There have been many diverse investigations leading to the current research. Generally the most relevant studies can broken down into three major categories: studies about graph perception, studies on physics graph concepts, and studies on auditory graphs. While the last category is most directly related to the current research, much of the subject material and questions were derived from studies in the first and second groups.
The usefulness of looking at how people perceive graphical information, and what attributes of graphs are important in the visual sense, can lead to important attributes that are also important for auditory graphs.
There is a large amount of literature devoted to the teaching and learning of graphical information. Enough has been published to warrant a review paper by Leinhardt, Zaslavsky, and Stein. [Lei90] Their paper investigated the research and theories related to the teaching and learning of functions, graphs, and graphing in high school mathematics. The functional relationships that are regarded as important constructs in the development of abstract knowledge has lead to a body of research upon which their paper is based.
In their review of the literature, they have the viewpoint that there is a fundamental link between graphs and the functions that they represent. Graph interpretation is thus colored by the viewpoint from which it is taught or learned with the result that often, students have difficulty translating their mathematical prowess to scientific graphs, even though graphs are a stable presence in science and social studies courses.
Leinhardt reports that there is no proven entry to the teaching of graphs and functions, although there are several preferred sequences. Also, it is noted that the use of technology dramatically affects the teaching and learning of functions and graphs. Misconceptions may arise from student's tendency for over generalization, poor inferences, or due to incomplete learning of the material.
The complexity of the domain of functions and graphing reflects the complexity of structure and kinds of demands it presents both to students and to teachers. Interpretation, construction, the setting and context in which the graph is presented, the variables utilized, and the focus of the data all play a role in determining the graph's overall complexity.
Interpretation of a graph can vary the complexity depending on the number, type, and location of features within a graph. Construction is the process of generating through inference, new parts of a graph that are not necessarily given in order for interpretation.
There are several factors that Leinhardt mentions as hindrances to student learning of graphical information. Students can be distracted by graphical features that can be seen as a pictorial representation of some aspect of the situation. Also, there is a tendency to become overwhelmed by the situational knowledge of new subjects, making abstraction of the data representation more difficult.
Some examples of common student problems that are mentioned are: confusion between interval and point representation, mistaking slope and height, and iconic interpretation of the graphs. Also, misconceptions are often seen when graphs contain pronounced features, such as a sudden rise or fall, or a discontinuous curve.
Some of the difficulties that students face in understanding graphical information is fundamentally due to a failure to view graphs as abstract formalisms. The results in giving more meaning to the graph's scale than is mathematically warranted, not comprehending the significance of the slope for a situation, and viewing a graph as a picture.
The way that a student correctly interprets a graph often involves some level of algebraic construction in order to provide a proper interpretation. However, direct comparison of displayed points is also necessary.
When there are problems learning from graphical information, it is usually in the form of three broad areas: a desire for regularity, a pointwise focus, and difficulty with the abstractions of the graphical world. There also is a disproportionate emphasis on single points, such as maximum and minimum values, which distracts from other salient features such as intervals and especially slope.
The remainder of the research reviews in this section are points not particularly covered in the Leinhardt paper, and of research that bears a more direct emphasis on subject matter necessary in the development and application of the conducted studies.
3.Some of the earliest studies on graph perception were by Vernon. [Ver45, Ver52a, Ver52b] The initial study concentrated on the ability of adult subjects to understand and acquire information about problems where the information was presented in graphical formats. The information was presented in the form of charts, diagrams, or pictures rather than through verbal statements to both college students and soldiers. This initial study lead the investigator to conclude that acquisition of information becomes muddled and uncertain when the learner does not have a definite, or clear, contextual basis of ideas and background knowledge about the problem.
When the knowledge base is limited to a random collection of ideas, slogans, clichés or prejudices, the presented factual information either is unassimilated, changed to conform to the preconceived ideas, or else remains an isolated fact that neither influences, nor is modified by, the main basis of the world view. Vernon came to the conclusion that people will tend to ignore facts when their ideas are mainly directed by emotionally toned opinions. When their knowledge of the issue is vague, they are more readily guided by preconceived ideas and prejudices. Thus, proper analysis of data is dependent upon sufficient education and impartiality about the subject matter presented.
