Technology Experiences for Prospective Secondary Teachers


by

Azita Manouchehri
Maryville University
(A Paper in Progress. Not to be cited without permission from the author.)

Recent mathematics reform strongly endorses the use of technological tools in instruction (NCTM, 1989, 1991; MSEB, 1991; MAA, 1991). The reform documents envision that mathematics teachers will use current technological tools to facilitate the cognitive development of learners. Proficiency in the use of technology, and the ability to design curriculum and instruction that takes full advantage of the available technological resources is one of the main goals for mathematics teachers. Similar messages have been manifested by educators in other content areas (NCTS, 1994; NCTE, 1997; NCTSS, 1996).

As a response to the national calls for establishing standards for technology use in school curriculum and instruction, efforts have been made to constitute compatible standards in teacher preparation. Recently published documents such as the second report of the Holmes Group (1990), the National Board for Professional Teaching Standards (1991), the National Council for Accreditation of Teacher Education (1997), and the Commission's Report (1996) have proposed standards for teacher preparation and they require that future teachers should develop adequate knowledge of the technological tools that facilitate the improvement of teaching at pre-college levels. The recommendations of the Commission explicitly address the need for teachers' proficiency in technological skills in order to support student learning and their own professional growth (p. 77). According to the Commission, education of future teachers should include modeling of how to teach for understanding, how to continually assess and respond to student learning, and how to use new technologies in doing so.

Although current reform documents have proposed recommendations towards the use of technology in teacher education, the perspectives offered on how they could be realized are not well developed. The reform does not define the program of study or the pathways to achieving technology standards in teacher education.

In this article, I will present a review of perspectives offered on how technology should be used in teacher preparation and highlight how these perspectives shape the challenges for teacher education. By discussing an example from one institution and describing experimentation with some strategies, I will consider possible resources for meeting these challenges.

Technology in Teacher Preparation: Messages From Literature

A review of available literature on technology and teacher preparation highlight the notion that effective infusion of technology in schools will take place if some form of modeling is provided for teachers in the course of their preparation. For instance, Barron and Goldman (1994) proposed that teacher educators must use technology as tools themselves, modeling for their students the many ways that technology can enhance teaching, learning, and scholarship. According to the authors, the notion of technology's use as a cognitive tool which engages students in authentic and challenging tasks should become one of the foci of teacher education. Knapp and Glenn (1996) maintained a similar perspective as they argued that effective modeling of how technological tools could enhance teaching and learning should be an integral constituent of teacher education programs. According to them, the key component in fostering change is for teacher educators to model appropriate technology use in classroom and curricula, and for future teachers to have frequent opportunities to practice using technology as both the learning and teaching tools. Hoffmann (1996) argued that schools of teacher education can better prepare teachers if university education faculty model technology use throughout the teachers' own education.

There has been some debate over whether requiring a course on computer based technology in teacher preparation would provide an appropriate vehicle for establishing technology competencies among preservice teachers (Handler 1993; Wetzel 1993a; Fox, Thompson, & Chan 1996; Robleyer & Barron 1993). The opponents of a required course in instructional technology argue that a single core course might convey the message that the computers were add-on and not an integral part of instructional resources (Fox, Thompson, & Chan 1996; Thomas, et. al. 1996; Wildmer & Amburgey 1994). These educators, have suggested that technology should be integrated across all teacher education courses. Brooks and Kopp (1990) state explicitly that competence with technological enhancements to instruction should begin with faculty research and instructional modeling of the technological enhancements through the preservice curriculum.

The findings of research conducted by Thomas and colleagues (1996), as well as Wildmer and Amburgey (1994), substantiate the proposition that training of teachers in the use of technology should incorporate modeling by instructors in all their courses. Moreover, they specifically suggest that the technological training of teachers should be incorporated in content related assignments. According to Thompson and Schmidt (1994) training in the use of technology is valued by students when it is connected to subject matter or has immediate instructional purpose. A number of other research studies have documented the importance of modeling and demonstration of technology assisted instruction in the context of subject matter preparation of future teachers (Brownell & Brownell 1991; Manouchehri & Pagnucco in press; Manouchehri, et. al 1997; Schmidt, et. al 1994).
Manouchehri and Pagnucco (in press), in their examination of ways in which technology could be integrated in education of future mathematics teachers, claimed that using technology in mathematics instruction requires a body of knowledge that is specialized in nature. This knowledge includes pedagogical presentations of mathematical concepts using appropriate software, and problem posing and questioning techniques that motivate productive use of technology within the learning environment. The authors asserted further that technology use in instruction with the purpose of enriching students' mathematical learning is valued by future teachers if they are convinced of their impact on their own learning of the content. Therefore, in order for the technology to become a part of a teacher's functioning and practice, experiences provided for them in the course of their own mathematics learning should assist them in constructing an image of the teaching and learning that is enhanced by technology.

