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Using Students' Portfolios to Assess Mathematical UnderstandingAuthor(s): Harold AsturiasSource: The Mathematics Teacher, Vol. 87, No. 9 (December 1994), pp. 698-701Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27969111 .
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sessment Standards for iool Mathematics
Harold Asturias
Using Students1 Portfolios to Assess Mathematical Understanding
I
Portfolios in progress
become
opportunities for
midcourse
adjustments
1989, NCTM published the Curriculum and Eval uation Standards for School Mathematics, which
presented the mathematics profession with a broad view of the important mathematics that should be
taught in schools. Two years later, the Professional Standards for Teaching Mathematics gave teachers the opportunity to address the pedagogical issues inherent in teaching a broad-based, thinking cur
riculum as described in the curriculum standards. The next link, assessment, though part of the first
document, required specific attention. Assessment Standards for School Mathematics, currently in
progress, will present the criteria for judging the
appropriateness and quality of assessment tools and systems.
Teachers continuously assess their students'
progress in the course of day-to-day instruction. This internal assessment is not valued when such traditional tools as standardized testing are used for external?and high-stakes?assessment. The Assessment Standards document will outline a
framework that values the role teachers play in
assessing their students' work and will recommend
ways in which the internal assessment can be used as part of external systems of assessment. The assessment standards are based on the assumption that teachers develop a useful, uniquely detailed sense of what their students know and can do
through constant interaction with their students. One assessment tool that weighs the work each stu dent does and the progress made, while illustrating the whole classroom for external assessment, is the student's portfolio.
Portfolios have long been used successfully to evaluate a student's work in the arts and writing. In recent years, mathematics teachers have used
portfolios in their classrooms to make instructional decisions. The mathematics education community is currently trying to define what it means to use
mathematics portfolios as a way to assess what stu dents are learning.
WHY USE PORTFOLIOS IN THE MATHEMATICS CLASSROOM? The use of portfolios to assess students' progress toward important goals offers many advantages.
As students begin to work on portfolios, they take an active role and assume some responsibility in their own assessment. When they judge the quality of their work while selecting the pieces to be includ
ed, they begin to reflect on their own learning and on ways to improve it. Portfolios contrast with on demand assessment in that they allow students to include work at different stages of completion as
drafts, revisions, and final versions. In so doing, students can include pieces of longer, sustained work completed in and outside the classroom.
As a work in progress, portfolios show students and their teachers concrete evidence of the
progress made toward preestablished goals and
yield more information about what and how stu dents learn than do other, more traditional forms of assessment. Continually examining the contents of the portfolios in progress presents opportunities for midcourse adjustments that students and teachers can make in their instructional interaction.
Finally, collectively and as finished products, portfolios are a way for students, teachers, parents, and external assessors to communicate and share
expectations about students' learning. This process is the foundation of a standards-based educational
system. Portfolios are a permanent record of stu dents' understanding and accomplishments at vari
Edited by Linda Wilson University of Delaware Newark, DE 19716
Harold Asturias is affiliated with the New Standards Project, Oakland, CA 94617. He is interested in the devel opment of assessment systems that are classroom based
and show evidence of the breadth, depth, and quality of students' work. Linda Wilson teaches at the University of Delaware. She is interested in assessment issues in the mathematics classroom and is assistant project director of NCTM's Assessment Standards.
698 THE MATHEMATICS TEACHER
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ous points in their school career. They also give an
accurate picture of the mathematics program in which students are involved and, in so doing, help teachers and external assessors make critical deci sions about the program's effectiveness.
CREATE A PORTFOLIO CULTURE IN THE CLASSROOM
Making portfolios an integral part of their instruc tional day remains a challenge for many mathemat ics teachers. The following suggestions can help teachers create a portfolio culture and make portfo lio use a worthwhile classroom routine for them selves and their students.
Make students responsible for keeping their
portfolios up to date and organized and contain
ing the work they consider most representative of themselves as learners of mathematics.
Use portfolios on an ongoing basis rather than only at the end of a period of time or on specific days.
View portfolios as part of the learning process rather than merely as record-keeping tools, as a
way to stimulate and enhance students' learning of mathematics. Portfolios allow students to include work on topics in which they have a par ticular interest or extended pieces of work they have been creating over time.
Create a shared, clear purpose for using portfo lios. Students should clearly understand
whether they are creating their portfolios for self-reflection about their learning, for a parent teacher spring conference, or as a special exhibit for an external assessor.