The primary impetus of Vernon's later papers is to view how education affects a person’s ability to interpret information presented in a graphical format. The first paper relies on the assumption that it is generally believed that graphs and charts of various kinds can not only present such data accurately and vividly; but also that the people who see them can understand and assimilate the information which the graphs give. [Ver52a, p. 22]
Vernon also claimed that a verbal argument seems necessary to provide a meaningful setting in most cases. Graphs are valuable only in so far as they can be perceived to corroborate or extend the facts upon which the argument if the text is based. [Ver52b] Useful conclusions about information portrayed in graphs or charts without written explanation can be made only when subjects have had a fair amount of education relating to the subject matter presented.
The current author believes that perhaps the most important conclusion that these three studies conducted on students and soldiers demonstrated was that education played a role in how well a person could derive factual information from graphs and charts when they were presented without written text. Another important conclusion was that there seemed to be no advantage in using pictorial charts rather than graphs to portray the information. Even when specific factual data are understood, it is often difficult for people to incorporate this new information into their general body of knowledge.
In the current studies, efforts were made to choose subjects who were compatible with the subject matter that was presented in the graphs. This was accomplished by designing a test and graphs from material that first year university physics students might encounter in the course of their studies.
The use of graphical material as a testing medium is not unwarranted, Wainer asserts that "graphs work well because humans are very good as seeing things, they are so basic to our understanding that we cannot easily imagine a world without them." [Wai92, p. 15] However, one must be careful in the presentation of information. Evaluating performance based on information that is presented in a flawed format can be misleading. When data is presented in a properly displayed graphical format, most common questions can be easily answered, and deeper analysis of the data can then follow. The reasoning for creating alternate displays of information is that a better graph of the same data should make interpretation easier.
3.To find what information should be contained in a new data display format, it is helpful to study the logical reasoning in graph construction. Wavering [Wav89] conducted such a study in an attempt to determine the logical reasoning necessary to construct line graphs with the premise that once the reasoning is known, student understanding of graphs can be improved as graph construction and misinterpretation issues can be addressed. The primary purpose of the study was to infer mental manipulations students use to construct line graphs and to propose connections to theoretical mental structures.
Wavering’s research design consisted of having the students from grades six through 12, construct graphs from given numbers. The subjects wrote down information on what they were doing as they constructed the graph and why they were doing it. Students were then asked to identify any patterns in the graphs and to state any relationships. Three types of graphs were developed which were referred to as the Research Instruments: positive slope, negative slope, and an exponential curve. It was implied that all students received all instruments.
Wavering classified the responses to the graphing task into nine categories, determined from patterns in the responses. These categories broke down roughly into ability to draw, label, and state the relationships between variables. The ability to produce and accurately describe the relationship was placed into a Piagetian stage development construction. The categories for responses were designed to coordinate with this view of theoretical mental structures.
There are several implications from this study. First, that the response categories appeared to be valid with the three types of graphs. Second, that student response patterns for grades six through 12 were similar for all instruments. Third, that students in higher grades demonstrated an increasing ability to provide more complete responses.
The response categories were composed in Piagetian terms, with the lower categories representing Concrete Operational reasoning, and the upper categories representing Formal Operational reasoning.
It is the current author’s belief that the data showed some credibility to Wavering's claim, although the trend was most pronounced when comparing the 6-9 to 10-12 groups. While the results are interesting, a much larger sample size was needed to provide better statistics as in some grades there were only 25 students spread into 18 graphing response categories ( nine graph by two genders.) Aside from some random fluctuations, the results should be fairly general for other similar school populations.
The important point from Wavering's study is that by grade 10, a majority of students were able to interpret graphs at a level that is consistent with formal operational and early correlational reasoning. They thus have the abilities needed for pattern recognition and recognition of relationships between variables.
