Beginning secondary mathematics teachers: a snapshot across one state.

Introduction

The National Council of Teachers of Mathematics (NCTM) has called for reform in the teaching of mathematics (1989; 2001). Beginning secondary mathematics teachers are being educated in mathematics methods courses to teach by NCTM-oriented standards. Yet, we know little as to how beginning mathematics teachers teach or under what conditions they teach. Ball (1996) called for us to look more critically at how mathematics teachers teach, and questioned if we really know what we think we know about mathematics teachers.

Literature Review

Not many studies directly research the teaching practices or conditions of beginning mathematics teachers at the secondary level (LaBerge and Sons, 1999). Some studies that exist examined particular elements of slightly more experienced teachers. Wohlhuter (1996) studied the decision-making of geometry teachers with at least 3 years of classroom experience. Factors influencing teachers' decisions included past geometry experiences, professional development experiences, articulated course goals, advanced planning decisions, teachers' beliefs, the geometry textbook and other materials, teachers' school settings, and students' needs and actions. Teachers were also influenced by whether they viewed the process of becoming an effective teacher as a life-long process.

Another study examined the qualifications of current mathematics teachers, as well as their work environments and how they continue to learn (Lewis, Parsad, Carey, Bartfai, Farris, and Smerdon, 1999). The study examined teacher preparation and some aspects of the teaching environment among 3,560 teachers whose experience ranged from one to thirty years. They found that less than 50% of the teachers felt that they were "very well" prepared for teaching by their undergraduate experience. Virtually all of them participated in professional development once they were teachers. Approximately 19% had mentors, and most believed that they had the support of their fellow teachers, but not of parents.

Student teaching was the setting for other studies that looked at mathematics teaching. Stahlhut (1992) showed that student teachers are quite dependent on the mathematics textbook and tend not to use technology. Valli and Agostinelli (1993) examined the teaching of a high school mathematics teacher both before and after student teaching. In this case study, the authors described a teacher whose teaching quickly became like that of an expert teacher. The authors suggested that "his professional preparation facilitated and reinforced movement toward good classroom management, sound planning, professional relations with students, and instructional strategies conducive to student learning" (Valli and Agostinelli, 1993, p. 117) and suggested that to improve the quality of mathematics teachers, teacher educators should increase the connection between academic courses and the experience of student teaching.

Skott (2001) directly studied the practices of a mathematics teacher from his student teaching experience through the first eighteen months of his career, concentrating on the relationship between the teacher's beliefs about mathematics and his classroom practices. The researcher found a strong relationship between beliefs and practices, but a very complex one as well. For example, although the reform movement greatly influenced the teacher, the complexities of how the teacher acted out the influence differed based on the classroom reality, including the role the students were playing at any given time. Along this same line, the role that the teacher played at any given time determined which beliefs would dominate.

Adams and Krockover (1997) found that mathematics teachers with less than 2 years experience were concerned with class assignments, curriculum development, time management, classroom management, and presentation of the content. In addition, these mathematics teachers thought that their content course work while earning a teaching degree was sometimes too specific; some of the pedagogical coursework was of limited usefulness; and there should be more field experiences provided.

LaBerge and Sons (1999) studied how the NCTM standards influenced how first year mathematics teachers taught. This study found that first year mathematics teachers were concerned with "keeping order and control in the classrooms (LaBerge and Sons, 1999, p. 151). In addition, these first year teachers most commonly used "traditional activities (working on exercises from the textbook and taking notes while the teacher talks)" as the instructional method (p. 151).

The current study expands the literature by examining the roles of beginning mathematics teachers, the roles of the students of beginning mathematics teachers and what influenced beginning mathematics teachers' development as teachers. A statewide group of mathematics educators designed a project to see what and how beginning mathematics teachers taught.

Research Questions

The project's broad research questions were:

* What are new mathematics teachers' current practice, knowledge, and beliefs about teaching mathematics?

* What are the conditions in which new mathematics teachers teach mathematics?

This article presents partial results of the study and focuses on a subset of these questions.

* What are the roles of beginning mathematics teachers?

* What are the roles of the students as perceived by the beginning mathematics teachers?

* What influenced beginning mathematics teachers' development as teachers?

These three questions generate a snapshot of what occurs in beginning teachers' classrooms. Comprehending how teachers see themselves and their students act out their roles, as well as identifying what influences their conditions may generate a context for better understanding of beginning mathematics teachers' practices and conditions.

