The Ontario Institute for Studies in Education
University of Toronto
Paper presented at the
8th International Congress on Mathematics Education
July 20, 1996
Teacher Learning Needs in Computer-Based Geometric Environments
This paper reports on the experiences of four mathematics teachers who replaced compass and straightedge construction tools with a computer program. The study examines, using a case study approach, how teachers change when placed in an exploratory computer-based classroom environment. In the paper, I briefly outline the method used to gather the data and the context in which it is set. The case studies are then described and findings resulting from the analysis of the case studies are presented. These findings enlighten the links between teaching practice and the issues of control surrounding the use of computers in mathematics education.
An important problem facing mathematics education is the implementation of recommendations contained in two National Council of Teachers of Mathematics (NCTM) documents (NCTM, 1989, 1991). These reform movement recommendations in mathematics education support the use of computers in the classroom. However, a critical problem is the translation of those recommendations into actual practice in schools. The NCTM vision requires "significant and worthwhile changes" in teaching practice of mathematics teachers (Richardson, 1990; NCTM, 1991). Unlike other curriculum changes in mathematics, this change does not come from within the mathematics community as a consequence of certain cultural developments of the discipline, but as the consequence of the great changes in the social and economic reality provoked by the impact of new information technologies (Bottino and Furinghetti, 1994). Research on the reactions of teachers facing curricula innovations by which teachers reorganize their pedagogical practice and beliefs is still in its infancy (Boufi, 1994; Bottino and Furinghetti, 1994). An assumption for many new educational programs has been the premise that teachers will adapt to change and that we need only to instruct the teachers on the nature of the new change: be they curriculum, teaching techniques or assessment methods. However, in order to bring desirable changes to the system, we need to find out what is actually happening when teachers undertake changes in their teaching practice.
In the traditional classroom, the teacher's role has been one of "telling and describing" where teachers present ideas and directions to the class as a whole. If students encounter difficulty with the subject material, the students ask the teacher to confirm the correctness of their solution. In certain classrooms, the student may also ask for assistance from the other students in the class.
Romberg (1985) has pointed out that the job of teaching is to "assign lessons to a class of students, start and stop lessons according to some schedule, explain the rules and procedures of each lesson, judge the actions of students during the lesson, and maintain order and control throughout" (p. 5). Romberg believes that the mathematics curriculum is something that needs to be covered, and that few teachers see student learning and understanding as the primary goal of mathematics education.
Management of the learning environment becomes an important issue for teachers trying to make changes in their teaching practice. The teaching role has been seen as one where the teacher controls the learning environment. That control can be restrictive: directing, ordering, telling, and demanding. How teachers use this control within their classroom will clearly influence the learning environment. It is true that teachers make demands of students. Teachers tell students where to sit, when to listen, when to talk, when to work, when to move, and even when to 'learn'. Students are expected to listen to others, ask mathematical questions, complete their homework, and search for mathematical relationships.
Against this backdrop, teachers also expect students to think creatively, independently, and not be told what to think--essentially to take responsibility for their own learning. Students receive mixed messages from teachers as they try to cope with the controlled environment of the traditional classroom and the open exploration of the creative classroom.
The management of the learning environment must allow for students to construct their own knowledge and to take responsibility for their own learning. Students also need the freedom to discover, through exploration, different ways to build solutions. They need to spend time working with problems and searching for solutions. This process may be organized and recorded according to the predetermined plan of the teacher whose role is to facilitate the student's exploration (Burns, 1992). As such, it is important that teachers provide students with the opportunity to explore, analyze, and demonstrate their skills.
The use of the computer has been heralded as one teaching tool suitable to mathematics teachers to encourage exploration of mathematics. The expanded use of computers in mathematics education may create a shifting of roles for teachers. Assuming that teachers are willing to utilize computers in their classrooms (NCTM, 1989, p. 67) to encourage the students to explore mathematical concepts, there is a need to investigate how this utilization can be implemented. Even though the NCTM Standards (1991) have provided the impetus to change the curriculum, teaching, professional development methods, and assessment practices, there are many more factors to consider when teachers and curricula change.
