Offering Prospective Teachers Tools to Connect Theory and Practice: Hypermedia in Mathematics Teacher Education
Lloyd, Gwendolyn M., Wilson, Melvin, Journal of Technology and Teacher Education
This article describes a program in which prospective secondary mathematics teachers develop hypermedia stacks to demonstrate their understanding of important principles of mathematics, teaching, and learning. Initial project creation occurs in methods classes where students create hypermedia projects in small groups. Later, during student teaching, interns construct individual, portfolio-like hypermedia projects. Intern-created stacks include hypertext links, quick-time movies from video-taped segments of practice teaching, other video and audio components, and links to a variety of applications. The hypermedia format encourages prospective teachers to build explicit connections within their own experiences, allowing them to personalize reform themes in terms of their own teaching. In addition, it offers them experiences with computer technology, cooperative learning activities, and alternative assessment strategies. Several project descriptions illustrate these ideas.
Recent efforts to overhaul the secondary curriculum in the United States have drawn increased attention to themes of inquiry, cooperation, and problem solving in the mathematics classroom (National Council of Teachers of Mathematics (NCTM), 1989, 1991, 2000). Many prospective teachers possess weak knowledge and narrow views of mathematics and mathematics pedagogy that include conceptions of mathematics as a closed set of procedures, teaching as telling, and learning as the accumulation of information (Ball, 1991; Brown, Cooney, & Jones, 1990; Even, 1993; Frykholm, 1996; Thompson, 1992; Wilson, 1994). These conceptions are bolstered by years of experience as students in traditional classrooms (Brown & Borko, 1992; Lortie, 1975; Zeichner & Gore, 1990). If reform themes are to be enacted in the mathematics classrooms of future teachers, conceptions of mathematics and teaching need to be challenged and developed in ways that will support meaningful and lasting change (Knapp & Peterson, 1995; Prawat, 1992; Richar dson, 1990; Smith, 1996).
In light of the differences in the instructional methods prospective teachers will be expected to use in schools and those they likely experienced as students of mathematics, teacher education programs face significant challenges. They are faced with the task of creating opportunities for prospective teachers to critically consider important teaching and learning ideas so that more flexible conceptions of mathematics and teaching may develop.
Computer technology has significantly impacted teacher education programs in many colleges and universities. Hypermedia applications in particular have grown in popularity in recent years. Common uses of hypermedia in teacher education focus on having prospective teachers work in predesigned computer environments to explore and respond to various teaching events and issues (Daniel, 1996; Hatfield, 1996). In contrast, the program described in this article involves hypermedia creation by the prospective teachers themselves.
In the program, prospective teachers develop hypermedia stacks to communicate their understandings of important issues in mathematics, teaching, and learning. Initial hypermedia project creation occurs in small groups during a secondary mathematics methods course (and associated practice teaching). Later, during student teaching, interns construct portfolio-like hypermedia projects. Student-created hypermedia stacks include links to written documents (e.g., teaching philosophy, resume) and other hypertext components. The stacks also include quick-time movies from video-taped segments of interns' practice teaching. A more recent extension of activities has included the World Wide Web (WWW or Web). In fact, interns are now required to create electronic documents that can be placed on the Web. The reader is encouraged to notice ways in which this medium can easily be used to support student creation of hypermedia projects. (1) The examples discussed in this article were created before the Web became popular and HyperStudio (1994) software was used, but almost any hypertext application or web composing software will work.
Project creation encourages prospective teachers to personalize reform themes in terms of their own teaching. It also helps them to become familiar with computer technology and to participate in cooperative learning activities and alternative assessment strategies. Perhaps most importantly, these hypermedia activities assist prospective teachers in reflecting upon and building explicit links within their own experiences. The following section elaborates the two prominent themes motivating our hypermedia program for prospective secondary mathematics teachers: connection and reflection.
