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.     

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.     

Students’ reasoning about relationships between variables in a real-world problem

This paper reports on part of an investigation of fifteen second-semester calculus students’ understanding of the concept of parametric function. Employing APOS theory as our guiding theoretical perspective, we offer a genetic decomposition for the concept of parametric function, and we explore students’ reasoning about an invariant relationship between two quantities varying simultaneously with respect to a third quantity when described in a real-world problem, as such reasoning is important for the study of parametric functions. In particular, we investigate students’ reasoning about an adaptation of the popular bottle problem in which they were asked to graph relationships between (a) time and volume of the water, (b) time and height of the water, and (c) volume and height of the water. Our results illustrate that several issues make reasoning about relationships between variables a complex task. Furthermore, our findings indicate that conceiving an invariant relationship, as it relates to the concept of parametric function, is nontrivial, and various complimentary ways of reasoning are favorable for developing such a conception. We conclude by making connections between our results and our genetic decomposition.     

An investigation of preservice teachers’ use of guess and check in solving a semi open-ended mathematics problem

Open-ended problems have been regarded as powerful tools for teaching mathematics. This study examined the problem solving of eight mathematics/science middle-school teachers. A semi-structured interview was conducted with (PTs) after completing an open-ended triangle task with four unique solutions. Of particular emphasis was how the PTs used a specific heuristic strategy. The results showed that the primary strategy PTs employed in attempting to solve the triangle problem task was guess and check; however, from the PTs’ reflections, we found there existed misapplications of guess and check as a systematic problem-solving strategy. In order to prepare prospective teachers to effectively teach, teacher educators should pay more attention to the mathematical proficiency of PTs, particularly their abilities to systematically and efficiently use guess and check while solving problems and explain their solutions and reasoning to middle-school students.     

Instructional Supports for Representational Fluency in Solving Linear Equations with Computer Algebra Systems and Paper‐and‐Pencil

This research addresses the issue of how to support students' representational fluency—the ability to create, move within, translate across, and derive meaning from external representations of mathematical ideas. The context of solving linear equations in a combined computer algebra system (CAS) and paper‐and‐pencil classroom environment is targeted as a rich and pressing context to study this issue. We report results of a collaborative teaching experiment in which we designed for and tested a functions approach to solving equations with ninth‐grade algebra students, and link to results of semi‐structured interviews with students before and after the experiment. Results of analyzing the five‐week experiment include instructional supports for students' representational fluency in solving linear equations: (a) sequencing the use of graphs, tables, and CAS feedback prior to formal symbolic transpositions, (b) connecting solutions to equations across representations, and (c) encouraging understanding of equations as equivalence relations that are sometimes, always, or never true. While some students' change in sophistication of representational fluency helps substantiate the productive nature of these supports, other students' persistent struggles raise questions of how to address the diverse needs of learners in complex learning environments involving multiple tool‐based representations.     

Evaluation model of problem solving

Problem solving is a style of thinking, which transforms a given problem to the goal state through a so-called PS (problem solving) path. Different from the traditional GPS (General Problem Solver) approach, the focus in this paper is placed on how to judge the performance of PS paths, that is, the evaluation problem of problem solving.A series of PS paths point from the given source problem to the destination goal, then form a PSN (PS Network). This paper proposes an elaborated CPSN (Coordinate Problem Solving Network) as the evaluation model of problem solving. In CPSN, each problem is assigned a unique coordinate and then each PS path can have an evaluation vector. Several examples show such arrangement can give more insight to PS paths.Furthermore, an incremental learning algorithm is developed for the update of CPSN. When a new PS path is obtained, it is not necessary to recalculate the whole CPSN. Examples show such algorithm provides a more efficient way in finding new PS paths.     

Mathematics in the making: Mapping verbal discourse in Pólya's “Let Us Teach Guessing” lesson

This paper describes a detailed analysis of verbal discourse within an exemplary mathematics lesson—that is, George Pólya teaching in the Mathematics Association of America [MAA] video classic, “Let Us Teach Guessing” (1966). The results of the analysis reveal an inductive model of teaching that represents recursive cycles rather than linear steps. The lesson begins with a frame of reference and builds meaning cyclically/recursively through inductive processes—that is, moving from specific cases, through recursive cycles, toward more general hypotheses and rules. Additionally, connections to specific forms of talk and verbal assessment, as well as to univocal (conveying meaning) and dialogic (new meaning through dialogue) discourse, are made.     

Prospective elementary teachers use of representation to reason algebraically

We used a teaching experiment to evaluate the preparation of preservice teachers to teach early algebra concepts in the elementary school with the goal of improving their ability to generalize and justify algebraic rules when using pattern-finding tasks. Nearly all of the elementary preservice teachers generalized explicit rules using symbolic notation but had trouble with justifications early in the experiment. The use of isomorphic tasks promoted their ability to justify their generalizations and to understand the relationship of the coefficient and y-intercept to the models constructed with pattern blocks. Based on critical events in the teaching experiment, we developed a scale to map changes in preservice teachers’ understanding. Features of the tasks emerged that contributed to this understanding.     

Affect control theory: An assessment*

This paper reviews affect control theory's major strengths, the contributions of recent work to its growth, and the most promising avenues for future work. Affect control theory's strengths include (1) the precision of its mathematical statement and empirical base (especially when compared with earlier interpretive sociologies), (2) its ability to link the internal processing that generates social action to the socio‐cultural system upon which that action is based, and (3) the generality that allows a parsimonious explanation of a wide range of processes and previous research findings. Recent advances provide (1) new, more accurate impression‐change formulas, (2) the expansion of the theory to encompass settings, emotions, and traits, (3) new dictionaries of evaluation, potency and acitivity meanings and (4) tests of the theory using likelihood judgments, verbal scenarios and actual behavior of naive experimental subjects. Further work must include links to cognitive structures that will further delineate definition of situation and behavior selection processes. In addition, integration of affect control theory with new sociological work on the development of shared social knowledge and on institutionalized production systems expand the theory in useful ways. Finally, new work must find innovative and convincing ways to test simulation outcomes using both verbal accounts and behavior.     

