Theoretical Framework

Problem solving is central to doing and learning mathematics (Ball, Ferrini-Mundy, Kilpatrick, Milgram, Schmid, & Scharr, 2005; Kilpatrick, Swafford, & Findell, 2001).  Effective problem solvers have been shown to (1) read and understand the problem, (2) develop a situation model, (3) make a mathematical model, (4) employ a set of procedures and arrive at a result, (5) interpret the result, and (6) report the solution (Verschaffel, Greer, & De Corte, 2000) whereas ineffective problem solvers tend to skip some steps while solving problems, which leads to incorrect results (Pape, 2004; Verschaffel et al., 2000).  Thus, learners should have instruction that encourages them to engage in each step of the problem-solving process.  This study responds to the National Council of Teachers of Mathematics (NCTM) (NCTM, 2000, 2006, 2009) and Chief Council of State School Officers (2010) advocacy for problem solving as an instructional priority and instructional scaffolding that supports students during problem solving. 

Mathematics instruction that draws on open and complex problems and encourages student-to-student mathematics dialogue helps students develop connections between concepts and procedures and effective problem-solving behaviors (English, 2009; English & Sriraman, 2010).  Teaching mathematics through problem-solving contexts has led to better student achievement as well as improved problem-solving performance among fifth-grade students (Verschaffel & De Corte, 1997; Verschaffel et al., 1999).  These studies informed designing and implementing an intervention, and effects and implications of this intervention are this session’s topic.

Methods of Inquiry

The intervention involved teaching mathematics through problem-solving contexts.  Three research questions will be discussed at this presentation.  Does teaching mathematics through problem-solving contexts influence students’ problem-solving performance?  Does problem-solving performance differ between students from the intervention and comparison groups?  Does unit test performance differ between students from the intervention and comparison groups?  

This study employed a nonequivalent control-group quasi-experimental research design (Gall, Gall, & Borg, 2007).  Three sections of sixth-grade mathematics were randomly selected from one school.  The researcher conducted the instructional intervention for one month in a randomly selected section (i.e., intervention group) while two remaining sections continued to learn from their classroom teacher (i.e., comparison group).  Components of the intervention included: (1) establishment of new social and sociomathematical norms; (2) prompts to facilitate mathematical discourse; (3) encouragement to ask other students questions; (4) tasks that were (a) designed based on state standards and curricular materials and (b) open, complex, and realistic; and (5) homework exercises and problems.  The intervention’s instructional format was (1) checking homework and discussing homework, (2) independently working on a problem, (3) small-group discussion, (4) whole-class discussion, and (5) reflecting on the problem.  Roughly 25-50 minutes of the block period was dedicated to discussions, depending on the task. 

Data Sources

Students completed a problem-solving pretest and posttest consisting of open, complex, and realistic problems modified from Verschaffel et al.’s (1999) instruments and a unit test selected from the textbook.  Students’ gender, ethnicity, fifth-grade mathematics and reading Florida Comprehensive Assessment Test scores, free-and-reduced lunch status, and years attending this K-12 school were gathered for use as covariates during analyses.  Two coders scored students’ responses on the pretest and posttest as correct/incorrect and calculated interrater agreement using r_wg (r_wg = 1; James, Demaree, & Wolf, 1984).  The classroom teacher scored participants’ responses on the unit test as correct/incorrect.  Internal consistency for the pretest, posttest, and unit test was r = .79, .72, and .82, respectively.  Alternate-forms reliability for the pretest and posttest was r = .77, which exceeded the minimum to link participants’ pretest and posttest scores (Ary, Cheser-Jacobs, Sorenson, & Razavieh, 2009).  A chi-square test using the covariates was conducted to check for group similarity prior to the intervention.  Repeated measures t-tests were employed to explore whether there were within-group differences between participants’ pretest and posttest problem-solving performance.  Poisson multiple regression analysis using backwards selection procedures was used to examine between-group differences, that is, how the intervention and covariates impacted problem-solving performance.  Finally, ordinary least squares multiple regression was used to investigate the intervention’s effect on participants’ unit test scores.