There are many societal influences that affect teachers and their teaching practice. The integration of the computer into our society, the beliefs and images of teachers, and how teachers change influence mathematics teaching practices. Computer technology is visible in almost every facet of our society. Its entry into schools, however, has been slow and this is especially true for mathematics classrooms (Kilpatrick and Davis, 1993). Many reasons exist for this lack of integration into the mathematics curriculum: teachers' view of knowledge acquisition (Hannafin & Freeman, 1995), lack of availability of computer hardware (Becker, 1990), teacher anxiety towards computers (Rosen & Weil, 1995; Berebitsky, 1985), and teacher control in classrooms (Cohen, 1989; Schoefeld & Verban, 1988). In fact, the interaction between computers, teachers and mathematics is complex (Noss, 1991).
Building on recent research on teacher change, this study investigated the use and implementation of geometric construction software in mathematics classrooms. This study examined the reactions of teachers as their students explored geometric constructions in a probing for understanding milieu.
Context and Methodology of the Research
The setting for the study was four independent schools in a Canadian city. All participants were teaching Grade 8 students (age 13-14) and were selected in the following way: a male teacher in a boys-only school, a female teacher in a girls-only school and two other teachers (one male and one female) from two co-ed schools.
A case study approach was used. The data was collected through three primary sources: classroom observations, interviews and journal entries by participants. More specifically, the data was gleaned from my field notes, transcripts of interviews with teachers, students and Head of School, a questionnaire, transcripts of classroom conversations between students and teachers, and participant's journals.
Each teacher was asked to teach the geometric construction unit, normally taught using compass and straight-edge tools, using the Geometer's Sketchpad computer program. I spent approximately three weeks with each teacher, observing their interaction with students and the computer software.
All interviews and classroom visits were audiotaped and I made field notes of my observations. The teacher was asked to keep a daily journal to record his feelings, concerns, successes, failures and other teaching and learning experiences. Data collection was ongoing throughout the study. I identified myself as a researcher and did not instruct the teacher on how to teach this unit although I did assist them in selecting appropriate curriculum materials to match their current curriculum unit on geometric constructions.
The teachers were interviewed at least four times during the study: twice before the first classroom session, at the midpoint of the in-class sessions, and at the conclusion of the classroom visits. A follow-up interview was held when the transcripts were delivered to the teachers.
A questionnaire on teacher beliefs and attitudes was given to the teacher before the class sessions to provide me with additional information about the teacher. Additional questions were asked during interview sessions based on the responses to the questionnaire and on my field notes.
Three students in each school were questioned about their interest, attitude and feelings in the areas of mathematics courses, content of this course, computers, teachers of mathematics, geometry, the software, their freedom to explore, and their anxiety about mathematics.
Each Head of School was interviewed. Questions focused on the types of supports teacher's receive in the school in the areas of computer hardware, computer training, teacher professional development opportunities, and teacher change.
Analysis of the Case Studies
Cathy Karen Simon Mike No. of Years Teaching Math 11 3 3 14 Math Experience extensive little extensive extensive Computer Experience extensive limited extensive medium Math Exploration extensive none none extensive Transition to Taken-as-Shared integrated beginning transition beginning transition progressing Perceived Role of the Teacher
Facilitator Transmitter of Knowledge Provider of Information Provider of Information Perceived Role of the Teacher
Facilitator Transmitter of Knowledge Facilitator Student-focused Guide Control Needs
low high moderate low Control Needs
low moderate moderate low Control Techniques designing worksheets increased own proficiency evaluation of students designing worksheets Control of Technology comfortable nervous / concerned comfortable comfortable
Table 1: A Cross-Case Summary of the Four Teachers
Case Study 1 The 1995-96 school year was Cathy's seventeenth year of teaching. She works in an all-girls independent school in a large metropolitan city in Canada. Cathy had experience teaching in a government school in the Bahamas and eleven years at another all-girls, independent school in this same city before beginning to teach at St. Francis School six years ago.
The case of Cathy demonstrates that the need for control over the teaching environment is based on a personal philosophy towards instruction. Cathy's control over her environment required her to be organized and to develop her own classroom activities. Cathy permitted her students to explore the software and share in the formulation of definitions and the discovery of relationships.
Cathy is a reflective practitioner who, through written and mental practice, makes adjustments to her pedagogical techniques before, during and after each lesson.
Certain characteristics of Cathy's efforts to implement the geometry curriculum unit were noteworthy:
- Cathy is a reflective practitioner.
- Cathy is an organized teacher.
- Cathy felt comfortable in a "taken-as-shared meaning" (Wood, Cobb & Yackel, 1991) classroom setting.
- Cathy's loss of classroom control was temporary and quickly resolved.
