Harrison’s article, Zero means nothing (1993) gives the reader an insight into the difficulties children face when trying to represent the number ‘zero’. A Year 1 pupil tells a fellow classmate to represent a ‘zero’ in her telephone number with a multilink cube, just as she had done for all the other numbers in the sequence. This appears to be a common problem for young children. In a video demonstrating representation in Key Stage 1 mathematics, our module class saw a young girl explaining why the number ‘fifty-five’ should be placed in the middle of ‘fifty’ and ‘sixty’ on a number line; “ because I know that ‘five’ is in the middle of ‘one’ and ‘ten’ so ‘fifty-five’ must be in the middle of ‘fifty’ and ‘sixty’. The girl appears to be incorrect, five, is as the teacher clarifies in her reply, in the middle of ‘zero’ and ‘ten, rather than ‘one’ and ‘ten’. The teacher does not however, tell the child she was incorrect, perhaps because she did appear to understand the concept that five is half of ten. We can only assume therefore that she could not find a way of representing the ‘zero’ as a unit in her head, but had no problem when the ‘zero’ became a symbolic ‘ten’ in the ‘fifty’.
In contrast to these findings, it has been reported by Jones (1981) in a convincingly large-scale study of Edinburgh preschool children that most children of this age can in fact represent zero;
“…despite frequent claims that the concept of zero is inherently difficult, the children studies here do not appear to have any particular difficulty with the idea.”
Whilst the results have since been confirmed by more studies internationally; Sinclair, Siegrist and Sinclair (1983), Litwin (1984), Potts (1983), it should perhaps be noted that the observations where made in a ‘play’ environment where a game similar to that of ‘Tins’ in Jones’ study was used. It involved children having to identify an empty tin by means of representation on the lid. This does not demonstrate that children have no difficulties understanding the concept of zero in written calculations or mental arithmetic.
Representing the operations of addition and subtraction etc.
Young children find representation operations (addition and subtraction first) more difficult in school mathematics and this is emphasized particularly when children begin to learn strategies for mental arithmetic. (Beishuizen, Gravemeiger and Leishout, 1997)
In class we discussed some of the ways models are used in classroom environments, representing numbers using the HTU (place value) system which heavily influences current British curricula, indicated by the CGI programme and also by Anghileri (2000) in her book, Teaching Number Sense;
“Before they can operate with abstract numbers, children will need to learn to ‘model’ situations, with their fingers or some form of apparatus that can represent objects.”
However, potential problems adding to rather than alleviating representational difficulties amongst young children may occur when using many of the different models available, according to the experts, particularly those4 conducting comparative research. For example, Staker (1996) voiced concerns after studying methods of British representation in early mathematics;
“…[it is important to decide on] exactly what methods should be taught and in what order.”
In more recent research, Harries and Suggate (2006) agree that pupils should be introduced to a limited variety of representations.
The number track, which British children will often begin to use (in the earliest years of formal learning) in guidance with the new ELG’s set by the Government (date?) is very much a counting activity, representing numbers as objects in their own right;
A Basic Number Track:
A Number Track where units are represented as single objects (of a familiar nature):
Children in Harries and Suggate (2006) conducted recent research exploring links across representations of numbers using a computer game with young children in Y1, Y2 and Y3. He found that the use of the number line was understood by all of the children. He also found counters, figures and arrow cards (as shown below), all of which represent numbers as ‘whole’ objects produced the highest percentage of correct responses.
Most incorrect responses were with the more complex number lines that include subtraction, minuses and halve representations and illustrate lengths and distances also. Beishuizen (1993) proposes that this is why children have difficulties representing mental number problems, i.e. it is the context in which they are taught number problems which leads to confusion.
Focusing on Empty Number Line (ENL) programmes, practised in Dutch schools, Beishuizen produces comparisons that show children using sequential representation (i.e. counting along a number line, crossing-ten and N10 jumping), encouraged by the empty number line, produce more correct answers and develop representational strategies quicker than children using the partitioning (by way of HTU) method.
Beishuizen concludes that problems with HTU include children ‘hanging onto materials’ and passively reading off the answer from the blocks when doing sums. So whilst Multilink and Dienes block are used widely in schools throughout the United Kingdom and are helpful for the representation of abstract number structures (on a small scale), they are weak in the representation of number operations when they become more complicated.
Hart (1989) in Beishuizen (1997) had previously voiced critique, suggesting;
“…this manipulation of blocks has little connection to the intended (written) algorithm…”
Conclusion
Whilst this essay focuses on just two of the main difficulties children face when learning formal mathematics, there is literature to suggest that many of the problems (including the above) are centred on the language barrier that exists between adult and child, and the contexts children have been exposed to numbers in outside of school. As noted in the introduction, more focus on what children can do, supported by Munn, and setting aside time for observing methods that children use before being moulded by the Curriculum, may be of use in understanding more about the difficulties young children face with number.
References
Anghileri, J. Teaching number sense (2000) Continuum International Publishing Group Ltd
Aubrey, C. Mathematics Teaching in the Early Years: An Investigation of Teachers' Subject Knowledge (1997) Routledge Falmer.
Beishuizen, M. The empty number line as a new model (????) Journal for research in mathematics education
Harries, T & Suggate, J. Exploring Links across Representations of Numbers with Young Children (2006) International Journal for Technology in Mathematics Education, Volume 13, No 2
Harrison, J. Zero means nothing in Young children talking about mathematics (1993) MT144 September 1993
Hughes, M. Children and Number: Difficulties in Learning Mathematics (1986) WileyBlackwell.
Maclellan, E. (1997). The importance of counting. In I.Thompson (Ed.). Teaching and learning early number . Buckingham, UK : Open University Press.
Munn, P. Children's Beliefs about Counting in Teaching & Learning Early 1, ed. Ian Thompson (1997) Buckingham: Open University Press
Wing, T. Working towards mental arithmetic and (still) counting (1996) MT131 December 1996.
