# Problem Solving

"The ability to solve problems is at the heart of mathematics ..... for many pupils this will require a great deal of discussion and oral work before even very simple problems can be tackled in written form".

(Cockcroft, 1982).

For more information on this resource please contact staff at the Centre for Peer Learning.

### About

**What is "Problem-Solving"?**

Peer tutoring of reading is now commonplace in both in primary (elementary) schools and high schools. For both same-age and cross-age tutoring, the structured “Paired Reading” method has been intensively researched and found to be effective, yielding substantial gains in reading ability and in motivation for both tutors and tutees if organised correctly. The national "Read On" project in Scotland extended this to give a more conscious focus on higher order reading and thinking skills. The main thrust was cross-age tutoring between whole classes of older and younger pupils (students) in the primary (elementary) school, with some schools also developing a parallel home-based parental involvement programme.

Subsequently BP provided funding to extend a similar model to learning in mathematics. This project had a particular focus on problem-solving, an area considered extremely important by the Scottish executive in the light of international comparisons. For instance the recent report: "Standards and Quality in Primary Schools: Mathematics 1998-2001; A Report by HM Inspectorate of Education" lists "problem-solving and enquiry" as the first area needing improvement in mathematics education in Scotland.

The pilot phase of "Problem-Solving with BP" took place in the school year 2001-2002. Almost all of the 32 local education authorities (school districts) in Scotland participated in the in-service training, involving well over 150 schools. Subsequently, many of these schools developed pilot projects. Training for pilot teachers took place through November 2001. In January 2002, the pilot teachers launched their own in-school programmes of intensive activity, in most cases concluding before Easter. The remainder of the school year involved follow-up, development, reflection, evaluation, and planning for the future.

The Aims of the Problem-Solving project are to:

- Raise mathematical achievement in problem-solving
- Deepen mathematical understanding
- Improve mathematical competence in school and in everyday life
- Raise the mathematical self-confidence of pupils
- Increase motivation and enthusiasm for maths in and out of school
- Strengthen the partnership between home and school
- Strengthen the partnership between home/school and industry
- Maintain close links to the 5-14 curriculum

The Options in Problem-Solving with BP:

Peer Tutors | Parent Tutors | |
---|---|---|

Maths Games | ||

Maths Curriculum (Duolog Maths) |

Some schools combine these options in all kinds of creative ways, while other schools take different options for different groups of pupils at different times.

Both Read On and Problem-Solving are based on these principles:

- integrate with existing national strategies and complement school development plans
- structured, organised and manageable within existing school resources
- evaluated to ensure aims and targets are met
- best value and excellent cost-effectiveness is demonstrated and sustained
- designed to be sustainable and replicable

A DVD is available at a nominal charge (£15 + £5 p&p). Order from the Centre for Peer Learning, School of Education & Social Work, University of Dundee, Dundee DD1 4HN, tel 01382 383000, email k.j.topping@dundee.ac.uk. Payment by cheque (made payable to the University of Dundee) is required with the order.

The Problem-Solving project is supported by BP and The Scottish Executive, in addition to virtually all the local education authorities in Scotland and a great many schools, parents and pupils.

### Parent Guide

**Peer Tutoring:** **A Brief Guide for Parents**

Children are often excited to be involved in peer tutoring, and talk about it at home. Rather than leave parents to ask questions, schools often inform them before the project starts. A "brief guide" to peer tutoring is given below. This is a reproducible taken from "Peer Assisted Learning: A Practical Guide for Teachers" by Keith Topping (Cambridge MA: Brookline Books, 2001).

**Peer Tutoring: A Brief Guide**

Part of life at school asks children to try to do better than other children. But another very important part of school life is helping other people. Children learn well in both ways.

**History**

Peer tutoring means having children help other children to learn. Sometimes older children help younger children, and sometimes more able children help less able children of the same age. The idea is a very old one, first noted hundreds of years ago. In Britain, Bell and Lancaster used peer tutoring a lot about 200 years ago. By 1816, 100,000 children were learning in this way. Peer tutoring then caught on in quite a few parts of the world.

As more and more schools for everyone were set up, paid for out of taxes, peer tutoring was used less and less. Helping children learn was taken over by paid adult teachers. However, in the 1960’s it began to be used on a large scale again, especially in the United States. Teachers came to grasp that peer tutoring was a great ‘boost’ or extra help for all children. Today, peer tutoring is again spreading rapidly in many parts of the world.

**Effects**

Many peer tutor projects focus on reading, the most vital skill of all. But a wide range of other subjects have also been peer tutored, including mathematics, spelling, writing, languages and science. The tutors are not just being ‘used’, because they gain as much, if not more, than the tutees. To be able to tutor a subject, you have to really get to understand it well. So tutoring helps the tutors learn faster, too.

There is no doubt that peer tutoring ‘works’. There is a lot of research showing that in peer tutoring projects, the tutors improve as much, if not more than, the tutees. Many studies show that peer tutoring also improves how both tutor and tutee feel about the subject area - they get to like it more. Also, in many cases the tutor and tutee grow to like each other more, and get on better. There are many reports of both tutor and tutee showing more confidence and better behaviour. The research clearly shows that peer tutoring is one of the most effective ways of using school time.

Some projects have tutors and tutees of the same age, and some have older children as the tutors. Any difference in age does not seem to matter, as long as the tutor is more able in the subject area than the tutee. If the tutors and tutees are not too far apart in age and ability, there may be even more chance of the tutor gaining as a result. Some schools are also now tutoring with pairs of the same ability, where the job of tutor switches from one to the other (this needs very careful planning).

**Planning**

Peer tutoring takes time and care to set up properly, and it is the professional teacher who has the skill to do this. Plans must be made about matching child pairs, finding the right sort of materials, training tutors and tutees, and lots of other points of organisation. However, this time is worthwhile, for peer tutoring is very effective. It is very cost-effective considering what is gained from a quite small amount of professional teacher time. For many pairs, peer tutoring also has good spin-off in terms of better social harmony and more interest in other subject areas. Teachers often start peer tutoring in reading, but then become more confident in using the method in other subject areas.