Manouchehri, Enderson, Pagnucco, and Jiang (1997) hypothesized that the education of teachers with the use of technology should be viewed as a gradual process achieved over a long period of time. Technological knowledge required of mathematics teachers can not be met as the result of exposure to the technological tools in the course of one generic course in instructional technology. Instead, technological education of future teachers should take place within both the teachers' content and methods courses. That is, those responsible for the education of teachers should use the technological tools currently available to foster teachers' learning of mathematics, learning to teach mathematics, and their enculturation into the world of practice. The authors encourage those responsible for educating future teachers to view technology as a means to an end and not an end in itself. Technology should be used to approximate future teachers towards the reform's visions of learning and teaching through developing personal and intimate experiences with technology.

Challenges

The common denominator of the research cited above is that teacher educators should incorporate technology in preparation of teachers and provide them with concrete examples and models of their integration into curriculum and instruction. Moreover, this training should occur on a timely fashion and not just in the course of one class. While these messages offer a general framework for teacher education, they also bring into light several challenges for teacher educators to grapple with and study. These relate to both the context and content of teacher education.

Challenge 1: The issue of modeling in teacher education: Who are the "teacher educators?"

A major influence on future teachers' professional development and teaching is the instructional practices of the professors of content. This is of no surprise for the coursework in pedagogical studies generally represents only about one-fifth of a secondary teacher's required program and only a third of an elementary teacher's program of study. Secondary teachers, for instance, take up to four or five professional education courses, only one or two of which address content specific pedagogy. Recognizing this major influence, the recommendations towards technology integration in teacher preparation call for collaborative efforts, joint planning, and modeling of technology use on the part of both professors of pedagogy and content. This collaboration should be built on shared vision and responsibility towards creating environments in which future teachers experience the use of technology as both a learning and a teaching tool. A component of this shared planning is deciding what technologies to use, and how and when to use them. The major challenge, however, arises from the fact that the current state of affairs within the institutions of higher education is not indicative of such collaboration. Communications among the professors of content and pedagogy is exception rather than the norm. This, coupled with the lack of modeling of technology use, or appropriate use of technology, by content area professors (Taylor 1994, Wetzel 1993b) makes accomplishing the standards identified earlier rather ambitious. At this point, this lack of communication serves as an obstacle in creating a unified and comprehensive technologically enriched plan for teacher preparation. Two major questions that are of concern to teacher educators and need to be addressed are:
· How can professors of pedagogy adequately meet the technology standards in teacher preparation when content preparation of the future teachers does not permit an awareness of the potential of technology for their learning?

· How can professors of pedagogy assist teachers in developing a disposition to use technology for their own learning of the content as well as the skills in using them in teaching school curriculum?
Challenge 2: Long term Planning: "Why?" and "when?" vs. "how?" and "what?"

Although the issue of improving the technological competencies of future teachers is important and certainly in need of attention, of greater importance is understanding "how" technology should be used in teacher education to improve teacher preparation. The use of technology in teacher preparation can not be considered without a complete examination of the aims of teacher preparation in a broader context of educational reform. This broader context involves understanding the goals, motivations, roles, and impact of teacher education. In essence, the use of technology in teacher education should lead to future teachers' learning of the professional knowledge base for their practice, and thus, improve the conditions of teacher preparation.

With the widespread enthusiasm towards using technology in instruction, in higher education the tendency has been to use technology as a source for providing information or a presentation device. There is a tendency to confuse the availability of technological tools and information with learning. The issue that has too often been ignored is that technologies are only a means for providing information, and not a learning media. In a general sense, many within the higher education community including teacher educators, are concerned about "what " technology to use whereas the first critical considerations should include "why" and "for what purpose," technology should to be used in their instruction. Within teacher education programs attention needs to be devoted to first determining the goals, investigating "why " and "when" technology should be used in the course of professional preparation of teachers to best assist in accomplishing those goals, and finally evaluating "what" technologies would be most appropriate for achieving those goals.
Such long term planning requires an extensive knowledge of both the capabilities of the available technologies and ways in which they can potentially improve teacher preparation. This knowledge, however, is not well developed in teacher education.

Challenge 3: Ambiguities associated with technology enhanced curriculum and instruction

Although in recent years, the research on the impact of technology on cognitive development of learners has blossomed, research studies on the impact of technology on teaching, and models of instruction in the presence of technology are rare (Thompson 1989; Lappen 1994). Moreover, while it is a shared consensus that the technology will play a major role in decisions about what to teach and what not to teach, affecting the curriculum in a fundamental way (Seeley, 1995), the education community knows very little about the long range impact of technology on both school and college level curricula. Indeed, what we have so far in the way of technology based materials are mostly bits and pieces, and small isolated components of materials. Full learning and integration however requires whole courses, full curricula, and a theoretical basis for practice.

