IMPROVING ONLINE MATHEMATICAL THINKING Zekeriya Karadag Ontario Institute for Studies in Education University of Toronto, CANADA


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1 IMPROVING ONLINE MATHEMATICAL THINKING Zekeriya Karadag Ontario Institute for Studies in Education University of Toronto, CANADA Abstract: This case study is aimed to explore and investigate students mathematical thinking processes in computer environment. Each action done by the participant is recorded by screencasting software, and the data is microanalyzed. As a result of this close analysis of data, information revealed is as follows: (1) Students may use online cognitive tools effectively in order to reduce their cognitive load, (2) An adjustable frame analysis method could be used for collaborating as well as feedback, (3) Both traditional and openended math problems could be effectively used in online studies, and (4) Online math contests aiming to assess students solution processes could be possible. Main Theme: New technologies in the learning (mathematical thinking) of mathematics Despite the diversity in computer based learning environments, some being online whereas others offline, they can be classified into three main categories based on their pedagogical approaches: Dynamic Learning Systems (i.e. Geometer s Sketchpad, Capri, and Virtual Manipulatives), Computer Algebra Systems (i.e. Maple), and Intelligent Tutoring Systems (i.e. Cognitive Tutors). Dynamic Learning Systems and Computer Algebra Systems aim improving the cognitive abilities of learners while Intelligent Tutoring Systems try to integrate artificial intelligence principles into the mathematics education (Heid, 2005; Izydorczak, 2003; Kieran & Drijvers, 2006; McDougall, 1997; and VanLehn et al, 2003). Dynamic Learning Systems promote learning by discovery whereas Computer Algebra Systems shift learners focus from procedure to thinking. It is no doubt that each of these systems provides unique contribution to mathematics education. However, besides these unique contributions, new questions emerge to the era of mathematics education: Do we miss traditional perspective of mathematics education? Do we also need to integrate traditional solution methods (i.e. factorization of polynomials or finding the domain of a function as in the paperandpencil methods) into these systems? More importantly, do we miss communicating, commenting, and providing feedback on students work and solution processes while studying online? Because studying in computer environment, whether online or not, is usually performed by individuals, the processes are hardly observed or shared with peers or teachers. Do we miss mathematical thinking processes which are highly promoted by interaction and collaboration (peer collaboration, teacherstudent interaction, or expertnovice interaction)? Online collaborative learning environments such as Knowledge Forum, Sakai, and Wikis provide special features for promoting collaboration and support to improve students learning as well as knowledge building. However, tracking students mathematical thinking processes demand more features than the ones provided by these environments. These learning environments provide either limited or no support for mathematical notation in communication as well as a tool for recording and sharing solution processes. In this case study, I explore one high school student s mathematical thinking by tracking her solution processes in computer environment. After reviewing related literature, I will describe the methodology, data collection, and analysis process in detail. 1
2 Mathematical thinking in online environments Mathematical thinking is usually defined as a set of mathematical and mental activities, such as abstracting, problem solving, conjecturing, generalizing, reasoning, deducting, and inducting (Tall, 1991 and Harel et al, 2006), although Sternberg (1996) concludes that there is no consensus on what mathematical thinking is because scholars define the term depending on their own perspective (p.303). In contrast, there is universally strong agreement on having mathematical thinking as the main goal of mathematics education. For example, the Ontario Curriculum (2006) suggests that problem solving, reasoning and proving, reflecting, selecting tools and computational strategies, connecting, representing, and communicating are the mathematical processes that support effective mathematics learning. Although there is a rapid increase in the number of studies related to classroom teaching applications, very little research has been conducted in mathematical thinking in online learning environments. Abstraction, as vertically reorganizing previously constructed mathematics into a new mathematical structure by either inducting or deducting (Tall, 1991; Dreyfus, 1991; Dubinsky, 1991; Hershkowitz et al., 2001; and Hazzan & Zazkis, 2005); generalization, as abstracting properties and relations and conjecturing new hypothesis and structures (Sriraman, 2004); representation, as displaying mathematical relationships pictorially, graphically, or symbolically (Greenes & Findell, 1999); and reasoning, as a tool for understanding abstraction (Russell, 1999) all demonstrate dynamic characteristics of mathematical thinking. Because of its dynamic characteristics, mathematical thinking needs specific tools, such as paper, pencil, calculator, compass, and protractors, as well as a twoway communication. Each technology regardless if it is a pencil or sophisticated software is used as a cognitive