PEDAGOGICAL EXPERIMENT WITH ONLINE VISUALIZATION OF MATHEMATICAL MODELS IN MATH TEACHING ON ELEMENTARY SCHOOL

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1 PEDAGOGICAL EXPERIMENT WITH ONLINE VISUALIZATION OF MATHEMATICAL MODELS IN MATH TEACHING ON ELEMENTARY SCHOOL R. Špilka, F. Popper University of Hradec Králové (CZECH REPUBLIC) Abstract Technological progress opens up new paths in the pedagogical research. More and more emphasis is placed on the use of online learning materials in the educational process. This growing pressure opens up the discussion to change education reality in school classrooms. One of the ways to change the educational realities in the educational facilities is the flipped classroom. Especially in math teaching which uses modelling reality using mathematical models, has a chance to use online visualization in education process. Learning mathematics requires a certain degree of concentration, which is quite difficult for today's generation of students of elementary school. The article brings the results of pedagogical experiment, when visualization of mathematical models was tested on the upper-primary schools. For the purposes of the experiment has created an animated educational visualizations. In post-test results was recorded statistically significant difference between the control and experimental group. The article also deals with the theoretical background and the degree of interaction of the students in the use of educational visualization. Keywords: Online visualization, pedagogical experiment, math teaching, student-computer interaction, flipped classroom. 1 INTRODUCTION The role of visualization in mathematics learning has been the subject of much research over the last few decades (e.g. Presmeg, 1986; Habre, 1999; Arcavi, 2003; Stylianou and Silver, 2004; van Garderen, 2006). Although research is indeed ambiguous it seems like visualization is being accepted as an important part in mathematical reasoning. Significant impact on current trends, toward strengthening the influence of visualization in mathematics, has the integration of technology in education. 1.1 Visualization in mathematics learning Mathematical visualization is the process of forming images (mentally, or with pencil and paper, or with the aid of technology) and using such images effectively for mathematical discovery and understanding (Zimmerman and Cunningham, 1991). This definition emphasizes two things: (1) the physical and mental aspect of visualization, (2) visualization can be a powerful tool that students could use to solve mathematical problems and better understand mathematical concepts. The use of visual models such as diagrams, sketches and animations in mathematics education has become an area of renewed interest. This interest may stem from the fact that visual aids have always played an important role in the understanding of various mathematical concepts. In addition, suitable diagram or image can be used as visual proof of some mathematical characteristic or theorem. Some researchers even suggest that visualizations have important contributions beyond the introduction of ideas to students. For instance, Barwise and Etchemendy (1996) or Dreyfus (1994) who stated that the status of visualization in mathematics education can and should be upgraded from that of a helpful learning aid to that of a fully recognized tool for mathematical reasoning and proof. However, despite the advantages, some studies identify difficulties associated with visualization. According to study by Diezmann and English (2001), when mathematical problems include information embedded in text and diagrams, students tend to disregard the information presented in the diagrams. Abrahamson (2006) proposed that mathematical representations are conceptual composites (i.e., they include two or more connected ideas). The composite nature of mathematical representations is often covert one can use these representations without appreciating which ideas they enfold and how

