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National Perspectives in Mathematics Education

The National Council of Teachers of Mathematics (NCTM) has led the movement for mathematics education reform. Unlike the top-down education reform efforts of the 1960s, NCTM began a grassroots discussion of issues in the mid-1970s. Subsequently, An Agenda for Action (1980) reflected input from all parties interested and involved in mathematics education, and set in motion much-needed discussion and debate. NCTM later published a series of documents which, appropriately, set criteria for excellence rather than singular prescriptions for implementing "new best ways." They included the NCTM Curriculum and Evaluation Standards (1989), the NCTM Teaching Standards (1991), and the NCTM Assessment Standards (1995). Collectively, these standards advocate methods that emphasize mathematical power: conceptual understanding, problem solving, reasoning, connection building, communicating, and self-confidence developing. Efforts continue throughout mathematics education community as standards evolve and are revised.

The NCTM Standards era was characterized by numerous efforts advanced by traditional publishers and researchers to improve mathematics curriculum. Virtually all mathematics curriculum projects, curriculum reform documents, and mathematics syllabi since 1989 have used the NCTM Standards as a basis or point of reference for their efforts. NSF funded five comprehensive middle-school mathematics curricula that are now widely available (Connected Mathematics Project, Seeing and Thinking Mathematically, Mathematics in Context, Six Through Eight Mathematics, Middle-school Mathematics Through Applications Project). These materials (and some commercially-published materials) promote applications and problem solving, conceptual understanding, building connections across mathematics domains and uses, alternative assessment strategies, and mathematics communications skills.

More recently, work has begun on a National Voluntary 8th Grade Mathematics Test (NVMT). The draft recommendations (Management, Planning, and Research Associates, 1997) to the National Test Panel underscore the need for assessments that reflect the emerging consensus of mathematical power in the standards.

Meaningful reform and genuine change will result only if it is manifested within mathematics classrooms. The role of the teacher is essential. The growing nearly universal availability of technology tools provides a grand opportunity to assist teachers in teaching well and in improving the mathematics experiences of students.

Technology can provide mechanisms to sustain assistance to mathematics teachers in their use of technology to implement mathematics education reforms in their classes. Technology enables mathematics education reform, but it is not reform per se (cf. Kaput, 1992). We must provide mathematics teachers extended opportunities to experience and do mathematics in an environment supported by diverse technologies (Dreyfus & Eisenberg, 1996). The heart of our approach is the development of mathematical power-- understanding, using, and appreciating mathematics. Our interest is in empowering teachers through the use of technology in mathematics exploration, open-ended problem solving, interpreting mathematics, developing understanding, and communicating about mathematics (see Bransford, et al, 1996; Schoenfeld, 1982, 1989, 1992; Silver, 1987).

Goals and Objectives

• To use technology tools to model and demonstrate standards-referenced mathematics content and pedagogy.
• To enable mathematics teachers to experience and enhance school mathematics using various technology tools for exploring real world applications, engaging in problem solving and problem formulation, and communicating about mathematics investigations.
• To use technology to develop an appreciation of the distinction between demonstration and proof in mathematics and to emphasize the value of each in the understanding of mathematics.
• To use technology tools to engage in mathematics explorations, to form mathematics ideas, and to solve mathematics problems
• To use technology tools to construct new ideas of mathematics for yourself.
• To engage in mathematical investigations using software applications.
• To use general tools such as word processing, paint programs, spreadsheets to
• To facilitate mathematics investigations and communication about mathematics investigations.

• To develop effective mathematics demonstrations using appropriate technology tools.
• To engage in some independent investigations of mathematics topics from the school curriculum or from mathematics appropriate for that level.
• To enable better communication of mathematics ideas that arise from technology enhanced investigations.

• To enable mathematics teachers to develop and adapt materials and goals from standards based curriculum through the use of technology.
• To model and explore collaborative instructional strategies.
• To develop mechanisms and expectations of sharing instructional ideas, materials, and information among mathematics teachers.
• To disseminate information supporting comprehensive standards-based school mathematics curricula and the implementation of Quality Core Curriculum objectives.
• To realize the use of technology tools in the implementation of alternative assessment strategies.

• To develop comfort and confidence in the use of technology by mathematics teachers. as they explore, practice, reflect, and become in technology-enhanced teaching and learning of mathematics
• To enable and encourage mathematics teachers to collaborate by using technology support.
• To support professional development opportunities for mathematics teachers and other key personnel through a network of peer teachers.