Methodology

SciMath[.sup.MN] is a statewide, public/private partnership that advocates and supports standards-based systemic improvements in the teaching and learning of K-12 science and mathematics. A suborganization within SciMath[.sup.MN] is Transforming Teacher Education, a program to encourage revitalization of programs to prepare teachers for lifelong teaching and learning. Teams of college/university faculty from ten undergraduate institutions in Minnesota have been working to examine the best practices nationally and then looking inward to their own programs. This article reports on data gathered by a team (The Teacher Research Network) consisting of invited mathematics educators from each of the undergraduate institutions that have a secondary mathematics education program.

Each member of the research team recruited two teachers that fit the guidelines (mathematics teachers within the first three years of teaching). The research network developed or revised several instruments to observe, interview, and survey new teachers of mathematics resulting in seven instruments to collect data. (The interested reader should see Davis, Simpson, Johnson, and Wallace, 2002 for more information on the research instruments.) The instruments gathered demographic data, mathematics teaching beliefs data as well as information on what is happening within a mathematics teacher's classroom. Each researcher followed a given protocol using the research instruments per each year of involvement. Students in the teacher's class filled out a survey describing his/her classroom. Teachers filled out demographics forms, a learning/teaching styles survey, a teaching beliefs survey, and pre- and post-observation forms. Researchers observed each teacher twice and interviewed the teachers. Each researcher analyzed the data and prepared a profile for each teacher.

One of the researchers analyzed the set of profiles. (Note that the analysts for the two years of data being reported on in this paper were the co-authors.) A protocol was carefully followed with categories of knowing content, knowing students, knowing pedagogy, establishing a learning environment, and developing professionally. Two other project researchers reviewed the analysis and provided feedback to the analyst. The analyst then made changes to the analysis. Finally, the entire Teacher Research Network held large group discussions. Thus, the data collection and analysis process included multiple formats (paper-and-pencil surveys, interviews, and classroom observations) and involved multiple researchers reviewing the procedures.

Sixteen secondary mathematics teachers participated in the project. Some subjects were one-year participants and others were two-year participants. All subjects were either preservice teachers who were student teachers or licensed secondary mathematics teachers within their first three years of teaching. None of the subjects taught at the same school. The ethnicities of the teachers' students were predominately white, with some presence of African Americans, Asian Americans and Native Americans. Demographic information is given in Table 1.

Results

The analyzed data generated a snapshot view of beginning mathematics teachers' views and practices of teachers' roles and students' roles. The discussion of what our teachers believed contributed to their development as a teacher and provided a platform for better understanding teachers' views and practices. In addition, the discussion may contribute to the question of what can be done so that beginning mathematics teachers stay in the profession.

Teachers' Roles

At least one teacher specifically mentioned or researcher clearly observed teachers' roles as provider of opportunities, provider of content, facilitator (coach), classroom manager, and assessor (gradekeeper). Descriptions of these roles and a brief discussion of situations that constrained teachers in carrying out their roles provide the focus of this section.

The teachers all saw one of their roles as the provider of opportunities. Teachers showed considerable variety when conducting mathematics lessons. One teacher guided an exploration in a computer lab while several guided explorations in the regular classroom. Several teachers included the use of graphing calculators. At least three times, teachers lectured followed by problem-solving as a class. Also, several times, teachers had students work on small group activities.

In providing opportunities, teachers acknowledged that students learn differently and were concerned that the opportunities differ. "I feel the students will learn well by hearing, seeing, and doing the new material. So many areas need to be touched in order to provide a learning method for each kid." One teacher expressed this by saying, "I try to find activities and problem-solving things that would meet the needs of all students. Every day in here, and I think you saw that, I have some students trying things at a fairly high level and they weren't just trying to get through it." Many teachers tried to use students' life experiences to facilitate all students learning mathematics, "I try to think about where is a seventh grader at in his life experiences, and try to intersperse things like sports and music." One teacher remarked that I think math is one of the only classes where you actually get to problem-solve and figure things out, and that is an essential role of what math is." Teachers often moved beyond simple manipulation of numbers or procedures. One teacher's thoughts summarized many of the teachers' reasons for using these activities. "The whole point is not to tell them. The whole point is for them to figure it out ... which goes back again to problem solving...." Another teacher, however, remarked that engaging students in problem-solving activities takes too much time.