**Scope**

Some primary (elementary) schools are now offering all young children the chance to be a tutee, and all the older children the chance to be a tutor. This helps to settle the young children into the school socially, and gives a boost to the older children, who feel very grown-up and responsible. In high schools parents can lose touch with what their children are doing, but peer tutoring is often more and more popular with children as they move up through the school. Like any other way of effective teaching or managing learning, setting up peer tutor projects needs enthusiasm, careful planning and hard work on the part of the teacher. It would be a great mistake to think of peer tutoring as an easy option.

### Teacher Guide

Schools were asked what advice they would give to other schools contemplating operation a Problem Solving project, yielding the following responses:

- go ahead and try it!
- easier the next time around
- be flexible and enjoy the experience
- don't under-estimate the time required for preparation and organisation
- selection of materials and timing their use is of vital importance
- make sure the tutors know the rules of the games in advance
- need for involvement of adults at the planning stage
- make sure you have enough games in all categories
- match pupils carefully - it is not wise to let them choose their own partner
- join in and enjoy the games and you will learn a lot about the pupils
- budget for games in advance
- after the initial outlay there are few ongoing costs
- make sure you try games before you buy
- go for a variety of games
- introduce new games later in the project to keep up momentum
- give yourself a good block of time to collect and organise games
- teachers of all classes involved need to have an input on resources
- requires staff time for organising problems into levels and categories
- involve parent and get them on board
- arrange an informal parents' evening to demonstrate the games and show their mathematical content is worthwhile set aside time for pupils to talk about benefits and difficulties
- keep an eye on progress and relationships between pupils

Also see the evaluation section.

### Children's Voices

**MARYBURGH PRIMARY SCHOOL**

The following are selected verbatim evaluative comments from children at Maryburgh Primary School:

**Games**

- I liked the Strategy games like Kono because you use your brain quite a lot.
- I liked strategy games because it's good when you have time to think of what move goes where and you had to watch the other people and where they go. It's just enjoyable.
- I liked the puzzles because they were really challenging.
- I liked the strategy games because you had to put a lot of thought into it.

**Instructions**

- Some games got confusing and you had to read the instructions two or three times to understand them. It was confusing if you just read them once.
- Some of the instructions were quite hard to read so you had to read them quite a few times to understand them. Once you thought you understood them there was always another bit to the game.
- If you couldn't understand them you would get quite bored. But the ones you did understand you wouldn't get bored.

**Talking**

- I talked about the game with my mum at home while we were playing.
- Me and mum sometimes took away some rules or we added some just to make it more interesting.

**Relating**

- Well at first we didn't get on that well but now we get on quite well. We agree a lot more.
- We're friends now because we never used to play with each other but now we do.
- It helped us a lot and we made new friends.

**Attitudes**

- I enjoy maths now. I used to hate maths but now I like it a lot.
- I used to hate maths. Now it's OK - I still don't like it but it is better than it was.
- I'm really enjoying maths now.
- It makes maths more fun.
- I didn't get bored at all - not even one second.
- It was really great and interesting.
- At first I found maths quite boring but now I see what fun it is it's quite good.

**Make you think?**

- When you're playing the games - well, I think it helped me a lot because for homework I had to read over what it says.
- Before, if I had problems I would just put down any answer, but now thinking it over with the games, it has just made it that bit easier.
- Oh yes it did make us think, especially the strategy games.
- Two brains are better than one.

**Learn maths?**

- The project definitely helped me learn maths and I would recommend it to another school or class.
- The project helped me learn maths because my dad keeps on putting things on the computer and I keep on getting them right.
- At first my partner always used to beat me but last time we played I beat her.

**Other comments**

I would like to work with someone from another class. We don't know the P3 and I think it would build up a friendship. We are in the classes during wet break but we don't get a chance to bond with other classes. It would also be better for them when they get to this stage. Some people don't like maths, but with the games it makes it easier. I would like to see the games go on longer, instead of six weeks maybe 10 weeks. I think is easier to play two sessions 30 minutes long. Or else the sessions are too short by the time you get the game out and read the instructions. It makes following instructions easier because at the start you felt oh these instructions are hard but now I feel that I can read them a lot better.

I would like to see the games going on for an hour. Working in pairs means you can think of an answer and your partner thinks of an answer - that way you are getting a right answer from your partner and a right answer from yourself.

**DUNROSSNESS PRIMARY SCHOOL **

The following are some verbatim evaluative comments from children at Dunrossness Primary School:

I thought that Paired Problem Solving was great fun. It was also good to find out more about your partner. It was nice to learn how to do maths quicker for my partner and I did not understand maths as much as I do now and I will soon get better. My partner and I are really good friends and we play together at break times and in Paired Problem Solving. I have really enjoyed it.

I thought it was good to have three different rooms because there were games in one, problems in one and puzzles in the other. I liked them all and I got to know my partner more.

I thought Mrs Hay's room was the best because my partner and I both liked exactly the same things and she liked me helping her.

I thought it was good to move around because if you get a bit bored then you can always look forward to moving to another class the next week.

I thought that my partner gave up on the problems too quickly sometimes but others he worked really well on.

I thought Paired Problem Solving helped you to get to know the Primary 4 pupils. It was good and as well as them it helped me with my maths. It was a very good idea and you should do it next year as well.

I thought it was very interesting and I think it should be done next year as well. It was good fun. Overall I enjoyed it.

I enjoyed Paired Problem Solving. The bit I liked was when we got to move to different rooms. I enjoyed the foam puzzles and the cubes.

I enjoyed Paired Problem Solving because you had lots of different problems to solve. I also enjoyed the different things to do and I liked meeting the Primary four pupils. My partner and I got along fine. I really enjoyed it.

I enjoyed Paired Problem Solving because you did not just stay in one room. I got on well with my partner. Sometimes I needed her to help me understand the game as well.

I enjoyed Paired Problem Solving because I got on well with Lynda. Some of the games were quite fun but some of them I did not like. Overall I thought it was quite good.

I really enjoyed Paired Problem Solving. I liked working with Lucy. It was a good idea to change rooms because you would get bored if you stayed in one room.

