Constructing Graphical Representations: Middle Schoolers' Intuitions and Developing Knowledge About Slope and Y‐intercept

Middle‐school students are expected to understand key components of graphs, such as slope and y‐intercept. However, constructing graphs is a skill that has received relatively little research attention. This study examined students' construction of graphs of linear functions, focusing specifically on the relative difficulties of graphing slope and y‐intercept. Sixth‐graders' responses prior to formal instruction in graphing reveal their intuitions about slope and y‐intercept, and seventh‐ and eighth‐graders' performance indicates how instruction shapes understanding. Students' performance in graphing slope and y‐intercept from verbally presented linear functions was assessed both for graphs with quantitative features and graphs with qualitative features. Students had more difficulty graphing y‐intercept than slope, particularly in graphs with qualitative features. Errors also differed between contexts. The findings suggest that it would be valuable for additional instructional time to be devoted to y‐intercept and to qualitative contexts.     

Stochastic kronecker graphs

A random graph model based on Kronecker products of probability matrices has been recently proposed as a generative model for large‐scale real‐world networks such as the web. This model simultaneously captures several well‐known properties of real‐world networks; in particular, it gives rise to a heavy‐tailed degree distribution, has a low diameter, and obeys the densification power law. Most properties of Kronecker products of graphs (such as connectivity and diameter) are only rigorously analyzed in the deterministic case. In this article, we study the basic properties of stochastic Kronecker products based on an initiator matrix of size two (which is the case that is shown to provide the best fit to many real‐world networks). We will show a phase transition for the emergence of the giant component and another phase transition for connectivity, and prove that such graphs have constant diameters beyond the connectivity threshold, but are not searchable using a decentralized algorithm. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 38, 453–466, 2011     

The n‐ordered graphs: A new graph class

For a positive integer n, we introduce the new graph class of n‐ordered graphs, which generalize partial n‐trees. Several characterizations are given for the finite n‐ordered graphs, including one via a combinatorial game. We introduce new countably infinite graphs R⁽ⁿ⁾, which we name the infinite random n‐ordered graphs. The graphs R⁽ⁿ⁾ play a crucial role in the theory of n‐ordered graphs, and are inspired by recent research on the web graph and the infinite random graph. We characterize R⁽ⁿ⁾ as a limit of a random process, and via an adjacency property and a certain folding operation. We prove that the induced subgraphs of R⁽ⁿ⁾ are exactly the countable n‐ordered graphs. We show that all countable groups embed in the automorphism group of R⁽ⁿ⁾. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 204–218, 2009     

Implicit Assumptions and Communication in Statistics

Statistics is becoming an increasingly important component of the school curriculum. The subject provides students with opportunities to explore situations that they read about or face in their everyday lives. This article describes a statistics project that focuses on communicating conclusions and on understanding implicit assumptions that may affect conclusions based on the information. The ability to make a table and draw graphs and charts is an important component of statistical education. However, the ability to communicate ideas to others and understand implicit assumptions associated with data are key objectives of educating students for a world that increasingly relies on statistical information as a means of mass communication.     

Semi‐transitive graphs

In this paper, we first consider graphs allowing symmetry groups which act transitively on edges but not on darts (directed edges). We see that there are two ways in which this can happen and we introduce the terms bi‐transitive and semi‐transitive to describe them. We examine the elementary implications of each condition and consider families of examples; primary among these are the semi‐transitive spider‐graphs PS(k,N;r) and MPS(k,N;r). We show how a product operation can be used to produce larger graphs of each type from smaller ones. We introduce the alternet of a directed graph. This links the two conditions, for each alternet of a semi‐transitive graph (if it has more than one) is a bi‐transitive graph. We show how the alternets can be used to understand the structure of a semi‐transitive graph, and that the action of the group on the set of alternets can be an interesting structure in its own right. We use alternets to define the attachment number of the graph, and the important special cases of tightly attached and loosely attached graphs. In the case of tightly attached graphs, we show an addressing scheme to describe the graph with coordinates. Finally, we use the addressing scheme to complete the classification of tightly attached semi‐transitive graphs of degree 4 begun by Marus?ic? and Praeger. This classification shows that nearly all such graphs are spider‐graphs. © 2003 Wiley Periodicals, Inc. J Graph Theory 45: 1–27, 2004     