tool and serves to develop our cognitive capacity. Solomon and Perkins (2005) describe this cognitive development in three models: (1) effects with, (2) effects of, and (3) effects through technology in terms of their magnitude and emerging periods. Modern cognitive tools such as dynamic learning systems and computer algebra systems have been developed to solve the first challenge (a need for specific tool), and provided the opportunity for effective use of students capability. Many scholars have been interested in investigating effectiveness of these cognitive tools, and many researches have been done (i.e. Forster, 2006; Galbraith, 2006; Kieran, and Drijvers, 2006; and Threlfall et al, 2007). Regarding with the use of these tools, Goldenberg (2000) suggests that technology should develop students thinking capacity by addressing the use of technology as a cognitive tool. As a consequence of the studies related with integration of these contemporary tools, understanding and assessing the process of understanding has also become as a major interest. Hosein et al (2007) have employed remote observation method to explore students understanding while studying in computer environment. The issue of students understanding and assessment of this understanding are discussed by Barmby et al (2007). However, developing communication tools and combining these tools are less considered. The second challenge (twoway communication) in mathematics, such as sharing mathematical work, getting feedback, and reflecting on work, remains challenging online mathematical thinking. What I mean by sharing is to share the process of mathematical thinking and solution processes. Although there are many online examples created either by video recording in classrooms and by employing 2
3 smart technologies, these examples have usually been done by teachers or some experts and their purpose seem far from establishing an online learning environment for students. Current collaborative learning environments such as Knowledge Forum, Sakai, and Wikis lack the support for mathematics education although Smart Technologies provide significant support in terms of mathematical notation. Methodology In order to explore and track students mathematical thinking process, I conducted a case study supported with two software, Geogebra, a free online cognitive tool, and Wink, a free screencasting software to capture screenshots and movements of mouse and to export as flash movies. In addition, wink allows users annotating notes on any of the frames, which teachers can provide justontime feedback for students as well as students can share their comments on peers work. After setting the stage for the research, I started to investigate students performance with these tools and did all communication online. The participant is a typical grade 12 student in an Ontario high school, recently immigrated and almost high achiever both in mathematics and computer use. Topics were chosen from current high school curriculum included functions, and training for using software was provided before starting the study, which was lasted in three main sessions each having two subsessions. One of the main sessions had traditional problems, which could also be studied in paperandpencil environment, whereas the other two contained openended problems to be explored by using the cognitive tool, Geogebra. Each session was performed in computer environment, and Wink, screencasting software, was used to capture student s work. The work captured was collected as qualitative data and microanalyzed by focusing on each half second of the work it was my decision to set up the rate of capture as 2 frames per second (means 2 screenshots per second). In order to avoid repetitions and not to go beyond the scope of the paper, I will provide a general view of one typical session and focus on some particular moments frames which correspond to seconds were recorded in this session, and four openended questions were provided to the student. Each change on the screen was taken as one unit of analysis and varied in length from 3 to 92 frames, and 66 units were microanalyzed in this session. Microanalysis was done by two different perspectives, based on what is seen and what is interpreted. What is seen stage is the description of exactly what is seen on the screen and provides me to have a general view of the work as well as preliminary categories emerged from data. What is interpreted analysis (see table 1) is more subjective analysis and based on my personal perception. This phase of analysis lets me get closer to the data and think as does the student. Actions Frames Duration Initial Coding Focused Coding Start Finish Frames Seconds (Interpreting what I see) (Categorizing) Reading the question 1. Reading question Examining the movement of the point by manipulating the slider. Analyzing the situation Reflecting on her writing because her writing rate is very slow and making some small changes during writing. Her writing rate and mistyping let me interpret that her mind could be busy. Busy mind Examining the points on the function either to assess Analyzing the situation and what she has written or what she should write more. synthesizing Writing the rest of her answer The point A is a point of function f. It is moving on the function Thinking on either what she has written or what she should write more. Reflecting Her mind seems busy because her writing rate looks slow. Busy mind Assessing her answer. Assessing Screenshots are blank; it might be because the Thinking 3
