[en] Research focusing on algebra from primary to early secondary school level has made several major advances over the past decades. Students’ difficulties have been identified and supportive teaching and learning environments have been set up (Cai & Knuth, 2011; Kieran, 2007; Radford, 2008, 2014). The effectiveness of these environments relies on the teachers’ ability to pay careful attention to students’ thinking, which then guides their instructional decisions. This inevitably raises the crucial question of the teachers’ knowledge for managing these types of situations in the classroom. In this context, this paper focuses on the mathematical knowledge for teaching figural pattern activities of 100 teachers at the primary and early secondary school levels. The results show that many primary teachers lack the essential knowledge for teaching these types of activities: they do not have a clear idea of their goal, they do not consider non-standard algebraic generalisations to be correct and they generally seem unable to help students to improve their arithmetical generalisation. Secondary school teachers also seem unable to give adequate feedback to improve students’ arithmetical generalisation. Although they seem to recognise that these activities are intended to improve algebraic thinking, they do not have a clear perception of the role that primary school learning can play in the development of this algebraic thinking
Disciplines :
Education & instruction
Author, co-author :
Demonty, Isabelle ; Université de Liège - ULiège > Département des Sciences de l'éducation > Analyse des systèmes et pratiques d'enseignement
Vlassis, Joëlle; Université du Luxembourg
Fagnant, Annick ; Université de Liège - ULiège > Département des Sciences de l'éducation > Didactique générale et intervention éducative
Language :
English
Title :
Algebraic thinking, pattern activities and knowledge for teaching at the transition between primary and secondary school.
Alavi, S. M. (2005). On the adequacy of verbal protocols in examining an underlying construct of a test. Studies in Educational Evaluation, 31(1), 1–26
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, 133–180
Bishop, J. W., & Stump, S. L. (2000). Preparing to teach in the new millennium: Algebra through the eyes of preservice elementary and middle school teachers. In Proceedings of the 22nd annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Tucson, AZ: ERIC
Cai, J., & Knuth, E. (2011). Early algebraization. A dialogue for multiple perspective. New York: Springer
Cai, J., Wang, N., Moyer, J. C., Wang, C., & Nie, B. (2011). Longitudinal investigation of the curricular effect: An analysis of student learning outcomes from the LieCal Project in the United States. International Journal of Educational Research, 50(2), 117–136
Carraher, D. W., & Schliemann, A. D. (2007). Early algebra and algebraic reasoning. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (Vol. 2, 669–705). Greenwich, CT: Information Age Publishing
Cooper, T. J., & Warren, E. (2011). Years 2 to 6 students’ ability to generalise: Models, representations and theory for teaching and learning. In J. Cai & E. Knuth (Eds.), Early algebraization. A dialogue for multiple perspective (pp. 187–214). Nex York: Springer
Demonty, I. (2015). L’enseignement des suites de nombres en primaire et au début de l’enseignement secondaire [The teaching of numeric pattern in primary school]. Unpublished research report
Demosthenous, E., & Stylianides, A. (2014). Algebra-related tasks in primary school textbooks. Proceedings of PME 38 and PME-NA, 36(2), 369–376
Depaepe, F., Torbeyns, J., Vermeersch, N., Janssens, D., Janssen, R., Kelchtermans, G., & Van Dooren, W. (2015). Teachers’ content and pedagogical content knowledge on rational numbers: A comparison of prospective elementary and lower secondary school teachers. Teaching and Teacher Education, 47, 82–92
Depaepe, F., Verschaffel, L., & Kelchtermans, G. (2013). Pedagogical content knowledge: A systematic review of the way in which the concept has pervaded mathematics educational research. Teaching and Teacher Education, 34, 12–25
Geraniou, E., Mavrikis, M., Hoyles, C., & Noss, R. (2011). Students’ justification strategies on the equivalence of quasi-algebraic expressions. Proceedings of PME, 35(2), 393–400
Hill, H. C., Sleep, L., Lewis, J. M., & Ball, D. L. (2007). Assessing teachers’ mathematical knowledge: What knowledge matters and what evidence counts. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (Vol. 1, 111–155). Greenwich, CT: Information Age Publishing
İmre, S. Y., & Akkoç, H. (2012). Investigating the development of prospective mathematics teachers’ pedagogical content knowledge of generalizing number patterns through school practicum. Journal of Mathematics Teacher Education, 15(3), 207–226
Jacobs, V. R., Franke, M. L., Carpenter, T. P., Levi, L., & Battey, D. (2007). Professional development focused on children’s algebraic reasoning in elementary school. Journal for Research in Mathematics Education, 38(3), 258–288
Kieran, C. (2007). Learning and teaching algebra at the middle school through college levels: Building meaning for symbols and their manipulation. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (Vol. 2, pp. 707–762). Greenwich, CT: Information Age Publishing
Kleickmann, T., Richter, D., Kunter, M., Elsner, J., Besser, M., Krauss, S., et al. (2013). Teachers’ content and pedagogical content knowledge: The role of structural differences in teacher education. Journal of Teacher Education, 64, 90–106
Krauss, S., Brunner, M., Kunter, M., Baumert, J., Blum,W., Neubrand, M., & Jordan, A. (2008). Pedagogical content knowledge and content knowledge of secondary mathematics teachers. Journal of Educational Psychology, 100, 716–725
Lannin, J. K. (2005). Generalisation and justification: The challenge of introducing algebraic reasoning through patterning activities. Mathematical Thinking and Learning, 7(3), 231–258
Mason, J. (1996). Expressing generality and roots of algebra. In N. Bernarz, C. Kieran, & L. Lee (Eds.), Approaches to algebra (pp. 65–86). Netherlands: Springer
Menzel, B., & Clarke, D. (1998). Teachers interpreting algebra: Teachers’ views about the nature of algebra. In C. Kanes, M. Goos, & E. Warren (Eds.), Teaching Mathematics in New Times (pp. 365–372). Gold Coast: MERGA
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. A dialogue for multiple perspective (pp. 277–301). New York: Springer
Radford, L. (2008). Iconicity and contraction: A semiotic investigation of forms of algebraic generalizations of patterns in different contexts. ZDM, 40(1), 83–96
Radford, L. (2014). The progressive development of early embodied algebraic thinking. Mathematics Education Research Journal, 26, 257–277
Rivera, F. D. (2013). Teaching and learning patterns in school mathematics: Psychological and pedagogical considerations. New York, NY: Springer
Rivera, F. D., & Becker, J. R. (2011). Formation of suite generalisation involving linear figural suites among middle school students: Results of a three-year study. In J. Cai & E. Knuth (Eds.), Early algebraization A dialogue for multiple perspective (pp. 323–366). New York: Springer
Senk, S. L., Tatto, M. T., Reckase, M., Rowley, G., Peck, R., & Bankov, K. (2012). Knowledge of future primary teachers for teaching mathematics: An international comparative study. ZDM Mathematics Education, 44, 307–324
Shulman, L. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57(1), 1–23
Stephens, A. C. (2008). What “counts” as algebra in the eyes of preservice elementary teachers? The Journal of Mathematical Behavior, 27(1), 33–47
Warren, E., & Cooper, T. J. (2009). Developing mathematics understanding and abstraction: The case of equivalence in the elementary years. Mathematics Education Research Journal, 21(2), 76–95
Warren, E., Trigueros, M., & Ursini, S. (2016). Research on the learning and teaching of algebra. In A. Gutiérrez, G. Leder, & P. Boero (Eds.), The second handbook of research on the psychology of mathematics education. The journey continues (pp. 73–108). Rotterdam: Sense
