* The theory of CMG *
The theory of Children's Mathematical Graphics
A usage-based theory of language acquisition: the emergence of the abstract symbolic language of mathematics Both oral and graphical languages - including the abstract written language of mathematics - are In learning to talk, babies, toddlers and young children observe and listen to those around them, taking in everything they can from their worlds, and absorbing the social rules of spoken language to communicate as they babble and gradually begin to try out words for familiar people, things and actions. In somewhat similar ways, toddlers and young children watch and listen to adults and peers as they use graphical marks and signs (drawing, writing, something mathematical), gradually building understanding of the cultural purposes and ways in which graphical signs are used. Language is socially learned. Of particular significance are socially interactive situations that allow children's growing cultural knowledge and understandings to develop. Sensitive adults and peers treat babies' and toddlers’ babbling and early words, and their early graphical marks and signs as meaningful. This acceptance, and their communicative interactions support toddlers and young children's developing confidence in speech and their use of graphicacy to communicate. In each instance, understanding and learning emerge, as they infuse their babbling and scribble-marks with intentions. Their fledging speech and graphical signs develop over time, enabling them to become proficient users of speech and of standard alphanumerical languages. Democratic and open learning cultures empower children, fostering agency as they communicate their thinking through their Mathematical Graphics, and enabling teachers to hear the voice of the child. (Worthington, 2025). |
communicative languages.
Emergent learning
All young children are emergent learners, at home, in pre-school, nursery school, kindergarten and in school. The word emergent indicates that “the child’s understanding is gradually developing, and is being produced through interactions within a culture, their understandings and knowledge expanding through activity" (Gillen, 2003, p. 19).
Emergent learning has been especially evident in emergent writing, our interest first triggered 
considerably by Marie Clay's work (1975). Having seen the power of children's emergent writing in our own classrooms, it was from this strong foundation that we began to explore children's graphical marks and signs when communicating their mathematical thinking, and originated the term Children's Mathematical Graphics. Our work is strongly grounded in cultural-historical and social-semiotic theories (e.g., Vygotsky, 1978; Kress, 1997).
Since our early beginnings, our work has been informed by many others' research findings and learning theories including children's difficulties with mathematics (Hughes, 1986); semiotics (Buchler/Peirce, 1955; ); play (e.g., Wood 2019); pretend play (Vygotsky, 1978); 'funds of knowledge' (Moll et al., 1992) and language acquisition (e.g. Tomasello, 2005; Lancaster, 2014) and research related to the pedagogy of Children's Mathematical Graphics.
We established the theory of Children's Mathematical Graphics early in the 1990s, developing a taxonomy that charted the various aspects of the children's developing graphical use and growing understandings of the abstract symbolic language of mathematics, and its associated pedagogy. Over 30 years later, we continue to research and publish our findings. Our work has been featured in a number of government and independent reviews and reports, and our extensive research into Children's Mathematical Graphics, and our numerous children's examples are all new to science.
- The voice of the child
- The learning environment
- Taxonomy
- Government and independent reviews and reports featuring Children's Mathematical Graphics
References
Clay, M. (1975). What did I write? Beginning writing behaviour.Heinemann.
Buchler, J. (Ed.) (1955). Philosophical writings of Peirce. Dover.
Gillen, (2003). The Language of Children. Routledge.
Hughes, M. (1986). Children and number: Children's difficulties in learning mathematics. Blackwell.
Lancaster, L. (2003). Moving into literacy: How it all begins. In N. Hall, J. Larson, & J. Marsh (Eds.),
Handbook of early childhood literacy (pp. 145-153). Sage.
Moll, L., Amanti, C., Neff,D., & Gonzalez, N. (1992). Funds of knowledge for teaching: Using a qualitative approach to connect homes and classrooms. Theory into Practice 3(2), 132-141.
Tomasello, M. (2003). Constructing a language: A usage-based theory of language acquisition. Harvard University.
Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes. Harvard University.
Wood, E. (2019). Unbalanced and unbalancing acts in the Early Years Foundation Stage: a critical discourse analysis of policy-led evidence on teaching and play from the office for standards in education in England (Ofsted). Education 3-13, 47(7), 784–795.