How can teachers help their most able mathematicians? Lynne McClure, consultant for the Mathematical Association discusses the problems and offers some solutions.
Joe is nine and a half, in Year 5 of a composite Year 5/6 class in a small primary school. He is enthusiastic and able in mathematics, and scored highly in the Year 4 assessments. The Year 5/6 teacher realises that he is able and so puts him into the Year 6 group for maths. He enjoys this and races through the work. By the time he enters Year 6 he has already met most of the Year 6 work, so the teacher, who is not very confident in her own mathematics, gives him Level 5 and 6 work taken from a variety of sources, including KS3 text books she has borrowed from a secondary teacher she knows. Most of the time Joe works alone. His interaction with the teacher is restricted to the time when she marks his answers. He is then told to go and try the next exercise and come back to the teacher if he gets stuck. He usually works it out for himself.
Joe is invited to join a group of Year 5 and 6 students meeting weekly at the local secondary school with one of the maths staff. Here the students work on ‘hard’ mathematical topics such as simultaneous equations and vectors. Joe enjoys this and feels satisfaction and pride at getting ahead of his peers and being able to describe mathematical topics of which they have never heard.
Joe moves to secondary school with a Level 5 in his maths SATs, looking forward to his maths lessons. He is placed in a mixed-ability class for the first term. He makes new friends but is bored with the ‘four rules of number’ that is the main content of the lessons. After Christmas he is put into the top set. Some of the topics he has already covered but has to repeat. He finds this boring and doesn’t see the point. The teacher realises this and so gives him the next textbook in the series so that he is kept occupied.
In Year 8 he has a new maths teacher. He expects the class to work quickly through the scheme of work as they are aiming for early entry to GCSE. He teaches the class as a whole so all the topics that Joe has already covered are revisited and he does the same exercises from the same textbook again. By the time he gets to Year 10 he spends much of his maths lessons doing past papers and he has come to the conclusion that maths is easy and boring. He achieves an A* in early entry. In Year 11, along with the rest of the early entry students, he takes an A-level statistics module. He achieves a B in this and is undecided whether to study maths in the sixth form.
The above story typifies many of those told to me by able mathematicians. You may want to consider what your reaction to it is, and perhaps think about what view Joe has of mathematics, apart from being bored by it. My sense would be that he perceives it as a solitary subject where speed is of the essence, and success is measured by how quickly one can get through the syllabus. On paper he is a success story for his school – early entry A* and a good grade in an A-level module. However, my contention is that his mathematics experience has been muddled and inappropriate, and we in the teaching profession need to think far more deeply about what we offer to those students who should be the next generation of mathematicians, but who probably won’t be.
Those of us who have been working in the G&T world for some time are well used, when trying to convince others that providing for able pupils is not elitist, to declaiming such phrases as ‘What’s good for G&T students is usually good for all’ or ‘The heart of G&T is good teaching’. We know that what we want to do is to encourage colleagues to offer appropriate experiences and challenge to all pupils so that they have a sense of achievement, and that includes the most able.
So what do we actually mean by ‘appropriate experiences and challenge’? What does good mathematics teaching look like when offered to G&T? And how does it impact on the rest of the students? I’d like to consider two aspects of this – the mathematical content itself and the way in which that content is offered.
Acceleration, unless carefully planned can lead to repetition or, perhaps even worse, to gaps
The content In the National Numeracy Project’s Framework for Teaching Mathematics (DfEE, 1999) the suggestion is that there are three ways of making the content of mathematics lessons more challenging:
- by offering the same content within a shorter time (acceleration)
- by offering additional content outside the regular curriculum (broadening)
- or by offering increased cognitive challenge within the same curriculum area (deepening).
I would predict that if you talk to any group of maths teachers in any school you would find that acceleration is the predominant strategy to increase challenge. Broadening, after all, requires additional preparation by hard-pressed teachers so is unlikely to find universal favour. Non-specialist colleagues may lack the confidence (or competence) to ask deeper questions requiring higher-order levels of thinking, so acceleration by default becomes the most common strategy. Certainly for Joe it was the most frequent offer.
