# Mathematical and Pedagogical Knowledge amongst First- and Second-Grade Inservice and Preservice Mathematics Teachers

## Abstract

The study investigates the mathematical and the pedagogical content knowledge among inservice and preservice first- and second-grade mathematics teachers. The sample of 300 subjects consisted of 150 first- and second-grade inservice teachers and 150 preservice teachers studying in a college of education, 75 of whom were first-year students and 75 third- and fourth-year students. The data was collected using two tools—a mathematical content knowledge (MCK) test and a mathematics pedagogical content knowledge (MPCK) test, the items on each of these representing the four sub-domains studied in first- and second-grade: numbers, arithmetic operations, geometry/measurements, and word problems. The principal findings of the study indicate that all three groups possess limited knowledge of both MCK and MPCK in all sub-domains. The inservice teachers achieved the highest mean in the MCK and MPCK and the differences between the inservice teachers and the two groups of preservice teachers were statistically significant in general MCK and MPCK and in all the domains. Moreover, the lowest MCK mean was in the geometry/measurements domain, and the lowest MPCK mean was in the word-problems domain## References

Akerson, V. L. (2004). Designing a science methods course for early childhood preservice teachers. Journal of Elementary Science Education, 16(2), 19-32.

Amarto, S. A., & Watson, A. (2003). Improving student teachers’ understanding of multiplication by two digit numbers. In N. A. Pateman, B. J. Dougherty, & J. Zilliox (Eds.), Proceedings of the 27th Conference of the International Group for the Psychology of Mathematics Education and the 25th Conference of PME-NA. Honolulu, HI: PME.

Ball, D. L. (1988). Knowledge and reasoning in mathematical pedagogy: Examining what prospective teachers bring to teacher education. Unpublished doctoral dissertation, Michigan State University.

Ball, D. L., & Bass, H. (2000). Interweaving content and pedagogy in teaching and learning to teach: Knowing and using mathematics. In J. Boaler (Ed.), Multiple perspectives on mathematics teaching and learning (pp. 83-104). Westport, CT: Ablex.

Ball, D. L., Hill, H., & Bass, H. (2004). Knowing and using mathematical knowledge in teaching: Learning what matters. Invited plenary address to the Southern African Association of Mathematics, Science, and Technology Education. Cape Town, South Africa.

Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching what makes it special?.Journal of teacher education, 59(5), 389-407.

Ball, D. L, Lubienski, S., & Mewborn, D. (2001). Research on teaching mathematics: The unsolved problem of teachers’ mathematical knowledge. In V. Richardson (Ed.), Handbook of Research on Teaching (4th ed.) (pp. 433-456). New York: Macmillan.

Battista, M. T. (2006). Understanding the development of students' thinking about length. Teaching Children Mathematics, 13(3), 140.

Baturo, A., & Nason, R. (1996). Student teachers’ subject matter knowledge within the domain of area measurement. Educational Studies in Mathematics, 31(3), 235-268.

Boonen, T., Van Damme, J., & Onghena, P. (2014). Teacher effects on student achievement in first grade: which aspects matter most? School Effectiveness and School Improvement, 25(1), 126-152.

Carpenter, P. T., Fennema, E., Peterson, P. L., & Cary, A. D. (1988). Teachers’ pedagogical content knowledge of students’ problem solving in elementary arithmetic. Journal for Research in Mathematics, 19(5), 385-401.

Carter, K. (1990). Teachers’ knowledge and learning to teach. In W. R. Houston & M. H. J. Sikula (Eds.), Handbook of research on teacher education (pp. 291–310). New York: Macmillan.

Cheang, W. K., Yeo, J. K. K., Chan, M. E., Lim-Teo, S. K., Chua, K. G., & Ng, L. E. (2007). Development of mathematics pedagogical content knowledge in student teachers. The Mathematics Educator, 10(2), 27-54.

Clarke, B., Doabler, C. T., Nelson, N. J., & Shanley, C. (2015). Effective Instructional Strategies for Kindergarten and First-Grade Students at Risk in Mathematics. Intervention in School and Clinic, 50(5), 257-265.