Share the criteria that will be used to assess the work in the portfolios as well as ways in which the results are to be used. Students should be familiar with the rubrics used to assess their
work, how each piece counts, and how the scores
they get in their portfolios affect their overall evaluation.
Give students access to their portfolios. Students will then take responsibility for their portfolios, and teachers can implement their use daily. The main responsibility for managing the portfolios should rest with the students.
Create multiple opportunities for feedback on the use of the portfolios from student to student, teacher to student, and so on. During these peri odic conferences, students can discuss with each other the value of different pieces of work they want to place in their portfolios. Another exam
ple includes conferences in which teachers can
discuss the use of portfolios with their students.
HOW TO ASSESS THE PORTFOLIOS On the basis of individual needs, decide ahead of time how students' work included in their portfolios
will be assessed and communicate that decision to them. One portfolio-assessment effort under devel
opment involves a joint venture by New Standards and Balanced Assessment. (The New Standards is an effort to set national standards for assessment; Balanced Assessment is a National Science Foun dation-funded project to create "balanced" assess ment packages to be used by teachers, schools, dis
tricts, states, and so on.) The following, though a
work in progress, illustrates one way to evaluate the portfolios to promote balance?a representative variety rather than coverage?in the work included in the portfolio. Using A Framework for Balance
(Dar? 1993), which defines this balance, figure 1
suggests dimensions to be assessed in the portfolios.
New Standards/Balanced Assessment Portfolio Assessment: Dimensions for Balance
Communication
Students will be expected to demonstrate their ability to communicate their mathematical ideas effectively by using a range of tools to do so. Some of these tools might include, but not be limited to, a picture, dia
gram, sketch, table, chart, spreadsheet, coordinate graph, equation, for
mula, prose, oral discussion, model, map, manipulatives, network, tree, matrix.
Problem solving Four main aspects of problem solving to be assessed in the portfolios are
understanding, approach, decisions, and generalizations.
Content
The content standards for grades 9-12 are outlined in the Curriculum and Evaluation Standards (NCTM 1989). The goal is not to include a
piece of work in each and every standard but rather to present a collec tion of pieces of work that show the various topics that students have studied in their mathematics classes. Some pieces of student work will
exemplify more than one of those standards.
Circumstances of performance Another dimension that should be balanced across the portfolio is the various circumstances under which students do mathematical work. Three main aspects of the actual work follow:
Time?two to three class periods, one to two weeks, three weeks or
longer
People?individual, group, interacting with teacher, peer feedback
Resources?information and tools
Reflection Reflection is an essential part of the portfolio. It renders insight into stu dents* thinking as they do their work and justifies the choices of work that students include in their portfolios. Reflecting is also a tool for self
assessment, which allows students to judge the quality of their own work as well as to demonstrate how they have grown as learners of mathematics.
Fig. 1
(Adapted from Dar? 1993)
The use of portfolios offers many
advantages
Vol. 87, No. 9 ? December 1994 699
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Self assessment
makes
students aware of
areas
that need improvement
Investigations: Kinds of work to include in the portfolio. Students must choose three investigations from the five described here.
Statistical survey studies Students identify, investigate, and draw conclu sions about a social question or situation of inter est to them or their community. This activity entails making decisions about how to collect and
display data as well as drawing conclusions and
making recommendations after interpreting those data.
At the high school level, students could investi
gate the attendance patterns of students at their own school and neighboring schools and compare them. In carrying out the investigation they would summarize, analyze, and transform the data to test hypotheses and draw inferences. In their report they would include the raw data; a representation of the data in charts, tables, or
graphs; a description of the adjustments they made as they collected the data; the conclusions
they reached; and how tests help validate their inferences. This investigation would last three to six weeks during which students revise their
projects. Those revisions are included as part of their report and taken into consideration when
drawing conclusions from the data collected.
Mathematical model of experimental data
Students choose a physical phenomenon to
investigate. They identify and collect the rele vant data and develop a mathematical model for that phenomenon using the data they have col lected. Finally, they draw conclusions about the
applicability and limitations of their model.
At the high school level, students might investi
gate the geometry of shadows and create a math ematical model to represent their findings.
Designs for physical structures
Students design and produce an item from the areas of architecture, engineering, art, or city planning. Students' work on the design should focus on two perspectives: 1. Product?setting Or meeting the criteria for a
design and demonstrating an understanding of how the finished work meets these criteria. 2. Process?communicating the design and the
process of carrying it out clearly enough so that another person could understand the design and
reproduce it or use it.