I thought it was a very good idea and good fun. I think it was good that we had to move because some rooms were boring when you were in there for too long. I think the Primary 4 room was the best because the puzzles were fun.

I think it should be done next year. It was fun but some bits were boring but good overall.

I thought it was good because in all the problems my partner tried his hardest at all of them and all the games, some he thought were boring but I liked it and I thought it was a good idea.

Paired Problem Solving was quite fun because it was not just maths because there were puzzles like the cubes and foam puzzles which were really fun. There were also games in which you learn but you were not actually thinking that you were learning. Some of the stuff was boring but most of it wasn't.

I liked Paired Problem Solving but in Primary 7 it got a bit boring because Erin was doing it all by herself. The Primary 6 class should be with the Primary 3 class next year.

I liked Paired Problem Solving especially in Primary 4 room but I did get a bit bored in Primary 7 because it was too easy for Chloe.

I think next year they should do Paired Problem Solving. It helps the Primary 4 pupils and it would have been good if I had done it in Primary 4.

I think it was a good idea to help the Primary 4 pupils with their maths. I thought some of it was a bit boring but overall it was fun.

I enjoyed Paired Problem Solving as it was a good change from normal maths.

I liked the games and I feel free and confident about maths. It is better with a partner because when you need help they help you. Thank you.

I liked Paired Problem Solving because you got a choice of problems and games. I also liked it because the games were good and fun. I think it is better when you work with a partner because it is easier.

I liked Paired Problem Solving because it gives you courage to say your answer and get rid of your fear of getting answers wrong.

I liked Paired Problem Solving because I feel confident that I can get the question right. It is a lot easier with a partner.

I think Paired Problem Solving is fun because we get to choose what problem we want to do. Jacqui is a great help to me. My best part was in P4 because everyone was really kind especially Jacqui.

I think Problem Solving is great fun to do and the different variety of games and maths make it better and it was easier.

I like Paired Problem Solving because you get a choice of games to play like Bingo and Pentominoes and David Laignel was a great partner.

I really liked Problem Solving because I have learned a lot. I really like working with a partner. I enjoyed working with Alister and Bobby. I like working with Clixi in P4.

I liked Paired Problem Solving because it is good fun working with a partner and it is easier. I like all the games and problems. Some of them are funny. It is quicker with a partner. It is fun and I am looking forward to it when I help the little ones.

It was really good to have extra help. My partner is really friendly. I would recommend Paired Problem Solving to anyone because it makes maths easier. Connie was an excellent partner because she is quick at maths but doesn't say the answer right away.

It made me confident. It made me think I can do maths. It made me like maths more. Andy was very kind to me as he helped. I know more.

I enjoyed Paired Problem Solving because of the variety of problems. I think working with a partner makes me confident because I have some help. It makes maths easier.

It's fun because of all the things and games my partner helped me and the best bit is the games.

I feel so confident and I don't make so many mistakes. It makes maths easier with a partner and there are lots of short problems. I really enjoy the stuff we do in our classroom.

In P4 my best things were the cubes. In the TV room I liked most of the games. In P7 I liked the problem when you had to find the school. Mhairi helped me a lot. She was an excellent partner. Some of the problems were hard and some were easy. Problem Solving is very good.

I really enjoyed Paired Problem Solving because it was interesting working with a partner. It was more fun and you got to choose any problems. My partner was great.

I would like to say that Paired Problem Solving has been really fun, doing it with a partner and maths is fun now with a partner by your side.

I feel more confident with Paired Problem Solving. It makes maths easier and it is better to work with a partner and I really like the games room best.

I thought Paired Problem Solving was fun. I liked the games room best. Magnus is fun to work with and he helped me when I was stuck. Now I am more confident with my maths.

I really enjoyed Paired Problem Solving because there was a variety of problems and there was always someone to help. I especially liked doing 3D cubes. I think Paired Problem Solving is great fun. I think it is better to have a partner because it is easier and it is much better than being on your own.

I liked Paired Problem Solving because it makes maths funnier. Working with a partner is lots of fun and it is easier. I like it because you can learn maths in different ways.

I thought Paired Problem Solving was great fun. If I did not have a partner it would be hard. My partner is very nice because Lindsey helps me.

I liked Paired Problem Solving because if I did a sum or something quite slowly Elsa would tell me how to do it a quicker way. The thing I liked best was the games. It was better with a partner because they help you when you are stuck.

### Evaluation

**Introduction**

In almost all cases, the evaluation involved gathering the perceptions and observations of the main participants, with feedback from a whole school questionnaire and separate questionnaires for individual teachers, parents and pupils.

**The Whole School Questionnaire**

The whole school questionnaire invited comments on:

- The usefulness of the draft resource pack coupled with suggestions for amendment
- The usefulness of the supplementary material with suggestions for amendment
- The aims and targets the school set for their project
- A description of the implementation of the project (numbers, stages, time scales)
- Outcomes of the project - positive features/benefits + negative features/difficulties
- The overall time scale for the project - too long/about right/too short
- Advice participating schools would give to others contemplating involvement
- The school's overall assessment of the added value of their project.

**Teacher Questionnaires**

There were two teacher questionnaires - one relating to Maths Games, the other to Duolog Maths. In both cases the questions related to the organisation of the activity, teacher observations of the children's' involvement, and some general comments on benefits, drawbacks and areas for improvement.

**Pupil Questionnaires**

There were two pupil questionnaires. The first, "What do you think?" about Maths Games, asked the pupils some 22 questions about their project experiences, after the event. The second questionnaire focused more generally on the pupil's Attitude to Mathematics and co-operation with other pupils.

**Parent Questionnaire**

The parents' questionnaire invited them to give their observations of their child's response to the project, their opinions of the value of the project and suggestions for improvements.

**Response Rates**

By the deadline, completed school questionnaires were returned by 45 of the 81 schools that took part in the project (56% response rate). Individual questionnaires were also returned from 76 teachers (46 doing Maths Games and 30 Duolog Maths), 1210 pupils (What do you Think of the project or Attitude to Maths) and 30 parents.

### Evaluation Results

**Results: Whole School Questionnaire**

All responding schools agreed that the draft co-ordinator's resource pack was easy to use and appropriate for the project.

