Electronic Technology and NCTM Standards

A Synthesis of Technology and Standards 2000 Conference

Alfinio Flores

Arizona State University

Preface

The Technology and Standards 2000 Conference was held in early June 1998. Experts from the United States and four other countries gathered to share experiences and points of view about the use of electronic technology, such as calculators, computers, and the internet, in the teaching and learning of mathematics. Of course, other technologies are also important for the development of mathematical ideas, however, the focus of this conference was on electronic technology. In this synthesis, accordingly, when the word technology is used, it should be understood as electronic technology.

The presentations included several kinds of papers. Some were global surveys of recent research on different aspects of the effect of using technology in the teaching of mathematics. Other papers were presentations of particular experiences of the use of technology in a few classrooms, or descriptions of particular research projects. There were also papers discussing theoretical and philosophical issues about what do we need to teach in mathematics, and caveats about and the potential of newly developed technologies. There were even papers that in places read as a short science fiction stories.

The conference was organized by themes: 1) lessons from research; 2) lessons from practice; 3) Possibilities for using the internet; 4) hand-held devices; 5) new mathematical content and new technologies; 6) dynamic mathematics; and 7) possibilities for Standards 2000 in electronic format.

Rather than using the themes of the conference, this synthesis follows in broad terms the outline of the Standards 2000 first draft. Because the main purpose of this synthesis is to serve as a working document for the Standards 2000 project, rather than being a polished scholarly paper, no effort was made to refer results to original sources. Therefore, when research results, findings, and recommendations that are summarized in the conference papers are presented here, they are referred to by the name of the author of the paper, and not the original source. Interested readers may find the original sources used by the conference authors in their corresponding papers at http://mathforum.org/technology/papers/ (username: nctmtech, password: nctmtech). For a synthesis from a different perspective, see Wilson (1998).

Introduction

In learning mathematics, as in many other realms of life, things that are worthwhile do not come easy. Learning occurs many times through perturbation of existing schemes. The purpose of using technology is not to make the learning of mathematics easier, but richer and better. The focus when using technology should not be on what the tool does, but on the way children think. Despite the palpable importance of hardware, content, and pedagogy of the software should receive more emphasis (Clements, 1998).

Clements (1998) mentions three research implications for the selection and use of software. First, use an appropriate combination of off- and on- computer activities. Second, consider technology more as a mathematical tool, rather than as a pedagogical tool. Third, consider technology as a thinking tool. One way to accomplish this is to use computer programs that can be extended, for long periods of time, across topics, to engage students in meaningful problems and projects, rather than providing a variety of applications with no internal coherence.

We need to develop a coherent vision to develop an approach to using technology in the teaching of mathematics that evolves consistently as the technology changes (Goldenberg, 1998). General trends affecting both personal computers and the internet are the ever more graphic oriented and user-friendly interfaces. This will without doubt also affect children as they learn mathematics in these new environments.

From a practical point of view, the sharp distinction between computers and calculators tends to be blurred, as computer options include not only more powerful machines but also portable and less expensive models. Hand-held devices are not limited to stand-alone calculators. Increasingly, such devices can perform telecommunication functions, and are compatible with computers and network systems. Calculators and computers need to be viewed as cooperating technologies, rather than competing technologies (Kaput, 1998). For example, the teacher could have a laptop computer with classroom display, and each student a hand-held device that can communicate with the computer. The real potential of technology in classrooms will become apparent when the devices in classrooms are linked.

 

Guiding principles for mathematics teaching and learning

Equity and opportunity

The impressive progress in electronic technology that is available for the teaching of mathematics will not automatically bridge the equity and opportunity gap present in our schools. In fact, one of the most salient aspects of the savage inequalities that exist among schools is the access or lack of it that students have to modern technologies. Coordinated efforts need to be made to provide access for all students to these tools.

However, simply having access to the tools does not either guarantee equitable opportunity either. The way the tools are used is crucial for this purpose. For example, computers should be used as an integral part of the curriculum, and not only to provide variety, rewards, enrichment. When computers are offered only to students who have finished their seat work, often slower students never get to use the computer. On the other hand, when computers are used as part of an environment that fosters longer mathematical investigations, they provide opportunity to all students to focus on high-level goals, and participate at equal levels (Clements, 1998).

It is important that teachers make a conscious effort to ensure that all students feel encouraged to use computers (Klawe, 1998). What works best for boys is not always what works best for girls and vice versa. It is important for teachers to be aware of the potential differences and provide flexibility in how computers are used (Klawe, 1998).

