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4 Assessing to Support Mathematics Learning
Pages 67-90

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From page 67...
... THE LEARNING PRINCIPLE Assessment should enhance mathematics learning and support good instructional practice. This principle has important implications for the nature of assessment.
From page 68...
... To satisfy the learning principle, assessment must change in ways consonant with the current changes in teaching, learning, and curriculum. In the past, student learning was often viewed as a passive process whereby students remembered what teachers told them to remember.
From page 69...
... Assessment, therefore, must reflect and reinforce this view of the learning process. This chapter examines three ways of making assessment compatible with the learning principle: ensuring that assessment directly supports student learning; ensuring that assessment is consonant with good instructional practice; and enabling teachers to become better facilitators of student learning.
From page 70...
... The assessment challenge we face is to give up old assessment methods to determine what students know, which are based on behavioral theories of learning and develop authentic assessment procedures that reflect current epistemological beliefs both about what it means to know mathematics and how students come to know.; Current research indicates that acquired knowledge is not simply a collection of concepts and procedural skills filed in long-term memory. Rather the knowledge is structured by individuals in meaningful ways, which grow and change over time.8 A close consideration of recent research on mathematical cognition suggests that in mathematics, as in reading, successful learners understand the task to be one of constructing meaning, of doing interpretive work rather than routine manipulations.
From page 71...
... If students have been using graphing calculators to explore trigonometric functions, giving them tests on which calcula tors are banned greatly limits the questions they can be asked and LEARN I NO Assessment must reflect the value of group interaction for mathematics.
From page 72...
... Using physical materials and technology appropriately and effectively in a critical part of instruction is a critical part of learning today's mathematics and, therefore, must be part of today's assessment. tociay's Reflecting Development of Competence As students mathematical progress through their schooling, it is obvious that the content of their assessments must change to reflect their growing mathematical instruction, such sophistication When students encounter new topics in mathemat ics, they often cannot see how the unfamiliar ideas are connected to tools must be anything they have seen before.
From page 73...
... The extensive experience that writing teachers have been accumulating in teaching and assessing writing through extended projects can be of considerable assistance to mathematics teachers seeking to do similar work. During the two-stage testing in the Netherlands, students reflected on their work, talked with others about it, and got information from the library.
From page 74...
... scoring guides to Such information, when shared between teacher and student, has critically important implications for the learning process. communicate 74 Teachers can appropriately communicate the features of the goals of scoring rubrics to students as part of the learning process to illus trate the types of performance students are striving for.
From page 75...
... It is hoped that students can, over time, develop an inner sense of excellent performance so that they can correct their own work before submitting it to the teacher. Incorporating a Scoring Rubric Directions for Students Today you will take part in a mathematics problem-solving assessment.
From page 76...
... Conversely, Intrinsic sources assessments to judge the effectiveness of an educational program where results are often not reported on an individual basis carry low of motivation stakes for the student and may not inspire students to excel. These extrinsic sources of motivation, although real, are not always consooffer a fruitful nant with the principle that assessment should support good instruc tional practice and enhance mathematics learning.
From page 77...
... For example, one group of sixth-grade students interviewed an elementary school principal who said that when cafeteria lunch prices went up, fewer students bought their lunches in the cafeteria. The students used a quadratic function to model the data, orally reported to their classmates, and wrote a report for their portfolios.
From page 78...
... A survey in Israel of junior high students' attitudes toward different types of tests showed that although they thought essay tests reflected their knowledge of subject matter better than multiple-choice tests did, they preferred the multiple-choice tests.23 The multiple-choice tests were perceived as being easier and simpler; the students felt more comfortable taking them. и иии~ииииOиOиииииииииииииииии ASSESSMENT IN SUPPORT OF INSTRUCTION , и If mathematics assessment is to help students develop their powers of reasoning, problem solving, communicating, and connecting mathematics to situations in which it can be used, both mathematics assessment and mathematics instruction will need to change in tandem.
From page 79...
... In addition to providing good records of individual student work, portfolios might also be useful in providing forma tive evaluation information for program development. Before they can be used as components of large-scale assessment efforts, however, consistent methods for evaluating portfolios will need to be developed.26 Of course the quality of student work in a portfolio depends largely on the quality of assignments that were given as well as on L E A R N I N G instructional activities from assessment activities.