REFLECTION AND CONNECTION IN MATHEMATICS TEACHER DEVELOPMENT
For prospective teachers to embrace reform-oriented views and practices, they need to participate actively and self-consciously in the process of developing and refining their personal conceptions of mathematics pedagogy. Much of the literature on teaching in general, as well as on mathematics teaching in particular, stresses the importance of encouraging teachers to become reflective practitioners (Calderhead, 1989; Roth, 1989; Schon, 1983, 1987; Valli, 1992; Zeichner & Liston, 1987). Being a reflective practitioner involves not only spontaneous and immediate reflection on the teaching task, but also serious thought about teaching experiences from a distance. Such creative and critical reflection about beliefs and practice can powerfully impact a teacher's view of his or her role in mathematics teaching and learning. For instance, ongoing reflective action can greatly enhance teachers' development of theories of practice (Smyth, 1989). Habits of reflection are necessary if prospective teachers are to see th emselves as decision-makers who will engage in career-long learning about students, mathematics, and pedagogy.
To effectively promote professional development, reflection-oriented activities should carefully take into account the conceptions, current practices, and needs of the particular teacher (Lasley, 1992). Beginning teachers, who often lack confidence in their abilities to develop and maintain effective classroom practices and who may not fully appreciate the complexities of mathematics pedagogy, may particularly benefit from reflection on and focused analysis of very specific issues, concepts, or practices. Such focused activity allows prospective teachers to transform tacit or unexamined beliefs into more explicit views, which they can draw on and adjust as they enter the mathematics classroom (Fenstermacher, 1979).
One of the most important aims of incorporating critical reflection into teacher education courses and programs is to promote teachers' building of several different types of connections within their beliefs and experiences. Prospective teachers need to make connections between their views of mathematics pedagogy, which largely grow out of their own experiences as students of mathematics (likely in traditional instructional environments), and the more formally articulated theories of content, teaching, and learning that form the core of the methods course activities. For example, many prospective teachers' views of what it means to teach are compilations of positive and negative images of prior teachers (Knowles, 1992). By connecting a past experience with a new pedagogical conception, reflection in a teacher education context may guide the reinterpretation of such images into more realistic and useful models of teaching.
Prospective teachers also need to make connections among theories and issues of content, teaching, and learning. For example, a teacher begins to translate the knowledge developed as a student of mathematics into pedagogical content knowledge (Shulman, 1987) as he or she revisits content from new perspectives based on theories of student learning and methods of instruction. Facilitating this complex type of connection, and the resulting development of new conceptions, necessitates extensive reflection and analysis on the part of the prospective teacher.
Another important connection that can be encouraged by reflection is between different aspects of the learning-to-teach experience. Prospective teachers frequently do not interpret their classroom experiences in ways that are consistent with the theory emphasized in teacher education courses. Rarely do the worlds of theory and practice meet-the university supervisor is typically the only person (other than the prospective teacher) to participate in both experiences. As a result, prospective teachers may view their teacher education programs as a series of disjointed courses and activities lacking a common theme or goal: "Most preservice teachers are unable to grasp the whole while experiencing the myriad of disconnected parts of the curriculum. Many teacher education programs fail to challenge their students to understand how ideas are connected and/or related to field experiences" (McIntyre, Byrd, & Foxx, 1996, pp. 171-172).
As prospective teachers attempt to make sense of their practice or student teaching experiences, it is of utmost importance that they receive support in using their university knowledge to do so. Conversely, field experiences provide a context for prospective teachers to deepen and receive validation about theories discussed in methods courses. Reflection on experiences in field and university settings enables prospective teachers to develop conceptions that incorporate diverse ideas and examples and thus include richer, more useful information that will better support their process of learning to teach.
HYPERMEDIA TECHNOLOGY IN TEACHER EDUCATION
Motivated by efforts to provide prospective teachers with opportunities for significant reflection and connection, teacher educators have turned increasingly to computer technology in recent years. Because they enhance teacher development in ways that may not be possible using other instructional tools, hypermedia applications in particular have grown in popularity among teacher educators. As indicated previously, the program discussed in this article involves hypermedia creation by the prospective teachers themselves.
