Arthur E. Levine, president of Teachers College at Columbia University, recently lectured at the University of Georgia on "The Remaking of the American University." He predicted that as a result of increased educational opportunities, each student will develop an educational "passport," a portfolio, that records ones own varied educational experiences, replacing the traditional diploma. The purpose of this portfolio is to provide a clearer and more detailed explanation of the talents and accomplishments acquired throughout the educational career of each student. In the discipline of mathematics, portfolios provide an authentic avenue for achievement in compliance with the NCTM Standards. The contents of the portfolio dictate the extent to which these goals and standards are met.
NCTM is on the eve of releasing an updated comprehensive set of standards that are guided by the mission to prepare students for the broad educational and career opportunities available in the 21st century. Mathematicians, educators, administrators, business leaders, and all interested individuals have been encouraged to review and provide suggestions to the rough draft of the standards, which has been available online and in document form since 1998. A consensus has been reached that the mathematical needs of all students are rapidly changing. The United States is now officially an "information society"; therefore, the capacity to utilize information and the technological tools used in obtaining, analyzing, manipulating, and dispensing it has become our most important commodity. The growth of the information economy has led to an expansion in choices available, demanding an increase in reasoning and problem solving skills for individuals to make informed, educated decisions. The powers of analysis and of evaluation are essential skills in todays society. NCTM and those professionals that have chosen to participate in the development of the new standards feel that these important "life" skills must play a vital role within the mathematics curriculum.
NCTM has identified ten main student achievement standards, three of which are communication, connections and representation. A portfolio is defined as "the purposeful collection of student work that exhibits the students efforts, progress, and achievements. The collection must include student participation in selecting [the] contents, the criteria for selection, the criteria for judging merit, and evidence of student self-reflection" (Lankes 1). By definition, a portfolio addresses these standards in a holistic sense, developing students that actively engage in and take personal responsibility for their mathematics education.
The communication standard emphasizes the ability of students to organize, consolidate and express their mathematical thinking to others using mathematical language that is personally understood. Portfolios accomplish the objectives of this standard as the students create a mathematical resume of work that has been completed through the year. Portfolios not only require students to choose a particular piece of work, but also to reflect meaningfully on why they chose the work, so that they might prepare a clear presentation based upon the prescribed criteria. This provides students with the opportunity to contemplate material that is "worth documenting either because the ideas are important, or because the work has come a long way" (Wolf 2).
An additional aspect of the communication standard is for students to extend their mathematical knowledge through observations and interactions with others. Portfolios can provide the forum for this communication through teacher, peer and self evaluations. These help create and maintain an atmosphere that is conducive to the exchange of ideas in better understanding and solving problems. Portfolios help students to "reflect on what makes some work better, and use this information to make improvements in future work" (Sweet 3). The process of choosing and developing a work into a portfolio piece directly focuses on the goal of improving mathematical communication.
The connections standard emphasizes the ability of students to recognize that mathematical concepts behave in two important but different ways. Mathematical ideas build upon one another, from simple to complex. They also are connected in a network schema. The ability to visualize the relatedness of concepts leads to the ability to use a variety of methods to solve problems. The reflection and evaluation process inherent in the use of portfolios helps to stimulate critical thinking. Students must look for and use mathematical connections in order to provide a clear and concise explanation of a work and why it was selected to be a part of the portfolio. "Portfolios present the opportunity for a student to learn about learning" (Coates 2). Through the development of a portfolio, students learn to actively seek, understand, and use connections within mathematics.
The connections standard also refers to the ability of students to recognize, use, and learn about mathematics in a broader context than just formal mathematics. Portfolios should be structured to place emphasis on exploring these connections. For example, an aspect of each entry could be for students to research and explain how the mathematical concept models a real-world case. Alternatively, the portfolio could contain one entry that has the criteria to explore the connection between a mathematical idea and various real-world problems. A natural aspect of a portfolio is for students to demonstrate their knowledge of the connection between the mathematics they have learned and the world around them.
The representation standard focuses on the ability of students to create and use a variety of representations to model, interpret and communicate mathematical ideas and phenomena. If students are expected to learn mathematics to this level, it is necessary that assessments are designed to determine the proficiency to which students have achieved this goal. No traditional test can truly provide authentic assessment of this standard. Project-type work is essential for establishing the significance of mathematical representation for the students, but this work should be encompassed into a portfolio for full development. Portfolios are a "clear departure from the old write, hand in, and forget mentality, where first drafts were considered final products" (Sweet 1). Students must use critical thinking skills in determining how best to represent the mathematical work of an entry, through writing, tables, charts, diagrams, models, graphs, or some combination of methods. Producing a portfolio allows a student to evaluate how former entries could be better represented and suggest ways that future entries might be best represented.
