exploring beliefs and practices of teachers of secondary mathematics who participated in a...

Exploring Beliefs and Practices of Teachers of Secondary

Mathematics who Participated in a Standards-Based

Pre-Service Education

Overview of Presentation

Introduction and Definitions Review of Literature Design of the Study Results of the Study Summary

Introduction

Definitions

"Standards-based practices" refers to teaching practices that are based on the guidelines of the National Council of Teachers of Mathematics (1989, 1991, 1995, & 2000) Standards documents.

Teachers’ beliefs are defined as integrated systems of personalized assumptions about the nature of mathematics, the nature of students, and about learning and teaching (Artzt, 1999).

Purpose

The purpose of the study was to contribute to an understanding of the relationship between the beliefs of secondary mathematics teachers who have participated in a Standards-based pre-service education and their teaching practices.

Research Questions

1. To what extent are secondary mathematics teachers incorporating the Standards-based approach that was promoted in their pre-service education?

2. How consistent are the secondary teachers' beliefs with their teaching styles using a Standards-based framework?

3. To what extent are other factors impacting secondary mathematics teachers' beliefs and practices toward Standards-based mathematics?

4. To what extent do teachers change their teaching style based on student demographics such as socioeconomic status, race, gender, and ability level?

Review of Literature

The review of literature examined the research on teachers’ beliefs and practices in four areas:

1. Affect

2. Other Influences

3. Equity

4. Standards-Based Practices

Affect

Negative experiences with mathematics can have far-reaching effect on students’ beliefs of their ability to learn mathematics (Ma, 1999; McLeod, 1992).

Students’ attitudes toward mathematics can change with the proper atmosphere (Hannula, 2002).

Other Influences

Time (Adams & Krockover, 1997) Curriculum (Frykholm, 2004; LaBerge &

Sons, 1999) Colleagues (Cwikla, 2004; Kitchen, 2003) Administrators (Kitchen, 2003) Professional development (Cwikla, 2004) High-stakes testing (Cady et al., 2006)

Equity

Some teachers lowered their expectations of students based on student demographics such as SES and perceived ability level of students (Sztajn, 2003; Thomas et al., 1998).

Some teachers equated equality and “color blindness” with equity (Rousseau & Tate, 2003).

The Algebra Project (Silva & Moses, 1990) and QUASAR Project (Silver & Stein, 1996) were systemic urban projects where teachers maintained high expectations for all students to participate in challenging activities and challenging coursework.

Standards-Based Principles

An effective pre-service program can promote beliefs in alignment with Standards-base guidelines for some but not all teachers (Cooney et al., 1998; Van Zoest & Bohl, 2002).

An effective Standards-based curriculum does not guarantee that teachers will conform their beliefs and practices to align with the Standards, but it helps pre-service teachers with Standards-base beliefs make the transition to the classroom (Frykholm, 2004; Lloyd, 1999).

A Standards-based professional development model influenced mathematics teachers to change their beliefs and ultimately their teaching practices as well (Hart, 2002).

Design of the Study

Overview of Methodology

Case Study Five teachers were selected for a purposeful,

maximal sample Observation field notes, Interviews, Reform

Teaching Observation Protocol, and Teachers’ Practices and Beliefs Survey were instruments used as sources for data

Overview of Case Study Participants

Teacher Years

Taught

Level of Ed.

Systemic Project

School

Grades

Taught

Level of

Class

SB

Beliefs

SB

Prac-

tices

Expect in Light

of Equity

Mr.

Easterly

5 Bachelors No 9-12 Differing Low Low Low

Ms. Danforth

2 Alternative

Masters

Yes 12 Same Low High Low

Mr.

Barry

1 Bachelors No 7-8 Differing High Low High

Ms.

Anthony

2 Alternative

Masters

Yes 7-8 Differing High High High

Ms.

