In order to well understand and use the abstract symbolic

language of mathematics, above all, young children need to be free to communicate their mathematical thinking in their own ways.

The 'Goldilocks principle

          This refers to babies and young children's 'preference to attend to events that are neither too simple nor too complex, according to their current representation of the world' (Kidd et al., 2012). Rather than teachers planning children's pretend play, or planning what and how young children should write or represent

their mathematics, providing opportunities to explore their thinking

in ways of their own choosing, can allow children to engage in learning that is 'just right'.

    For young children, free pretend play provides ideal contexts in which children can explore their cultural knowledge (i.e., their 'funds of knowledge' - Moll et al., 1992). Often their play will include aspects of mathematics that they communicate with their peers, through speech and graphicacy.

   A second potentially valuable context, is in open, adult-planned small groups. In these the adult might, for example provide something such as an artefact with mathematical potential, to trigger children's interest.

  For example, one teacher provided several bathroom scales for the group to explore. Isaac (4 years, 5 months) used his emergent understanding of a variety of measuring units to talk about David’s weight. As he intentionally made circular scribble-marks on paper (above right), Isaac explained “David weighs 700 kilos, he’s 60 metres heavy.”  Next David (4 years, 1 month) stood on the scales, and looking at the dial he announced, “I’m 15, so I need to write it down.” He made some letter-like signs (as numerals) on the whiteboard (not shown here). The children's graphics were subsequently shared within the group, individuals and their teacher commenting on various examples and what they showed.

    In kindergarten or school (depending on your country's curriculum) in mathematics lessons, children may be expected to write or represent something mathematical that they are working on. Rather than copying something the teacher has given them or completing worksheets, blank paper provides the very best opportunities for children to explore their own thinking in their own ways, and make personal sense of mathematical signs, symbols and meaningful ways of representing, such as problem solving, calculations or data handling. 

Note: The article below relates directly to children's 'recording' in mathematics lessons:

References

Kidd, C., Piantadosi, S. T., Aslin., & R. N. (23 May 2012). "The Goldilocks Effect: Human Infants Allocate Attention to Visual Sequences That Are Neither Too Simple Nor Too Complex". PLOS ONE. 7 (5).

Moll, L. Amanti, C., Neff, D., & Gonzales, N. (1992). Funds of knowledge for teaching: Using a 

qualitative approach to connect homes and classrooms. Theory into Practice, 31(2), 132-141. 

Worthington, M., & van Oers, B. (2016). Pretend play and the cultural foundations of mathematics. European Early Childhood Education Research Journal 24(1), 51-66.