Challenges facing pre-service secondary teachers.

Abstract

The development of a sense of community is important in secondary classrooms if students are going to be able to explore and develop sophisticated mathematical reasoning. This longitudinal study was conducted over eight years and examined the kinds of challenges that pre-service secondary mathematics teachers face as they enter the practical realm of a secondary classroom. Analysis led to four categories of challenges and suggested topics that need to be addressed in methods courses and monitored during student teaching.

Introduction

The National Council of Teachers of Mathematics (NCTM, 1991) crafted a new vision for secondary mathematics classroom in which teachers help students work together to make sense of mathematics and independently determine whether something is mathematically correct using mathematical reasoning. These classrooms are described as mathematics learning communities and help students learn to conjecture, invent, and solve problems. Researchers (Arbaugh, 2003; Lampert & Ball, 1998; Grossman et al., 2001; Olson & Kirtley, 2005; Pepin, 1999) investigated different types of supports theorized to help teachers develop mathematical communities.

Arbaugh (2003) examined the use of study groups with in-service secondary geometry teachers. Results of this research indicated that the teachers felt support in several areas including: (a) building a mathematical community, (b) creating professional relationships, and (c) making connections between theory and practice. Small groups of pre-service teachers gained content and pedagogical knowledge by analyzing student work and video-tapes of themselves teaching (Lampert & Ball, 1998). Grossman et al. (2001) underscored the importance of teachers working collaboratively to deepen their understanding of content because it would be unreasonable to "expect teachers to create a vigorous community of learners among students if they have no parallel community to nourish themselves" (p. 993). These studies indicate that collaborative work among secondary mathematics teachers supports educational reform. However, Pepin (1999) found that institutional and societal constraints influenced teachers' pedagogy and interpretation of students' responses. Olson & Kirtley (2005) also suggested that helping secondary teachers implement reform practices can be challenging. They found that secondary and elementary teachers experienced cognitive dissonance when they collaboratively solved non-routine problems using multiple representations. This cognitive dissonance provoked a secondary teacher to re-examine her beliefs about teaching and learning. This re-examination led her to change her beliefs and work toward establishing a mathematical community within her classroom in which students questioned each other and posed conjectures.

Research on the professional development of secondary teachers is sparse and limited to a case-study design. The development of mathematical communities is theorized to be a critical component in secondary mathematics classroom to increase student achievement. Research is needed to investigate how the development of a sense of community among secondary teachers might transfer into their own classrooms to help them create an environment in which students explore and develop sophisticated mathematical reasoning. To investigate this phenomenon within the realm of pre-service secondary mathematics teachers, a longitudinal study was conducted over 8 years to examine the kinds of challenges that pre-service secondary teachers face as they confront curricula defined by a textbook and students accustomed to listening rather than thinking for themselves. Specifically, this study sought to identify challenges that pre-service teachers face as they move from the theoretical realm of professional education courses to the practical realm of the secondary mathematics classroom.

Theoretical Perspective and Methods

The apprenticeship model described by Lave and Wenger (1991) as legitimate peripheral participation portrays learning as an enculturation of novices into a way of thinking and doing that reflects the practices of an expert. Novices learn by watching, listening, doing, and making mistakes while under the tutelage of experienced practitioners. These experts guide and support the newcomers as they struggle to acquire skills and knowledge necessary for success. As novices gain skills and competencies, they may invent new practices which in turn can have an impact on the skills and practices of the expert. In this study, Hartter assumed the role of an expert who was a researcher/participant as she supervised and mentored pre-service teachers. The pre-service teachers were novice teachers and their learning community included practicing teachers, their students, and other mathematics pre-service teachers. The pre-service teachers gathered weekly to discuss the challenges that they faced. Hartter was an outside expert who closely observed pre-service teachers teach in classrooms and interact with other pre-service teachers during the discussions.

To investigate the challenges that pre-service teachers face as they move from the theoretical realm of professional education courses to the practical realm of the secondary mathematics classroom, four sets of data were collected: field notes of classroom observations, reflection of the observed lesson, field notes of pre-service teachers' discussions, and reflective notes made by pre-service teachers. These data were analyzed using constant comparative analysis to categorize the types of situations that arose during teaching that caused a challenge for pre-service teachers to interpret or make an instructional decision.

The pre-service teacher participants had previously completed a content methods course taught by Hartter and were fulfilling their student teaching requirement for licensure at a Midwestern university. Hartter supervised the pre-service teachers using a cognitive coaching model. Prior to each observation, the lesson plan was discussed. Then, Hartter took field notes while observing the class. After the lesson, Hartter wrote a reflection on the implementation of the observed lesson and the pre-service teacher wrote a reflection addressing the following questions:

What surprised you about this lesson?

What went well during this lesson?

What were you disappointed about?

What would you change if you could re-do this Lesson?

