Harder, Faster, Higher? – Supporting More Able Mathematical Learners
The new curriculum, we are told, is a mastery curriculum. This means there is an expectation that instead of pushing our more able learners on to ever higher level curriculum content, the focus is much more on making sure all our children are secure with the core content for each year group. This leaves us with a challenge for our more able learners, but also with a great opportunity. With less pressure (we assume – the assessment procedures are not yet clear) to push these children through the levels, we have to think of different ways to challenge them – not so much higher and faster as broader and richer content. These are a few ideas about how we can support these children.
Open ended questions
Challenging more able children should not be primarily about moving them on to ‘harder’ questions with higher numbers. We want to extend their thinking by asking more open ended questions which challenge them to apply their knowledge in new ways. We also want to develop their reasoning skills by asking them to explain their reasoning. Using Bloom’s question stems can be helpful in planning this. Ask children to explain how to find the answer to a problem and decide which approaches would be best to use. Ask them to explain the rule for a growing pattern or to explain what would happen if … Get them to think of other ways of doing things and compare approaches to decide which is best. Challenge them to explain their reasoning so that a younger child could understand it.
Mathematically rich activities
Children need to learn to think mathematically and to apply their skills. The Nrich website provides lots of games, challenges and activities to encourage this. They aim for activities to be ‘low threshold, high ceiling’ ie. accessible to as many children as possible but with enough to challenge more able children. Their curriculum mapping documents are very useful in identifying activities which link to different curriculum areas.
Investigations can help children to extend their mathematical thinking in a more open-ended way. Typically in an investigation, children are given a starting-point and some ideas of how to get started, but they won’t know what the answer will look like. They need to look for patterns and identify what is happening. The Maths Warriors site has a number of interesting investigations suitable for primary aged children.
More able children often respond well to challenges. The ‘Mathematical Challenges for more able pupils’ have a number of challenges divided by age group. Another good source of challenges for KS2 are the Challenge cards on the Maths Warriors site.
Whenever a new skills is taught and learned, make sure the children have the opportunity to practise their skills in a real context by applying them to solving problems. It can be particularly meaningful to give the problems a context from another curricular area. As well as regular opportunities to solve problems as part of their maths lessons, children often also respond well to the challenge of a ‘Problem of the Week’ which can be displayed in class for a set time. The Nrich site is a good source of suitable problems, some of which are available as posters. The Numeracy Strategy Logic Problems also have problems at a range of levels.
Missing Number questions
For calculation in particular, once a calculation process is learned (eg. column addition), presenting questions with missing digits can extend children’s thinking about the process they have been using.
Other useful websites
Mathpickle has some interesting videos and other activities.
Mathsticks has lots of useful resources. There are lots of great activities which are free to download, and some premium resources if you can stretch to a membership.
7puzzle posts a new puzzle every day. The site also categorises the archived puzzles into Easy, Medium and Hard etc.
My Pinterest board has lots of other ideas for investigations, puzzles and challenges.
The new primary maths curriculum has been criticised for its focus on fact fluency and traditional written methods. However, of the three key aims at the beginning of the document, only one focuses on fluency. The other two are that we should ensure that all pupils:
“reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language.”
“ can solve problems by applying their mathematic to a variety of routine and non-routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions.”
It’s very important that we don’t lose sight of these very important aims in the new drive for increased fluency in recall and calculation. Regular use of rich mathematical tasks in our maths classes can really contribute to both of these.
So what is a rich task? In her very helpful Nrich article, Jennifer Piggott describes the characteristics in some detail and some of these are:
- accessibility – they should offer different levels of challenge to learners of different ability, giving opportunities for early success but also scope to extend learning for the more able (low threshold, high ceiling)
- encouraging growing confidence and independence, often by working collaboratively
- potential to link with other areas of maths or to introduce entirely new areas of maths
- encouraging different approaches and creative solutions to problems
- allow learners to pose their own problems and ask questions
Jennifer Piggott also makes the important point that a mathematical task, although it may have the potential to do many of these things, does not become rich unless it is well led by the teacher, asking timely questions and supporting the children just enough to start to construct their own mathematical understanding whilst avoiding ‘spoon-feeding’ them. In practice, this can be difficult to do. In a busy classroom, it can be very tempting to wade in when a child is stuck and show them how to do it, but if we can restrain ourselves and instead offer a hint or a question that might open up a new avenue to explore, the experience will ultimately be much more satisfying and beneficial to our learners.
One good example of the sort of activity that could be used in this way is the ‘Sticky Triangles’ activity from Nrich. Children are presented with a growing pattern of triangles as above. These can be made from lolly sticks or pencils or similar or just sketched. You might like to present just the first two steps to start with and see if the children can suggest how to extend the pattern. Then get them to work on their own or in pairs or groups to explore the patterns. It’s probably best not to ask too many questions to start with. Children often naturally start to notice things about eg. how many triangles are in each row, how many lolly sticks are needed to make each pattern. It can be very interesting to watch the children and see how they approach things. Do they work systematically? Do they record anything? After a while, you might want to suggest some possible avenues for exploration. Can you see any patterns in the way the number of lolly sticks increases with each new row? Can you predict how many triangles will be in the next row? How many triangles would be in the tenth row? How many lolly sticks would be needed by this stage? What about the 100th row? Can you suggest any good ways of recording your findings? Encourage children to explain the patterns they see to each other and to you, and encourage the use of accurate mathematical vocabulary as they do this. The notes on the activity on the Nrich site also offers some other possible ways of extending the task even further.
The Nrich site offers lots of these sorts of activities at all sorts of different levels. As a teacher, I’ve found their curriculum mapping documents for KS1 and KS2 very helpful in identifying activities which might be linked to our other current work. Another source of helpful activities is the BEAM resources which can now be found in the elibrary of the National STEM Centre. You do need to register to access these resources, but registration is free and well worth while as there are a great wealth of resources in the elibrary.
For other suggestions for mathematical investigations, puzzles and challenges, have a look at my Pinterest board.