- Cathy believes that student's becoming active independent learners is quite important.
Case Study 2
The 1995-96 school year was Karen's eleventh year of teaching. Karen works in a Grade K-12 independent school in a large metropolitan city. The school is a co-educational environment, attracting students from various areas of the city.
The case of Karen demonstrates that the perceived need for control over the teaching environment is an important issue in teacher change. Karen's perception of the role as a teacher created a tension between having a structured classroom where the meanings are teacher-directed and a flexible classroom where taken-as-shared meanings are formulated.
The variation of the teacher's need for and actual control over their teaching environment emerged from the data. Karen's transition from a traditional, structured and controlled environment to lack of control in the new environment and, finally, to maintaining a new type of control was a significant finding.
As I studied Karen's efforts to implement the geometry unit, several characteristics began to emerge:
- Karen was a structured teacher.
- Karen found it difficult to reflect, in written form, on her teaching practice.
- Karen was beginning to show some movement from teacher- imposed meanings towards a "taken-as-shared meanings" (Wood, Cobb & Yackel, 1991) with her students.
- Karen's perceived role of the teacher had a major influence on her instructional techniques.
- The degree of control Karen felt she needed in her teaching practice created a tension that was not resolved.
- Karen felt a lack of control over her use of the technology.
Case Study 3
My work with Simon revealed a number of characteristics:
- Simon is becoming a reflective teacher.
- Simon is a structured teacher.
- Simon is aware that, in the process becoming an advocate of using taken-as-shared meanings with his students, he was often very tense in the classroom from a more traditional approach.
- Simon uses evaluation of students to maintain control over his classroom environment.
Case Study 4
The 1995-96 school year was Mike's eighteenth year of teaching and his fifth year at Stevenson College. Mike teaches in a boys-only independent school in a large metropolitan city in Canada. Prior to moving to Stevenson College, he taught thirteen years at an all-girls independent school in the same city. Mike changed his view of the role of the teacher from that of a "provider of information" to that of a guide. He believes that Middle School students should have the same opportunities to explore and share experiences in mathematics as those currently afforded his senior students. Mike had allowed them to take control of their own learning. This new class of 'explorers' and 'experts' became the focus of Mike's use of the computer software. Throughout my study, certain characteristics of Mike's teaching practice were noteworthy:
- Mike has a flexible approach to classroom management.
- Mike is an organized teacher.
- Mike allows the students to participate in taken-as-shared meanings within his classroom.
- Mike's perception of the role of the teacher allowed for students to take a more active role in their own learning.
- There is a strong sense of exploration and discovery in Mike's teaching practice.
The issue of control permeated the entire study. The teachers wanted their students to learn the geometric relationships and they wanted to be able to evaluate how well their students grasped the concepts. Jaworski (1994) recognized the issue of control in her research. In the present study, how the teacher reacted to this control issue contributed to their perceived degree of success of the program. Teaching strategies are closely linked to classroom management strategies (Keller, 1996). Teachers in Keller's study noted that students are more on task and self-managed in a computer classroom than in the regular mathematics classroom. They also noticed that there was an increased noise level in the computer lab. These changes may not be directly linked to the use of the computer but as a result of other changes that were made to accommodate their use. Keller's teachers took more interest in their students' successes and allowed the students the freedom to explore learning materials. These changes were enacted by teachers 'letting go of the reins'.
All four teachers in the present study faced the issue of control. Karen, Simon, and Mike made explicit reference to their temporary loss of control while Cathy made implicit reference to this issue. Of the four teachers, Karen's experience was most significant and will be used to develop a new understanding of the issue of teacher control.
Karen perceived a significant loss of control early in the study. She felt uncomfortable with the software and felt that she was not in control of the learning environment. Karen's students, although
they were conditioned to ask and expect Karen to answer their questions, recognized this loss of control. Her students began to take responsibility for their own learning. They began to share conjectures and to help each other develop the skills necessary to explore geometric relationships using the software. Karen regained her sense of control by the third day of the study through her successful interaction with the software the previous evening. By this time, however, her students were feeling comfortable with the software and helping each other. By the sixth day, Karen regained her confidence in her ability to teach geometric constructions using the software. Both Karen and her students were transformed by the experience. The students felt in control of their own learning and Karen regained her sense of control over the learning environment and her confidence as a teacher.
