The objective of the study was to characterise how pre-service kindergarten teachers used a Hypothetical Learning Trajectory on length and its measurement to notice children’s mathematical thinking. A total of 64 pre-service kindergarten teachers enrolled in an Early Years Education mathematics teaching course were asked to notice teaching situations focusing on kindergarten children’s learning of length. On the one hand, three profiles of pre-service kindergarten teachers were found according to the use they made of the Hypothetical Learning Trajectory. These profiles were characterised by three ways of learning to use the Hypothetical Learning Trajectory based on the type of mathematical elements they identified: only mathematical elements related to the magnitude; only mathematical elements related to the measurement of length; or both magnitude and measurement elements. On the other hand, when considering the three skills of professional noticing, by identifying the mathematical elements required to solve the proposed task, a group of pre-service kindergarten teachers within each profile were able to notice the thinking of these elements by children and to suggest activities. Our findings provide learning opportunities to pre-service kindergarten teachers who use a Hypothetical Learning Trajectory. It provides them with ‘check-points’ to answer the proposed questions, in order to learn the specialised knowledge for teaching length and its measurement as well as to develop the skill of noticing student’s mathematical thinking.
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Alsina, A. (2011). Educación matemática en contexto de 3 a 6 años. Barcelona: ICE-Horsori.
Authors (2017). Journal of Mathematics Teacher Education.
Barnhart, T., & van Es, E. (2015). Studying teacher noticing: Examining the relationship among pre-service science teachers’ ability to attend, analyze and respond to student thinking. Teaching and Teacher Education, 45, 83–93.
Barrett, J. E., Sarama, J. & Clements, D. H. (2017). Children’s measurement: A longitudinal study of children’s knowledge and learning of length, area, and volume. Journal for Research in Mathematics Education Monograph Series (Vol. 16). Reston, VA.
Baturo, A., & Naso, R. (1996). Student teachers’ subject matter knowledge within the domain of area measurement. Educational Studies in Mathematics, 31, 235–268.
Castle, K., & Needham, J. (2007). First graders’ understanding of measurement. Early Childhood Education Journal, 35(3), 215–221. https://doi.org/10.1007/s10643-007-0210-7.
Chappell, M. F., & Thompson, D. R. (1999). Perimeter or area? Which measure is it? Mathematics Teaching in de Middle School, 5, 20–23.
Clements, D. H. (2010). Teaching length measurement: Research challenges. School Science and Mathematics, 99(1), 5–11.
Clements, D. H., & Bright, G. (2003). Learning and teaching measurement: 2003 Yearbook. Reston, VA: National Council of Teachers of Mathematics.
Clements, D. H., & Sarama, J. (2004). Learning trajectories in mathematics education. Mathematical Thinking and Learning, 6(2), 81–89.
Clements, D. H., & Sarama, J. (2009). Learning and teaching early math: The learning trajectories approach. New York, NY: Routledge.
Congdon, E. L., Kwon, M. K., & Levine, S. C. (2018). Learning to measure through action and gesture: Children’s prior knowledge matters. Cognition, 180, 182–190. https://doi.org/10.1016/j.cognition.2018.07.002.
Ellis, S., Siegler, R. S., & Van Voorhis, F. E. (2003). Developmental changes in children’s understanding of measurement procedures and principles. Paper presented at the Society for Research in Child Development, Tampa, FL.
Fernández, C., Llinares, S., & Valls, J. (2012). Learning to notice students’ mathematical thinking through on-line discussions. ZDM. Mathematics Education, 44, 747–759.
Gómezescobar, A., Fernández-Cézar, R., & Guerrero, S. (2018). Numbers and space intervals in length measurements in the Spanish context: Proposals for the transition to measuring with the ruler. International Journal of Science and Mathematics Education, 16(7), 1375–1386.
Gómezescobar, A., Guerrero, S., & Fernández-Cézar, R. (2020). How long is it? Difficulties with the conventional ruler use in children aged 5 to 8. Early Childhood Education Journal, 48, 693–701. https://doi.org/10.1007/s10643-020-01030-y.
