The aim of this study is to explore pre-service teachers’ procedural and conceptual understanding of pupils’ understanding of mean value, before and after an intervention. Participants are 66 pre-service teachers, and two teacher educators. The data consists of five video-recorded lessons and pre-service teachers’ pre- and post-test discussions. Based on variation theory, the analysis aimed to identify pre-service teachers’ expressed knowledge of procedural and conceptual understanding, focusing four categories of expressions: procedural (P), procedural didactics (PD), conceptual (C), and conceptual didactics (CD). The pre-service teachers’ expressed understanding of pupils’ knowledge developed differently in the five teacher student groups. A shift from procedural to conceptual focus occurred in two groups. The number of procedural related words expressed in the first group decreased from 104 (pre-test: 65 P and 39 PD) to 16 (post-test: 9 P and 7 PD), at the same time as the conceptual related words increased (pre 4 C and 0 CD and post 11 C and 16 CD). In the second group, 87 procedure-related words (36 P and 51 PD) identified during pre-test decreased after the intervention (3 P and 9 PD). Instead, concept-related words increased from 16 (12 C and 4 CD) to 29 (10 C and 19 CD). The changes occurred among participants who had got, except of theoretical instruction, challenging hands-on tasks about the Mathematical content in focus. The findings from this study contribute to understand how design of instruction for pre-service teacher contributes to enhance teacher students’ didactic perspective on pupils’ learning.

• 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. https://doi.org/10.1177/0022487108324554
• Brante, G., Holmqvist Olander, M., Holmquist, P. O., & Palla, M. (2015). Theorising teaching and learning: pre-service teachers’ theoretical awareness of learning. European Journal of Teacher Education, 38(1), 102-118. https://doi.org/10.1080/02619768.2014.902437
• Chapman, O. (2013). Investigating teachers’ knowledge for teaching mathematics. Journal of Mathematics Teacher Education, 16(4), 237-243. https://doi.org/10.1007/s10857-013-9247-2
• Cobb, P., & Yackel, E. (2011). Chapter 4. Introduction. A journey in mathematics education research—Insights from the work of Paul Cobb, 33-40. https://doi.org/10.1007/978-90-481-9729-3_4
• Goulding, M., Rowland, T., & Barber, P. (2002). Does it matter? Primary teacher trainees’ subject knowledge in mathematics. British Educational Research Journal, 28(5), 689-704. https://doi.org/10.1080/0141192022000015543a
• 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: Erlbaum.
• Hill, H. C., Rowan, B., & Ball, D. L. (2005). Effects of teachers’ mathematical knowledge for teaching on student achievement. American educational research journal, 42(2), 371-406. https://doi.org/10.3102/00028312042002371
• Holmqvist, M. (2004). En främmande värld: om lärande och autism [A foreign world: about learning and autism]. Studentlitteratur.
• Holmqvist, M. (2011). Teachers’ learning in a learning study. Instructional science, 39(4), 497-511. https://search.ebscohost.com/login.aspx?direct=true&db=edsjsr&AN=edsjsr.23882877&lang=sv&site=eds-live&scope=site
• Holmqvist, M. (2017). Models for collaborative professional development for teachers in mathematics. International Journal for Lesson and Learning Studies, 6(3), 190-201. https://doi.org/10.1108/IJLLS-12-2016-0051
• Holmqvist, M. (Ed.). (2006). Att teoretisera lärandet. [To theorize learning]. In M. Holmqvist (Ed.), Lärande i skolan: Learning study som skolutvecklingsmodell [Learning in School: Learning Study as School Development Model] (pp. 9-24). Studentlitteratur.