The lack of knowledge about technology enhanced teaching and curriculum directly influences the activities of teacher education. On the one hand, the uncertainties associated with the outcomes of technology use makes its impact of little perceived value for teacher educators. Therefore, the future teachers' education is not centrally oriented around investigating the relevance of technology for their practice. On the other hand, due to lack of models of technology enhanced instruction that could be immediately used in teacher education itself, teacher educators either do not use technology or do use it in a manner that is not meaningful to the content and context of what they teach. The reality of limited faculty models in the application of technology in instruction by professors of pedagogy testifies to this assertion (Taylor 1994, Wetzel 1993a,b).

The reluctance to use technology in teacher preparation by teacher educators is a manifestation of a lack of conceptualization about how technology could enhance their work in ways other than using technology as an information provider and a presentation media. Teacher educators have not yet conceptualized the implications of technologies for their own curriculum and their usefulness in facilitating the growth of the pedagogical knowledge base necessary for future teachers. They are not convinced of the potential of technologies for their instruction and of their value in achieving their instructional goals and objectives. A challenge facing teacher educators is defining a mechanism for meeting the goals of teacher preparation in the presence of technology.

Challenge 4: School conditions

The ultimate goal and purpose of teacher education is to prepare professionals that understand the purpose of schooling within our society, learn to function productively within them, and contribute to their professional development. An important function of teacher education is providing future teachers with an adequate knowledge base about, and an understanding of, the context of schools and their conditions. Moreover, teacher education should enable teachers with strategies that allow them go beyond an "existence mode" within schools and act as active professionals. These goals are not always addressed or achieved in teacher education. The issue of the technology competencies of future teachers and their technological preparation can not be viewed independent of these goals and functions. The major challenge facing teacher educators is not only preparing teachers to use technology effectively but also using it within the contexts of schools and resources available (or unavailable) to them.

It has been well documented that the conditions of schools do not always allow for beginning teachers' practices of what they are taught in their professional college preparation (Wilson & Ball 1996; Thomas, Larson, Cliff & Levin 1996). Models of instruction provided for them or the textbooks used by teachers do not provide future teachers with contexts to experience the recommendations towards innovative curriculum and instruction. Future teachers' awareness of lack of use of innovative instructional tools (including technology) within schools creates, on their part, some skepticism towards the usefulness of acquiring technological competencies for their practice. A challenge facing teacher educators is enabling future teachers to work within the disabling context of schools. A component of which includes the use of technology in instruction.

An Example: Meeting the Challenges!

As it is the case at many universities and colleges across the county, at Maryville University-St. Louis, the secondary mathematics preservice teachers are required to take a computing course for state certification. This course is taken usually at the beginning of their sophomore year and consists of a programming language. The general nature of the this course is such that students spend some time learning the syntax of the language, and write a few computer programs. Most students take this class to fulfill the requirement and too often fail to connect the ideas explored in the computer course to instances within their specific field of study.

At our institution, the secondary preservice mathematics teachers obtain a bachelors degree in the content area and are required to take an additional 25 units of education courses and field experiences. Two of the required education courses (8 units) address content specific pedagogy. These are "Mathematics Topics in Secondary Curriculum," and "Methods of Teaching Mathematics in the Secondary Schools." The students take the first course in the spring and the second course in the fall semester of the final year of their program. Concurrent to taking the sequence, students do field experiences. Upon completion of this sequence, students enter their student teaching phase.

The purpose of the first course is to familiarize students with the content of the secondary curriculum. The second course concentrates on introducing preservice teachers to theories of learning mathematics, and the instructional tools and techniques useful for teaching the subject. In the past, these two courses were taught by two different faculty members with varying degrees of emphasis on technology use in their instruction and different sets of course objectives. In a general sense, the first course focused on helping teachers develop a working knowledge of graphing calculators in exploring topics from the secondary curriculum. The second course familiarized teachers with educational software, and engaged them in review and evaluation of educational videotapes appropriate for teaching topics in secondary curriculum. Neither one of the two courses examined in great depth the use of technology in mathematics instruction. Neither one required students to write lesson plans that integrated technology, to investigate the cognitive and affective elements associated with their use in learning and teaching mathematics, or to study their impact on school mathematics curriculum.

During the 1996-1997 school year, the same faculty member (the author) was assigned to teach both courses. I intended to teach the two courses as a sequence and on a continuum. Rather than setting different objectives for each class, my intention was to develop a set of objectives that could be extended over the entire year. In this way, the opportunity to tackle both the content and pedagogical issues from a deeper perspective was more feasible. Through informal conversations with colleagues in the department of mathematics as well as those in the college of education, and through the review of a number of different course syllabi our students had completed in previous semesters, it became evident that none of the students had used computer technologies in their instruction beyond word processing purposes. In a statistics course, students had used calculators but none had exposure to graphing calculators or used them.