Research on hand-held calculators points to positive benefits for groups of students who in the past have done less well with traditional school mathematics than the general population (Dunham, 1998). The use of such tools provides alternative points of entrance to higher-order thinking activities. No longer need students to be held back because of lack of traditional computational skills.

Computer tools can provide children with opportunities for cognitive play and subsequent mathematical activity that can follow from that play under the careful guidance of a teacher (Olive, 1998). Children can progress from cognitive play to teacher directed mathematical activity, and then to independent mathematical activity. Social interaction between students and between teacher and students is a vital component of these activities. Different students have different ways to be good in mathematics. Technology can give more students the opportunity to do what they are able to do.

 

High quality mathematics

The new tools make new mathematics accessible to students, and the discussion needs to transcend a preoccupation with performing old tasks with new tools (Dick, 1998). However, we need to ask ourselves what it is we value and how to use, or not use, technology to support our goals (Goldenberg, 1998). The temptation when revising curriculum with technology in mind is to include topics that can be done well with the technologies, without regard to their importance (Goebel & Teague, 1998). New electronic technologies make possible to have access to, and gather more information. We need to make sure it is not at the cost of systematization of ideas or the fostering and formalization of reasoning (Goldenberg, 1998).

Technology makes it possible for students to have access to certain important mathematical ideas in intuitive forms earlier, potentially laying a valuable foundation for the later formalization of these ideas. But sometimes the ideas are genuinely more subtle than they appear, and early access can trivialize or distort them (Goldenberg, 1998). Also, many ideas in mathematics have more than one role. If we use technology to replace one approach, we need to make sure that the concomitant benefits do not get lost (Goldenberg, 1998).

Technology tools should be used to help describe inner structure of mathematical concepts. Students should be able to create another representation or another expression for the same mathematical object, but changed in form to reveal some property of interest (Goldenberg, 1998). Computer technologies can make the abstract more concrete. However, we need to remember that mathematics is about abstraction. Its value in the ‘real world’ derives from the fact that it applies across wildly different domains (Goldenberg, 1998).

According to Hoyles (1998), the computer can help bridge the gap between action and expression and can help connect different modes of expression. However, in order for this to happen the following are required: 1) students and teachers must appreciate what they wish to accomplish and how the technology might help them; 2) the technology must be carefully integrated into the curriculum and not simply added; and 3) the focus of all activity is kept on mathematical knowledge and not on the hardware or software. Spending most of the computer time learning computer-specific skills or using mostly drill-and-practice software does no contribute to attaining the goals of the Standards (Clements, 1998). Programming can help students develop higher levels of mathematical thinking.

Technology will provide students with easier access to courses, especially those that use web sites and interactive CD-ROMs. However, students will still need guidance, structure, and external evaluation to ensure that they know what the profession thinks they should know (Goebel & Tague, 1998). Working in isolation, it is difficult for students to know what ideas are really important and which are secondary. Students can also form misconceptions if they have no one to correct them.

 

Teaching and learning environment

It is not enough to have the new technology available. Teachers need to develop a teaching and learning environment that is conducive to learn mathematical ideas. The kind of software, and the way it is used are crucial to develop this environment. Software can help students learn mathematics better when the following conditions are satisfied (Klawe, 1998): a) provide a vast amount of examples and problems; b) facilitate visualization and manipulation, and link visualization with symbolic representations; c) Provide adaptive sequencing and feedback, so that students assume gradually greater cognitive responsibility; d) Provide sustained contextualization in a meaningful and engaging application. Most importantly, students need to have conscious awareness of the concepts, structures, and algorithms they encounter and use in educational software. When multimedia activities are used with supporting classroom activities such as related pencil and paper worksheet activities, large and small group discussions, and journal writing, students show significant increases in interest, enjoyment, and achievement in learning the mathematics concepts involved (Klawe, 1998).

By computer environment, Battista (1998) means not only the computer tools made available to students, but also the instructional tasks in which students are to apply these tools. He describes four characteristics of fertile computer environments. First, computer environments should support problem centered inquiry. Second, computer environments should promote students’ learning in ways that are consistent with the way students construct mathematical ideas according to research. Third, computer environments should engender students’ development and use of appropriate mental models of mathematical phenomena. Finally, computer environments should support reflection on and abstraction of mental operations necessary for properly conceptualizing and reasoning about mathematization of the phenomena. When students make predictions before acting, they consider their actions in the context of their theories.

The nature of the mathematics classroom can be drastically changed through the use of linked technology. Through the internet students can have a world wide classroom where they can hold conversations about mathematics; receive individualized and personal support; participate in collaborative problem solving with students all around the world; deal with problems in context, applications of mathematics, relevant problems; pursue their own interests and inquire independently; and construct and grow by creating and interacting (Weimar, 1998).

