In England the use of the words recording (or written maths) are used extensively to refer to copying symbols and written strategies provided by the teacher, or writing something they have already done following a practical activity with objects. Recording what they did following a practical activity has limited value for children and involves lower levels of thinking. Children do not need to record mathematics if they can do it mentally; neither do they need to record something they have worked out in a practical context.
We found that recording is of very limited value for the child, placing the emphasis on signs, symbols, drawings and problem-solving strategies as products to be formally assessed, and requires a lower level of cognitive demand (thinking) in mathematics. The difference between representing mathematical thinking and recording is one of quality and depth of thinking. Most significantly, recording fails to enable the child to build deep understandings of semiotic representation for mathematics.
In children’s own Mathematical Graphics, the emphasis is on the children's emerging understandings and on the processes of mathematical thinking (creative thinking, reasoning, meanings, understanding, problem-solving, flexible thinking, negotiation and co-construction of understanding) rather than products (recording something done practically). Authentic, meaningful social contexts for their mathematics allows children to connect their existing cultural knowledge and make greater sense of their mathematics when they explore their thinking through their own representations.
For children, one of the benefits of using paper (or another drawing/writing surface) to explore their thinking, is that their own marks, signs and symbols support understanding by allowing them to see some of their emerging understanding of ‘written’ mathematics, helping them to 'translate' between their early informal marks and the standard symbols and the formal, abstract written language of mathematics. Children need to be free to choose how they will represent their mathematical thinking that best fits their purpose, the particular mathematical context or calculation they are exploring, or the problem they wish to solve.