My overarching goal for the sequence was to familiarize students with the recommendations of the mathematics reform for curriculum, teaching, and assessment as manifested by the NCTM Standards (1989, 1991, 1995), and involve them in a critical examination of both the content and context of those recommendations. In order to do so, I intended to:
provide students with learning opportunities in which they were active learners, did mathematics, solved mathematical problems, and engaged in mathematical explorations from a standpoint that was in accordance with the recommendations of the mathematics reform;

motivate students to reflect on and analyze the content of secondary curriculum first hand as learners and explore the potential of a variety of teaching tools including manipulatives and the available educational software for teaching those topics;
·
engage students in an examination of a variety of teaching techniques such as lecture presentations, small and large group cooperative activities, individual and group concept discovery methods;
·
assist students to develop knowledge about problem solving as a method of instruction;

provide students the opportunity to examine theories of learning and teaching mathematics and their implications for their practice;

assist students to develop the necessary skills and knowledge for planning, organizing , and assessing their instruction;

increase students' awareness of the issues and complexities associated with learning and teaching mathematics;

familiarize students with cognitive obstacles associated with learning certain mathematical topics; and the difficulties associated with teaching them;

familiarize students with the context of schools including learner attributes, teacher responsibilities, and educational resources.
It was my intention to make the use of technology explicit in achieving these goals when appropriate. In particular, I intended for students to:
·use technology to solve mathematical problems,

realize the capabilities of graphing calculators, and a variety of computer software appropriate for teaching high school mathematics;

develop knowledge about different methods in which technology could be integrated in their instruction ;

realize the potential of technology in restructuring school curriculum, and framing their planning in light of this knowledge;

gain experience in designing tasks, simulations, and explorations that lend themselves to technology based investigations;

learn how to design and use technology based explorations and computer simulations for whole class investigations and small group activities;

use technology for on-line research and locating resources;

familiarize students with methods of design and use of multi-media presentations;

demonstrate how technology could be used to facilitate students' understanding of mathematical concepts through linking their multiple representations .
Due to the fact that no technology was used in students' past mathematical training, it was evident that a significant amount of time needed to be spent on convincing students of the value of technology throughout the course of their own learning that year. Therefore, it was essential to select mathematical contexts that would allow them to realize the potential of technology for learning topics from a standpoint that went beyond algorithmic manipulations and fostered meaningful conceptual explorations of mathematical ideas. Moreover, it was clear that while the use of technology in their own learning was a necessary condition for them learning about its usefulness, it was by no means sufficient in preparing them to use such tools in their instruction. Therefore, time needed to be devoted to assisting them translate those learning experiences into teaching actions. By focusing students' reflection on their own learning, I intended to provide them with contexts to develop an understanding of the pedagogical aspect of the use of technology in the classroom. In order to have concrete examples for our classroom discussion, I decided to videotape our sessions and use them in class for analysis of learning, teaching, and curriculum over the entire year. Moreover, since students were also scheduled to be enrolled in a field based experience and therefore would have the opportunity to observe teachers and work with small groups of students, I intended to use their practicum experiences to investigate issues concerning learning, curriculum, and teaching in school settings. In particular, it was my intention to address the existing technological constraints within the schools and work with students towards developing strategies for productive functioning within those constraints.

The Sequence of Learning Activities

Both courses were designed to be problem based. The major area of emphasis during the first course was on investigation and analysis of mathematical topics and problems with the aid of technology and other manipulative materials. In particular, mathematical modeling and problem solving constituted the core of instruction. Due to wide availability of TI-83s in schools, all students were required to purchase a graphing calculator. Students were also required to purchase all the NCTM Addenda Series for grades 9-12. A number of interactive software appropriate for secondary and college level mathematics curriculum such as GSP, Microsoft Excel, X-Functions, Algebra-Xpressor, Theorist, and Math lab were also selected to be used through the year. Learning about the capabilities of each piece of software occurred within problem contexts and as students explored different problems. During the first course, students spent about half of their class sessions in the computer lab. During the rest of the sessions, access to a computer was possible via a computer workstation and LCD panel in the classroom.

Whole Group Activities and Assignments

Approximately once a week I conducted a whole group technology based presentation which included either the use of CBL for collecting data, or a computer based simulation for motivating the discovery of certain mathematical relationships (For examples, see Manouchehri 1997; Manouchehri & Pagnucco in press, Manouchehri, et. all; 1997; Wilson 1997). These were done to illustrate two major points. On the one hand, I intended to illustrate the value of visual media on providing insight and building intuition towards understanding mathematical concepts and their problem solving ability. On the other hand, I wanted to demonstrate how a single piece of technology could be used to facilitate exploration of topics within the learning environment. My rationale for using these demonstrations were also regularly shared with students in class.