Of course, there are advantages for a teacher in accelerating pupils through the curriculum. There is no need to invent activities or acquire different resources as everything is already mapped out for the older pupils. It’s an easy option but can set up possible problems for the subsequent teacher and, as we saw from Joe’s experience, unless carefully planned can lead to repetition or, perhaps even worse, to gaps. For the student it may well offer increased self-esteem but it may also be lonely and seem like a random and disjointed set of experiences.
Thinking like a mathematician
For me, the saddest part of Joe’s story is that he is not inspired to continue his mathematical learning. He has no idea of what it is to behave mathematically because his perception is that maths is the acquisition of a set of skills, a set of exercises to be completed, a speed test. Essential though these skills are, they are just the starting point. I want able students to ‘think like mathematicians’ – to enquire, to generalise, to question and seek proof. I don’t want them to accept everything I offer or be able to regurgitate pre-digested algorithms. I want them to engage with the thinking and find some things difficult. I want them to know that mathematicians see things in lots of different ways: that maths is creative and although there may be accepted elegant methods of solving particular problems or calculations, there are other equally acceptable ways of doing the same. I want them to take time to deepen their understanding rather than skating through the curriculum at high speed.
Writers on the subject (Krutetskii, Kennard) say that these forms of behaviour (generalising, seeking proof and so on) are characteristic of able mathematicians. This is what they do naturally. The teacher’s role is to offer opportunities, activities, investigations for able students to display these characteristics, and to model such behaviour themselves and hence help students to refine their thinking. Spending time deepening understanding in this way is time well spent for able pupils, and can have knock-on effects on the rest of the class who are exposed to behaviour that they might otherwise never experience.
I want able students to think like mathematicians– to enquire, to generalise, to question and seek proof
Where to look for more help
The good news is that, of course, there are teachers who do this and an increasing number of initiatives and resources to support them in doing it. One of the most exciting is the Primary Strategy Learning Network initiative in which groups of schools bid for funding for action-research projects. Several networks have taken the theme of increasing challenge for able pupils, or problem solving, or using and applying, all of which require teachers to deepen their knowledge of the primary maths curriculum. You can read more about them on the NCSL website.
Various associations offer Inset or day conferences for primary and secondary teachers on the theme of increasing challenge. For example, the Mathematical Association, Association of Teachers of Maths, and London GT all offer teacher events where the focus is on thinking mathematically. NAGTY offers a free accredited PGCE+ course for beginning secondary teachers and the Further Maths Network offers support for teachers of further maths, especially in areas where there is little CPD.
As well as professional development opportunities, there are some excellent resources which both primary and secondary teachers will find useful. Best of the bunch is the NRICH website which has thousands of investigations for all ages and stages, each offering opportunities for participants to think like a mathematician. Student solutions (not answers) to the monthly problems are posted on the website in the following month.
If you are looking for a paper-based resource, the Mathsinsight investigations, based on the World Class Test questions are excellent and are available for primary and early secondary students. RISPS is a new website offering rich activities for A-level students and the strategy websites also carry enrichment activities for primary and KS3.
For the students themselves there are advanced learning centres, supported by the National Primary Trust and offering weekly meetings for like-minded able students to work together. Similar to the Royal Institution Masterclasses, the idea is offer activities and situations that real mathematicians encounter. And for those who enjoy rising to the challenge, there are local and national (and international) competitions such as the Mathematical Association’s Primary Maths Challenge or the secondary UK Mathematics Trust (UKMT) competitions that feed into the Maths Olympiad.
The bad news
The bad news is that there is still a tension between the mathematics education community and various government agencies for whom targets are important, whether or not meeting them serves the needs of the individual pupils. The knock-on effect of this is obvious. One of my ITT students said to me, ‘We haven’t got time to think about it – we’ve got to get through the syllabus because we have to get so many Level 4s’. For whose benefit?
The new National Centre for the Excellence in Teaching Mathematics (NCETM) will, one hopes, ensure a united voice putting forward a strong case for all pupils to take time to appreciate the wonder and awe of mathematics. We wouldn’t call a Grade 8 violinist a ‘musician’ if the only pieces ever learned and played were examination pieces. There’s so much more music in the world. Why then do we think the mathematical equivalent acceptable? It’s too important to not get right!
Lynne McClure is an education consultant. She co-edited Curriculum Provision for the Gifted and Talented in the Primary School and edits the Primary Maths journal for the Mathematical Association and the Maths Coordinator File for pfp Publishing.