Explain this
Why Do We Calculate Backwards? Left to Right is Better
How the publisher describes it:
“Taking a different view of calculation”
Review by Steph Harvey
In brief:
We have a nation of adults willing to admit they were no good at maths, and I suggest that this is in part due to an historical teaching belief that there was a ‘one way solves all’ solution for calculating within the four operations. Kenneth Williams is advocating exactly this and I suggest that for as many people who like his procedure there will be an equal number for whom it makes little sense. Without the understanding that is implicit in the left to right calculating which he advocates and the right to left method I was taught, we will continue to produce a nation of adults who happily confess to being rubbish at maths.
“His explanation of vedic formulaes do offer opportunities for us to explore and explain how the formulaes work”
One of the premises of the National Numeracy Strategy was to find ways of supporting children to work in a mathematically more efficient way which frequently advocated calculating in ways other than from right to left — and not just in the opposite way from left to right. I therefore take issue with Kenneth Williams and his assertion that ‘no alternative to right to left (calculating) has appeared until now.’
Despite the author criticising the ‘right to left’ tradition that we are ‘thoroughly accustomed and habituated to’ the ‘left to right’ calculating he advocates continues in the ‘tradition’ from pre Numeracy Strategy in simply providing us with an alternative procedure. In his examples he advocates partitioning numbers into their composite place value parts and then recombining – (an objective in year 2 from the NNS); and soon I found myself losing my sense of the size of number I was calculating. Many primary practitioners recognise that increased accuracy is obtained when the largest number in a calculation is kept whole and subsequent smaller numbers are partitioned and added according to their value, with the largest value being added first — and these methods are supported in primary school with the use of the number line, providing children with an image to support their mental calculations.
Despite the author advocating the need to work with the largest value first in order to help with approximating his procedures treat each digit as though it were a single digit. For example in 23 x 4 we are advised to first get 2 x 4 immediately diminishing the value of the 2 within the number 23. Without a secure knowledge of the place value system I would suggest that calculating mentally left to right is just as prone to error as right to left procedures are, and a more efficient strategy for multiplying by 8 would be to use doubling. Indeed once the author moves us into written methods there are numerous pitfalls awaiting the procedural worker who follows rules without understanding; the example of using bar numbers in multiplication is one such pitfall.
Whilst the author is right that tradition does continue to raise the questions from parents about ‘why don’t you teach it the way we were taught?’ primary practitioners have been encouraged to adopt a wider range of strategies to support children’s more efficient calculating which invariably include keeping a sense of the size of the numbers — by implication left to right calculating — to support mental methods. Also implicit within current primary practice is the need to think/reason about the numbers involved before deciding on a strategy. Within the context of reasoning and communicating about our mathematical understanding Kenneth Williams explanation of vedic formulaes do offer opportunities for us to explore and explain how the formulaes work.
In his conclusion Kenneth Williams reiterates again why we should calculate from left to right, though I suggest it is a weakness in his argument to say that it makes sense to work from left to right in order to be consistent with division; I am left wondering if the vedic mathematicians didn’t need a method for dividing since this formula is conspicuous by its absence. I would suggest also that any person who adopts vedic formulae for solving calculations mentally already has a level of numeracy whereby they can retain multiple steps in their head.
We have a nation of adults willing to admit they were no good at maths, and I suggest that this is in part due to an historical teaching belief that there was a ‘one way solves all’ solution for calculating within the four operations. Kenneth Williams is advocating exactly this and I suggest that for as many people who like his procedure there will be an equal number for whom it makes little sense. Without the understanding that is implicit in the left to right calculating which he advocates and the right to left method I was taught, we will continue to produce a nation of adults who happily confess to being rubbish at maths.
The author comments
I would like to take issue with some of the points raised in this review. It is stated in the book that it is aimed at the general reader, but it seems to have been examined only in terms of primary education. Consequently the fact that left to right calculation is more natural because we read and write from left to right, the way this unifies computation because then all four basic operations can be carried out from left to right, and the way this leads to the possibility of calculations that would otherwise be impossible, have been missed. In the book I have tried to present a bigger picture than is implied from the review.
Contrary to what the reviewer says, I do not advocate that only left to right calculation should be taught. Please see page 93 of the book, where this is made clear. The reviewer also says that the procedure for division ‘is conspicuous by its absence’, but as explained, the procedure is the standard well-known one and so there seems to be little point in including it. An example on division can however be found on page 51.
Steph Harvey • Mathematics Consultant, Devon LDP
Paperback: 112 pages
Publisher: Inspiration Books (10 Jan 2010)
ISBN-10: 1902517237
ISBN-13: 978-1902517230