Each week students were assigned to read particular chapters from the graphing calculator manual. To reinforce practice and use of those functions, several related mathematical activities were also assigned for their investigation. At the beginning of each session, we spent time discussing those activities as students shared their solution methods. In addition, questions related to the mechanics of graphing calculators were also answered in small and large groups.

Homework assignments also included students writing lesson plans about specific topics which utilized graphing calculators, computer software, and manipulative materials. In-class time was assigned for presentation of these lessons during which students taught their lessons to their peers. These presentations were also videotaped to be used with individual students as they examined their own teaching actions. In small groups, students analyzed the content of the lesson, the sequence of activities that were included, the questions that were asked, and the appropriateness of the tools selected for teaching the topic of the lesson.

Internet Activities

At the beginning of the first course students were introduced to the Internet and began using it as a source of data collection for problem solving, and research in specific content areas. I provided students with a list of useful cites to review. Throughout the semester, students were assigned to research and study specific mathematical topics. Frequently, as the result of their electronic search, students accessed cites that provided pedagogically useful activities such as lesson plans for teaching of the same topics. They used these activities to help them better understand the concept they were assigned to study. The use of the Internet was later extended to include students finding specific types of lesson plans and activities. They were required to analyze these lessons and identify how they aliened with the NCTM Standards. The objective was for students to realize that (1) there were a number of resources available to them, and (2) when locating resources, they should be analytical about what they find, and think carefully about how to use them.

Students were also assigned to subscribe to electronic listserves and to participate in group discussions that revolved around professional issues. These were bulletin boards where teachers posed questions and shared teaching ideas. This was assigned for two main purposes. I envisioned that such participation would facilitate their understanding of the "real" issues associated with teaching mathematics and provide them with a professional network thorough which they learned to articulate professional matters. Moreover, I hoped that as the result of their conversations with other experienced teachers they would realize in greater depth the complexities associated with teaching and learning and develop a disposition to appreciate the value of professional discourse in resolving these problems.

Curriculum Analysis

The analysis of curriculum occurred on several levels throughout the sequence. It began by students reviewing the explorations, problems and activities they used themselves during the sequence. They evaluated how mathematical ideas were embedded in problem contexts, and in what ways the connections among the geometric, numerical, and algebraic representations of ideas were established within a sequence of interconnected activities. They determined how these problem situations related to other mathematical structures. They examined also how certain concepts were visited and developed throughout the semester rather than at a different point in the course in a disconnected fashion.

This initial curriculum analysis was then extended to include their investigation of different textbooks. Primarily, I selected those textbooks which were being used by teachers in their practicum setting. Students evaluated the content of each chapter, assessed the methods in which ideas were organized, evaluated the methods of presentation of topics and how connections among different chapters were established. They examined the content of the problems that were assigned in different sections, and the quiz and tests that accompanied each chapter. Moreover, they assessed the nature of suggested instructional tools for teaching topics in each chapter of the book. Students were specifically assigned to address how these textbooks aliened with the curriculum and assessment Standards for school mathematics. To further assist students conceptualize how technology could be used to enhance the textbook, they were asked specifically to select a chapter and discuss:
What were the big mathematical ideas in the chapter and what knowledge bases were being explored?

What were good questions to ask in order to facilitate students' inquiry and understanding of those big ideas?

How could the chapter be organized to incorporate technology and manipulative materials? What other pieces of content become relevant as the result of this restructuring?

How would student learning be assessed?

What would a follow up chapter look like?
In order to guide students' planning in light of the resources that would be available to them within the schools, I also assigned students to research what technologies were available at the school where they did their field experience, along with how they were used by teachers. This included both the software and hardware availability and other equipment. They were asked to prepare a list of available software and equipment in both the science and mathematics departments. Our classroom discussions then centered around how they could productively work with the types of technologies that were available in order to enrich the pre-existing curriculum at the schools.

In addition, as the final course project in each class, students were asked to write a book chapter on a specific mathematical topic. The topics were selected carefully so to assure continuity among them. My goal was for the totality of the chapters to cover the content of high school Algebra, Geometry, and Precalculus courses. The basic criteria for this project included their use of a variety of teaching tools and simulations that utilized computer software, and a rationale as to why the chapter was organized as suggested. The rationale was expected to include a detailed account of how the chapter organization addressed the NCTM Standards for secondary mathematics curriculum. The chapter had to be accompanied with an appropriate assessment instrument. Students were also required to use multi-media systems for organizing and presentation of the chapter. Prior to final submission of the chapter, each project was reviewed and evaluated by at least another class member. This initial peer evaluation resulted in major revisions. Students chose to rewrite parts of their chapters in light of what others had included in their sections. This assisted in creating continuity among different parts of the final book product.

