When should formal recording (numerals and symbols) begin?
Worthington and Carruthers (2003) presented evidence that, by working with personalised mathematical graphics, practitioners can support children’s development as “informal marks are gradually transformed into standard symbolism” (2003:77). This could be one way for practitioners to think about working at ‘greater depth’ in mathematics. Certainly, it is important for children to learn standard written notation and we obviously expose them to numerals, alongside developing their confidence in making their own mathematical marks, so they can begin to appreciate the benefits of number symbols as efficient tags for amounts. These can be in the form of wooden or plastic numerals, as well as on number cards. This helps them to understand the meaning of the symbols, which is crucial. The move from informal to formal can be seen as a shift in balance, where children continue to use their own jottings and these sit alongside standard ways of representing as their recording gradually adopts standard written forms (Gifford, 1997:76). This takes time and requires a firm foundation rooted in experience. We cannot underestimate the leap of understanding and the depth and range of experience that contributes to a child being able to confidently make one mark, ‘5’, to stand for five separate objects. We also need to help children to learn to write the symbols, which in the case of 5, can be tricky (to start with children tend to reverse numerals, but gradually correct this by the age of 6 or 7).
Using concrete objects as manipulatives is a key aspect of early primary education and forms an important part of preschool mathematics (Clements and Sarama, 2009, 2012). Most 6-year-olds seem to be able to represent at least some addition and subtraction word problems with concrete objects (Carpenter, Hiebert & Moser, 1981; Fuson, 1992; Lindvall & Ibarra, 1980). Younger children appear to have greater difficulty. Dowker (2005) with her students, Mark Gent and Louisa Tate, gave 4, 5 and 6 year old children written (e.g. “2 + 5 = 7”; “6−2 = 4”) and orally presented word problems (e.g. “Paul had 4 sweets; his mother gave him 3 more; so now he has 7 sweets”; Alice had 5 buns; she ate 3 buns; so now she has 2 buns”. They were asked to show how to find the answer with the counters or “show me the story with the counters”. Responses were classified as ‘complete’ (clearly showing the addition or subtraction); ‘incomplete’ (showing just the result, or e.g. showing the addends and the result without showing how to get from one to the other) or incorrect (showing the wrong operation; an irrelevant response; or no response. According to the above classification, 44% of the 6-year-olds’ responses, 3% of the 5-year-olds’ responses and 1% of the 4-year-olds’ responses were complete; 28% of the 6-year-olds’ responses, 57% of 5-year-olds’ responses and 40% of 4-year-olds’ responses were incomplete; and 28% of the 6-year-olds’ responses, 40% of the 5-year-olds’ responses and 60% of the 4-year-olds’ responses were incorrect. Thus, the age differences were striking. It is not clear how much of this was due to increasing familiarity with numerals and number words, and how much to their increasing ability to translate between different formats, but this suggests that young children’s ability to represent oral and written (formal notation) addition and subtract problems using manipulatives develops considerably between 4 and 6 years of age. It is therefore important to use graphics and manipulatives rather than formal notation or verbal explanation alone for children to at least six years of age, perhaps using formal notation alongside other representations and not moving to formal notation alone too quickly as under 6s may find it challenging to completely understand the problem. Furthermore, work with somewhat older children suggests that, as Hughes’ work might indicate, children do not always make spontaneous links between numerals and concrete representations. They may learn most effectively to make the connections if they are simultaneously shown the symbols and concrete representations, and explicitly encouraged to link them (Fuson, 1986; Fuson & Burghardt, 2003; Hart, 1989; Hiebert, J. & Wearne, 1992). Overall, research underlines the huge challenge for young children in making sense of symbolic mathematical representations. As Herbert Ginsberg points out:
‘If children get off on the wrong symbolic foot, the result may be a nasty fall down the educational stairs.’ Ginsberg (2021)