2 these ideas are coordinated. As a result, students can use representations procedurally without fully understanding them. Abrahamson suggests that classroom discussions moderated by teacher can help students understand the ideas embedded in mathematical representations. Presmeg (1986) notes three problems associated with the use of visualization: (1) The one case concreteness of an image or diagram may tie thought to irrelevant details, or may even introduce false data, (2) An image of a standard figure may induce inflexible thinking which prevents the recognition of a concept in a non-standard diagram, (3) An uncontrollable image may persist, thereby preventing the opening up of more fruitful avenues of thought. Anyway, the importance of visualization in mathematical reasoning is widely recognized. In the last decades interest in visualization has grown up leading to both new researches and educational applications. 1.2 Computer-based visualization There has been a remarkable increase in the number of studies which focused on computer visualizations since the early 1980s. This trend is, to a significant degree, associated with technology development. It seems that computers have the potential to assist students in the formation of mathematical images. Computer based visualizations can engage the students' interest in mathematics and motivate him or her to additional study, and to provide the student with a way to think about mathematics that is different from the more traditional symbolic approach (Cunningham, 1994). Computers impart the advantage of flexibility to visual reasoning (Dreyfus, 1991). Computer-based environments with visually compelling displays, together with facilities for interaction, can provide the setting for more dynamic, powerful experiences. These environments are filled with stimuli, which encourage rich constructions, by students (Nelsen, 2001). Powerful, multiple representation software can be used to encourage the learner to construct meaning for different representations and their interrelations. The relationship between representations lies at the heart of much mathematics (O Reilly et al., 1997). As noted above, there exists much research which highlights the role and importance of visualization in the mathematical reasoning. However, some research point out the limitations and difficulties around visualization. Multirepresentational software could contribute towards misunderstanding and confusion amongst students; any difficulty experienced with one particular representation could be intensified by the presence of other forms of representations (O'Reilly et al, 1997). Smart (1995) believe that students may become too dependent on technology, regarding the solutions generated as irrefutable. Aspinwall et al. (1997) noted that visualization has the possibility of confusing mathematics students by leading them to focus on unnecessary detail. Vinner (1997) points out that in some cases the intuitive mode of thinking just misleads us. It seems that computers open up further possibilities of using visualization in school. However, visualization cannot be isolated from the rest of mathematics. As Zimmermann and Cunningham (1991) noted, computer-based visualization, whether static, dynamic, or interactive, is only one facet of the role of computers in mathematics. Visualization must be linked to the numerical and symbolical aspects of mathematics to achieve the greatest results. Some of the most interesting and important applications of visualization involve problems which use numerical and/or symbolic processing as well as graphics. 1.3 Effectivity of visualization Important question is, how to create effective visualization of mathematical models, which help student better understand the curriculum. Unfortunately the situation is not so clear. Indeed, it is not certain what makes a visualization good or bad, or even what the definition of visualization should be. Phillips, Norris & Macnab (2010) after surveying related articles to visualization in science and mathematics education, states: Perhaps the most defining feature of the current state of empirical research on visualization is the lack of consensus about the most elemental issues that surround it, including settling on a definition for visualization and deciding how to document both short term and long-term effectiveness. After evaluating the literature Phillips, et al. (2010) do identify five characteristics of effective visualizations. Their descriptions are summarized here. 1. Colour. There is some evidence to suggest colourful images may be more effective in triggering student learning than simple black and white images. 2. Simplification. Abstracts line diagrams that focus on the essential details of a concept may be more effective than overly complicated images which include unnecessary detail.

3 3. Relevance. This can refer to cultural relevance, for example of geometric designs in art, or in the relevance of the images to the problem at hand is the visualization really necessary to help solve the problem, or does it serve as a distraction? 4. Interactivity. The ability to control and interact with the visualization seems to be an effective way to stimulate student learning; this is similar to the use of physical manipulatives in a classroom. 5. Animation. Many mathematical concepts depend on a changing parameter which can be represented as time. Animations can provide a more accurate representation of such ideas than a static image. Having identified these characteristics, one might hope we have an algorithm for making an effective visualization: create a colourful image, relevant to the problem at hand, which includes only the essential details, and allow the image to vary with time as appropriate, perhaps through the control of the student. Unfortunately, these guidelines might be helpful, but they provide no guarantee about the effectiveness of the resulting image. The problem is that incorporating these five features of effective characterizations is not a simple yes or no proposition. All of these characteristics live on a spectrum, and there are choices to be made. For example, colour can certainly make an image more visually arresting, but too many colours, or clashing colour combinations, could be detrimental. All of these issues must be balanced, and the right balance depends not only on the subject matter but on the viewer and no two viewers are exactly the same. 1.4 Animated video Mathematics is a systematic way of thinking that creates solutions to real events. In teaching math teacher try to model reality through simplification. In mathematical educational videos just animation allows to simplify and allows students to focus on understanding the nature of mathematics. Widely used option is screencasting (Špilka, Maněnová, 2013), the creator of the video is recorded using the software of your notes and display the records in your comment. Another option is to create a method of direct video animation without disturbing the cursor. The method of direct animation, there are sophisticated commercial programs or you can use a combination of freeware software resources. These technologies in education, enabling integration into an active method of teaching. Thus, the use of new teaching-learning tools, as podcasting and networked educational videos (Fernandez et. 2009), are tools in expansion within the academic setting. But the speed with which these technologies have appeared and progressively consolidated, lead to get first evidences in this moment and draw real possibilities scenario, in order to identify more efficient and effective learning methods and improve teaching quality. Caspi, Gorsky and Privman (2005) divide educational videos into three categories depending on use and purposes: demonstration videos, narrative videos and lecture sessions videos. First of these categories, demonstration videos, are a really good tool for explaining the technical and natural sciences in order to allow and improve autonomous learning, becoming more effective than other methods based on more traditional teaching, such as books and written or oral manuals. 1.5 Flipped classroom model One of the ways to use visualization in teaching at the elementary school is flipped classroom model. The concept of Flipped / Flip / invert classroom appeared in educational research a few years ago. Due to the limited amount of research there is little consensus on the complete definition of this concept. Lage (2000) defines the inverse of the class as follows: "Upside class means that the events that traditionally took place in the classroom, takes place outside the classroom and vice versa." This explanation captures the reasons for use of terminology flipped classroom. This definition would mean that the flipped classroom represents only change the arrangement of learning activities. Most research deals with the inverse class activity methods in the classroom. There are quoted on a student-oriented learning theory based on the works of Piaget (1967), Vygotsky (1978). Theoretical studies range is wide and yet can t work with the general conclusions. Likewise, there are big differences in what is considered "homework". The flipped classroom most uses asynchronous online courses, which are shared via the web interface study materials, most educational videos. From this point of view, the inverse class rather extension of the curriculum, rather than just a new way of working. Since 2013, the academic work is the concept of inverse model class. Parent category to flipped classroom is blended learning, which can be translated as computer-aided instruction. Skater (2012) defines a "blended learning" as an educational program in which the student learns partly by on-line learning materials and individually checked their education and partly educated in school under the supervision of a teacher.

