Dealing with the left overs
A A TES website maths forum about "bottom-set maths" raised some important issues about this concept.
Sometimes, what we perceive to be straightforward maths, or a very simple mathematical concept, can be even more difficult to teach because of its simplicity, as often the methods and language to teach the concept are incorrect and lead to confusion for the pupil - particularly those unable to make the necessary conceptual leap.
One of the first comments illuminated that fact. Dr Daniel says: "Confusion probably arises because 85 is not 1r3!"
He goes on to explain that it is 1 and 35. "The implication in the answer 1r3 is that the 1 and 3 are both whole numbers. The 1 actually represents 1 whole group of 5 and the 3 represents 35 of a group of 5. For example, 8 counters divided by 5 or 8 counters put into groups of 5, will give one whole group of 5 in which each counter is 15 of the group. As there are now 3 counters left over it follows that these 3 are in fact 35 of the group of 5."
Dr Daniel then puts the posting in a workplace context concerning a boy who has to order some five metre-long titanium tubes. These tubes are only available in eight metre lengths. "Billy recalls his lessons on fractions and quickly remembers that he was taught that 85 is 1r3. Feeling pretty pleased with himself he pauses and thinks: '1r3, what does it mean? I can't ask for help I'll look right stupid'. So little Billy reckons it must mean he can get four rods from one eight-metre length because 1 + 3 = 4."
I read this posting, first in disbelief until I was working with a bottom-set maths Year 11 student. When I asked him what the 1r3 meant, he did indeed conclude that it must equal 4. Dr Daniel made an important point and illuminated why pupils find short division such a difficult concept.
This discussion was used as the basis for a workshop with maths teachers at the Specialist Schools Trust Seminar for maths and science teachers held at the University of Warwick.
When teaching short division, which inevitably leads to fractions, we have to use the correct language and ensure that the remainder is, in fact, left over.
For instance, if I pose the question, "I have 8 sweets and wish to share them between 5 people. How many will each get?", then the response is, of course, that each receives 1 sweet with 3 sweets left in the bag. We don't necessarily have to divide those sweets into sections to use them all up.
An alternative case could be where all the 8 items should be used. More will be learnt and understood if pupils are allowed to work on the problem themselves in small groups to stimulate discussion, rather than trying to teach through exposition.
Create some work cards with a series of problems on them, called "pizza problems" or similar. Give the students lots of cardboard pizzas and scissors. Once they have finished they can report their solutions back to the whole group for discussion. This can then lead to more abstract examples and consolidation practise.
The problems should be staged, for example: 1. A group of 4 friends bought 2 pizzas to eat. How much would each eat if all had the same amount and they ate both pizzas?
2. Three students had enough money to buy 1 pizza after a visit to the cinema. How much of the pizza would each get if they shared it equally?
3. Eight pizzas were bought for a party of 5. They were shared equally. How much pizza will each have?
One group suggested we should ban the division sign and replace it with a fraction line, as this would help to avoid confusion. Thus 7V3 would instead be written as 73.
The introduction to the workshop used about the first 20 postings from the website. They were read as a play to stimulate the discussion. This would be a great way to start an inservice training workshop, allowing primary and secondary colleagues to work together.
Thanks to those who provided the illuminating discussion on The TES website and to teachers from the Specialist Schools Trust Seminar for their contribution to this column. More details of their contribution, with diagrams, can be found at www.mathagonyaunt.co.uk under the archive of The TES columns for September 5.
Wendy Fortescue-Hubbard is a teacher and game inventor. She has been awarded a three-year fellowship by the National Endowment for Science, Technology and the Arts (NESTA) to spread maths to the masses.
www.nesta.org.uk Email questions to Mathagony Aunt at email@example.com or write to TES Teacher, Admiral House, 66-68 East Smithfield, London E1W 1BX