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The article has been published in the Joensuun yliopisto kasvatustieteiden tiedekunnan selosteta,
University of joensuu belletins of the facultu of education №99. Karl Sormunen (ed.).
The Bologna in Scince and Mathematics Higher Educatijn in North-Eastern Europe:
Tendenciens, Perspectives and Problems - Joensuun ylioristo University of joensuu, 2006

Material is provided for free personal use with the obligatory reference to the authors (according to Art. 1229 Civil Code RF)

           Developing graphic culture of students and their graphical language which used in science, technologies, manufacturing and other industries is vital; that is why it occupies such a significant place in the teaching process. The basics of graphical culture are formed at the middle stage of education (7th-8th forms) and developed during the whole period of school training. However it is commonly known that even school-leavers often are not able to read and build graphs, have difficulties in building graphs and often make them incorrectly or inaccurately. Linear graph transformations cause most difficulties as school manuals either do not explain them or give separate examples of building point-by-point, without clear explanations of common algorithms which allow good learning of common laws of parallel shift and extension of the graphs along the axis of the Cartesian coordinates.

           Most present-day computer-based tutorials expose students to linear graphs of elementary functions through different graph plotters which are rather convenient f or visual analysis of graph variations depending on specific values of parameters of this function or another. However with using these programs it is difficult to focus the students' attention of on certain significant issues of graph building. For development of graphical culture of the students we apply such kind of computer-based training aids as slide-show. It is a gradually revealed topic presented as a row of slides. We refuse though the so commonly used simple transfer of the textbook to the computer screen as this approach makes the teacher's work still more complicated rather than simplifies it.

           We believe that the theory should be visualized on the screen which means the student should readily see the connection between concepts and visually percept the logic of proof or argumentation (Bashmakov, Pozdnyakov & Reznik 1997). That is why when developing a slide-show the following rule is observed: the presentation of the information new to the students should be based on the simplest examples which allows gradually accumulate the knowledge. The films are composed of a number of slides, so that each portion of new information available within the one step-slide is sufficient and reasonable.

           For the teacher to be able to combine the contents of the computer “pages” as easy as information from the common textbooks we develop subject collections presented as series of slide-shows. Like pieces of mosaic, they can be easily combined into various sets for learning purposes, built into lessons scenarios or given out to students for homework combining with other training aids.

           We aim at maximum simplicity of our product installation and operation of its delivery to the computer screen. We remember that the end users of the program are teachers who may not have advanced computer skills. The best option for us is the “projector-film” (exe-file) which can be reproduced on computers with Windows 95 (or higher) without any additional software. The teacher only need to double click on the file's icon, and use only two keys for control (up and down) which allows stopping on the slides of special importance.

           The students of 8-9th forms are already familiar with the functions y=x² and y=1/x, that is why their acquaintance with the subject naturally begins with these examples. First two slide-shows “The guiding rectangles of parabola and hyperbola” (Figure 1) present the gradual building of parabola and hyperbola (Reznik 2002).

           The slides provide clear perception of the structure and learning configuration of the graphs. At this stage the explanation is slow. The students are getting used to the new way of information presentation, analyze all the nuances and details of graph building. Everything is discussed but notes are not made - the students learn from seeing and hearing.

           The following slide-shows present parallel transfers of graphs along the y-axis. The definition of the shift of parabola or hyperbola is implemented on the screen.

           Then by shift of their guiding rectangles graphs of the functions y=x²-1 and y=1/x+1, are built (Figure 2).

           The culmination is the analysis of the formulae specifying the functions y=x²+B(y=x²-B) and
y=1/x+B(y=1/x-B), (Figure 3, up). In the end of each slide-show the general algorithm is shown as an information scheme (Figure 3, down) and afterwards the students are asked to build graphs of the studied functions by themselves and make a certain transformation. Now the children try to do the drawings as carefully, accurately and neatly as the slide-show character commenting the show.

           Extensions and compressions of the graphs along the y-axis are presented in the following slide-shows. Here, too, the students first study the drawings, analyze the formulae, answer the questions. The discussion goes faster this time. They can predict what the next step is, how the guiding rectangles will change and what will happen to the function graph itself (Figure 4).

           Parallel shifts of graphs of functions along the x-axis are more difficult for learning. That is why the slide-show “Shifts of the guiding rectangles of parabola along the x-axis” helps organize the research activity of the students. First, a standard graph is shifted to the right by one unit which should cause certain changes in the structure of the formula of the function. To define the nature of these changes the students should put forward a hypothesis of the formula of the new function (Figure 5).

           The first and quite natural one is the hypothesis y=(x+1)². The proof is done by substitution of certain values of variables in the formula. A new graph is built on the corresponding points with coordinates (x;y) after that the formulae written in general are analyzed and the common and different is defined in the formulae of these functions. Then the rule is formulated which allows to define the shift direction of the function graph for the given formula depending on the parameters in the formula (Figure 5, down).

           Now the question is reasonable whether the resultant conclusion will be valid for other graphs of the functions and whether the formula will change according the same laws or a similar proof will be required for every specific case?

           To prove the truth of the general conclusion the students can be suggested working with analogous transformation of hyperbola but this time on their own, with the help of paper-based visual didactic materials (Figure 6). The latter are based on the visual tasks during which visual perception is activated and the wordily language transforms into the language of drawings and formulae. Each at their own pace, the students can put forward and prove their hypotheses of the function formula and positioning of the graph on the coordinate space.

           In 2004 lessons with the use of the above-described didactic aids: computer-based training aids (slide-shows) and series of visual tasks (presented in the hard copy) - were given in the Lyceum (secondary school) No1, form 8. The results showed that the students completely understand and learn the topic - the progress was estimated as 100% while the quality of knowledge (good and excellent results) was 70%. In three months time the students of the same group were given a random test (with no announcement or special preparation) containing tasks like: “Build the graph of the function: y=(6/(1-x))+4; y=(4/(x-3))-1; y=3-(x-2)²; y=x²+1; y=(x+5)²”. The results were better than immediately after the presentation of the topic - 98% of the students successfully completed the task, 90% with the evaluation “good” or “excellent”. This autumn which is in one year's time and again without any specific preparation the students were given a remaining knowledge test for the three year mathematics course (based on the materials of forms 6-8). Of the students given the task to build graphs, 100% of the students successfully completed the tasks.

           As it is seen from the practical experience the material presented in the form of computer slide-shows was learnt successfully and for a long time. The initial time investment necessary for developing skills on function graph building was completely compensated by fast acquiring of the skills of graph transformations, graph reading, function formulae analysis and summarizing the acquired knowledge using it for new unfamiliar formulae and function graphs. For this, both the teacher and the student didn't have to pay extra effort for familiarization with the new computer-based training aid.

References

 2. Reznik, N.A. 2002. Visual Algebra. Equations and Graphs. St. Petersburg: Informatizatsia Obrazovania.

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© Natalya Reznik (nareznik@yandex.ru): the chief of the project “The visual school”
© Natalia Yezhova (naegova@yandex.ru): the methodologist-elaborator of the website
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