Review of Graph Comprehension Research: Implications for Instruction

Graphs are commonly used in textbooks and educational software, and can help students understand science and social science data. However, students sometimes have difficulty comprehending information depicted in graphs. What makes a graph better or worse at communicating relevant quantitative information? How can students learn to interpret graphs more effectively? This article reviews the cognitive literature on how viewers comprehend graphs and the factors that influence viewers' interpretations. Three major factors are considered: the visual characteristics of a graph (e.g., format, animation, color, use of legend, size, etc.), a viewer's knowledge about graphs, and a viewer's knowledge and expectations about the content of the data in a graph. This article provides a set of guidelines for the presentation of graphs to students and considers the implications of graph comprehension research for the teaching of graphical literacy skills. Finally, this article discusses unresolved questions and directions for future research relevant to data presentation and the teaching of graphical literacy skills.     

Challenges with graph interpretation: a review of the literature

With the growing emphasis on the development of scientific inquiry skills, the display and interpretation of data are becoming increasingly important. Graph interpretation competence is, in fact, essential to understanding today’s world and to be scientifically literate. However, graph interpretation is a complex and challenging activity. Graph interpretation competence is affected by many factors, including aspects of graph characteristics, the content of the graph and viewers’ prior knowledge. For instance, the prior theory and expectations that students have may lead to biases and misinterpretation of graphs. One basic controversy that remains unanswered, for example, is what should we teach first in order to make students scientific literate, how to graph or how to interpret a graph? If it is the case that the ability to interpret a graph be developed prior to the ability to create, then it is important to understand what graph interpretation entails. This paper reviews current literature on graph interpretation competence and argues that it should be explicitly taught given its importance and its complexity.     

Interpretations of graphs by university biology students and practicing scientists: Toward a social practice view of scientific representation practices

Using graphs is a key social practice of professional science. As part of a research program that investigates the development of graphing practices from elementary school to professional science activities, this study was designed to investigate similarities and differences in graph‐related interpretations between scientists and college students engaged in collective graph interpretation. Forty‐five students in a second‐year university ecology course and four scientists participated in the study. Guided by domain‐ specific concerns, scientists' graph‐related activities were characterized by a large number of experience‐based, domain‐specific interpretive resources and practices. Students' group based activities were characterized by the lack of linguistic distinctions (between scientific terms) which led to ambiguities in group negotiations; there was also a lack of knowledge about specific organism populations which helped field ecologists construct meaning. Many students learned to provide correct answers to specific graphing questions but did not come to make linguistic distinctions or increase their knowledge of specific populations. In the absence of concerns other than to do well in the course, students did not appear to develop any general interpretive skills for graphs, but learned instead to apply the professor's interpretation. This is problematic because, as we have demonstrated, there are widely differing viable interpretations of the graph. Suggestions for changes in learning environments for graphing that should alleviate this problem are made. © 1999 John Wiley & Sons, Inc. J Res Sci Teach 36: 1020–1043, 1999     

含有Hamilton路的图的联图

Hamilton图是图论中重要的一类特殊图.主要证明了两个图的联图是Hamilton图,从而进一步证明了n个图的联图也是Hamilton图.     

Interpreting unfamiliar graphs: A generative, activity theoretic model

Research on graphing presents its results as if knowing and understanding were something stored in peoples' minds independent of the situation that they find themselves in. Thus, there are no models that situate interview responses to graphing tasks. How, then, we question, are the interview texts produced? How do respondents begin and end utterances? And, what is the relation between words and gestures used as part of the communication? Based on a database developed in two studies with research scientists (N = 37), we developed a theoretical framework using cultural-historical activity theory for understanding the texts produced during interviews. Our framework addresses three major findings, whose implications are discussed in the paper. First, the interview text is the contingent and situated product of the entire activity system, including interviewer and other aspects of the setting; the interview text can therefore not be reduced to the cognitive properties of the individual interviewee. Second, the interpretation unfolds in time, shaping the way the graph itself is perceived; unlike a written text that accompanies a graph, the verbally produced interview text therefore has to be analyzed through a moving interpretive window without recourse to subsequently produced talk. Third, gestures, speech, and the perceptual aspects of the graph currently salient to the interviewee, have to be understood as expressions that are irreducible to one another, requiring comprehensive research reports to present all modes of concurrent expression.     

四角系统的Z-变换图的Hamilton路

如果G表示一个四角系统,则G的Z-变换图Z(G)指如下定义的图:图Z(G)的所有顶点对应于四角系统G中的所有完美匹配,且Z(G)中的两个顶点有一条边相连当且仅当它们在G中对应的两个完美匹配的对称差恰好形成G的一个四角形.利用图同构的方法,证明了两类四角系统(L-四角系统和Z-四角系统)的Z-变换图必含有一条Hamilton路.     