The teachers did view themselves as providers of content, as well. The 16 teachers most often (but not always) were accurate in their mathematical content knowledge. Many were able to ask questions of students that forced the students to rethink and restate their conclusions accurately when the original conclusions were inaccurate. Yet, several teachers made mathematical mistakes. Some were unable or unwilling to let students explore, but provided a procedure for solving problems so that there was one correct method and one correct answer.

Teachers saw their roles as more than just presenter of information. Many teachers viewed themselves as a facilitator, or a coach. One teacher stated, "Sometimes you kind of have to be a coach.... If you're doing group work, the teacher should be cycling around, this is your coach mode, and make sure they're on task." Another teacher described this as "If I see they are really going and understanding, then I quit talking, and walk around the room, and watch the thing evolve. This one talking to this one, and this one helping, that one, then they become the teachers ... then my role, what I try to do, is just assure them that they are on the right track, or try to point them in another direction."

Teachers were also classroom managers, which included creating a comfortable classroom atmosphere. "It's my personal philosophy to make sure that the students feel comfortable and safe in my classroom. I don't want any of my students feeling like they don't belong, that they are stupid, that they are never going to get it." Another teacher stated that she wanted to "make them feel wanted (a feeling most lacked)." Students need to be kept on track and to behave. "One of the very biggest parts of my job is teaching kids how to get here, get here on time, have their materials ready ... how to behave in class, how to talk respectfully to adults and other children." The teachers seemed to manage well by walking around the room, being aware of students, modifying lessons, establishing rapport, and being respectful and demanding respect. Only one of the 16 teachers was not able to handle the class well.

Finally, teachers thought of themselves as assessors (or gradekeepers). It appeared that all teachers used homework and tests/quizzes to assess. At least six teachers used projects to assess (e.g., posters). At least three teachers informally assessed students multiple times during the class period. Assessment methods included students working on the blackboard, calling on students, directly asking students if they understand, or what they think the answer is, or just observing students to see if they look like they understand. One teacher stated that "Class discussion gives me an immediate assessment of students' understanding of the material."

All of the teachers were often constrained by outside curriculum decisions, standards, and frameworks. These constraints removed much of the freedom from teachers to present the mathematical content of their choice in the manner of their choice. This loss of freedom also caused the teachers to be less able to carry out their roles as they defined them. One teacher commented that 90% of his decisions on what to cover came from the textbook and that he had been told to cover the book. Another teacher remarked that she tried to incorporate problem-solving into her course, but content choice was mainly determined by the text which emphasized routine procedural skills. She felt as a new teacher she must "cover" that material before introducing other content and activities. Additional constraints included the pressure felt to have high student performance on exams and high student future ability to perform in college. Although some teachers stated that they made decisions about mathematics content based on their students, they did so within these constraints. Still, teachers used feedback on assessments to determine which content and how the content should be addressed.

Students' Roles

Although teachers described students' roles in various manners, all descriptions came down to one broad term: active participant. Engaging in mathematics seemed to include communicating mathematically, participating in activities, and asking appropriate questions. The teachers tended to describe students' roles in general terms (e.g., "to learn, to enjoy mathematics and to understand a little more about mathematics so they can go on in math, to do well in math in high school, or go to college and do well there"), but when pursued, teachers gave more specific answers (e.g., "paying attention, taking notes, and doing their assignments," "describe findings," "making conjectures," and "enter into discussions"). One teacher succinctly stated the apparent thoughts of the teachers: "The student's role is to learn, whether by watching, listening, asking, writing, doing, explaining, or all of the above."

Students were responsible for entering the process of learning mathematics and for communicating mathematically. The teachers viewed mathematics as a subject for all students and indicated that mathematics should be learned by all. Most of the teachers worked to do things that would indeed include all students so that no student was excused from his/her roles. However, some of the teachers again felt they could not always make the decisions that would result in mathematics for all due to curriculum standard constraints. For example, at one point a teacher told the researcher that she knew a certain activity was too hard for some of her students, but she could not afford the time it would take to make the activity more understandable. Rather, she conducted the activity and allowed some students to remain confused. Nevertheless, she could state that she had "covered" that particular curricula standard.

