Beads, Racks and Counting.. Ian Sugarman


 Bennett Jacobs
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1 Beads, Racks and Counting.. Ian Sugarman Ian believes there is an ambiguous and sometimes uneasy relationship between Numbers and the Number System and Calculations elements of the primary strategy, and that that teachers of this phase would appreciate some guidance on how these apparently contradictory objectives can be effectively reconciled. Here is his attempt at doing just that. During the two years of Key Stage 1, pupils are expected to be developing their counting skills whilst at the same time abandoning Counting as a major strategy with which to mentally calculate. The key to reconciling these two objectives lies in the way we help children develop images of numbers that will aid their understanding of number relationships, and provide them with a real alternative to counting when they are adding and subtracting. Recognising these spatial arrangements of dots instantly (subitising), releases pupils from having to count each dot separately. Combining them promotes the practice of counting on the one or two extra dots. Eventually, some combined images will be subitised without needing to count on. Using the image of 5 dots in a line as an intermediate grouping for 10 reinforces the partitionings of the numbers 6, 7, 8, and 9 as 5 and something. In both of the images displayed here, 5 is set as a maximum number of dots to be placed together. Beyond 5, the units either change colour or form a new line below. The Fives rack allows the equivalence of different partitionings to be shown, e.g. 7 as 4,3 and 5,2. The mental movement of a single dot from one row to the other shows the equivalence of 4 and 3 with 5 and 2. Primary Mathematics Autumn 2005 Page 1 of 6
2 becomes When two numbers are placed together for adding, it is possible to solve the problem in two distinct ways. by extracting the two 5s and then adding on the remaining ones. by filling up the spaces on one of the numbers to make it up to 5  bridging the ten and adding on the remaining ones. BRIDGING THROUGH TEN Until this point, the emphasis has been upon numbers as quantities of objects their cardinal aspect. The alignment of 10 (or 20) beads on a string also suggests a position on a line can be modelled as 7 add 3, add This builds on the knowledge pupils will have gained of 7 being 3 less than 10 This +4 operation will later be modelled on an empty number line: Primary Mathematics Autumn 2005 Page 2 of 6
3 Subtraction from numbers between 10 and 20 can be modelled very easily on the bead rack, and the process involved links strongly with the nonpractical methods that we would wish pupils to move on to later. To use it to solve a calculation like 13 5, 13 beads are first isolated from the others. It is then easy to see that to take away 5, there are 3 there to take away, leaving 10. The remaining 2 must then be subtracted from 10. This approach is ideally suited to any calculation where the unit number is larger in the number being subtracted e.g. 178, During Year 1, most pupils are taught to count on or to count back to solve calculations like 8+5, 7+4, 114, 125, and But if these images of numbers have been strongly embedded, children can relate to the structural features of the numbers and can undertake mental transformations. Investing time in familiarising children with these images of Numbers through flashcards or their Power Point equivalent equips them with the means to respond creatively to calculation problems at this level of difficulty and at subsequent levels. Without this investment, children will need to draw on laborious and potentially errorfraught counting strategies which may endure for several years when they attempt to use the same approach with larger numbers. The practical approaches involving the manipulation of beads on strings or on racks is structurally more valuable than the counting of individual cubes or counters. But for most calculations involving numbers up to 20, the images alone, of numbers with their fives structure highlighted, should be sufficient for children at Key Stage 1 to solve mentally. But empty number line jottings act as a bridge between practical methods and purely mental ones and can be offered to provide another story of the procedure adopted But it is wise not to assume that an item of structured equipment will automatically be used in a way that is conducive to building a mental strategy. It may equally reinforce a strategy that increases the child s dependence on the equipment. In the 138 example shown, it is relevant to look at the options once the thirteenth bead has been identified. What is the best way to proceed with the subtraction of 8? A A B 8 as 5,3 Primary Mathematics Autumn 2005 Page 3 of 6
4 In method A1, the child starts with the thirteenth bead and counts back each bead, stopping at 8, to identify the 5 red beads remaining. In method A2, the child uses its knowledge of 8 as being 5 and 3. First the 3 red beads are identified and then the remaining 5 (whites), to reveal the 5 at the beginning of the line. In method B, the child identifies 8 beads to remove by looking at the beginning of the row (left hand side). this action corresponds to the mental act of subtracting 8 from 10 (in this case the 10 beads at the end of the row. If this operation were undertaken with interlocking cubes it would like this: cubes remove 3 remove 5, from the 10 The trouble with method A1 is that the process of counting 8 is locked into the presence of the beads. Without the beads there is nothing to count. In this respect it is equivalent to the practice of counting back on fingers starting at 13 and stopping at the 8 th finger: If the objective is to attempt to move children on from counting in ones and to encourage them to use their number knowledge (in this case the bonds or partitionings of 10) then methods A2 and B are to be preferred to method A When it comes to numbers up to and beyond 100, the earliest idea of addition that is offered to pupils is that of combining two numbers as quantities rather than as positions on a number line. This can be modelled with structured apparatus such as Dienes blocks. The stages in this approach are: 1. Partition both numbers and add the tens: 20+10=30 2. Add the units values: 4+7=11 3. Add both totals: 30+11= Primary Mathematics Autumn 2005 Page 4 of 6
5 However, some pupils see the sense in attaching the larger of the two unit numbers to the multiple of ten and then adding the smaller unit number = = = 41 Unlike the previous method, this particular algorithm has the added advantage of being a continuous process that can be modelled on an empty number line: Towards the end of Key Stage 1 some pupils will be capable of managing without equipment but the opportunity will exist for the imagery to be used to provide a reference point when the equipment is not there. For example, in 37+24, just drawing representations of the equipment may be as good as actually manipulating it The image of combining quantities like this may remain with many pupils as their main algorithm for adding numbers of this magnitude. Others however will proceed to the journey model adding partitionings of one of the numbers to the whole of the other one and recording each stage of the journey as jumps on an empty number line: A strong argument to recommend the empty number line or journey approach over the practical approach is that starting with one of the numbers, rather than partitioning them both, can work just as effectively with subtraction as with addition. Problems are often caused because pupils try to apply the dual partitioning approach inappropriately to subtractions, e.g = = = = = = 11 A misconception Although it is not until the end of Year 4 that pupils are expected to be able to solve subtraction calculations where the number subtracted crosses the tens boundary e.g ; 5327; 6224, the Framework leaves it open as to the method that is taught for solving the less problematic cases, such as or Clearly, the Primary Mathematics Autumn 2005 Page 5 of 6
6 practical approach using equipment, or simply their representations, is a sufficiently robust method when both column values of the number being subtracted are smaller. However, it is extremely unlikely that during Years 2 and 3 cases will not ever arise involving the more problematic cases. Primary Mathematics Autumn 2005 Page 6 of 6
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