Problem-based Learning in Mathematics

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-based Learning in Mathematics

Kyeong Ha RohApril 2003

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 SolvingSince 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 ConstructivismThe 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 DIGEST

EDO-SE-03-07considered 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 EnvironmentThe 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 EnvironmentWithin 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