Why do U.S. and Chinese students think differently in mathematical problem solving?: Impact of early algebra learning and teachers’ beliefs

This paper reports two studies that examined the impact of early algebra learning and teachers’ beliefs on U.S. and Chinese students’ thinking. The first study examined the extent to which U.S. and Chinese students’ selection of solution strategies and representations is related to their opportunity to learn algebra. The second study examined the impact of teachers’ beliefs on their students’ thinking through analyzing U.S. and Chinese teachers’ scoring of student responses. The results of the first study showed that, for the U.S. sample, students who have formally learned algebraic concepts are as likely to use visual representations as those who have not formally learned algebraic concepts in their problem solving. For the Chinese sample, students rarely used visual representations whether or not they had formally learned algebraic concepts. The findings of the second study clearly showed that U.S. and Chinese teachers view students’ responses involving concrete strategies and visual representations differently. Moreover, although both U.S. and Chinese teachers value responses involving more generalized strategies and symbolic representations equally high, Chinese teachers expect 6th graders to use the generalized strategies to solve problems while U.S. teachers do not. The research reported in this paper contributed to our understanding of the differences between U.S. and Chinese students’ mathematical thinking. This research also established the feasibility of using teachers’ scoring of student responses as an alternative and effective way of examining teachers’ beliefs.     

Toward a student-centred process of teaching arithmetic

This article describes a way toward a student-centred process of teaching arithmetic, where the content is harmonized with the students’ conceptual levels. At school start, one classroom teacher is guided in recurrent teaching development meetings in order to develop teaching based on the students’ prerequisites and to successively learn the students’ arithmetic. The students are assessed in interviews. Two special teachers participate and their current models of each student's arithmetic are tested when assessing the students. The students’ conceptual diversity and the consequent different content in teaching are shown. Further, the special teachers’ assessments and the class teacher's opinion of the new way of teaching are reported. A wide range both of the students’ conceptual levels and of the kinds of relevant problems was found. The special teachers manage their duties well and the classroom teacher has so far been satisfied with the new teaching process.     

Promoting productive mathematical classroom discourse with diverse students

Productive mathematical classroom discourse allows students to concentrate on sense making and reasoning; it allows teachers to reflect on students’ understanding and to stimulate mathematical thinking. The focus of the paper is to describe, through classroom vignettes of two teachers, the importance of including all students in classroom discourse and its influence on students’ mathematical thinking. Each classroom vignette illustrates one of four themes that emerged from the classroom discourse: (a) valuing students’ ideas, (b) exploring students’ answers, (c) incorporating students’ background knowledge, and (d) encouraging student-to-student communication. Recommendations for further research on classroom discourse in diverse settings are offered.     

The mathematical beliefs and behavior of high school students: Insights from a longitudinal study

There is a documented need for more research on the mathematical beliefs of students below college. In particular, there is a need for more studies on how the mathematical beliefs of these students impact their mathematical behavior in challenging mathematical tasks. This study examines the beliefs on mathematical learning of five high school students and the students’ mathematical behavior in a challenging probability task. The students were participants in an after-school, classroom-based, longitudinal study on students’ development of mathematical ideas funded by the United States National Science Foundation. The results show that particular educational experiences can alter results from previous studies on the mathematical beliefs and behavior of students below college, some of which have been used to justify non-reform pedagogical approaches in mathematics classrooms. Implications for classroom practice and ideas for future research are discussed.     

Students’ reflections on their learning experiences: lessons from a longitudinal study on the development of mathematical ideas and reasoning

This paper characterizes the views on mathematical learning of five high school students based on the students’ reflections on their mathematical experiences in a longitudinal study that focused on the development of mathematical ideas and reasoning in particular research conditions. The students’ views are presented according to five themes about learning which describe the students’ views on the nature of knowledge and what it means to know, source of knowledge, motivation to engage in learning, certainty in knowing, and how the students’ views vary with particular areas of mathematical activity. The study addresses the need for more research on epistemological beliefs of students below college age. In particular, the results provide evidence that challenge the existing assumption that, prior to college, students exhibit naïve epistemological beliefs.     