“I’m not very good at solving problems”: An exploration of students’ problem solving behaviours

This paper reports one aspect of a larger study which looked at the strategies used by a selection of grade 6 students to solve six non-routine mathematical problems. The data revealed that the students exhibited many of the behaviours identified in the literature as being associated with novice and expert problem solvers. However, the categories of ‘novice’ and ‘expert’ were not fully adequate to describe the range of behaviours observed and instead three categories that were characteristic of behaviours associated with ‘naïve’, ‘routine’ and ‘sophisticated’ approaches to solving problems were identified. Furthermore, examination of individual cases revealed that each student's problem solving performance was consistent across a range of problems, indicating a particular orientation towards naïve, routine or sophisticated problem solving behaviours. This paper describes common problem solving behaviours and details three individual cases involving naïve, routine and sophisticated problem solvers.     

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.     

Generalization and extension of a problem from an American mathematics competition

ABSTRACT

The purpose of these notes is to generalize and extend a challenging geometry problem from a mathematics competition. The notes also contain solution sketches pertaining to the problems discussed.     

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.     

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.     

Comparing representations and reasoning in children and adolescents with two-year college students

Problem solving and justification of a diversified group of two-year college students was compared with approaches of younger elementary and secondary school students working on the same tasks. The students in this study were engaged in thoughtful mathematics. Both groups found patterns, justified that their patterns were reasonable and, utilized similar strategies for their solutions and methods of justification. They were also able to make connections and build isomorphisms among the various problems.     

Understanding the complexities of student motivations in mathematics learning

Student motivation has long been a concern of mathematics educators. However, commonly held distinctions between intrinsic and extrinsic motivations may be insufficient to inform our understandings of student motivations in learning mathematics or to appropriately shape pedagogical decisions. Here, motivation is defined, in general, as an individual's desire, power, and tendency to act in particular ways. We characterize details of motivation in mathematical learning through qualitative analysis of honors calculus students’ extended, collaborative problem solving efforts within a longitudinal research project in learning and teaching. Contextual Motivation Theory emerges as an interpretive means for understanding the complexities of student motivations. Students chose to act upon intellectual-mathematical motivations and social-personal motivations that manifested simultaneously. Students exhibited intellectual passion in persisting beyond obtaining correct answers to build understandings of mathematical ideas. Conceptually driven conditions that encourage mathematical necessity are shown to support the growth of intellectual passion in mathematics learning.     

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.     

A problem-solving approach to product design using decision tree induction based on intuitionistic fuzzy

Customer complaint problem is a product design used to understand customer requirements. Furthermore, product design corresponding to customer requirement does not feel adequately solved for a cause of problem. The cause of the problem affecting product design is solved to prevent customer complaint from reoccurring. However, the problems by customer may have observation uncertainty and fuzzy. Fuzzy concept considers not only the degree of membership to an accept set, but also the degree of non-membership to a rejection set. Therefore, we present a new approach for problem solving using decision tree induction based on intuitionistic fuzzy sets in this paper. Under this approach, we first develop the problem formulation for the symptoms and causes of the problem based on intuitionistic fuzzy sets. Next, we identify the cause of the problem using intuitionistic fuzzy decision tree by the problem formulation. We then provide the approach to find the optimal cause of the problem for the consideration of product design. A numerical example is used to illustrate the approach applied for product design.     

Analyzing the word-problem performance and strategies of students experiencing mathematics difficulty

The purpose of this study was to examine the word-problem performance and strategies utilized by 3rd-grade students experiencing mathematics difficulty (MD). We assessed the efficacy of a word-problem intervention and compared the word-problem performance of students with MD who received intervention (n = 51) to students with MD who received general education classroom word-problem instruction (n = 60). Intervention occurred for 16 weeks, 3 times per week, 30 min per session and focused on helping students understand the schemas of word problems. Results demonstrated that students with MD who received the word-problem intervention outperformed students with MD who received general education classroom word-problem instruction. We also analyzed the word-problem strategies of 30 randomly-selected students from the study to understand how students set up and solve word problems. Students who received intervention demonstrated more sophisticated word-problem strategies than students who only received general education classroom word-problem instruction. Findings suggest students with MD benefit from use of meta-cognitive strategies and explicit schema instruction to solve word problems.     

Control decisions and personal beliefs: their effect on solving mathematical problems

The main purpose of this paper is to discuss how college students enrolled in a college level elementary algebra course exercised control decisions while working on routine and non-routine problems, and how their personal belief systems shaped those control decisions. In order to prepare students for success in mathematics we as educators need to understand the process steps they use to solve homework or examination questions, in other words, understand how they “do” mathematics. The findings in this study suggest that an individual’s belief system impacts how they approach a problem. Lack of confidence and previous lack of success combined to prompt swift decisions to stop working. Further findings indicate that students continue with unsuccessful strategies when working on unfamiliar problems due to a perceived dependence of solution strategies to specific problem types. In this situation, the students persisted in an inappropriate solution strategy, never reaching a correct solution. Control decisions concerning the pursuit of alternative strategies are not an issue if the students are unaware that they might need to make different choices during their solutions. More successful control decisions were made when working with familiar problems.