How schools rated the value of the supplementary material is shown below as a percentage of schools (Table 1):

very useful | useful | not useful | |
---|---|---|---|

Poster (general aims of the project) | 8 | 88 | 4 |

Leaflet (question and answer leaflet for parents) | 21 | 71 | 8 |

Stickers (sets of stickers for pupil use) | 58 | 38 | 4 |

Certificate (for pupils completing the programme) | 79 | 21 | 0 |

Game Cards (for older pupils to take home & try with parents) | 33 | 54 | 13 |

Suggestions for amendment or addition to the draft Resource Pack included:

- a video
- problem resource suggestions for Duolog Maths
- clearer instructions needed for certain of the games
- simpler examples of games needed to get children started
- include a pack of games ready for use
- some activities needed to be more challenging

Suggestions for amendment or addition to the Supplementary Material included:

- more game cards
- larger game cards
- games cards to present to parents on completion
- positive reinforcement stickers for children

**Aims and Targets**

The specific aims/targets set by schools for their Problem Solving project included some of the following (from most frequent responses at the top to least frequent at the bottom):

- to improve problem solving strategies
- to develop problem solving skills
- to promote active & co-operative learning
- to encourage mathematical understanding through discussion and sharing
- to raise the confidence and self esteem of pupils
- to improve enthusiasm for maths both in and out of school
- to develop a positive attitude to maths
- to encourage children to talk using appropriate maths vocabulary
- to increase understanding of maths
- to develop positive attributes through praise
- to promote decision making
- to encourage reflection
- to improve determination and perseverance
- to work together amicably.

**Implementation**

Schools implemented the project in very various ways to suit their very different needs and contexts. The schools varied from small rural schools with a total roll of some 15 children to large urban schools with at least 2 classes in each year. Hence the number of pupils involved per school varied from 5 up to 120 in some larger schools. The vast majority of schools had pupils from P6 or P7 as tutors and from P3 or P4 as tutees. The projects ran on average for 7 to 8 weeks, with the children meeting together 2 or 3 times weekly, for 20-25 minutes per session. Once matched, the children generally remained with their partner throughout the project. Schools that decided to incorporate Duolog Maths into their project generally used Maths Games as an introduction for the first few weeks, before moving on to Duolog Maths for the remainder of the project - usually 4 or 5 weeks.

**Outcomes - Teacher Perceptions**

The following comments give a good indication of the positive features and benefits of the project and more negative features or difficulties schools encountered (from most frequent responses at the top to least frequent at the bottom):

**Positive Features/Benefits**

- aims achieved
- gave older children an opportunity to recall strategies
- pupils encouraged to focus on strategy rather than outcome
- helped to reinforce strategies P7 used for problem solving
- helped younger children to problem solve systematically/appreciate importance of recording
- gains in social and interpersonal skills
- very useful discussion between peers
- the confidence and self esteem of older children was boosted
- children enjoyed working together
- younger children benefited from one-to-one
- children were organised in their approach to problem solving
- pairs able to communicate in maths terms in order to play games and solve problems
- increased motivation for both sets of children
- very good for small school situation with composite classes
- games encouraged speedy mental calculation as well as strategy seeking
- children looked forward to their sessions
- confidence building for less able older pupils
- enthusiastic approach to particular games
- children took time to praise and encourage each other
- good relationships build up between classes
- children made new friends
- maths now considered a bit more 'cool
- talking and listening improved
- involvement of parents
- pupils said how much fun Duolog Maths had been

**Negative Features/Difficulties:**

- not sufficient time between in-service and implementation
- initial organisation and preparation time consuming
- the need for co-operation between staff
- the need to sell the programme to parents
- pupils found it difficult initially but improved with practice
- resources costly (£100)
- some timetabling problems
- games in the resource pack not attractive enough for children
- some of the instructions for the games were unclear
- time was needed to prepare older pupils for tutoring
- collecting sufficient suitable games for certain categories
- some games need instructions to be re-written and simplified
- lack of interesting games for shape and pattern
- noise can build up when playing games - use dinner or PE hall

A number of these issues concern the investment in time and resources needed for start-up. However, participating schools who had made that investment were now in a strong position to continue with the project much more easily in the future.

Schools were asked whether in retrospect what they thought of the overall timescale for the project: 91% felt it was "about right", 3% too long and 6% too short. Similarly regarding location of the pupil project activity between January and March, 97% felt it was "about right", 0% too early and 3% too late.

The following statements represented the participating schools' overall assessment and conclusion about their Problem Solving Project:

- successful project - targets achieved
- very valuable activity which we will repeat
- all targets met with a very positive response from both sets of pupils
- in general project a success, not everything perfect but would do it again next year
- happy with the results - now using it with P4 and P5 pupils
- project worthwhile - flexibility in length of session proved valuable
- very worthwhile project - achieved set aims - we will certainly repeat it next session when we will be more "geared up".
- 'good feeling' - highly motivating for both groups - seen as better than normal maths
- pupils' confidence in tackling problems enhanced; they stay with problems to the end
- children's motivation and enthusiasm for maths improved
- some children demonstrated ability above their level of maths learning
- high degree of co-operation between pupils
- all children enjoyed the pairing arrangement
- children thought that working with a partner to solve problems was a good idea
- no change of pairing and no behavioural difficulties even although some children had a history of poor behaviour
- children look at problem solving in a new light
- listening and questioning skills of tutors improved significantly
- children more confident in their approach and keen to be involved
- pupils look forward to the session
- 12 parents came to the presentation - a high turn out for this area!
- parental involvement was less enthusiastic than anticipated

**Results: Teacher Questionnaires**

There were two Teacher Questionnaires - one related to the use of Maths Games and one to the use of Duolog Maths. Both questionnaires explored:

- organisation
- observation of children's involvement, and
- some general issues

**Maths Games**

The responses from the teachers are encapsulated in the following tables giving percentages of teacher responses and in the summaries of their comments.

