References
- Alex, J., & Roberts, N. (2019). The need for relevant initial teacher education for primary mathematics: Evidence from the Primary Teacher Education project in South Africa. In N. Govender, R. Mudaly, T. Mthetwa, & A. SinghPillay (Eds.), Book of proceedings, 27th annual conference of the Southern African Association for Research in Mathematics, Science and Technology Education (SAARMSTE) (pp. 59–72). Durban: University of KwaZulu–Natal.
- Askew, M. (2012). Transforming primary mathematics. Abingdon: Routledge.
- Askew, M., & Brown, M. (2003). How do we teach children to be numerate? A professional user review of UK research undertaken for the British Educational Research Association.
- Barmby, P., Bolden, D., & Harries, T. (2011). A representational approach to developing primary ITT students’ confidence in their mathematics. In Proceedings of the British Society for Research into Learning Mathematics, 31(1), 31–36.
- Bass, H. (2015). Quantities, numbers, number names and the real number line. Primary Mathematics Study on Whole Numbers: ICMI Study 23. Springer.
- Bowie, L., & Reed, Y. (2016). How much of what? An analysis of the espoused and enacted mathematics and English curricula for intermediate phase student teachers at five South African universities. Perspectives in Education, 34(1), 102–119.
- Bowie, L., Venkat, H., & Askew, M. (2019). Pre-service primary teachers’ mathematical content knowledge: An exploratory study. African Journal of Research in Mathematics, Science and Technology Education, 23(3), 286–297.
- Brombacher, A., & Roberts, N. (2022). The NumberSense Programme: A focus on Shikaya intervention in South Africa. In N. Spaull & S. Taylor (Eds.), Early grade reading and mathematics interventions in South Africa. Oxford: Oxford University Press.
- Butterworth, B. (2005). The development of arithmetical abilities. Journal of Child Psychology and Psychiatry, 46(1), 3–18.
- Carpenter, T., & Fennema, E. (1999). Children’s mathematics: Cognitively guided instruction (1st ed.). London: Heineman.
- Deacon, R. (2012). The Initial Teacher Education Research Project: The initial professional development of teachers: A literature review. Roger Deacon.
- Dehaene, S. (1992). Varieties of numerical abilities. Cognition, 44(1), 1–42.
- Ensor, P., Hoadley, U., Jackline, H., & Kuhne, C. (2009). Specialising pedagogic text and time in foundation phase numeracy classrooms. Journal of Education, 47, 5–30.
- Fonseca, K., Maseko, J., & Roberts, N. (2018). Students mathematics knowledge in a Bachelor of Education (foundation or intermediate phase) programme. In R. Govender & K. Junqueira (Eds.), Proceedings of the 24th annual national congress of the Association for Mathematics Education of South Africa (pp. 124–139). AMESA.
- Gelman, R., & Gallistel, C.R. (1978). The child's understanding of number. Cambridge, MA: Harvard University Press.
- Goldin, G. (2020). Mathematical representations. In S. Lerman (Ed.), Encyclopedia of mathematics education. Cham: Springer.
- Graven, M. (2022). A decade of the South African Numeracy Chair (SANC). In H. Venkat & N. Roberts (Eds.), Mathematics education in the early grades. Oxford: Oxford University Press.
- Graven, M., Venkat, H., Westaway, L., & Tshesane, H. (2013). Place value without number sense: Exploring the need for mental mathematical skills assessment within the Annual National Assessments. South African Journal of Childhood Education, 3(2).
- Hoadley, U. (2012). What do we know about teaching and learning in South African primary schools? Education as Change, 6(2), 187–202.
- Lakoff, G., & Núñez, R. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York: Basic Books.
- Moloi, Q., Roberts, N., & Thomo, J. (2022). Using NumberSense workbooks and formative assessment to improve learning outcomes in early grade mathematics. In N. Spaull & S. Taylor (Eds.), Early grade reading and mathematics interventions in South Africa. Oxford: Oxford University Press.
- Mulligan, J., Verschaffel, L., Baccaglini-Frank, A., Coles, A., Gould, P., He, S., … Yang, D. (2018). Whole number thinking, learning and development : Neuro-cognitive, cognitive and developmental approaches. Springer.
- Naidoo, D., & Venkat, H. (2013). Coherent discourse and early number teaching. Journal of Education, 57, 55–77.
- Primary Teacher Education (PrimTEd) Assessment workstream (2019) Attainment in PrimTEd mathematics assessments across South African universities, PrimTEd: Johannesburg.
- Porteus, K. (2022). Improving rural early grade mathematics: Design principles and patterns of improvement. In H. Venkat, & N. Roberts (Eds.), Early grade mathematics in South Africa (pp. 97–118). Oxford: Oxford University Press.
