# Development of Arithmetical Thinking: Evaluation of Subject Matter Knowledge of Pre-Service Teachers in Order to Design the Appropriate Course

## Abstract

One of the key courses in the mathematics teacher education program in Israel is arithmetic, which engages in contents which these pre-service mathematics teachers (PMTs) will later teach at school. Teaching arithmetic involves knowledge about the essence of the concept of “number” and the development thereof, calculation methods and strategies. properties of operations on different sets of numbers, as well as the properties of the numbers themselves. Hence, the question arises: how to educate PMTs in order to supplement their mathematical knowledge with the required components? The present study explored the development of arithmetic thinking among pre-service teachers intending to teach mathematics at elementary school. This was done by matching the van Hiele theory of the development of geometric thinking to arithmetic. Analysis of findings obtained both in the present study and in many studies of geometry teaching indicates that this approach to considering the learners’ level of thinking development might lead to meaningful learning in arithmetic course for PMTs.

## Keywords

Prospective mathematics teachers (PMTs) Mathematical knowledge for teaching Development of mathematical thinking Arithmetic## Notes

### Acknowledgments

This paper presents part of my doctoral thesis written under the supervision of Prof. Shlomo Vinner. I express my deep personal and professional appreciation to Prof. Vinner and am grateful for the honor of having been his research student. I also wish to thank Dr. Marita Barabash for her constructive comments during my work on this paper.

