The Tabular Mode: Not Just Another Way to Represent a Function

Understanding mathematical functions as systematic processes involving the covariation of related variables is foundational in learning mathematics. In this article, findings are reported from two investigations examining students' thinking processes with functions. The first study focused on seven middle school students' explorations with a dynamic physical model. Students were videotaped during the 20‐ to 45‐minute sessions occurring two or three times per week over a period of 2 months, and students' written work was collected. The second investigation included 19 preservice elementary and middle school teachers enrolled in a course focusing on a combination of mathematical content and pedagogy. Participants' written problem‐solving work and reflective writing were collected, and participants were individually interviewed in 50‐minute videotaped sessions. Results from both investigations indicated that students often relied on a table, or some variation of a table, as a cognitive link advancing the development of their reasoning about underlying function relationships.     

An Investigation of Relationships between Seventh-Grade Students' Beliefs and Their Participation during Mathematics Discussions in Two Classrooms

As mathematics teachers attempt to promote classroom discourse that emphasizes reasoning about mathematical concepts and supports students' development of mathematical autonomy, not all students will participate similarly. For the purposes of this research report, I examined how 15 seventh-grade students participated during whole-class discussions in two mathematics classrooms. Additionally, I interpreted the nature of students' participation in relation to their beliefs about participating in whole-class discussions, extending results reported previously (Jansen, 2006) about a wider range of students' beliefs and goals in discussion-oriented mathematics classrooms. Students who believed mathematics discussions were threatening avoided talking about mathematics conceptually across both classrooms, yet these students participated by talking about mathematics procedurally. In addition, students' beliefs about appropriate behavior during mathematics class appeared to constrain whether they critiqued solutions of their classmates in both classrooms. Results suggest that coordinating analyses of students' beliefs and participation, particularly focusing on students who participate outside of typical interaction patterns in a classroom, can provide insights for engaging more students in mathematics classroom discussions.     

Compacton and generalized kink wave solutions of the CH-DP equation

The CH-DP equation is investigated by using the bifurcation method of planar systems and simulation method of differential equations. The bifurcation phase portraits are drawn in different regions of parameter plane. The planar graphs of compactons and generalized kink waves are simulated by using mathematical software Maple. Exact explicit parameter expressions of compactons and implicit expressions of generalized kink wave solutions are given, and the dynamic characters of these solutions are investigated.     

Understandings of Solutions to Differential Equations Through Contexts,Web‐Based Simulations,and Student Discussion

As in the case of elementary mathematics, the instruction of high‐level mathematical concepts can often be sacrificed at the expense of a focus on algorithmic procedures. Computer‐based simulations can expand an undergraduate mathematics instructor's opportunity to explore high‐level mathematical concepts in an applied environment. This study describes one instructor's approach to incorporating simulations and classroom discussions in a differential equations course and the subsequent effects on student learning attitudes and outcomes. Students made modest gains in the area of conceptualizing and applying ideas regarding solutions to differential equations in this learning environment. Implications of the study include the identification of specific gains relative to computer‐mediated learning environments and recommendations for using simulations to support conceptual development.     

Hadamard-type fractional differential equations for the system of integral inequalities on time scales

ABSTRACT

This paper investigates some system of integral inequalities of one independent variable on time scales. The conclusion can be obtained by using Hadamard-type fractional differential equations and Greene's method which bring together and expand some integral inequalities on time scales. The established inequalities give explicit bounds on unknown functions which can be utilized as a key in examining the properties of certain classes of partial dynamic equations and difference equations on time scales. As an application, a system of fractional differential equations is considered to explain the value of our results.     

Locating starting points in differential equations: a realistic mathematics education approach

The paper reports on ongoing developmental research efforts to adapt the instructional design perspective of Realistic Mathematics Education (RME) to the learning and teaching of collegiate mathematics, using differential equations as a specific case. This report focuses on the RME design heuristic of guided reinvention as a means to locate a starting point for an instructional sequence for first-order differential equations and highlights the cyclical process instructional design and analysis of student learning. The instance of starting with a rate of change equation as an experientially real mathematical context is taken as a case for illustrating how university students might experience the creation of mathematical ideas. In particular, it is shown how three students came to reason conceptually about rate and in the process, develop their own informal Euler method for approximating solution functions to differential equations.     

It's not the done thing: social norms governing students’ passive behaviour in undergraduate mathematics lectures

Students often play a passive role in large-scale lectures in undergraduate mathematics courses: they observe the lecturer demonstrate mathematical procedures, but they rarely engage in authentic mathematical activity themselves. This study uses semi-structured interviews of undergraduate students to investigate the implicit and explicit social norms and expectations that influence students to maintain their passive roles during lectures. Students were aware that their passivity was influenced by social norms, but perceived these norms as necessary for allowing the lecturer to get through the content in the allotted lecture time, while enabling students to avoid being publicly embarrassed in the lecture. However, the students appreciated opportunities to work on examples in small groups during lectures. We argue that the success of small group interactions during large-scale lectures depends on students and lecturers establishing supportive social norms, and adjusting their lecture goals from ‘covering the content’ to ‘developing mathematical understanding’.     