Karen's experience is similar to the experiences Frobisher (1994) noticed happening to teachers when a problem-centred classroom diverges from the traditional model. The sense of insecurity that teachers experienced using the computer software is consistent with the experiences of the teachers in a problem-based mathematics classroom.
The issue of teacher control can be viewed from three perspectives: control-as-management of the learning environment, control-as-personal, and control-as-professional.
The most significant effect on the teaching pedagogy of the teachers involved in this study relates to their interaction with their students. The teachers were, to various degrees, inculturated into the traditional teaching paradigm where the teacher structures the classroom so that the teachers can be the authority. Berebitsky (1985) found that elementary school mathematics teachers have a low level of mathematical background. There are a number of problems inherent in this situation. Teachers are not confident in their mathematical ability and, therefore, the textbooks are taken as the authority for mathematics. Steffe (1990) suggests that the mathematical concepts and how they are taught seldom get questioned.
Teachers having low level of mathematics background or those who depend on the textbook as if it were the curriculum, tend not to respond favourably to suggestions that they teach in an exploratory mode. They perceive this environment as being too difficult of control and that it requires the teacher to tolerate uncertainty about what the students are learning (Schoefeld & Verban, 1988; Cohen, 1989).
The role of the teacher in mathematics education also influences the control mechanisms the teacher places on the classroom environment. When teachers chose various situations for their classroom, they make judgments about the relevance of the situation to their students and how likely the students are to "bump" into the appropriate mathematics in the course of investigating the problem (Lappan & Briars, 1992). These activities will vary depending on the level of control and the tolerance level a teacher has within the classrooms. These levels of control and tolerance levels may restrict the use of cooperative/collaborative learning activities used by the teacher. Johnston, Johnston and Stanne (1986) found that students working in cooperative learning groups had increased achievement within computer-based environments. Even with these findings, teachers may choose to have students work one to one with a computer simply to minimize the noise. In so doing, the teacher may inadvertently lessen the opportunities for students' discourse and shared meanings within a small mathematical community.
Teaching can be an isolated activity within many schools. The teacher is expected to teach around 30 students, maintain control, and inspire the class to learn (Cuban, 1986). Compound this problem by introducing a computer software tool and tensions develop between the teachers perception of their role within the classroom and their personal control needs for perceived control of the learning environment.
These personal control needs are expressed in many forms. The need for mutual trust within the learning environment between the students and the teacher, the need of being the authority within the mathematics classroom and the ability to freely admit mistakes are within this category. How a teacher perceives herself within the classroom and how the teacher reacts to personal, rather than professional, change has an impact on the degree to which change is accepted by the teacher.
Karen provides an interesting backdrop for the role personal control plays in an elementary school mathematics teacher's practice. While she had concerns over the content of the geometry program, she was most concerned with her personal control over her environment. She wanted to maintain a personal presence in the classroom and expects respect from her students. She felt that this respect was synonymous with her personal control within the classroom. Karen made some changes in her teaching practice. These changes, however, were closely related to her feeling comfortable with and in control of her personal acceptance of the need to share the authority of mathematics. She also felt comfortable to share her lack of complete understanding of the software with her students. It was at this point that she began to recognize her control over the teaching environment increase.
Simon, as a new teacher, had similar hopes in the classroom. He wanted students to recognize him as an individual. He was not concerned about making mistakes but he was unable to freely inform his class that mistakes were part of life. Simon's willingness to share his 'authority' with his students began the change process. He realized that he didn't have to know everything and that sharing knowledge with the students actually allowed his personal control to increase.
Mike and Cathy appear to be very comfortable with their role within and without the classroom. They have a personal interest in investigating mathematics and freely admit to their students that they make mistakes. They do not need to be the centre of attention and, perhaps as they have both taught for a number of years, felt confidence in their abilities to make changes within their teaching practice without creating a loss of personal control over their environment.
The issue of personal control is important to new teachers to the profession and those new to teaching mathematics. The need to be the centre of attention and to be the mathematics authority in the classroom does influence how a teacher reacts to change within their classroom. Students benefit from seeing teachers as evolving, learning members of the mathematics community. Rather than providing students with information and then determining if they have captured the concepts, knowledge and skills, teachers will need to become a part of a learning community and act as a model and a participant.
Prospective teachers enter a profession steeped in tradition and history. As a profession, teachers are well regarded in some communities and not in others and may experience some trepidation about their role within the community. Within the independent school system, teachers are usually well regarded for their hard work and dedication to the profession.
