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Computers in Mathematics Education: An Introduction Sergei Abramovich
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State University of New York at Potsdam , Potsdam , New York , USA
To cite this article: Sergei Abramovich (2013): Computers in Mathematics Education: An Introduction, Computers in the Schools, 30:1-2, 4-11 To link to this article: http://dx.doi.org/10.1080/07380569.2013.765305
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Computers in the Schools, 30:4–11, 2013 Copyright © Taylor & Francis Group, LLC ISSN: 0738-0569 print / 1528-7033 online DOI: 10.1080/07380569.2013.765305
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INTRODUCTION Computers in Mathematics Education: An Introduction SERGEI ABRAMOVICH State University of New York at Potsdam, Potsdam, New York, USA
It has been more than 50 years since the use of computers in large-scale scientific computations gave rise to the teaching machine movement (Skinner, 1958), with mathematics education always being at its frontier. Just like advances in pure and applied mathematics have often been contingent on the available methods of calculation (and vice versa), the effectiveness of mathematical education theories and success of teaching methods depend on our knowledge and understanding of how the appropriate use of computing technology can support mathematics teaching and learning. This special issue is a collection of 10 scholarly reports from Australia, Canada, the Czech Republic, Israel, Serbia, and the United States on the effective use of computers and calculators within the wide range of experiences, grade levels, educational programs, and curricular topics. In this issue, as in the very title of the journal, the word computer means any digital computing device including handheld calculators. Whereas the word technology in the context of education can be interpreted broadly to include any external support system, in this issue technology refers to digital computational tools and those made possible by the use of the computer. Using technology in mathematics education has great potential to demonstrate to learners the duality of mathematics learning and computer use in the sense that, whereas computers do enable an easy path to mathematical knowledge, the concepts of mathematics can be used to improve (even at a rather elementary level) the efficiency of computations, something that, in turn, enables better access to new mathematical ideas. Using the computer as a teaching tool makes it possible to communicate the presence of momentous mathematical ideas within a seemingly mundane curricular topic and, by the same token, to study traditionally difficult and conceptually rich topics in a new way. The appropriate use of computers in the schools can promote an experimental approach to mathematics that draws on their
Address correspondence to Sergei Abramovich, State University of New York at Potsdam, New York 13676, USA. E-mail:
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power to perform numeric and symbolic computations as well as graphic and geometric constructions. Furthermore, the approach makes it possible to balance informal and formal teaching and learning of mathematical ideas. It is experimentation with mathematics that can lead to the discovery of new knowledge, something that can be construed as what Dewey (1938) termed collateral learning. In most cases, knowledge so discovered in the context of education is considered “new” only by the students and their teachers alike. Yet, there are instances when using computers as educational tools brings about knowledge that is new for professional mathematicians as well. Consequently, computers enable the appropriate revision of advanced mathematics courses, including capstone courses for secondary teachers, to show to the motivated learners of mathematics a mathematical frontier. By responding to the call for papers, the authors of this special issue shared their expertise in and enthusiasm for teaching mathematics with technology to different populations of learners: middle and high school students, pre-service and in-service teachers, military cadets, non-traditional distance learning students, first-year calculus students, future mechanical and civil engineers, and students of architecture and art. Likewise diverse are technology tools discussed by the authors: GeoGebra, Rhinoceros, the Graphing Calculator (GC), Mathematica, Maple, Minitab, SimCalc MathWorlds, Geometer’s Sketchpad, Fathom, NonEuclid, interactive diagrams from VisualMath, the Internet, Web 2.0 tools, and handheld calculators. Two articles in this issue are devoted exclusively to GeoGebra—a free online software application for the study of geometry, algebra, and calculus at different grade levels and instructional settings. In the article by Jeffrey Hall and Gregory Chamblee (United States), multiple uses of GeoGebra are discussed. These include teaching basic concepts of plane geometry, algebra, analytic geometry, and regression analysis, including the construction of a line of best fit. GeoGebra can be used in the context of Web-2.0-related activities, and the authors provide information about Web sites where GeoGebra files can be uploaded from as well as downloaded to. Although the authors discuss the use of GeoGebra in the mathematical preparation of teachers for middle and secondary schools, the existence of a special version of GeoGebra to be used with young children, GeoGebraPrim, is mentioned also. This, without a doubt, makes the software attractive for all populations of teacher candidates and their mathematics educators. The authors noticed that the widespread avoidance of computer algebra system (CAS) tools at the lower level algebra curriculum is due to a concern that the use of technology can result in students’ reluctance to learn algebraic techniques. This, in particular, raises the issue of reformulating traditional algebraic problems to allow their meaningful use in a technological paradigm. In fact, the ubiquity of CAS opens a whole new direction in mathematical problem-posing, calling for the development of new curriculum materials to be used meaningfully with powerful CAS tools.
