A new vision
Assessment a critical issue in the teaching and learning of mathematics and one that requires careful consideration by teachers and preservice teachers alike. The assessment experiences for many students in the classroom is still one that is based on a behaviourist approach where discrete facts and skills are tested, where grading and ranking are the primary goals (Niss, 1993). Kilpatrick argues that an alternative vision is necessary for today's classrooms:
The challenge for the 21st century, as far as mathematics educators are concerned, is to produce an assessment practice that does more than measures a person's mind and then assign a mind treatment. We need to understand how people, not apart from but embedded in their cultures, come to use mathematics in different social settings and how we can create a mathematics instruction that helps them use it better, more rewardingly, and more responsibly. To do that will require us to transcend the crippling visions of mind as a hierarchy, school as a machine, and assessment as engineering. (Kilpatrick, 1993, p. 44)
This view changes the focus of assessment from summative assessment where students are assessed principally to determine an overall measure of achievement, to the more supportive role of formative assessment where students' achievements result in action plans, for both teacher and student, in the pursuit of further learning.
The need for change
The need to change assessment practices must be seen in the wider context of changes to society, and changes to the way we view mathematics, teaching mathematics and learning mathematics. Today's society has moved from an industrial to an information-based society that relies on a far greater use and application of technological understanding and has goals that promote equal opportunity for mathematics learning for all its citizens (NCTM, 1989). Mathematics itself is no longer seen as hierarchical and discrete with the consequent belief that this is the way it is learnt (Stephens, 1992). Instead, influential reports on the mathematics education show a vision of mathematical knowledge that is different to many preconceived beliefs about the subject. The accepted view of mathematics as basic arithmetic skills has given way to a broader view that emphasises mathematics as general processes, or ways of thinking and reasoning (NCTM, 1989), as an important form of communication (DES, 1982), and as a science of patterns (AEC, 1991).
Current theories of learning mathematics suggest that students are not passive receivers knowledge but actively construct knowledge consensual with social and cultural settings (von Glasersfeld, 1991). These changing views of mathematics and the way students learn have broadened the ways in which mathematics is taught. instead of a view of teaching that relies on teacher exposition followed by student practice, modern teaching practices involve strategies such as problem solving, investigations, practical activity, (DES, 1982); group work, projects (NCTM, 1989); and applications of relevant technologies (AEC, 1991).
What is needed
Rather than relying on approaches that provide assessment solely for the purposes of grading, ranking and credentialling, assessment practices are needed that integrate with learning activities, that support students' construction of knowledge and that reflect the diversity found in the curriculum and in the learners themselves.
As well as the variety of mathematical topics found in any mathematics curriculum, the expected learning outcomes for each topic vary from routine mathematical facts and skills, to conceptual understanding, strategic knowledge, appreciations and awareness, personal attitudes and qualities (Swan 1993a). The learners themselves bring rich experiences that reflect different social, cultural and gender groupings in society as well as varying ages, developmental levels and maturity (NCTM, 1989). The recognition of such diversity has necessitated 'a shift in the vision of evaluation toward a system based on evidence from multiple sources and away from relying on evidence from a single test as well as a shift toward relying on the professional judgements of teachers and away from using only externally derived evidence' (NCTM, 1995, p. 2).
Multiple sources of assessment will involve different ways of presenting tasks to students as well as different ways of probing assessment information so that valid inferences about students' progress can be made. Tasks can include a variety of formats: written, oral, practical; can be closed or open-ended; real life or abstract; completed individually or as a group (Swan, 1993b). Teachers and students alike will make inferences about learning based on information gained through broad approaches like observing, questioning and testing. More fine-grained strategies can also be employed. Observational strategies include such techniques as anecdotal record keeping, annotated class lists and checklists. Questioning approaches may involve structured or open-ended interviews, self questioning, using higher-order questions or fact recall. Reporting may take the form of oral reports given to the class, written reports on a project or investigation, portfolios, journals and diaries. Testing procedures may be formative in nature such as diagnostic tests or have a more summative purpose as in examinations (Clarke, 1988; Mitchell & Koshy, 1993; NCSM, 1996; Stenmark, 1991).
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