Teachers' Considerations of Students' Thinking During Mathematics Lesson Design

Teachers' abilities to design mathematics lessons are related to their capability to mobilize resources to meeting intended learning goals based on their noticing. In this process, knowing how teachers consider Students' thinking is important for understanding how they are making decisions to promote student learning. While teaching, what teachers notice influences their decision‐making process. This article explores the mathematics lesson planning practices of four 4th‐grade teachers at the same school to understand how their consideration of Students' learning influences planning decisions. Case study methodology was used to gain an in‐depth perspective of the mathematics planning practices of the teachers. Results indicate the teachers took varying approaches in how they considered students. One teacher adapted instruction based on Students' conceptual understanding, two teachers aimed at producing skill‐efficient students, and the final teacher regulated learning with a strict adherence to daily lessons in curriculum materials, with little emphasis on student understanding. These findings highlight the importance of providing professional development support to teachers focused on their noticing and considerations of Students' mathematical understandings as related to learning outcomes. These findings are distinguished from other studies because of the focus on how teachers consider Students' thinking during lesson planning. This article features a Research to Practice Companion Article . Please click on the supporting information link below to access.     

Commentary: Linking epistemology and semio-cognitive modeling in visualization

To situate the contributions of these research articles on visualization as an epistemological learning tool, we have employed mathematical, cognitive and functional criteria. Mathematical criteria refer to mathematical content, or more precisely the areas to which they belong: whole numbers (numeracy), algebra, calculus and geometry. They lead us to characterize the “tools” of visualization according to the number of dimensions of the diagrams used in experiments. From a cognitive point of view, visualization should not be confused with a visualization “tool,” which is often called “diagram” and is in fact a semiotic production. To understand how visualization springs from any diagram, we must resort to the notion of figural unity. It results methodologically in the two following criteria and questions: (1) In a given diagram, what are the figural units recognized by the students? (2) What are the mathematically relevant figural units that pupils should recognize? The analysis of difficulties of visualization in mathematical learning and the value of “tools” of visualization depend on the gap between the observations for these two questions. Visualization meets functions that can be quite different from both a cognitive and epistemological point of view. It can fulfill a help function by materializing mathematical relations or transformations in pictures or movements. This function is essential in the early numerical activities in which case the used diagrams are specifically iconic representations. Visualization can also fulfill a heuristic function for solving problems in which case the used diagrams such as graphs and geometrical figures are intrinsically mathematical and are used for the modeling of real problems. Most of the papers in this special issue concern the tools of visualization for whole numbers, their properties, and calculation algorithms. They show the semiotic complexity of classical diagrams assumed as obvious to students. In teaching experiments or case studies they explore new ways to introduce them and make use by students. But they lie within frameworks of a conceptual construction of numbers and meaning of calculation algorithms, which lead to underestimating the importance of the cognitive process specific to mathematical activity. The other papers concern the tools of mathematical visualization at higher levels of teaching. They are based on very simple tasks that develop the ability to see 3D objects by touch of 2D objects or use visual data to reason. They remain short of the crucial problem of graphs and geometrical figures as tools of visualization, or they go beyond that with their presupposition of students' ability to coordinate them with another register of semiotic representation, verbal or algebraic.     

Triangle‐free graphs that are signable without even holes

We characterize triangle‐free graphs for which there exists a subset of edges that intersects every chordless cycle in an odd number of edges (TF odd‐signable graphs). These graphs arise as building blocks of a decomposition theorem (for cap‐free odd‐signable graphs) obtained by the same authors. We give a polytime algorithm to test membership in this class. This algorithm is itself based on a decomposition theorem. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 204–220, 2000     

Using Sociocultural Theory to Teach Mathematics: A Vygotskian Perspective

This study describes an elementary teacher's implementation of sociocultural theory in practice. Communication is central to teaching with a sociocultural approach and to the understanding of students; teachers who use this theory involve students in explaining and justifying their thinking. In this study ethnographic research methods were used to collect data for 4 1/2 months in order to understand the mathematical culture of this fourth‐grade class and to portray how the teacher used a sociocultural approach to teach mathematics. To portray this teaching approach, teaching episodes from the teacher's mathematics lessons are described, and these episodes are analyzed to demonstrate how students created taken‐as‐shared meanings of mathematics. Excerpts from interviews with the teacher are also used to describe this teacher's thinking about her teaching.     

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.     

Preparing K‐8 Preservice Teachers in a Content Course for Standards‐Based Mathematics Pedagogy

Many K–8 preservice teachers have not experienced learning mathematics in a standards‐based classroom. This article describes a mathematics content course designed to provide preservice teachers experiences in learning mathematics that will help build a solid foundation for a standards‐based methods course. The content course focuses on developing preservice teachers' mathematical knowledge, as well as helping them realize what it means to learn mathematics that is taught using the pedagogy in the Principles and Standards for School Mathematics ( National Council of Teachers of Mathematics, 2000 ). Furthermore, findings are presented from a study on this course that describe students' pre‐ and postcourse beliefs, attitudes, and perceptions of what it means to learn and teach mathematics. These findings provide evidence that the students in the study are beginning to understand what is meant by a standards‐based classroom. Data were collected from surveys and interviews. Quotes from the students who aspire to be elementary teachers are used throughout the article to support the points.     