From page 80...
... The state of Vermont, for example, has been devising a program in which the mathematics portfolios of fourth- and eighthgrade students are assessed;27 other states and districts are experimenting with similar programs. Some problems have been reported in the portfolio assessment process in Vermont.28 The program appears to hold sufficient merit, however, to justify efforts under way to determine how information from portfolios can be communicated outside the classroom in authoritative and credible ways.29 The trend worldwide is to use student work expeditiously on instructional activities directly as assessment.
From page 81...
... Elaborate schemes are not necessary, but some system is needed. A few carefully selected tasks can give a reasonably accurate picture of a student's ability to solve a range of tasks.33 An example of a task constructed for this purpose appears above.34 USING ASSESSMENT RESULTS FOR N S T R U C T ~ O N The most typical form of assessment results have for decades been based in rankings of performance, particularly in mandated assessment.
From page 82...
... 82 think about To be effective In Instruction, assessment results need to be timely.35 Students' learning is not promoted by computer printouts mathematics sent to teachers once classes have ended for the year and the students have gone, nor by teachers who take an inordinate amount of time to grade assessments. In particular, new ways must be found to give teachers and students alike more immediate knowledge of the students' performance on assessments mandated by outside authorities so that those assessments as well as the teacher's own assessments can be used to improve learning.
From page 83...
... The Classroom Assessment in Mathematics (CAM) Network, for example, is an electronic network of middle school teachers in seven urban centers LEARN I N G assistance in assessments consonant with today's vision of mathematics instruction.
From page 84...
... These Collaborations collaborations can start locally or be developed through and spon sored by professional organizations. Publications are beginning to with others appear that can help teachers assess mathematics learning more thoroughly and productively.40 can assist There are indications that using assessments in professional mathematics development can help teachers improve instruction.
From page 85...
... Particularly when fairly complex tasks have been used, the wider audience will benefit more from a few samples of actual student work than they will from detailed descriptions and analyses of anticipated student responses. Teachers are also playing an active role in creating and using assessment results.
From page 86...
... The Pittsburgh schools, for example, recently piloted an auditing process through which portfolios developed for instruc tional uses provided "publicly acceptable accountability informa tion." 45 Audit teams compsing teachers, university-based research ers, content experts, and representatives of the business community evaluated samples of portfolios and sent a letter to the Board of Education that certified, among other things, that the portfolio process was well defined and well implemented and that it aimed at success for all learners, challenged teachers to do a more effective job of supporting student learning, and increased overall system accountability. There is reason to believe, therefore, that the learning principle can be honored to a satisfactory degree for both internal and external assessments.
From page 87...
... 'a Nancy S Cole, "Changing Assessment Practice in Mathematics Education: Reclaiming Assessment for Teaching and Learning" (Paper presented at the Conference on Partnerships for Systemic Change in Mathematics, Science, and Technology Education, Washington, D.C., 7 December 1992~.
From page 88...
... The Report of Vermont's Mathematics Portfolio Assessment Program (Montpelier, VA: Author, 1991~; Jean Kerr Stenmark, Assessment Alternatives in Mathematics: An Overview of Assessment Techniques that Promote Learning (Berkeley, CA: University of California, EQUALS, 1989~.
From page 89...
... 2~3 Daniel Koretz etal., The Reliability of Scores from the 1992 Vermont Portfolio Assessment Program, CSE Technical Report 355 (Los Angeles, CA: University of California, National Center for Research on Evaluation, Standards, and Student Testing, 1993~. 29 Pamela A
From page 90...
... .K. Stenmark, Mathematics Assessment Myths' Models, Good Questions, and Practical Suggestions (Reston, VA: National Council of Teachers of Mathematics, 1991~; Assessment in the Mathematics Classroom; Measuring Up; Assessing Higher Order Thinking in Mathematics; California Assessment Program, A Sampler of Mathematics Assessment (Sacramento, CA: California Department of Education, 1991~; Judy Mumme, Portfolio Assessment in Mathematics (Santa Barbara, CA: California Mathematics Project, University of California, Santa Barbara, 1990~.


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