Mathematical Thinking Involved in U.S. and Chinese Students' Solving of Process-Constrained and Process-Open Problems

Abstract

This study examined U.S. and Chinese 6th-grade students' mathematical thinking and reasoning involved in solving 6 process-constrained and 6 process-open problems. The Chinese sample (from Guiyang, Guizhou) had a significantly higher mean score than the U.S. sample (from Milwaukee, Wisconsin) on the process-constrained tasks, but the sample of U.S. students had a significantly higher mean score than the sample of the Chinese students on the process-open tasks. A qualitative analysis of students' responses was conducted to understand the mathematical thinking and reasoning involved in solving these problems. The qualitative results indicate that the Chinese sample preferred to use routine algorithms and symbolic representations, whereas the U.S. sample preferred to use concrete visual representations. Such a qualitative analysis of students' responses provided insights into U.S. and Chinese students' mathematical thinking, thereby facilitating interpretation of the cross-national differences in solving the process-constrained and process-open problems.

Using Open-Response Fraction Items to Explore the Relationship Between Instructional Modalities and Students’ Solution Strategies

“…We employed a conversion mixed methods approach (Teddlie & Tashakkori, 2006) to analyze open-response items, which we coded and quantitized for quantitative and qualitative analysis. Open-response items provide windows into students' thinking processes and strategies for solving mathematics tasks (Cai, 2000;Cai, Magone, Wang, & Lane, 1996;Lane, 1993). This study complements and extends previous studies by using open-response items to examine these phenomena in depth using qualitative analysis with a large sample of participants (n = 355).…”

“…Researchers used the open-response items to assess students' responses and representations, identify students' errors, and examine students' strategies. Using open-response items to assess students' mathematical reasoning and strategies reveals aspects of students' thinking beyond the correct/incorrect information provided by multiple-choice questions (Cai, 1995;Cai, 2000;Silver, 1992).Each open-response fraction item selected for this analysis was a matched question (i.e., similar or the same) that appeared on the pretest and posttest. There were two sets of matched questions for third grade and two sets of matched questions for fourth grade.…”

Invited Lectures from the 13th International Congress on Mathematical Education

“…Such studies can help to identify and describe the mathematical activities performed by children in their home context, a context that has been little investigated with relation to mathematical knowledge. Vergnaud, G. (1997 symbol-based strategies and algebraic solutions and U.S. students prefer concrete, pictorial-based strategies in problem solving (e.g., Cai 2000). A corresponding difference in the approach of teachers is that U.S. teachers put more emphasis on the use of concrete examples to aid student understanding, while Chinese teachers tend to emphasize the abstract reasoning beyond the concrete after presenting concrete examples (e.g., Cai and Wang 2006).…”

Hong Kong and U.S. Teachers’ Perceptions of Mathematical Disagreements and their Resolution Processes

“…Similarly, Cai and his colleagues (Cai, 2000(Cai, , 2005Cai & Wang, 2006) conducted a series of comparative studies between Mainland China and the U.S. on teachers' construction of representations. Both groups of teachers used concrete representations for developing the concepts of ratio and average.…”