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Another article about GeoGebra, written by Natalija Budinski and Djurdjica Takaˇci (Serbia), reflects on training seminars for in-service teachers organized by the Department of Mathematics and Informatics at the University of Novi Sad. The topic of one such seminar was the use of the logarithmic scale in representing data about earthquakes. The authors emphasize that a big idea underlying the very use of the logarithmic scale is the property of logarithms to bring large numbers closer to each other and small numbers farther away. Another major focus of the article is the value of modeling perspective in the teaching of mathematics with technology, in particular, with GeoGebra. The value of the use of the Internet by in-service teachers and their students alike in collecting data for modeling activities has been highlighted as well. The authors share and analyze reflections by several in-service teachers on the activities introduced through the seminar. These reflections point to the teachers’ appreciation of computer-enabled learning of high school mathematics in context. Whereas GeoGebra is a powerful software tool for exploring concepts in plane geometry, its use is limited to two-dimensional problems only. The article by Svˇetlana Tomiczkov´a and Miroslav L´aviˇcka (Czech Republic) concerns the joint use of GeoGebra and the less commonly known three-dimensional computer application Rhinoceros in teaching a descriptive geometry course at the tertiary level. Descriptive geometry is a branch of mathematics in which the methods of representation of three-dimensional objects in the plane are studied. Knowing these methods is especially important for future professionals in the areas of design, civil engineering, architecture, and art. Rhinoceros is typically used in engineering design; however, it can also be used for didactic purposes. However, unlike GeoGebra, Rhinoceros is not a dynamic tool and therefore, as the authors suggest, the joint use of the two-dimensional dynamic and three-dimensional static programs can create a powerful medium for the teaching of sophisticated geometric ideas to future engineers. The article shows how this multiple application environment allows for the construction of surfaces of revolution, helical and envelope surfaces. The authors argue that using technology in the construction of surfaces of revolution, helical and envelope surfaces commonly used in applications helps engineering students develop strong conceptual understanding of the algorithms involved rather than simply memorizing different techniques of descriptive geometry. Because uses of dynamic geometry tools such as GeoGebra are mostly associated with mathematics education at the pre-college level or with tertiary level courses for teacher candidates, this article contributes to our knowledge and understanding of how such tools can be used in teaching university-level mathematical courses outside the field of teacher education. The article by Elena Naftaliev and Michal Yerushalmy (Israel) concerns the creation of interactive e-textbooks made possible by the use of a computer application known as VisualMath. The idea behind the creation of an
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interactive electronic textbook is that a student can use it as a cognitive partner, a feature that a traditional (print) textbook hardly provides. Recent investigations (Woody, Daniel, & Stewart, 2012) of students’ preferences for print textbooks and e-textbooks (not specific to mathematics education) suggest that there are significant design challenges for the latter format to become a viable alternative to the former format. In fact, these investigations point at the importance of work on addressing such challenges. In the article, the authors propose the inclusion of interactive diagrams into an e-textbook, and they suggest that such a textbook might be a medium of choice for mathematics instruction in the near future. Thus, the authors provide a theoretical foundation for their educational innovation. They argue that in order for students to use diagrams in a meaningful way, a diagram should be interactive in the sense that one must be able to modify it both mentally and physically. Such diagrams designed and used by the authors with middle school students are software tools created to support an exploration of a particular concept in a way that students’ interaction with a diagram is controlled by the authors’ didactic design and mathematical intent. Furthermore, the diagrams determine available options for explorations and establish boundaries within which those explorations may take place. The authors’ use of the word boundary has a positive connotation in the sense that when learning of mathematics follows a guided discovery mode, boundaries for explorations provide intellectual guidance rather than set limitations on students’ creative thought. A similar theme of students’ intelligent partnership with technology tools underpins the article by Dragana Martinovic, Eric Muller, and Chantal Buteau (Canada). Drawing on several theoretical constructs on intelligent humancomputer interaction introduced in the 1990s and aligning the theory with the current Ontario mathematics curriculum which emphasizes students’ learning in the technological paradigm, the authors describe the Mathematics Integrated with Computers program developed at the Brock University Department of Mathematics. The goal of this program was to revise the department’s undergraduate core curriculum in order to encourage students’ mathematical creativity, intellectual growth, and in doing so, to promote Type II applications of technology (Maddux & Johnson, 2005). The authors study conditions under which using computers creates residual mental power that can be used when technology is not available. This brings to mind the notion of learning in the zone of proximal development (ZPD) by Vygotsky (1987), who argued that things “the child is able to do in collaboration today, he will be able to do independently tomorrow” (p. 220) as ZPD creates such a learning residue. The article supports the idea of technology use as an agent of mathematical activities when rather sophisticated computer programming is required to investigate complicated mathematical phenomena. A notable outcome of the revision of the core curriculum by the authors to address students’ intelligent partnership with technology is the significant increase in
