ERIC/CSMEE

A Student Watcher's Guide to Performance in Mathematics David L. Haury May 1999 (Updated June 2003)

ERIC

Clearinghouse for Science, Mathematics, and Environmental Education

DIGEST

After all is said and done with international comparisons of student achievement, national education goals, and national standards for mathematics education, most parents and teachers simply want to know how their children are doing in class. Are they learning? Do they know what they are supposed to know? Are they ready for the mathematics in the next grade or class? Will they pass the state proficiency examinations? These are healthy concerns, and schools are becoming better at assessing student performance in ways that provide more helpful feedback to those wanting to help students perform at their best.

How To Know What You Are Seeing

Interpreting grades and test results seemed easier in the traditional math classrooms of years past. After each unit, students would take a test, usually with rows of math problems or pages of true-or-false and multiple-choice items, and they would earn some number of points for choosing the correct answers. Scoring 90 out of 100 points was very good; scoring 49 out of 100 was not. At the end of the grading period, the points earned on tests, homework, and other assignments would be added, percentages would be calculated, and grades would be assigned. Assessment was somewhat narrow in focus-with an emphasis on knowing certain important facts, or rules and procedures-and deceptively straight-forward: you either mastered the content or you didn't. How you got the right answers-whether by actually knowing, memorizing, guessing, or being lucky-was seldom examined.

Though it is still considered important for students to know fundamental concepts, rules, and procedures, national and state standards place increasing emphasis on active learning and the cognitive skills associated with critical thinking, solving open-ended problems, inquiry, and decision making. Assessing performance in these areas requires alternatives to traditional forms of assessment, and the array of assessment strategies being used in today's classrooms can be confusing. Furthermore, there is also more of an attempt these days to embed assessment into the curriculum in ways that provide students, teachers, and parents with information about how to improve performance.

So, one of the first questions a student watcher must answer is, "What are we trying to see?" For mathematics, state standards and local programs are being strongly influenced by the national standards formulated by the National Council of Teachers of Mathematics (NCTM, 1989), standards that are being revised for the year 2000. Along with increased attention to some content areas, such as estimation and statistics, the math standards emphasize attention to problem solving, critical thinking and reasoning, and expressing mathematical ideas through words, pictures, diagrams, and other forms of communication.

In addition to the assessment activities currently in place, many schools will soon become involved in the Voluntary National Tests (VNT) being developed by the National Academy of Sciences under Public Law 105-78. The VNT will include an 8th grade mathematics test of proficiency. The content strands will include: (a) number sense, properties, and operations; (b) measurement; (c) geometry and spatial sense; (d) data analysis, statistics, and probability; and (e) algebra and functions. Test items will conform to objectives of the National Assessment of Educational Progress (NAEP) framework, and they will assess conceptual understanding, procedural knowledge, problem solving ability, mathematical reasoning, ability to communicate conclusions, and the ability to connect mathematical ideas in one context (algebra, for instance) to ideas in another context (geometry, for instance).

Given the recent changes in both the use of assessment activities to guide learning, and the increased spectrum of knowledge, skills, and habits of mind to be learned in math, classroom assessments are becoming more varied and complex. Student watchers must look for more than the bottom line of how many points were scored on a test; they must look for the links between various forms of classroom assessment and the broad array of learning outcomes sought.

Box 1: Forms of Performance Assessment

Unlike traditional tests where students select answers from a set of alternatives, performance assessments require students to perform a task, generate their own responses to questions, or create products that demonstrate both their knowledge and their cognitive or procedural skills. Here are some of the more common forms of performance assessment being used.

Computer Adaptive Testing

Student responses to questions presented by a computer allow testing to be adjusted to a student's ability level.

Concept Mapping

A structured method for presenting ideas and conceptual relationships in pictorial form. (See http://users.edte.utwente.nl/lanzing/cm_home.htm)

Constructed-Response Questions

Students construct their own answers rather than select from a set of possible answers.

Essays

Essays allow a student to demonstrate his or her ability to describe, analyze, explain, or summarize ideas or events.

Experiments or Investigations

Experiments generally require students to plan and conduct research, test hypotheses, use skills of measurement and estimation, and report findings orally or in written form.

Interviews

Students respond to verbal questions, sometimes posed according to a standard protocol.

Observations

A student performs a task or procedure while being observed and rated using a rubric, an agreed-upon set of scoring criteria.

Portfolios

Collections of student work representing a selection of products or works-in-progress. Items that may be included in math portfolios have been listed in an ERIC Digest, Portfolio Assessment in Mathematics Education (http://www.ericse.org/digests/dse98-2.html).

For more information, see http://www.ed.gov/pubs/OR/ConsumerGuides/classuse.html or http://www.ncrel.org/. Elementary and middle-school teachers may also find the following resource useful: Jorgensen, M. (1996). Rethinking Portfolio Assessment: Doc-umenting the Intellectual Work of Learners in Science and Mathematics. Columbus, OH: ERIC/CSMEE Publications.

Projects (Individual or Group)

Projects usually require a broad range of skills, are often interdisciplinary, and are completed over an extended period of time. Group projects require students to work collaboratively in teams that plan, discuss, prepare, and present their findings or product. Projects are often evaluated using a rubric that informs participants of the expected standards before they begin.

