The word ‘divide’ comes from Latin: dividere “to force apart, cleave, distribute,”
Division as a concept is fiendishly difficult to teach. Most children can understand ‘sharing’ objects and move on to sharing on ‘plates’ but does this naturally lead to more formalised methods of division?
According to Nunes and Bryant (1996), division as an operation in not the same as sharing and a notion of ‘sharing’ does not ensure an understanding of the inverse co-variation between division terms.
Moreover, Bus Stop method LOOKS weird. The written algorithm works left to right and does anyone ever explain WHY it’s called Bus Stop?! A picture can help.
Division builds on previous understanding in mathematics, involving all of the operations. Furthermore, additional skills and knowledge are required such as:
- estimation and times table fluency
- division is not commutative, unlike multiplication and addition
- it is the inverse or opposite of multiplication
- strong working memory, as information has to be held in mind whilst attending to something else
Williams and Shuard outline 3 stages in understanding division:
- using grouping and sharing as different operations; solving problems using concrete apparatus.
- relating sharing to grouping
- using knowledge of multiplication to deal with both types of division by the same numerical procedure.
Vergnaud (1990, 1997) : ‘Understanding the concept of division is often confused with skill in operating algorithms’.
What students need to understand:
- that parts must be the same size
- the size of the whole is the number of parts multiplied by the size of the parts (plus remainder)
- the inverse co-variation between the size of parts and number of parts (e.g. more parts = smaller size)
- the whole must be distributed until the remaining elements are insufficient
- the remainder CANNOT be larger than, or equal to the size or number of the parts
Selva (1998) outlined 3 difficulties for students:
- Type of division problem
- Difficulties understanding the inverse c0-variation between the terms when the dividend remains constant
- Difficulties dealing with the remainder
Partitive question type:
Charles bought 15 pencils to give to each of his 3 friends, how many will they each get?
This question is easier because it involved the action schema of sharing which is understood from a young age.
Quotative question type:
Charles bought 15 pencils. He wants to give 3 pencils to each friend. How many friends will get the pencils?
This type of understanding appears to be acquired later, through teaching.
Haylock and Cockburn (2008) make some great observations about division and language in their excellent book Understanding Mathematics for Young Children.
12 divided by 3 = equal sharing structure.
In this example the language of ‘share equally between’ is appropriate.
However, just as valid in answer to the same question: how many 3’s make 12?
This is the inverse of multiplication and involves grouping into sets of 3. This reinforces the earlier points made by Williams and Shaurd. Haylock and Cockburn feel that perhaps there is an overemphasis on sharing when introducing young children to division. This results in a tendency to attach the words ‘share’, ‘shared between’ etc to the symbol. They propose that in the long term, the sharing structure of division is limited and of less significance than the inverse of multiplication structure.
What to do:
- Take a Big Picture view of how division is taught across the school from Early Years to Year 6.
- What are the teachers’ views and understanding of division?
- Is the language used consistent?
- Ensure that concrete manipulatives are used across the school: counters, cubes, cuisenaire, tape measures
- Ensure that whilst ‘sharing’ can be a useful concept (and one easily understood) it is not overly emphasised.
- Explain what Bus Stop is, point out explicitly the differences between it and other algorithms
- Encourage children with low working memory to jot down times table facts, think out loud and annotate during division as this will ease the cognitive burden.
- Practice, practice, practice…to automaticity! Division is great because it’s an opportunity to practice an array of maths skills and to build fluency.
Read the excellent blog by Laurence Holmes @LHteaching on the importance of practice in maths here:
Excellent study looking at difficulties hindering division practice:
Williams, E. and Shuard, H. (1994) Primary Mathematics Today. Longman
Derek Haylock and Anne Cockburn (2008) Understanding Mathematics for Young Children SAGE