3.A study conducted by Berg and Phillips [Ber94] also investigated the relationship between thinking structures and the ability of secondary school students to construct and interpret line graphs. Graphing abilities were assessed through construction and interpretation of graphs with varying content and difficulty. This study again showed that students who utilized logical thinking structures, were better able to interpret questions based on the graphs, such as choosing the part of the graph with the greatest rate of change.
Other implications of the Berg study are that without "cognitive development, students are dependent upon their perceptions and low-level thinking" {Ber94, p.340] and their responses revert to cueing off words used in the questions. With the development of mental structures such as proportional reasoning, logic overrides perception and students will no longer just see graphs as pictures, but can use them to make inferences.
An important note about the study conducted by Berg and Phillips is that they question the validity of studies that use multiple-choice instruments for determining how students learn about graphs. They advocate a testing process where students can supply their own answers, and reasons for their answers. They also suggest that researchers should use a number of questions that address elements of graphing and conflict with perceptual cues.
It is the current authors viewpoint that the consequence of Wavering and Berg's research studies is that it is not unreasonable to expect first year college students to answer questions based on complex relationships of graphical data, as the subjects have had full cognitive development since at least grade 10.
While the current study did rely on multiple choice responses to questions, the goal was slightly different from determining students' thinking processes. The goal instead was to determine how well students can answer questions based on graph differences. Thus, reliance on free response answers was greatly reduced. Also there was a large number of questions and subjects used to try to determine where any differences may reside.
3.Continuing the concerns about the validity of using multiple choice questions to examine the ability of students to construct and interpret line graphs, Berg and Smith [Ber94] compared the results between student's answers for multiple choice or free response tests.
The purpose of the Berg and Smith paper was to challenge the validity of using multiple choice instruments to assess graphing abilities and was a report of two studies on students from seventh through eleventh grades, that addressed this issue. The first study utilized numerous graphs to examine the subjects’ abilities to construct and interpret graphs. The second study continued to investigate the questions of the first study with the addition of attempting to learn about the differences in assessment when subjects drew their own graphs as opposed to selection from a multiple-choice instrument. This was a comparison of the results for three graphing questions that asked students to either choose between, or draw, graphs representing various situations.
Their first study utilized three graphing questions that had already been examined in other research studies using multiple choice formats. These questions were used in studies by Barclay [Bar86], and Mokros & Tinker. [Mok87] There were three basic questions which were modified to meet clinical interview standards which was the method of data collection. The questions involved a distance vs. time graph of a person walking from and to a wall, and speed vs. time graphs of a ball rolling down a varied surface and a bike traveling over a hill.
The second study constructed graphing instruments which consisted of the three graphing scenarios used in the first study, but had the subjects complete either a free response instrument which had them draw a graph which best represented the situation, or a multiple choice instrument where they chose a graph to best represent the given situation. Student responses were categorized into either correct, picture, or other responses. The response time for answering each of the questions was also analyzed.
The results of from the first study showed substantial differences in the percentages of answer types between the free-response answers and those reported in the literature. This difference is what prompted the second study. Also, the responses to the first study provided categories of possible answers for scoring answers in the second study. In the second study, a direct comparison between the answers of the two groups could be accomplished as the samples taking the two tests were nearly identical.
The end result of the second study was that the type of instrument used directly affected the response rate of correct answers in two of the three graphing questions studied. The free-response students drew significantly more correct responses on the Walk-Wall and Ball-Hill graphs, while the multiple-choice students chose more correct responses for the Bike-Hill graph, although the data was not presented in a clear manner. There was demonstrated that the percentage of "Picture Response" graphs was significantly and greatly reduced in the free-response choices.
From the results of the first study, Berg and Smith [Ber94a] concluded that the multiple-choice format used in studies might not encourage students to think through graphing questions in more than a superficial sense. During the interview method, it was noted that the students would often answer a question quickly, but then change their answer as they explained their reasoning. Thus, the authors claim that the multiple-choice instruments often do not assess much more than superficial, first-reaction thoughts.