**Organisation (Table 2)**

Involving the children with Maths Games was:

1 | 2 | 3 | 4 | 5 | ||
---|---|---|---|---|---|---|

enjoyable | 63 | 24 | 13 | 0 | 0 | not enjoyable |

interesting | 55 | 34 | 8 | 3 | 0 | not interesting |

hard work | 13 | 16 | 32 | 26 | 13 | easy |

useful | 21 | 45 | 18 | 16 | 0 | not useful |

difficult to organise | 12 | 13 | 40 | 21 | 14 | easy to organise |

Comments: Organisation was time consuming but once done in advance the project went smoothly. Not long enough between in-service and the beginning of the project (i.e. Nov-Jan), Some instructions for games unclear, More games needed - more adult support.

**Observation (Table 3)**

majority of children | some children | a few children | |
---|---|---|---|

Improved co-operation with partner | 68 | 29 | 3 |

Increased discussion of tasks | 61 | 36 | 3 |

Increased enthusiasm & interest | 66 | 34 | 0 |

Increased concentration span | 45 | 55 | 0 |

Improved attitude to maths | 40 | 49 | 11 |

Comments: Some of the above were difficult to assess. Children loved it, thought they were not working.

**Major Benefits**

- Children working together was fun
- Helped confidence, co-operation and communication skills
- Concentration improved
- Beneficial social interaction
- Challenging / more interesting
- Older pupils took more responsibility

**Drawbacks**

- Shortage of suggestions for games
- Organisation of classes/timetables
- Some older children had difficulty - deciding how much to tell tutees.

**Suggested Improvements**

- Quality of games essential
- Clearer instructions on game cards
- More funding for games
- More training for older pupils
- Need for more problem solving activities in the pack
- Material should be categorised
- Before and after test

Final Comments: Provision of more information/catalogues where desirable resources can be found for each age group. Staff to be made aware of implications for time. It was enjoyable for all/worked well. Will do it again. Pupils and staff requested a repeat, agreeing that benefits outweighed effort.

**Duolog Maths**

The responses from the teachers are encapsulated in the following tables giving percentages of teacher responses and in the summaries of their comments.

Organisation (Table 4)

Involving the children in Duolog Maths was:

1 | 2 | 3 | 4 | 5 | ||
---|---|---|---|---|---|---|

enjoyable | 18 | 41 | 36 | 5 | 0 | not enjoyable |

interesting | 23 | 59 | 14 | 4 | 0 | not interesting |

hard work | 14 | 32 | 23 | 14 | 17 | easy |

useful | 14 | 59 | 27 | 0 | 0 | not useful |

difficult to organise | 14 | 23 | 27 | 32 | 4 | easy to organise |

Comments: Once organised, daily running was very easy to implement. Time consuming. A lot of thought required to organise the problem solving activities.

**Observation (Table 5)**

majority of children | some children | a few children | |
---|---|---|---|

Improved co-operation with partner | 86 | 14 | 0 |

Increased discussion of tasks | 68 | 32 | 0 |

Increased enthusiasm & interest | 55 | 45 | 0 |

Increased concentration span | 50 | 32 | 18 |

Improved attitude to maths | 23 | 63 | 14 |

Comments: P4s appreciated help as it gave them an opportunity to try problem solving in a less threatening way. Less fear of failure. More benefit for younger pupils than older. Staff found difficulty in gauging problems and difficulty.

**Major Benefits**

- Significant social benefits - improved confidence
- Communication and co-operation
- Helped develop strategies for problem solving
- Encouraged younger ones to keep trying
- Gave children more responsibility

**Drawbacks**

- Organising time between classes
- Production of sufficient materials
- Getting started!
- Trying to keep an eye on each pair

**Suggested Improvements**

- Pre-post assessment activities needed
- Provision of problems graded to suit different ages and abilities
- Extend Duolog concept to other areas of maths
- Children should be asked for their comments throughout the project

**Final Comments**

- Children responded positively
- Challenging and fun at the same time
- Ability of older pupils as tutors was encouraging
- Need for a bank of resources from BP
- Need for more time in school for planning
- Once ground work is done it would be worth repeating

**Results: Pupil Questionnaires**

There were two pupil questionnaires - one headed 'What do you Think?' about maths games; the other related to pupils' Attitude to Maths.

**Maths Games**

In the first questionnaire the pupils were asked to circle the answer they thought was right for them or put in their ideas where asked. The following summarises the pupils' views:

Which games did you like best?

Snakes & Ladders, Shape, Mazes, Chinese Chequers, Chess, Battleships, Race to Treasure, Bingo, Belly Buttons, Connect 4, Tunnels, Guess Who, Rockets, Parachutes, Fox, Geese, Race Track, Fill The Bucket, Pathways, Honey Bears, Climb The Rigging, Go, Palm Tree, Strategy, Dominoes, Uno, Jenga, BP card games, Tumbling Monkeys.

Was one box of games specially:

a) Boring? Which box? Number, Shape

b) Interesting? Which box? Strategy, Spatial Relations

Was the number of games:

Too Many 2%; Enough 78%; Not Enough 20%

Was it hard or easy to choose good games?

Hard 40%; Easy 60%

Were the rules and instructions too long or too hard for some games?

No 29%; Yes for a Few games 57%; Yes for a Lot 7%; Yes for Most 7%

Was it easy or hard to sort out which partner did what?

Easy 65%; Hard 14%; Varied 21%

Did you get bored?

No 44%; A Bit 44%; A Lot 12%

Did your partner get bored?

No 44%; A Bit 31%; A Lot 25%

Did you TALK about the games?

No 14%; A Bit 40%; A Lot 46%

Did you change the rules of any games?

Yes 24%; No 76%

Do you now get on with your partner:

Better 56%; Worse 11%; the Same 33%

Overall, was the Problem Solving project:

Fun? Yes 89%; No 11%

Interesting? Yes 82%; No 18%

Did the Problem Solving project:

Make you think? Yes 85%; No 15%

Help you learn maths? Yes 80%; No 20%

Are you now better at:

Maths Games? Yes 85%; No 15%

All kinds of maths? Yes 61%; No 39%

Do you like maths more now?

Yes 65%; No 35%

Do you put maths mistakes right by yourself now?

Yes 84%; No 16%

Could you find or invent some maths games of your own?

Yes 62%; No 38%

Did any other good things come out of doing the Problem Solving project?