Course Components

The courses are organized using sets of open-ended problems grouped into Study Guides (Assignments). The course information, syllabus, and study guides are implemented the model developed by the author for secondary mathematics teacher education (<http://jwilson.coe.uga.edu>). Participants can access these Web resources via Netscape or Explorer from any site with Internet access, including home and school sites. Mathematics topics relevant to School Mathematics are used and problems posed to enable teachers to experience mathematics with technology as a learner. The extension into the next course shifts the focus from experiencing mathematics as a learner to communicating, organizing, and teaching mathematics with the aid of technology tools.

These courses present mathematics problems, open-ended mathematics investigations, challenges to organize and communicate information from these investigations, and selected technology tools to support such tasks. The mathematics content is consistent with the focus of school mathematics curriculum, but not necessarily from a specific curriculum.

The underlying psychological theme is the use of visual reasoning in mathematical discourse (Zimmerman & Cunningham, 1991; Presmeg, 1993; Parzysz, 1988). The technology tools enable students to construct visual and symbolic representations of ideas and incorporate these into their approaches and thinking about problems. The technology-enabled visualizations are not the end-product but rather a means to facilitate student's mental images that help them to form, relate, and organize mathematical concepts.

The underlying pedagogical theme is to experience mathematics as problem solving, communication, reasoning, and building connections (President’s Committee of Advisors on Science and Technology, 1997). Conceptual knowledge and procedural knowledge are emphasized together and technology tools reinforce their mutual development.

The Course Content

The mathematics content reflects topics relevant to the school mathematics curriculum: number, number relations, and their graphs; estimation and computation; number systems; patterns and functions; algebra, including graphs of relations; data display, graphs, and statistics; measurement; geometry of the plane and space, including construction and locus problems; and probability.

The Technology

The technologies available span from low-end, hand-held calculators through high-end multimedia workstations. Technology is available and supported both in course implementation sites and in the participants’ schools. The courses can be implemented in technology-rich laboratories with modern student workstations (Macintosh Power PC or Windows Pentium) and teacher workstations with projection capability. All computers have high-speed Internet access to support individual workstation, local network, and web-based mathematics activities and applications. The laboratory also has ready access to non-computer technologies, including graphing calculators and manipulative materials.

Software Tools

The software tools are of several types. The emphasis is on functionality and recognition of use across computer platforms versus platform-specific applications. Some technology tools will be used primarily for mathematics exploration; others will be for communication and presentation.

Spreadsheet. Workshop and follow-up activities will not focus on using a spreadsheet per se, but rather doing mathematics problems where the spreadsheet can enable and reinforce investigation, conjecture, and problem solving. Microsoft Excel is widely available, but other general purpose applications, such as Clarisworks, provide good spreadsheet functionality to support middle-grades mathematics.

Hand Held Calculators. The desired functionality of the HHC include the range of computations, simple function keys such as square root and square, making graphs. The TI-81, TI-82, or TI-83 support more than this level of capability.

Graphing. Graphics programs of enormous sophistication are available to support middle-grades mathematics programs. We opt for the use of the spreadsheet and the hand-held calculator for part of graphing activities and building connections to data display. An elementary and user friendly graphing program such as Algebra Xpresser (Hoffer, 1990) will also be available. Function graphing environments such as Theorist, Mathematica, Maple, or MatLab are available, but likely too sophisticated for middle grades use. The "graphic calculator" software bundled with PowerPC computers provides useful functions and relation graphers with simple animation.

Dynamic Geometry. Dynamic geometry programs (such as Geometer's Sketchpad, Cabri, or Geometric Supposer) provide exploration tools with rich potential for the middle grades. The can be used to explore relationships of and among geometric objects in a plane. Geometer’s Sketchpad (Jackiw, 1992) is our tool of choice, but the problems we develop should be explorable with any software that allows the manipulation of geometric objects depicted on the computer desktop.

Communication Tools. A range of communication tools include word processors, (e.g. Microsoft Word, Clarisworks, etc.), spreadsheets (e.g. Microsoft Excel, Clarisworks, etc.), Internet browsers (e.g. Netscape Navigator, Microsoft Explorer, etc.), web page tools (e.g. Adobe PageMill, Netscape Gold, Clarisworks HTML, etc.), E-mail (e.g. Netscape Mail, Claris Mail, Eudora, etc.), and other utility software.

The Website. The website is integral to both the courses as participants build their own functional web pages on the server. The website also is integral to building continuing support of the mathematics teachers after the courses. The website is user-friendly, featuring simple but elegant navigation. This helps to promote understanding of the concepts as well as the skills needed to optimize use of the Internet as a teaching-learning tool.

Course Procedures

Each participant is provided an e-mail account and file space via the Website. Materials for the Study Guides reside on the Website where participants learn to access and use web materials. Beginning with the first meeting, the participants build their personal web page using artifacts and productions from the workshops (e.g. write-ups, final projects, instructional units, or items of personal interest such as useful links) to compile an electronic portfolio.

