Graphical Representations of Speed: Obstacles Preservice K‐8 Teachers Experience

This study examines the difficulties college students experience when creating and interpreting graphs in which speed is one of the variables. Nineteen students, all preservice elementary or middle school teachers, completed an upper‐level course exploring algebraic concepts. Although all of these preservice teachers had previously completed several mathematics courses, including calculus, they demonstrated widespread misconceptions about the variable speed. This study identifies four cognitive obstacles held by the students, provides excerpts of their graphical constructions and verbal interpretations, and discusses potential causes for the confusion. In particular, misconceptions arose when students interpreted the behavior and nature of speed within a graphical context, as well as in situations where they were required to construct a graph involving speed as a variable. The study concludes by offering implications for the teaching and learning of speed and its interpretation within a graphical setting.     

Interpreting a graph and constructing its derivative graph: stability and change in students’ conceptions

This present study investigated engineering students’ conceptions and misconceptions related to derivative, particularly interpreting the graph of a function and constructing its derivative graph. Participants were 147 first year engineering students from four universities enrolled in first year undergraduate calculus courses with or without the incorporation of computers for the purposes of seeing the power of visualization, investigating worked examples given in steps and solving various questions related to the worked examples, assisting conceptual understanding, and/or providing feedback besides lectures in the classroom. Students were tested before and after being exposed to instruction on differentiation and integration by a diagnostic test measuring their understanding of major aspects of calculus. Follow-up interviews were conducted with 18 students. Analyses of the results revealed that A-level student's performance was improving more than non-A-level students, particularly in computer groups. The analyses of the students’ written and oral responses in all groups indicated that prototypes, poor understanding of the notion of limit, confusion between the process and the product, and difficulties in using graphical information to give meaning to symbolic representation account for the errors and the misconceptions identified.     

Focuses of awareness in the process of learning the fundamental theorem of calculus with digital technologies

This paper brings together three themes: the fundamental theorem of the calculus (FTC), digital learning environments in which the FTC may be taught, and what we term “focuses of awareness.” The latter are derived from Radford’s theory of objectification: they are nodal activities through which students become progressively aware of key mathematical ideas structuring a mathematical concept. The research looked at 13 pairs of 17-year-old students who are not yet familiar with the concept of integration. Students were asked to consider possible connections between multiple-linked representations, including function graphs, accumulation function graphs, and tables of values of the accumulation function. Three rounds of analysis yielded nine focuses in the process of students’ learning the FTC with a digital tool as well as the relationship between them. In addition, the activities performed by the students to become aware of the focuses are described and theoretical and pedagogical implementations are also discussed.     

Vertex-magic labelings of regular graphs II

Previously the first author has shown how to construct vertex-magic total labelings (VMTLs) for large families of regular graphs. The construction proceeds by successively adding arbitrary 2-factors to a regular graph of order n which possesses a strong VMTL, to produce a regular graph of the same order but larger size. In this paper, we exploit this construction method. We are able to show that for any r≥4, every r-regular graph of odd order n≤17 has a strong VMTL. We show how to produce strong labelings for some families of 2-regular graphs since these are used as the starting points of our construction. While even-order regular graphs are much harder to deal with, we introduce ‘mirror’ labelings which provide a suitable starting point from which the construction can proceed. We are able to show that several large classes of r-regular graphs of even order (including some Hamiltonian graphs) have VMTLs.     

The function of graphs and gestures in algorithmatization

The purpose of this paper is to present evidence supporting the conjecture that graphs and gestures may function in different capacities depending on whether they are used to develop an algorithm or whether they extend or apply a previously developed algorithm in a new context. I illustrate these ideas using an example from undergraduate differential equations in which students move through a sequence of Realistic Mathematics Education (RME)-inspired instructional materials to create the Euler method algorithm for approximating solutions to differential equations. The function of graphs and gestures in the creation and subsequent use of the Euler method algorithm is explored. If students’ primary goal was algorithmatizing ‘from scratch’, they used imagery of graphing and gesturing as a tool for reasoning. However if students’ primary goal was to make predictions in a new context, they used their previously developed Euler algorithm to reason and used graphs and gestures to clarify their ideas.     

A Strategy for Writing Equations of Graphs

A strategy for writing equations of graphs is introduced to help students and teachers build strong conceptual connections between the symbolic representations of algebra and the spatial representations of geometry. The strategy helps students and teachers weave the conceptual fabric of equations and graphs by (a) moving from unknown graphs to known graphs rather than from known to unknown graphs and by (b) moving from spatial representations to algebraic representations rather than from algebraic to spatial representations. Beginning with the unknown graph is distinctly different from present practices and can lead to significant and useful change in curriculum and instructional practices.     

