Contrary to popular opinion, maths is for everybody and the excitement of Maths Week last week, with the House competition in the Prep school, really embodied the enthusiasm that Maths can create, at whatever level it is being studied. It has moved far beyond the correct/wrong answers of the past and today children learn to discuss, visualise, explore and even debate situations that present the concept in a deeper and thorough way. We use the Maths No Problem scheme in the Pre-Prep and Prep to Year 4; this lays superb ground work for Mastery of Maths. From Year 5 we continue to use the Mastery theories alongside a wide variety of additional resources in preparation for the range of senior schools to which the children are heading.
Maths Mastery has its roots in Singapore and has the aim of mastering a concept to the depth that it can be applied to a completely new and unfamiliar situation. This is far more reflective of the application skills needed in the world around us and still enables children to answer a standard question when asked.
For example, last week Year 5 studied common multiples. A standard question might be:
Find the common multiples of 3 and 8 between 50 and 100
In the Mastery scheme, the question may be presented more like this:
The pictorial aspect (which could be made concrete by using blocks) immediately enables every child to understand the problem presented, even if they are not yet secure with the vocabulary of common multiples. It also gives a reason for using common multiples that is understandable and relevant to the children’s experiences, rather than just an abstract idea. The discussion around this problem brings in the 3 and 8 times tables and the conversion between centimetres and meters, all the time referring to the vocabulary of ‘common’ and ‘multiple’ for reinforcement. The second question, compared to the first, provides much richer and deeper thinking for all levels of ability.
The Mastery approach often refers to the movement of understanding through the concrete, pictorial and abstract constructs of an idea and children become skilled at using different equipment at different stages. Short division is an area that children can struggle to visualise, particularly when still building up their tables. By using Numicon (concrete materials) Year 6 were easily able to divide a number and visually conclude what remainder was left.
They could then use a pictorial representation when approaching new problems and eventually move entirely into the abstract when only numbers were presented and solve questions they had never seen before.
Another core skill used in Mastery is that of the bar model. These are used throughout the school but really come into regular use from Year 3. They have the benefit of turning a question or problem into a pictorial version and therefore accessible to a wider range of abilities.
For example, they work particularly well with ratios. Consider this question that could be found on a Common Entrance paper:
The ratio of boys to girls in a group is 5:3.
There are 80 more boys than girls.
Work out how many girls there are?
Traditionally this would have needed a strong understanding of ratios and experience of similar examples. Children would have needed to be able to visualise and recognise that it is the difference of the ratios that is utilised to find out the number of girls. With the bar model method, children can access and have success with this kind of problem even if they have not seen one like it before.
Children would firstly draw as close to equal blocks or bars to represent the 5:3 ratio, labelling the boys and girls rows clearly.
By re-reading the question they would see that there are 80 more boys than girls and would show this on the extra bars the boys have.
Seeing that these 2 extra bars represent 80, children can visually see one bar represents 40 and all the separate bars can be shown as 40, as all parts are equal.
Counting up the girls bars shows there are 120 girls. Equally the children can easily see how many boys there are in total as well.
Who knew maths could be so interesting and such fun!?