Analysis of Learning and Teaching

As we continued our mathematical explorations with technology and other manipulatives, opportunities for analysis of learning and teaching of mathematics naturally emerged. This was a direct consequence of students' realization of the differences between what they had previously experienced in the course of their own mathematical education and their current experiences in the course. Specifically, we spent time examining instances of students' own learning throughout the sequence. Videotapes from our sessions were used to initiate discussions about the dynamics of large and small group activities, teacher behaviors during the session, and the impact of technology use on the nature of these activities and behaviors. For instance, in viewing an episode in which I conducted a whole group presentation using computer demonstration, students were asked to focus on the type of questions I posed, the nature of my interventions during the classroom discussions, the amount of time they spent on self investigations and the questions they asked me in the process, the amount of time I spent on the actual demonstration, and the nature of my responses to their questions. They discussed also how the demonstration assisted in formulation of the problem, and allowed them to gain insight into the conditions and constraints of the task.

These personal episodes allowed the students to reflect on classroom events and to articulate the teaching actions that enlived the progression of their own mathematical discussions from an informal to a more sophisticated level of analysis. By asking questions about my intentions for using certain strategies, students also gained a perspective on the basis for my pedagogical practices and my choice of instructional methods during the session. In addition, they suggested ways the same activity could be organized to optimize learning outcomes in high school classrooms.
The same videotapes were used to assist students in the articulation of their own patterns of growth in understanding of mathematical concepts in the presence of a variety of teaching tools including technology. In particular, I asked students to focus on their own actions as they tackled a problem, identify the obstacles they encountered as they were introduced to the concepts by classifying the type of questions they had asked, and examine the nature of their own mathematical discourse during the sessions. They also discussed how these patterns of classroom interactions were similar to or different from what they had previously experienced in the course of their mathematics education. In addition to using our own classroom videotapes, I used tapes of teaching episodes from regular classrooms to provide students with opportunities to investigate similarities and differences between their own learning and the children portrayed in them.

Students' practicum experiences were also used to examine specific learning issues. Students were asked to collect data on how learners solved problems, and to report on learners' cognitive difficulties associated with specific content pieces. They were asked to identify the exact learner behaviors that they thought were indicative of learners' understanding or lack of understanding of the mathematics under study. Moreover, they collected data on the affective consequences of students mathematical activities in classroom. For instance, they investigated how learners' mathematical difficulties impacted their general behavior during the session and influenced their participation and engagement in the classroom. The students were also asked to focus on the techniques used by the teachers to assist students overcome those obstacles and ways in which they addressed the affective elements that were played out in class. This data provided us with on-line real life pedagogical problems. These problems were used to also investigate how technology could potentially assist students in overcoming some of their difficulties and misconceptions concerning mathematical ideas. In addition, we discussed how the use of technology could assist in defying student passivity in class and their lack of engagement in mathematics learning. In this way explicit connections between the curriculum, learner achievement, and teaching actions were established. The specific questions guiding our discussion included:
What were students' sources of difficulty and obstacles as they explored the topics?

What are some difficulties associated with teaching these topics?

How can we help students better learn these ideas?

What tools and techniques could we use to help students overcome those obstacles? How do you know that would help?

How would you convince others that your ideas are appropriate? And, how would you argue that your suggestions are essential to what it is being studied?
In addition to our classroom discussions of these questions, students were assigned to extend their investigation of the same issues outside class by conducting on line research, and by finding resources and relevant research studies. Students were also assigned to test their hypothesis about the usefulness of their plans with particular students in the school setting and to report on their findings in class.

The Aftermath: Student Teaching Activities

During the spring semester of 1997, I was also responsible for supervising the student teaching phase of two of the students who were enrolled in the sequence. This allowed me to observe their teaching, and ultimately, to investigate the impact of the experiences provided for them during the one year on their practice. The student teachers were placed in the same high school. They both taught Geometry, Algebra I, and Algebra II courses. One of the student teachers taught an additional analysis course. Both student teachers were placed within classrooms where technology had not been used by either one of their supervising teachers in the past.

Each student teacher was observed once a week for a period of 16 weeks. The attempt was made to observe each one as she taught different subjects. In addition, each student teacher was asked to videotape sessions in order to trace progression of the lessons they designed over a number of teaching episodes. I also reviewed these videotapes on a weekly basis. While student teachers were not required to submit lessons plans prior to teaching them, regular phone conversations with them kept me updated on their regular planning procedures. In addition, conversations with their supervision teachers granted me additional information on their activities and progress, as well as their difficulties during their student teaching phase.

As students began their teaching activities, several unique and intriguing patterns of behaviors and planning emerged in their work. These related to not only their use of technology in instruction but also their general pedagogical practices. In what follows a summary of these features is presented.