Chandler

5 Bachelors

& Masters

Yes 10-12 Same High Low High

Results of the Study

Case 1 Mr. Easterly

His cooperating teacher incorporated traditional teaching methods

Fifth year of teaching Very authoritative and structured teacher

(coach) Taught at a very affluent high school in a

large metropolitan area Not associated with the systemic project

Mr. Easterly Beliefs and Practices

He held traditional views which were supported by his internship.

He also felt that part of his role as a teacher was to instill a good work ethic into his students.

He exhibited traditional, teacher-centered lessons during observations.

Mr. Easterly’s Other Influences

His administrators were pleased with his structured classrooms with “those lower-achieving kids.”

He had access to all of the latest technology and a disciplined school climate

The discipline was tightly controlled.

Mr. Easterly and Equity

Mr. Easterly defined equity as treating all students equally.

His teaching practices followed the same format in the algebra IA and algebra III class.

He believed that students’ ability to learn conceptually depended on the perceived ability level of students.

Case 2 Ms. Danforth

Second year of teaching (11 previous years of experience in the Middle East)

Taught in a high school where the majority of the students were classified as low socioeconomic status; all students were African-American

Taught in a school that participated in the systemic project and participated in quarterly meetings

Her internship was with a cooperating teacher who effectively modeled Standards-based guidelines

Ms. Danforth’s Beliefs

She experienced a traditional high school education in her native country

Her internship and pre-service education caused her to adjust her traditional beliefs to include learning mathematics conceptually

After she began teaching, she adjusted her beliefs again. She felt that Standards-based teaching was more effective for younger students—not juniors or seniors

Ms. Danforth’s Practices

Her teaching was characterized by having notes on the board for the students to copy

She taught algebraic connections, and she incorporated real-life topics like credit card interest, mortgages, and finding different averages.

The students were usually only given 5 to 10 problems to complete in class—usually no homework

Ms. Danforth’s Other Influences

The school climate was chaotic and students routinely wandered the halls

Resources were limited—textbooks, functional copying machines, and computer access for students

Ms. Danforth’s classroom management was weak

Ms. Danforth and Equity

She believed that every student should receive appropriate accommodations be made to promote attainment for all students.

She qualified that this statement is only theoretical, and the implementation of equity is virtually impossible.

She lowered her expectations of her students based on their socioeconomic and backgrounds.

Case 3 Mr. Barry

His cooperating teacher in his internship had emphasized traditional teaching

He was in his first year of teaching at his alma mater He was teaching at a middle school in a small town

whose school system had a reputation for academic excellence

The school was not associated with the systemic project

Mr. Barry’s Beliefs

He felt that his high school education was very effective and felt that a traditional teaching approach was effective.

He adjusted his beliefs during his pre-service education—he believed that mathematics should be learned conceptually.

He thought that Standards-based teaching practices was good to be used as a supplement to the curriculum.

Mr. Barry’s Practices

He used traditional teaching that was teacher-centered and involved whole group format.

He incorporated real-life examples and conceptual explanations when presenting new topics.

No calculators were ever used. Students routinely presented homework

solutions to the class.

Mr. Barry’s Other Influences

Curriculum—Saxon Math Administrators, tenure, high-stakes testing Fear that investigative activities would cause

the students to get out of control

Mr. Barry and Equity

He admitted a special bonding with his seminar class.

He had different expectations for his slower-paced classes.

His RTOP scores and questioning techniques showed disparity in his teaching practices between the different classes.

Case 4 – Ms. Anthony

Her internship was with a cooperating teacher who combined a Standards-based curriculum with the traditional textbook.

She was in her second year of teaching. She was teaching at a rural middle school. She was a school teacher leader for the

systemic project.

Ms. Anthony’s Beliefs

Her experiences in her pre-service methods classes influenced her to hold beliefs in alignment with the Standards.

She felt that a teacher needed to be a guide and not a dictator.

She strongly agreed that it was important for students to solve mathematics problems for themselves.

She adjusted her beliefs after she began teaching.

Ms. Anthony’s Practices

Ms. Anthony always assigned contextual homework problems and had students present their solutions to the contextual problems.

She encouraged engaging student-to-student discussions in which students justified their method of solving problems.