How does imposing a step-by-step solution method impact students’ approach to mathematical word problem solving?

This paper presents the second phase of a larger research program with the purpose of exploring the possible consequences of a gap between what is done in the classroom regarding mathematical word problem solving and what research shows to be effective in this particular field of study. Data from the first phase of our study on teachers’ self-proclaimed practices showed that one-third of elementary teachers from the region of Quebec require their students to follow a specific sequential problem-solving method, known as the ‘what I know, what I look for’ method. These results led us to hypothesize that the observed gap may have an impact on students’ comprehension of mathematical word problems. The use of this particular method was the foundation for us to study, in the second phase, the effect of the imposition of this sequential method on students’ literal and inferential understanding of word problems. A total of 278 fourth graders (9–10 years old) solved mathematical word problems followed by a test to assess their understanding of the word problems they had just solved. The results suggest that the use of this problem solving method does not seem to improve or impair students’ understanding. From a more fundamental point of view, our study led us to the conclusion that the way word problem solving is addressed in the mathematics classroom, through sequential and inflexible methods, does not help students develop their word problem solving competence.     

Ways of talking and ways of positioning: Students’ beliefs in an inquiry-oriented differential equations class

As part of developmental research for an inquiry-oriented differential equations course, this study investigates the change in students’ beliefs about mathematics. The discourse analysis has identified two different types of perspective modes - i.e., discourse of the third-person perspective and discourse of the first-person perspective - in the students’ mathematical narratives, depending on their ways of positioning themselves with respect to mathematics. In the third-person perspective discourse, the students positioned themselves as passive recipients of mathematics that has been established by some external authority. In the first-person perspective discourse, the students positioned themselves as active mathematical inquirers and produced mathematics by interweaving their own mathematical ideas and experiences. Over the semester, students’ mathematical discourse changed from third-person perspective narratives to first-person perspective narratives. This change in their discourse pattern is interpreted as an indication of change in their beliefs about mathematics. Finally, this article discusses the instructional features that promote the change.     

How do domains of knowledge integrate into mathematics teachers’ practice?

This study is a part of a research project that seeks to characterize the relationship between mathematics teachers’ knowledge and their practice. In this paper, we focus on identifying the characteristics of subject matter knowledge and pedagogical content knowledge that two teachers integrate in decisions they make about the introduction of specific mathematical content. Then, we examine the changes that arise in their classrooms as their plans are put in action. Data were obtained through audiotapes of several semi-structured interviews, through observations, and through videotapes. Although the two teachers in this study had similar backgrounds and experiences, our analysis shows differences in the characteristics of the domains of knowledge they integrated in their planning as well as differences in the adaptations that each made in the classroom. In this sense, this study contributes to better understanding the complexity of teachers’ professional practice.     

A geometry teacher's use of a metaphor in relation to a prototypical image to help students remember a set of theorems

This article asks the following: How does a teacher use a metaphor in relation to a prototypical image to help students remember a set of theorems? This question is analyzed through the case of a geometry teacher. The analysis uses Duval's work on the apprehension of diagrams to investigate how the teacher used a metaphor to remind students about the heuristics involved when applying a set of theorems during a problem-based lesson. The findings show that the teacher used the metaphor to help students recall the apprehensions of diagrams when applying several theorems. The metaphor was instrumental for mediating students’ work on a problem and the proof of a new theorem. The findings suggest that teachers’ use of metaphors in relation to prototypical images may facilitate how they organize students’ knowledge for later retrieval.     