With the wide array of dynamic software programs that facilitate mathematical learning, the NCTM Standards of communication, connections and representation can be more effectively addressed with an electronic, rather than traditional, portfolio. The rapid development of technology has made this increasingly possible and when appropriate, students should be strongly encouraged, and at times required, to build an electronic portfolio. National Public Radio recently reported the statistic that approximately ninety percent of schools are connected to the Internet. This trend will only continue; it is only matter of time before every classroom, and then every student, is connected to the Internetat least while at school. Teachers can have students use a basic web page editor that is provided with many computer software packages, or "there are several commercially available portfolio programs which offer teachers the ability to track student achievement" (Lankes 2) while providing platforms for web page development. Additionally, providing students with the opportunity to become proficient users of a variety of software applications better prepares them for their post-secondary education and future careers.
In order for students to truly reach the goals that have been set by the NCTM Standards, portfolios must become an important assessment tool. Basic skills must provide the foundation for mathematics literacy, but students must also be able to use and extend basic skills in meaningful ways. A portfolio is an effective vehicle to drive student mathematical understanding beyond the superficial, allowing students to work with mathematical concepts in accordance with the NCTM Standards.
Research shows that good portfolios place additional demands on teachers, students and school resources (Sweet 3). Therefore, it is important to know how effective portfolio programs have been. Relatively little research has been done in the specific area of mathematics portfolios; however, studies show that students benefit from an awareness of the processes and strategies involved in writing, problem solving, researching a topic, analyzing information, and describing their own observations. If students do not have assessments that authentically evaluate the processes and strategies that underlie effective performance of these types of work, most students will not learn them or will only minimally learn them. Also, without curriculum-specific experience in using critical thinking processes and strategies, students will experience little success in carrying these skills forward into new and appropriate contexts (Sweet 3). Vermont is considered to be a leader in portfolio assessment, with a program that focuses on the areas of writing and mathematics. Results indicate that portfolios have brought about significant improvement in teaching practices (Goals 3). It is clear that more research needs to be completed concerning mathematics portfolios, but it also seems clear that initial portfolio programs have had positive effects. In order for students to truly meet the NCTM Standards, teachers and schools need to willingly accept the demands of portfolio assessment. It is the only way that we can expect students to willingly accept the demands required by the critical thinking that allows students to achieve future success in an ever more demanding information society.
Coates, Diane. "Alternative Assessments to Reflect a Changing Mathematics Curriculum." (2/12/00). December 1995. http://www.frontiernet.net/~dcoates/altass.htm.
"Goals 2000: Reforming Education to Improve Student Achievement." (4/30/98). Department of Education. (2/10/00). http://www.ed.gov/pubs/G2Kreforming/g2ch3.html .
Lankes, Anna Maria D. "Electronic Portfolios: A New Idea in Assessment." (2/11/00) ERIC Digest. http://www.ed.gov/databases/ERIC_Digests/ed390377.html .
Paulson, L.F., Paulson P.R. & Meyer C. (1991) "What makes a portfolio a portfolio?" Education Leadership, 48(5), 60-63 qtd. in Anna Maria D. Lankes, "Electronic Portfolios: A New Idea in Assessment." (2/11/00) ERIC Digest. http://www.ed.gov/databases/ERIC_Digests/ed390377.html .
"Principles and Standards for School Mathematics: Electronic Version, Discussion Draft 1998." (1/15/1999). National Council for Teachers of Mathematics. (2/10/00). http://standards-e.nctm.org/1.0/normal/standards/standardsFS.html.
Sweet, David. "Student Portfolios: Classroom Uses." (2/12/00). Office of Research Education Consumer Guide. November 1993. http://www.ed.gov/pubs/OR/ConsumerGuides/classuse.html .
Wolf, Dennie Palmer, Paul G. LeMahieu and JoAnne Eresh. (2/12/00). "Evaluating Middle Schoolers Effectively." Education Digest on GALILEO database. Oct 92, Vol. 58 Issue 2, pp. 13-17. Internet pp. 1-5. http://www.galileo.peachnet.com .
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