She rarely used group activities. When she did, the groups were loosely organized with no structure.

Her beliefs and her practices were not in alignment.

Ms. Anthony’s Other Influences

7-month county curriculum policy and benchmarks

High-stakes testing, making AYP Classroom management Large class sizes

Ms. Anthony and Equity

She defined equity as “expecting greatness from every student and providing them with the opportunity and support that they need to accomplish it.”

Her basic mathematics class involved less student discussion and more direct teaching.

The combination of the lack of classroom control, large class sizes, and the perceived academic level of a class promoted inequitable teaching practices and lower expectations for students.

Case 5 Ms. Chandler

Ms. Chandler’s internship yielded positive experiences incorporating Standards-based teaching guidelines (especially in the area of technology) even though her cooperating teacher held traditional beliefs.

She was in her fifth year of teaching. She requested to teach students who were

perceived as lower-achieving in a large public high school.

She was a school teacher leader and a presenter for the systemic project.

Ms. Chandler’s Beliefs

During her pre-service education, Ms. Chandler developed strongly held beliefs that supported the Standards guidelines, and she continues to hold those beliefs.

Ms. Chandler’s Practices

She incorporated technology effectively to improve conceptual learning of mathematics.

She often used investigative activities to introduce topics that encouraged the students to “discover” concepts themselves.

Students routinely worked in groups and justified their findings to the class

The classroom was student-centered.

Ms. Chandler’s Other Influences

Technology and resources Curriculum Professional development—participation in

both systemic project and professional development opportunities in technology

Ms. Chandler and Equity

She incorporated a lot of accommodations in her classes to help her students experience success.

Her use of technology provided scaffolding to promote conceptual learning for her students.

She loved to teach mathematics conceptually because her students have been drilled and drilled in their past mathematics classes.

She promoted high expectations for all students by incorporating worthwhile opportunities.

Summary of Teachers’ Beliefs and Practices

Teacher Survey

Reform

Beliefs

Interview

Reform Beliefs

Survey Reform

Practices

RTOP Teacher Observe

Overall Reform

Practices

Survey Equity

Responses

Actual Equity

Practices

Mr. Easterly

Low Low Low Low Low Low Low

Ms. Danforth

Low Mixed High Low Low Low Low

Mr. Barry High Mixed Low Mixed Low High Low

Ms. Anthony

High High High Mid-High Mixed High Low

Ms. Chandler

High High Low High High High High

Summary Flowchart

Summary

The pre-service education impacted the beliefs of four case study participants, however, three adjusted their beliefs after they began teaching.

Only one of the case study participants was effectively incorporating Standards-based methods, and a second wanted to incorporate the Standards more effectively

Only one teacher had beliefs that were not in alignment with her practices. Three adjusted their beliefs so that they were in alignment with their practices.

Summary (cont’d)

Influences that affected participants teaching practices included time, administration, high-stakes testing, classroom management, technology, professional development, school climate, and curriculum.

Four teachers lowered expectations of their students based on students’ perceived ability level and socioeconomic background.

References

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

Artzt, A. (1999). A structure to enable preservice teachers of mathematics to reflect on their teaching. Journal of Mathematics Teacher Education, 2(2), 143-166.

AzTEC. (2002). About RTOP. Retrieved February 13, 2008, from http://physicsed.buffalostate.edu/AZTEC/rtop/RTOP_full/about_RTOP.html

Brown, C., & Borko, H. (1992). Becoming a mathematics teacher. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics (pp. 209-239). New York: Macmillan Publishing Co, Inc.

Cady, J., Meier, S., & Lubinski, C. (2006). Developing mathematics teachers: The transition from preservice to experienced teacher. Journal of Educational Research, 99(5), 295-305.

Cooney, T., Shealy, B., & Arvold, B. (1998). Conceptualizing belief structures of preservice secondary mathematics teachers. Journal for Research in Mathematics Education, 29(3), 306-333.