4 Flipped classroom uses implementation rotation-model in the learning process when certain procedures are repeated cyclically which means: 1. The teacher outside the school prepares on-line study materials instead of interpreting the new school curriculum. 2. Students will get acquainted with the new curriculum through on-line learning materials, and thus control their own education. 3. The teacher in school activities prepared in accordance with activation methods of teaching, during which students discuss and practice the new curriculum. 4. During lessons are used personalized and activization methods of teaching. So it works with the definition of flipped classroom as methods of teaching, which is cycled through the above points (1-4). Study of George Mason University and a Pearson company (2013) defines four pillars on which the reciprocal class built: 1. Teachers introducing reciprocal teaching class in its sole discretion, which combine different methods and forms of teaching according to students' needs. 2. Teaching is focused on students. The teacher becomes a creative activity in which students are actively manage their education. 3. The teacher uses appropriate on-line learning materials to help students understand curriculum. 4. The teacher's role is irreplaceable, while providing feedback and individual approach to students in the learning process. Furthermore, these studies describe the increase in interest the flipped classroom and presents qualitative data, according to which the majority of teachers and students with this way of teaching satisfied. In his dissertation Strayner (2007) describes the effect of the flipped classroom on learning environment on college students in course of statistics, which compares with the study environment in the traditional method of teaching. Works of Moravec et al. (2010) and Day and Foley (2006) dealing with academic performance of students using the flipped classroom model. In both studies, flipped classroom students achieved significantly significant better results than students taught by traditional method. So far it is not known to many research papers flipped classroom of an elementary school. 2 RESEARCH PROJECT Traditional teaching methods (such as explanation, dialogue, description, laboratory exercises, etc. and motivational teaching methods (such as dramatization, project training, field training, etc. (25) are used in conditions of secondary level of primary schools in the Czech Republic. The use of information and communication technology brings new possibilities, new procedures and methods. The project was focused on the application of flipped teaching method, when students learned some chapters of mathematics through animated visualisation. 2.1 Project Aims The aim of the research project was to implement training using the flipped classroom model and find out whether used animated visualisation can help to increase students' academic performance. Based on goals we have set the following hypotheses: H01: In the resulting average score of pre-test we do not expect statistically significant difference between the control and experimental students groups. H02: In the resulting average score of intermediate test we do not expect statistically significant difference between the control and experimental students groups. H03: In the resulting average score of post-test we do not expect statistically significant difference between the control and experimental students groups. 2.2 Methodology Long term classical pedagogical experiment was used to verify the functionality of created animated visualisation (26). We worked with the control and experimental group (always one class of the same school year). The control group of students progressed by traditional teaching methods, especially new exposition of the new curriculum took place during lessons. The experimental group had available