• Holmqvist, M., & Selin, P. (2019). What makes the difference?: An empirical comparison of critical aspects identified in phenomenographic and variation theory analyses. Palgrave Communications, 5, 1-8. https://doi.org/10.1057/s41599-019-0284-z
• Kullberg, A., Runesson, U., Marton, F., Vikström, A., Nilsson, P., Mårtensson, P., & Häggström, J. (2016). Teaching one thing at a time or several things together?–Teachers changing their way of handling the object of learning by being engaged in a theory-based professional learning community in mathematics and science. Teachers and teaching, 22(6), 745-759. https://doi.org/10.1080/13540602.2016.1158957
• Lewis, C., & Tsuchida, I. (1997). Planned educational change in Japan: The case of elementary science instruction. Journal of Education Policy, 12(5), 313-331. https://doi.org/10.1080/0268093970120502
• Liljekvist, Y. (2014). Lärande i matematik: Om resonemang och matematikuppgifters egenskaper. [About reasoning and the characteristics of mathematical problems] (Doctoral dissertation, Karlstads universitet).
• Lo, L. M., Chik, P., & Pang, M. F. (2006). Patterns of variation in teaching the colour of light to Primary 3 students. Instructional Science, 34(1), 1-19. https://doi.org/10.1007/s11251-005-3348-y
• Lo, M. L. (2012). Variation theory and the improvement of teaching and learning. Acta Universitatis Gothoburgensis.
• Marton, F., & Booth, S. (1997). Learning and awareness. Routledge.
• Marton, F., & Lo, M. L. (2007). Learning from “The Learning Study”. Tidskrif foer Laerarutbildning och Forskning [Journal of Research in Teacher Education], 2007(1), 31-44.
• Marton, F., Runesson, U., & Tsui, A. B. M. (2004). The space of learning. In F. Marton (Ed.), Classroom discourse and the space of learning (pp. 3-36). Lawrence Erlbaum Associates Publisher. https://doi.org/10.4324/9781410609762
• Pang, M. F., & Marton, F. (2003). Beyond ``lesson study’’: Comparing two ways of facilitating the grasp of some economic concepts. Instructional Science, 31(3), 175-194. https://doi.org/10.1023/A:1023280619632
• PRIM (2020). Examples of tasks for assessment in mathematics Grade 6. Stockholm University, Department of Mathematics and Didactics. https://www.su.se/primgruppen/matematik/%C3%A5rskurs-6/exempel-ur-tidigare-prov
• Radford, L. (2013). Three key concepts of the theory of objectification: Knowledge, knowing, and learning. REDIMAT-Journal of Research in Mathematics Education, 2(1), 7-44. https://search.ebscohost.com/login.aspx?direct=true&db=eric&AN=EJ1112191&lang=sv&site=eds-live&scope=site
• Rowland, T. (2014) The Knowledge Quartet: the genesis and application of a framework for analysing mathematics teaching and deepening teachers’ mathematics knowledge. SISYPHUS Journal of Education, 1(3), 15-43. https://doi.org/10.25749/sis.3705
• Rowland, T., Martyn, S., Barber, P., & Heal, C. (2000). Primary teacher trainees’ mathematics subject knowledge and classroom performance. Research in Mathematics Education, 2(1), 3-18. https://doi.org/10.1080/14794800008520064
• Shulman, L (1986) Those who understand, knowledge growth in teaching. Educational Researcher, 15(2), 4-14. https://doi.org/10.3102/0013189X015002004
• Skott, J., Mosvold, R., & Sakonidis, C. (2018): Classroom practice and teachers’ knowledge beliefs, and identity. In T. Dreyfus, M. Artigue, D. Potari, S. Prediger, & K. Ruthven (Eds.), Developing research in mathematics education. Twenty years of communication, cooperation, and collaboration in Europe (pp. 162-180). Routledge. https://doi.org/10.4324/9781315113562-13
• Star, J. R. (2005). Reconceptualizing procedural knowledge. Journal for research in mathematics education, 36(5), 404-411. https://search.ebscohost.com/login.aspx?direct=true&db=afh&AN=18830665&lang=sv&site=eds-live&scope=site
• Stigler, J. W., & Hiebert, J. (1999). The teaching gap: Best ideas from the world’s teachers for improving education in the classroom. Free Press.
• Tutak, F. A., & Adams, T. L. (2017). A study of geometry content knowledge of elementary preservice teachers. International Electronic Journal of Elementary Education, 7(3), 301-318. https://search.ebscohost.com/login.aspx?direct=true&db=edsdoj&AN=edsdoj.50746d225684271b5eb0766e964675e&lang=sv&site=eds-live&scope=site

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