4 software could not find any movement to capture means she is thinking Keeps thinking. Thinking Manipulates the slider and probably trying to understand if she did correct or if there is something missing. Adding another comment such that It can take every number like decimal ones. about the nature of the coordinates of the point A, but leaves her sentence uncompleted Looking at the coordinates of the point Coming back to the answer sheet and writing the abscissa of the last version of the point and Checking the ordinate of the point. Writing the ordinate and moving to the next step of the assignment. She seems to forget the ordinate of the point and checks it Reading the step Entering the coordinates of the point B Reading the rest of the instruction at step Making the point B colored and traceable Examining the movement of the point by manipulating the slider. Table 1: An excerpt from microanalysis of data Using geogebra as a cognitive tool Cognitive Load Theory: She is using her working memory for thinking rather than storing the data. Effective use of working memory (Cognitive Load Theory) Checking what is done After microanalysis stage, I focused particularly on the moments, which mathematical thinking had occurred. While data analysis was on progress, I provided feedback to the student to explore the development in her understanding and more importantly her benefits from annotating feature of the software, in other words to understand the effects of feedback in online environments. Figure 1: 2 frames from the solution process of the problem 1 The screenshots seen above are the two distinct scenes just after student s reading the first question. The first one (figure 1: frame 300) is a frame which demonstrates the student s examination of problem situation on geogebra screen whereas the second one (figure 1: frame 380) is an example of writing answer for the question. Figure 2: An empirical example for cognitive load theory Another visual excerpt from the analysis shown above demonstrates a significant example of cognitive load theory. She is writing the question by 4
5 exemplifying the coordinates of a point (see figure 2: frame 657), switching to the geogebra screen to look at the abscissa (see figure 2: frame 658), writing the abscissa (see figure 2: frame 671), and switching to the geogebra screen to look at the ordinate. This close and deep analysis to explore the pattern of student work revealed the fact that the student applied a general problem solving pattern very similar to the one described by Polya (1945) although the borders between processes were not so clear. A typical pattern of working on a problem was as follows: (1) reading and understanding the question, examining the situation with cognitive tool, (2) writing some initial comments, assessing what is written and trying to find what is next, (3) switching a couple times between cognitive tool and questionanswer page depending on difficulty of the problem, (4) adding some more comments, and assessing what is written by switching a few more times. Results and Discussion Results obtained from microanalysis of data shed light on three major issues: (1) support for cognitive load theory, (2) benefit from feedback on thinking process rather than product, and (3) confidence in studying traditional problems online and need for a new drawing tool. The results demonstrate that the student benefits from studying mathematics supported with technology because she tends to behave in favor of cognitive load theory and uses her working memory effectively (Sweller, 1999). One of the main purposes of using cognitive tools in mathematics education is defined as to reduce the amount of cognitive load stored in working memory so that the memory could be used for the process of information (LeFevre et al, 2005). I have documented many evidences for the existence of using this specific cognitive tool throughout the entire study even though for very little information pieces. Figure 3: Annotating on students' work for feedback I have also documented evidence on the effective use of this method in communication and collaboration. I have used the annotating feature of the capturing software (see figure 3), as many others have, in order to provide just on time feedback and to increase student s understanding at a higher level. Besides my way of using in this study, providing just on time feedback may lead students focus on a particular thinking process or solution procedure, and students may benefit from this opportunity in collaborating with each other or sharing their thoughts and studies with their peers. They may also comment on each other s work to elaborate their understandings. In order to explore the use of traditional problems, I asked three questions in a separate session related with finding domain and inverse of the functions. The 5