Problem solving heuristics,affect, and discrete mathematics

It has been suggested that activities in discrete mathematics allow a kind of new beginning for students and teachers. Students who have been “turned off” by traditional school mathematics, and teachers who have long ago routinized their instruction, can find in the domain of discrete mathematics opportunities for mathematical discovery and interesting, nonroutine problem solving. Sometimes formerly low-achieving students demonstrate mathematical abilities their teachers did not know they had. To take maximum advantage of these possibilities, it is important to know what kinds of thinking during problem solving can be naturally evoked by discrete mathematical situations—so that in developing a curriculum, the objectives can include pathways to desired mathematical reasoning processes. This article discusses some of these ways of thinking, with special attention to the idea of “modeling the general on the particular.” Some comments are also offered about students' possible affective pathways and structures.     

Mathematics and Science Teachers' Use of and Confidence in Empirical Reasoning: Implications for STEM Teacher Preparation

The recent trend to unite mathematically related disciplines (science, technology, engineering, and mathematics) under the broader umbrella of STEM education has advantages. In this new educational context of integration, however, STEM teachers need to be able to distinguish between sufficient proof and reasoning across different disciplines, particularly between the status of inductive and deductive modes of reasoning in mathematics. Through a specific set of mathematical conjectures, researchers explored differences between mathematics (n = 24) and science (n = 23) teachers' reasoning schemes, as well as the confidence they had in their justifications. Results from the study indicate differences between the two groups in terms of their levels of mathematical proof, as well as correlational trends that inform their confidence across these levels. Implications particularly for teacher training and preparation within the context of an integrated STEM education model are discussed.     

Characteristics of Second Graders' Mathematical Writing

This study compared the characteristics of second graders' mathematical writing between an intervention and comparison group. Two six‐week Project M2 units were implemented with students in the intervention group. The units position students to communicate in ways similar to mathematicians, including engaging in verbal discourse where they themselves make sense of the mathematics through discussion and debate, writing about their reasoning on an ongoing basis, and utilizing mathematical vocabulary while communicating in any medium. Students in the comparison group learned from the regular school curriculum. Students in both the intervention and comparison groups conveyed high and low levels of content knowledge as indicated in archived data from an open‐response end‐of‐the‐year assessment. A multivariate analysis of variance indicated several differences favoring the intervention group. Both the high‐ and low‐level intervention subgroups outperformed the comparison group in their ability to (a) provide reasoning, (b) attempt to use formal mathematical vocabulary, and (c) correctly use formal mathematical vocabulary in their writing. The low‐level intervention subgroup also outperformed the respective comparison subgroup in their use of (a) complete sentences and (b) linking words. There were no differences between groups in their attempt at writing and attempts at and usage of informal mathematical vocabulary.     

The analysis of the understanding of the three-dimensional (Euclidian) space and the two-variable function concept by university students

Conceptual understanding is being emphasized in mathematics education. Students often have difficulty understanding the multi-variable function, a key concept. Based on the APOS theory, which analyzes the cognitive structures formed by individuals in learning a mathematical concept and produces components related to that learning, this study analyzes the conceptual understanding of three-dimensional spaces and two-variable functions by university students. The genetic decomposition of these concepts proposed by Trigueros and Martinez-Planell is also considered. The analyzes results revealed that only one student constructed the concept of three-dimensional space as an object within the framework of genetic decomposition. Some students could not relate the concepts of two-variable function and three-dimensional space. Students who could perform algebraic operations had problems related to geometric representation. This study suggests the refinement of genetic decomposition to include, e.g., mental construction steps for writing algebraic equations of special surfaces whose graphs are given in R³.     

An Exploratory Study of College Students' Views of Mathematical Thinking in a Historical Approach Calculus Course

Heuristic training alone is not enough for developing one's mathematical thinking. One missing component is a mathematical point of view. This study reports findings regarding outcomes of a historical approach calculus course to foster Taiwanese college students' views of mathematical thinking. This study consisted of 3 stages. During the initial phase, 44 engineering majors' views on mathematical thinking were tabulated by an open-ended questionnaire, and 9 randomly selected students were invited to participate in follow-up interviews. Students then received an 18-week historical approach calculus course in which mathematical concepts were problematized to challenge their intuition-based empirical beliefs about doing mathematics. Near the end of the semester, all participants answered the identical questionnaire, and we interviewed the same students to pinpoint any shifts in their views on mathematical thinking. We found that participants were more likely to value logical sense, creativity, and imagination in doing mathematics. Further, students were leaning toward a conservative attitude toward certainty of mathematical knowledge. Participants' focus seemingly shifted from mathematics as a product to mathematics as a process.     

