G is for Goes into Division

Division, the mathematical operation that appears to bring dread to educators and to students trying to learn it. Why do we all find this operation so challenging? Is it because division builds upon children's comprehension of addition, subtraction, and multiplication and so if this understanding is not mastered division will be challenging? Is it because of the dreaded “bus stop” method and not knowing what to do? Is it because of the different language we use that makes an abstract concept even more abstract?  

There are three main ideas we employ when teaching division to children, often switching between them without considering the potential confusion this might create. 

Sharing 

Division is frequently introduced as "sharing". “If have 15 sweets and I want to share them between 3 people, how many sweets do they receive? In this context, we tackle problems involving distributing a set number of items among a group of individuals. Interestingly introduction through this concept means we do not always necessarily need to know the original quantity of items, or the number of people involved as it is more about the concept of sharing something out. Consequently, this conception doesn't readily lend itself to representing the situation as a mathematical expression like 15÷3=5.  

At this stage, children benefit from plenty of hands-on experience with physically sharing objects and articulating their actions verbally before progressing to symbolic representations. 

Something I’ve been wondering with having a 2-year-old, is the number of times I talk about sharing with her, whether this be sharing the time with a toy or sharing a set of objects. I’ve realised I’ve just used this word without really thinking about the mathematical link. In maths sharing, means distributing something equally but in the real world do we share on equal terms. Food for thought if this is one of the earliest moments where a misconception could form. 

Grouping 

Another way to think about division is through grouping, which is closely tied to the concept of successive subtraction.  

For example, 

15÷3 

15 – 3 = 12 

12 - 3 = 9 

9 - 3 = 6 

6 – 3 = 3 

3 – 3 = 0 

Therefore, there are 5 groups of 3 in 15 and the answer is 5. 

In this scenario, the problem revolves around determining how many groups of a given number can be formed from a set of items. For example, "How many groups of 3 are there in 15?" This approach connects division with inverse multiplication, providing children with ample opportunities to explore patterns in multiplication tables and express them in various ways, both verbally and symbolically. 

For instance, consider the expression 3×5=15, which generates multiple sentences and symbolic expressions: 

Three lots of five make fifteen 

5×3=15 

Five lots of three make fifteen  

3 x 5 =15 

If you share fifteen things between three people, they will have five things each 

15÷3=5 

If you share fifteen things between five people, they will have three things each 

15÷5=3 

It's crucial for children to understand the equivalence of these expressions to enhance their proficiency with division. How long do we spend in classrooms really playing with and exploring numbers. If I give you the number 32, what are all the different ways you can represent this number? 

Written Form 

Following on from above, the next challenge arises with the written algorithms for division. Unlike other standard algorithms for addition, subtraction, and multiplication, the standard algorithm for division is worked from left to right, with a significantly different layout.  

For example, 741÷3 

Using this method, you would initially calculate how many times 3 goes into 7 and so forth. 

There are pros and cons to the bus stop methods. The pros being when you come to more complicated division involving algebra you will have a useful procedure to follow. The cons being that people confused about what they are doing and that it holds no meaning as the 7 is actually 700 which can lead to confusion. 

Despite advice against introducing written algorithms too early, many children encounter them before they are fully prepared to grasp them and without sufficient opportunities to develop their own strategies for dividing large numbers. Before children embark on any formal recordings using the standard algorithm for division, they need ample familiarity with tables and number patterns within them, as well as practice with mental strategies such as chunking, doubling, and halving. 

References: 

 Anghileri, J. (2000) Teaching Number Sense. Continuum. 

Plunkett. S, (1979) 'Decomposition and all that rot'. Mathematics in School 8(3), pp. 2-5