Croninger, R. G., Rice, J. K., Rathbun, A., & Nishio, M. (2007). Teacher qualifications and early learning: Effects of certification, degree, and experience on first-grade student achievement. Economics of Education Review, 26(3), 312-324.

Davis, B., & Simmt, E. (2006). Mathematics-for-teaching: An ongoing investigation of the mathematics that teachers (need to) know. Educational Studies in Mathematics, 61(3), 293-319.

Delaney, S., Ball, D. L., Hill, H. C., Schilling, S. G., & Zopf, D. (2008). “Mathematical knowledge for teaching”: Adapting US measures for use in Ireland. Journal of Mathematics Teacher Education, 11(3), 171-197

Goulding, M. (2003). An investigation into the mathematics knowledge of primary teacher trainees. Proceedings of the British Society for Research into Learning Mathematics, 23(3), 73–78.

Hamlet, B. (2007). Mathematics content knowledge of pre- service primary teachers : Developing confidence and competence. Proceedings of the British Society for Research into Learning Mathematics, 27(3), 43-48.

Hill, H. C., & Ball, D. L. (2004). Learning mathematics for teaching: Results from California's mathematics professional development institutes. Journal for research in mathematics education, 330-351.

Hill, H. C., Ball, D. L., & Schilling, S. G. (2008). Unpacking pedagogical content knowledge: Conceptualizing and measuring teachers' topic-specific knowledge of students. Journal for Research in Mathematics Education, 39(4), 372-400.

Hill, C. H., Rowan, B., & Ball, D. L. (2005). Effects of teachers’ mathematical knowledge for teaching on student achievement. American Educational Research Journal, 42(2), 371-406.

Hill, H. C., Schilling, S. G., & Ball, D. L. (2004). Developing measures of teachers’ mathematics knowledge for teaching. Elementary School Journal, 10(5), 11-30.

Jones, K. (2000). Teacher knowledge and professional development in geometry. Proceedings of the British Society for Research into Learning Mathematics, 20(3), 109-114.

Kleickmann, T., Richter, D., Kunter, M., Elsner, J., Besser, M., Krauss, S., & Baumert, J. (2013). Teachers’ content knowledge and pedagogical content knowledge the role of structural differences in teacher education. Journal of Teacher Education, 64(1), 90-106.

Lim-Teo, S. K., Chua, K. G., Cheang, W. K., & Yeo, J. K. (2007). The development of diploma of education student teachers’ mathematics pedagogical content knowledge. International Journal of Science and Mathematics Education, 5, 237-261.

Livy, S., & Vale, C. (2011). First year preservice teachers' mathematical content knowledge: methods of solution to a ratio question. Mathematics teacher education and development, 13(2), 22-43.

Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum Associates.

Malzahn, K. A. (2002). Status of elementary school mathematics teaching. Report from the 2000 National Survey of Science and Mathematics Education. Chapel Hill, NC: Horizon Research: http://2000survey.horizon-research.com/ reports/#status teaching.

Markworth, K., Goodwin, T., & Glisson, K. (2009). The Development of Mathematical Knowledge for Teaching in the Student Teaching Practicum. In D. S. Mewborn&H. S. Lee (Eds.).AMTE Monograph, 6. Scholarly Practices and Inquiry in the Preparation of Mathematics Teachers. 67–83. San Diego, California: Association of Mathematics Teacher Educators.

Mewborn, D. (2001). Teachers’ content knowledge, teacher education, and their effects on the preparation of elementary teachers in the united states. Mathematics Education Research Journal, 3, 28-36.

Ministry of Education. (2006). Mathematics' curriculums. Retrieved from http://meyda.education.gov.il/files/Tochniyot_Limudim/Math/Yesodi/mavo1.pdf

Morris, H. (2001). Issues raised by testing trainee primary teachers’ mathematical knowledge. Mathematics Teacher Education and Development, 3, 37-47.

Nason, R., Chalmers, C., & Yeh, A. (2012). Facilitating growth in prospective teachers’ knowledge: teaching geometry in primary schools. Journal of Mathematics Teacher Education, 15(3), 227-249.

