EXPLORING PROSPECTIVE ELEMENTARY MATHEMATICS TEACHERS’ KNOWLEDGE: A FOCUS ON FUNCTIONAL THINKING

Hélia Oliveira; Irene Polo Blanco; Ana Henriques

doi:10.22342/jme.12.2.13745.257-278

EXPLORING PROSPECTIVE ELEMENTARY MATHEMATICS TEACHERS’ KNOWLEDGE: A FOCUS ON FUNCTIONAL THINKING

Hélia Oliveira, Irene Polo Blanco, Ana Henriques

Abstract

The importance of students being acquainted with algebraic ideas before secondary education has been revealed in the research literature. It is therefore essential that prospective elementary teachers (PTs) be prepared to instill an early algebra perspective in their teaching. However, PTs often show difficulties in algebra content knowledge, which need to be diagnosed aiming to assist them in developing the required knowledge to teach according to that perspective. This study aims to understand what aspects of functional thinking Spanish and Portuguese elementary PTs exhibit at the beginning of their teacher education program. The findings show that although PTs from both countries use different strategies to generalize functional relationships, the occurrence of successful strategies is low. Also, most participants provide local approaches in their interpretation of relationships between variables and reveal difficulties in understanding and connecting different representations of functions. These difficulties show that PTs lack important knowledge about functional thinking. By providing a framework concerning the functional thinking required for PTs to teach within an early algebra perspective, we shed light on a necessary step for teacher education programs to diagnose PTs’ functional thinking and to assist them in developing the needed mathematical knowledge to teach accordingly.

Keywords

Early Algebra; Functional Thinking; Generalization; Prospective Teachers’ Knowledge

Full Text:

PDF

References

Alajmi, A. H. (2016). Algebraic generalization strategies used by Kuwaiti pre-service teachers. International Journal of Science and Mathematics Education, 14(8), 1517–1534. https://doi.org/10.1007/s10763-015-9657-y

Apsari, R. A., Putri, R. I. I., Sariyasa, S., Abels, M., & Prayitno, S. (2019). Geometry representation to develop algebraic thinking: A recommendation for a pattern investigation in pre-algebra class. Journal on Mathematics Education, 11(1), 45-58. https://doi.org/10.22342/jme.11.1.9535.45-58

Ayalon, M., Watson, A., & Lerman, S. (2016). Progression towards functions: Students’ performance on three tasks about variables from grades 7 to 12. International Journal of Science and Mathematics Education, 14, 1153-1173. https://doi.org/10.1007/s10763-014-9611-4

Barbosa, A., & Vale, I. (2015). Visualization in pattern generalization: Potential and challenges. Journal of the European Teacher Education Network, 10, 57-70.

Blanton, M., & Kaput, J. (2011). Functional thinking as a route into Algebra in the elementary grades. In J. Cai & E. Knuth (Eds.), Early Algebraization, Advances in Mathematics Education (pp. 5-23). Berlin Heidelberg: Springer-Verlag. https://doi.org/10.1007/978-3-642-17735-4_2

Blanton, M., Levi, L., Crites, T., & Dougherty, B. (2011). Developing Essential Understanding of Algebraic Thinking for Teaching Mathematics in Grades 3-5. Reston, VA: NCTM.

Branco, N. C. V. (2013). O Desenvolvimento do Pensamento Algébrico na Formação Inicial de Professores dos Primeiros Anos. PhD Thesis, University of Lisbon, Portugal.

Brown, S., & Bergman, J. (2013). Preservice Teachers’ understanding of variable. Investigations in Mathematics Learning, 6(1), 1-17. https://doi.org/10.1080/24727466.2013.11790327

Carraher, D. W., & Schliemann, A. D. (2019) Early algebraic thinking and the US mathematics standards for grades K to 5. Infancia y Aprendizaje, 42(3), 479-522. https://doi.org/10.1080/02103702.2019.1638570

Confrey, J., & Smith, E. (1994). Exponential functions, rates of change, and the multiplicative unit. Educational Studies in Mathematics, 26, 135-164. https://doi.org/10.1007/BF01273661

Ellis, A. (2011). Algebra in the middle school: developing functional relationships through quantitative reasoning. In J. Cai & E. Knuth (Eds.), Early Algebraization, Advances in Mathematics Education (pp. 215-238). Berlin Heidelberg: Springer-Verlag. https://doi.org/10.1007/978-3-642-17735-4_13

Erickson, F. (1986). Qualitative methods in research on teaching. In M. Wittrock (Ed.), Handbook of Research on Teaching (pp. 119-161). New York, NY: MacMillan.