When your students left class, did you have a good idea of how well they understood the material you presented?

Results

Four basic categories emerged from the analysis: (a) Challenges regarding the need to "cover material;" (b) Challenges surrounding teaching philosophies of the pre-service teacher and the cooperating teacher and/or university supervisor; (c) Challenges related to questioning; and (d) Challenges addressing student thinking. In the following four sections, these challenges will be presented with illustrative examples. Then, the findings will be discussed with regard to the development of a community of learners and implications for mathematics methods courses.

Challenges regarding the need to "cover material"

Pre-service teachers often implement a completely different lesson when facing the actual classroom in spite of careful planning for interactive lessons and group activities. For instance, one pre-service teacher had planned a differentiated lesson during which the students would move to each of six different stations in order to review some basic properties of integers, such as the associative property and the additive identity. When implementing the lesson, the pre-service teacher simply reviewed these properties by writing each property on the board and providing an example; no student-student interaction took place and no learning stations were used. During the discussion between the researcher and the pre-service teacher, she stated, "Well, I knew that I needed to get this stuff covered; IStep [Indiana Statewide Testing for Educational Progress] review starts Monday." Another reason provided after a different lesson taught by another pre-service teacher was, "I still had three standards to cover before the exam coming up. I knew I couldn't take the time to do the group activity today." Pre-service teachers do not often realize that several standards might be addressed by a single activity or project; they believe that each standard must be individually addressed.

In contrast, other pre-service teachers showed no evidence of planning. They used the textbook as the sole source and carefully copied the solution strategy for the provided examples. Based on discussion with these pre-service teachers, it is apparent that they believe that "the textbook has the best plans for the way the material should be taught. After all, it was written by a teacher; why should I plan fancy lessons and group activities if this is what worked for them?" The lack of planning does not allow for any thought about alternative teaching strategies or special approaches for students who may have trouble understanding the material. In spite of discussions that have taken place within content methods courses, these pre-service teachers still believe that there is "one right way" to teach each mathematical concept.

Challenges surrounding teaching philosophies

Another challenge is one which arises due to the difference in teaching philosophies between the cooperating teacher and the pre-service teacher. If the cooperating teacher does not believe that cooperative learning is effective or that alternative assessments are valid, he or she will probably not be willing to allow the pre-service teacher to try any alternative teaching strategies. This produces a conflict between what the pre-service teacher has experienced in the content methods course and what the cooperating teacher expects. It is also difficult for the pre-service teacher to learn to choose appropriate teaching strategies for at-risk students. "My teacher said that I have to teach it this way; I even have to count the answer wrong if the student doesn't do it the teacher's way." Another pre-service teacher reported, "I have to use his (cooperating teacher's) lesson plans. I'm not allowed to change anything--not even the examples."

In other instances, cooperating teachers or university supervisors do not see the need for providing mentoring or feedback for the pre-service teacher. "My university supervisor has never observed a complete lesson; she's been in the room for 10 minutes and writes elaborate feedback based on those 10 minutes. She really has no idea about the actual learning environment." "My cooperating teacher has never said anything to me about my lessons. When I asked for feedback, he simply left the room and never stayed in the classroom again during any of my lessons." If one espouses the apprenticeship model described earlier, one would certainly argue that this is not the best way to help the pre-service teacher develop a learning community nor does it help the pre-service teacher learn to make better instructional decisions.

Challenges related to questioning

Perhaps the most difficult challenge to overcome is the challenge of asking appropriate questions to elicit student thinking. Pre-service teachers must become aware of the variety and breadth of intention behind classroom questions. Driscoll (1999) states that any effective lesson should use a blend of questions types: (a) managing; (b) clarifying; (c) orienting; (d) prompting mathematical reflection; and (e) eliciting algebraic thinking. Good questions are those which extend the students' thinking about a problem. Rarely do pre-service teachers pose challenging questions such as "compare and contrast the circumference and the area of a circle." More often, the pre-service teacher asks: "what is the circumference of a circle?" Or "what is negative 5 times negative 4?" is asked rather than "why is positive 20 the correct answer for the product of negative 5 and negative 4?"

Even the most basic question regarding understanding of material presented is addressed as: "Are there any questions?" as opposed to the better question "What questions do you have?" Most of the questions asked by novice pre-service teachers are those at the most basic level of Bloom's Taxonomy (1956) and are those which focus on procedural knowledge rather than conceptual knowledge. Many times, good questions are actually written in the lesson plan. For example, one pre-service teacher included the question "Why would it be more appropriate to use the law of cosines in solving this problem than using the law of sines?" During the actual implementation of the lesson, the pre-service teacher simply skipped over this question and told the students what to do. During reflection and discussion with the pre-service teacher, the reason provided was "I was just afraid that nobody would answer."

