The Study Of Graphs By Computers: Is Easier Better?

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The Study Of Graphs By Computers: Is Easier Better? Bracha Kramarski a

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Bar‐Ilan, Israel

Available online: 09 Jul 2006

To cite this article: Bracha Kramarski (1999): The Study Of Graphs By Computers: Is Easier Better?, Educational Media International, 36:3, 203-209 To link to this article: http://dx.doi.org/10.1080/0952398990360306

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The Study Of Graphs By Computers: Is Easier Better? Bracha Kramarski, Bar-Ilan, Israel

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Abstracts English: The purpose of the present study was to identify conceptions and alternative conceptions that students have with regard to the construction of graphs representing everyday situations in a computer learning environment using a GRAPHIC plotter, and to examine the resistance of these alternative conceptions to formal instruction. Following instruction, no significant improvement in students' performance was observed. The study further shows the kinds of alternative conceptions that were held by students prior to being exposed to formal instruction about graphing and the extent to which these alternative conceptions were resistant to change under formal instruction. The theoretical and practical implications are discussed. Français: le but de la présente étude était d'identifier les conceptions et les conceptions alternatives des étudiants par rapport à la construction de graphes représentant des situations quotidiennes dans un environnement informatique utilisant un traceur de courbes et pour examiner la résistance de ces conceptions alternatives à l'instruction formelle. En suivant les instructions on n'a pas pu observer d'amélioration significatife dans les performances des étudiants. L'étude montre par ailleurs les types de conceptions alternatives que connaissaient les étudiants avant d'être exposés à l'instruction formelle sur la fabrication de graphes et de déterminer à quoi ces conceptions alternatives étaient résistances à l'instruction formelle. Les implications théoriques et pratiques sont discutées. Deutsch: Zweck dieser Studie war, herauszufinden, welche Ideen und alternative Vorstellungen Studenten entwickeln, wenn sie Alltagssituationen unter Nutzung von Computern und Plottern graphisch darstellen sollen und dabei auch herauszufinden, welche Widerstände diese alternativen Vorstellungen dem üblichen Unterricht entgegensetzen. Eine signifikante Verbesserung der studentischen Leistungen wurde nicht beobachtet. Die Studie läßt ferner erkennen, welcher Art die alternativen Ideen der Studenten waren, bevor sie über graphische Darstellungen unterrichtet wurden und das Maß an Widerstand, den die alternativen Konzepte einer Anderung durch Unterricht entgegensetzen. Die theoretischen und praktischen Auswirkungen dieser Ergebnisse werden diskutiert.

Introduction Graphs are a very important factor in the teaching of mathematics. First, graphs represent one of the first points in mathematical studies by which the student uses mathematical language in various ways. Second, graphs enable powerful learning due to the fact that they can be used as a bridge between concrete thinking and abstract thinking (Piaget et al., 1968, 1977). Third, graphs are the link between high level mathematics - the world of functions - on the one hand and everyday mathematics. Although there is a growing awareness that graphing is a fundamental part of the curriculum and children are exposed to graphs at a formal level in school and a nonformal level in different media, research on the subject of graphs (Leinhardt et al., 1990; Kramarski and Mevarech, 1997) indicate that students' understanding of graphs is rather limited. Many students have difficulties when asked to shift between different modes of presentations, and others have misconceptions on this subject. Few empirical data have documented resistance to changes in students' alternative conceptions as a result of being exposed to formal instruction about graphing (Wainer, 1992; Arcavi, 1994; Mevarech and Kramarski, 1997). Why do so many students encounter difficulties in comprehending graphs? To address this issue, one has to analyse the structure of graphs and the cognitive processes that are activated in interpreting and constructing graphs. Similar to many other mathematical processes, graphing involves both interpretation and construction. Interpretation usually refers to students' ability to read a graph and make sense of it, or gain meaning from it. Leinhardt et al. (1990, p. 12) explained that 'construction is quite different from interpretation. Whereas interpretation relies on and requires reaction to a given piece of data (e.g. a graph, an equation, or a data set), construction requires generating new parts that are not given'. Graph users have to:

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understand the meaning of variables; know the principles of Cartesian system; be able to consider two variables simultaneously and comprehend the notion of covariation; and understand transformation processes between verbal, algebraic, and graphical representations.

The extensive exposure of students to graphs outside of school may generate a unique situation where students enter the learning situation with a substantial amount of knowledge about graphing. Current work in assessing what students know and do not know about graphing reflects a growing sensitivity to the importance of students' prior knowledge. The basic instance that underlies this approach assumes that: students do not enter the learning situation as a tabula rasa - through interaction with the physical and social worlds, students construct knowledge prior to being exposed for formal instruction; • sometimes that knowledge is in accord with the accepted meaning but in other cases it differs; and • students' prior knowledge plays a crucial role in subsequent learning (Confrey, 1990).

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The question of what students learn in enriched technological environments has been on the agenda since computers were introduced into the classrooms (Pea, 1985; Perkins, 1985). Since advanced technologies are aimed at assisting students in organizing, analysing, and displaying information, many researchers (e.g. Leinhardt et al., 1990) believe that learning with computer programs can facilitate the learning of various mathematical skills, such as graphing and statistics. This assumption is particularly important in the light of current constructivist theories of learning which emphasize that 'learners have to construct their own knowledge - individually and collectively. Each learner has a tool kit of conceptions and skills with which he or she must construct knowledge to solve problems presented by the environment. The role of community — other learners and teacher — is to provide the setting, pose the challenges and offer the support that will encourage mathematics construction' (Noddings, 1990, p. 3). Most of the studies have focused on interpretation (Wainer, 1992). Only a few empirical studies examined construction tasks (Kramarski and Mevarech, 1997). Few attempts have focused on the effects of computer learning environments on students' graphing comprehension (e.g. Yerushalmy, 1991) but even here studies rarely focused on students' misconceptions in a computer learning environment. The question of what kind of support should be offered when students use advanced technologies is still open. The purpose of the present study was, therefore, twofold: to identify the conceptions and alternative conceptions that students have with regard to the construction of graphs representing everyday situations in a computer learning environment using a GRAPHIC plotter; and to examine the resistance of these alternative conceptions to formal instruction.