Yes! Made friends, Helped each other, Liked working in pairs, Think more.

Would you recommend Problem Solving to others?

Yes 86%; No 14%

How could we make the Problem Solving project better?

New games, Change partner, Have a longer time

Do you want to (choose only one):

Stop Problem-Solving for now? | 15% |

Go on, but only twice a week? | 12% |

Go on as often as before? | 33% |

Do other maths work in pairs? | 40% |

**Attitude to Maths**

Almost all schools using this questionnaire asked pupils to complete it only at the end of the project (post-project only). Two schools used it on a pre-post basis. Another school devised their own questionnaires, one used in the middle of the project and the other (with different content) at the end.

The Attitude to Maths questionnaire had 16 statements about maths, schoolwork in general, how the respondent felt about their own work and about working with other children. Respondents were asked to circle the extent to which they agreed with each statement in relation to themselves from these options: Yes, Sometimes, Not Really, No. The following table summarises pupil responses in percentages:

Attitude to Maths Questionnaire Results (Table 6)

Yes | Sometimes | Not really | No | |
---|---|---|---|---|

1. I enjoy maths lessons | 41 | 41 | 11 | 7 |

2. I have lots of friends | 81 | 8 | 8 | 3 |

3. My teacher thinks my work could be better | 26 | 52 | 11 | 12 |

4. I like to keep my ideas to myself | 32 | 36 | 14 | 16 |

5. Maths games are boring | 9 | 28 | 17 | 46 |

6. I try my hardest at maths | 70 | 21 | 3 | 6 |

7. Working with a partner is stupid | 6 | 8 | 19 | 67 |

8. Some days I don't care if I do bad work | 15 | 16 | 15 | 54 |

9. Maths is a hard subject | 17 | 39 | 18 | 26 |

10. You can solve problems with a partner | 69 | 25 | 3 | |

11. My work is quite bad | 10 | 30 | 28 | 32 |

12. I can understand maths words | 60 | 31 | 5 | 4 |

13. Other children don't like to work with me | 12 | 18 | 20 | 50 |

14. The others think I am ace at maths | 22 | 28 | 24 | 26 |

15. I work well with my partner | 63 | 26 | 6 | 5 |

16. It's good to help others in lessons | 68 | 23 | 5 | 4 |

Pre-post attitude to maths questionnaire results were available from two primary schools: Dunrossness and Maryburgh. the Maryburgh results are embedded in a more intensive evaluation which is reported in greater detail below.

The Dunrossness responses showed a pattern often found in the use of pre-post attitude scales with young children. At pre-test, lacking any benchmark, pupils often rate themselves unrealistically high. At post-test, there is consequently very little head-room for any increase to be possible. Additionally, at post-test pupils are often more reflective and realistic as a result of their project experiences, so large gains are doubly unlikely. At pre-test, the Dunrossness pupils rated themselves very highly on items 1, 2, 5, 6, 7, 10, 12, 13, 15 and 16 (see items in Table 6, above). At post-test, slight gains were evident on items 1, 2 and 4, but other items showed little difference.

West Linton school devised its own mid-project and post-project questionnaires. At mid-project, 76% of the 29 pupils felt that only a few of the game instructions were too long and hard, but 62% reported being bored some of the time (the latter perhaps indicating a need for mid-project organisational or resource adjustment). However, only one pupil reported not enjoying the games. Asked if the games made them think about maths, the pupils were divided in their views (a common finding, as primary school pupils often do not consider the games to be "proper" - i.e. traditional - maths). A number of pupils expressed a wish to continue with the project, and several felt that the time allocated was too short. They were equally divided about whether to keep their existing partner or change.

Perhaps the effects of teacher adjustment were evident in the post-project responses - 61% voted the project "great fun" and a further 25% "quite enjoyable", while 85% would recommend the project to other pupils and 81% would be interested in same-age peer tutoring as a continuation. About 80% of pupils said they liked being able to choose a game, enjoyed working together, enjoyed playing most of the games and found the project "more fun than normal maths". A similar proportion reported having changed the rules of games, usually to make them more interesting. However, still only half the pupils felt they "learned something about maths ideas" while playing games - again raising questions about their definition of "maths".

Children commented on their liked and disliked games. Several expressed dislike because a game was "too hard" for them - although some but fewer children disliked games because they were too easy. This emphasises the importance of differentiation in games and careful selection by pairs to ensure appropriate levels of challenge. Some children liked games with an activity or movement component. Some games were disliked because they were seen as mechanical and repetitive. Some pupils liked games that were different from the usual version, while others complained about this novelty.

**Results: Parent Questionnaire**

There was only one parent questionnaire. Almost all responding parents had been involved with maths games. Parents were asked to circle the answer that was true for them, or put in a comment where asked. The following summarises the parents' views:

**Found it easy to take part in the project:**

Yes 92%; No 4%; Not sure 4%**Found the game instructions difficult to follow:**

Yes 10%; No 70%; Not sure 20%**Found using the maths words on the diary cards:**

Easy 50%; Difficult 0%; Not sure 50%**My child sometimes reminded me if I forgot our session:**

Yes 85%; No 15%**I became bored with the sessions:**

Very soon 0%; Occasionally 70%; Not at all 30%**My child became bored with sessions:**

Very soon 0%; Occasionally 60%; Not at all 40%**I will probably go on doing maths games:**

Every day 2%; Regularly 68%; Sometimes 30%**Did you use any other maths games?:**

Yes 30%; No 70%

If yes, what did you do?: Mental Arithmetic, Card/Board Games, Monopoly, Cluedo, Chess, Pitya Game, Dominoes, Magic Squares**I have recommended Problem Solving to others:**

Yes 18%; No 82%**I will recommend the project to other parents:**

Yes 18%; Probably 56%; Not Really 20%**What did you and your child learn from the project?**

More to maths than just sums

Enjoyable to have a game each night

Made time to spend together

Discussed taking turns

Different ways of solving problems

Made child stop and think

Good to be able to learn and play at the same time

How to play and involve the whole family

Patience and team work**Have you noticed any other changes in your child?**

Liked maths games

Enjoys maths more

More willing to sit and play

Follows instructions more easily

Thinking more logically and not just trying to win

Sulks when he loses

Got fed up with only 2 games after a week or so**Do you have any suggestions for improvement?**

Instructions: Could be made easier to understand, Should not be hand written.