Dede (1998) depicts futuristic vignettes of how technology can be used in the teaching of mathematics. They include distributed learning with technology, a gig in cyberspace (by Roschelle & Kaput), a pre-service observation experience, and a parent teacher conference. These vignettes, like science fiction stories, unobtrusively brief readers about the future, while they tell about educational goals achieved through effective action. They are examples of powerful ideas of how technological tools enable new types of teaching and learning that speak to problems in today’s classrooms and communities.

Computer manipulatives. Computer manipulatives (such as Shapes) offer students a medium that can be more consistent with the mental actions than the concrete manipulatives, and can offer multiple representations of concepts simultaneously (symbolic, and graphical). The computer environment can help link the manipulatives with the symbols and help students develop integrated-concrete knowledge (Clements, 1998). Computer manipulatives need to provide children with the opportunity of generative symbolic activity (Olive, 1998). Complex concepts need to be presented several times. Technology can provide another way to present material, in addition to concrete materials and the more formal, symbolic approach (Alejandre, 1998).

Computer software offers the possibilities of providing dynamic manipulation environments for students. Finzer & Jackiw (1998) characterize dynamic manipulation environments by three attributes: 1) manipulation is direct; 2) motion is continuous; and 3) the environment is immersive.

According to Hoyles (1998), there is the need to bridge between understanding developed by interaction with software and more conventional mathematical meanings. There are new objects and relationships to attend to, different things to do and representations to interpret. While there is also room for new misconceptions, there is also the potential for more engagement with mathematical ideas. Any technological implementation of mathematical ideas gives rise to unintended effects. Rather than trying to avoid completely all theses effects, we need to be able to say in as much detail as possible what they are. The features of the software, including the unintended ones are likely to turn into specific characteristics of the meaning constructed by students (Balacheff, 1998).

Integrating new technology. According to Laborde (1998), the technology should not be used for itself, but for supporting, improving and changing the learning of mathematics through explorations of a great number of cases, possible variations of the problems, and visual or numerical feedback. The computer may work as a catalyst for generating questions (what if...?), for generating counterexamples, and so fostering a more scientific approach.

Goebel & Tague point out at several lessons learned as their school integrated the use of technology into the curriculum. 1) A strategy needs to be developed to upgrade the computer equipment; 2) It takes time to learn to use the technology for teacher and students; 3) Differences in notations between computers and books can be difficult; 4) Students may loose the big picture when learning to use technology; 5) Students who document their work are better able to explain what they are doing; 6) Real world problems are more important with technology; 7) Students need to learn to make choices about which methods are appropriate.

Integration can be realized to less or more extent. Laborde characterizes high levels of integration when the teacher introduces mathematical content through technology and not only using technology on previously introduced mathematical notions, and in addition, the teacher institutionalizes the content to be learned by referring to the computer environment. Laborde underlines that the greater degree of integration require long term use of technology. A greater degree of integration is achieved if the students on their own have access to technology to solve problems given independently of the computer.

Computer based teaching of mathematics involves of course mathematical knowledge, but also knowledge or how to use the software. The created tension of the two demands can be solved only if the use of the technology is a long term use (Laborde, 1998). Computers are truly beneficial when they are integrated as part of the curriculum from the start. We need to plan for long term benefits.

Role of students. We need to focus on ways in which computer based environments can enhance children’s own construction of mathematics. Computer environments should not do the mathematics for the students, but they allow them to express their own mathematical ideas (Olive, 1998). Students need to personally manipulate computer representations and reflect on these actions. They need time to construct or reconstruct ideas. Students can also use computers to change and rearrange information. Students can construct several representations and comparisons, making qualitative differences in their educational experience (Clements, 1998).

Computers, specially when used to support an environment for intentional learning can help students build shared knowledge (Clements, 1998). Computers can also help students develop projects for knowledge design. Computers environments are conducive to wondering and playing with mathematical ideas. However, in order to empower students, they need to develop fluency and competency with the new tools (Goldenberg, 1998).

Students become more active in classrooms with graphing technology, do more group work, investigations, explorations, and problem solving (Dunham, 1998). Similarly, collaborative interaction with the computer has several benefits. In addition to the motivational effects of sharing a computer, the continuous dialogue between partners about how to interact helps students recognize and articulate the mathematical concepts of the software (Klawe, 1998). Also, children need to verbalize their thinking. Collaborative work, with the help of the technological tools can help diminish the dependency of the students on the teacher (Heid, 1998).