National Council for Accreditation of Teacher Education (NCATE). (2002). Professional Standards for the Accreditation of Schools, Colleges, and Departments of Education. Washington, DC: NCATE.

National Research Council. (2010). How People Learn: Brain, Mind, Experience, and School. Expanded Edition. Washington, DC: The National Academies Press.

Peterson, P. L., Fennema, E., Carpenter, T., & Loef, M. (1989). Teachers’ pedagogical content beliefs in mathematics. Cognition and Instruction, 6, 1-40.

Rogers, J. (2004). Autonomy and mathematical problem-solving: The early years, education. International Journal of Primary, Elementary and Early Years Education, 32(3), 24-31.

Sarama, J., & Clements, D. H. (2009). Early childhood mathematics education research: Learning trajectories for young children. New York: Routledge.

Shulman, L. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4-14.

Southwell, B., White, A. L., Way, J., & Perry, B. (2006). Attitudes versus achievement in preservice mathematics teaching. Refereed Proceedings of the Australian Association for Research in Education Annual Conference, Sydney:.

Stevens, C., & Wenner, G. (1996). Elementary preservice teachers’ knowledge and beliefs regarding science and mathematics. School Science and Mathematics, 96, 2.

Tatsuoka, K. K., Corter, J. E., & Tatsuoka, C. (2004). Patterns of diagnosed mathematical content and process skills in TIMSS-R across a sample of twenty countries. American Educational Research Journal, 41(4), 901-926.

Tatto, M. T., Schwille, J., Senk, S., Ingvarson, L., Peck, R., & Rowley, G. (2008). Teacher Education and Development Study in Mathematics (TEDS-M): Policy, practice, and readiness to teach primary and secondary mathematics—conceptual framework. East Lansing, MI: Teacher Education and Development International Study Center, College of Education, Michigan State University.

Theil, O. (2012). Investigating the structure, level and development of professional skills of preschool teachers in mathematics. In 12th International Congress on Mathematical Education. Retrieved from www.icme12.org/upload/UpFile2/TSG/0450.pdf

Turnuklu, E. B., & Yesildere, S. (2007, October). The pedagogical content knowledge in mathematics: Preservice primary mathematics teachers perspectives in Turkey. IUMPST: The Journal 1 (Content Knowledge): www.k-12prep.math.ttu.edu.

Tutak, F. A. (2009). A study of geometry content knowledge of elementary preservice teachers: The case of quadrilaterals. PhD dissertation, University of Florida: http://etd.fcla.edu/UF/UFE0041186/tutak_f.pdf.

Wilkins, L. M. J. (2008). The relationship among elementary teachers’ content knowledge, attitudes, beliefs, and practices. Journal of Mathematics Teacher Education, 11(2), 139-164.

Amarto, S. A., & Watson, A. (2003). Improving student teachers’ understanding of multiplication by two digit numbers. In N. A. Pateman, B. J. Dougherty, & J. Zilliox (Eds.), Proceedings of the 27th Conference of the International Group for the Psychology of Mathematics Education and the 25th Conference of PME-NA. Honolulu, HI: PME.

Ball, D. L. (1988). Knowledge and reasoning in mathematical pedagogy: Examining what prospective teachers bring to teacher education. Unpublished doctoral dissertation, Michigan State University.

Ball, D. L., & Bass, H. (2000). Interweaving content and pedagogy in teaching and learning to teach: Knowing and using mathematics. In J. Boaler (Ed.), Multiple perspectives on mathematics teaching and learning (pp. 83-104). Westport, CT: Ablex.

Ball, D. L., Hill, H., & Bass, H. (2004). Knowing and using mathematical knowledge in teaching: Learning what matters. Invited plenary address to the Southern African Association of Mathematics, Science, and Technology Education. Cape Town, South Africa.

Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching what makes it special?.Journal of teacher education, 59(5), 389-407.

Ball, D. L, Lubienski, S., & Mewborn, D. (2001). Research on teaching mathematics: The unsolved problem of teachers’ mathematical knowledge. In V. Richardson (Ed.), Handbook of Research on Teaching (4th ed.) (pp. 433-456). New York: Macmillan.