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the number of mathematics majors at the department. Not a small measure of this outcome was the program’s thoughtful support of ideas included in the Ontario mathematics curriculum so much needed for secondary teacher candidates’ self-confidence in their ability to teach mathematics according to the standards. At the secondary level, whereas the National Council of Teachers of Mathematics (2000) has posited that technology changes the way mathematics is taught and learned in the schools, some researchers argue that computers don’t have a large impact on classroom practices. The article ¨ by Helen Doerr, Jonas Arleb¨ ack, and AnnMarie O’Neil (United States) describes one specifically designed computer environment based on SimCalc MathWorlds for the in-depth study of the average rate of change, a conceptually rich topic of the secondary school mathematics curriculum. The authors show how important is the teacher’s role in the modern classroom, as he or she has to capitalize on students’ ideas, encourage them to generate ideas, and be able to bring these ideas to fruition. In other words, the teacher is the main agent of change in the classroom in terms of having knowledge of technology, mathematics, and technology-enabled mathematics pedagogy. The authors investigated three teaching practices observed in a technology-rich secondary mathematics classroom and argue for a need for teachers’ high-level mathematical competence, good understanding of students’ mathematical thinking, and intellectual courage to engage them into computer-motivated dialogic discourse. Several articles in this issue concern the use of technology in teaching calculus at the tertiary level. The article by Patrick Thompson, Cameron Byerley, and Neil Hatfield (United States) is devoted exclusively to the use of the GC (version 4.0, developed by Pacific Tech) in the teaching of introductory calculus. The authors’ basic assumption is that so far calculus reform has had little effect on students’ learning. With this in mind, they offer a new approach to calculus that helps students develop coherent meaning of the Fundamental Theorem of Calculus by reflexively connecting two core concepts involved, accumulation and rate of change, to enable one’s appreciation of the reciprocity of their importance for each other. The authors found that emphasizing the only interpretation of integral as area under curve often leads to students’ confusion when other meanings of integration focusing on the notion of accumulation are considered as well. Instead, the focus of the authors is on a functional approach to integration facilitated and made possible by the use of advanced graphing features of the GC, including animation in the plane through the change of parameters. Using this tool enables the authors to support their strategy of defining accumulation function from exact rate of change functions by using step functions the construction of which is made possible by the use of the tool. In doing so, the authors notably demonstrate how “software can embody a mathematical definition” (Conference Board of the Mathematical Sciences, 2001, p. 132).
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Such a powerful interplay between mathematics and technology enables the improvement of the efficiency of computations. The authors also point at the duality of technology and conceptual development: Whereas the former is used as a means to achieve the latter, the latter as a didactic intent determines the type of technology to be used to support the intent. Furthermore, the appreciation of the GC as a tool by students goes hand in hand with their construction of mathematical knowledge using this tool. Emphasizing practical aspects of mathematical situations that students encounter in calculus, the authors recommend avoiding unnecessary formalization both in terms of language and the level of rigor by hiding mathematical sophistication in technology. The article by Brian Winkel (United States) shares experience in using Mathematica and Maple when teaching calculus to cadets at West Point Military Academy. It emphasizes the use of real-life problems as a vehicle for one’s discovery learning opportunities within technology-rich undergraduate mathematics curricula. The article provides several examples of using technology in exploring an optimization problem, Fourier representation of signals, fizz experiment in chemistry, and other real-life situations. Technology used by the author allows for the investigation of mathematical models depending on parameters, something that is extremely important not only for the study of mathematics but of its authentic applications to engineering (or, more generally, real-life) subjects. Using multiple examples, the author convincingly demonstrates a bona fide educational phenomenon: Direction in which students’ reasoning goes in the technological paradigm is not necessarily determined by the intent of the instructor but, rather, it is a function of opportunities provided by the appropriate use of computers. The appreciation of this phenomenon implies that technology makes it possible not only to teach (and learn) undergraduate mathematics differently but its use can lead learners of mathematics to unexpected