To a large extent, prospective teachers' views of mathematics pedagogy grow out of their own experiences as students of mathematics. However it is primarily in methods courses that prospective teachers begin to integrate their understandings of content, teaching, and learning. Few educators will argue about the importance of helping learners use their own experiences to connect understandings to each other and to related ideas. Effective teaching emphasizes and provides opportunities for learners to build such links in their knowledge. By encouraging serious reflection on understandings and experiences, the hypermedia program aims to facilitate this integration.
Hypermedia environments allow prospective mathematics teachers to place together in a single document various media and aspects of learning to teach. For example, prospective teachers select video segments illustrating important mathematics education themes. As they design and create hypermedia projects, quick-time movies of these video segments are explicitly linked to textual or other representations of the chosen themes. Although teachers can view video segments and think about their teaching without hypermedia technology, hypermedia allows them to embed the teaching examples in documents with explicit links to other representations of the chosen theme. Physically connecting several representations of the learning-to-teach process increases the likelihood that prospective teachers will develop integrated understandings of what it means to teach mathematics.
A long-range project goal is to prepare mathematics teachers to practice effective teaching principles, routinely reflect on their own teaching practice, and work with other teachers to improve mathematics teaching and learning in the schools where they work. The examples that follow illustrate this goal. Selecting video segments for their projects, for example, requires prospective teachers to reflect upon their own teaching: as they analyze videos, the teachers make decisions about which segments best illustrate their chosen theme. Representing their views on important issues in mathematics education, or describing their philosophies of teaching and learning mathematics, requires that teachers reflect on, analyze, and learn about their beliefs. By using hypermedia creation to reflect on critical experiences and make explicit connections between parts of their experiences, preservice teachers can begin to construct coherent representations of themselves as teachers of mathematics.
The next section describes the program in more detail. The themes of connection and reflection are illustrated using several prospective teachers' hypermedia projects. Detailed analyses of students' projects illustrates the powerful role that hypermedia creation can play in informing our understanding of how preservice teachers think and learn about teaching mathematics.
The first phase of the hypermedia program has been implemented numerous times in secondary mathematics methods courses. The students in these courses enrolled concurrently in a prestudent teaching practicum. It was primarily from this practice teaching placement (i.e., prestudent teaching) that video segments for the projects were obtained. Prospective teachers worked two days per week in local schools and also did individual tutoring (of secondary students) at the university. Groups of two to four prospective teachers created hypermedia documents (stacks) illustrating their understandings of important teaching themes discussed in the literature and in class. Hypermedia documents had various points of emphasis but focused particularly on meaningful individual and group reflection on important issues in the teaching and learning of mathematics.
Hypermedia Projects: Examples from a "Methods of Teaching Mathematics" Course
The examples in this section were created during the fall semesters of 1995 and 1996, when interns made hypermedia projects representing their views about various topics, including mathematics anxiety, cooperative learning, alternative assessment, technology, diversity, equity, state-wide competency testing, mathematical communication, real-world applications, and multiple learning styles. The hypermedia format gave students the very powerful ability to create non-linear stacks as well as to represent the themes using various media. The composition of the hypermedia stacks is illustrated by discussing aspects of projects about alternative assessment, cooperative learning, and real-world applications. To provide a sense of how project creation relates to conceptions of mathematics teaching and learning, the final example also includes a discussion of one prospective teacher's conceptions prior to the project completion.
Alternative assessment project. One pair of students created a hypermedia document describing alternative assessment. To introduce the topic, the prospective teachers constructed a card displaying several examples of traditional test items and quick-time movies of high school students sitting at their desks quietly taking mathematics tests. After being introduced to the idea of and need for alternative assessment, users are taken to a card (see bottom of Figure 1) providing them with options to explore Purposes of alternative assessment activities, teacher Roles in alternative assessment environments, and Kinds or examples of alternative assessment activities.
