adapted from Polya ( 1957), general enough to pinpoint strengths and weaknesses reliably, yet specific enough that subsequent instruction can be prescribed: (i) understanding the problem, (ii) developing a plan, (iii) implementing the plan, and (iv) answering the question and checking the results. Ideas to consider with regard to changes in course content or teaching methods when student weaknesses are detected in these four areas are discussed in some detail in Charles et al. ( 1987).
A final way that assessment data are used is to assign grades. It is important that teachers understand that assessment is not synonymous with grading. Every teacher should have a plan for assessing progress in problem solving, whether or not grades are assigned. When teachers decide to assign a grade, the following guidelines may be useful: (i) advise students in advance when their work will be graded; (ii) use a grading system that considers the process used to solve problems, not just the answer; (iii) be aware that pupils may not perform as well when they are to be graded; (iv) use as much assessment data, and as many different techniques, as possible as a basis for assigning grades; and (v) consider using a testing format that matches your instructional format (e.g., consider testing performance in cooperative groups if this is the way students usually work on problems).
For many students, grades are a very motivating factor. Such students will gain considerably when the system used to assign problem-solving grades reflects the many facets of problem-solving performance.
In this chapter, we have described a model for the assessment of students' growth in mathematical problem solving and we have discussed a number of assessment techniques. Our intent has been to provide some much needed clarity to the current discourse about the role of assessment in mathematics instruction. Of course, if, as is recommended by the Curriculum and Evaluation Standards for School Mathematics ( NCTM, 1989), educators decide to make a serious effort to bring assessment into better alignment with contemporary curricular emphases and instructional practices, there will be reason for real optimism about the future of mathematics instruction in our schools. However, Silver and Kilpatrick ( 1988) point out that, despite our best efforts, many of the most important aspects of problem-solving growth are likely to remain intractable to assessment. Appreciation of an elegant solution and willingness to take risks as a problem solver are but two examples of a host of traits that teachers should want their students to develop. However, none of the assessment techniques we have considered adequately measures these traits. Furthermore, the assessment of higher order thinking, especially thinking processes associated with problem solving, is an extremely difficult task to do well. And, even the best instruments and techniques are only as good as the person using them -- namely, the teacher. If the mathematics education community expects teachers to develop the expertise needed to assess problem-solving performance, attitudes, and beliefs, it must begin now to help teachers learn not only to select and use existing assessment procedures wisely, but also to design and implement their own assessment techniques.
Portions of this chapter refer to research supported by National Science Foundation Grant No. MDR 85-50346. Any opinions, conclusions, or recommendations expressed are those of the authors and do not necessarily reflect the views of the National Science Foundation.
We are grateful to Peter Kloosterman for his helpful comments on an earlier version of this chapter
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