For several years it has been mandatory for those embarking on primary initial teacher education in the U.K. to have – as a minimum – a C grade in GCSE mathematics. More recently there has been the proposal that this should be raised (Williams Review, 2008). Certainly I would support this although it might make it more difficult to recruit sufficient numbers to meet the demand for primary teachers. Research (Ball, Thames, and Phelps, 2008) has demonstrated that one of the qualities of expert mathematics teachers is their sound subject knowledge. By this they do not simply mean the ability to achieve success in examinations but rather successful teachers need to have a thorough understanding of their subject and the implications of what they teach. In this short article I would like to draw your attention to three mathematical topics which, at first sight, appear straightforward but which, on closer inspection, are considerably more complicated. The first is the seemingly innocent symbol '0'. It is one thing to know that it does not actually mean 'nothing' but (a) how is that best conveyed to a young child and (b) what are the implications if they do not get the correct message? When discussing how best to introduce zero to young children with groups of early years' teachers from across Europe it soon became apparent that it required considerable thought and that it was vital to include the idea of zero as part of everyday life and learning as, for example, illustrated by two colleagues: Sandy: I think it is important that zero is displayed alongside the other numbers in the classroom in a wide variety of ways. Annabelle: Yes I agree. I have all kinds of numbers around the classroom: phone numbers (pretend of course!), house numbers, birthday and date charts, number lines... (Cockburn and Parslow-Williams, 2008, p. 9)[1] In exchanging further ideas it became clear that both Sandy and Annabelle presented zero as the absence of something rather than as nothing. Sandy explained (see figure 1), Someone suggested that I have a zero to ten 'washing line' with shirts on it labelled '0' to '10' but I think that's confusing as should one count the shirts or focus on the numbers? Instead I hang string bags from a number line. The first one – that is hanging from zero – is empty, the next one contains one cube, the next two and so one. (ibid) Figure 1 This prompted Ruth to say (see figure 2), I worry about showing empty palms to represent zero as they could be seen as showing 10 fingers. I much prefer showing my fists for zero and then raising one finger for one, two for two and so on... I quite like the standpoint of the same approach for each digit. I have always given my children a card with a number on it and asked them to get the same number of cubes as shown on the card. Now I could include a card with '0' on it and make sure that there is a table with nothing on it in the room and ask the children to show me zero and they've actually got to show me something. Of course they might not point to the table but if they pointed to a space on the carpet or showed their fists that would be fine. (Cockburn and Parslow-Williams, 2008, p. 10) Figure 2 Reflecting on the issue Annabelle realised that in the past she had encouraged her class of 4-year-olds to count up from one to ten but down from ten to zero, 'I'm now going to try counting up from 0. It may be unconventional but it is worth a try!' (ibid). If someone perceives zero as meaning 'nothing' rather than the absence of something their misconception may not become apparent for months or even years. Hints of it are likely to creep in when they begin work on place value. For example, on being asked to read '602', a child might respond 'sixty-two' rather than 'six hundred and two' perceiving '0' as having no function. Or, when multiplying 2.5 by 10, they might write '2.50' rather than '25' having been taught, but never questioned, 'when you multiply by ten you add a zero.' All manner of other problems can emerge as the numbers get larger and more complicated operations are required. If spotted they can be overcome but the earlier a misconception is established in life the harder it is to shift and therefore, although I am aware I only have a limited understanding of zero in mathematical terms, I appreciate that I must never refer to it as 'nothing'! The second potential minefield for early years' teachers is the apparently easy concept of subtraction and here I have a confession to make: it was not until I was learning to become a teacher that I realised why 'two minuses make a plus'. Prior to then I had always thought that subtracting was simply about 'taking away'. Perhaps my teachers had taught me about comparison, inverse addition and so on when I was young but, if they had, I soon forgot about them in my quest to successfully complete page after page of calculations! When I became a teacher educator myself I made sure that I discussed the various representations of subtraction with my students and these are detailed in a book Derek Haylock and I produced (Haylock and Cockburn, 2008). Part of the problem lies, I think, in the notion that subtraction is often perceived solely in terms of 'taking away' and, as such, is apparently so easy to demonstrate. Thus, for example, I might have three biscuits, I eat – or take away two – and I am left with one. You will note, however, that I used the word apparently at the start of this paragraph for sometimes people illustrate such a problem thus: 3 - 2 = ? ooo - oo = ? Where did the two extra 'biscuits' come from? A slightly more subtle version of this is when a child is given the following problem, 'If there were four birds sitting in a tree and one flies away how many are there now?' There are still four but, if one adds, '...on the tree?' the correct answer becomes three unless, of course, some other birds have settled on the tree in the meantime! (This may not be so unreasonable in the eyes of a child who is