- Porteus, K., & Mostert, I. (2022). Moving beyond counting in ones: A South African framework for number picture representations. In T. Mosiane, L. Mokotedi, D. Laubscher, & S. Mahlab (Eds.), Proceedings of the 27th annual national congress of the Association for Mathematics Education of South Africa (pp. 177–198). Nelson Mandela Institute.
- Porteus, K., Roberts, N., & Moloi, Q. (2021). Grade 3 backlogs in mathematics: research report. East London: Nelson Mandela Institute, University of Fort Hare.
- Ramadiro, B., & Porteus, K. (2017). Foundation phase matters: Language and learning in South African rural classrooms. East London: Magic Classroom Collective Press.
- Roberts, N. (2017). A historical and socio-political perspective on mathematics education in South Africa. In P. Webb & N. Roberts (Eds.), The pedagogy of mathematics: Is there a unifying logic? (pp. 41–65). Johannesburg: Mapungubwe Institute for Strategic Reflection (MISTRA)/Real African Publishers.
- Roberts, N. (2019). The standard written algorithm for addition: Whether, when and how to teach it. Pythagoras, 40(1), 1–17.
- Roberts, N., Mostert, I, Kakoma, L., Long, C., & Maseko, J. (2019) Maths Intensive Course, University of Johannesburg, Centre for Education Practice Research (CEPR): Soweto.
- Roberts, N. (2020). Maths Intensive first design trial in response to PrimTEd mathematics assessment. In P. Vale, L. Westaway, Z. Nhase, & I. Shudel (Eds.), Proceedings of the 28th annual conference of the Southern African Association for Research in Mathematics, Science and Technology Education (pp. 126–142).
- Roberts, N., & Maseko, J. (2022). From measuring impact to informing course design: The second design trial of the Maths Intensive course. Journal of Education, 87, 1–22.
- Roberts, N., & Moloi, Q. (2022). The quality of primary mathematics teacher preparation in SA: Findings from PrimTEd. Resep: University of Stellenbosch.
- Roberts, N., Porteus, K., Morrison, S., Jooste, Z., & Westaway, L. (2021). Emergent number sense: Maths 4 primary teachers lecturer guide.
- Sasman, M.C., & Linchevski, L. (1999). The influence of different representations on children’s generalisation thinking processes. Seventh annual conference of the Southern African Association for Research in Mathematics and Science Education.
- Schmittau, J. (2003). Cultural historical theory and mathematics education. In A.B.V. Kozulin & M.S. Cambridge (Eds.), Vygotsky’s educational theory in cultural context (pp. 225–245). Cambridge: Cambridge University Press.
- Schollar, E. (2008). Final report of the primary mathematics research project.
- Sfard, A. (2008). Thinking as communicating: Human development, the growth of discourses, and mathematizing. Cambridge: Cambridge University Press.
- Sinclair, N., & Coles, A. (2015). ‘A trillion is after one hundred’: Early number and the development of symbolic awareness. In X. Sun, B. Kaur, & J. Novotná (Eds.), The twenty-third ICMI study: Primary mathematics study on whole numbers. Macao, China: ICMI.
- Spencer-Smith, G., Hazell, E., & Kuhne, C. (2022). The R-Mathematics Project (R-Maths) in the Western Cape 2016–2019: Conceptualisation, development, implementation and development. In N. Spaull, & S. Taylor (Eds.), Early grade reading and mathematics interventions in South Africa (pp. 189–206). Oxford: Oxford University Press.
- Tall, D.O. (2004). Building theories: Three worlds of mathematics. For the Learning of Mathematics, 24(1), 29–32.
- Taylor, N. (2021). The dream of sisyphus: Mathematics education in South Africa. South African Journal of Childhood Education, 11(1), 1–12.
- Treffers, A., & Buys, K. (2008). Children learn mathematics. In M. van den Heuvel-Panhuizen (Ed.), Children learn mathematics: A learning teaching trajectory with intermediate attainment targets for calculation with whole numbers in primary school. Rotterdam/Tapei: Sense.
- Venkat, H. (2022). A decade of Wits Connect Primary Mathematics. In H. Venkat & N. Roberts (Eds.), Mathematics education in the early grades. Oxford: Oxford University Press.
- Venkat, H. (2013). Using temporal range to theorize early number teaching in South Africa. For the Learning of Mathematics, 33(2), 31–37.
- Venkat, H., & Roberts, N. (2022). Children doing mathematics with confidence by 2030: What will it take? In H. Venkat & N. Roberts (Eds.), Early grade mathematics in South Africa (pp. 208–224). Oxford: Oxford University Press.
- Venkat, H., & Spaull, N. (2015). What do we know about primary teachers’ mathematical content knowledge in South Africa? An analysis of SACMEQ 2007. International Journal of Educational Development, 41, 121–130.
- Wheatley, G. (1992). The role of reflection in mathematics learning. Educational Studies in Mathematics, 23(5), 529–541.