Method Participants Participants were 82 students (boys and girls) who studied in grade 8 of one Israeli junior high school. The school, located in the centre of die country, serves a student population of lower, middle, and higher socio-economic status families as denned by the Israel Ministry of Education. Students were assigned to four computer classes.

Treatments The students learned the unit 'graphing skills' with a computer graphic plotter MATHEMATICS as part of the topic 'Solving problems: algebraic and graphic solutions' twice a week for ten weeks. This software includes a graph plotter which displays a large number of graphs, extremely quickly with a high degree of accuracy. It is designed to release students from constructing graphs from scratch widi paper and pencil. It allows the learner to move quickly from one representation to the other, eg, from verbal representation to tables and graphs. The typical software is a very efficient tool for constructing graphs. It can manipulate data in many different ways: tables, graphs. The speed at which graph plotters operate provides students with unique experience for manipulating data and thereby examining a large number of graphs. These properties led researchers to hypothesize that the use of a graph plotter as an instructional tool would enhance students' ability to interpret graphs. The unit included the following topics: recognizing ordered pairs, reading and plotting points in a Cartesian system, organizing data in tables, interpreting and constructing graphs, and drawing conclusions from graphs. The linear graphs (y = ax + b) were introduced, the students were trained to find a graphic solution for an

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equation system which includes two variables, and they solved various mathematical problems (e.g. 'distance problems') on this topic. Students learned to construct graphs by using the graph plotter, to identify ordered pairs by 'walking on the graph' to understand the meaning of the slope, intersection points, and the meaning of different modes of two lines: intersection/parallel/perpendicular. They discussed with the teachers the importance of using graphs in mathematics and in daily life. At this point they did not learn about functions.

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Measurement Graph construction test Prior to and at the end of the study, all students were administered the Graph Construction Test (Mevarech and Kramarski, 1997). The test is composed of verbal description of four phenomena describing the relationships between the time students devote to preparing for tests and their grades on the tests. One phenomenon described an increasing function: 'The more time a student prepares for tests the better her grades on the tests are.' The second phenomenon described a constant function: 'No matter how much time a student prepares for tests, her grades on the tests are always the same.' The third phenomenon described a curvilinear function: 'When a student prepares for tests up to three hours, the more time she prepares for tests the better her grades on the tests are, but if she prepares more than three hours she becomes tired and her grades on the tests decline.' And finally, the fourth phenomenon described a decreasing function: 'The more time a student prepares for tests the lower her grades on the tests are.' Students claim was to construct two graphs representing each situation (e.g. a line graphs and a bar graph). The student was asked to represent each situation by two graphs because quite often students' alternative conceptions of graphing were reflected when they were required to construct multiple graphs (Mevarech and Kramarski, 1997). Students could construct the graphs with rulers or free-hand, but were not allowed to use advanced technology for constructing the graphs. Scoring Scoring of each graph was made on a scale from 0 to 2 points, with 0 points given to an incorrect graph, 1 point to one correct graph, and 2 points for two correct graphs. A graph was considered correct if it presented correctly the function described, regardless of the kind of graph the student elected to display. In addition, we carried out a qualitative analysis of students' responses by focusing on alternative concepts that students displayed in constructing the graphs.

Results Data were analysed by two methods: qualitative and quantitative. The qualitative analyses focused on identifying and classifying students' conceptions and alternative conceptions. The quantitative analysis examined students' knowledge about graphing prior to and following formal instruction with the computer. When analysing students' graphs, we classified the alternative conceptions which resulted into eight categories: • Constructing an entire graph as one single point. The one point graph was observed when some students constructed correctly the axes, but marked only one point, one bar, or one histogram. • Constructing a series of graphs, each representing one factor from the relevant data. • Conserving the form of increasing function under all conditions. As described above, students were asked to plot four kinds of functions: increasing, constant, curvilinear, and decreasing. Yet, some students conceived all linear graphs as a function of the form y = ax + b; a>0. some students conserved the shape of an increasing line by repairing the x or y axis: the scale was changed to include only one value, the value of the constant (eg, 'no matter how long she studies, she always gets the same grade). • Constructing a graph as a picture. Graphing daily objects like watches, arrows, books. • Tabulating the data. Representing the data only in table format. • Constructing a graphic solution, two graphs intersect in a whole axes system. • Constructing a graph as an angle, the length of the sides representing the variance of the variables. • Constructing a graph without the axes. Quantitative analysis: Students' pre-existing knowledge and conceptual changes Data analysis showed that on the pre-test, about 16% (13 students) constructed correctly two graphs in all four situations, 26% (22 students) constructed correctly at least one graph. This may be explained by the extensive

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exposure of eight-grade students to graphs outside of school. Following instruction, no significant improvement in students' performance was observed. The number of students who constructed two graphs in all four situations graphs correctly increased to 21% (18 students), and 34% (28 students) constructed correctly at least one graph. The mean score over all items didn't change 0.68 (SD = 0.67) on the pre-test to 0.7 (SD = 0.69) on the post-test (t-test for dependent samples = 0, p