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.     

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.     

Are beliefs believable? An investigation of college students’ epistemological beliefs and behavior in mathematics

College students’ epistemological belief in their academic performance of mathematics has been documented and is receiving increased attention. However, to what extent and in what ways problem solvers’ beliefs about the nature of mathematical knowledge and thinking impact their performances and behavior is not clear and deserves further investigation. The present study investigated how Taiwanese college students espousing unlike epistemological beliefs in mathematics performed differently within different contexts, and in what contexts these college students’ epistemological beliefs were consistent with their performances and behavior. Results yielded from the survey of students’ performances on standardized tests, semi-open problems, and their behaviors on pattern-finding tasks, suggest mixed consequences. It appears that beliefs played a more reliable role within the well-structured context but lost its credibility in non-standardized tasks.     

An exploratory framework for handling the complexity of mathematical problem posing in small groups

The paper introduces an exploratory framework for handling the complexity of students’ mathematical problem posing in small groups. The framework integrates four facets known from past research: task organization, students’ knowledge base, problem-posing heuristics and schemes, and group dynamics and interactions. In addition, it contains a new facet, individual considerations of aptness, which accounts for the posers’ comprehensions of implicit requirements of a problem-posing task and reflects their assumptions about the relative importance of these requirements. The framework is first argued theoretically. The framework at work is illustrated by its application to a situation, in which two groups of high-school students with similar background were given the same problem-posing task, but acted very differently. The novelty and usefulness of the framework is attributed to its three main features: it supports fine-grained analysis of directly observed problem-posing processes, it has a confluence nature, it attempts to account for hidden mechanisms involved in students’ decision making while posing problems.     

Reversibility of thought: An instance in multiplicative tasks

In line with current efforts to understand the piece-by-piece structure and articulation of children’s mathematical concepts, this case study compares the reversibility schemes of two eighth-grade students. The aim of the study was to identify the mechanism through which students reverse their thought processes in a multiplicative situation. Data collected through clinical interviews depict the precise strategies that the participants used to work back to find the missing values in an inverse proportional task. This study also illustrates how a conceptual template generated by one of the participants afforded him considerable flexibility in the multiplicative task. Another outcome of the study is that it shows how the numerical characteristics of the parameters in the problem affected the students’ ability to reverse their thought processes. We infer that there is a need for further research on how students might represent their reversibility schemes in the form of algebraic equations.     

From test cases to special cases: Four undergraduates unpack a formula for combinations

Through a case study of four elementary education undergraduates, we seek new analytic constructs that could help make clearer how arguments discovered or tested in quite special cases might come to support assertions that are understood to hold in general. We began analysis from a particular standpoint: to focus fundamentally on learners’ representations and on how the learners reason from them. We have found it helpful to distinguish two perspectives to guide the subsequent analysis. On the one hand, we direct detailed attention to how learners reason, most especially on how they organize the logic of their arguments. On the other hand, we seek to understand the learners’ representations based on the way they structure them, and through the ways such structures might be reshaped or reframed over time. Based on this analysis the student subjects offer further evidence of the depth and power that students often viewed as less mathematically inclined can demonstrate in learning situations that engage them deeply.     

Flexibility and coordination among acts of visualization and analysis in a pattern generalization activity

This study aims at exploring processes of flexibility and coordination among acts of visualization and analysis in students’ attempt to reach a general formula for a three-dimensional pattern generalizing task.The investigation draws on a case-study analysis of two 15-year-old girls working together on a task in which they are asked to calculate the number of blocks in a three-dimensional tower of different heights. The students’ activity was video- and audio-taped, fully transcribed and lasted for 50 min.The analysis discloses several instances of how the students were linking acts of visualization and analysis to reach a general formula. However, regarding flexibility, we found that it was more natural for the students to change visual format than to change analytical position and direction in their attempts to generalize the three-dimensional pattern of the task in a closed formula.     

Students’ images of problem contexts when solving applied problems

This article reports findings from an investigation of precalculus students’ approaches to solving novel problems. We characterize the images that students constructed during their solution attempts and describe the degree to which they were successful in imagining how the quantities in a problem's context change together. Our analyses revealed that students who mentally constructed a robust structure of the related quantities were able to produce meaningful and correct solutions. In contrast, students who provided incorrect solutions consistently constructed an image of the problem's context that was misaligned with the intent of the problem. We also observed that students who caught errors in their solutions did so by refining their image of how the quantities in a problem's context are related. These findings suggest that it is critical that students first engage in mental activity to visualize a situation and construct relevant quantitative relationships prior to determining formulas or graphs.     

Systematic approaches to experimentation: The case of Pick's theorem

In this paper two 10th graders having an accumulated experience on problem-solving ancillary to the concept of area confronted the task to find Pick's formula for a lattice polygon's area. The formula was omitted from the theorem in order for the students to read the theorem as a problem to be solved. Their working is examined and emphasis is given to highlighting the students’ range of systematic approaches to experimentation in the context of problem solving and aspects of control that are reflected in these approaches.     

Translation between external representation systems in mathematics: All-or-none or skill conglomerate?

