V.3 #4 Mathematics - Math Problem Solving
Solving real world word problems in math is one of the harder topics for students to learn regardless of ability or grade level. Word problems typically require students to comprehend information presented in a collection of sentences or embedded within a paragraph or story structure. Ideally, students develop a plan on how to solve the problem by identifying relevant information and appropriate calculations that lead to a solution. The level of difficulty in solving word problems is influenced by the specific wording of the problem (i.e., length, grammatical and semantic complexity, and the order of statements) (Briars & Larkin, 1984; Helwig, Rosek-Toedesco, Tindal, Heath, & Almond, 1999) the extent to which the problem is embedded in contextually realistic situations, the presence of extraneous information (Palmer, Cawley, & Frazita, 1996), and the number and type of calculations required to solve the problem.
Various approaches for solving word problems have been described in a previous Strategies for Successful Learning column (Agaliotis, 2008), and a collection of studies by Fuchs, Fuchs, and colleagues (Fuchs and Fuchs, 2007). It is the findings of the studies by Fuchs and Fuchs that will be discussed here. The Fuchs describe math problem solving (MPS) as being situations found in real world settings where extraneous information may be present, or the student may need to look up information needed to solve a problem, or where multiple calculations are required to obtain a solution. Fuchs and Fuchs examined whether teaching students to solve these types of problems using scheme theory (Fuchs and colleagues (2003a), metacognitive skills (Fuchs, and colleagues, 2003b), and in small group settings Fuchs and colleagues (2002) improves their MPS skills.
Fuchs, Fuchs, and colleagues found that teaching students to use schema theory, where they group problems into types that require the same solution (Chi, Feltovich, & Glaser, 1981; Gick & Holyoak, 1980; Mayer, 1992; Quilici & Mayer, 1996), and providing students with explicit instruction in transfer were both effective for most students to be able to solve transfer problems. For transfer to take place students must master rules for problem solving, develop categories for sorting problems that require similar solutions, and be aware that novel problems are related to previously solved problems (Cooper & Sweller, 1987). Fuchs and colleagues (2003a) found that using schema theory was effective for teaching MPS for students with and without math disabilities, but only students without math disabilities were able to transfer these skills to novel problem solving situations. A second study (Fuchs, and colleagues, 2003b) introduced the instruction of self-regulated learning strategies (e.g., goal setting and self-evaluation) in addition to the schema-based instruction for problem solving situations and transfer and found the combination of instructional strategies to have a positive effect for both students with and without math disabilities. In the final study, Fuchs and colleagues (2002) introduced the use of small group tutoring for schema-based instruction for students with math disabilities. It was the use of the small group instruction that was found to be most effective for students with math disabilities to be able to successfully engage in MPS and transfer these skills to solving novel problems. Thus small group instruction which facilitates opportunities to respond, seek clarification, and obtain guided feedback is more effective for students with math disabilities than whole-class instruction (Fuchs & Fuchs, 2007).
Overall, the series of studies by Fuchs and Fuchs (2007) indicate that students as young as 8 or 9 years old with or without math disabilities can benefit from MPS instruction, even when their calculation or foundational math skills are low or immature. Second, students need to master problem-solution methods on problems with low transfer demands in order to build the skills necessary to solve novel problems correctly. Finally, there is a need for explicit instruction on transfer that is designed to increase students’ awareness of the connections between novel and familiar problems. Although students may find learning how to solve word problems challenging, instruction that is delivered in small groups and incorporates schema theory and components of self-regulation is effective for student with math disabilities.
References
Agaliotis, I. (2008, March). Strategies for Successful Word Problem Solving by Students with Learning Disabilities. Strategies for Successful Learning, 1(4).
Briara, D. J., & Larkin, J. H. (1984). An integrated model of skill in solving elementary word problems. Cognition and Instruction, 1, 245-296.
Chi, M. T., H., Feltovich, P. J., & Glaser, R. (1981). Categorization and representation of physics problems by experts and novices. Cognitive Science, 5, 121-152.
Cooper, G., & Sweller, J. (1987). Effects of schema acquisition and rule automation on MPS transfer. Journal of Educational Psychology, 79, 347-362.
Fuchs, L. & Fuchs, D. (2007). Mathematical problem solving: Instructional interventions. (pp. 397-414). In D. B. Berch, M. M. M. Mazzocco. Why is Math so Hard for Some Children? The Nature and Origins of Mathematical Learning Difficulties and Disabilities. Baltimore, MD: Brooks Cole Publishing Co.
Fuchs. L. S., Fuchs, D., Hamlett, C. L., & Appleton, A. C. (2002). Explicitly teaching for transfer in small groups: Effects on the mathematical problem-solving performance of student with mathematics disabilities. Learning Disability Research and Practice, 17, 90-106.
Fuchs, L. S., Fuchs, D., Prentice, K., Burch, M., Hamlett, C. L., Owen, R., et al. (2003a). Enhancing third-grade students’ mathematical problem solving with self-regulated learning strategies. Journal of Educational Psychology, 95, 306-315.
Fuchs, L. S., Fuchs, D., Prentice, K., Burch, M., Hamlett, C. L., Owen, R., et al. (2003b). Explicitly teaching for transfer: Effects on third-grade students’ mathematical problem solving. Journal of Educational Psychology, 95, 293-304.
Gick, M. L., & Holyoak, K. J. (1980). Analogical problem solving. Cognitive Psychologist, 12, 306-355.
Helwig, R., Rosek-Toedesco, M. S., Tindal, G., Heath, B., & Almond, P. J. (1999). Reading as an access to mathematics problem solving on multiple-choice tests for sixth-grade students. Journal of Educational Research, 93, 113-125.
Mayer, D. P. (1998). Do new teaching standards undermine performance on old tests? Educational Evaluation and Policy Analysis, 15, 1-16.
Parmar, R. S., Cawley J. F., & Frizita, R. R. (1996). Word problem-solving by students with and without mild disabilities. Exceptional Children, 62, 415-429.
Quilici, J. L., & Mayer, R. E. (1996). Role of examples in how students learn to categorize statistics word problems. Journal of Educational Psychology, 88, 144-161.
Teresa Foley is an Instructor of Mathematics at Asnuntuck Community College in Enfield, Connecticut.