Proportionality is a huge part of mathematics, defining and linking areas of maths that include multiplication, division, ratio, fractions, percentages, scale, place value, circle properties, algebra, pattern… in fact anything that involves proportional relationships.
Some of the training sessions I ran for the MaST (Maths Specialist Teachers) Programme included workshops on ‘Proportionality’ – one of the ‘Big Ideas’ on the programme. It was the way that proportionality connected different areas of mathematics that made it such a powerful idea. Unfortunately this big idea is a little lost in the new NC programmes of study.
Within the introduction of the 2014 National Curriculum, the importance of making connections is clearly stated:
Within the introduction of the 2014 National Curriculum, the importance of making connections is clearly stated:
Mathematics is an interconnected subject in which pupils need to be able to move fluently between representations of mathematical ideas. The programmes of study are, by necessity, organised into apparently distinct domains, but pupils should make rich connections across mathematical ideas to develop fluency, mathematical reasoning and competence in solving increasingly sophisticated problems.
It is acknowledged here that the structure of the document (PoS with ‘distinct domains’) makes the inter-connected nature of mathematics difficult to present in a written form. It may have helped to have proportionality as a ‘domain’ - this may have led the way towards making connections. Perhaps it would also have been good to make use of technology to produce an application that helped teachers with making connections in maths.
However, it is left to teachers’ planning to give pupils opportunities to make rich connections. This relies on teachers’ subject knowledge to make the most of opportunities.
Connecting different aspects of maths isn’t easy when planning, particularly if using the current preference of an ‘objectives and assessment’ approach. Haylock and Cockburn (1989) identified two curriculum planning models:
However, it is left to teachers’ planning to give pupils opportunities to make rich connections. This relies on teachers’ subject knowledge to make the most of opportunities.
Connecting different aspects of maths isn’t easy when planning, particularly if using the current preference of an ‘objectives and assessment’ approach. Haylock and Cockburn (1989) identified two curriculum planning models:
‘Objectives and assessment’ approach
· Specify a sequence of objectives to be obtained by the pupils
· Assess the pupils’ current attainment against these objectives
· Design a learning programme to move them forward
‘Experiences’ approach
· Identify meaningful situations and purposeful activities
· Pupils’ apply and develop their mathematical skills
· Monitor learning outcomes and pupil experiences
· Specify a sequence of objectives to be obtained by the pupils
· Assess the pupils’ current attainment against these objectives
· Design a learning programme to move them forward
‘Experiences’ approach
· Identify meaningful situations and purposeful activities
· Pupils’ apply and develop their mathematical skills
· Monitor learning outcomes and pupil experiences
When we ran the MaST session on Proportionality, we showed a video of
‘The Powers of Ten’ – a dated but still powerful short film by Charles and Ray Eames of a family lying on a picnic mat in a park.
A camera zooms out by 101 every 10 seconds to 1026 and then moves back in to 10-18. It goes from the outer reaches of the universe through to the centre of an atom in the man’s hand. It is well worth a look at with your class – and obviously reinforces the idea of scale and proportionality.
‘The Powers of Ten’ – a dated but still powerful short film by Charles and Ray Eames of a family lying on a picnic mat in a park.
A camera zooms out by 101 every 10 seconds to 1026 and then moves back in to 10-18. It goes from the outer reaches of the universe through to the centre of an atom in the man’s hand. It is well worth a look at with your class – and obviously reinforces the idea of scale and proportionality.
For a more recent example, take a look at
The scale of the Universe 2. It is like a super-Prezi with its amazing zooming in and out, as well as being very informative.
You’ll be hooked!
So, how should you plan your maths?
There is no reason why these two models cannot be used together. With my own maths planning I favour more of an ‘experiences’ approach, but I also have very clear objectives as a focus for the unit of work.
When supporting schools with their planning I suggest starting with an ‘objectives and assessment’ approach to focus on the skills, concepts or procedures that are appropriate for the class. Once the unit of work is established (after perhaps 3-4 lessons) an ‘experiences’ approach can be adopted, using the skills, concepts and procedures learnt to solve problems, investigate a line of enquiry and connect to other areas of maths. This is why I find a context to ‘hang the maths on’ such an important part of the planning.
This takes a little confidence, moving away from the established objectives-driven approach of the old Primary Strategy, but it makes perfect sense to give the maths a real purpose by linking and combining maths topics. You can assess what the children have learnt by the end of the unit, based on the shared objectives and success criteria, but also based on any other aspects of the maths curriculum in which the children showed good understanding or knowledge.
There is no reason why these two models cannot be used together. With my own maths planning I favour more of an ‘experiences’ approach, but I also have very clear objectives as a focus for the unit of work.
When supporting schools with their planning I suggest starting with an ‘objectives and assessment’ approach to focus on the skills, concepts or procedures that are appropriate for the class. Once the unit of work is established (after perhaps 3-4 lessons) an ‘experiences’ approach can be adopted, using the skills, concepts and procedures learnt to solve problems, investigate a line of enquiry and connect to other areas of maths. This is why I find a context to ‘hang the maths on’ such an important part of the planning.
This takes a little confidence, moving away from the established objectives-driven approach of the old Primary Strategy, but it makes perfect sense to give the maths a real purpose by linking and combining maths topics. You can assess what the children have learnt by the end of the unit, based on the shared objectives and success criteria, but also based on any other aspects of the maths curriculum in which the children showed good understanding or knowledge.
Proportionality can be the link to connect areas of maths. For example, take the theme of ‘Doubling’:
• Shape: Draw a right-angled triangle. Now double the length of each side to make a ‘similar’ triangle. What do you notice about the angles? Try this with other shapes, doubling the lengths of sides.
• Measures: What is the area of this 2cmx2cm square? Now double the length of each side. What is the area of the new square? What pattern do you notice if you keep doubling the lengths of sides?
• Pattern: A grain of rice is put on a square of a chessboard. The next day 2 grains of rice are put on the next square, then 4, 8, 16… and so on. How many days are there until there are more that 1 million grains of rice?
• Fractions: 1/4 of the sweets in a pack of 24 are raspberry flavour. Double that number are lemon flavour. What fraction are lemon? What do you notice? What happens when you ‘double’ a fraction?
• Shape: Draw a right-angled triangle. Now double the length of each side to make a ‘similar’ triangle. What do you notice about the angles? Try this with other shapes, doubling the lengths of sides.
• Measures: What is the area of this 2cmx2cm square? Now double the length of each side. What is the area of the new square? What pattern do you notice if you keep doubling the lengths of sides?
• Pattern: A grain of rice is put on a square of a chessboard. The next day 2 grains of rice are put on the next square, then 4, 8, 16… and so on. How many days are there until there are more that 1 million grains of rice?
• Fractions: 1/4 of the sweets in a pack of 24 are raspberry flavour. Double that number are lemon flavour. What fraction are lemon? What do you notice? What happens when you ‘double’ a fraction?