A computer environment should not be considered independently from the action of the students. A "microworld" is established by students when actually working in the environments (Olive, 1998). The microworlds cannot be used without a teacher to pose problems and guide the children in their explorations. However, it is the action of the child that determines the effectiveness of an intervention rather than the intent of the teacher. The features of a microworld are products of the students’ actions within the microworld and do not exist independent of those actions (Olive, 1998).

Computers can give students the opportunity to reason about and manipulate mathematical ideas and relationships prior to fully mastering them, this helping them promote their learning (Heid, 1998). On the other hand, students need to develop a modeling relationship in which mathematics is a tool to make sense of what they experience with the computer. Mathematical exploration is a way to support learning as a construction of new knowledge, however, knowledge is necessary to guide and organize the mathematical exploration. Modeling and learning as construction of meaning cannot be separated during an interaction with a computer-based learning environment. Students need to construct a link between a mechanical world offered by the computer interface, and a theoretical world, which is the world of mathematics. The passage from the mechanical world to the world of mathematics, even if the need of this passage has been perceived by the student, is not obvious (Balacheff, 1998). Prediction tasks are particularly appropriate for computer environment. When the prediction of the student does not match the outcome, this is a good opportunity to ask why, and call for proof (Laborde, 1998). An additional aim of integrating new technology is to develop students’ ability to choose the appropriate tool to use and use it in a relevant way for solving a problem (Laborde, 1998).

Role of the teacher. The teacher plays a crucial role in a curriculum where computers are embedded. The teacher needs to choose, use, and infuse these programs. Teacher has a critical role mediating students’ computer work. Teacher can help bring the mathematics in computer work to a conscious level of awareness and extend the ideas encountered; focus attention on critical mathematics aspects of activities, emphasizing the need for consistency and mathematical language; facilitate the assimilation and accommodation of new ideas using the computer feedback as a catalyst; construct links or mappings between computer and non-computer work; and provoke reflection and prediction (Clements, 1998). Teachers have to ask questions to cause students to reflect on what they do. Group discussions with a computer and a large screen display are critical (Clements, 1998).

Generalizing a claim by Klawe (1998) about computer games, it could be said that students are more likely to see a software as helpful in learning mathematics and are more likely to interact with it, if they receive clear indications that the teacher believes this is the purpose of the software and expects them to interact with them.

Teachers need to know about the research findings, about the capabilities of calculators and computers, about how to operate them and ways to use them effectively (Dunham, 1998). Teachers must know the computer based learning environments from a didactical point of view (Balacheff, 1998). A key issue concerns that of teacher control of the learning situation, while leaving enough autonomy to the student so that a genuine learning process can develop (Balacheff, 1998). Even when using technology it is crucial that the teacher has a mathematical understanding of the topics, and effective classroom strategies to get students to reflect on, appreciate, and understand how mathematics can explain real world phenomena (Charischak, 1998).

The integration of technology is a long term process depending on several factors, such as the features of the computer environment, and the tacit learning hypothesis and beliefs of the teacher (Laborde, 1998).

Technology and teacher preparation.

Technology can play several roles in the preparation of teachers. It can be used as part of the methods courses to help them understand the complex situation that a classroom is. Technology can also play a role to help teachers learn mathematics in a different way and develop a set of beliefs that is in harmony with kind of mathematics learning advocated by the Standards. Technology can also play an important role in the ongoing professional development of teachers, serving as access to materials and other colleagues.

Methods courses based on cases for teacher education provide opportunities to learn about exemplary practices and to learn how to analyze and reflect about classroom contexts (Lehrer, Petrosino & Koehler, 1998). Hypermedia have the potential to support case-based learning, because they have the capacity to represent the complexities of teaching and learning. Hypermedia are well suited to learn a complex domain like teaching. Lehrer, Petrosino & Koehler, describe design principles for the selection of cases that include among others: development of big ideas; observe prototypical development; realism; facilitate remembering; are engaging; make learning visible; provide longitudinal timelines; provide greater detail and complexity; expand elements of the case; view student learning as mediated action; and view teachers as designers.

In order to capture the richness of cases, design of hypermedia should follow several principles too: ability of revisit cases; multiple points of view with respect to the case; modeling student’s conceptual development; making navigation easy; learning by example; developing and maintaining relationships among associated concepts; consistent visual cues; and give feedback for reader actions (Lehrer, Petrosino, & Koehler, 1998).

Multimedia can facilitate the representation of complexity of conducting classroom lessons and the integration of these within a theoretical perspective on mathematics education (Lehrer, Petrosino, & Koehler, 1998). A primary criterion for selecting video episodes is to highlight the complex nature of classroom interactions. This includes a focus on the role that norms play in supporting students’ mathematical development. They need to include episodes in which students engage in mathematical argumentation, the proactive role of the teacher in initiating and guiding the development of norms, and carefully sequenced instructional tasks (Lehrer Petrosino, & Koehler, 1998).

