Technology Experiences for Prospective Secondary Teachers
(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
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'
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
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
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
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
Challenge 2: Long term Planning: "Why?" and "when?"
vs. "how?" and "what?"
· 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?
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
Challenge 3: Ambiguities associated with technology enhanced curriculum
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
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;
It was my intention to make the use of technology explicit in achieving
these goals when appropriate. In particular, I intended for students to:
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
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.
·use technology to solve mathematical problems,
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.
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
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
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
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.
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
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.
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
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?
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.
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
How would student learning be assessed?
What would a follow up chapter look like?
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
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?
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.
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
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
· 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
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
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
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
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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