Within each of these branches or paths, the user can further explore a chosen sub-theme. In the Kinds branch of the stack, the user can explore examples of Geometer's Sketchpad (GSP) and other alternative assessment activities. The stack not only describes problem situations (to use in alternative assessment formats), but actually takes the user to documents supported by the Geometer 's Sketchpad software. The card shown in Figure 2 illustrates the organization of the Roles branch. Within this branch, the hypertext words Observe, Record, and Communicate allow the user, by clicking on the appropriate word, to follow the associated sub-path in the Roles branch. If users select the Observe subpath, they can see a quick-time movie from one prospective teacher's practice teaching and related text describing the situation. The movie and text emphasize the importance of teachers routinely doing informal observations and "catching students being right."
The reflection theme is evident throughout this stack. For example, the video segment illustrating one of the teachers "catching [her students] being right" evidences this group's recognition of the importance of the teacher's role in alternative assessment. The organization of the stack, in which some cards are accessible from various points, also demonstrates the interconnections in how the teachers viewed various alternative assessment themes. For example, because it contains both written and oral types of alternative assessment, one suggested "project" can be reached through two different sub-branches of the Kinds branch. Another evidence of connection in the stack is the existence and use of links to other applications (e.g., GSP). The application's (GSP) relationship to alternative assessment is not only acknowledged and described, but physically connected to the topic's description.
Cooperative Learning. Reflection and connection are also evident in other preservice-teacher-generated projects. For example, a hypermedia project about cooperative learning contains a video segment showing a prospective teacher attempting to bring students working in small groups together for a whole-class discussion. The group used this segment to illustrate one of the major difficulties of cooperative learning--getting students to refocus on a large-group activity following small group work. On the basis of a followup interview and the appropriateness of the segment in the project, the role of this intern's teaching experience in shaping her growing conceptions of mathematics learning and teaching was recognized. It is not uncommon for prospective teachers to struggle with management issues (e.g., class behavior) when attempting to implement nontraditional instructional strategies. What is less common, however, is reflection and use of video examples to illustrate this struggle and its relationship to effe ctive mathematics teaching. For this intern, issues related to cooperative learning had not only been explored and experienced in her own teaching, but these themes were explicitly connected to each other in the hypermedia document.
Another important part of the hypermedia program is oral presentation: preservice teachers present and lead class discussions about their projects. Because students know that peers will view and critique their projects, they are motivated to do their best. Presentation also provides a natural context for student-led class discussions about project topics. An additional positive outcome of such sharing is that students gain confidence as they become recognized among peers as the "experts" of their project topic.
Discussion with Alice about real-world applications. The following example focuses on another intern who created a hypermedia project during the methods class (before student teaching). This example will help the reader to better understand how teachers' conceptions of mathematics and mathematics teaching are refined during hypermedia creation. The case of Alice (the intern described) also demonstrates how student-created hypermedia projects can provide fruitful contexts in which to engage students in rich conversations about their personal conceptions and experiences.
At the beginning of the methods course, Alice described deciding to become a teacher because she enjoyed working with children, and particularly enjoyed studying their learning and development. She chose to teach mathematics not only because she liked the subject and was good at it, but because mathematics "applies to everything in the world." Although the fact that mathematics is everywhere "is not something that we are taught," she believes strongly that emphasizing real-world applications can help students to connect school mathematics with everyday mathematics:
When people say that they don't like math or that they can't do math, they are thinking of school math and they don't realize how much math they do every day that they don't even think about....If they realized how much they were actually doing, then they wouldn't say that they hate math or that they can't do it, because they are doing it.
In her view, "everyday math" and "school math" should be presented together. For example, she explained that students should be given opportunities to relate mathematical concepts to everyday activities such as figuring out restaurant tips, or prices of items that are on sale. She pointed out that even though "any teenager can say what 20% off their shirt is going to be," most students do not apply the same understanding when required to do a worksheet of 25 exercises involving percents.