Challenges addressing student thinking

The Professional Standards for Teaching Mathematics (NCTM 1991) suggests that teachers need to do more listening and students need to do more reasoning. "Students should engage in making conjectures, proposing approaches and solutions to problems, and arguing about the validity of particular claims" (p. 45). Students must learn to assume responsibility for their learning. In order to facilitate students' learning, classroom discourse must be developed and maintained (Johanning & Keusch, 2004). By maintaining a classroom filled with discourse, pre-service teachers can gain invaluable insight on student thinking. However, without the ability to ask appropriate questions, pre-service teachers learn little about what their students are actually thinking and what they actually understand about the mathematical concepts that are presented.

In addition to asking questions in order to elicit student thinking, pre-service teachers also have difficulty even understanding and interpreting the questions that students ask or the comments they might make in response to questions. For example, during a lesson on linear equations, the pre-service teacher was graphing the line y = 5x--4, using the slope of 5. A student asked if you could also use 4/1 as the slope. The pre-service teacher did not understand that the student really thought that either 5 or 4 could be used for the slope of this line and thought she was asking whether a line could have a slope of 4. The pre-service teacher proceeded to answer that "yes, a line could have a slope of 4" and graphed that new line. It is difficult for novice teachers to actively listen to each student and respond appropriately. Many times, they are too concerned about classroom management issues which may arise and are afraid to veer from their plans or the textbook.

Discussions and Implications

As reported in the results above, pre-service secondary mathematics teachers do face challenges other than classroom management as they move from the theoretical realm of professional education courses to the practical realm of the secondary mathematics classroom. As novice teachers, they have neither the experiences nor the expertise on which to draw in order to anticipate or react to these challenges. Some of these challenges may be found to actually prevent the pre-service teachers from implementing reform recommendations which they discussed at length in professional education courses. Depending upon the degree of collaboration among the community of learners in which they are working, the pre-service teachers may not even recognize the challenges they face without discussion with an observer/mentor or their peers.

In order to gain insight for making the instructional decisions necessary and for recognizing and addressing challenges they face, pre-service teachers need the support of a community of learners. If this support is not present in the school learning community in which the pre-service teacher is student teaching, it must come from an external community of learners--consisting of fellow pre-service teachers and a faculty mentor within the area of mathematics. As prior research with other constituents (Lampert & Ball, 1998) suggests, the awareness of such challenges might best be addressed by careful analysis of student work and self-analysis of video-tapes of actual lessons. This analysis could begin within the content methods course, thus helping the pre-service teacher anticipate issues that may be problematic prior to student teaching. In this way, pre-service teachers would reflect on these challenges in the presence of a supportive learning community and gain skills and knowledge necessary to create a supportive learning community within their own classrooms.

References

Arbaugh, F. (2003). Study groups as a form of professional development for secondary mathematics teachers. Journal of Mathematics Teacher Education 6(2), 139-163.

Bloom, B., Englehart, M., Furst, E., Hill, W., & Krathwohl, O. (1956). Taxonomy of educational objectives: The classification of educational goals: Handbook 1. The cognitive domain. White Plains, NY: Longman.

Driscoll, M. (1999). Fostering algebraic thinking: A guide for teachers grades 6-10. Portsmouth, NH: Heinemann.

Grossman, P., Wineburg, S. & Woolworth, S. (2001). Toward a theory of teacher community. Teachers College Record, 103(6), 942-1012.

Johanning, D. & Keusch, T. (2004). Teaching to develop students as learners. In

Rubenstein, R. & Bright, G (Eds.) Perspectives on the teaching of mathematics. Reston, VA: NCTM.

Lampert, M. & Ball, D. (1998). Teaching, multimedia, and mathematics: Investigations of real practice. The practitioner inquiry series. New York, NY: Teachers College Press.

Lave, J. & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. New York, NY: Cambridge University Press.

National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics. Reston, VA Author.

Olson, J. (2004). Evoking pedagogical curiosity: A coaching approach to support teacher's professional growth. The Mathematics Educator, 8(4), 84-94.

Olson, J. C., & Kirtley, K. (2005). The transformation of a secondary mathematics teacher: From a reform listener into a believer. In H. L. Chick & J. L. Vincent (Eds.), Proc. 29th Conf. of the Int. Group for the Psychology of Mathematics Education (Vol. 4, pp. 25-32). Melbourne: PME.

Pepin, B. (1999). Epistemologies, beliefs, and conceptions of mathematics teaching and learning: The theory, and what is manifested in mathematics teachers' work in England, France, and Germany. TNTEE Publications, 2(1), 127-146.

Beverly J. Hartter, Ball State University, IN

Jo Clay Olson, University of Colorado at Denver and Health Sciences Center, CO

Hartter, Ph.D., is Professor of Mathematics with a focus on secondary education, and Olson, Ph.D, is Professor of Mathematics Education.