BEAM’s Big Book of Word Problems for Years 3 and 4
Solving word problems in mathematics has often been an underdeveloped area in many schools and classrooms practitioners and working as a Numeracy Consultant it is invariably cited as an area of difficulty when I work with class teachers and maths subject leaders. This is certainly borne out by many personal analyses of SATs and Optional SATs papers and, in their reviews of National Curriculum tests, it is perennially cited by the QCA as an area requiring attention. For example in the leaflet summarising the implications for teaching and learning from the 2004 Key Stage 2 tests teachers are asked that, in order to improve pupil performance they should:
- continue to provide opportunities for pupils to identify appropriate calculations to solve a variety of word problems, particularly those involving multiplication and division
- encourage pupils to persist when solving problems which involve more than one step, for example by checking that the answer makes sense in the context of the problem
Traditional approaches to word problems are frequently unimaginative and often seem to perpetuate the difficulties. There are still many text books that have a few addition problems almost as an after thought for children to do after a page of addition calculations – so some never get as far as the word problems and some know that all they have to do is apply the preceding calculation strategy to a context so just pick out the numbers and go. A second problem is that of the actual contexts of the problems. For one Year 3 inner-city class I observed, a set of word problems involving bees and flowers and plants and bulbs was incomprehensible, a situation made even worse when the numbers of flowers and bees in the illustrations didn’t correspond to the numbers in the questions. Another frequent approach is to suggest children look for prompts or keywords. Whilst in a superficial way it is appropriate for children to recognise the vocabulary associated with each operation – hence the classroom posters with word lists linked to each of the four operations - the peculiarities of language structure mean that this cannot be used in a simplistic way to help solve word problems. So faced with problems like “3 pizzas are each divided into 6 pieces. How many pieces altogether?” or “A rubber is 2cm long and a pencil is 10cm long. How many times as long as the rubber is the pencil?” a pupil referring to the class poster of arithmetic vocabulary could be excused for giving answers of 2 and 20 respectively. Working in an LEA with 70% children speaking English as an additional language I have had many teachers claim that this too is a contributory factor to the difficulty however my gut feeling (and the evidence from QCA that this is a universally recognised area of difficulty) suggests that whilst word problems may present challenges for EAL pupils, the real difficulty is the traditional pedagogic approaches. For EAL children the approaches characterised above are simply going to make things even worse.
It was with these thoughts in mind that I was pleased to receive copies of a new BEAM series ‘BEAM’s Big Book of Word Problems’ last summer and I have had opportunity to work with some local schools in using this resource. The series consists of three big books (one for each of Years 1 and 2, 3 and 4 and 5 and 6 respectively). Each book comes with a detailed set of teachers’ notes and a CD with the contents of the big book. This was particularly welcome in our schools as it enables the contents to be used on an Interactive Whiteboard and the beautiful illustrations and often humorous contexts really engaged the pupils I worked with. Although there is an emphasis on vocabulary (and indeed some of the problems are quite wordy) rather than spotting keywords the emphasis is on recognising problem types. For example with addition and subtraction children are taken through a series of exercises where they learn to recognise situations where a change has occurred, which might be an increase or a decrease, “I have 10 jelly beans and I eat 6. How many are left?” or where items are combined, “Gran gave me 5p and Grandad gave me 10p. How much do I have altogether?” or compared, “I have 10p, Penny has 5p. How much more do I have?”
Each book includes materials for 24 lessons and each of the lessons have a similar structure. The initial page of the big book presents three problems, which the class read together and solve. I led first the lesson with a Year 4 class. The illustration and context immediately engaged the children. The lesson focussed on combine and change (increase) problems. The teachers’ notes encourage discussion (between teacher and pupils and between groups of pupils) about what is similar or what is different about problem situations and about appropriate ways to solve the problems. Some children (particularly more able children) simply wanted to give the answer and for the first few lessons the class teacher and I had to repeatedly throw this back and insist that the children discussed the context and type of problem first. This can be quite a hard thing for some. Some of the more able pupils were also bemused that the follow up activity sheets for group work had only six questions! But the final one of these was an opportunity for reflection and gradually the children became convinced that it was better to do a smaller number of questions with understanding and insight and to be sure (through various checking strategies) that these were correct.
The activity sheets are provided in two parallel sets with identical word problems with different numbers and there is also a third set provided with blanks for further extension or simplification. A particular success was the way that the contexts encouraged children to make up their own examples and then challenge each other to identify which problem type they were and they became more adept at this as the term progressed. (The class used the book once each week). Another positive aspect was the way children were able to apply a range of calculation strategies or to check using a different one – which led to some interesting discussion about which was the best method. In a pure calculation lesson I’ve never been convinced that children ever really see the point of comparing different methods (what is the point?) but in the context of solving a problem that they cared about the debate about using different methods or the use of a second method as a check seemed to make good sense.
This is a series which offers a real way forward in terms of addressing a difficult but vitally important aspect of the maths curriculum.
Jon Kurta
Jon Kurta is now Strategy Manager (Primary, Special and Foundation Stage) in Camden. He was Numeracy Consultant in Westminster when this work took place. Thanks are due to Catherine Ballantine and her Year 4 class at St. Gabriel’s Primary School, Westminster
The whole pack consists of a big book (A2 size), an A4 book which contains teachers' notes and more word problems in photocopiable format. There is also a CD on which there is the big book and photocopiable pages, plus blanks for teachers to insert their own numbers, but not the useful teachers' notes.
When I looked through the big book my first reaction was to the pictures which are varied and fun - every page being by a different artist. My second reaction was the thought that the print was very small for a large group (it would be better with one problem rather than three to a page), but that the book was too big to be used by a small group of 4 - 6 children. My third reaction was to take the whole package into a nearby school and hand it over. The school in Snape is small with two classes, one of Years 1 and 2 and the other of Years 3 and 4 taught by the head, Angela Skinner. Here are her thoughts:
I trialled the use of these materials with a mixed-age Y3 and Y4 class.
The children enjoy using the big book although the text is rather small for a whole class to see, even when gathered round on the carpet. A better approach for whole-class situations is to use a digital projector to display the version of the big book included on the CD Rom. The illustrations are humorous and memorable; they help children focus on the “root situations for addition, subtraction, multiplication and division&rdsquo;.
The analysis of these situations on pages 8 - 10 in the teachers' notes is very succinct and clear. It really does help children if they have labels for the situations. Shared labels allow teachers and their pupils to interact effectively, discussing possible approaches when solving a range of problems.
The BEAM materials allow for effective differentiation, with related, but harder material in the second half of the big book. The harder and easier versions of the unit problems sheets (together with the blank problems sheets) are an effective way of meeting a wide ability range and yet keeping the whole class focused on the same root situations. Giving explanations and listening to explanations are important for learning; a whole-class focus increases opportunities for communication between class members.
Jenny Murray, Independent Maths Consultant, Suffolk
& Angela Skinner, Snape Community Primary School, Suffolk
BEAM’s Big Book of Word Problems for Years 3 and 4
Mike Askew
BEAM
£44.95 +VAT (£7.87)
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