Alice's comments about the prevalence of mathematics in everyday life are consistent with her group's choice of topic for the hypermedia project, "Bringing Real-World Applications into the Classroom." The table of contents (Figure 3) shows how the project was structured around various realworld problems and activities that can be used in middle and high school mathematics classes.
Alice described the group's work in constructing and organizing the project:
For each problem, we presented either our solution or some of the students' solutions. Then--we all did it a little differently--but I went through and talked about why I thought it was a good problem and for what age group....Then [we] went through each Standard that we picked out and described a little bit about how applications could meet that Standard.
As her comment indicates, the project discusses how the four problems and the idea of applications in general support many recommendations of the NCTM (1989) Standards.
The specific Standards that the group chose to discuss are outlined in Figure 4. The group's elaboration of the "Learning to Value Mathematics" Standard stated,
The NCTM Standards stress that students recognize the existence of mathematics in our society. The use of real-world applications will present the students with concrete examples of mathematical situations that can be observed in everyday life. When students see that math DOES exist outside of the classroom, they will develop a greater VALUE for mathematics in general.
The text on this card coincided very closely with statements Alice had made in her initial interview.
One of the problems developed and described in the project was based on a Connected Mathematics Project (Lappan, Fey, Fitzgerald, Friel, & Phillips, 1996) activity used in the middle school class where Alice assisted as a prestudent teacher. The problem is stated as follows and in Figure 5: "When Tupelo Township was founded, the land was divided into sections that could be farmed. Each section is a square that is 1 mile long on each edge--that is, each section is 1 square mile of land. There are 640 acres of land in a 1-square-mile section."
In talking about this problem during our discussion of her project, Alice reflected on her experience in the classroom where this problem situation was explored:
It was a really involved problem. I liked the way that the teacher set it up. She was talking about how she grew up on a farm and was relating it to how her father had to buy all his land and decide how much he wanted to plant and how much more land he was going to need. She Set it up saying that it is something that really happens every day with farmers and the kids were really interested in it. Watching them work in groups and stuff you could see that everyone was involved and was talking about the problem and even if they didn't understand the answer they had the words to ask their questions because they could talk in English about a problem that they could picture. So then I heard them talking in English about adding and subtracting fractions and then figuring out that this is what they are supposed to be doing.
The previous quote alludes to many of the potential benefits for students of real-world problems that Alice mentioned in her first interview. However, the above comments advance her previous ones by including the classroom context: Reflecting on her practice teaching, she was able to articulate a specific example of how a teacher structured an exploration of this real-world problem and how students responded to the situation. This problem in the hypermedia project represents an explicit link, between Alice's conceptions and classroom experiences, that provides a richer perspective from which she can view and appreciate the role of real-world applications in mathematics teaching.
This comment also indicates the role perceived by Alice of how real-world problems can set the context for the type of mathematical activity that includes extensive exploration, communication, and sense-making by students. Through participation in her prestudent teaching placement, observations of other classes using application problems, and her tutoring work, Alice developed an appreciation for having students "really talking in class and talking about math." The Fraction Problem in the hypermedia project incorporates not only a meaningful real-world context, but also her appreciation for the extensive student cooperation that she observed and participated in during her field experiences. A related issue identified by Alice was how to determine whether mathematical ideas and solutions "make sense" given a particular situation or real-world context. That this issue was on her mind as she constructed the hypermedia project is evidenced by her explanation of why she had decided to include a particular quick-t ime movie from a video segment (showing her leading a discussion about solving one of the project's problems): "I chose that clip because of the discussion about how we knew whether that answer made sense or not.... I included it because I thought it really showed the discussion of the problem and figuring out how to find out if our answer was right or not.