Competent Workplace Mathematics: How Signs Become Transparent in Use

Past research has shown that many scientists, when asked to interpret unfamiliar graphs that have nevertheless been culled from introductory undergraduate courses in their own field, experience problems and cannot give the standard answer accepted in the field. Yet, these same scientists turn out to be highly competent when it comes to graphs from their immediate domain of research. In this research, which is based on ethnographic studies among scientists and technicians, I show how graph interpretation in one biology laboratory initially required tool (computer)mediation. After scientists had become familiar with the phenomenon, data collection,and resulting graphs, they interpret the latter correctly without requiring prior transformation. Furthermore, in the course of their work, they established what they understood to be a one-to-one correspondence between graphs and some aspect of the natural world. As a result of scientists' embodied laboratory work, talk about the graphical representation and talk about the object represented are often indistinguishable. The process of developing competency in graph use is equivalent to that of a tool that becomes transparent to the consciousness of its user;I describe this process in terms of activity theory as a transition from tool-mediated action to operation. This revised version was published online in July 2006 with corrections to the Cover Date.     

What are the last digits of …?

We propose a class assignment where students are asked to construct and implement an efficient algorithm to calculate the last digits of a positive integral power of a positive integer. The mathematical prerequisites for this assignment are very limited: knowledge of remainder calculus and the binary representation of a positive integer. The periodicity of the last digits is studied by means of the Euler totient function and the Carmichael function.     

Nested hierarchies in planar graphs

We construct a partial order relation which acts on the set of 3-cliques of a maximal planar graph G and defines a unique hierarchy. We demonstrate that G is the union of a set of special subgraphs, named ‘bubbles’, that are themselves maximal planar graphs. The graph G is retrieved by connecting these bubbles in a tree structure where neighboring bubbles are joined together by a 3-clique. Bubbles naturally provide the subdivision of G into communities and the tree structure defines the hierarchical relations between these communities.     

Contrasting Cases of Calculus Students' Understanding of Derivative Graphs

This study adds momentum to the ongoing discussion clarifying the merits of visualization and analysis in mathematical thinking. Our goal was to gain understanding of three calculus students' mental processes and images used to create meaning for derivative graphs. We contrast the thinking processes of these three students as they attempted to sketch antiderivative graphs when presented with derivative graphs. These students constructed different and idiosyncratic images and representations leading to different understandings of derivative graphs. Our results suggest that the two students whose cognitive preferences were strongly visual or analytic and who did not synthesize visual and analytic thinking experienced different difficulties associated with their preferred modes for mathematical representation and thinking. Even the student who did synthesize these modes to some extent, to good effect, experienced difficulty when he did not do so. We discuss pedagogical implications for these results in a final section.     

Fifth graders’ additive and multiplicative reasoning: Establishing connections across conceptual fields using a graph

This paper looks at 21 fifth grade students as they discuss a linear graph in the Cartesian plane. The problem presented to students depicted a graph showing distance as a function of elapsed time for a person walking at a constant rate of 5 miles/h. The question asked students to consider how many more hours, after having already walked 4 h, would be required to reach 35 miles. To answer this question, the students needed to extend the graph that was presented, either mentally or on paper, as the axes did not go up to 7 h or 35 miles. They also needed to be able to consider not only the total number of hours to reach 35 miles, but also the interval of time after 4 h. The purpose of this paper is to consider the student responses from the viewpoint of multiplicative and additive reasoning, and specifically within Vergnaud's framework of multiplicative and additive conceptual fields and scalar and functional approaches to linear relationships (Vergnaud, 1994). The analysis shows that: some student answers cannot be classified as either scalar or functional; some students combined several kinds of approaches in their explanations; and that the representation of the problem using a graph may have facilitated responses that are different from those typically found when the representation presented is a function table.     

Shortest paths in linear time on minor-closed graph classes, with an application to Steiner tree approximation

We generalize the linear-time shortest-paths algorithm for planar graphs with nonnegative edge-weights of Henzinger et al. (1994) to work for any proper minor-closed class of graphs. We argue that their algorithm can not be adapted by standard methods to all proper minor-closed classes. By using recent deep results in graph minor theory, we show how to construct an appropriate recursive division in linear time for any graph excluding a fixed minor and how to transform the graph and its division afterwards, so that it has maximum degree three. Based on such a division, the original framework of Henzinger et al. can be applied. Afterwards, we show that using this algorithm, one can implement Mehlhorn’s (1988) 2-approximation algorithm for the Steiner tree problem in linear time on these graph classes.     