The study described herein represents an initial, exploratory attempt to understand what it means to translate between external representation systems. Researchers have traditionally considered translation as an all-or-none activity. We hypothesize that translation is comprised of both knowledge and skill components, and accordingly construe translation as an activity that our framework allows us to define as partial or complete. We examined the effects of part-whole knowledge in change unknown subtraction situations and the structure of three different external representations on first grade students’ ability to translate in a partitioning task. Based on three interviews conducted with each student, results from our mixed methods analysis showed that translation skill was affected by what students knew about subtraction and the structure of the ER. Further, whereas high knowledge students demonstrated greater difficulty with translating to the ERs, low knowledge students had difficulty translating from the problem statement as well as translating to the ERs.     

Affect and graphing calculator use

This article reports on a qualitative study of six high school calculus students designed to build an understanding about the affect associated with graphing calculator use in independent situations. DeBellis and Goldin's (2006) framework for affect as a representational system was used as a lens through which to understand the ways in which graphing calculator use impacted students’ affective pathways. It was found that using the graphing calculator helped students maintain productive affective pathways for problem solving as long as they were using graphing calculator capabilities for which they had gone through a process of instrumental genesis (Artigue, 2002) with respect to the mathematical task they were working on. Furthermore, graphing calculator use and the affect that is associated with its use may be influenced by the perceived values of others, including parents and teachers (past, present and future).     

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.     

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.     

Promoting students’ graphical understanding of the calculus

Our purpose in this paper is to report on an observational study to show how students think about the links between the graph of a derived function and the original function from which it was formed. The participants were asked to perform the following task: they were presented with four graphs that represented derived functions and from these graphs they were asked to construct the original functions from which they were formed. The students then had to walk these graphs as if they were displacement-time graphs. Their discussions were recorded on audio tape and their walks were captured using data logging equipment and these were analysed together with their pencil and paper notes. From these three sources of data, we were able to construct a picture of the students’ graphical understanding of connections in calculus. The results confirm that at the start of the activity the students demonstrate an algebraic symbolic view of calculus and find it difficult to make connections between the graphs of a derived function and the function itself. By being able to ‘walk’ an associated displacement time graph, we propose that the students are extending their understanding of calculus concepts from symbolic representation to a graphical representation and to what we term a ‘physical feel’.     

Students’ use of technological tools for verification purposes in geometry problem solving

Despite its importance in mathematical problem solving, verification receives rather little attention by the students in classrooms, especially at the primary school level. Under the hypotheses that (a) non-standard tasks create a feeling of uncertainty that stimulates the students to proceed to verification processes and (b) computational environments - by providing more available tools compared to the traditional environment - might offer opportunities for more frequent usage of verification techniques, we posed to 5th and 6th graders non-routine problems dealing with area of plane irregular figures. The data collected gave us evidence that computational environments allow the development of verification processes in a wider variety compared to the traditional paper-and-pencil environment and at the same time we had the chance to propose a preliminary categorization of the students’ verification processes under certain conditions.     

Traditional instruction in advanced mathematics courses: a case study of one professor’s lectures and proofs in an introductory real analysis course

It is widely accepted by mathematics educators and mathematicians that most proof-oriented university mathematics courses are taught in a “definition-theorem-proof” format. However, there are relatively few empirical studies on what takes place during this instruction, why this instruction is used, and how it affects students’ learning. In this paper, I investigate these issues by examining a case study of one professor using this type of instruction in an introductory real analysis course. I first describe the professor’s actions in the classroom and argue that these actions are the result of the professor’s beliefs about mathematics, students, and education, as well as his knowledge of the material being covered. I then illustrate how the professor’s teaching style influenced the way that his students attempted to learn the material. Finally, I discuss the implications that the reported data have on mathematics education research.     

Grade 3 students’ mathematization through modeling: Situation models and solution models with mutli-digit subtraction problem solving

In considering mathematics problem solving as a model-eliciting activity ( [Lesh and Doerr, 2003], [Lesh and Harel, 2003] and [Lesh and Zawojewski, 2008]), it is important to know what students are modeling for the problems: situations or solutions. This study investigated Grade 3 students’ mathematization process by examining how they modeled different types of multi-digit subtraction situation problems. Students’ modeling processes differed from one problem type to another due to their prior experiences and the complexity of the problems. This study showed that students make their own distinctions between solution and situation models in their mathematization process. Mathematics curricula and teaching should consider these distinctions to carefully facilitate different model development of and support student understanding of a content topic.     

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.     

Probing for reasons: Presentations, questions, phases

This paper reports on a research study based on data from experimental teaching. Undergraduate dance majors were invited, through real-world problem tasks that raised central conceptual issues, to invent major ideas of calculus. This study focuses on work and thinking by these students, as they sought to build key ideas, representations and compelling lines of reasoning. Speiser and Walter's psychological and logical perspectives (see Speiser, Walter, & Sullivan, 2007) provide opportunities to focus not just on the students’ thinking, but perhaps most especially, through detailed examination of important choices, on their exercise of agency as learners. Close analysis of student data through these lenses triggered the development of two new analytic categories—logic of agency and logic of proof. The analysis presented here treats students as active shapers of their own experience and understanding, whose choices open opportunities for continued growth and learning, not just for themselves but also for each other.