Enhancing Calculation Skills: A Case Study on a Child's Numeracy Strategies

Case study, Pages 4 (989 words)

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In this report, I aim to analyse the calculation strategies used by one child, Cheryl, for a number of addition and subtraction questions. With reference to literature and my own observations of Cheryl, I will discuss the strengths and weakness in Cheryl’s procedural and conceptual understanding by evaluating her choice and execution of calculation strategies. Using this information I will then aim to suggest targets to develop the teaching of calculation to Cheryl in the future.

Haylock (2010) suggests that one of the vital components of numeracy is “the ability to calculate mentally using a range of strategies.

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Haylock (2010, p.98). The first mental strategy that we see Cheryl perform is that of compensation. Haylock underpins one of the reasons why we apply such strategies by stating, ‘the trick in the strategy is always be on the lookout for an easier calculation than the one you have to do.”(Haylock, 2010, p.102). Cheryl uses her knowledge of multiples of 10 to covert 19 to 20, in her words, ‘I pretended 19 was 20 and I counted up in 10s, 26, 36, 46, then took away the 1”.

Cheryl demonstrates she can count in multiples of 10, Montague and Price identify that, ‘counting in 10s and 100s is an essential element in developing calculation skills’, (Montague and Price, 2012, p.28), Cheryl then correctly adjusts her answer to compensate the supplementary 1 to give the final answer of 45.

We could leave this here and move on as Cheryl has given the correct answer, however, The National Centre for Excellence in the Teaching of Mathematics, (NCEMT), would argue that, “getting the correct answer does not support conceptual understanding".

We see why this is relevant when Cheryl continues to use the compensation method for the subtraction 53-19. It is only when the teacher asks Cheryl why she added the 1 this time that we become aware that she may be relying on procedural knowledge of the strategy and not actually making the connections with the number facts that is required for conceptual understanding.

The NCEMT Calculation Guidance (2015), encourages teachers to use effective questioning to develop mathematic reasoning and this is a perfect example of why and when this is needed. Cheryl seemed to have a complete understanding until probed further by her teacher, she had shown she knew the how something happens here, but not the why it happens which is the important difference between procedural and conceptual understanding, Hiebert and Lefevre (1986). Along with effective questioning to assist and assess Cheryls progression, I would recommend Cheryl receive further support from the use of visual aids, such an open number line, as this should help her identify number relationships more easily, strengthening her mathematical reasoning.

In the next video, Cheryl is presented with the calculation of 81-78. Her procedural efficiency does not seem as fluent here as in the first section. She appears confused and hesitant but attempts the calculation by putting the numbers on top to perform a vertical layout of subtraction. Consumed by using an algorithm for this calculation, Cheryl fails to recognise that counting up and finding the difference would have proven most efficient. Cheryl does what Hope (1986) argues to be one of the dangers of teaching written algorithms to primary children, “children choosing to use a formal written algorithm when it is not necessary,” (Hope, 1986 quoted in Clarke, 2005, p.94).

Haylock also reflects on the disadvantage of teaching algorithms too early by asserting that, “this leads children to treat the digits in the numbers as though they are individual numbers and then to combine them in all kinds of bizarre and meaningless ways.” (Haylock 2010, p.99). Cheryl demonstrates she is not secure with her knowledge of column value when she says 1-8 is 7. Harris warns that it is only when, “children are completely confident in partitioning numbers as tens numbers plus a teen number, should they be using any written subtraction method, “(Harris 2014, p. 155).

Therefore, in future for Cheryl to be able choose the most efficient strategy and know why, improving her procedural fluency and conceptual understanding, it would be valuable for her practice a variety of mental calculations and to be given time to explore partitioning numbers, counting on or back in twos, fives, tens, hundreds for example using an array of manipulatives, open number lines, number beads, a hundred square, place value grids etc. until, as Clarke advocates, she finds the strategy that makes the most sense to her, Clarke (2005).

In the final video we see Cheryl struggle again with a subtraction calculation. This highlights how Cheryl is failing to use the knowledge from the last calculation, where the teacher introduced her to using an open number line for support, and transfer it into this new situation. Cheryl’s inclination again is to put the numbers on top and do a vertical layout subtraction. She also makes an error with commutative law thinking that 3-7 and 7-3 give the same answer. Similarly to the previous video, we see Cheryl attempting to use a method she has been taught but applying it inappropriately, Hansen (2011).

The teacher correctly supports Cheryl by reminding her of the open umber line again and Cheryl seems comfortable with this aid and she seems to be able to identify the number relationships more effectively with this. Lawton and Hansen (2011) recommend another way teachers can help children making these errors is by having the child estimate answers by approximating and checking their results are reasonable by thinking about the context of the problem, Hansen (2011, p.69) this will also ensure students are covering expected requirements of making estimations set by the National Curriculum.

In conclusion, calculation strategies are highly personal, Cheryl’s number experiences will not be the same as another child’s and the strategies that work for her may not for someone else. To support young mathematicians in their numeracy journeys educators then, must ensure that children are continuously stimulated and questioned in a number rich environment, with plentiful opportunity to explore mathematics and represent it ways that are meaningful to them.