Gupta, D., Soto, M., Dick, L., Broderick, S. D., & Appelgate, M. (2018). Noticing and deciding the next steps for teaching: A cross-university study with elementary pre-service teachers. In G. Stylianides & K. Hino (Eds.), Research advances in the mathematical education of pre-service elementary teachers. An international perspective (pp. 261–275). Cham: Springer.
Hiebert, E. H. (1981). Developmental patterns and interrelationships of preschool children’s print awareness. Reading Research Quarterly, 16(2), 236–260.
Hines, E., & y McMahon, M.T. (2005). Interpreting middle school students’ proportional reasoning strategies: Observations from preservice teachers. School Science and Mathematics, 105(2), 88–105.
Irwin, K. C., Ell, F. R., & Vistro-Yu, C. P. (2004). Understanding linear measurement: A comparison of Filipino and New Zealand children. Mathematics Education Research Journal, 16(2), 3–24. https://doi.org/10.1007/BF03217393.
Ivars, P., Fernandez, C., & Llinares, S. (2017). Pre-service teachers’ uses of a learning trajectory to notice students’ fractional reasoning. In B. Kaur, W. K. Ho, T. L. Toh, & B. H. Choy (Eds.), Proceedings of the 41st Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 25–32). Singapore: PME.
Jacobs, V. R., Lamb, L. C., & Philipp, R. (2010). Professional noticing of children’s mathematical thinking. Journal for Research in Mathematics Education, 41(2), 169–202.
Kamii, C. (1995). Why is the use of a ruler so hard? Paper presented at the 17th annual meeting of the North American Chapter of the International group for the Psychology in Mathematics Education, Columbus, OH. Retrieved from https://files.eric.ed.gov/fulltext/ED389558.pdf
Kotsopoulos, D., Makosz, S., Zambrzycka, J., & McCarthy, K. (2015). The effects of different pedagogical approaches on the learning of length measurement in kindergarten. Early Childhood Education Journal, 43(6), 531–539. https://doi.org/10.1007/s10643-014-0686-x.
Lee, J. (2010). Exploring kindergarten teachers’ pedagogical content knowledge of mathematics. International Journal of Early Childhood, 42(1), 27–41.
Lehrer, R. (2003). Developing understanding of measurement. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 179–192). Reston, VA: National Council of Teachers of Mathematics.
Lobato, J., & Walkers, C. D. (2017). A taxonomy of approaches to learning trajectories and progressions. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 74–101). Reston, VA: National Council of Teachers of Mathematics.
Mason, J. (2002). Researching your own practice. The discipline of noticing. London: Routledge-Falmer.
Menon, R. (1998). Preservice teachers’ understanding of perimeter and area. School Science and Mathematics, 98, 361–368.
Mojica, G. F., & Confrey, J. (2009). Pre-service elementary teachers’ understanding of an equipartitioning learning trajectory. In S. L. Swars, D. W. Stinson, & S. Lemons-Smith (Eds.), Embracing diverse perspectives: Proceedings of the 31st Annual Meeting of the North American Chapter of the International Group of the Psychology of Mathematics Education. Georgia State: Atlanta, GA.
Nunes, T., & Bryant, P. E. (1996). Children doing mathematics. Cambridge, MA: Blackwell.
Panorkou, N., & Kobrin, J. L. (2017). Enhancing teachers’ formative assessment practices through learning trajectory-based professional development. Mathematics Teacher Education, 5(2), 178–201.
Parks, A. M., & Wager, A. A. (2015). What knowledge is shaping teacher preparation in early childhood mathematics? Journal of Early Childhood Teacher Education, 36(2), 124–141.
Sarama, J., & Clements, D. (2009). Early childhood mathematics education research. Learning trajectories for young children. London and New York: Routledge.
Sherin, M. G., Jacobs, V. R., & Philipp, R. A. (2011). Situating the study of teacher noticing. In M. G. Sherin, V. R. Jacobs, & R. A. Philipp (Eds.), Mathematics teacher noticing: Seeing through teachers’ eyes (pp. 3–13). New York: Routledge.
Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26(2), 114–145.
Simon, M. A. (2006). Key developmental understandings in mathematics: A direction for investigating and establishing learning goals. Mathematical Thinking and Learning, 8(4), 359–371.
