Assessing Higher Order Thinking in Mathematics

By Gerald Kulm | Go to book overview

Three of the students' projects would be submitted for external assessment. At least one of the three would be a practical project, such as designing packaging for candies. A second must be an investigational project in which the student engages in mathematically reflective activity. The internally assessed (school), but externally moderated (examination board) record is intended to promote many of the practices attempted in pilot efforts ( Wharm by , 1986): (i) a modular approach, (ii) practical work, (iii) extended project work, (iv) written assignments, (v) oral assessment, (vi) written assessment, (vii) assessment as an integral part of the learning process, (viii) greater involvement of the teacher in the assessment process, (ix) a cumulative profiling of students' mathematical achievement, and (x) an implicit intent to send all students -- not just the brightest and most mathematically able -- into the adult world with some mathematical understanding and confidence.

Graded Assessment in Mathematics (GAIM) also stresses assessment tasks that are good instructional tasks and therefore have curricular validity. The curriculum is divided into progressive levels ( Brown, 1986) determined by the facility hierarchies identified in the Concepts in Secondary Mathematics and Science Project ( Hart, 1980). Practical and investigative activities in this project are shorter (20 minutes to 1.5 hours). They start from specific but open-ended activity cards and are linked to student and class profiles.

In summary, the thrust of the effort in Great Britain is toward a much wider variety of teaching and learning strategies, with the assessment process regarded as a catalyst. Proposals for national assessment include a score-weighted battery of written tests, performance tests, tests of mental arithmetic, an oral interview, and a review of project work. Ongoing experiments address integrated assessment tasks that simultaneously assess students in a performance task that includes science, mathematics, and written and verbal communication. These are multifaceted strategies that have the potential for providing a more flexible and much more detailed picture of student achievement that simultaneously contributes to their understanding of the subject.


Summary and Conclusion

Traditional assessment practices have consistently used a content-by-behavior matrix as their theoretical framework and relied heavily on independent, multiple-choice items. Cost efficiency almost eradicated other approaches to group testing. However, the mathematical psychological, sociological and pedagogical theories embedded in the model are, quite simply, inadequate. Consequently, it is important to replace the matrix model with one more capable of handling complexity and one that will stimulate change. Unfortunately, the cohesive power of the matrix model exerts a powerful influence which subliminally impedes change.

It is essential that the new model be powerful and have both tight internal coherence and congruence with current trends in mathematics, science, and society. It is also important that the key indicators and instruments for measuring be equally coherent and congruent, the cohesive force being purpose -- namely, the development of each student's mathematical power.

Because the intent is to assess the creation of knowledge and the processes involved rather than to measure the extent to which students have acquired a coverage of the field of mathematics, a much wider variety of measures, many of them qualitative, are needed. Considerable effort is needed to find instruments adequate for the purpose.

Only a few of a wide variety of approaches to assessment have been discussed here. They were selected as representative of the range of instruments that might form a coherent repertoire. Some are theory driven by recent research in cognition which would support the new world view, and others are the result of practical insights by educators with a new world view of mathematics. Both approaches are appropriate at this stage. The urgent need is for a much greater variety of learning and assessment tasks, a coherent body of tools that will precipitate curricular change.


References

Bell, A. W. ( 1985). "Some implications of research on the teaching of mathematics". In A. Bell, B.Low

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