References

Cwikla, J. (2004). Less experienced mathematics teachers report what is wrong with their professional support system. Teachers & Teaching, 10(2), 181-197.

Fey, J., & Graeber, A. (2003). From the New Math to the Agenda for Action. In G. Stanic, G & J. Kilpatrick (Eds.), A history of school mathematics (pp. 521-558). Reston, VA: National Council of Teachers of Mathematics.

Frykholm, J. (2004). Teachers' tolerance for discomfort: Implications for curricular reform mathematics. Journal of Curriculum & Supervision, 19(2), 125-149.

Green, T. (1971). The activities of teaching. New York: McGraw-Hill.Gutierrez, R. (1999). Advancing Urban Latina/o Youth in Mathematics: Lessons

from an Effective High School Mathematics Department. Urban Review, 31(3), 263-281.

Hannula, M. (2002). Attitude towards mathematics: emotions, expectations and values. Educational Studies in Mathematics, 49(1), 25-46.

Hart, L. (2002). A four year follow-up study of teachers' beliefs after participating in a teacher enhancement project. In G. Leder, E. Pehkonen & G. Torner (Eds.), Beliefs: A hidden variable in mathematics education? (pp. 161-176). Dordrecht, Netherlands: Kluwer Academic Publishers.

References

Kitchen, R. S. (2003, April). Transforming Mathematics Education: Barriers to Reform in High Poverty, Diverse Schools. Paper presented at the Third International Conference on Mathematics Education and Society, Helsingor, Denmark.

LaBerge, V., & Sons, L. (1999). First-year teachers' Implementation of the NCTM Standards. Primus, 9(2), 139.

Lambdin, D., & Walcott, C. (2007). Changes through the years: Connections between psychological learning theories and the school mathematics curriculum. In W. G. Martin & M. E. Strutchens (Eds.), The learning of mathematics, Sixty-Ninth Yearbook, (pp. 3-26). Reston, VA: National Council of Teachers of Mathematics.

Leatham, K. (2006). Viewing mathematics teachers' beliefs as sensible systems. Jounal of Mathematics Teacher Education, 9(1), pp. 91-102.

Lloyd, G. (1999). Two teachers' conceptions of a reform-oriented curriculum: Implications for mathematics teacher development. Journal of Mathematics Teacher Education, 2(3), 227-252.

Ma, X. (1999). A meta-analysis of the relationship between anxiety towards mathematics and achievement. Journal for Research in Mathematics Education, 30(5), 520-540.

References

McLeod, D. (1992). Research on affect in mathematics education: A reconceptualization. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics (pp. 575-596). New York, NY: Macmillan Publishing Co, Inc.

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

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

National Council of Teachers of Mathematics. (1995). Assessment standards for school mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.

References

Rokeach, M. (1968). Beliefs, attitudes, and values: A theory of organization and change. San Francisco: Jossey-Bass.

Rousseau, C., & Tate, W. (2003). No time like the present: Reflecting on equity in school mathematics. Theory into Practice, 42(3), 210-216.

Silva, C., & Moses, R. (1990). The Algebra Project: Making middle school mathematics count. Journal of Negro Education, 59(3), 375-391.

Silver, E., & Stein, M. (1996). The QUASAR project: The "revolution of the possible" in mathematics instructional reform in urban middle schools. Urban Education, 30, 476-521.

Strauss, A., & Corbin, J. (1998). Basics of qualitative research: Techniques and procedures for developing grounded theory (2nd ed.). Thousand Oaks, CA: Sage Publications.

References

Sztajn, P. (2003). Adapting reform ideas in different mathematics classrooms: Beliefs beyond mathematics. Journal of Mathematics Teacher Education, 6(1), 53-75.

Van Zoest, L., & Bohl, J. (2002). The role of reform curricular materials in an internship: The case of Alice and Gregory. Journal of Mathematics Teacher Education, 5(3), 265-288.

Wilkins, J., & Brand, B. (2004). Change in preservice teachers' beliefs: An evaluation of a mathematics methods course. School Science & Mathematics, 104(5), 226-232.