Why 'Children's Mathematical Graphics' Matter
What's the problem?
A number of researchers have identified the abstract written language of mathematics as the aspect of maths that causes young children the greatest difficulties, yet it is rare for early childhood curricula to address this important area of the mathematics curriculum.
We present a new theory of early 'written' mathematics. Children's Mathematical Graphics are not an alternative curriculum, but represent its missing element - mathematical literacy. They represent only a part of the mathematics curriculum, but they are nevertheless an important part, since 'written' mathematics is integral to mathematics throughout the school system.
Bridge the gap! 
We developed Children's Mathematical Graphics to help young children become mathematically literate, and to come to understand the formal 'written system of mathematics.
Our experiences in our classrooms with emergent writing, was itriggered considerably by Marie Clay's work (1975) and developed by researchers and teachers during the 1980s (e.g., Bissex, 1982; Ferriero & Teberosky, 1979; Goodman & Goodman, 1992; Newman, 1984); Smith, 1982; Teale & Sulzby, 1992) and others. This led to interest in what was known as whole language (e.g., Cambourne, 1988; Goodman, 1994). More recently, Gunter Kress (1997), highlighted the multimodal underpinnings of young children's early writing, followed by Pahl (1999) and others.
Our other significant influence was the seminal work of Martin Hughes (1986) who wrote of his research into the difficulties young children experience with the abstract symbolic language of mathematics. Through at first using their own, intuitive ways of representing their mathematical thinking, this helps 'close the gap'* of which Hughes wrote. This gap refers to the gap between children's concrete knowledge of maths, and their understanding of the abstract and formal written language of mathematics. Children's Mathematical Graphics points a way forward.
Hughes urged teachers to "build on children's own strategies' and to 'respect children's invented symbolism" (pp.176/177), and these are at the heart of Children's Mathematical Graphics. Our extensive research into Children's Mathematical Graphics, and our numerous children's examples are all new to science.
Children's Mathematical Graphics has no programmes, interventions or instruments, but is founded on our belief in young children, the importance of holistic experiences and a recognition of the power of emergent learning. 'Children's Mathematical Graphics' (CMG) is the term we originated to describe the marks, signs and symbols that children freely (and often spontaneously) choose to use to communicate their mathematical thinking.
The children's own marks, signs and symbols originate in very young children’s marks for drawing
(e.g., Lancaster, 2003; 2014), and we have included the word Children's to emphasise the significance of the children's role in their mathematics.
Children's Mathematical Graphics are not a scheme or a step-by-step approach and need no special programmes, interventions, instruments or published materials. They are founded on our belief in the amazing capacity, creativity and strengths of young children to make meaning and communicate their thinking in ways that make sense to them through holistic, social experiences and through their emergent learning.
Our research shows that the more children freely use their own graphics to communicate their mathematical thinking, the more readily they will come to understand the (formal) abstract symbolic language of mathematics, and this is what our research has found (Worthington, 2021).
Note: Children’s mathematical signs and representations are variously termed external representations; inscriptions; notations; cultural, psychological or symbolic tools; emergent models; schematisations; visual signs, and (from Worthington & Carruthers, 2003), Children’s Mathematical Graphics.
References
Clay, M. (1975). What did I write? Beginning writing behaviour. Heinemann.
Bissex, G. (1982). GNYS AT WRK: A child learns to write and read. Harvard.
Cambourne, B. (1988). The whole story: Natural learning and the acquisition of literacy in the classroom. Ashton Scholastic.
Carruthers, E., & Worthington, M. (2006). Children's Mathematics: Making Marks, Making Meanings. (2nd ed.). Sage.
Ferriero, E. & Teberosky, A. (1979). Literacy before schooling. Heinemann.
Goodman, Y., & Goodman, K. (1992). Vygotsky in a whole-language perspective. In L. Moll (Ed.). Vygotsky and education: Instructional implications and applications of sociohistorical psychology (pp. 223-250). Cambridge University Press.
Goodman, K. (1994). What's whole in whole language? Scholastic.
Hughes, M. (1986). Children and Number: Difficulties in learning mathematics. Basil Blackwell.
Kress, G. (1997). Before writing:Rethinking the paths to literacy. Routledge.
Lancaster, L. (2003). Moving into literacy: How it all begins. In N. Hall, J. Larson & J. Marsh (Eds.). Handbook of Early Childhood Literacy. Sage.
Lancaster, L. (2007). Representing the ways of the world: How children under three start to use syntax in graphic signs. Journal of Early Childhood Literacy 7 (2).
Lancaster, L. (2014). The emergence of symbolic principles: The distributions of mind in early sign making. Biosemiotics, 7(1), 19-47.
Newman, J. (1984). The craft of children's writing. Scholastic.
Pahl, K.(1999). Transformations: Children's meaning making in a nursery. Thentham.
Smith, F. (1982). Writing and the writer. Heinemann.
Teale, W. H.& Sulzby. E. (Eds.). (1992). Emergent literacy: Writing and reading. Ablex.
Worthington. M. (2009). Fish in the water of culture: Signs and symbols in young children's drawing. Psychology of Education Review 33(1), 37-46.
Worthington M. (2020). Mathematical signs and their cultural transmission in pretend play. In A. MacDonald, Danaia. L, & S. Murphy (Eds.), STEM education across the learning continuum (pp. 45-65). Springer Nature.
Worthington, M. (2021). The emergence and development of young children’s personal mathematical inscriptions: The evolution of graphical signs explored through children’s spontaneous pretend play. [Doctoral Dissertation]. Vrije Universiteit, Amsterdam.
Worthington, M. & Carruthers, E. (2003). Children's Mathematics: Making Marks, Making Meanings. (1st ed.). Paul Chapman.
Worthington, M., & van Oers, B. (2017). Children's social literacies: Meaning making and the emergence of graphic symbols in pretence. Journal of Early Childhood Literacy 24(1), 1-29.