tool. As an assessing tool, the tasks incorporating into everyday instruction, decisions about task appropriateness were often related to students’ communication of their thinking, or the students’ problem-solving strategies displayed in classroom. The mathematics concepts to be taught at a grade level became as an elementary element of designing assessment tasks integrated into instruction. Other decisions concerning the appropriateness of a task were relevant with teaching events students encountered in everyday lesson.
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