Games: Button game unsuitable for parents with younger children, A bit repetitive, Games should be larger and more colourful, Lotto type game with answers.**Which game did you like best?:**Yahtzee, Connect 4, Incy Wincy Spider, Billy Buttons, Rummikub; Pitya, Fractions, Rebel, Chinese Chequers, Multiplication Puzzle, Uno, Guess Who?, Dot to Dot, Magic Squares, Number Games, Palm Tree, Strategy, 57 Game.**Any other comments?**

Looked forward to each night, whole family involved

Useful exercise, made you sit with your child

3 times per week for longer would be better as some games took a while to complete

Difficult to fit in each night due to family commitments but worthwhile

Good idea

### Case Study

**An Intensive Case Study Evaluation**

At Maryburgh Primary School (Ross & Cromarty, Highland), the co-ordinating class teacher (Teresa/Terry Kerr) conducted a more intensive evaluation using multiple methods, which is summarised below as a coherent case study. It echoes many of the findings outlined above.

**Abstract**

Raising mathematical achievement in problem solving and self-esteem in mathematics was the main focus of this study. It involved enhancing mathematical self-confidence through co-operative learning, and improving language, motivation and cohesion between "school" maths and "life" maths. The utility of same-age peer collaboration and parent partnership using mathematical games was explored in an investigation of process and pre-post impact on pupils. The experimental group were all the 10/11-year-olds in a lower maths set (n= 17) in a small rural primary school. The comparison group were the higher set (n=15). Randomly matched pairs of pupils worked collaboratively with maths games in school. Parents were encouraged to "problem solve" using mathematical games at home. Twelve parents participated in this aspect of the project. The project ran for 6 weeks, with two gaming sessions of 30 minutes each week in class and a daily gaming session at home for a maximum of 20 minutes. Pre- and post-intervention, the experimental group completed questionnaires on attitudes to mathematics. They also participated in small group subjective feedback discussions post-intervention. The class teacher and classroom assistant recorded their structured observations and perceptions. Results of a mathematical test given pre- and post-intervention showed a greater increase for the intervention group. On the attitude questionnaire there were gains in sharing of ideas within class and attitude towards mathematics. On the parent questionnaire, all parents commented that it was easy to take part in the project and that they would recommend it to others, as they enjoyed the experience and felt it was of value to their children. Improvements in social interactions in class were evident through observation. Overall, the subjective feedback from participants, teachers and parents was very positive. This small-scale project appeared very successful in increasing self-esteem for participants and strengthening the partnership between home and school, as well as increasing motivation and enthusiasm for maths in and out of school. Recommendations for future research and practice were made.

**Aim of the Study**

This study investigated whether a structured problem-solving programme using mathematical games would:

- have a positive impact on attitude towards mathematics
- raise the mathematical self-confidence of the participating pupils
- have a positive impact on mathematical skills and knowledge
- strengthen the partnership between home and school.

Using mathematical games, it deployed paired same-age peer collaboration in class and parental involvement at home in parallel. The study was part of a national project and evaluation on Problem Solving sponsored by BP.

**Context and Sampling**

This project was located in a small rural primary school in the Highlands of Scotland, with a roll of 124 pupils, 6 members of staff and a class-committed Head teacher. It catered for the children of the local village and surrounding area. The ethos of the school was good and there were no major discipline problems. Well-established contact with parents resulted in almost 100% attendance at parents' evenings. A focus of school development at the time of the project was raising attainment in mathematics. Primary classes 6 and 7 were given NFER-Nelson maths tests in June 2001 and 'set' into two ability groups as a result. The intervention group for the project was purposively selected: the Primary 6/7 pupils (10/11-year-old) pupils (n=17; 8 girls, 9 boys) in the lower set. The comparison group was the more able set (15 pupils; 8 girls, 7 boys). The comparison group received normal mathematical instruction from their class teacher. Before the project, relationships between pupils in the experimental class were seen to be unsatisfactory, the group lacking cohesion. The nature of the ability setting in the school led to the targeting of the development of greater positivity in self-concept as a mathematician and attitude to maths.

**Materials**

Some commercially produced mathematical games were already in stock at the school. The researcher gathered several more games from a section of reproducible mathematical games in the Paired Maths Handbook (Topping & Bamford, 1998a), and some from the Heinemann Mathematics Scheme. The games were selected because of their clear instructions and use of mathematical vocabulary. Two or more players could play games, while puzzles presented a problem for one person. The games and puzzles were then colour-coded and labelled: red for number, yellow for strategy, purple for space and blue for puzzles. Participants were intended to choose from the different categories in weekly rotation. Number included counting, bonds, sequence, order, computation and probability. Strategy games involved a large element of deduction and prediction leading to the generation of strategies. Space games incorporated a large element of visual-spatial perception and organisation and an appreciation of shape, pattern and geometry. Puzzles were included because pupils were able to play them individually should their partner be absent and because they led to children working together to find a solution. The researcher gathered materials such as counters, dice, packs of cards, and simplified the instructions for some of the games if necessary. The researcher herself played most of the games and solved most of the puzzles provided. This familiarisation was time consuming but found to be essential. Some games were discarded during the project as some of the participants found them too difficult to understand.

**Matching Pairs**

The usual procedure for matching tutors and tutees is to rank tutors in order of mathematical competence, rank tutees likewise, and then match the most able tutor with the most able tutee, and so on. However, in this school context this was felt unnecessary, as the difference in mathematical ability between pupils in the set was not great. Therefore pairs were matched by random allocation. Subsequently the pairs worked co-operatively with no role definition. However, later it became apparent that the matching had resulted in some dysfunctional pairings. Pairings were switched in these circumstances, and also to accommodate temporary absence of a partner.