6 problems were solved in computer environment as done in the other sessions, and actions on the screen were captured. The student effectively used the equation editor feature of MS Word to write mathematical notations, however, she did hesitate when finding intersection of the sets. Although there is drawing tool in many word processors, their effectiveness in drawing a number line and using this number line for set operation is quite limited so this result leads me conclude that there is need for including this specific feature into the mathematics learning environments. Despite the advantages of macro and micro analysis in frame analysis method, this method of analysis is significantly time consuming, that is why; I suggest an adjustable frame analysis method, which allows microanalysis as needed. An adjustable frame analysis which allows switching between macro and micro analysis could be beneficial both for researches and online learning environments as well as online math contests. In fact, online math contests are needed a specifically designed software or platform, including cognitive and recording tools as well as writing and drawing tools designed for mathematical symbols. As conclusion, in addition to the mathematical ability of a learning environment, or online math contest environment, this environment should allow students study math problems and record their own solutions. Furthermore such an environment should be designed flexible in reviewing recorded files so that reviewers (experts, teachers, or peers) could focus on a particular moment as needed. References Barmby, P., Harries, T., Higgins, S., and Suggate, J. (2007). In Woo, J. H., Lew, H. C. Park, K. S. & Seo, D. Y. (Eds). Proceedings of the 31 st Conference of the International Group for the Psychology of Mathematics Education. Seoul: PME, Vol. 3, Dreyfus, T. (1991). Advanced Mathematical Thinking Processes. In D. Tall. (Ed). Advanced mathematical thinking. Boston: Kluwer academic publishers. Dubinsky, E. (1991). Reflective Abstraction in Advanced Mathematical Thinking. In D. Tall. (Ed). Advanced mathematical thinking. Boston: Kluwer academic publishers. Forster, P. A. (2006). Assessing technologybased approaches for teaching and learning mathematics. International Journal of Mathematical Education in Science and Technology. 37:2, Galbraith, P. (2006). Students, mathematics, and technology: assessing the present  challenging the future. International Journal of Mathematical Education in Science and Technology. 37:3, Goldernberg, E. P. (2000). Thinking (and Talking) about Technology in Math Classrooms. Issues in Mathematics Education Greenes, C. & Findell, C. (1999). Developing Students Algebraic Reasoning Abilities. In Stiff, L. V. & Curcio, F.R. (Eds). Developing Mathematical Reasoning in Grades K12. Reston, VA: The National Council of Teacher of Mathematics, Inc. Harel, G., Selden, A., &Selden, J. (2006). Advanced Mathematical Thinking: Some PME Perspectives. In A. Gutierrez, P. Boero. (Eds), Handbook of Research on the Psychology of Mathematics Education: Past, Present and Future Sense Publishers. Hazzan, O. & Zazkis, R. (2005). Reducing Abstraction: The Case of School Mathematics. Educational Studies in Mathematics, 58,
7 Heid, M. K. (2005). Technology in Mathematics Education: Tapping into Visions of the Future. In W. J. Masalski and P. C. Elliot (Eds), TechnologySupported Mathematics Learning Environment. Reston, VA: The National Council of Teachers of Mathematics, Inc. Hershkowitz, R. et al. (2001). Abstraction in Context: Epistemic Actions. Journal for Research in Mathematics Education, 322, Hosein, A., Aczel, J., Clow, D., and Richardson, J. T. E. (2007). In Woo, J. H., Lew, H. C. Park, K. S. & Seo, D. Y. (Eds). Proceedings of the 31 st Conference of the International Group for the Psychology of Mathematics Education. Seoul: PME, Vol. 3, Izydorczak, A. E. (2003). A Study of Virtual Manipulatives for Elementary Mathematics. Unpublished PhD Thesis. Buffalo, USA: State University of New York. Kieran, C. and Drijvers, P. (2006). The CoEmergence of Machine Techniques, Paper andpencil Techniques, and Theoretical Reflection: A Study of CAS Use in Secondary School Algebra. International Journal of Computers for Mathematical Learning. 11: LeFevre, J., DeStefano, D., Coleman, B. and Shanahan, T. (2005). Mathematical Cognition and Working Memory. In J. I. D. Campbell (Ed). Handbook of Mathematical Cognition. UK: Taylor & Francis, Inc. McDougall, D. E. (1997). Unpublished PhD Dissertation. Toronto, Canada: Ontario Institute for Studies in Education. Polya, G. (1945). How to Solve it. USA: Princeton University Press. The Ontario Curriculum (2006). Grade 11 Mathematics. Ontario: Ministry of Education. Russell, S. J. (1999). Mathematical Reasoning in the Elementary Grades. In Stiff, L. V. & Curcio, F.R. (Eds). Developing Mathematical Reasoning in Grades K12. Reston, VA: The National Council of Teacher of Mathematics, Inc. Solomon, G. and Perkins, D. (2005). Do technologies make us smarter? Intellectual amplification with, of, and through technology. In R. J. Sternberg and D. D. Preiss (Eds). Intelligence and Technology: The Impact of Tools on the Nature and Development of Human Abilities. NJ: Lawrence Erlbaum Associates, Inc. Sriraman, B. (2004). Reflective abstraction, uniframes and the formulation of generalizations. Journal of Mathematical Behavior, 23, Sternberg, R.J. (1996). What is Mathematical Thinking? In R. J. Sternberg, T. Ben Zeev (Eds). The Nature of Mathematical Thinking. Mahwah, NJ: Lawrence Erlbaum Associates, Publishers. Sweller, J. (1999). Instructional Design in Technical Areas. Australia: The Australian Council for Educational Research Ltd. Tall, D. (1991). The Psychology of Advanced Mathematical Thinking. In D. Tall. (Ed).Advanced mathematical thinking. Boston: Kluwer Academic Publishers. Threlfall, J., Pool, P., Homer, M. and Swinnerton, B. (2007). Implicit aspects of paper and pencil mathematics assessment that come to light through the use of the computer. Educational Studies in Mathematics. 66: VanLehn, K. et al. (2003). Why do only some events cause learning during human tutoring? Cognition and Instruction. 21(3),
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