The importance to Alice of students making sense of solutions and communicating their thoughts was deepened by her interactions with the other group members regarding the problems incorporated in the hypermedia project:
[In the hypermedia project] we talked about assessment a little bit, that if you can solve the problem, it doesn't really mean that you know what is going on. Usually the real-world problems are set up so that you have to write about your answer and explain your thinking. We thought that was a stronger part of the problems [included in our project] because we really got to know what was going on. One student in the class--we used her sample of work [in our hypermedia project]--she had a column of the answers that she came up with and then a column labeled "How I know." She reasoned through everything that she did.
As this comment suggests, project creation enabled Alice and her peers to discuss important issues such as assessment and problem solving, and share in each other's ideas. It also required them to determine how to relate specific examples of student work to the theme of their project.
As has been illustrated, there was considerable value in having Alice, during her prestudent teaching practicum, assist and tutor students from classes where her project theme and other reform themes (e.g., cooperative learning) were effectively enacted. Observing and participating in innovative activities is extremely important in learning to teach in innovative ways. But by creating the hypermedia project, Alice was able to even more deeply connect these experiences with her own conceptions about how real-world examples help students to value and understand mathematics.
Most of the prospective teachers enrolled in the methods courses in 1995 and 1996 participated in student teaching during the following semester and created hypermedia portfolios. One of these portfolios, created by Rosie, an individual who participated in the construction of the cooperative learning stack discussed previously, is described in the following section.
Student Teaching Portfolios: An Example
Figure 6 shows the Table of Contents from Rosie's student teaching portfolio. The table of contents, which followed an introductory title card, outlines the main branches of the hypermedia portfolio. Within each of the six branches, the user can click on a button at the bottom of the page to return to the table of contents. Both the Student Teaching and Prestudent Teaching branches contain quick-time movies of this student teacher's practice teaching. The Tutoring branch contains photographs of some of the students this individual tutored. The Future Plans branch allows the user to open a word-processing document showing a resume. Other textual discussions are contained in additional branches of the portfolio. For example, the Personal Statement branch allows users to read Rosie's teaching philosophy.
Rosie's personal statement, and its links to other parts of her portfolio, suggests the power of the hypermedia portfolio for assisting prospective teachers in constructing and representing personal theories of mathematics instruction. Rosie's personal statement began with a tribute to the teachers she had as a student of mathematics, "Many teachers have positively affected my development both in academics and in my personal life. I attribute my choice of profession, and goals as a teacher, to the dedication of those teachers." This genuine respect and appreciation for the work of mathematics teachers is illustrated in another branch of her portfolio, where she presents a photograph of the display case that she created in the hallway at the school where she student-taught. As seen in her portfolio, this display case showed the words "Making Math Come Alive" above photographs and names of the mathematics teachers in the school. In these two ways (textual and pictorial), users can recognize this Rosie's high esteem for the mathematics teachers with whom she has interacted.
Additionally, Rosie presented some of her teaching goals in her personal statement:
The major objectives I would like to pursue in my teaching career focus on three major areas: classroom environment, student understanding, and application. First, I believe that it is a teacher's primary responsibility to create an environment where learning can take place. To accomplish this, I would take a genuine interest in students' lives, provide clear objectives for classroom conduct and learning, and establish links with home support and other educators. Next, I think that it is important for each student to be able to explain a concept to other students in their own words, to apply it to other disciplines, and to exhibit the building up of additional concepts upon it. Finally, one of the main objectives will be to foster application of concepts to everyday life.
Each of these goals is further elaborated in other branches of Rosie's hypermedia portfolio. For example, she illustrates her commitment to creating a positive classroom environment by including numerous photographs and descriptions of her students, and by writing reflections about the various teaching strategies she implemented during her student teaching. The hypermedia stack structure allowed Rosie to support the main points of her Personal Statement by including non-textual materials (e.g., video clips, photographs) that could be accessed from multiple points in the hypermedia portfolio.
