Along with an emphasis placed on problem solving in school mathematics education, problem posing has emerged as a part of the mathematics education reform vision for promoting mathematics as a cognitive activity based on a constructivist perspective (Cai, 1998; Silver & Cai, 1996). The National Council of Teachers of Mathematics’ [NCTM] Principles and Standards for School Mathematics (2000) stated that through the problem solving process, a teacher could help students extend what they know for developing their mathematical fluency (NCTM, 2000).  In addition, NCTM supports teachers’ encouragement of student discourse by choosing and posing mathematical tasks engaging students in higher-order thinking. As problem solving and problem posing are central to the learning of mathematics (NCTM, 1989, 1991, 2000), teachers have a crucial role in helping students develop a repertoire of associations for proficient problem posing and effective problem solving (Moses, Bjork, & Goldenberg, 1990).

 

Perspectives or Theoretical Framework for the Research

Generally, there is scant research in the literature emphasizing the link between mathematical problem solving and problem posing with respect to teacher education (Chen, Van Dooren, Qi, & Verschaffel, 2010; Silver, Mamona-Downs, Leung, & Kenney, 1996). Many previous studies have focused these topics on school-age students (e.g., Cai, 2003; Cai & Hwang, 2002; Chen et al., 2007). To date, no researcher has yet explored how preservice teachers (PTs) solve a mathematical task and reformulate the given task sequentially during the same time setting. In this present study, we adapted Stoyanova’s (1999) framework of structured situation in which the problem posing was based on a specific problem that was solved prior to reformulation of new mathematical tasks.

 

Methods, Techniques, or Modes of Inquiry for the Research

Using Teddlie and Tashakkori’s (2006) general typology of mixed methods research, this study was categorized as a conversion mixed design. The mixing of the qualitative and quantitative methods occurred during the data analysis and data interpretation stages (Nastasi, Hitchcock, & Brown, 2010). Two research questions were used to guide this study:

(1) How did PTs solve mathematical problems and reformulate new tasks based on the given ones?

(2) What was the relationship between PTs’ mathematical problem solving and problem posing?

 

Data Sources or Evidence for the Research

We gathered qualitative data through a convenience sampling of 106 middle school PTs who were juniors, mostly female (95%) with ages ranging from 18 to 22 years, and enrolled in a problem-solving course during the 2010-2011 academic year. The professor based the course on Polya’s four-step problem solving process and occasionally used problem-posing activities as part of her class instruction. The instrument used was based on Cai and Lester’s (2005) The Block Pattern Problem with some minor changes (Appendix A) and was validated by three mathematics content and mathematics education experts. Participants were given multilink cubes and had 35 minutes to complete the task during a regular class session. Researchers coded ten written responses and achieved 70% inter-rater agreement. Any disagreement was discussed and resolved before analyzed the remaining data.

A sequential mixed methods analysis was utilized (Onwuegbuzie & Teddlie, 2003) wherein the data were transformed quantitatively based on scoring rubrics, analyzed quantitatively and qualitatively, and then qualitized through narrative discussion (Tashakkori & Teddlie, 1998). In Step 1, we removed 16 incomplete written responses because participants did not reformulate any new problems for the problem posing tasks. Then, we adapted a 5-dimensional problem-solving rubric (Appendix B) in which 1 to 6 points were awarded with a total of 30 possible points. Also, we categorized the preservice teachers’ problem solving strategies (O’Connell, 2007).

In Step 2, the researchers discussed and developed a problem-posing rubric consisting of four important elements with 1-4 points each (Appendix C). Descriptive statistics pertaining to preservice teachers’ performance were computed using Statistical Package for Social Science (SPSS) version 17.0 (2008). Then, total points were used to find the differences and relationship between preservice teachers’ problem solving and problem posing. The data collected were adequate to a detect statistically relationship and mean differences at a medium Cohen’s d effect size with .80 power at the 5% level of significance (Collins, Onwuegbuzie, & Jiao, 2007). 

 

Results and/or Conclusions

Research Question 1

The middle school preservice teachers’ performance was consistent in all five dimensions of the problem solving processes as measured by the rubric. A majority (76% - 98%) scored at least 4 points in making sense of the task, representing and solving the task, communicating reasoning, accuracy, and reflecting and evaluating processes. PTs used a variation of problem solving strategies. Specifically, 63.3% PTs drew a picture or diagram of a staircase of 5-steps when finding the blocks needed whereas 64.4% preferred to find a pattern when they were asked the number of blocks for a staircase of 20 steps. In contrast, preservice teachers’ performance in reformulating new problems was different from their problem solving abilities. Even though 75% posed two or more middle-grade level appropriate tasks, only 27% were categorized above average in term of mathematical content, mathematical language, and originality that was similar to the findings from Silver et al. (1996). 

 

Research Question 2

The results of the Pearson correlation demonstrated no statistically significant relationships between problem solving and problem posing.  Meanwhile the confidence intervals showed a large difference in performance, PTs’ scores were higher on the problem-solving task.

In summary, if PTs are to become autonomous problem solvers and problem posers, they should have substantial educational experiences (Kilpatrick, 1987; Silver et al., 1996) that providing them more opportunities to engage and analyze their own posed problems in term of feasibility and quality (Silver et al., 1996).

 

Educational or Scientific Importance of the Research

This study was unique is two ways. First, it is one of the only studies to examine PTs’ problem solving and problem posing on the same mathematical task. Second, it likely represents the first study to use mixed analysis techniques to understand the complex connection between these mathematical activities. Thus, the present study led to a combination that yielded “complementary strengths and nonoverlapping weaknesses” (Johnson & Turner 2003, p. 299).