Department of Mathematics Education

Dr. J. Wilson, EMAT 6690

**The Importance of Using Portfolios****within the Mathematics Curriculum**by David Wise

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 one’s 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 21^{st} 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 today’s 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 student’s 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 Internet—at 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**
.

If you have any comments that would be useful, especially for
use at the high school level, please send e-mail to **esiwdivad@yahoo.com**.

**Return**
to my homepage.