Jinfa Cai

Generalized and generative thinking in US and Chinese students’ mathematical problem solving and problem posing

Abstract This study examined US and Chinese 6th grade students’ generalization skills in solving pattern-based problems, their generative thinking in problem posing, and the relationships between students’ performance on problem solving and problem posing tasks. Across the problem solving tasks, Chinese students had higher success rates than US students. The disparities appear to be related to students’ use of differing strategies. Chinese students tend to choose abstract strategies and symbolic representations while US students favor concrete strategies and drawing representations. If the analysis is limited to those students who used concrete strategies, the success rates between the two samples become almost identical. With regard to problem posing, the US and Chinese samples both produce problems of various types, though the types occur in differing sequences. Finally, this study revealed differential relationships between problem posing and problem solving for US and Chinese students. There was a much stronger link between problem solving and problem posing for the Chinese sample than there was for the US sample.

Investigating Parental Roles in Students' Learning of Mathematics from a Cross-National Perspective

This study investigates the roles parents in the United States of America and parents in the People’s Republic of China play in their children’s mathematics learning. It also examines the relationship between parental involvement and students’ mathematical problem-solving performance. In the study, 232 US sixth-grade students and 310 Chinese sixth-grade students along with their parents were surveyed. The results of this study support the argument, from a broader cross-national perspective, that parental involvement is a statistically significant predictor of their children’s mathematics achievement. Cross-nationally, Chinese parents seemed to play a more positive role than do the US parents.

An investigation of U.S. and Chinese students’ mathematical problem posing and problem solving

This study explored the mathematical problem posing and problem solving of 181 U.S. and 223 Chinese sixth-grade students. It is part of a continuing effort to examine U.S. and Chinese students’ performance by conducting a cognitive analysis of student responses to mathematical problem-posing and problem-solving tasks. The findings of this study provide further evidence that, while Chinese students outperform U.S. students on computational tasks, there are many similarities and differences between U.S. and Chinese students in performing relatively novel tasks. Moreover, the findings of this study suggest that a direct link between mathematical problem posing and problem solving found in earlier studies for U.S. students is true for Chinese students as well.

Generating Multiple Solutions for a Problem: A Comparison of the Responses of U.S. and Japanese Students.

A task involving simple mathematics, yet complex in its call for the generation of multiple solution methods, was administered to about 150 U.S. students, most of whom were in fourth grade. Written responses were examined for correctness, evidence of strategy use and mode of explanation. Results for the U.S. sample were also compared to those obtained from about 200 Japanese fourth-grade students. Students in both countries (a) produced multiple solutions and explanations of their solutions, (b) exhibited almost identical patterns and frequency of strategy use across response occasions, and (c) used the same kinds of explanations, with a majority of the responses involving solution explanations that combined both visual and verbal/symbolic features. Nevertheless, Japanese students tended to produce explanations involving more sophisticated mathematical ideas (multiplication rather than addition) and formalisms (mathematical expressions rather than verbal explanations) than did U.S. students.

The teaching of equation solving: approaches in Standards-based and traditional curricula in the United States

This paper discusses the approaches to teaching linear equation solving that are embedded in a Standards-based mathematics curriculum (Connected Mathematics Program or CMP) and in a traditional mathematics curriculum (Glencoe Mathematics) in the United States. Overall, the CMP curriculum takes a functional approach to teaching equation solving, while Glencoe Mathematics takes a structural approach. The functional approach emphasizes the important ideas of change and variation in situations and contexts. It also emphasizes the representation of relationships between variables. The structural approach, on the other hand, requires students to work abstractly with symbols and follow procedures in a systematic way. The CMP curriculum may be regarded as a curriculum with a pedagogy that emphasizes predominantly the conceptual aspects of equation solving, while Glencoe Mathematics may be regarded as a curriculum with a pedagogy that emphasizes predominantly the procedural aspects of equation solving. The two curricula may serve as concrete examples of functional and structural approaches, respectively, to the teaching of algebra in general and equation solving in particular.

A Protocol-Analytic Study of Metacognition in Mathematical Problem Solving.