Berg & Smith’s second study [Ber94b] resulted with the conclusion that there was a clear disparity between the results of the multiple choice and free response graphing instruments in term of both correct responses and those relating to "picture as event" distracters. The results were that there was a statistically significant, 19% disparity between the two methods.
It is the current authors opinion that some of the limitations of the Berg & Smith studies were that only three graphing questions were investigated, and in one of those cases the reported data was obscured, suggesting that results may not have completely agreed with their conclusions. Also, the free response answers were grouped in categories, with "correct" answers being rather loosely defined, thus allowing percentages to be manipulated to the authors advantage. Their conclusion that the disparity in testing methods may be important in test construction, but may be not as relevant with identification of processes. Also, while there was a reported difference in success rate with response time and instrument utilized, it was not demonstrated that this was a significant effect.
3.Part of the difficulty in assessing the ability to understand and interpret graphical information is that such an activity is a practiced skill. Roth & McGinn [Rot97] made several assertions to that effect in their research survey paper. They mentioned that graphs act as semiotic objects, whose relationship to the phenomena they represent is established through considerable work. The relationship holds because of convention. Students may misinterpret graphical information, not because they have not developed sufficient cognitive processes, but because they have not fully learned the conventions. Often, questions developed to provide objective responses, showed signs of being socially constructed, and thus could not be pure measures of a subjects cognitive abilities. The assessment of subjects' competencies is affected by social factors (linguistics, motivation, testing climate, etc.) and can not be held as an isolated measurement.
Roth & McGinn go on to point out that graphing ability is also a matter of practice. Since graphing is "one of an array of signing practices such as talking, writing, gesturing, drawing, or acting used extensively in scientific communities," [Rot97, p. 96] the more exposure one has to graphs, the better one can interpret their meaning. To develop graphing competence, students need to actively participate in the development of graphing practice.
3.There have been several studies concerning the methods that people use to interact with graphs in order to draw information from the images. One model of how people encode information is the Mixed Arithmetic-Perceptual model proposed by Gillan & Lewis [Gil94] which states that common processes that people use to analyze and respond to graphical information are searching for indicators, encoding the values of those indicators, performing mathematical operations on the values, and performing spatial comparison between indicators. They performed two experiments to investigate a proposed linear relationship between response time and the number of processing steps used to analyze a graph.
Gillan & Lewis' investigation began with a questionnaire given to scientists and students asking them to recall recently used graphs and the purpose of their use. From the responses, they found that the uses for the graphs were often for quantitative purposes. To develop their categories for the perceptual and arithmetic processes, they conducted a series of task analyses of people interacting with graphs. These studies consisted of detailed verbal reports as users were asked to perform a task, and observations of people answering questions about information presented in graphs. Tasks included identifying values of graphical elements, comparing the amounts of two or more indicators, summing, negating, taking the ratio, determining the average values of indicators, and determining a trend.
Based on the task analyses, Gillan & Lewis decided that there was a limited set of component processes when performing frequently used tasks. The processes are searching for the location of a data point of interest, encoding the value from the axis or associated label, performing arithmetic operations, comparing spatial relations between several indicators, responding with the answer. Their model predicted, and was generally supported by their testing results, that there is a linear increase in time to complete a task dependent on the number of processing steps as well as a difference in the number of processing steps to complete a task for different types of graphs.
Gillan & Lewis conclude their paper with suggestions for reducing the time for users to make calculations relating to graphs. Of particular note are the suggestions to "organize the task so that users do not have to keep many partial results in working memory," [Gil94, p.439] and to design graphs to minimize the number of arithmetic operations that one needs to do for interpretation of the data.
3.Another related study by Milroy & Poulton [Mil74] concerns the use of labeling graphs to improve reading speed. This study looked at three techniques for annotating graphed data (placing the key in the graph field, direct labeling of the lines, and placing the key below the graph) and the resulting time and accuracy of reading those graphs. Their study indicated that for line graphs, direct labeling tended to produce the quickest readings. The authors speculated that this could be an effect that direct labeling, as opposed to use of keyed labeling, tended to reduce the amount of information that subjects had to commit to short term memory.