Changes in Students' Science Ability Produced by Multimedia Learning Environments: Application of the Linear Logistic Model for Change

The purpose of this study was to measure changes in students' science proficiency produced by a multimedia learning environment, Astronomy Village: Investigating the Solar System, developed at Wheeling Jesuit University's Center for Educational Technologies with funding from the National Science Foundation. The inquiry‐based design of Astronomy Village supports middle school students in learning fundamental concepts in life, earth, and physical science. Astronomy Village was compared to an alternative treatment that simulated elements of traditional science instruction using web site access to background materials and content in Astronomy Village. The results indicate sizable treatment effects for two groups of Astronomy Village students, as well as for the alternative treatment group. Differences in the treatment effect sizes among the three treatment groups reveal the relative merits of different approaches to using technology. The Linear Logistic Model for Change applied in this study is beneficial for comparing alternative uses of technology, since it separates effects due to treatments from natural trend effects and eliminates drawbacks of traditional statistical designs for pretest‐posttest changes.     

Using Emerging Technologies to Help Bridge the Gap Between University Theory and Classroom Practice: Challenges and Successes

This article describes and illuminates the challenges that the authors faced as we integrated a web‐supported professional development system into elementary science methods courses housed at three different universities. Using a design experiment framework, the challenges and difficulties encountered while attempting to develop and sustain effective discussions about inquiry‐based teaching are discussed. Three main issues were identified through this analysis: (a) creating meaningful interactions for preservice teachers, (b) supporting preservice teacher reflection and articulation of their belief systems, and (c) technical, social, and institutional challenges of using a World Wide Web based professional development system. The article closes with recommendations concerning the implementation of a web‐based professional development system into elementary methods science courses and describes what appear to be successful strategies for fostering a collaborative atmosphere between teacher educators, preservice teachers, and in‐service teachers.     

Enhancing Teaching Assistants' (TAs') Inquiry Teaching by Means of Teaching Observations and Reflective Discourse

Recent education reform efforts advocate teaching the process of science (inquiry) in undergraduate lecture and laboratory classes. To meet this challenge, professional development for the graduate student instructors (teaching assistants, or TAs) often assigned to teach these classes is needed. This study explored the implementation of an observation protocol designed to support peer observation and reflection among TAs teaching inquiry‐based undergraduate biology laboratories. The researchers of this study and TA‐peers who are experienced in teaching inquiry used the protocol to observe novice TAs at the beginning, mid‐point, and end of a semester‐long teaching assignment. Novice TAs used a modified version of this protocol to observe an experienced and a novice TA. Analysis of post‐semester interview data indicated engaging in both sets of observations and post‐observation discussions facilitated by the protocol gave novice TAs new ways to teach content, guidance on implementing pedagogical theory, and means to improve communications with students and classroom management skills. Researchers also used quantitative data collected from the observations to document frequency of teaching and student behaviors associated with teaching inquiry and how these frequencies changed during the semester. Considerations for how to use both sets of data to inform changes in TA professional development are discussed.     

Vertex Cuts

Given a connected graph, in many cases it is possible to construct a structure tree that provides information about the ends of the graph or its connectivity. For example Stallings' theorem on the structure of groups with more than one end can be proved by analyzing the action of the group on a structure tree and Tutte used a structure tree to investigate finite 2‐connected graphs, that are not 3‐connected. Most of these structure tree theories have been based on edge cuts, which are components of the graph obtained by removing finitely many edges. A new axiomatic theory is described here using vertex cuts, components of the graph obtained by removing finitely many vertices. This generalizes Tutte's decomposition of 2‐connected graphs to k‐connected graphs for any k, in finite and infinite graphs. The theory can be applied to nonlocally finite graphs with more than one vertex end, i.e. ends that can be separated by removing a finite number of vertices. This gives a decomposition for a group acting on such a graph, generalizing Stallings' theorem. Further applications include the classification of distance transitive graphs and k‐CS‐transitive graphs.     