Multimedia can be used to provide in-service and pre-service teacher preparation, helping teacher answer questions like why they need ongoing professional development; how technology can enhance learning; when and how to teach for understanding (Lehrer, Petrosino, & Koehler, 1998).

Integrating computer environments in the preparation of teachers helps teachers to plan tasks that are likely to enhance children’s mathematical understandings, to interact with students towards this end, and to reflect on their plans and interactions (Olive, 1998). Teachers’ own beliefs about mathematics and their own comfort with the use of technology as a teaching and learning tool influences their use of technology with their students (Olive, 1998). Pre-service and in-service teachers need support to develop the knowledge and skills needed to make sense of and build on children’s meaning making. Teachers need available help with technical problems, and also with choosing software, teaching with technology, and organizing projects (Clements, 1998).

The World Wide Web has grown dramatically over the last five years. It provides many sources and materials for students and teachers. As teachers become learners, they can both learn from other teachers and share experiences through the internet (Alejandre, 1998). However, the number, size, and complexity of many web sites makes it difficult for teacher to locate curriculum resources in a timely and effective way. Recent advances in web protocols and standards may greatly increase the effectiveness of searching for curriculum resources (Simutis, 1998).

Materials to enhance teacher preparation will also take advantage of hypermedia formats that can be explores using web browser software. Most relevant to Standards 2000 project is of course the electronic version of the Standards. The electronic version includes examples, case studies, student tasks, vignettes, classroom excerpts, and other depictions of mathematics learning, teaching, and assessment. The electronic format can also serve as a forum for discussion (Galindo, 1998)

 

Assessment

Technology can play an important role in assessing what students have learned, both for summative evaluation purposes and for ongoing assessment to inform day to day teaching.

Dick (1998) mentions several aspects that need to be considered when using technology during assessment: 1) Providing a tool that students do not have enough experience may confound assessment of their mathematics performance; 2) technology should play a significant role in assessment to encourage its use in the classroom; and 3) technology-active assessment tasks are not easy to create, it is a challenge to create authentic technology tasks.

We need to distinguish between the means by which assessment are presented and the quality of the assessment tasks. For example, low-tech tasks such as basic skills questions can be delivered through a high-tech medium such a computer linked to the internet (Dick, 1998).

In the classroom, computers provide mirrors to the mathematical thinking of students. Difficulties and misconceptions of students emerge. On the other hand, computers provide a fruitful setting to learn to take the student’s perspective when dealing with mathematical situations, and for discovering abilities of students to construct sophisticated ideas (Clements, 1998).

Technology and content standards preK-12

Number and Operations

Technology can play a role to help students develop number and operations concepts; view computation in a new light; and help them in the transition from arithmetic thinking to algebraic thinking.

With computer programs like Shapes students can manipulate base ten blocks. Students’ actions are represented as changes in the pictures, but also in their symbolic representation. Students can break computer base-ten blocks into ones, or glue ones into tens, which is more consistent with the mental actions we want them to learn (Clements, 1998).

Computer programs that represent fractions and operations on fractions visually (such as The Visual Calculator) can help student understand that multiplying by 3/7 is equivalent to the application of two operations in succession, times 3, and divided by 7 (Carraher & Schlieman, 1998). Such software can also help students unite number line and part-whole models of fractions, which research has shown difficult for students.

Software tools such as Sticks and Fraction Bars, developed for Tools for Interactive Mathematical Activity, together with possible actions that children can perform on those manipulatives, involve fundamental operations with both discrete and continuous quantities, such as uniting, unitizing, fragmenting, segmenting, partitioning, splitting, disembedding, replicating, iterating, and measuring (Olive, 1998).

In the past much of school mathematics was dominated by paper-and-pencil computation, with numbers in the early grades, and algebraic symbols in the higher grades. Present day hand-held calculators make it possible (at a reasonable price) for many students to have access to tools that perform exact arithmetic and algebraic symbolic manipulation. These tools permit us to rethink a more balanced approach to selecting the content for school mathematics at the higher levels (Waits & Demana, 1998), the same way that common calculators permit a more balanced approach in the elementary grades.

Research has shown that calculator use does not harm students’ computational skills. In addition, students who use calculators and computers make significant gains on estimation skills, understanding of number concepts, and mental arithmetic and calculation algorithms (Heid, 1998). Students who learn paper-and-pencil skills in conjunction with the use of calculators, and are tested without calculators perform as well or better than students who do not use technology in instruction (Dunham, 1998).