5 animated visualisation that was specially created for the purpose of the experiment. For distribution educational videos were created websites (prevracenatrida.cz). Fig. 1 Screenshot from animated visualization There we also explain, what flipped classroom teaching model is. Students study animated visualisation during home preparation. Each student was assigned a login name and password. Students had also opportunity to comment each visualisation and discuss the problematic part of the matter on the social network. Brief summary of the topic and explanation of the problematic parts was performed in classes. Emphasis was placed on independent work and deepening knowledge. At the beginning of the experiment the control and experimental group went through a didactic test (pre-test). In the middle of experiment students pass intermediate test. At the end of the experiment both groups then passed another didactical test (post-test). Twenty-five educational videos were created that cover the mathematics curriculum first half of the eighth grade. The researcher was also a math teacher for the experimental group. At the end of pedagogical experiment students of the experimental group filled out a simple questionnaire, which consisted of three closed questions. The questionnaire was chosen as a fast feedback of students to the new method. Pedagogical experiment was conducted from September 2013 to January Statistical software NCSS and Excel was used for data processing. Basic values of descriptive statistics were calculated for testing hypotheses, then Student t-test and the Mann-Whitney nonparametric test whereas the normality tests did not confirm unequivocally normal distribution of the collected data. Hypotheses were tested at a significance level α = 0, Research sample Pedagogical experiment was attended by 54 students, 27 in the control and experimental class. The average age of students in the control group was 13.4 years (standard deviation 0.96) in the experimental group was 13.2 (standard deviation 1.15) The research results of student academic performance The basic task to enter the pedagogical experiment was to compare the input knowledge of students. The students finished entering pre-test, the descriptive results are shown in and Table 1 and then Figure 4 shows the distribution of the input results in both groups. Table 1 Descriptive Statistics for pre-test Mean 14,3 14,4 Standard deviation 5,45 6,1 Mode 19 - Median 16 16

6 Minimum 3 3 Maximum Range Fig. 2 Box plots for data of pre-test To compare the level of knowledge of mathematics in the control and experimental groups of students, we drew on the formulation of the null hypothesis: H01: In the resulting average score of pre-test we do not expect statistically significant difference between the control and experimental students groups. The test results are shown in Table 2. Table 2 Results for T-test and Mann-Whitney test for pre-test t-test u-value Hypothesis H01-0,2265-0,3201 accept Based on the results of the Student's T-test and Mann-Whitney test (see tab. 2) has been accepted and the null hypothesis was thus fulfilled the basic requirement of pedagogical experiment that the input is no difference between the control and experimental groups in the observed variables. For this testing and subsequent testing of intermediate and output simultaneously with the parametric Student's t-test, nonparametric Mann I-Whitney test, because the data obtained clearly not a normal distribution (normality of data was tested by Kolmogorov Smirnov test, D'Agostino Skewness tests, D'Agostino Kurtosis and D'Agostino Omnibus). Given that pedagogical experiment continued five months, we performed an experiment in intermediate continuous testing. Descriptive statistical results of both groups are shown in box plot Fig. 3 and Table 3. Table 3 Descriptive Statistics for intermediate test Mean 21,0 16,7 Standard deviation 7,41 9,66 Mode 12 -

7 Median Minimum 11 1 Maximum Range Fig. 3 Box plots for data of intermediate test When comparing the results of the experimental and control groups, we drew on the formulation of the null hypothesis: H02: In the resulting average score of intermediate test we do not expect statistically significant difference between the control and experimental students groups. In intermediate test (after approximately 2.5 months of experimental teaching) were not statistically significant differences in student performance. Null hypothesis was accepted.testing was carried out using the same statistical tests as input and test results are reported in Table 4. Table 4 Results for T-test and Mann-Whitney test for intermediate test t-test u-value Hypothesis H02 1,8035 1,6124 accept At the end of January (at the end of the fifth month of experimental teaching) was lower the output test. The descriptive results are shown in and Table 5. From Fig. 4 can be seen that the experimental group students achieved higher test scores than the control group. Table 5 Descriptive Statistics for post-test Mean 18,2 12,6 Standard deviation 7,87 7,02 Mode - - Median 16 15