Complexity of networks I: The set‐complexity of binary graphs

The balance between symmetry and randomness as a property of networks can be viewed as a kind of “complexity.” We use here our previously defined “set complexity” measure (Galas et al., IEEE Trans Inf Theory 2010, 56), which was used to approach the problem of defining biological information, in the mathematical analysis of networks. This information theoretic measure is used to explore the complexity of binary, undirected graphs. The complexities, Ψ, of some specific classes of graphs can be calculated in closed form. Some simple graphs have a complexity value of zero, but graphs with significant values of Ψ are rare. We find that the most complex of the simple graphs are the complete bipartite graphs (CBGs). In this simple case, the complexity, Ψ, is a strong function of the size of the two node sets in these graphs. We find the maximum Ψ binary graphs as well. These graphs are distinct from, but similar to CBGs. Finally, we explore directed and stochastic processes for growing graphs (hill‐climbing and random duplication, respectively) and find that node duplication and partial node duplication conserve interesting graph properties. Partial duplication can grow extremely complex graphs, while full node duplication cannot do so. By examining the eigenvalue spectrum of the graph Laplacian we characterize the symmetry of the graphs and demonstrate that, in general, breaking specific symmetries of the binary graphs increases the set‐based complexity, Ψ. The implications of these results for more complex, multiparameter graphs, and for physical and biological networks and the processes of network evolution are discussed. © 2011 Wiley Periodicals, Inc. Complexity, 17,51–64, 2011     

Deterministic random walks on finite graphs

The rotor‐router model, also known as the Propp machine, is a deterministic process analogous to a random walk on a graph. Instead of distributing tokens to randomly chosen neighbors, the Propp machine deterministically serves the neighbors in a fixed order by associating to each vertex a “rotor‐router” pointing to one of its neighbors. This paper investigates the discrepancy at a single vertex between the number of tokens in the rotor‐router model and the expected number of tokens in a random walk, for finite graphs in general. We show that the discrepancy is bounded by O (mn) at any time for any initial configuration if the corresponding random walk is lazy and reversible, where n and m denote the numbers of nodes and edges, respectively. For a lower bound, we show examples of graphs and initial configurations for which the discrepancy at a single vertex is Ω(m) at any time (> 0). For some special graphs, namely hypercube skeletons and Johnson graphs, we give a polylogarithmic upper bound, in terms of the number of nodes, for the discrepancy. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 46,739–761, 2015     

Learning to See: Making Sense of the Mathematics of Change in Middle School

In this article, we will describe the results of a study of 6th grade students learning about the mathematics of change. The students in this study worked with software environments for the computer and the graphing calculator that included a simulation of a moving elevator, linked to a graph of its velocity vs. time. We will describe how the students and their teacher negotiated the mathematical meanings of these representations, in interaction with the software and other representational tools available in the classroom. The class developed ways of selectively attending to specific features of stacks of centimeter cubes, hand-drawn graphs, and graphs (labeled velocity vs. time) on the computer screen. In addition, the class became adept at imagining the motions that corresponded to various velocity vs. time graphs. In this article, we describe this development as a process of learning to see mathematical representations of motion. The main question this article addresses is: How do students learn to see mathematical representations in ways that are consistent with the discipline of mathematics? This revised version was published online in July 2006 with corrections to the Cover Date.     

Sequential and non-sequential acceptance sampling plans for autocorrelated processes using ARMA(p,q) models

This paper presents variable acceptance sampling plans based on the assumption that consecutive observations on a quality characteristic(X) are autocorrelated and are governed by a stationary autoregressive moving average (ARMA) process. The sampling plans are obtained under the assumption that an adequate ARMA model can be identified based on historical data from the process. Two types of acceptance sampling plans are presented: (1) Non-sequential acceptance sampling: In this case historical data is available based on which an ARMA model is identified. Parameter estimates are used to determine the action limit (k) and the sample size(n). A decision regarding acceptance of a process is made after a complete sample of size n is selected. (2) Sequential acceptance sampling: Here too historical data is available based on which an ARMA model is identified. A decision regarding whether or not to accept a process is made after each individual sample observation becomes available. The concept of Sequential Probability Ratio Test (SPRT) is used to derive the sampling plans. Simulation studies are used to assess the effect of uncertainties in parameter estimates and the effect of model misidentification (based on historical data) on sample size for the sampling plans. Macros for computing the required sample size using both methods based on several ARMA models can be found on the author’s web page .     

An exploratory study on student understandings of derivatives in real-world,non-kinematics contexts

Much research on calculus students’ understanding of applied derivatives has been done in kinematics-based contexts (i.e. position, velocity, acceleration). However, given the wide range of applications in science and engineering that are not based on kinematics, nor even explicitly on time, it is important to know how students understand applied derivatives in non-kinematics contexts. In this study, interviews with six students and surveys with 38 students were used to explore students’ “ways of understanding” and “ways of thinking” regarding applied, non-kinematics derivatives. In particular, six categories of ways of understanding emerged from the data as having been shared by a substantial portion of the students in this study: (1) covariation, (2) invoking time, (3) other symbols as constants, (4) other symbols as implicit functions, (5) implicit differentiation, and (6) output values as amounts instead of rates of change.