Results

Chi-square results indicated there were no between-group differences prior to the intervention.  Intervention participants improved from pretest to posttest, p = .02, d = .48 whereas their peers did not, p = .61.  Poisson multiple regression analyses indicated that pretest performance and intervention status alone best predicted students’ posttest performance χ²(2) = 33.06, p < .001, R² = .64.  The intervention group had a 0.37 log unit advantage over the comparison group.  The effect size, d = .26, is 18% larger than the mean effect size on achievement measures for students in fourth- through sixth-grade (Hill, Bloom, Black, & Lipsey, 2008).  Finally, there was a significant difference in unit test performance between groups, F(2, 50) = 17.45, p < .001, R² = .41.  Comparison group students experienced a 0.34 SD advantage over their matched intervention peers.  

Importance of the Research

Teaching mathematics through problem-solving contexts in the manner conducted in this study was linked to both positive and negative outcomes.  Intervention participants had better posttest problem-solving performance than their peers but they did not perform as well as their peers on the unit test.  This study informs theory (e.g., instructional components impacting students’ outcomes and suggestions for future instructional interventions) and practice (e.g., ideas for constructing and implementing instruction that teaches mathematics from the standards through problem-solving contexts).  

Organization

Three minutes for examining relevant literature, four minutes for methods section, and eight minutes for discussing results and conclusions.  Roundtable discussions will focus on this investigation’s implications for problem-solving research and teaching the Standards for Mathematics Content in ways that align with the Standards for Mathematical Practice (CCSSO, 2010) as well as means for supporting students to become effective problem solvers.

References

Ary, D., Cheser-Jacobs, L., Sorenson, C., & Razavieh, A. (2009). Introduction to research in education (8^th ed.). Belmont, CA: Wadsworth.

Ball, D., Ferrini-Mundy, J., Kilpatrick, J., Milgram, J., Schmid, W., & Schaar, R. (2005). Reaching for common ground in K-12 mathematics education. American Mathematical Society, 52, 1055-1058.

Council of Chief State School Officers. (2010). Common core standards for mathematics. Retrieved from http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf

English, L. (2009). Promoting interdisciplinarity through mathematical modeling. ZDM: The International Journal on Mathematics Education, 41, 161-181.

English, L., & Sriraman, B. (2010) Problem solving for the 21^st century. In. L. English & B. Sriraman (Eds.) Theories of mathematics education: Seeking new frontiers (pp. 263-290). Berlin, Germany: Springer

Gall, M., Gall, J., & Borg, W. (2007). Educational research: An introduction (8^th ed.). Boston: Pearson.

Hill, C., Bloom, H., Black, A., & Lipsey, M. (2008). Empirical benchmarks for interpreting effect sizes in research. Child Development Perspectives, 2, 172-177.

James, L., Demaree, R., & Wolf, G. (1984). Estimating within-group interrater reliability with and without response bias. Journal of Applied Psychology, 68, 85-98.

Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics. (2006). Curriculum focal points for Prekindergarten through grade 8 mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics. (2009). Focus in high school mathematics: Reasoning and sense making. Reston, VA: Author.

Pape, S. (2004). Middle school children’s problem-solving behavior: A cognitive analysis from a reading comprehension perspective. Journal for Research in Mathematics Education, 35, 187-219.

Verschaffel, L., & De Corte, E. (1997). Teaching realistic mathematical modeling in the elementary school: A teaching experiment with fifth graders. Journal for Research in Mathematics Education, 28, 577-601.

Verschaffel, L., De Corte, E., Lasure, S., Van Vaerenbergh, G., Bogaerts, H., & Ratinckx, E. (1999). Learning to solve mathematical application problems: A design experiment with fifth graders. Mathematical Thinking and Learning, 1, 195-229.

Verschaffel, L., Greer, B., & De Corte, E. (2000). Making sense of word problems. Lisse, Netherlands: Swets & Zeitlinger.