At the high school level, students might design, make, and advertise a proposed fund-raising game based on probability. Their product would include the game itself or a picture of it, a full
description of how it works, and clear communi cation about how to build it and use it. Students could devise, test, and vary the game according
to cost per play, number of players, probability of winning, size of prize, or payoff. They could
predict the amount of funds the game would raise under certain conditions then enact those conditions.
Resource planning and managing Students work on projects that allow them to show what they can do in the areas of risk
analysis, planning and scheduling, design of games, software design, and optimization.
At the high school level, students could investi
gate the system design of manufacturing facto ries and evaluate the effectiveness of the system by testing different alternatives. They would define constraints, goals, alternative plans of
production, recommendations, and justifications for their choices.
Pure mathematical investigation Students formulate the mathematical questions about which they are curious. They pose a ques tion, note a phenomenon, identify a pattern, or make an observation. Their exploration of the mathematical idea may or may not lead to a sin
gle, definitive solution; the investigation may proceed along several fronts or in an idiosyncrat ic sequence; real-life applications may not be
apparent to the student. Students do, though, record the discoveries and connections they have made as well as their analysis of the question, phenomenon, or pattern with which they began. Many explorations will take students up blind
alleys, just as with mathematicians. Exploring blind alleys are valid investigations.
At the high school level, students could explore the limits of the magic-square problem ("Using the numbers 1-9 once each, arrange them in a 3x3 grid so that the sum of each row, each column, and each diagonal is the same"). Stu dents could produce a series of mathematical
questions, that vary the conditions of the magic square, present solutions or explain why a solution is impossible, and relate the mathe matical significance of their findings. Ques tions students might pose and test include the
following: Does more than one way exist to solve it, or only one? Can one construct a
magic square with the sequence 1,3, 5, 7,... ? Can one do it with 2,4,6, 8,... ? Will every sequence work? With the original problem, the sum of every row, column, and diagonal is 15. Could a set of workable numbers sum to 21? Could they sum to 22? Can one construct
magic squares of 4 4? Magic squares of 5 5? If so, what are their properties?
Fig. 2
700 THE MATHEMATICS TEACHER
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WHAT TO INCLUDE IN THE PORTFOLIO The items that students include in their portfolios are tied to the general purpose for which it is creat ed. The New Standards portfolio asks students to
give evidence of the various topics they are studying by using (1) investigations and (2) gap fillers. Five
investigations are defined: statistical survey studies, mathematical model of experimental data, designs for physical structures, resource planning and man
aging, and pure mathematical investigations. In Figure 2, Students are asked to include three
different investigations out of the five described. Even though the investigations clearly address
more than one particular standard, other standards or topics are not sampled. When students collaborate with their teachers to identify gaps in the content of their portfolios, they will be asked to include medium and small-sized pieces of work to fill those gaps.
CONCLUSION Portfolio use is a powerful tool that helps students become responsible for their own learning. It stimu lates students' thinking by connecting other areas
of study with their lives outside the classroom. The self-assessment aspect makes students aware of areas that need improvement. Students gain a
deeper understanding of the concepts they are
learning and are able to communicate better math
ematically. Portfolios are instrumental in working with students to meet high expectations and per form to the NCTM's Standards. In short, portfolio use is one of the best venues through which educa tors can ensure mathematical power for all students.
BIBLIOGRAPHY Dar?, P. A Framework for Balance. Oakland, Calif.: The New Standards Project, 1993.
Kuhs, Therese M. "Implementing the Curriculum and Evaluation Standards: Portfolio Assessment: Mak
ing It Work for the First Time." Mathematics Teacher 87 (May 1994):332-35.
Mumme, Judith. Portfolio Assessment in Mathematics. Santa Barbara, Calif.: University of California, 1990.
National Council of Teachers of Mathematics. Cur riculum and Evaluation Standards for School Math ematics. Reston, Va.: The Council, 1989.
-. Professional Standards for Teaching Mathe matics. Reston, Va.: The Council, 1991.
-. Assessment Standards for School Mathematics.
Working draft. Reston, Va.: The Council, 1994.
Petit, Marge. Getting Started: Vermont Mathematics
Portfolio?Learning How to Show Your Best! Cabot, Vt.: Cabot School, 1992.
Stenmark, Jean Kerr, ed. Assessment Alternatives in Mathematics: An Overview of Assessment Techniques That Promote Learning. Berkeley, Calif.: University of California, 1989.
-. Mathematics Assessment: Myths, Models, Good
Questions, and Practical Suggestions. Reston, Va.: National Council of Teachers of Mathematics, 1991. ?|
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