The metacognitive behaviours of two subjects having a high level of mathematical experience and two subjects having a low level of mathematical experience were compared within each of the four cognitive processes of mathematical problem solving: orientation, organisation, execution, and verification. The results showed that the high-experience subjects engaged in self-regulation during the problem-solving process and that the low-experience subjects did not. Also, the high-experience subjects had stronger awareness about what they knew and how they should use this knowledge, and were able to sequentially monitor their goal-changing and decision-making activities in order to implement their goal. Another important finding was that the high-experience subjects spent the majority of their time on orientation and organisation rather than on execution, while low-experience subjects spent the majority of their time on execution rather than on orientation and organisation. Finally, the high-experience subjects accurately evaluated their strategies, actions and intermediate results. The results suggest that individual differences between the high- and low-experience subjects are unlikely to emerge either from the subjects’ selection of solution strategies or from the level of mathematical knowledge required for solving the problem. Therefore, the results from this study support the argument that metacognitive behaviours have important influences on subjects’ problem-solving success. This study also suggests that a complex, difficult, or novel task appears to function well as a task for examining metacognitive behaviours because such a task results in the subjects being unable to arrive at closure quickly.

International Comparative Studies in Mathematics

Comparing is one of the most basic intellectual activities. We consciously make comparisons to understand where we stand, both in relation to others as well as to our own past experiences. International comparative studies have completely transformed the way we see mathematics education. The focus of this ICME-13 Topical Survey is to discuss the ways international comparative studies can be used to improve students’ learning. We take a strong position that the main purpose of educational research is to improve student learning. International comparative studies are not an exception. From many possibilities, we have particularly selected four lessons that we can learn from international comparative studies: (1) understanding students’ thinking, (2) examining the dispositions and experiences of mathematically literate students, (3) changing classroom instruction, and (4) making global research locally meaningful. We decided to focus on these four aspects because of their potential impact on students’ learning. Throughout the paper, we point out future directions for research.

Problem-Posing Research in Mathematics Education: Some Answered and Unanswered Questions

This chapter synthesizes the current state of knowledge in problem-posing research and suggests questions and directions for future study. We discuss ten questions representing rich areas for problem-posing research: (a) Why is problem posing important in school mathematics? (b) Are teachers and students capable of posing important mathematical problems? (c) Can students and teachers be effectively trained to pose high-quality problems? (d) What do we know about the cognitive processes of problem posing? (e) How are problem-posing skills related to problem-solving skills? (f) Is it feasible to use problem posing as a measure of creativity and mathematical learning outcomes? (g) How are problem-posing activities included in mathematics curricula? (h) What does a classroom look like when students engage in problem-posing activities? (i) How can technology be used in problem-posing activities? (j) What do we know about the impact of engaging in problem-posing activities on student outcomes?

Examining Students’ Algebraic Thinking in a Curricular Context: A Longitudinal Study

This chapter highlights findings from the LieCal Project, a longitudinal project in which we investigated the effects of a Standards-based middle school mathematics curriculum (CMP) on students’ algebraic development and compared them to the effects of other middle school mathematics curricula (non-CMP). We found that the CMP curriculum takes a functional approach to the teaching of algebra while non-CMP curricula take a structural approach. The teachers who used the CMP curriculum emphasized conceptual understanding more than did those who used the non-CMP curricula. On the other hand, the teachers who used non-CMP curricula emphasized procedural knowledge more than did those who used the CMP curriculum. When we examined the development of students’ algebraic thinking related to representing situations, equation solving, and making generalizations, we found that CMP students had a significantly higher growth rate on representing-situations tasks than did non-CMP students, but both CMP and non-CMP students had an almost identical growth in their ability to solve equations. We also found that CMP students demonstrated greater generalization abilities than did non-CMP students over the three middle school years.

CONTRIBUTIONS FROM CROSS-NATIONAL COMPARATIVE STUDIES TO THE INTERNATIONALIZATION OF MATHEMATICS EDUCATION: STUDIES OF CHINESE AND U.S. CLASSROOMS

Cross-national studies offer a unique contribution to the internationalization of mathematics education. In particular, they provide mathematics educators with opportunities to situate the teaching and learning mathematics in a wider cultural context and to reflect on generalization of theories and practices of teaching and learning mathematics that have been developed in particular countries. In this chapter, we discuss a series of cross-national studies involving Chinese and U.S. students that illustrate to how cultural differences in Chinese and U.S. teachers’ teaching practices and beliefs affect the nature of their students’ mathematical performance. We do this by showing that the types of mathematical representations teachers present to students strongly influence the choice of representations students use to solve problems. Specifically, the Chinese teachers overwhelmingly used symbolic representations of instructional tasks, whereas the U.S. teachers relied almost exclusively on verbal explanations and pictorial representations, illustrating that mathematics teaching is local practice which takes place in settings that are both socially and culturally constrained. These results demonstrate the social and cultural nature of teachers’ pedagogical practice