Station Activities

Students move from station to station, individually or in groups, performing a sequence of tasks during a prescribed period of time. Open-ended questions may be used to elicit specific thinking skills or process skills.

Online Sources of Additional Information:

o Assessment Alternatives (http://www.nwrel.org/nwedu/fall_96/article9.html).

o Authentic Assessment in Mathematics (http://mathforum.org/sum94/project2.html/).

o Performance assessment in the Third International Mathematics and Science Study (TIMSS) (http://labnet.terc.edu/archive/ra-ed-reform/00000002.htm).

o ERIC Digest: Creating Meaningful Performance Assessments (http://ericae.net/db/edo/ED381985.htm).

What New Forms of Assessment Can I Expect to See?

Various terms have been used to describe the newer forms of classroom assessment, from alternative assessment and authentic assessment to performance-based assessment. Though there may be subtle nuances in the meanings of the terms, what they all connote is a break with past assessment practices. As schools implement national standards and seek a broader spectrum of learning outcomes, assessment activities are becoming more diversified. The first sign of change, then, that parents and other student watchers will notice is the increased use of less traditional forms of assessment. A sampling of what might be noticed is presented in Box 1. Parents will see an increased emphasis on "products" that will be scored according to "rubrics" (see Box 2), and there will likely be increasing evidence of conformity to the NCTM content standards.

Box 2: What is a Rubric?

A rubric is a set of scoring criteria that a teacher or other person uses to evaluate responses to a performance assessment task. Since most performance tasks require open-ended responses or individual initiative, a scoring rubric ensures that the responses of different students will be judged on the same merits in a fair and consistent way. As an example, here is a simple rubric for assigning points to a portfolio:

Portfolio Rubric

10 Portfolio well organized and documented. Format is accurate, complete, and easy to follow. Excellent quality; well designed portfolio; indicates superior effort.

8 Portfolio fairly well organized and documented. Format is mostly accurate, complete, and easy to follow. High quality; clear design; indicates excellent effort.

6 Portions of the portfolio are poorly organized or inaccurately documented. Not complete or possibly somewhat difficult to follow. Average quality; adequate design; indicates acceptable effort.

4 Portfolio unorganized, poorly docu-mented, significantly incomplete or inaccurate. Difficult to follow. Poor quality; little or no design; indicates insufficient effort.

2 Portfolio incomplete, incorrect, or inadequate. Work indicates little or no effort.

0 Portfolio not submitted.

Note. This is a simple example to illustrate how scoring rubrics are used. The actual rubrics used in classrooms may be consider-ably more detailed. For more information about rubrics, see the following website hosted by The Discovery Channel: http://discoveryschool.com/schrockguide/assess.html.

What Can I Do to Help a Child Do Their Best on Assessments?

Parents and others wanting to interpret feedback from classroom assessments in mathematics may want to begin by looking for evidence of the three qualities identified as being essential for the productive use of assessments: validity, fairness, and credibility (Land, 1997). Valid assessment measures are well aligned with classroom learning objectives, the state curriculum framework, and national standards for mathematics. Parents, then, can help students by becoming familiar with local and national curriculum standards, and helping students focus on topics and skills emphasized by the standards.

Fairness is a serious, but difficult, issue. Math reform efforts in recent years have focused on serving the needs of all students. Yet, there are some inherent difficulties in fairly assessing performance when students come to the classroom with very different life histories and experiences: different interests and expectations, different learning styles, different levels of proficiency with the English language, different economic and ethnic identities, and different abilities and disabilities. Parents and others can help by being alert to difficulties that individual students may be having as a result of these situational factors, and by taking note of reported differences in performance by identifiable groups. Though there will be a broad spectrum of performance levels among members of any particular group, concerns should be raised if members of a particular group are systematically disadvantaged by the nature of a particular assessment activity.

Finally, assessment activities should seem credible to you, whether you are a teacher, parent, or public administrator. It should be expected that the purposes of each assessment can be clearly and openly explained to you, and there should be a public forum for voicing any concerns you may have regarding student assessments, and for providing constructive input as practices are refined. For other questions to consider in examining the quality of classroom assessments, please see "What does research say about assessment?" at http://www.ncrel.org/ncrel/sdrs/areas/stw_esys/4assess.htm.

Resources

Land, R. (1997). Moving up to complex assessment systems. Evaluation Comment, 7(1), 1-21. (Available online through http://www.cse.ucla.edu/)

National Council of Teachers of Mathematics. (1989). Curriculum and Evaluation Standards for School Mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. Reston, VA: Author (Available online http://standards.nctm.org/)

Content Knowledge: The McREL Standards Database (Online at http://www. mcrel.org/standards-benchmarks/) This website provides an extensive compendium of K-12 standards for all subject areas. Included are links to standards-based instructional activities and other standards-related resources.

For more information about testing in general, visit the ERIC Clearinghouse on Assessment and Evaluation, online at http://ericae.net/.

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This digest was funded by the Office of Educational Research
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Opinions expressed in this digest do not necessarily reflect the positions
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