The role of operating premises and reasoning paths in upper elementary students’ problem solving

The validity of students’ reasoning is central to problem solving. However, equally important are the operating premises from which students’ reason about problems. These premises are based on students’ interpretations of the problem information. This paper describes various premises that 11- and 12-year-old students derived from the information in a particular problem, and the way in which these premises formed part of their reasoning during a lesson. The teacher’s identification of differences in students’ premises for reasoning in this problem shifted the emphasis in a class discussion from the reconciliation of the various problem solutions and a focus on a sole correct reasoning path, to the identification of the students’ premises and the appropriateness of their various reasoning paths. Problem information that can be interpreted ambiguously creates rich mathematical opportunities because students are required to articulate their assumptions, and, thereby identify the origin of their reasoning, and to evaluate the assumptions and reasoning of their peers.     

An ecological reading of mathematical language in a Grade 3 classroom: A case of learning and teaching measurement estimation

In our work in teacher education and professional development, we aim to help teachers to learn to participate in, and create, classroom ecologies that support students’ learning. In this article we focus on the challenges of developing a classroom ecology that provides mathematical sustenance for students. We pay particular attention to the ways in which classroom language can impede the development of a classroom ecology—one where all students are heard and where knowing is understood as participatory. We present recommendations for teaching practice drawn from an ecological reading of the classroom discourse during a series of lessons on measurement in a Grade 3 classroom.     

Studying teachers’ mathematical argumentation in the context of refuting students’ invalid claims

This study investigates teachers’ argumentation aiming to convince students about the invalidity of their mathematical claims in the context of calculus. 18 secondary school mathematics teachers were given three hypothetical scenarios of a student's proof that included an invalid algebraic claim. The teachers were asked to identify possible mistakes and explain how they would refute the student's invalid claims. Two of them were also interviewed. The data were analysed in terms of the content and structure of argumentation and the types of counterexamples the teachers generated. The findings show that teachers used two main approaches to refute students’ invalid claims, the use of theory and the use of counterexamples. The role of these approaches in the argumentation process was analysed by Toulmin's model and three types of reasoning emerged that indicate the structure of argumentation in the case of refutation. Concerning the counterexamples, the study shows that few teachers use them in their argumentation and in general they underestimate their value as a proof method.     

Using interactive algebra software to support a discourse community

In this work we studied the impact of using NuCalc, an interactive computer algebra software, on the development of a discourse community in a college level mathematics class. Qualitative and quantitative data were collected over the course of 3 weeks of instruction. We examined the influence of the software on: group interactions; the mathematical investigations of learners; and the teacher’s interactions with students. Data points to four distinct ways in which the presence of NuCalc positively impacted the learning community we studied: (1) it served as a tool for extending students’ mathematical thinking, (2) it motivated students’ engagement in group discourse, (3) it became a tool for mediating discourse, (4) it became a catalyst for refining the culture of classroom, shifting the patterns of interactions between the teacher and learners.     

Learning trajectories in teacher education: Supporting teachers’ understandings of students’ mathematical thinking

Recent work by researchers has focused on synthesizing and elaborating knowledge of students’ thinking on particular concepts as core progressions called learning trajectories. Although useful at the level of curriculum development, assessment design, and the articulation of standards, evidence is only beginning to emerge to suggest how learning trajectories can be utilized in teacher education. Our paper reports on two studies investigating practicing and prospective elementary teachers’ uses of a learning trajectory to make sense of students’ thinking about a foundational idea of rational number reasoning. Findings suggest that a mathematics learning trajectory supports teachers in creating models of students’ thinking and in restructuring teachers’ own understandings of mathematics and students’ reasoning.     

Likelihood and sample size: The understandings of students and their teachers

The aim of this study was to consider the match of student statistical understanding and teacher pedagogical content knowledge in relation to sample size and likelihood. Students were given two contexts within which to compare the likelihood of events for different sample sizes. Teachers were presented with one of the contexts and asked what their students would do and how they would remediate incorrect responses. The data also provided the opportunity for a detailed hierarchical analysis of students’ and teachers’ understandings. Analysis of student solutions revealed a wide range of reasoning, some of which was apparently unfamiliar to teachers.     

