**ERIC Identifier:** ED482725

**Publication Date:** 2003

**Author: **Roh, Kyeong Ha

**Source: **ERIC Clearinghouse for Science Mathematics and Environmental
Education

**Problem-Based Learning in Mathematics. ERIC Digest. **

Problem-Based Learning (PBL) describes a learning environment where
problems drive the learning. That is, learning begins with a problem to
be solved, and the problem is posed is such a way that students need to
gain new knowledge before they can solve the problem. Rather than seeking
a single correct answer, students interpret the problem, gather needed
information, identify possible solutions, evaluate options, and present
conclusions. Proponents of mathematical problem solving insist that students
become good problem solvers by learning mathematical knowledge heuristically.

Students' successful experiences in managing their own knowledge also
helps them
solve mathematical problems well (Shoenfeld, 1985; Boaler, 1998). Problem-based
learning is a classroom strategy that organizes mathematics instruction
around problem
solving activities and affords students more opportunities to think
critically, present their own creative ideas, and communicate with peers
mathematically (Krulik & Rudnick, 1999; Lewellen & Mikusa, 1999;
Erickson, 1999; Carpenter et al., 1993; Hiebert et al., 1996; Hiebert et
al., 1997).

**PBL AND PROBLEM SOLVING**

Since PBL starts with a problem to be solved, students working in a
PBL environment
must become skilled in problem solving, creative thinking, and critical
thinking.
Unfortunately, young children's problem-solving abilities seem to have
been seriously
underestimated. Even kindergarten children can solve basic multiplication
problems
(Thomas et al., 1993) and children can solve a reasonably broad range
of word
problems by directly modeling the actions and relationships in the
problem, just as
children usually solve addition and subtraction problems through direct
modeling.

Those results are in contrast to previous research assumptions that
the structures of
multiplication and division problems are more complex than those of
addition and
subtraction problems. However, this study shows that even kindergarten
children may
be able to figure out more complex mathematical problems than most
mathematics
curricula suggest. PBL in mathematics classes would provide young students
more
opportunities to think critically, represent their own creative ideas,
and communicate
with their peers mathematically.

**PBL AND CONSTRUCTIVISM**

The effectiveness of PBL depends on student characteristics and classroom
culture as
well as the problem tasks. Proponents of PBL believe that when students
develop
methods for constructing their own procedures, they are integrating
their conceptual
knowledge with their procedural skill.

Limitations of traditional ways of teaching mathematics are associated
with

teacher-oriented instruction and the "ready-made" mathematical knowledge
presented
to students who are not receptive to the ideas (Shoenfeld, 1988). In
these
circumstances, students are likely to imitate the procedures without
deep conceptual
understanding. When mathematical knowledge or procedural skills are
taught before
students have conceptualized their meaning, students' creative thinking
skills are likely
to be stifled by instruction. As an example, the standard addition
algorithm has been
taught without being considered detrimental to understanding arithmetic
because it has
been considered useful and important enough for students to ultimately
enhance
profound understanding of mathematics. Kamii and Dominick(1998), and
Baek (1998) have shown, though, that the standard arithmetic algorithms
would not benefit

elementary students learning arithmetic. Rather, students who had learned
the standard
addition algorithm seemed to make more computational errors than students
who never learned the standard addition algorithm, but instead created
their own algorithm.

**STUDENTS' UNDERSTANDING IN PBL ENVIRONMENT**

The PBL environment appears different from the typical classroom environment
that
people have generally considered good, where classes that are well
managed and
students get high scores on standardized tests. However, this conventional
sort of
instruction does not enable students to develop mathematical thinking
skills well.

Instead of gaining a deep understanding of mathematical knowledge and
the nature of mathematics, students in conventional classroom environments tend to
learn inappropriate and counterproductive conceptualizations of the nature
of mathematics. Students are allowed only to follow guided instructions and to obtain
right answers, but not allowed to seek mathematical understanding. Consequently, instruction
becomes focused on only getting good scores on tests of performance. Ironically,
studies show that students educated in the traditional content-based learning environments
exhibit lower achievement both on standardized tests and on project tests dealing
with realistic situations than students who learn through a project-based approach
(Boaler, 1998).

In contrast to conventional classroom environments, a PBL environment
provides students with opportunities to develop their abilities to adapt and
change methods to fit new situations. Meanwhile, students taught in traditional mathematics
education environments are preoccupied by exercises, rules, and equations that
need to be learned, but are of limited use in unfamiliar situations such as project
tests. Further, students in PBL environments typically have greater opportunity to
learn mathematical processes associated with communication, representation, modeling,
and reasoning (Smith, 1998; Erickson, 1999; Lubienski, 1999).

**TEACHER ROLES IN THE PBL ENVIRONMENT**

Within PBL environments, teachers' instructional abilities are more
critical than in the
traditional teacher-centered classrooms. Beyond presenting mathematical
knowledge to students, teachers in PBL environments must engage students
in marshalling
information and using their knowledge in applied settings.

First, then, teachers in PBL settings should have a deep understanding
of mathematics
that enables them to guide students in applying knowledge in a variety
of problem
situations. Teachers with little mathematical knowledge may contribute
to student failure in mathematical PBL environments. Without an in-depth
understanding of mathematics, teachers would neither choose appropriate
tasks for nurturing student problem-solving strategies, nor plan appropriate
problem-based classroom activities (Prawat, 1997; Smith III, 1997).