Mathematical communication included the process of students participating in oral discussions. Some oral discussion was limited to students' answering questions and identifying the next step in a process (or problem). Other oral discussion included students talking with each other in groups and presenting ideas to the class. Communication through forms of representation was seen in the use of numerical tables, graphs, symbols, and two- and three-dimensional models. Students also had to communicate their mathematical reasoning. For example, lessons included students working in groups to determine a process for graphing equations of a line, students making conjectures based on collected data and on numerical patterns, and students focusing on developing their spatial reasoning skills as they worked with two- and three-dimensional models. Teachers encouraged students to use a variety of approaches and established expectations that there would be different solutions from different groups and that they would be asked to support their choices.

Although teachers indicated that active participation meant students engage in activities, it was not always easy to tell what this meant. The definition of activities itself seemed to be rather broad, but included the physical (or hands-on) involvement of a student for the purpose of learning a concept. Yet, it was not always a particularly extensive involvement; one teacher said that involvement included anything "where they're not just sitting listening to me." Yet, it is interesting that many of the teachers considered students listening to lectures or reading the textbook as involvement, but emphasized that balance must be used. Only 4 of the 16 teachers used lecture predominantly and each of them expressed wanting to use other formats. Even though lectures were used, much of the lectures still involved students. "Even on lecture days, I would say it was probably 60% me to 40% student discussion."

The degree to which the student's role was passive versus active differed in the classrooms. At times, all teachers expected the students to be somewhat passive as they listened to the teacher, for example. At other points, students were partly responsible for directing classroom discourse by asking questions or even setting the direction of the discussion. Many of the teachers' words agreed with one teacher's words: "if you're doing group work, be part of the group, don't just sit and watch things happen, make them happen." A few teachers expressed lack of knowledge in how to use groups well.

Development as a Teacher

The teacher role and student role descriptions provided insight into the beginning teachers' classroom. Ideas that affect how individuals develop as teachers offer additional information about how teachers decide on these roles. When teachers were directly asked what contributed to their professional development they all named past teaching experience whether this was student teaching, practicum experiences, summer teaching, tutoring, or the first year or two of teaching. All but one of the teachers named their university courses as helping them to develop as a teacher. About a third of the teachers named life experience, past work experience, or experiences as a parent as fostering their development as a teacher.

Many of the teachers stated that talking with other teachers was a factor that influenced their own development as a teacher. Two teachers indicated attending professional development opportunities, such as conferences, were factors. Other items that influenced teachers' development included enthusiastic teachers from the past, love for mathematics, recognition of the need for mathematics in the world, love of learning, family members, school administrators, and input from students.

By the examples teachers gave, it was clear that their development affected their decision-making about what goes on in the classroom. For example, when teachers talked about their development, they related it to how they teach now. Past teaching experience is the greatest influence on future teaching experience. In other words, the teachers were able to be reflective about their teaching and were interested in the reflection influencing what they did next. Another example is that when teachers talked with other teachers or attended conferences, it was for the explicit purpose of improving their own teaching.

Conclusions

A conservative conclusion might be these beginning mathematics teachers at least consistently tried to teach in NCTM-oriented manners. Teachers saw and demonstrated their roles as being provider of opportunities, provider of content, facilitator (coach), classroom manager, and assessor (gradekeeper). The teachers were most often but not always mathematically correct in their teaching. They demonstrated a variety of knowledge in pedagogy, as they expected their students to be active participants in their learning of mathematics.

This first conclusion is in line with previous research that also found teachers in alignment with NCTM-standards (Skott, 1998) and offers hope when compared to previous research which found "that more traditional activities (working on exercises from the textbook and taking notes while the teacher talks) are still the most commonly used instructional methods in mathematics classrooms" (LaBerge and Sons, 1999, p. 151).

When the teachers were not able to teach in manners consistent with NCTM, they often felt quite constrained by outside influences such as state curricula requirements. As a result, they were not always making decisions about their roles in the manner that they would have liked to. This observation may include their dependence on the textbook (Stahlhut, 1992) but seems to go beyond this to a dependence on outside influences. Regardless, secondary education programs should look closely at this issue of constraints and address it in the process of educating future teachers.

Still, in a positive manner, beginning mathematics teachers attributed their professional development to methods courses as well as to their past experience. Their beginning professional development was closely tied to how they make decisions about teaching. At least in the beginning years, these mathematics teachers were spending time reflecting on their professional development and finding ongoing methods to continue to develop (e.g., attending conferences, mentoring, searching for further resources, continued involvement with NCTM). Since this study reveals that these beginning teachers were open to ongoing professional development, this, too, is an area which our secondary education programs could have greater influence.