Complex calculators in the classroom: theoretical and practical reflections on teaching pre-calculus

University and older school students following scientific courses now use complex calculators with graphical, numerical and symbolic capabilities. In this context, the design of lessons for 11th grade pre-calculus students was a stimulating challenge.In the design of lessons, emphasising the role of mediation of calculators and the development of schemes of use in an 'instrumental genesis' was productive. Techniques, often discarded in teaching with technology, were viewed as a means to connect task to theories. Teaching techniques of use of a complex calculator in relation with 'traditional' techniques was considered to help students to develop instrumental and paper/pencil schemes, rich in mathematical meanings and to give sense to symbolic calculations as well as graphical and numerical approaches.The paper looks at tasks and techniques to help students to develop an appropriate instrumental genesis for algebra and functions, and to prepare for calculus. It then focuses on the potential of the calculator for connecting enactive representations and theoretical calculus. Finally, it looks at strategies to help students to experiment with symbolic concepts in calculus.This revised version was published online in September 2005 with corrections to the Cover Date.     

Homology Theory of Graphs

Higher homotopy of graphs has been defined in several articles. In Dochterman (Hom complexes and homotopy theory in the category of graphs. arXiv math/0605275 v2,28/09/2006, 2006), the authors asked for a companion homology theory. We define such a theory for the category of unoriented reflexive graphs; it exhibits a long exact sequence for a pair of graphs (G, A), satisfies an excision property and a Hurewicz theorem. This allows us to compute the top homology of the graphical n-spheres showing that the theory is not trivial and is able to detect n-dimensional holes in a graph. The long-term objective is to compare the homotopy of the topological and graphical spheres.     

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.     

Interactive video: A bridge between motion and math

This paper examines the characteristics of interactive digitized video as a medium in which motion is presented to students learning graphical representations. We situate graphs of motion as early topics in learning calculus, the bugaboo of many math students. In comparing video to both everyday perceptions and mathematical representations, we construct a conceptual framework that compares these three contexts along several dimensions: object extent, scale, time, and space. We then examine one student's experience constructing graphs of her own design from a video image and describe her work in the context of the our conceptual framework. To further specify the unique characteristics of video, we compare it as a medium with that of computer simulations of motion, in particular as studied by diSessa et al. (1991).The authors are listed in alphabetical order.     

Algorithms for recognition of regular properties and decomposition of recursive graph families

This paper focuses on two themes within the broad context of recursively definable graph classes. First, we generalize the series-parallel operations and establish exactly how far they can be extended subject to some consistency conditions. We show explicitly how Halin graphs are included in the extension. Second, for recursively constructed graphs in general, we construct a predicate calculus within which graph problems can be stated and for those so stated, a linear time algorithm exists and can be automatically generated. We discuss some issues related to practical automatic generation.     

Core-free,rank two coset geometries from edge-transitive bipartite graphs

It is known that the Levi graph of any rank two coset geometry is an edge-transitive graph, and thus coset geometries can be used to construct many edge transitive graphs. In this paper, we consider the reverse direction. Starting from edge-transitive graphs, we construct all associated core-free, rank two coset geometries. In particular, we focus on 3-valent and 4-valent graphs, and are able to construct coset geometries arising from these graphs. We summarize many properties of these coset geometries in a sequence of tables; in the 4-valent case we restrict to graphs that have relatively small vertex-stabilizers.     

The role of the design of interactive diagrams in teaching–learning the indefinite integral concept

This study was designed to micro-analyze the role of the design of interactive diagrams in learning-teaching the concept of the indefinite integral. The presented case study focuses on the engagement of one pair of 17-year-old students with an interactive diagram for graphically learning-teaching the indefinite integral concept. The authors performed three rounds of analysis to detect the mathematical elements involved in finding connections between the function graphs and the antiderivative function graphs and to identify how the design features of the interactive diagram functioned in the students’ engagement processes. The data analysis identified how the pedagogical functions designed in the interactive calculus diagram were reflected in the students’ learning-teaching processes with the diagram.     

Exploring Pi Using the Computer in Middle School Mathematics

The National Council of Teachers of Mathematics (NCTM) in its Curriculum and Evaluation Standards for School Mathematics encourages a middle school curriculum that integrates technology. It recommends that students should be able to identify and use functional relationships and make connections among seemingly diverse concepts and topics. In this activity of exploring the derivation of Pi, students take a constructivist role by collecting data and making conjectures. Using the data they construct graphs and tables and discourse about appropriate algebraic representations. The computer is used as an instructional aid enabling the students to view the data in a variety of forms. They are encouraged to communicate about the connections among the various representations.