Battista, M. T. (2006). Understanding the development of students' thinking about length. Teaching Children Mathematics, 13(3), 140.

Baturo, A., & Nason, R. (1996). Student teachers’ subject matter knowledge within the domain of area measurement. Educational Studies in Mathematics, 31(3), 235-268.

Boonen, T., Van Damme, J., & Onghena, P. (2014). Teacher effects on student achievement in first grade: which aspects matter most? School Effectiveness and School Improvement, 25(1), 126-152.

Carpenter, P. T., Fennema, E., Peterson, P. L., & Cary, A. D. (1988). Teachers’ pedagogical content knowledge of students’ problem solving in elementary arithmetic. Journal for Research in Mathematics, 19(5), 385-401.

Carter, K. (1990). Teachers’ knowledge and learning to teach. In W. R. Houston & M. H. J. Sikula (Eds.), Handbook of research on teacher education (pp. 291–310). New York: Macmillan.

Cheang, W. K., Yeo, J. K. K., Chan, M. E., Lim-Teo, S. K., Chua, K. G., & Ng, L. E. (2007). Development of mathematics pedagogical content knowledge in student teachers. The Mathematics Educator, 10(2), 27-54.

Clarke, B., Doabler, C. T., Nelson, N. J., & Shanley, C. (2015). Effective Instructional Strategies for Kindergarten and First-Grade Students at Risk in Mathematics. Intervention in School and Clinic, 50(5), 257-265.

Croninger, R. G., Rice, J. K., Rathbun, A., & Nishio, M. (2007). Teacher qualifications and early learning: Effects of certification, degree, and experience on first-grade student achievement. Economics of Education Review, 26(3), 312-324.

Davis, B., & Simmt, E. (2006). Mathematics-for-teaching: An ongoing investigation of the mathematics that teachers (need to) know. Educational Studies in Mathematics, 61(3), 293-319.

Delaney, S., Ball, D. L., Hill, H. C., Schilling, S. G., & Zopf, D. (2008). “Mathematical knowledge for teaching”: Adapting US measures for use in Ireland. Journal of Mathematics Teacher Education, 11(3), 171-197

Goulding, M. (2003). An investigation into the mathematics knowledge of primary teacher trainees. Proceedings of the British Society for Research into Learning Mathematics, 23(3), 73–78.

Hamlet, B. (2007). Mathematics content knowledge of pre- service primary teachers : Developing confidence and competence. Proceedings of the British Society for Research into Learning Mathematics, 27(3), 43-48.

Hill, H. C., & Ball, D. L. (2004). Learning mathematics for teaching: Results from California's mathematics professional development institutes. Journal for research in mathematics education, 330-351.

Hill, H. C., Ball, D. L., & Schilling, S. G. (2008). Unpacking pedagogical content knowledge: Conceptualizing and measuring teachers' topic-specific knowledge of students. Journal for Research in Mathematics Education, 39(4), 372-400.

Hill, C. H., Rowan, B., & Ball, D. L. (2005). Effects of teachers’ mathematical knowledge for teaching on student achievement. American Educational Research Journal, 42(2), 371-406.

Hill, H. C., Schilling, S. G., & Ball, D. L. (2004). Developing measures of teachers’ mathematics knowledge for teaching. Elementary School Journal, 10(5), 11-30.

Jones, K. (2000). Teacher knowledge and professional development in geometry. Proceedings of the British Society for Research into Learning Mathematics, 20(3), 109-114.

Kleickmann, T., Richter, D., Kunter, M., Elsner, J., Besser, M., Krauss, S., & Baumert, J. (2013). Teachers’ content knowledge and pedagogical content knowledge the role of structural differences in teacher education. Journal of Teacher Education, 64(1), 90-106.

Lim-Teo, S. K., Chua, K. G., Cheang, W. K., & Yeo, J. K. (2007). The development of diploma of education student teachers’ mathematics pedagogical content knowledge. International Journal of Science and Mathematics Education, 5, 237-261.

Livy, S., & Vale, C. (2011). First year preservice teachers' mathematical content knowledge: methods of solution to a ratio question. Mathematics teacher education and development, 13(2), 22-43.

Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum Associates.

