Problem-Based Learning in Mathematics
ERIC Identifier: ED482725
Publication Date: 2003-00-00
Author: Roh, Kyeong Ha
Source: ERIC Clearinghouse for Science Mathematics and Environmental Education Columbus OH.
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).
This digest was funded by the Office of Educational Research and Improvement, U.S.
Department of Education, under contract no. ED-99-CO-0024. Opinions expressed in
this digest do not necessarily reflect the positions or policies of OERI or the U.S.
Department of Education. ERIC Digests are in the public domain and may be freely reproduced.
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.
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