Let's Measure What's Worth Measuring
Education reform revolves around three central issues: what children should learn, how they should be taught, and how progress should be measured. Ask people in the education community which of those three areas is the least developed, and my suspicion is that most would answer the last. In contrast to curriculum and teaching standards, the state of the art for assessment standards is primitive and the needed departures from past practice profound.
Mathematics education is no exception. The National Council of Teachers of Mathematics has articulated and built support for some of the most expert and enlightened thinking about curriculum and teaching standards. But in assessing achievement, the mathematics community is still in a period of intense and diverse experimentation governed by new and yet to be fully understood principles.
Curriculum, teaching, and assessment must be integrated and mutually supportive elements in the educational process, each serving common goals. Assessment, in particular, must cease to be an autonomous enterprise that is designed principally for efficiency and economy of administration without sufficient regard for the significance of what is measured or its contribution to students' learning.
In mathematics as in other disciplines, the new and enlightened aims of assessment can be captured in three broad educational principles, as spelled out in a recent publication of the Mathematical Sciences Education Board, "Measuring What Counts.''
THE CONTENT PRINCIPLE: Assessment should reflect themathematics that is most important for students to learn.
THE LEARNING PRINCIPLE: Assessment should support good instructional practice and enhance mathematics learning.
THE EQUITY PRINCIPLE: Assessment should support every student's opportunity to learn important mathematics.
Despite their benign appearance, these principles contain the seeds of revolution. Few assessments given to students in America today reflect any of these vital principles. But they are essential if educational reform is to succeed.
The crucial role of assessment is captured by the folk wisdom that you get what you assess and you do not get what you do not assess. Despite common rhetoric to the contrary, both students and teachers take their cues about what is most important and most valued from what is assessed. Topics on the test are what teachers most seriously teach and what students most seriously try to learn. In these times, with so many practical, material, and emotional pressures, such educational pragmatism is all the more pronounced.
Assessment reform does not scorn such attitudes as cynical or unwholesome. Quite the contrary, reformers recognize in them the expression of the powerful leverage that assessment exercises over the educational process. The aim of assessment reform is to tap the very power that in the past has deformed and subverted educational aims, and to make it now an engine that propels and guides education reform. The three assessment principles--on content, learning, and equity--express the ways in which this may be accomplished.
- Content. Efficiency and economy of administration coupled with inappropriate application of psychometric notions have produced assessment instruments far removed from central issues of education. Mathematics tests are dominated by multiple-choice questions, focused on tasks involving routine algorithmic procedures, and limited in available time for solution. Such instruments clearly have the capability of discerning only a very narrow range of knowledge and skills. Such tests are structurally incapable of probing the full range of a student's mathematical abilities.
The standards for school mathematics emphasize that students must acquire the capabilities to cope with open-ended problems, to analyze and interpret data, to construct and communicate substantial webs of reasoning, to formulate questions and conjectures, and to carry out extended projects both alone and in groups. Unless these kinds of knowledge and skills are consistently and convincingly sought in assessment, these laudable goals will be dismissed as empty rhetoric.
- Learning. If assessment is going to support learning, then assessment tasks must provide genuine opportunities for all students to learn. Too often a sharp line has been drawn between assessment and instruction. Teachers teach, then instruction stops and assessment occurs.
In mathematics the curricular goals inherently require a greatly enriched and more diverse kind of teaching. Students must have opportunities to construct their own mathematical knowledge, and assessment must be capable of monitoring whether this construction is taking place. The best way to provide such opportunities is through assessment tasks that resemble learning tasks in that they promote strategies such as analyzing data, drawing contrasts, and making connections. Thus assessment in mathematics must not only support revitalized teaching, but also become more integrated with instructional practice.
The results of assessment should yield information that can be used to improve students' access to subsequent mathematical knowledge. The results must be timely and clearly communicated to students, teachers, and parents. School time is precious. When students are not informed of their errors and misconceptions, let alone helped to correct them, the assessment may both reinforce misunderstandings and waste valuable instructional time.
- Equity. What distinguishes the present reform movement from all precedents is its truly revolutionary premise that all students can and must learn. Assessments should be used to determine what students have learned and what they still need to learn. They should never be used to filter students out of educational opportunity.
The idea that some students can learn mathematics and others cannot must end. Mathematics is not reserved for the talented few, but is required of all to live and work in the 21st century. The economic and social forces that give education reform its high national priority demand that the benefits of reform must effectively reach those populations--women and minorities--that represent the greatest infusion into the workforce and for whom education in mathematics and science has historically been least successful. (See the related Commentary on page 23.)
This poses a formidable challenge in the design of assessment instruments. They must not operate as a cultural filter, as has previously been the case. Instead, they must be sensitively crafted to accommodate diverse forms of authentic communication and they should assess only what students have had a fair opportunity to learn.
Equity implies that every student must have an opportunity to learn the important mathematics that is assessed. Students who have experience reflecting on the mathematics they are learning, presenting and defending their ideas, or organizing, executing, and reporting on a complex piece of work will be better prepared when called upon to do so in an assessment situation. Especially when assessments are used to make high-stakes decisions on matters such as graduation and promotion, the equity principle must be honored. Students cannot be assessed fairly on mathematics that they have not had an opportunity to learn.
The principles of content, learning, and equity place quite extraordinary demands on assessment reform. While the desired outcomes are clearly spelled out, the effective means of achieving them entail a host of largely unanswered questions. These are the arena of a grand and exciting experiment that will passionately engage students, teachers, parents, politicians, and the public over the next decade.
Hyman Bass is a professor of mathematics at Columbia University and the chairman of the Mathematical Sciences Education Board of the National Research Council.