3.A study on the acquisition and retention of quantitative information from a line graph was conducted by Price, Martuza, & Grouse. [Pri74] Their study was particularly concerned with three aspects of learning from graphs: the nature of informational units, the relationship between the number of informational units and performance, and the relationship between study time and acquisition of information from the graph.
Multiple-line graphs of fictitious stock data was constructed from semi-random data which would show increasing, constant, or decreasing trends. A criterion test consisting of six sub-tests, each having eight questions was used. Three of the sub-tests were based on point information and the others on slope information. The test question items were constructed using several rules to ensure balanced wording of comparatives (increased/decreased, more/less, etc.) and truth value. Two groups differing is the length of time given to study the graph formed the basis of the comparative study.
The number of correct responses for each item was averaged for all subjects in the separate groups. The data were then analyzed as a one-between, three-within factorial analysis of variance. Study time was the between factor and the information type, number of informational units, and wording of logical opposites were considered the within factors. This analysis showed that all four main effects were significant but that none of the interactions were. The mean score of the eight minute was higher than that of the two minute group. The data was analyzed with study time and logical opposite pair wording having significant results.
Price, Martuza, & Crouse explained the overall pattern of the results as not supporting their initial hypothesis that the point and slope items included in the criterion test measured distinct constructs. The final statement was that the amount of data point information seems to be a more important factor than informational type in determining a subject’s performance level. They also conjectured that Slope question items are more difficult than point items, and that subjects reconstructed slope information from recalled points.
It occurred to the current author that most of the study had to do with giving students variable time to memorize data. Perhaps much of the effect that was seen was the ability to remember the smallest useful unit of data. The more time that was allocated allowed the students to remember more of the data points.
3.In an attempt to find out if college students could reproduce graphs shown in classes, Cohn & Cohn [Coh94] conducted an experiment with an economics course at the University of Southern Carolina. A second purpose of this study was to tell whether the accuracy of the graphs in students’ notes affected their success on tests in which graphs were included. Lastly, the extent to which instructor handouts containing unique graphs presented only in lecture facilitated learning was also presented as a purpose of the study.
The general design of the study had the students complete a one page questionnaire, attend an experimental lecture, and complete a post-test. Copies of the students notes were obtained for comparative analysis. The design was essentially a post-test only control group design on a single class of students. The comparison was having graphs provided in lecture versus the students writing their own graphs to determine which was most beneficial to the students. The authors associated student’s scores on exams given in class, and SAT and GPA scores as being equivalent to a sort of pretest. The post-test results were compared to these values. Prior to the lecture, the students completed a general background questionnaire consisting of questions about scholastic standing, socio-economic background, and a self-assessment of their ability of read and interpret graphs.
Students were randomly assigned envelopes consisting of a handout for taking lecture notes and requested to take their class notes on this handout. In all cases, the handout contained an outline of the lecture. In half of the cases the handouts also included reproduction of the two diagrams shown in class. Following the lecture, students reviewed their notes for 10 minutes, and then completed a 15 item multiple choice test. This procedure allowed for a good comparison to test for the effect of teacher supplied graphs as compared to presented graphs on short term student learning.
The post-test consisted of two definition questions, and 13 items to test student understanding relating to the lecture. Reliability was mentioned in that all of the test questions significantly and correctly discriminated between the upper and lower quartiles of the class. Graphs drawn on the lecture notes were assessed for accuracy.
From this study, Cohn & Cohn claimed that many students had a tendency to draw inaccurate graphs, and that students who drew more accurate graphs performed at a significantly better rate compared to the rest of the class. When instructor supplied graphs for their notes were provided, students with the tendency for drawing inaccurate graphs had increased test scores. However, students who could drew accurate graphs tended to perform best with their own notes.
It is the current authors opinion that the study could have been strengthened if a delayed post-test had also been included. This would have been easily accomplished as test items could have been included on the course’s final exam. It was not mentioned if the post-test had been pilot tested or checked for validity, but it is probable that it had not been.
Copyright 1999 Steven Sahyun