Malzahn, K. A. (2002). Status of elementary school mathematics teaching. Report from the 2000 National Survey of Science and Mathematics Education. Chapel Hill, NC: Horizon Research: http://2000survey.horizon-research.com/ reports/#status teaching.

Markworth, K., Goodwin, T., & Glisson, K. (2009). The Development of Mathematical Knowledge for Teaching in the Student Teaching Practicum. In D. S. Mewborn&H. S. Lee (Eds.).AMTE Monograph, 6. Scholarly Practices and Inquiry in the Preparation of Mathematics Teachers. 67–83. San Diego, California: Association of Mathematics Teacher Educators.

Mewborn, D. (2001). Teachers’ content knowledge, teacher education, and their effects on the preparation of elementary teachers in the united states. Mathematics Education Research Journal, 3, 28-36.

Ministry of Education. (2006). Mathematics' curriculums. Retrieved from http://meyda.education.gov.il/files/Tochniyot_Limudim/Math/Yesodi/mavo1.pdf

Morris, H. (2001). Issues raised by testing trainee primary teachers’ mathematical knowledge. Mathematics Teacher Education and Development, 3, 37-47.

Nason, R., Chalmers, C., & Yeh, A. (2012). Facilitating growth in prospective teachers’ knowledge: teaching geometry in primary schools. Journal of Mathematics Teacher Education, 15(3), 227-249.

National Council for Accreditation of Teacher Education (NCATE). (2002). Professional Standards for the Accreditation of Schools, Colleges, and Departments of Education. Washington, DC: NCATE.

National Research Council. (2010). How People Learn: Brain, Mind, Experience, and School. Expanded Edition. Washington, DC: The National Academies Press.

Peterson, P. L., Fennema, E., Carpenter, T., & Loef, M. (1989). Teachers’ pedagogical content beliefs in mathematics. Cognition and Instruction, 6, 1-40.

Rogers, J. (2004). Autonomy and mathematical problem-solving: The early years, education. International Journal of Primary, Elementary and Early Years Education, 32(3), 24-31.

Sarama, J., & Clements, D. H. (2009). Early childhood mathematics education research: Learning trajectories for young children. New York: Routledge.

Shulman, L. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4-14.

Southwell, B., White, A. L., Way, J., & Perry, B. (2006). Attitudes versus achievement in preservice mathematics teaching. Refereed Proceedings of the Australian Association for Research in Education Annual Conference, Sydney:.

Stevens, C., & Wenner, G. (1996). Elementary preservice teachers’ knowledge and beliefs regarding science and mathematics. School Science and Mathematics, 96, 2.

Tatsuoka, K. K., Corter, J. E., & Tatsuoka, C. (2004). Patterns of diagnosed mathematical content and process skills in TIMSS-R across a sample of twenty countries. American Educational Research Journal, 41(4), 901-926.

Tatto, M. T., Schwille, J., Senk, S., Ingvarson, L., Peck, R., & Rowley, G. (2008). Teacher Education and Development Study in Mathematics (TEDS-M): Policy, practice, and readiness to teach primary and secondary mathematics—conceptual framework. East Lansing, MI: Teacher Education and Development International Study Center, College of Education, Michigan State University.

Theil, O. (2012). Investigating the structure, level and development of professional skills of preschool teachers in mathematics. In 12th International Congress on Mathematical Education. Retrieved from www.icme12.org/upload/UpFile2/TSG/0450.pdf

Turnuklu, E. B., & Yesildere, S. (2007, October). The pedagogical content knowledge in mathematics: Preservice primary mathematics teachers perspectives in Turkey. IUMPST: The Journal 1 (Content Knowledge): www.k-12prep.math.ttu.edu.

Tutak, F. A. (2009). A study of geometry content knowledge of elementary preservice teachers: The case of quadrilaterals. PhD dissertation, University of Florida: http://etd.fcla.edu/UF/UFE0041186/tutak_f.pdf.

Wilkins, L. M. J. (2008). The relationship among elementary teachers’ content knowledge, attitudes, beliefs, and practices. Journal of Mathematics Teacher Education, 11(2), 139-164.

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2017-05-14

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Articles