具有小直径平面图的电力控制数

The problem of monitoring an electric power system by placing as few measurement devices in the system as possible is closely related to the well-known vertex covering and dominating set problems in graph theory. In this paper, it was shown that the power domination number of an outerplanar graph with the diameter two or a 2-connected outerplanar graph with the diameter three is precisely one. Upper bounds on the power domination number for a general planar graph with the diameter two or three were determined as an immediate consequences of results proven by Dorfling, et al. Also, an infinite family of outerplanar graphs with the diameter four having arbitrarily large power domination numbers were given.     

Teaching the design and interpretation of graphs through computer-aided graphical data analysis

Graphs are one of the primary means of exploration and communication in the practice of science, but students in science laboratories are customarily taught only the low-level mechanics of constructing a single kind of graph when given a table of information. The use of a microcomputer can relieve the drudgery of plotting, allowing students to pursue higher-level issues in the design and interpretation of graphs through repeated “thought experiments.” We introduced computer-assisted graphical data analysis to inner-city high school students with weak math and science backgrounds, emphasizing the dynamic manipulation of various kinds of graphs to answer specific questions. Drawing on extensive recordings and classroom observations, we describe examples of the performance of these students on open-ended problem-solving tasks in which graphs can be used to arrive at meaningful answers to applied data analysis problems.     

Meaning in constant flow – university teachers’ understanding of examination tasks

Effective feedback presupposes that students understand the task on which feedback is given. But what about the teachers formulating and assessing the task? Do they always understand it as intended? And if so, feedback on what? The purpose of this study is to examine how university teachers individually understand tasks distributed to students. Does interpretation differ if the teachers themselves try to solve the task, discuss the solution with other teachers, as well as try to formulate better versions of the task? The theoretical framework rests upon a hermeneutic understanding of reality. There is thereby reason to doubt the possibility of information transfer and the understanding of feedback as a strict rational process. The empirical material was collected in connection with development work, and sections where the participants expressed uncertainty considering the interpretation of the task were transcribed. The empirical material shows that teachers interpret a task somewhat differently when examining it more carefully, on their own and together with other teachers. It also shows that the same teacher vacillates in their interpretation of a task when examined more thoroughly. Consequently feedback given to students also differs. The drift of meaning is probably quite minor, but still noteworthy.     

Symmetry properties of tetraammine platinum(Ⅱ) with C2v and C4v point groups

Let G be a weighted graph with adjacency matrixA=[aij]. An Euclidean graph associated with a molecule is defined by a weighted graph with adjacency matrix D=[dij], where for i≠j, dij is the Euclidean distance between the nuclei i andj. In this matrix dij can be taken as zero ifall the nuclei are equivalent. Otherwise, one may introduce different weights for different nuclei. Balasubramanian (1995) computed the Euclidean graphs and their automorphism groups for benzene, eclipsed and staggered forms of ethane and eclipsed and staggered forms of ferrocene. This paper describes a simple method, by means of which it is possible to calculate the automorphism group of weighted graphs. We apply this method to compute the symmetry of tetraammine platinum(Ⅱ) with C2v and C4v point groups.     

一些简单有限连通图的连通包数

通过图的连通包集和连通包数的定义,得到了6类常见连通图（路、圈、树、完全二部图、轮图、蛛网图）的连通包数,并确定了Petersen图的连通包数。     

Can juniors read graphs? a review and analysis of some computer-based activities

Abstract

This article reviews three published studies that appear to show that children as young as seven or eight can understand line graphs and scatter graphs to a greater extent than was previously supposed. These results are contrasted with frequent reports of poor graphical interpretation by students in secondary school. Given time and support, junior age children can carry out a wide range of interpretation tasks with graphs. In predicting their performance, the context of the graph seems to be more important than its syntax. An analogy is drawn between the acquisition of graphing skills and learning to play board games. It is argued that it is more important that the curriculum provides a rich and varied diet of graphical experiences, than a detailed teaching of graphical syntax. Information handling activities with computers are seen as an important way to achieve this.     

Lecturing graphing: What features of lectures contribute to student difficulties in learning to interpret graph?

Some studies suggest that individuals having completed undergraduate science programs are often poorly prepared to use graphs in ways typical of their disciplines. Science and technology studies have identified competency in graphing as being of central importance to the practice of a scientific discipline. Given the centrality of graphing to the practice of science, an important aspect of becoming enculturated into the practices of a scientific discipline is being able to use and interpret graphs in ways that are typical to that discipline. For example, competency in this usage is important to reading, interpreting and understanding journal articles in a discipline. Undergraduate science students spend a considerable amount of time in lectures where graphical representations play a major role in the presentation of subject matter. To gain an understanding of the use of graphs in lectures and how this use contributes to student understanding, this paper provides a microanalysis of graph use in lectures drawn from artifacts compiled from videotaping all lectures and seminars in a thirteen week ecology course. This analysis focused on both the text and the geestural references made in the reading of a graph in an ecology lecture. We conclude that the common ground existing amongst scientists that help them reach an agreed upon interpretation of a graph is missing from the present lectures and then discuss the constraints this places on students, learning about graphs in lectures.     