Her statement also included some important contradictions that are indicative of the preliminary nature of her conceptions, and her initial attempts to connect her beliefs to her classroom experiences. For example, she makes this remark at the beginning of her personal statement: "From as far back as I can remember, I've always wanted to be a teacher. Although many years have passed since I taught my cats and dolls how to add two plus two, the desire to pass on my experience and knowledge has remained."
However at the end of her statement, she suggests the following: "I believe that the purpose of education should not be to simply pass information to future generations. Education is the means to empower and engage students' minds. Teachers hold a great deal of responsibility. I accept that responsibility, and welcome the challenge it will present to me."
This struggle with her beliefs about student learning, and the issue of passing on information" versus "empower[ing] students' minds" is one that she struggled with over the course of her student teaching experience. For instance, she grappled with the issue of whether students would understand important concepts during small group activities if she did not explicitly tell them. However, she also believed that students would develop critical understandings through communication with one another without her intervention. These components of her beliefs are further reflected in two quick-time movies in other parts of her portfolio: in one video clip, she leads the class in creating a calculator graph; in another clip, she circulates between small groups asking and responding to questions as students work on a class activity.
Many of the connections referred to above were explicitly represented in the stack and could be accessed from several points. However, Rosie did not make some connections explicit that were clear to us. Such omissions, together with apparent contradictions (e.g., what are appropriate places for different teaching strategies), made it unclear whether Rosie recognized all of the connections herself. Her struggle about the roles of cooperative and teacher-centered activities in mathematics teaching and learning had not been resolved during her introductory methods class, when she and other students created a hypermedia project about it. Neither was it resolved while constructing her student-teaching portfolio. However, she mentioned to us after completing the portfolio that hypermedia creation had forced her to think a lot about what she believed in and how it made sense in relation to her student teaching experience.
IMPLICATIONS AND DISCUSSION
The preceding examples illustrate some of the powerful features of the hypermedia format. The use of multiple media (e.g., videotape, word processors, computer software) permits prospective teachers to experience powerful activities and to reflect on their experiences. Further, the hypermedia format allows individuals to incorporate all of these items into a single document. With hypermedia, prospective teachers have greater opportunities to connect these components to each other as well as to their own growing understanding of mathematics teaching.
As illustrated in the preceding sections, prospective teachers use hypermedia software to assist them in creating and organizing components of their projects. Participation in the creation of hypermedia projects helps prospective teachers become acquainted with computers and other technological tools while at the same time providing them with a context to discuss, reflect upon, and improve their own teaching. As prospective teachers study and make decisions about which artifacts best illustrate the group's or individual's chosen theme, they come to understand how critical analysis of one's own and others' teaching leads to confidence-building and the ability to continually refine teaching philosophies and practices.
The Curriculum and Evaluation Standards (NCTM, 1989), the Teaching Standards (NCTM, 1991), the Assessment Standards (NCTM, 1995) and the Principles and Standards for School Mathematics (NCTM, 2000) emphasize that it is not enough to change the type of mathematics taught and the methods of teaching. Teachers must also change the way they assess mathematical learning. The Standards documents encourage teachers to move away from exclusive use of traditional forms of assessment such as tests and quizzes, and include activities such as writing, computer presentations, projects, and portfolios. Our hypermedia program gives prospective teachers experiences learning and teaching in a non-traditional setting, one consistent with the recommendations of the Standards volumes. The program provides a context for participating prospective teachers to gain experience with an alternative form of assessment--hypermedia projects and the associated presentations.
In the process of creating projects, prospective teachers choose, discuss, and reflect upon important pedagogical and mathematical outcomes. They determine not only which learning outcomes to include, but also how to effectively illustrate these outcomes. By selecting (or creating) segments that illustrate the chosen outcomes, prospective teachers are able to observe and critically think about their own and their peers' teaching. A willingness and ability to so reflect is crucial to the continuing development of effective teachers (Jones, 1990; Schon, 1983). Critical reflection such as that experienced in creating and sharing hypermedia projects helps to create a culture of scholarship among the prospective teachers--a culture they can nurture as classroom teachers because they come to understand how the processes of observing, reflecting upon, and discussing specific classroom events can improve their own practice. Furthermore, teacher educators using hypermedia are able to more easily emphasize themes not always emphasized in mathematics methods courses (e.g., alternative assessment strategies, communication, and cooperation).