Furthermore, it is important that teachers in PBL environments develop
a broader range of pedagogical skills. Teachers pursuing problem-based
instruction must not only supply mathematical knowledge to their students,
but also know how to engage students in the processes of problem solving
and applying knowledge to novel situations. Changing the teacher role to
one of managing the problem-based classroom environment is a challenge
to those unfamiliar with PBL (Lewellen & Mikusa, 1999). Clarke (1997),
found that only teachers who perceived the practices associated with PBL
beneficial to their own professional development appeared strongly positive
in managing the classroom instruction in support of PBL.

Mathematics teachers more readily learn to manage the PBL environment
when they
understand the altered teacher role and consider preparing for the
PBL environment as
a chance to facilitate professional growth (Clarke, 1997).

**CONCLUSIONS**

In implementing PBL environments, teachers' instructional abilities
become critically
important as they take on increased responsibilities in addition to
the presentation of
mathematical knowledge. Beyond gaining proficiency in algorithms and
mastering
foundational knowledge in mathematics, students in PBL environments
must learn a
variety of mathematical processes and skills related communication,
representation,
modeling, and reasoning (Smith, 1998; Erickson, 1999; Lubienski, 1999).
Preparing
teachers for their roles as managers of PBL environments presents new
challenges
both to novices and to experienced mathematics teachers (Lewellen &
Mikusa, 1999).

**REFERENCES**

Boaler, J. (1998). Open and closed mathematics: student experiences
and

understandings. "Journal for Research on Mathematics Education," 29
(1). 41-62.

Carpenter, T., Ansell, E. Franke, M, Fennema, E., & Weisbeck, L.
(1993). Models of problem solving: A study of kindergarten children's problem
solving processes. "Journal for Research in Mathematics Education," 24
(5). 428-441.

Clarke, D. M. (1997). The changing role of the mathematics teacher.
"Journal for
Research on Mathematics Education," 28 (3), 278-308.

Erickson, D. K. (1999). A problem-based approach to mathematics instruction."Mathematics Teacher," 92 (6). 516-521.

Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K., Human, P., Murray,
H., Olivier, A., & Wearne, D. (1996). Problem solving as a basis for
reform in curriculum and instruction: The Case of Mathematics. "Educational
Researcher," 12-18.

Hiebert, J. Carpenter, T. P., Fennema, E., Fuson, K., Human, P., Murray,
H., Olivier, A., & Wearne, D. (1997). Making mathematics problematic:
A rejoinder to Prawat and Smith. "Educational Researcher," 26 (2). 24-26.

Krulik, S., & Rudnick, J. A. (1999). Innovative tasks to improve
critical- and

creative-thinking skills. In I. V. Stiff (Ed.), "Developing mathematical
reasoning in grades K-12." Reston. VA: National Council of Teachers of
Mathematics. (pp.138-145).

Lewellen, H., & Mikusa, M. G. (February 1999). Now here is that
authority on
mathematics reform, Dr. Constructivist! "The Mathematics Teacher,"
92 (2). 158-163.

Lubienski, S. T. (1999). Problem-centered mathematics teaching. "Mathematics
Teaching in the Middle School," 5 (4). 250-255.

Prawat, R. S. (1997). Problematizing Dewey's views of problem solving:
A reply to
Hiebert et al. "Educational Researcher." 26 (2). 19-21.

Schoenfeld, A. H. (1985). "Mathematical problem solving." New York:
Academic Press.

Smith, C. M. (1998). A Discourse on discourse: Wrestling with teaching
rational
equations. "The Mathematics Teacher." 91 (9). 749-753.

Smith III, J. P. (1997). Problems with problematizing mathematics: A
reply to Hiebert et al. "Educational Researcher," 26 (2). 22-24.

**SELECTED ERIC RESOURCES**

The ERIC database can be electronically searched online at:

http://www.eric.ed.gov/searchdb/index.html. To most effectively find
relevant items in
the ERIC database, it is recommended that standard indexing terms,
called "ERIC
Descriptors," be used whenever possible to search the database. Both
"problem based
learning" and "problem solving" are ERIC descriptors, so these would
be good terms to use in constructing an ERIC search. Following are some
sample items that are included in the ERIC database:

Delisle, R. (1997) "How to use problem-based learning in the classroom."
Alexandria,
VA: Association for Supervision and Curriculum Development. [ED 415
004]

This book shows classroom instructors how to challenge students by
providing them with a structured opportunity to share information, prove their knowledge,
and engage in independent learning.

Ulmer, M. B. (2000). "Self-grading: A simple strategy for formative
assessment in
activity-based instruction." Paper presented at the Conference of the
American
Association for Higher Education, Charlotte, NC. [ED 444 433]

This paper discusses the author's personal experiences in developing
and implementing
a problem-based college mathematics course for liberal arts majors.
The paper
addresses concerns about increased faculty workload in teaching for
critical thinking
and the additional time required for formative assessment.

van Biljon, J. A., Tolmie, C. J., du Plessis, J. P.. (1999, January).
Magix-An ICAE
System for problem-based Learning. "Computers & Education," 32
(1), 65-81. [EJ 586 410]

Discussion focuses on Magix, a prototype ICAE system for use in problem-based
learning of linear mathematics for 10- to 12-year olds. The system
integrates the
principles of constructivism, user-driven interaction, knowledge-based
systems, and
metacognition.

Erickson, D. K. 1999, September). A problem-based approach to mathematics
instruction. "Mathematics Teacher," 92 (6), 516-21. [EJ 592 083]

This article describes preparation for instruction using a problem-based
approach as
part of a teaching-strategy repertoire. Suggestions of ways that mathematics
teachers
can get assistance in successfully implementing a problem-based teaching
approach
are included. Research results indicate what students are likely to
accomplish in such
classes.