4-正则图的纵横扩张优化

针对4-正则图的平面嵌入的纵横扩张的特殊性，某些4-正则图类的最小折数纵横扩张已经有了线性算法．本文通过基纵横扩张，提供了从一个4-正则图扩充为另一个4-正则图的方式。使得从原图的最小折数基纵横扩张自然导出扩充图的最小折数基纵横扩张．     

Physical sciences and the school curriculum

The difficulties encountered by pupils and students when learning physics can often be explained by the differences that exist between their spontaneous ideas about the real world and how the scientist models this reality. How can children and adolescents be helped to better understand scientific ideas that could be of use to them? What is the role in learning and in teaching of: peer group interaction, different forms of representation (pictorial analogies, schemes, graphs), intelligent tutoring systems; etc? In this special edition there are a number of pieces of recent research of interest both to the researcher and to educator concerned with the development of knowledge and the teaching of the experimental sciences.     

There’s more to the multimedia effect than meets the eye: is seeing pictures believing?

Textbooks in applied mathematics often use graphs to explain the meaning of formulae, even though their benefit is still not fully explored. To test processes underlying this assumed multimedia effect we collected performance scores, eye movements, and think-aloud protocols from students solving problems in vector calculus with and without graphs. Results showed no overall multimedia effect, but instead an effect to confirm statements that were accompanied by graphs, irrespective of whether these statements were true or false. Eye movement and verbal data shed light on this surprising finding. Students looked proportionally less at the text and the problem statement when a graph was present. Moreover, they experienced more mental effort with the graph, as indicated by more silent pauses in thinking aloud. Hence, students actively processed the graphs. This, however, was not sufficient. Further analysis revealed that the more students looked at the statement, the better they performed. Thus, in the multimedia condition the graph drew students’ attention and cognitive capacities away from focusing on the statement. A good alternative strategy in the multimedia condition was to frequently look between graph and problem statement, and thus to integrate their information. In conclusion, graphs influence where students look and what they process, and may even mislead them into believing accompanying information. Thus, teachers and textbook designers should be very critical on when to use graphs and carefully consider how the graphs are integrated with other parts of the problem.     

Differences between experts’ and students’ conceptual images of the mathematical structure of Taylor series convergence

Taylor series convergence is a complicated mathematical structure which incorporates multiple concepts. Therefore, it can be very difficult for students to initially comprehend. How might students make sense of this structure? How might experts make sense of this structure? To answer these questions, an exploratory study was conducted using experts and students who responded to a variety of interview tasks related to Taylor series convergence. An initial analysis revealed that many patterns of their reasoning were based upon certain elements and actions performed on elements from the underlying mathematical structure of Taylor series. A corresponding framework was created to better identify these elements and how they were being used. Some of the elements included using particular values for the independent variable, working with terms, partial sums, sequences, and remainders. Experts and students both focused on particular elements of Taylor series, but the experts demonstrated the efficiency and effectiveness of their reasoning by evoking more conceptual images and more readily moving between images of different elements to best respond to the current task. Instead of moving between images as dictated by tasks, students might fixate on “surface level” features of Taylor series and fail to focus on more relevant features that would allow them to more appropriately engage the task. Furthermore, how experts used their images, supports the idea that they were guided by formal theory, whereas students were still attempting to construct their understanding.     

On the Road to Graphicacy: The learning of graphical representation systems

This article examines the learning of different types of graphic information by subjects with different levels of education and knowledge of the content represented. Three levels of graphic information learning were distinguished (explicit, implicit, and conceptual information processing) and two experiments were conducted, looking at graph and geographical map learning. The graph study (Experiment 1) examined the influence of the variables' numerical relationship structure on adolescent students with different levels of education and knowledge of social sciences and also assessed their proportional reasoning skills. The map study (Experiment 2) looked at the learning of a geographical map studied spontaneously by secondary school and university students with different geographical knowledge (experts and novices) and also assessed their spatial skills. The results of both studies show that graph and map learning performance improves with the subjects' educational level. The groups' differential performance varied according to the type of information involved (explicit, implicit, or conceptual). The subjects' knowledge of the domain in question determined the level at which they processed the information. Verbal and superficial processing of graphic information were also found to predominate. This has important educational implications, suggesting the need for differential treatment in teaching different types of information. The results of the study also raise interesting issues regarding the type of expertise involved in learning graphic information: expertise related to the content represented, to knowledge of the syntax (graphicacy), and/or the system of knowledge graphically represented – spatial in the case of maps, numerical in the case of graphs.     

Results on diameter-stable graphs related to girth

A graph G is said to be an （l,d）-graph （with respect to edges） if d（G-E）≤d,E E（G） such that ｜ E ｜≤l-1.The l-diameter-stable graphs are （l,d）-graphs with diameter d.In this paper some new results on diameter-stable graphs are obtained.