*Credits: All modules are presented by Terezinha Nunes (University of Oxford) except Module 0 presented by Gabriel Stylianides & Louise Matthews (University of Oxford). All videos are filmed by Miguel Mocho (Mocho Cinematic) with a contribution from The Open University.*

**General Introduction
**

This unit was designed to highlight moments in the path that, as a psychologist, I followed as I became more engaged with how children learn mathematics. This path starts with the encounter with findings from my own research that challenged conceptions of ability and pedagogy that were predominant at the time, and that I espoused. There were no theories in psychology that helped me to understand the within-individual differences described in the work on street and school mathematics, first published in Portuguese in 1982, but I found some direction in the work of Ubiratan d'Ambrosio and Gérard Vergnaud when I attended ICME in 1984 in Adelaide, Australia. My personal trajectory was hugely influenced by what I learned there and in many subsequent PME meetings.

A theme that emerges at every module is the cultural nature of mathematics and the difficulties of trying to coordinate the idea of logical invariants with culture. Over the four decades during which I have been working on how children learn mathematics. Freudenthal’s ideas and the work of researchers from the Freudenthal Institute have helped me to understand that children learn mathematics as use actions schemas that capture invariants in the situations that they mathematize and learn to use the mathematical tools for representing and processing information. This is the short version of the story and I hope it will motivate you to watch the modules and find out about the full story

**Bibliography**

Official citation for the 2017 Felix Klein ICMI Award

https://www.mathunion.org/icmi/awards/hans-freudenthal-award/2017-hans-freudenthal-award

Research always takes place in an intellectual context and in this module I try to evoke some of the issues being debated at the time the research was carried out. These issues included why working-class children performed less well in school than their peers from more advantaged socio-economic backgrounds and how children could be taught mathematics with understanding of concepts and of their relevance outside school. The study of street mathematics highlighted within-individual differences in children’s performance in and out of school and the existence of two different cultural practices, oral and written arithmetic, which are some of the visible differences between street and school mathematics.

**Bibliography**

Carraher, T. N., Carraher, D. W., & Schliemann, A. D. (1985). Mathematics in the streets and in schools. British Journal of Developmental psychology, 3(1), 21-29.

Carraher, T. N. & Schliemann, A. D. (1985). Computation routines prescribed by schools: Help or hindrance? Journal for Research in Mathematics Education, 16, 1, 37-44. d'Ambrosio, U. (1985). Ethnomathematics and its place in the history and pedagogy of mathematics. For the Learning of Mathematics, 5(1), 44-48.

Freudenthal, H. (1983). Didactical Phenomenology of Mathematical Structures. Dordrecht: D. Reidel.

Gay, J., & Cole, M. (1967). The New Mathematics and an Old Culture. A Study of Learning among the Kpelle of Liberia. New York: Holt, Rinehart and Winston.

Nunes, T., Schliemann, A. D., & Carraher, D. W. (1993). Street Mathematics and School Mathematics. New York: Cambridge University Press.

Vergnaud, G. (1997). The nature of mathematical concepts. In T. Nunes & P. Bryant (Eds.), Learning and Teaching Mathematics. An International Perspective (pp. 1-28). Hove (UK): Psychology Press.

Arithmetic is visible; the reasoning that lies beneath arithmetic is not. Module 2 focuses on proportional reasoning, a form of quantitative reasoning considered by developmental psychologists Piaget and Inhelder as a hallmark of advanced cognitive development and by mathematics educators a challenge for many secondary school students. The module focuses on proportional reasoning used outside school. I suggest that the origins of multiplicative (proportional) reasoning are not in repeated addition, but in the action schema of one-to-many correspondence, which children and adults use in many everyday activities. Drawing on Vergnaud’s ideas, I explore the forms of reasoning about proportions that emerge without schooling and speculate about the role of schooling in promoting proportional reasoning.

**Bibliography**

Brink, J. V. D., & Streefland, L. (1979). Young Children (6-8): Ratio and Proportion. Educational Studies in Mathematics, 10, 403-420.

Carraher, T. Nunes. (1986). From drawings to buildings: Working with mathematical scales. International Journal of Behavioral Development, 9, 527-544.