This study started to answer the "question" of how and what beginning teachers taught. The results were based on various methods of data collection including observations. Observing subjects twice or four times generated a beginning look at teachers' practices, knowledge, and beliefs. Future studies should include visiting the teachers' classrooms more often. In addition, we looked at a subset of the possible questions of interest by looking at teachers' and students' roles. Examination of other aspects of the classroom (e.g., How are the physical aspects of the classroom managed? What do the problem-solving activities look like?) will provide a more detailed picture of the mathematics classroom.

As we better understand the beginning teacher's environment, we can better address concerns such as those raised in the Glenn Commission Report, Before It's Too Late. The Glenn Commission set forth three goals, one of which is "Improve the working environment and make the teaching profession more attractive for K-12 mathematics and science teachers" (U.S. Department of Education, 2000, p. 9) so that not so many beginning mathematics teachers drop out of teaching. It is not likely one will make improvements if one does not know what improvements are necessary. For these reasons, it is important that we continue to understand the environment of our beginning mathematics teachers.

Table 1. Demographic Information

A                               B                 C

One-year      Two-year          Subjects taught   School type
observation   observation
5 preservice  1 preservice/1st  7th grade math     3 private
3 1st         2 1st/2nd         8th grade math    13 public
1 2nd         1 2nd/3rd         algebra
3 3rd                           remedial math
                                Core-Plus         10 urban
                                calculus           6 rural

REFERENCES

Adams, P. E., & Krockover, G. H. (1997). Concerns and perceptions of beginning secondary science and mathematics teachers. Science Education, 81(1), 29-50.

Ball, D. L. (1996). Teacher learning and the mathematics reforms: What we think we know and what we need to learn. Phi Delta kappan, 77, 500-508.

Davis, G., Simpson, P., Johnson, B., & Wallace, A. (2002). Getting to the fourth year: The instruments and protocols used to study the practice of beginning k-12 science teachers. Proceedings of the 2002 Annual International Conference of the Association for the Education of Teachers in Science. Retrieved June 1, 2002, from http://aets.chem.pitt.edu

LaBerge, V. B. & Sons, L. R. (1999). First year teachers' implementation of the NCTM standards. Primus, 9, 139-156.

Lewis, L., Parsad, B., Carey, N., Bartfai, N., Farris, E., & Smerdon, B. (1999). Teacher quality: A report on the preparation and qualifications of public school teachers. National Center for Education Statistics, US Department of Education: Washington D.C. (ERIC Document Reproduction Service No. ED 427 009)

National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics. (2001). Principles and standards. Reston, VA: Author.

Skott, J. (2001). Emerging practices of a novice teacher. Journal of Mathematics Teacher Education, 4, 29-53.

Stahlhut, R. G. (April, 1992). Math student teachers: How well prepared are they? Paper presented at the national conference of the National Council of Teachers of Mathematics, Nashville TN.

United States Department of Education. (2000). Before it's too late: A report to the nation from the National Commission on Mathematics and Science Teaching for the 21st Century. Author: Washington, DC.

Valli, L., & Agostinelli, A. (1993). Teaching before and after professional preparation: The story of a high school mathematics teacher. Journal of Teacher Education, 44, 107-118.

Wohlhuter, K. A. (1996). Decision-making in a mathematics reform context: Factors influencing geometry teachers' planning and interactive decisions. Unpublished doctoral dissertation, Oregon State University.

Carmen M. Latterell and Kay A. Wohlhuter with the Teacher Research Network (1)

(1) The Teacher Research Network is a suborganization of SciMath[.sup.MN]. The specific researchers include the authors and the following: Cyndy Crist, SciMath[.sup.MN] higher education project director; George Davis, Minnesota State University Moorhead and Patricia R Simpson, St. Cloud State University, Teacher Research Network co-directors; John Bauman, College of St. Scholastica; David Cline, Saginaw Valley State University; Alice Mae Guckin, College of St. Scholastica; Lynn Hartshorn, University of St. Thomas; Jean Hoff, St. Cloud State University; Bruce Johnson, University of Arizona; Michele Koomen, Gustavus Adolphus College; Robert McClure, St. Mary's University; Jeff Pribyl, Minnesota State University-Mankato; Lon Richardson, Southwest State University; Teresa Shume, Minnesota State University Moorhead; Chery Takkunen, College of St. Scholastica; Tom Tommet, University of St. Thomas; Dorrie Tonnis, West Bend, Wisconsin; and Alison Wallace, Minnesota State University Moorhead.