Observing mathematical fluency through students’ oral responses

‘Procedural’ fluency in mathematics is often judged solely on numerical representations. ‘Mathematical’ fluency incorporates explaining and justifying as well as producing correct numerical solutions. To observe mathematical fluency, representations additional to a student’s numerical work should be considered. This paper presents analysis of students’ oral responses. Findings suggested oral responses are important vantage points from which to view fluency – particularly characteristics harder to notice through numerical work such as reasoning. Students’ oral responses were particularly important when students’ written (language) responses were absent/inconsistent. Findings also revealed the importance of everyday language alongside technical terms for observing reasoning as a fluency characteristic. Students used high modality verbs and language features, such as connectives, to explain concepts and justify their thinking. The results of this study purport that to gain a fuller picture of students’ fluency, specifically their explanations or reasoning, students’ oral responses should be analyzed, not simply numerical work.     

Mathematical Dimensions of Students' Use of Proportional Reasoning In High School Physics

This study examines the mathematical processes used by students when solving physics tasks requiring proportional reasoning. The study investigates students' understanding and explanations of their mathematical processes. A qualitative and interpretive case study was conducted with 6 students from a coeducational urban high school for 5 months. Students were engaged with some high school physics tasks requiring proportional reasoning, during which a hermeneutic dialectic design was used to investigate their processes, understandings, and difficulties. Research techniques such as interviews, dialectical discourses, journal dialogue, and video and audio recordings were employed to generate, analyze, and interpret data. Results of the study indicate that the students employed mathematical proportional reasoning patterns and algorithms which they could not explain. Students also had difficulties translating physics tasks into mathematical statements, symbols, and relations. Students could not perform mathematical operations that were not directly obvious from the physics tasks, and some had difficulty with division. Students did not have adequate understanding of the mathematical processes involved in proportional reasoning.     

Student performance and attitudes in a collaborative and flipped linear algebra course

Flipped learning is gaining traction in K-12 for enhancing students’ problem-solving skills at an early age; however, there is relatively little large-scale research showing its effectiveness in promoting better learning outcomes in higher education, especially in mathematics classes. In this study, we examined the data compiled from both quantitative and qualitative measures such as item scores on a common final and attitude survey results between a flipped and a traditional Introductory Linear Algebra class taught by two individual instructors at a state university in California in Fall 2013. Students in the flipped class were asked to watch short video lectures made by the instructor and complete a short online quiz prior to each class attendance. The class time was completely devoted to problem solving in group settings where students were prompted to communicate their reasoning with proper mathematical terms and structured sentences verbally and in writing. Examination of the quality and depth of student responses from the common final exam showed that students in the flipped class produced more comprehensive and well-explained responses to the questions that required reasoning, creating examples, and more complex use of mathematical objects. Furthermore, students in the flipped class performed superiorly in the overall comprehension of the content with a 21% increase in the median final exam score. Overall, students felt more confident about their ability to learn mathematics independently, showed better retention of materials over time, and enjoyed the flipped experience.     

Asymptotic existence theorems for formal power series whose coefficients satisfy certain partial differential recursions

We study power series whose coefficients are holomorphic functions of another complex variable and a nonnegative real parameter s, and are given by a differential recursion equation. For positive integer s, series of this form naturally occur as formal solutions of some partial differential equations with constant coefficients, while for s=0 they satisfy certain perturbed linear ordinary differential equations. For arbitrary s?0, these series solve a differential-integral equation. Such power series, in general, are not multisummable. However, we shall prove existence of solutions of the same differential-integral equation that in sectors of, in general, maximal opening have the formal series as their asymptotic expansion. Furthermore, we shall indicate that the solutions so obtained can be related to one another in a fairly explicit manner, thus exhibiting a Stokes phenomenon.     

Writing as a Tool to Demonstrate Mathematical Understanding

In this study, I examine how using a writers' workshop model in mathematics creates a space for students to write about their mathematical thinking and problem solving and how their writing impacts instruction. This case study of one classroom with one teacher spanned 6 weeks and included 18 implementations of an adapted version of the Writers' Workshop (WW) in a fourth‐grade mathematics class. On a biweekly basis, the data were reviewed and changes made to the model. The analysis of the students' writing revealed (a) their understandings and misunderstandings of the mathematical content, (b) their readiness for more challenging tasks, and (c) their connections to prior knowledge. Students used writing to demonstrate their understanding of mathematics and show their mathematical processes. In some cases, examining only the numerical work failed to illuminate the students' understanding, their writing provided deeper insight. Students recognized writing as a tool for learning; this was evident in interview responses.     

Difference and differential equations in the mathematics of finance

Many problems arising in the mathematics of finance involve identical money flows at regular time intervals and are typically solved by appropriate valuation at a focal date or by setting up an equation of value. It is shown here that such problems can be viewed as special cases of a certain class of first‐order difference equations. Problems relating to continuous money flows can be viewed analogously as special cases of a certain class of ordinary first‐order differential equations. Students should thus be encouraged to view the concepts and techniques of the mathematics of finance as not being inherently different from those prevalent in more traditional applied mathematics.     

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.