Freudenthal, H. (1968). Why to Teach Mathematics So as to Be Useful. Educational Studies in Mathematics, 1, 3-8.

Inhelder, B., & Piaget, J. (1958). The growth of logical thinking from childhood to adolescence. New York: Basic books.

Nunes, T., & Bryant, P. (1996). Children Doing Mathematics. Oxford: Blackwell.

Nunes, T., Schliemann, A. D., & Carraher, D. W. (1993). Street Mathematics and School Mathematics. New York: Cambridge University Press.

Schliemann, A. D. & Nunes, T. (1990) A situated scheme of proportionality. British Journal of Developmental Psychology, 8, 259-268.

Vergnaud, G. (1983). Multiplicative structures. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 128-175). London: Academic Press.

The theme of this module is school mathematics. I argue that understanding school mathematics requires making a distinction between quantitative reasoning and arithmetic and that quantitative reasoning does not receive yet the focus it merits; it remains invisible in the classroom. The pedagogical belief that teaching arithmetic is sufficient for applying mathematics has not led to successful mathematics learning for many people. The alternative approach proposed by Freudenthal, which is to help children to mathematize situations by focusing on the logical invariants while learning to use mathematical tools leads to the construction of quantitative reasoning systems. New cultural tools are being created continuously but, so far, they do not replace reasoning; rather, they need to be incorporated into a person’s reasoning system. I write this as AI becomes a new and more powerful cultural tool that can be used in mathematics; we do not know yet if AI will also become a tool to be incorporated into reasoning systems or if it will replace reasoning.

**Bibliography**

Brink, J. V. D., & Streefland, L. (1979). Young Children (6-8): Ratio and Proportion. Educational Studies in Mathematics, 10, 403-420.

de Corte, E., Greer, B., & Verschaffel, L. (2000). Making sense of word problems: CRC Press.

Guedj, D. (1998). Numbers. A universal language. London: Thame and Hudson.

Hoyles, C., Noss, R., & Pozzi, S. (1999). Mathematising in practice. In C. Hoyles, C. Morgan, & G. Woodhouse (Eds.), Rethinking the mathematics curriculum (Vol. 10, pp. 48-62). London: Falmer Press.

Luria, A. R. (1973). The working brain. An introduction to neuropsychology. Harmondsworth (UK): Penguin. Nunes, T. (2004). Teaching Mathematics to Deaf Children. London: Wiley/Blackwell (translated into Greek).

Nunes, T. (2002). The role of systems of signs in reasoning. In T. Brown & L. Smith (Eds.), Reductionism and the Development of Knowledge (pp. 133-158). Mawah (NJ): Lawrence Erlbaum.

Nunes, T., Bryant, P., Gottardis, L., Terlektsi, M.-E., & Evans, D. (2015). Can we really teach problem solving in primary school? Mathematics Teaching, 246, 44-48.

Thompson, P. W. (1993). Quantitative Reasoning, Complexity, and Additive Structures. Educational Studies in Mathematics, 3, 165-208.

Verschaffel, L., & De Corte, E. (1993). A decade of research on word problem solving in Leuven: Theoretical, methodological, and practical outcomes. Educational Psychology Review, 5(3), 239-256. doi:10.1007/BF01323046

In Module 4 I present a synthesis of theoretical ideas that help me to conceptualize mathematics learning and teaching. I start by rejecting a currently popular hypothesis that number knowledge originates from associations between number words and specific numerosities. In line with a distinct psychological tradition, I argue that number knowledge is intrinsically related to quantitative reasoning and that, in teaching, it is best not to leave this coordination to chance, but to build the connections between quantitative reasoning and number by helping children to mathematize situations.

**Bibliography**

Nunes, T., & Bryant, P. (1996). Children Doing Mathematics. Oxford: Blackwell.

Nunes, T., & Bryant, P. (2022). Using Mathematics to Understand the World. How Culture Promotes Children's Mathematics. London: Taylor & Francis.

Nunes, T., & Bryant, P. (2022). Number systems as models of quantitative relations. In G. K. Akar, İ. Ö. Zembat, S. Arslan, & P. W. Thompson (Eds.), Quantitative reasoning in science and mathematics education (pp. 71-106): Springer.

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