Pedagogical representations to teach linear relations in Chinese and U.S. classrooms: Parallel or hierarchical?

This study investigates Chinese and U.S. teachers’ construction and use of pedagogical representations surrounding implementation of mathematical tasks. It does this by analyzing video-taped lessons from the Learner's Perspective Study, involving 15 Chinese and 10 U.S. consecutive lessons on the topic of linear equations/linear relations. We examined patterns of pedagogical representations that Chinese and U.S. teachers construct over a set of consecutive lessons, but also investigated the strategies of using representations to solve mathematical problems by Chinese and U.S. teachers. It was found that multiple representations were constructed simultaneously to develop the connection of relevant concepts in the U.S. classrooms while selective representations were constructed to develop relevant concepts in the Chinese classrooms. This study is significant because it contributes to our understanding of the cultural differences involving Chinese and U.S. students’ mathematical thinking and has practical implications for constructing pedagogical representations to maximize students’ learning.     

Neither even nor odd: Sixth grade students’ dilemmas regarding the parity of zero

This study investigates two sixth grade students’ dilemmas regarding the parity of zero. Both students originally claimed that zero was neither even nor odd. Interviews revealed a conflict between students’ formal definitions of even numbers and their concept images of even numbers, zero, and division. These images were supported by practically based explanations relying on everyday contexts. By using mathematically based explanations that rely solely on mathematical notions, students were able to correctly conclude that zero is an even number. Extending the natural number system in elementary school to include zero can be used as springboard to encourage the use of mathematically based explanations.     

Students’ use of technological features while solving a mathematics problem

The design of technology tools has the potential to dramatically influence how students interact with tools, and these interactions, in turn, may influence students’ mathematical problem solving. To better understand these interactions, we analyzed eighth grade students’ problem solving as they used a java applet designed to specifically accompany a well-structured problem. Within a problem solving session, students’ goal-directed activity was used to achieve different types of goals: analysis, planning, implementation, assessment, verification, and organization. As we examined students’ goals, we coded instances where their use of a technology feature was supportive or not supportive in helping them meet their goal. We categorized features of this applet into four subcategories: (1) features over which a user does not have any control and remain static, (2) dynamic features that allow users to directly manipulate objects, (3) dynamic features that update to provide feedback to users during problem solving, and (4) features that activate parts of the applet. Overall, most features were found to be supportive of students’ problem solving, and patterns in the type of features used to support various problem solving goals were identified.     

An “inverse” relationship between mathematics identities and classroom practices among early career elementary teachers: The impact of accountability

This qualitative case study guided by portraiture examines the relationships between three early career elementary teachers’ beliefs about themselves in relation to mathematics (mathematics identities) and their classroom practices. Through autobiographical inquiry, reflective practice, classroom observations, interviews, and artifacts, findings show that all three second grade teachers appeared to have an “inverse” relationship between their mathematics identities and their classroom practices. In this relationship, as negative as they felt about themselves with regards to mathematics, they expended that much more effort to ensure that their students would have positive experiences with it and not be stigmatized by it as they had been. Accountability to schools, students, and parents, to increase student achievement appeared to play an important role in this relationship. Implications for preservice teacher education, inservice professional development, and research on beliefs and practices are discussed.     

Geometry students’ hedged statements and their self-regulation of mathematics

Statements conveying a degree of certainty or doubt, in the form of hedging, have been linked with logical inference in students’ talk (Rowland, 2000). Considering the current emphasis on increasing student autonomy for effective mathematical discourse, I posit a relationship between hedging and student autonomy. In the current study, high school Geometry students’ frequency of producing hedged mathematical statements were correlated with their perceived mathematical autonomy to determine if a relationship existed. Results found a strong and statistically significant correlation, providing support for a connection between students’ hedging and their perceived autonomy. However, additional analysis revealed that perceptions of mathematical competence and social relatedness were also influential to hedging. Implications of these results are discussed.