When problem posing is used in the classroom as a means to engage students in learning important concepts and skills and to enhance their problemsolving competence, then teachers might reasonably consider also using generative activities as part of classroom assessment related to those concepts, skills, and competencies. For example, consider the task shown in figure 1. This task might be used on a classroom assessment following an instructional unit that deals with division with remainders.
A student might propose a problem that has a correct answer of 13 (for example, “If you have 540 CDs, and you place 40 in each case, how many cases can you fill?”) and another that has an answer of 14 (“If you have 540 CDs, and you place no more than 40 in each case, what is the smallest number of cases needed to hold all the CDs?”). Another student might pose a problem with an answer of 13 and 20 left (“If you have 540 CDs, and you place exactly 40 in each case, how many cases can you fill and how many CDs will be left over?”) or another with an answer of 20 (“If you have 540 CDs, and you place exactly 40 in each case, how many CDs will be left over after you fill as many cases as you can fill?”). Yet another student might pose a problem that has an answer of 13 1/2 (“If Joshua earns $40 for each day worked in his summer job, how many days would he need to work to earn exactly $540?”), and perhaps problems with other reasonable answers. Thus, the problem could be a good way of assessing whether students understand that any division statement can be used to model many problem situations and that the answer to a situated division problem depends to a great extent on the features of the situation and the nature of the quantities involved. Of course, this knowledge might also be tested in different ways, but a problem-posing task of this type seems especially well suited to this testing goal.
Good assessment is about expanding the assessment repertoire because no single form is sufficient. Since every assessment tool faces different issues related to reliability and validity, each of them has its strengths, weaknesses and its place in educative assessment. More importantly, good assessment practice is about applying the assessment repertoire to provide students with multiple opportunities, in varying contexts, to demonstrate what they know, understand and can do in relation to the stipulated learning outcomes.
Despite the recent development in incorporating problem posing as an instructional tool, far less attention has been paid to problem posing and assessment. Cai and Silver (2005) suggested the possibility of expanding the role of problem posing as a classroom assessment to evaluate, students’ understanding and proficiency. It is the use of problem posing activities to inform teachers about students’ learning that forms the impetus for this action research on exploring the use of problem posing as a formative assessment tool.
In order to effectively use problem posing in the classroom as a generative activity from which information about the students’ engagement, competencies and areas of improvement can be drawn, it is essential for teachers to understand its value as a formative assessment. There should be a common understanding of the main goal of formative assessment. These assessments, whichever form that they take, aim to help both teachers and learners find out how much learning has taken place, and then use the results to chart the process of the learners’ learning. Teachers can then point directions to where their learners’ strengths can be further developed and at the same time identify those areas of improvement. It is this interest in the process of gathering data about students’ growth in learning mathematics for improving classroom instruction and learning that drives this study.
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