· Attention to student learning

A prevalent feature of students' practice was the high degree of their attention to learners' needs and the frequency of their reflection on student outcomes. Both students attributed significant importance to student learning. Moreover, they paid great attention to students' conceptualization of ideas rather than their mastery of algorithms. They investigated students' understanding and thinking by frequently questioning them on how they thought about ideas. In cases where learners were not clear on specific concepts, student teachers used alternative teaching techniques which included providing visual media, animated graphics, or physical models. While at times they were not capable of balancing the conditions of the learning environment to accommodate all student abilities during the session, their reflections on classroom events assisted them in writing their subsequent lessons. This attention to student outcomes allowed them to plan their instruction in a manner that built on multiple resources including technology.

· Technology use in instruction

Due to the fact that both student teachers were aware of the technological resources that would (or would not) be available to them during their student teaching, they planned prior to starting their teaching phase what resources they could use to prepare their lessons and units of instruction. Upon starting their placement they downloaded several pieces of useful software from the Internet, and planned to "borrow" other resources from the science department. These included the CBL equipment, and other software that were being used by the physics faculty in the school.
Both student teachers used technology in their instruction. Computer and graphing calculator based explorations were utilized in all courses they taught. They frequently began their instructional period by using a computer or calculator based demonstration. They introduced new concepts within problem contexts that built on graphical representations and visual models. In addition, computer based explorations were used for a sustained period of time as they taught topics in geometry and algebra. They used CBL simulations to engage students in collecting data for their study of graphs and function, assigned students to collect data from different Internet cites for independent project activities, and worked with the resource personnel in assisting students in developing multi-media presentation of those projects.

The student teachers' use of technology was also extended to address other pedagogical issues in their practice. For instance, both student teachers began storing solutions to the daily homework assignments on the computers available in class. Rather than spending a large amount of time in class reviewing exercises, they assigned students to check their answers to homework problems on the computer if needed. This strategy provided them with a larger amount of instructional time, and the freedom to spend more time on mathematical investigations.

· Reflective and collaborative planning

A major component of students' work was their attempts at organizing their instruction in light of student understanding and performance. They regularly videotaped their sessions and reviewed them in order to better focus on what had occurred in class during their instruction. Moreover, both teachers used their planning time to observe one another and provide each other with feedback on their instruction. Their enthusiasm for examining the impact of their instruction on student learning extended their conversations beyond their own classrooms. Supervising teachers reported that both teachers were asking other faculty members questions about the type of problems students usually encounter as they learn a certain topic, the kind of responses and explanations that tend to assist them, as well as additional resources that they had found useful for teaching certain ideas.

According to the supervising teachers, both student teachers had initiated an ongoing dialogue among the faculty concerning teaching and student learning. Their willingness to pose questions, to seek out advice , and to involve others in their own planning of instruction was intriguing to other mathematics teachers and thus, lead to creation of a greater support structure within the school setting for them.

· Critical evaluation of textbooks and content integration

Since the textbooks used in all the courses they taught were traditional in nature and did not make use of open ended questioning or rich real life applications, students attempted to design lessons in light of a variety of other resources. Each of the student teachers taught the "book chapter" she had designed. Moreover they used a number of other related lessons and activities that we had developed and used in our sequence. The use of these resources facilitated their efforts in integrating technology in their instruction.

An extremely powerful and significant manifestation of students' content and pedagogical growth revealed through the quality and substance of questions they asked in class. They seemed comfortable asking open ended questions, moving spontaneously from one question to another, allowing room for student exploration and inquiry. Both students relied heavily on student responses for posing subsequent problems and questions. They used project oriented activities for students' in class activities and for assessment purposes.

Final comments and a few conjectures

At the beginning of this paper I identified a few challenges associated with implementing the recommendations of research and reform towards integrating technology in teacher preparation. The technological and pedagogical experiences provided for secondary mathematics teachers in the course of a year described in this paper appeared to have met the these challenges. They impacted teachers' professional knowledge base as well as their practices. These experiences increased their flexibility in posing open ended questions. They fostered their skills to critically analyze the textbooks and materials they used in their teaching. Teachers also became more concerned about the impact of their lessons on student learning and revised their plans in light of their understanding of their own students' understanding. They attempted to individualize instruction by using a variety of activities that built around the use of manipulatives and technology. These experiences also seemed to have prepared them to work collaboratively within the school setting, and increased their tolerance towards the conditions/constraints under which they worked. Although at the present time I can not make definite claims about the specific nature of those factors that resulted in such profound impact on student teachers' activities a number of conjectures can be made.