The value of many traditional paper and pencil algorithms is not just to obtain an answer, which nowadays can be readily obtained with more efficient tools. For example, the long division algorithm provides as side benefits estimation, front-end strategies, understanding of repeating decimals (akin to constructible proofs) (Goldenberg, 1998). Proficiency is to be measured in the ability to use the algorithm as illustration of, or explanation of, other mathematical processes and ideas (Goldenberg, 1998). Technology can help in analyzing and exploring algorithms, which is an important part of mathematical thinking. Tools like Function Machines, and programming languages let students construct models of the mathematics they’re learning, and let students build, apply, and analyze and tinker with the algorithms they are studying, and explore their interconnections (Goldenberg, 1998).

Battista (1998) describes the use of spreadsheets to help students grow through three levels of sophistication in thinking about a numerical procedure. At the first level students can complete the sequence of operations that constitute the procedure when it is in full view and with a specific number. At the second level, students can imagine the sequence while not actually performing it. At the third level students can reflect on, decompose, and analyze a numerical procedure. Thinking about a procedure at this level is the beginning of algebraic thought.

 

Patterns, functions, and algebra

Uses of technology permit a major emphasis on the concept of function, developed through graphing, spreadsheets, and programming. Algebra systems that include symbolic manipulation can help a more balanced approach to algebra. Qualitative and numerical approaches to solving equations become increasingly feasible and important. Through the appropriate use of technology, concepts such as change are more accessible, and the sequencing of topics and skills can be revised.

Students can have reliable graphs of functions with computers. Graphs and functions can become one of the organizing features of school mathematics (Heid, 1998). Students can concentrate on creating functions that model phenomena. A part of this modeling process involves also gathering data and being able to fit a function to that data. The use of graphing calculators can improve students’ understanding of functions and graphs in aspects such as read and interpret graphical information; obtain more information of graphs; understand global features of functions; and understand connections among graphical, numerical, and algebraic representations (Dunham, 1998). Students in a computer-intensive algebra course that focused on the development of algebraic concepts such as function, families, equivalence, and system consistently outperformed students in traditional courses in concepts, applications, and problem solving, without diminished skills (Heid, 1998). However, as with any other tool, students need to understand what the different commands accomplish, and need a careful selection of examples and non-examples to avoid develop forming misconceptions as they interact with the graphing calculator (Ward, 1998).

Research found that in an introductory algebra course students in computer intensive algebra approach were better in overall understanding of functions, and at the components of modeling, interpreting, and translating (Heid, 1998). The effects of graphing technology on conceptual understanding of graphs include higher levels of understanding; better interpretation of graphs; relating graphs to their symbolic representation; deeper understanding of functions; and connection among a variety of representations (Heid, 1998).

When students use a computer algebra system throughout calculus so that the course can focus on concepts and applications, students do better on conceptual tasks with no loss in symbolic manipulation skills (Heid, 1998). Research has found that in a calculus course students using computer algebra system showed greater facility with graphical and numerical representations and with the relationships among graphical, numerical, and symbolic representations (Heid, 1998). Recent changes in technology include the availability of hand-held graphing calculators with capabilities of doing exact arithmetic, and performing symbolic algebra, in addition to their former capabilities for graphing functions, and solve equations numerically (Waits & Demana, 1998).

Computer programming helps students use powerful tools of algebra such as variables, develop concepts of ratio and proportion, and form more generalized and abstract views of mathematical objects (Clements, 1998). The use of spreadsheets aimed at enabling students express the generality of symbolic relationships, improved students in their understanding of functions (Heid, 1998).

Students need to learn to solve some equations so that they know what solutions mean. They should not run to the computer to solve linear equations, or equations like x2 = 3x, or dx/dt=3x, but they do not need to spend long hours learning specialized tricks to solve equations. Qualitative and numerical issues regarding equation solving are now much more important. Students can now view the graph of a polynomial and can estimate where the roots are. Numerical methods play an important role now that students have means to make the thousands of computations involved in algorithms. Students need to develop understanding of the parameters involved; understand what the algorithm does and does not accomplish; provide the required data appropriately; and be aware that the algorithm may fail (Devaney, 1998).

The idea of change has always been important. Technology enables us to teach this concept more effectively (Goebel & Tague, 1998). Many important ideas such as rate, ratio, proportion, slope, area of geometric figures, arithmetic of signed numbers, fractions, and algebraic representations of functions can be learned in context, organized, and motivated by studying the mathematics of change and variation (Kaput, 1998).