Simon, M. A., & Blume, G. W. (1994). Building and understanding multiplicative relationships: A study of prospective elementary teachers. Journal for Research in Mathematics Education, 25, 472–494.
Skoumpourdi, C. (2015). Kindergartners measuring length. In K. Krainer & N. Vondrová (Eds.), Proceedings of the Ninth Conference of the European Society for Research in Mathematics Education (CERME9) (pp. 89–1995). Prague, Czech Republic: Charles University.
Smith III, J. P., van den Heuvel-Panhuizen, M., & Teppo, A. R. (2011). Learning, teaching, and using measurement: Introduction to the issue. ZDM Mathematics Education, 43, 617–620.
Solomon, T. L., Vasilyeva, M., Huttenlocher, J., & Levine, S. C. (2015). Minding the gap: Children’s difficulty conceptualizing spatial intervals as linear measurement units. Developmental Psychology, 51(11), 1564–1573. https://doi.org/10.1037/a0039707.
Stahnke, R., Schueler, S., & Roesken-Winter, B. (2016). Teachers’ perception, interpretation, and decision-making: A systematic review of empirical mathematics education research. ZDM Mathematics Education, 48(2), 1–27.
Stephan, M., Bowers, J., Cobb, P., & Gravemeijer, K.P.E. (2003). Supporting students’ development of measuring conceptions: Analyzing students’ learning in social context. Journal for Research in Mathematics Education Monograph Series, 12.
Stephan, M., & Clements, D. H. (2003). Linear and area measurement in prekindergarten to grade 2. In D. H. Clements & G. Bright (Eds.), Learning and teaching measurement. 2003 Yearbook (pp. 3–16). Reston, VA: National Council of Teachers of Mathematics.
Stockero, S. L. (2014). Transitions in prospective mathematics teacher noticing. In J. J. Lo, K. R. Leatham, & L. R. Van Zoest (Eds.), Research trends in mathematics teacher education (pp. 239–259). Cham: Springer.
Strauss, A., & Corbin, J. (1994). Grounded theory methodology. In N. K. Denzin & Y. S. Lincoln (Eds.), Handbook of qualitative research (pp. 217–285). Sage Publications: Thousand Oaks.
Szilagyi, J., Clements, D. H., & Sarama, J. (2013). Young children’s understanding of length measurement: Evaluating a learning trajectory. Journal for Research in Mathematics Education, 44(3), 581–620.
van den Heuvel-Panhuizen, M., & Buys, J. (Eds.). (2005). Young children learn measurement and geometry. Amersfoort: Freudenthal Institute.
Van Es, E. A., & Sherin, M. G. (2008). Mathematics teachers’ “learning to notice” in the context of a video club. Teaching and Teacher Education, 24(2), 244–276.
Walkoe, J. (2015). Exploring teacher noticing of students algebraic thinking in a video club. Journal of Mathematics Teacher Education, 18(6), 523–550.
Wilson, P. H., Mojica, G., & Confrey, J. (2013). Learning trajectories in teacher education: Supporting teachers’ understanding of students’ mathematical thinking. Journal of Mathematical Behavior, 32, 103–121.
Wilson, P. H., Sztajn, P., Edgington, C., & Confrey, J. (2014). Teachers’ use of their mathematical knowledge for teaching in learning a mathematics learning trajectory. Journal of Mathematics Teacher Education, 17, 149–175.
Zacharos, K. (2006). Prevailing educational practices for area measurement and students’ failure in measuring areas. Journal of Mat hematical Behavior, 25, 224–239.S.
Mari Luz passed away when we were reviewing this paper to IJSME. We would like this article in rememberance of her.
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Callejo, M.L., Pérez-Tyteca, P., Moreno, M. et al. The Use of a Length and Measurement HLT by Pre-Service Kindergarten Teachers’ to Notice Children’s Mathematical Thinking. Int J of Sci and Math Educ (2021). https://doi.org/10.1007/s10763-021-10163-4
- Hypothetical Learning Trajectory
- Length and measurement
- Pre-service kindergarten teachers
- Professional noticing