During methods class and related field experiences, many prospective secondary mathematics teachers begin to translate some of their prior experiences as mathematics students into pedagogical content knowledge (Shulman, 1987). For example, Alice recognized that most of the mathematics she had learned about was not connected to the real world. Hypermedia creation, as well as other methods class activities, gave her the opportunity to revisit and reflect on this experience from a perspective based on theories of teaching and learning, express and represent her own view of the importance of real world problems, and connect it to her growing conceptions of mathematics teaching.
Effective teacher preparation allows teachers to articulate (and build upon) their beliefs about teaching, learning, and mathematics (Liston & Zeichner, 1988). The examples illustrate how Alice and the other interns used the hypermedia project to do this. The project enabled her to take into account her existing conceptions and needs, and it provided her with opportunities to focus on a very specific issue: real-world applications. Such specific focus enabled Alice and the other prospective teachers to consider, represent, and perhaps transform their beliefs, which were probably held tacitly up to the point of teacher education activity (Fenstermacher, 1979), so that they could be accessed and used explicitly in future classroom teaching.
As this article demonstrates, the pedagogical benefits of the hypermedia program are numerous. Hypermedia activity requires prospective teachers to reflect on and build explicit connections between their field experiences and their developing understandings of mathematics and mathematics teaching. Because video segments of practice teaching, for example, are explicitly linked to textual or other representations of the chosen themes, hypermedia environments allow prospective mathematics teachers to place together in a single document multiple representations of the learning-to-teach process. The teachers recognize the hypermedia format's powerful ability to represent their themes coherently. In Alice's words, "You could have shown the same information in another way but you would be giving a speech having either overheads or would be writing on the board. You would have to bring in a TV and a VCR to show the clips. It really made an efficient way of bringing it all together."
Although teachers can view video segments and think about their teaching without hypermedia technology, physically connecting several representations of the learning-to-teach process increases the likelihood that prospective teachers will develop integrated understandings of what it means to teach mathematics.
Finally, the hypermedia program has tremendous potential to assist us in conducting research and in helping us understand what prospective teachers think about mathematics and pedagogy. Prospective teachers generally possess meaningful and powerful beliefs, but it is sometimes difficult to establish a context that will encourage them to reveal their beliefs (Cooney, 1985). The projects discussed in this article provide such a context. For example, which video segments prospective teachers choose for their projects and why they choose them can be very revealing about what they think it means to teach mathematics. The fact that Alice's project was so closely aligned with her existing beliefs about mathematics and mathematics teaching illustrates very clearly that projects can serve as excellent contexts for discussions with prospective teachers about their beliefs. Rosie's inability to resolve her personal struggles with the issue of cooperative learning is another example illustrating the potential for discussions about projects' relationships to beliefs. Hypermedia offers an effective way to learn about preservice teachers based on their own examples of teaching and learning situations.
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(1.) Sample stacks can be viewed on the Web at
Authorship on this paper is alphabetical; both authors contributed equally. Gwendolyn M. Lloyd, Department of Mathematics; Melvin (Skip) Wilson, Department of Teaching and Learning.
The program described in this paper was supported by the Eisenhower Foundation and the University of Michigan Office of Instructional Technology.…
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Publication information: Article title: Offering Prospective Teachers Tools to Connect Theory and Practice: Hypermedia in Mathematics Teacher Education. Contributors: Lloyd, Gwendolyn M. - Author, Wilson, Melvin - Author. Journal title: Journal of Technology and Teacher Education. Volume: 9. Issue: 4 Publication date: Winter 2001. Page number: 497+. © Not available. COPYRIGHT 2001 Gale Group.