Conjecture 1: Technology based explorations and mathematical learning

I hypothesize that a significant influence on student teachers' understanding of teaching mathematics and their use of technology and open ended questioning techniques was the problem situations and the mathematical explorations that occurred in class. These were the medium that created the environment for discourse and the use of manipulatives and technology. The learning the occurred within mathematical contexts allowed for ongoing "convincing" of the usefulness of the recommendations of reform for their classroom practices. These problems and explorations not only assisted them in deepening their own understanding of mathematical ideas, but also provided them with models of how curriculum goals could be met within contexts that were removed from standard algorithmic manipulations. Moreover, the extensive use of technology in their own work as they set up and solved problems, and investigated mathematical relationships granted them with detailed images of how such tools could facilitate learning.

Conjecture 2: Analysis of learning and teaching through self reflections

Student teachers' great attention to the cognitive needs of their students, and their concerns for the substance of learning that occurred in their classrooms was the result of the ongoing analysis of learning and teaching that occurred in the course of their own learning in the sequence. Extensive use of episodes from their own learning, and their investigation of the patterns of growth of their own mathematical understanding increased their sensitivity towards learners' actions and performances. It is likely that as they were confronted by their own understanding, they became aware of the importance of extending the learner's thinking. As the result of their group activities and group reflections, they learned to appreciate the importance of listening to others' processes, hearing and learning how others were thinking, and trying to make sense of others' difficulties associated with learning topics.

Conjecture 3: Relevancy of course materials and assignments

What facilitated student teachers' use of technology in their own classes, and allowed them to go beyond the boundaries of traditional textbooks was the relevance and appropriateness of assignments and activities used in our own courses. Their extensive examination of different curricular materials, their experiences with writing lessons and building curricular models that could easily be adopted in secondary school mathematics setting played a major role in fostering their ability to be discriminating in their choice of materials and selective in what they used with their students. Moreover, the opportunity to "rehearse" those lessons and materials in class, and with students during their field experiences gave them the confidence and experience they needed to use those lessons in their own classes. Furthermore, by examining the impact of those materials on the cognitive development of their peers, it is likely that students were more prepared to deal with the consequences of the use of such materials in their own instruction. While modifying the materials was necessary for their successful use, the students analysis of teaching during the course of one year gave them the flexibility and the disposition to examine curricula in light of student understanding as well as their connections to a broader mathematical context.

Conjecture 4: Practicum experiences

Student teachers' field experiences that accompanied their professional coursework served as a powerful medium for making immediate connections between what we discussed in class and what they experienced in the school setting. This assisted them in realizing the usefulness of theoretical perspectives for solving problems of practice. While the models of teaching provided for them within the school setting did not resemble innovative and progressive practice, the subsequent classroom discussions about teachers' practices and what could be learned from them assisted teachers in developing strategies for using alternative methods of instruction in their own teaching. Moreover, the real life data on learners' activities they collected as they observed classrooms provided genuine problem situations that we explored in a practical fashion. In addition, their knowledge of the context of school and the teaching that took place within classrooms, their awareness of resources that would be available to them during the course of their own student teaching as well as the obstacles they needed to resolve, allowed them to be more prepared and articulate about what they wanted to accomplish as they started their own practices.

Conjecture 5: Time for maturation and adoption

The learning about teaching in general and learning about teaching in the presence of technology occurred over two semesters of instruction. As the description of course activities outlines, a significant amount of time was spent on investigating "why" and "how-to" questions associated with the use of alternative methods of instruction and technological tools in class. Having planned the instruction for the entire year, provided us with ample opportunities to tackle issues more in-depth, revisit ideas in timely fashion, and engage in meaningful dialogues about mathematics, mathematics learning, mathematics teaching, and curriculum. The time devoted to understanding the role of technology and its relevance for school curriculum also occurred over the entire year and on a continuum within appropriate contexts. The sustained and long term experiences with both learning and teaching in the presence of technology resulted in a type of professional maturation that could not possibly occur otherwise.

Conjecture 6: Professional discourse

One of the most salient features of student teachers' practices was their need for and enthusiasm towards learning about learning and teaching. It was evident that they recognized the complexities associated with student learning, and were determined to develop a better understanding of learners' cognitive activities. Moreover, their willingness to constantly examine their own practice and evaluate the impact of their planning and instruction in collaboration with colleagues was a driving force in their productive functioning within their classrooms. A factor that increased their awareness of the importance of such inquiry for improving teaching was their active participation on professional listserves and their dialogues with other veteran teachers across the country. The modeling provided for them by other teachers, and the value placed on professional discourse by them shaped their thinking and behaviors during the course of their own teaching and practice.
Another contributing element was their extensive use of group activities throughout the sequence. Small and large group problem solving along with small and large group planning and peer evaluations constituted the core of their activities throughout the year. The students relied on each other for researching ideas, for sharing resources, for designing lessons, and for receiving critical feedback on their work. This certainly demonstrated to them the value of collaborative work and its positive impact on the quality and substance of learning outcomes.

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