discoveries. Another interesting aspect of using technology in mathematics education mentioned in this article is that it enables one to deal with open problems and attack a problem without knowing if his/her effort would result in success. Nonetheless, even in the case of failure, as Euler noted in the context of reasoning by induction in number theory, regardless of whether something can be achieved, “we ´ may learn something useful” (cited in Polya, 1954, p. 3). Rich in citations, the article includes many interesting comments by mathematicians, educators, and psychologists regarding technology uses in mathematics education. The article by Sandra Herbert (Australia) describes the use of handheld calculators as a support system for the alternative teaching of introductory calculus with emphasis on the multiple representations of functions. This approach allowed the author to alter the order of presentation of the basic concepts of calculus. In particular, the author reflects on a course that begins with the concept of integration (traditionally interpreted as finding area under curve) and then introduces the idea of differentiation (finding
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instantaneous rate of change). Many mathematicians argued for postponing if not avoiding (as suggested in the article by Thompson, Byerley, & Hatfield) the introduction of the theory of limits when teaching calculus (especially to non-mathematics majors) because this theory is only needed for a formal demonstration of the existence theorems (Rohlin, 2012). Such an alternative approach to calculus, the rudiments of which can be traced back to Archimedes, is sometimes called non-standard analysis (Robinson, 1974). The author describes how simple experiments with calculating area under curve are conducive to recognizing patterns which, in turn, can be used to formulate generalizations in constructing rules for the integration of elementary functions. Using empirical (classroom-based) evidence, the author includes evaluation of the results of the alternative order in the presentation of introductory calculus concepts, arguing that the use of CAS enables a better understanding of the ideas of instantaneous rate of change within an introductory calculus course. One notable role of technology in the modern educational environment is to serve the needs of students who are able to take online courses only. The article by Shannon Guerrero and Terry Crites (United States) reflects on an online mathematics education program at the master’s level serving non-traditional students within a large county in the State of Arizona who are either secondary or two-year college mathematics teachers. While distance education opportunities have been offered to non-traditional students for many decades, it is the advent of computers and other technological tools into the world of education that has radically changed how one can learn and may be taught without participating in a face-to-face instruction. When developing the program, the authors integrated different learning theories with multiple tools of technology. The main emphasis, however, was on interaction and its various aspects (student-student, student-content, student-instructor, student-system) each of which can be supported both by system-based (e.g., print materials) tools and non-system-based tools (e.g., mathematical software). The authors demonstrate how the joint use of all technologies involved contributes to the program’s objective by maximizing the didactic power of online learning environments for non-traditional students that are much different from just being replicas of face-to-face instruction. To conclude, I thank Managing Editor Mary L. Johnson for providing me with much needed assistance during the review process. I also greatly appreciate the professionalism of many anonymous reviewers who, via Mary, helped me in my work with the authors toward the improvement of their original submissions. Finally, I am grateful to Editor D. LaMont Johnson and Associate Editor—Research Cleborne D. Maddux for the invitation to put together a special issue on the use of computers in mathematics education. This issue is now offered to the judgment of the readers in the hope that the authors’ teaching ideas presented in the following 10 articles will encourage
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others in the mathematical education community to use these ideas in their own teaching of mathematics with computers in the schools.
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REFERENCES Conference Board of the Mathematical Sciences. (2001). The mathematical education of teachers. Washington, DC: The Mathematical Association of America. Dewey, J. (1938). Experience and education. New York, NY: MacMillan. Maddux, C. D., & Johnson, D. L. (2005). Type II applications of technology in education: New and better ways of teaching and learning. Computers in the Schools, 22(1/2), 1–5. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author. ´ Polya, G. (1954). Induction and analogy in mathematics (Vol. 1). Princeton, NJ: Princeton University Press. Robinson, A. (1974). Non-standard analysis. Amsterdam, The Netherlands: NorthHolland. Rohlin, V. A. (2012). A lecture about teaching mathematics to non-mathematicians. Part I. Retrieved from http://mathfoolery.wordpress.com/2011/01/01/a-lectureabout-teaching-mathematics-to-non-mathematicians/ Skinner, B. F. (1958). Teaching machines. Science, 128(3330), 969–977. Vygotsky, L. S. (1987). Thinking and speech. In R. W. Rieber & A. S. Carton (Eds.), The collected works of L. S. Vygotsky (Vol. 1, pp. 39–285). New York, NY: Plenum. Woody, W. D., Daniel, D. B., & Stewart, J. M. (2012). Students’ preferences and performance using e-textbooks and print textbooks. In S. Abramovich (Ed.), Computers in Education (Vol. 1, pp. 43–58). New York, NY: Nova Science.