## Supplementary material

## References

- Artzt, A., Sultan, A., Curio, F. R. & Gurl, T. (2012). A capstone mathematics course for prospective secondary mathematics teachers.
*Journal of Mathematics Teacher Education, 15*(3), 449–462.CrossRefGoogle Scholar - Ausubel, D. P. (1968).
*Educational psychology: A cognitive view*. New York, NY: Holt, Rinehart and Winston.Google Scholar - Balacheff, N. (1988). Aspects of proof in pupils’ practice of school mathematics. In D. Pimm (Ed.),
*Mathematics, teachers and children*(pp. 216–235). London, England: Hodder and Stoughton.Google Scholar - Ball, D. L. (1990). The mathematical understandings that prospective teachers bring to teacher education.
*Elementary School Journal, 90*, 449–466.CrossRefGoogle Scholar - Ball, D. L., Hill, H.C. & Bass, H. (2005). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade, and how can we decide?
*American Educator, 29*(1), 14–17, 20–22, 43–46.Google Scholar - 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.CrossRefGoogle Scholar - Blömeke, S., Suhl, U. & Döhrmann, M. (2013). Assessing strengths and weaknesses of teacher knowledge in Asia, Eastern Europe, and Western Countries: Differential item functioning in TEDS-M.
*International Journal of Science and Mathematics Education, 11*(4), 795–817.CrossRefGoogle Scholar - Battista, M. T. (2007). The Development of Geometric and Spatial Thinking. In F. K. Lester (Ed.),
*Second Handbook of Research on Mathematics Teaching and Learning*(pp. 843–908). Reston, VA: National Council of Teachers of Mathematics & Charlotte, NC: Information Age Publishing.Google Scholar - Baumert, J., Kunter, M., Blum, W., Brunner, M., Voss, T., Jordan, A. & Tsai, Y. M. (2010). Teachers’ mathematical knowledge, cognitive activation in the classroom, and student progress.
*American Educational Research Journal, 47*(1), 133–180.CrossRefGoogle Scholar - Borko, H., Eisenhart, M., Brown, C. A., Underhill, R. G., Jones, D. & Agard, P. C. (1992). Learning to teach hard mathematics: Do novice teachers and their instructors give up too easily?
*Journal for Research in Mathematics Education, 23*(3), 194–222.CrossRefGoogle Scholar - Clements, D. H. & Battista, M. T. (1992). Geometry and spatial reasoning. In D. Grouws (Ed.),
*Handbook of research on mathematics teaching and learning*(pp. 420–464). New York, NY: Macmillan.Google Scholar - Clements, D. H., Battista, M. T. & Sarama, J. (Eds.). (2001).
*Journal for research in mathematics education monograph 10: Logo and geometry*. Reston, VA: NCTM.Google Scholar - Crowley, M. L. (1990). Criterion-referenced reliability indices associated with the van Hiele geometry test.
*Journal for Research in Mathematics Education, 21*(3), 238–241.CrossRefGoogle Scholar - Guberman, R. (2008). A framework for characterizing the development of arithmetical thinking.
*Proceedings of ICME-11–topic study group 10; research and development in the teaching and learning of number systems and arithmetic*, (pp. 113–121). Monterrey, Mexico.Google Scholar - Guberman, R. & Leikin, R. (2012). Interesting and difficult mathematical problems: Changing teachers’ views by employing multiple-solution tasks.
*Journal of Mathematics Teacher Education, 16*(1), 33–56.CrossRefGoogle Scholar - Hiebert, J. & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.),
*Conceptual and procedural knowledge: The case of mathematics*(pp. 1–27). Hillsdale, NJ: Erlbaum Associates.Google Scholar - Hill, H.C., Ball, D.L., Sleep, L. & Lewis, J.M. (2007). Assessing teachers” mathematical knowledge: What knowledge matters and what evidence counts? In F. Lester (Ed.),
*Handbook for research on mathematics education*(2nd ed), (pp. 111–155). Reston, VA: NCTM & Charlotte, NC: Information Age Publishing.Google Scholar - Isksal, M. & Cakiroglu, E. (2011). The nature of prospective mathematics teachers ‘pedagogical content knowledge: The case of multiplication of fractions.
*Journal of Mathematics Teacher Education, 14*, 213–320.CrossRefGoogle Scholar - Lannin, J. K., Webb, M., Chval, K., Arbaugh, F., Hicks, S., Taylor, C. & Bruton, R. (2013). The development of beginning mathematics teacher pedagogical content knowledge.
*Journal of Mathematics Teacher Education, 16*(6), 403–426.CrossRefGoogle Scholar - Leikin, R. (2006). Learning by teaching: The case of sieve of Eratosthenes and one elementary school teacher. In R. Zazkis & S. Campbell (Eds.),
*Number theory in mathematics education: Perspectives and prospects*(pp. 115–140). Mahwah, NJ: Erlbaum.Google Scholar - Livy, S. & Vale, C. M. (2011). First year pre-service teachers’ mathematical content knowledge: Methods of solution for a ratio question.
*Mathematics Teacher Education and Development, 13*(2), 22–43.Google Scholar - Ma, L. (1999).
*Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States*. Hillsdale, NJ: Erlbaum Associates.Google Scholar - Mewborn, D. (2001). Teachers content knowledge, teacher education, and their effects on the preparation of elementary teachers in the United States.
*Mathematics Teacher Education and Development, 3*, 28–36.Google Scholar - Moreira, P. C. & David, M. M. (2008). Academic mathematics and mathematical knowledge needed in school teaching practice: Some mathematics professional literacy conflicting elements.
*Journal of Mathematics Teacher Education, 11*, 23–40.CrossRefGoogle Scholar - Nathan, M. J. & Petrosino, A. J. (2003). Expert blind spot among preservice teachers.
*American Educational Research Journal, 40*(4), 905–928.CrossRefGoogle Scholar - Rowland, T., Huckstep, P. & Thwaites, A. (2005). Elementary teachers’ mathematics subject knowledge: The knowledge quartet and the case of Naomi.
*Journal of Mathematics Teacher Education, 8*(3), 255–281.CrossRefGoogle Scholar - Sabar Ben-Yehoshua, N. (1999).
*Qualitative research in teaching and learning*. (5th ed.). Tel Aviv: Modan publishers. (in Hebrew).Google Scholar - Salomon, G. & Perkins, D. N. (1996). Learning in wonderland: What computers really offer education. In S. Kerr (Ed.),
*Technology and the future of education NSSE Yearbook*(pp. 111–130). Chicago: University of Chicago Press.Google Scholar - Saxe, G. B., Gearhart, M. & Nasir, N. (2001). Enhancing students’ understanding of mathematics: A study of three contrasting approaches to professional support.
*Journal for Research in Teacher Education, 4*, 55–79.Google Scholar - Senk, S. L. (1989). Van Hiele levels and achievement in writing geometry proofs.
*Journal for Research in Mathematics Education, 20*(3), 309–321.CrossRefGoogle Scholar - Skemp, R. R. (1976). Relational understanding and instrumental understanding.
*Mathematics Teaching, 77*, 20–26.Google Scholar - Sowder, J. (2007). The Mathematical Education and Development of Teachers. In F. Lester (Ed.),
*Second Handbook of Research on Mathematics Teaching and Learning*(pp. 157–223). Reston, VA: NCTM & Charlotte, NC: Information Age Publishing.Google Scholar - Tatto, M. T., Schwille, J., Senk, S., Ingvarson, L., Rowley, G., Peck, R., et al. (2012).
*Policy, practice, and readiness to teach primary and secondary mathematics in 17 countries*. Retrieved from http://www.iea.nl/fileadmin/user_upload/Publications/Electronic_versions/TEDS-M_International_Report.pdf. - Thanheiser, E., Browning, C., Edson, A. J., Kastberg, S. & Lo, J. J. (2013). Building a knowledge base: understanding prospective elementary teachers’ mathematical content knowledge.
*International Journal for Mathematics Teaching & Learning*. Retrieved from http://www.cimt.plymouth.ac.uk/journal. - Tirosh, D. (2000). Enhancing prospective teachers’ knowledge of children’s conceptions: the case of division of fractions.
*Journal for Research in Mathematics Education, 22*, 125–147.Google Scholar - Usiskin, Z. (1982).
*Van Hiele Levels and Achievement in Secondary Sschool Geometry*(Final report). Chicago: Department of Education, University of Chicago, IL: Retrieved from http://www.psych.stanford.edu/~jlm/pdfs/Usiskin82AssessingvanHiele.pdf - Van Hiele, P. M. (1986).
*Structure and insight: A theory of mathematics education*. New York, NY: Academic.Google Scholar - Van Hiele, P. M. (1999). Developing geometric thinking through activities that begin with play.
*Teaching Children Mathematics, 5*(6), 310–316.Google Scholar - Verschaffel, L., Greer, B. & De Corte, E. (2007). Whole number concepts and operations. In F. K. Lester (Ed.),
*Second handbook of research on mathematics teaching and learning*(pp. 557–628). Reston, VA: NCTM & Charlotte, NC: Information Age Publishing.Google Scholar - Wirszup, I. (1976). Breakthroughs in the psychology of learning and teaching geometry. In J. I. Martin & D. A. Bradbard (Eds.),
*Space and geometry: Papers from a research workshop*. Columbus, Ohio: ERIC Center for Science, Mathematics and Environment Education.Google Scholar