The use of technology also has implications for the way material is sequenced in the courses. Traditionally, students first learned the procedure, then learned to apply it or to acquire an understanding of underlying concepts. Studies suggest that symbolic manipulation skills may be learned more quickly after students have developed conceptual understanding through the use of computer algebra systems (Heid, 1998).

 

Geometry

Technology makes possible an approach to geometry that permits students make discoveries on their own, conjecture and provide convincing evidence. At the same time technology can help students make progress in their levels of geometrical thinking.

Dynamic geometry software offers microworlds in which abstract objects and relations can be visualized and physically manipulated. Such environments offer the possibility for students to construct knowledge by action and not only by having to use language (Laborde, 1998). Dynamic geometry programs that link multiple representations can encourage students to make conjectures and provide insight into the reasons those conjectures might be true or false (Clements, 1998).

Battista (1998) describes Shape Makers, a Geometer’s Sketchpad based microworld where students explore quadrilaterals. Through the interaction with the computer and the teacher, students progress from focusing on the shapes as a whole to thinking about their parts, to thinking about different quadrilaterals in hierarchical relationships.

 

Measurement

Technology provides new tools for students to measure quantities in ways that are more efficient, and also provides ways to measure quantities that were difficult to obtain directly. Technology is also a means for students to display and analyze their measurements. Calculator based laboratories provide opportunities for students to measure movement, temperature, and other attributes of objects at the same time that data are recorded and displayed.

In addition, telecommunication capabilities gives students the opportunity to share their data and to participate in joint projects where measurements at different locations can help have a better understanding of a situation. For example, students at different latitudes can measure the angle of the sun’s rays at noon, and share their measurements to compute the circumference of the Earth (Charischak, 1998).

Data analysis, statistics, and probability

One of the most obvious impacts of the new technology is the increased ability of students to have access to or generate large sets of data; and the facility to display, represent, transform, and analyze those data.

In addition, technology can also provide tools for better understanding of the concepts and processes. For example, a computer learning environment for data analysis, like Fathom, can reveal how the algorithm for computing least squares regression line works (Finzer & Jackiw, 1998).

Technology can also provide students with large numbers of probabilistic experiments to help them develop their intuitions in this field, which cannot be generated through limited number of experiments.

 

Process standards and technology

Problem solving/posing

Computer programming helps students develop problem-solving abilities such as problem decomposition, and systematic trial and error (Clements, 1998). With computers, students may play with certain mathematical ideas creatively earlier than is currently possible with other means. Students can create and be in control of solving their own mathematical problems (Clements, 1998). Computers also provide students with an environment to test their ideas and receive immediate feedback on those ideas. Computers can provide problem solving scaffolding that allows students to build on their initial intuitive visual approaches and construct more analytic approaches.

Calculators can lead to improved problem solving because they provide more tools to solve problems, and change the perception of the students of problem solving because they free students from the burden of computation and can focus on formulating the strategies to solve the problem and analyze the solution (Dunham, 1998). On the other hand, students are more likely to want to take advantage of instructional modules, or practice needed skills when they are stuck in the middle of a highly motivating challenge (Klawe, 1998).

 

Reasoning and proof

With computers, students can test ideas on their own and receive feedback. Computers can help students to move from naive to empirical to logical thinking, and also encourage them to make conjectures, thus developing autonomy, rather than relying on authority.

Computer environments encourage the manipulation of specific screen objects in ways that assist students in viewing them as mathematical objects and as representatives of a class of objects. Such activities develop students’ ability to reflect on the properties of the class of objects and to think in more general and abstract manner (Clements, 1998).

Generalizations are best made by abstraction from experience. Students can use courseware to see examples of mathematical concepts as many times as they want. They see for themselves what the issues are before the technical words are used and generalizations are made. Courses can emphasize conceptual questions. As students interact with the computer, the instructor answers questions as they arise (Uhl, 1998).

Students have to develop a need to explain, a need to prove, as part of the constructive process of using the computer to make the transition from particular to general cases (Hoyles, 1998). Students construct mathematical objects on their own, conjecture about the relationships among them, and check the truth of their conjectures. They also need reflection guided by the teacher, and the introduction of mathematical proof as a particular way to express their convictions and communicate them to others (Hoyles, 1998).

Recording and replaying student’s actions is most completely realized in computer programming (Clements, 1998). A programming language such as Logo helps students to think about doing a drawing. When students write programs they construct mathematical processes and objects. The formal language of programming offers generality and power.

Through dynamic manipulation environments students can experience problems and situations in which continuity between one state and another allows them to reason about intermediate states (Finzer & Jackiw, 1998).