Write-ups and Final Projects, reflecting student explorations, synthesis, and communication are submitted electronically for course credit. The purpose and focus of a write-up is to communicate and synthesize investigations involving exploration, solving a problem, or working with an application. The key points include synthesis, communication, correct mathematical ideas, interpretation, and utility. Typically, participants are expected to conduct a range of explorations from a study guide. Participants will explore all, complete some, and write-up one. Feedback is provided and iteration for improvement encouraged. Indeed, the Write-ups are basically instructional components rather than assessment components. Final Projects, focusing on a technology-enhanced mathematics implementation of the individual participant’s determination, are submitted and discussed at the end of the course. Student productions are placed on the Web page for public sharing.

The instructor presents demonstrations and explanations, clarifies problems, and demonstrates alternative solutions using a projected image from the teacher’s workstation. A typical session might allocate one-third of the time in whole-group mode, with the balance providing direct support for participants working on their projects or units, either individually or in groups. The website will enables participants to do some of the work at their home or school sites.

The EMT 669 extension promotes the use of technology to enhance mathematics teaching in participants’ home schools and to extend each participant’s expertise. Each participant’s web page contributions includes links to related teaching-learning resources in order to establish a highly connected contribution. Further they learn to place their own productions on the Web, to develop ways for their students to produce and publish web resources of their own Web, and to organize two teaching units using technology tools.

Unique Features

Several pedagogical and curriculum features, as well as the delivery system of the Web site, are noteworthy. The delivery system uses direct guidance in the form of laboratory workshops and site-based support, distance learning technologies, and sustained support through web site resources which can be continually updated. This model allows individuals to move along a continuum of professional growth and models new uses of technology to deliver instruction that teachers may incorporate into own classrooms.

Linking to and Extending Standards. The National Council of Teachers of Mathematics have presented standards for mathematics curriculum, evaluation, professional teaching, and assessment in documents from 1989 to 1995. The curriculum standards emphasize conceptual understanding, problem solving, reasoning, communication, building connections, and building self-confidence. These courses provide examples for implementing these curriculum standards for students.

Cooperative Learning. People learn via interaction in groups. We have long recognized the value of have students develop reading comprehension in groups but have traditionally clung to doing mathematics in groups of one. Cooperative learning approaches are appropriate for mathematics students at all levels and for the preparation of mathematics teachers. Cooperative learning in mathematics means thinking about what we value in mathematics differently that the traditional curriculum.

Alternative Assessment/Portfolio Assessment. Assessment standards emphasize important mathematics, learning, equity, openness, valid inference, and coherence. Electronic student folders generated in this course are both a documentation of progressions in understanding and a collection of best work--the essential features of a developmental portfolio. The write-ups and final projects call for summary, synthesis, explanation, presentation, and communication. These processes can be linked to the assessment standards.

Demonstration and Proof. Technology often provides convincing demonstrations of ideas, helps to generate hypotheses, and encourages exploration. Demonstration, however, does not replace the need for proof. Nor does constructing a proof rule out the use of technology. This contrast of demonstration and proof will be emphasized and must be appreciated by participants.

Technology and Curriculum Refocusing. Mathematics content and pedagogy are enhanced through technology. It is important to recognize, however, that as technology tools become available and our insight in using them expands, the very nature of the workshops and what is emphasized will need to change. The technology impacts not just what we want to select from mathematics, but also fundamentally at the substance of the mathematics we teach.

World-Wide Web Technology. The website provides a unique method that provides a mechanism for creating a community of learners who recognize each other as powerful resources in the teaching-learning endeavor; serves as a vehicle for delivery of continuous instruction related to content, pedagogy, assessment and use of the technology itself; provide tools which can be accessed any-time, any-where and allows for instantaneous updating promoting continual renewal opportunities for all members of the learning community; serve as a source of annotated linkages to other exemplary websites and to other databases which can, in turn, promote collaborative project-based learning using real-time data

Animation. Animation is a powerful tool embedded in many mathematics tools and programs. This feature creates new ways for students to explore and develop understanding. It also raises important research questions about what students learn and how they learn it.

Teacher-Innovator Community. For many educators, professional isolation is a barrier to implementing classroom innovation and reform. Many teachers simply do not have the support base needed to sustain changes over time. A principal goal of these course is the creation of a community of middle-grades mathematics innovators. This will be both a real and virtual community, who can draw upon the wisdom and experience of one-another as they advance often-difficult reform efforts.

Zimmermann, W., & Cunningham, S. (Eds.) (1991). Visualization in teaching and learning mathematics. Washington, D.C.: The Mathematical Association of America.