All four of the teachers in this study agree that there are many roles for the teacher within the classroom. They agree that being good in mathematics is important but not essential. The ability to motivate students is a key factor, according to Simon, while Cathy believes that teachers should ask questions to encourage students to explore mathematics. Both Cathy and Simon, by the end of the study, saw the teacher's role as that of a facilitator while Mike used the word guide to describe his role in a more student-focused classroom environment. Karen continued to believe that her role was to 'teach'. That is, she should provide an environment where she is the transmitter of knowledge to the students. In each case, the perceived role of the teacher dictated the types of questions posed, the distribution of the worksheets, and the interaction between teachers and students.
How a teacher perceives the role of the teacher will contribute to the type and degree of control used in the classroom. A teacher who believes that the teacher should be a facilitator will naturally maintain a different form of control over the classroom. A facilitator will have less difficulty with open-ended activities and will invite questions from the class that will be different in scope and depth than from a teacher who believes that students need to be told what to learn and under what conditions. The transmission-type teacher will be less likely to open the students to new questions and interaction, the building blocks of a mathematical community.
Teachers play a central role in establishing the mathematical quality of the learning environment for students and in establishing norms for mathematical aspects of students' activity (Yackel and Cobb, 1996). This implies that the teacher does not take a passive role in the constructive perspective but plays a critical role as a representative of the mathematical community. Given this central role, what influences come to bear on the role of the teacher in mathematics education?
The role of the teacher in mathematics education is influenced by teachers' individual mathematical agenda. When teachers choose various situations for their classrooms, they make judgments about the relevance of the situation to their students, judge how well the task represents the concepts to be taught, and how likely the students are to "bump" into the appropriate mathematics in the course of investigating the problem (Lappan & Briars, 1995). Steffe (1990) writes about "provocations" that are caused by the teacher that induces students to experience perturbations. These perturbations, according to Steffe, are normally connected to the actions of the teacher. Therefore, teachers can provide exploratory activities in their classrooms that will challenge the current understandings of the students.
The teacher's view of learning about mathematics and mathematics teaching clearly affects how teachers present the course material. The teacher needs to do more than just change the nature of the classroom task from teacher-directed to student-directed. Social constructivism implies that students need to communicate with each other. This communication could cause anxiety for teachers who feel that classrooms should be quiet, or that only one person should be talking at a time.
The use of the word 'control' conjures up different images for each of us. It is an emotional word that can be used negatively to suggest that the teacher is not giving students any freedom to develop their own thoughts (Jaworski, 1994). It can also mean that the students take responsibility for their own learning. Classroom control is important for teachers and is used to influence the way students think and behave within the classroom. Teachers use of their inherent control within the classroom will influence the type and form of activities that take place within the classroom. This control can be used to limit interaction between student by reducing the noise level to a minimum or nil and by insisting on individual work. However, as Jaworski found in one of her case studies, control can also create an environment in which mathematics thinking is fostered.
We need to develop a careful understanding of the settings that encourage teachers to learn to use these new teaching environments and materials. We need to determine the real costs of teachers learning to teach geometry. We also need to empower teachers to create an experimenting environment in their classrooms. Teachers need to be observed in computer exploratory environments so that we can determine their learning needs so they can provide this educational experience with their students.
Teachers experienced an initial loss of control in this environment. As the teachers gained confidence in their own use of the software and recognized that students were experiencing success, teachers began to regain their sense of control. The investigation also reveals that teacher control can be expressed in one of three categories: control by management, control by individual, and control by professional.
The implication for teacher education is that preservice and inservice teachers should be given a mentor or coach to reinforce the premise that, although the teacher will experience a temporary loss of control, increased confidence in mathematics and experience in other software packages, will be helpful for teachers attempting to introduce dynamic geometric software packages into their classrooms. The implication for mathematics education is that students thrive in dynamic geometric software environments when teachers maintain control over the management of learning, their own personal expectations, and their role as a professional.
Teacher control as a professional is a reality in middle school mathematics teaching. Teacher educators can assist teachers to maintain a level of control over their professional lives by providing them with the tools to be mathematical explorers. Teachers need to be placed in learning environments where they can explore mathematics, interact with their peers though discussion and case studies, and work with dynamic computer environments. These dynamic computer environments provide an environment where teachers and students can interact and share their conjectures and findings with each other. Teacher educators should provide opportunities within their curriculum for teacher exploration in these computer-based tools.
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