 

Communication

Computers can encourage positive social interaction. Students cooperate more, and they cooperate in learning. Students disagree about ideas more, but are more likely to resolve these disagreements successfully. Technology can play a role in facilitating the externalization of internal representations of mathematical concepts. It can also provide an open setting in which mathematical disagreements can be brought to the fore and discussed (Heid, 1998).

Students have to formalize their ideas to communicate them to the computer. Students formalize more often as they use computers. When intuition is translated into a form that computers "understand" it becomes more accessible to reflection (Clements, 1998). However, students need to develop fluency with the technology in order to be able to communicate their ideas (Goldenberg, 1998).

 

Connections

A computer- or a calculator-based laboratories permit students to represent something that they have experienced (Kaput, 1998). Computer- and calculator- based laboratories allow students to acquire, and display, data in real time, and facilitate their analysis. Students can explore, measure, and learn from the material environment. Conversely, students can create and study physical phenomena using mathematically defined functions to control devices such as cars, pumps, and laser-pointers. Students can also create cybernetic phenomena or simulations (Kaput, 1998).

The process of finding the model helps students understand the phenomenon and the predicting steps are more meaningful. The mathematics needed to find the model is more important than the mathematics required to manipulate final form of the model. (Goebel & Tague, 1998). Realistic problems require graphical, numerical, and analytical solutions and involve more than one simple concept (Goebel & Tague, 1998). Also, when using computers to make a mathematical model of a phenomenon in another field, students need also to develop background understanding and knowledge (Goldenberg, 1998).

Dynamic manipulation environments offer a context where students can solve problems and address challenges by building interactive, manipulable, mathematical models (Finzer & Jackiw, 1998). As students drag one object on the screen, the objects that are linked to it change as well. According to Finzer & Jackiw (1998), students can think of these linkages sometimes as dependencies, sometimes as causality, or sometimes as an implication.

Teachers need to help students make the connection between the mathematics they are doing on the computer, and the mathematics they learn in class (Heid, 1998). Also, teachers need to help students see connections between different branches of mathematics. For example, computers can play a critical role in the integration of ideas from algebra and geometry (Clements, 1998).

 

 

Representations

Computers can promote the connection of formal representations with dynamic visual representations, supporting the construction of mathematical ideas out of initial intuitions and visual approaches. For example, newer versions of Logo permit a two way connection between visual and symbolic representations (Clements, 1998).

The appropriate interface with the computer can also help students focus on the concepts being learned. For example, computer interfaces in which students manipulate a mathematical representation of the transformation being applied to the shape are more effective than interfaces where students manipulate the shape being transformed (Klawe, 1998). Interfaces can also help students focus their attention on ideas. For example, if students need to think about the choice of a numeric value, having them type in the value is better than just clicking at the value (Klawe, 1998). It is important that the software uses a representation that reflects what the students should think about. Also, that the same representation is used as in other accompanying modes of mathematics education (textbook, lectures, worksheets). That will help students transfer and integrate understanding between different modes (Klawe, 1998).

As with any other representation tool, research shows that some students may interpret what they see in ways that are not intended by the designer of the tool, or by the teacher (Dunham, 1998). Students may form incomplete concept images or misconceptions if instruction does not bring students ideas to the forefront to be discussed and analyzed.

Technology also extends the manipulativity that makes mathematics so powerful to several forms of representations. Previously, the syntactical manipulativity was almost done exclusive through character strings. With electronic technology, the manipulativity is available through graphically editable systems (Kaput, 1998).

 

Concluding remarks

Many of the examples shared during the conference were not specific to mathematics. They illustrate exemplary ways of using this technology that could also be used in other fields. The use of electronic mail to provide a forum to share ideas among students and the instructor (Banchoff, 1998), or the use of the internet to make it possible that students in different geographical locations work on a common project (Ramírez, 1998) are such examples. Teachers of mathematics need to get acquainted with ways of integrating technology that teachers in different fields have devised and that have the potential to serve in the teaching of mathematics as well.

On the other hand, many of the teaching principles illustrated during the conference, were also not specific to the use of technology, but rather were examples of good mathematics teaching. Many of the principles that good teachers use to enhance the mathematical thinking of their students, and promote reflection are found when they work both with or without electronic technology. We need to keep this good teaching as we use the new tools. Furthermore, as Heid (1998) points out, technology is one of the most important catalysts for change in the teaching of mathematics.

However, if we want to learn how to make the most of this tool to enhance the learning of mathematics of our students, in addition to using ways that are helpful across the curriculum, and using general principles of good teaching of mathematics, we will need to develop effective ways that are unique of using technology for the teaching of mathematics. Furthermore, we need to understand the specific interactions of learning mathematics with the help of electronic technology. We need to understand why calculators and computers make a difference in students’ learning of mathematics.

 

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