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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).
Assessment approaches
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).
You'll find there's a lot of detailed information in resources, so here's
a few hints:
- Take notes as you go so that you're not overwhelmed by the material.
At the end of each computer session write additional notes.
- Always keep in mind your aim is solve a problem or to investigate
a task.
Copyright information
References
Australian Education Council. (1991). A national
statement on mathematics for Australian schools. Carlton, Vic: Curriculum
Corporation.
Clarke, D. (1988). Assessment alternatives in mathematics.
Canberra: Curriculum Corporation.
Department of Education & Science. (1982). Mathematics
counts: Report of the committee of inquiry into the teaching of mathematics
in schools under the chairmanship of Dr W H Cockcroft. London: HMSO.
(The Cockcroft Report).
Kilpatrick, J. (1993). The chain and the arrow: From
the history of mathematics assessment. In M. Niss. (Ed.), Investigations
into assessment in mathematics education. An ICMI Study. (pp. 31-46).
Dordrecht: Kluwer.
Mitchell, C., & Koshy, V (1993). Effective teacher
assessment: Looking at children 's learning in the primary classroom.
London: Hodder & Stoughton.
National Council of Supervisors of Mathematics. (1996).
Great tasks and more!! A source book of camera - ready resources on
mathematics assessment. Golden, CO: Author.
National Council of Teachers of Mathematics. (1989).
Curriculum and evaluation standards for school mathematics. Reston,
VA: Author.
National Council of Teachers of Mathematics. (1995).
Assessment standards for school mathematics. Reston, VA: Author.
Niss, M. (1993). Assessment in mathematics education
and its effects: An introduction. In M. Niss. (Ed.), Investigations
into assessment in mathematics education: An ICMI Study (pp. 1-30).
Dordrecht: Kluwer.
Stenmark, J.K. (Ed.). (1991). Mathematics assessment:
Myths, models, good questions, and practical suggestions. Reston,
VA: NCTM.
Stephens, M. (1992). Foreword. In M. Stephens & J.
Izard. (Eds.), Reshaping assessment practices: Assessment in the mathematical
sciences under challenge (pp. vi-xii). Hawthorn, Vic: Australian
Council for Educational Research.
Swan, M. (1993a). Assessing a wider range of students'
abilities. In Webb, N.L. & Coxford, A.F. (Eds.). Assessment in
the mathematics classroom. Reston, VA: NCTM.
Swan, M. (1 993b). Improving the design and balance of
mathematical assessment. In M. Niss. (Ed.), Investigations into assessment
in mathematics education. An ICMI Study (pp. 195-216). Dordrecht:
Kiuwer.
von Glasersfeld, E. (Ed.). (1991). Radical constructivism
in mathematics education. Dordrecht: Kluwer.
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