being encouraged to use his/her imagination.) The other most common form of subtraction introduced to young children is comparison. Indeed they tend to compare quite naturally when they start reflecting on heights, ages, the number of sweets each person has been given. Practically comparison is relatively easy to demonstrate but great care must be taken if anything is committed to paper particularly if, at a young age, you were one of the many people taught that '-' is 'take away'. Again, I cannot explain all the mathematical details as to why it is important that young children learn that '-' can involve taking away, comparing, reducing (as in cutting down a hedge etc), the inverse of addition (e.g. Sam has £1. He wants to buy a model car for £3, how much more does he need?) and so on. Suffice it to say, however, there are times in life where one has to subtract which cannot be illustrated by simply taking away. You cannot take four teddy bears away if you only have three of them; just as you cannot buy a book for £15 if you have an overdraft of £20. Sadly simply comparing 15 and 20 will not work in this case either! My third concept – equality – may also appear deceptively simple and yet, in a study of first year undergraduates in Italy, it was found that a significant percentage of the students had successfully completed their schooling without a clear understanding of the issue (Marchini et al, 2010). Over the years I have seen equality taught in a wide variety of ways. Those which I would deem successful have been notable in their acknowledgement of children's imaginations, their logic and their wonderful ability to interpret: to some children four add five is simply not the same as nine. How could they be when there are two of them and they are both an entirely different shape to '9'? I have also noticed that some children become quite puzzled when they are told that '=' sometimes means 'makes' and yet at other times they are informed that it means 'leaves'. There has been much written about the equals sign and researchers frequently report that young children view '=' as an invitation to act (Jones and Pratt, 2006). Thus, for example, '4 + 5 =?' would make sense to them (i.e. you have to do something – in this case add – to four and five) whereas '? = 4 + 5' would not as there is no indication as to what has to be done to the question mark. 9 = 9 is also perfectly acceptable mathematically speaking and yet, again, no action is implied. I appreciate that young children do not usually encounter mathematics written in such a formal manner as they embark on their early educational careers. Nonetheless, however, I think it is important that they become familiar with the idea as to how quantities can be built up and broken down: four apples and five apples would give a total of nine apples which could, in turn, be re-distributed as six and three apples or even two and three and four apples! Failure to understand that '=' is a symbol of equivalence rather than action often goes undetected for many years as indicated above and yet, if children acquire such a misconception, they are likely to encounter serious difficulties when they embark on more advanced mathematics and, in particular, algebra: the mathematical statement 'x = 4 + 5' would almost certainly appear nonsensical viewing '=' as a call for action. The above are all illustrations of the need to introduce young children to the richness of basic mathematical concepts. What we learn when we are young can become very routinised and fixed. Think, for example, how you do complex calculations. My strong suspicion is that, however you teach them now, you still do long multiplication in the way you were taught when you were at school. In many ways, if it works, that's fine. But what happens when you – or the children with whom you work – encounter more advanced mathematics and you do not have the breadth of understanding to interpret what is being asked of you? I am not suggesting that we all need to become expert mathematicians in order to teach young children but I am advocating a greater appreciation and flexibility when introducing such fundamental concepts as those discussed in this article. I would also urge you to check out your own mathematical understanding from time to time by noticing when children appear to pick up the wrong end of the stick, talking to colleagues and going on courses which look as if they might prove stimulating and interesting. Finally, Haylock's Connections Model (2010) is a useful reminder as to how one might explore a new concept in a range of ways in order to develop a deeper understanding of it and its relationship to a wider mathematical and, indeed, everyday context. References Ball, D. L., Thames, M., and Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59, 389-407. Cockburn, A.D. and Parslow-Williams, P. (2008) Zero: understanding an apparently paradoxical number. In Cockburn, A.D. and Littler, G. (eds.) Mathematical Misconceptions. (pp. 7-22) London: Sage Publications Haylock, D. (2010) Mathematics Explained for Primary Teachers (4th ed.) London: Sage Haylock, D.W. and Cockburn, A.D. (2008) Understanding Mathematics for Young Children. London: Sage Publications Jones, I. and Pratt, D. (2006) Connecting the equals sign. International Journal of Computers for Mathematical Learning, 11, 1382-3892 Marchini, C., Cockburn, A.D., Parslow-WilliamS, P. and Vighi, P. (2010) "Equality relation and structural properties – a vertical study." Proceedings of the Sixth Conference of European Research in Mathematics Education. (pp 569 – 578). Paris: Institut National de Recherche Pédagogique. May be accessed at www.inrp.fr/editions/cerme6 Williams, P. (2008) Independent Review of Mathematics Teaching in Early Years Settings and Primary Schools. London: Department for Children, Schools and Families. [1] This study was funded by the British Academy (award # LRG-42447)
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