Hart, K. M. (1981). Children’s Understanding of Mathematics: 11-16. London: John Murray.

Hill, H., Ball, D. L., & Schilling, S. G. (2008). Unpacking pedagogical content knowledge: Conceptualising and measuring teachers’ topic-specific knowledge of students. Journal for Research in Mathematics Education, 39(4), 372-400. http://www.jstor.org/stable/40539304

Hohensee, C. (2017). Preparing elementary prospective teachers to teach early algebra. Journal of Mathematics Teacher Education, 20, 231–257. https://doi.org/10.1007/s10857-015-9324-9

Kaput, J. (2008). What is algebra? What is algebraic reasoning? In J. Kaput, D. Carraher, & M. Blanton (Eds.), Algebra in the Early Grades (pp. 5-18). Mahwah, NJ: Lawrence Erlbaum/Taylor & Francis Group & NCTM. https://doi.org/10.4324/9781315097435

Kieboom, L., Magiera, M. T., & Moyer, J. C. (2014). Exploring the relationship between K-8 prospective teachers’ algebraic thinking proficiency and the questions they pose during diagnostic algebraic thinking interviews. Journal of Mathematics Teacher Education, 17, 429-461. https://doi.org/10.1007/s10857-013-9264-1

Kieran, C., Pang, J., Schifter, D., & Ng, S. F. (2016). Early Algebra. Research into its Nature, its Learning, its Teaching. Cham, Switzerland: Springer. https://doi.org/10.1007/978-3-319-62597-3

Kusumaningsih, W., Darhim, D., Herman, T., & Turmudi. T. (2018). Improvement algebraic thinking ability using multiple representation strategy on realistic mathematics education. Journal on Mathematics Education, 9(2), 281-290. https://doi.org/10.22342/jme.9.2.5404.281-290

Lannin, J. K., Barker, D. D., & Townsend, B. E. (2006). Recursive and explicit rules: How can we build student algebraic understanding? The Journal of Mathematical Behavior, 25(4), 299-317. https://doi.org/10.1016/j.jmathb.2006.11.004

Leinhardt, G., Zaslavsky, O., & Stein, M. K. (1990). Functions, graphs, and graphing: Tasks, learning, and teaching. Review of Educational Research, 60(1), 1-64.

Magiera, M. T., van den Kieboom, L., & Moyer, J. C. (2013). An exploratory study of pre-service middle school teachers’ knowledge of algebraic thinking. Educational Studies in Mathematics, 84, 93-113. https://doi.org/10.1007/s10649-013-9472-8

McAuliffe, S., & Vermeulen, C. (2018). Preservice teachers' knowledge to teach functional thinking. In C. Kieran (Ed.), Teaching and Learning Algebraic Thinking with 5- to 12-Year-Olds: The Global Evolution of an Emerging Field of Research and Practice (pp. 403-426). Cham, Switzerland: Springer. https://link.springer.com/chapter/10.1007/978-3-319-68351-5_17

Morales, R., Cañadas, M. C., Brizuela, B. M., & Gómez, P. (2018). Relaciones funcionales y estrategias de alumnos de primero de educación primaria en un contexto funcional [Functional relationships and strategies of first grade students in a functional context]. Enseñanza de las Ciencias, 36(3), 59-78. https://doi.org/10.5565/rev/ensciencias.2472

Moss, J., & McNab, S. L. (2011). An approach to geometric and numeric patterning that fosters second grade students’ reasoning and generalizing about functions and co-variation. In J. Cai & E. Knuth (Eds.), Early Algebraization, Advances in Mathematics Education (pp. 277-301). Berlin Heidelberg: Springer-Verlag. https://doi.org/10.1007/978-3-642-17735-4_16