8 Minimum 0 1 Maximum Range Fig. 4 Box plots for data of post-test Again we used the same procedure and taking the formulation the null hypothesis, we tested using Student's t-test and the nonparametric Mann-Whitney test. H03: In the resulting average score of post-test we do not expect statistically significant difference between the control and experimental students groups. Results of testing the output of the test are shown in Table 6. Table 6 Results for T-test and Mann-Whitney test for post-test t-test u-value Hypothesis H03 2,6763 2,2093 reject From the results shown in Table 6 shows that we reject the null hypothesis and we can conclude that he was found statistically significant difference in test results output in the control and experimental groups of students. The results of the questionnaire showed that 96 % of students well understood content, to 89 % of students visualisation helped to understand the new mathematic matter and 96 % of students would like to continue teaching mathematics by flipped classroom model. 3 CONCLUSION After evaluating the long term pedagogical experiment we can conclude, that there was significant difference in achievement (evaluated based on post-test) between students of experimental and control groups in the selected thematic unit of mathematics. Animated visualization applied by flipped classroom model, when students are studying a new educational material through on line media, did significantly affect academic performance of students. Creative animated visualization were evaluated positively. We assumed that the new method of teaching students interested, especially because the use of modern technology. Which was confirmed.

9 Research studies met goals. Animated visualisation was tested by flipped classroom model in educational practice and based on long term pedagogical experiment. Using simple reflection questionnaire, was received useful result of testing this method in teaching. REFERENCES [1] Abrahamson, D. (2006). Mathematical representations as conceptual composites: Implications for design. Paper presented at the 28th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Merida, Mexico. [2] Arcavi, A. (2003). The role of visual representations in the learning of mathematics. Educational Studies in Mathematics, 52, [3] Aspinwall, L., Shaw, K. L., Presmeg, N. C. (1997). Uncontrollable mental imagery: Graphical connections between a function and its derivative. Educational Study in Mathematics, 133(3), [4] Barwise, J., & Etchemendy, J. (1996). Visual information and valid reasoning. In G. Allwein & J. Barwise (Eds.), Logical reasoning with diagrams, New York: Oxford University Press. [5] Cunningham, S. (1994). Some Strategies for Using Visualization in Mathematics Teaching, invited article for the Zentralblatt für Didaktik der Mathematik 26:3, [6] Diezmann, C. M., English, L. D. (2001). Promoting the use of diagrams as tools for thinking. In A. A. Cuoco & F. R. Curcio (Eds.), The Roles of Representation in School Mathematics, Reston, Virginia: National Council of Teachers of Mathematics. [7] Dreyfus, T. (1991). On the status of visual reasoning in mathematics and mathematics education In: Furinghetti, F. (Ed.), Proceedings of the 15th Conference of the International Group for the Psychology of Mathematics Education, Assisi - vol.1, [8] Dreyfus, T. (1994) Imagery and reasoning in mathematics and mathematics education. Selected Lectures from the Seventh International Congress on Mathematical Education (pp ). Les Presses de l'universite Laval: Sainte-Foi, Quebec. [9] Habre, S. (1999). Visualization enhanced by technology in the learning of multivariable calculus. International Journal of Computer Algebra in Mathematics Education, 8(2), [10] Nelsen, R. B.(2001) Proofs without Words II: More Exercises in Visual Thinking. The Mathematical Association of America. [11] O Reilly, D., Pratt, D., Winbourne, P. (1997). Constructive and Instructive Representation, Journal of Information Technology for Teacher Education, 6, No 1, pp [12] Phillips, L. M., Norris, S. P., Macnab, J. S. (2010). Visualization in Mathematics, Reading and Science Education. Springer. [13] Presmeg, N.C. (1986). Visualisation in high school mathematics. For the Learning of Mathematics, 6 (3), [14] Smart, T. (1995) Visualisation, Confidence and Magic: The Role of Graphic Calculators. In Burton, L. & Jaworski, B. (Eds), Technology in Mathematics Teaching, [15] Stylianou, D.A., & Silver, E.A. (2004). The role of visual representations in advanced mathematical problem solving: An examination of expert-novice similarities and differences. Mathematical Thinking and Learning, 6(4), [16] van Garderen, D. (2006). Spatial visualization, visual imagery, and mathematical problem solving of students with varying abilities. Journal of Learning Disabilities, 39(6), [17] Vinner, S. (1997). From intuition to inhibition Mathematics, education and other endangered species. In E. Pehkonen (Ed.), Proceedings of the 21st Conference of the International Group for the Psychology of Mathematics Education, 1, Helsinki, Finland: PME. [18] Zimmermann, W., Cunningham, S. (1991). Editors introduction: What is mathematical visualization? Visualization in teaching and learning mathematics: 1-7.

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