Oliveira, H., & Mestre, C. (2014). Opportunities to develop algebraic thinking in elementary grades throughout the school year in the context of mathematics curriculum changes. In Y. Li, E. Silver & S. Li (Eds.), Transforming Mathematics Instruction: Multiple Approaches and Practices (pp. 173-197). Dordrecht: Springer. https://doi.org/10.1007/978-3-319-04993-9_11

Patterson, C. L., & McGraw, R. (2018). When time is an implicit variable: an investigation of students’ ways of understanding graphing tasks. Mathematical Thinking and Learning, 20(4), 295-323. https://doi.org/10.1080/10986065.2018.1509421

Radford, L. (2008). Iconicity and contraction: a semiotic investigation of forms of algebraic generalizations of patterns in different contexts. ZDM Mathematics Education, 40, 83–96. https://doi.org/10.1007/s11858-007-0061-0

Radford, L. (2011). Grade 2 students’ non-symbolic algebraic thinking. In J. Cai & E. Knuth (eds.), Early Algebraization, Advances in Mathematics Education (pp. 303-322). Berlin Heidelberg: Springer-Verlag. https://doi.org/10.1007/978-3-642-17735-4_17

Richardson, K., Berenson, S., & Staley, K. (2009). Prospective elementary teachers use of representation to reason algebraically. Journal of Mathematical Behavior, 28(2-3), 188-199. https://doi.org/10.1016/j.jmathb.2009.09.002

Rodrigues, R. V., Cyrino, M. C., & Oliveira, H. (2019). Percepção profissional de futuros professores sobre o pensamento algébrico dos alunos na exploração de um caso multimídia. Quadrante, 28(1), 100-123. https://doi.org/10.48489/quadrante.22975

Stephens, A. C., Ellis, A. B., Blanton, M., & Brizuela, B. M. (2017). Algebraic thinking in the elementary and middle grades. In J. Cai (Ed.), Compendium for Research in Mathematics Education (pp. 386-410). Reston, VA: NCTM.

Strand, K., & Mills, B. (2014). Mathematical content knowledge for teaching elementary mathematics: A focus on Algebra. The Mathematics Enthusiast, 11(2), 384-432. https://scholarworks.umt.edu/tme/vol11/iss2/8

Thompson, P. W., & Carlson, M. P. (2017). Variation, covariation, and functions: Foundational ways of thinking mathematically. In J. Cai (Ed.), Compendium for Research in Mathematics Education (pp. 421-456). Reston, VA: NCTM.

Wilkie, K. J. (2016). Learning to teach upper primary school algebra: changes to teachers’ mathematical knowledge for teaching functional thinking. Mathematics Education Research Journal, 28, 245-275. https://doi.org/10.1007/s13394-015-0151-1

Yemen-Karpuzcu, S., Ulusoy, F., & I??ksal-Bostan, M. (2017). Prospective middle school mathematics teachers’ covariational reasoning for interpreting dynamic events during peer interactions. International Journal of Science and Mathematics Education, 15, 89-108. https://doi.org/10.1007/s10763-015-9668-8

Zazkis, R., & Liljedahl, P. (2002). Generalization of patterns: The tension between algebraic thinking and algebraic notation. Educational Studies in Mathematics, 49(3), 379-402. https://doi.org/10.1023/A:1020291317178

DOI: https://doi.org/10.22342/jme.12.2.13745.257-278

Refbacks

There are currently no refbacks.

This work is licensed under a Creative Commons Attribution 4.0 International License.

Journal on Mathematics Education

Doctoral Program on Mathematics Education

Faculty of Teacher Training and Education, Universitas Sriwijaya
Kampus FKIP Bukit Besar
Jl. Srijaya Negara, Bukit Besar
Palembang - 30139

email: jme@unsri.ac.id

p-ISSN: 2087-8885 | e-ISSN: 2407-0610

Journal on Mathematics Education (JME) is licensed under a Creative Commons Attribution 4.0 International License.

View My Stats

Username
Password
Remember me