**Implementation**

All the lower set of 19 Primary 6/7 pupils took part in the project. It operated for six weeks, with two 30-minute gaming sessions per week in class. Initially there were three 20-minute sessions, but the researcher found that a lot of time was being taken up with setting up and tidying away, so it was changed to two 30-minute sessions. Division of the games and activities into categories allowed the pairs to make a choice from a different area each week. The games sessions were organised and supervised by the teacher. She circulated around the groups while they were playing, encouraging and helping any confused pairs to follow their game instructions. Towards the end of the session the teacher instructed the pairs to put away the games. This routine was adhered to at all the games sessions, sometimes followed by discussion between the teacher and the children about the games and any difficulties or successes they had experienced. Children doing puzzles worked largely individually, and this provided a change of pace.

Twelve parents took part in the parental involvement aspect of the project. The main features for this were a commitment to playing maths games for a short period of time for several sessions per week over a period of six weeks. Parents were given a practical demonstration of a few of the games at a launch meeting, with clarification of the mathematical concepts and language involved. During the course of the meeting the co-ordinating teacher circulated among the pairs to help with any problems of understanding or implementation. At subsequent game exchange time the teacher was available to provide the parents with support and encouragement. Parents were asked to keep a weekly record of the games played and the time spent at each session. There was also a space on the diary record for any comments that they chose to make. At the end of the project, parents were asked to attend a follow-up meeting to exchange feedback between parents and the teacher, to evaluate the project and to discuss future planning. At this meeting parents completed a Parent Evaluation Questionnaire.

**Instrumentation**

The assessment tools selected for this project were:

- Feelings Questionnaire (a reproducible Attitude To Maths questionnaire from Topping & Bamford, 1998a) for all the experimental pupils pre- and post-intervention. This required each child to respond to 16 statements about themselves in terms of four options: Yes, Sometimes, No, Not Sure.
- A What Do You Think? questionnaire (a reproducible feedback questionnaire from Topping & Bamford, 1998a) for all the experimental pupils post-intervention. This required each child to respond to 22 statements about what they thought about the project.
- Audio recording and subsequent content analysis of semi-structured interviews between the teacher/researcher and the participant children.
- A Parent Questionnaire (a reproducible questionnaire from Topping & Bamford, 1998a) for all the participating parents post-intervention. This required each parent to respond to 9 statements about the project. A further five statements allowed the parents to make any comments they wish.
- Parental views gathered at meetings with the teacher occurring before and after the project, and their completion of diary record cards.
- The focus of this project was problem solving. It was felt that traditional maths tests would not be very sensitive to that focus or to short-term gains. However it was decided to use NFER-Nelson 9/10 tests at the end of the project with the experimental and control groups. The school normally completed these tests late in the summer term and they had already used them in June 2001.

**RESULTS**

**The Attitude Questionnaire**

On the Attitude Questionnaire, a number of the components showed very little significant change (a not uncommon finding, since pupils have no benchmark against which to judge their pre-test responses). Those that did change mainly related to improved self-esteem and relationships in class. There was a 10% rise in answering Yes to item 1: "I enjoy maths lessons", although the overall positive response to that item remained the same. Item 4 showed a significant shift away from "Keeping ideas to myself". No-one thought maths games were boring (item 5) and everyone tried their hardest at maths by post-test (item 6). At pre-test, 80% thought that maths was hard all or some of the time, but post-project that figure had gone down to 47%. There was a small rise in the percentage that "Can now understand maths words" (item 12). In item 13 "Other children don't like to work with me" there was a significant rise in the percentage of No responses, from 32% to 68%.

**Group Discussions with Pupils**

In general, participant pupils were enthusiastic about the project. Some children reported finding the instructions for games too long and therefore hard to understand on a first reading, but when they read them 2-3 times they became clearer. One participant commented that this helped him a lot in his homework, because he had to read over what it said two or three times, something he didn't mind doing now. Most participants thought that the shorter instructions were easier to follow, but they were able to ask for help if required. Some felt that they could now follow instructions more easily. On first reading, they used to always think that instructions were hard, but now they felt that they could "read them a lot better". When invited to share what they regarded as the positive aspects of the project, the majority of children reported that they found it enjoyable and interesting. Some reported that they made new friendships and enjoyed working together playing the games. Others commented that they would like to work with someone from a different class "to build up a friendship". This was because they have experience of monitoring younger classes but never get a chance to "bond with them". They also reported that it would be better for the younger ones, because it would make maths "easier for them by playing the games".

Participants appreciated learning new games and felt they could adapt them for future use. They reported that the gaming sessions were more successful if played twice a week for 30 minutes, rather than three 20-minute sessions. The 20-minute sessions were too short by the time "you got the game out and read the instructions". When asked about their favourite games box, the majority of participants chose the Strategy or Puzzle boxes. One pupil commented that this was "because you had to use your brain a lot". Another stated that it was good because "you had time to think of what moves you could make", and you had to watch your partner's moves too. Most participants felt that the Strategy games made them "think - think a lot!" When asked if it was hard to choose games, some reported that it was hard because there were so many good games.

**Parent Feedback**

The parents' reactions were very positive and suggested that they had found it very easy to take part in the project. The game instructions were generally not difficult to follow. The parents reported that the children were enthusiastic to play the games and reminded the parent if the parent forgot. Only occasionally did either parents or children become bored with the session. Many of the parents said that they would continue to play maths games with their children regularly and that they would probably recommend Paired Maths to other parents.

In discussion with the teacher/researcher the majority of parents reported that both they and the children had found that "maths can be fun", even interesting sometimes. Some noticed an increase in confidence in maths homework and mental calculations. One parent commented that the repetition of the game improved strategies and confidence. Time spent in the daily sessions was also valuable - "we could learn together and from each other". One parent felt their child had learned a deeper understanding of number in everyday life, although he was initially resistant to the project. Parents thought that more commercially produced games might be perceived by children as less school orientated, but in general they thought there was a good variety of games. All parents reported that it would be helpful to reduce the number of project days at home to 5 per week (i.e. exclude weekends), to retain enthusiasm and to fit in with daily routine. Other things happened at weekends and it was more inconvenient to fit in maths games.

Some parents felt that letters, rather than attendance at a meeting, could have conveyed more information about the project, because some parents could not attend the initial meeting. More flexibility with game exchange days might be useful, as some parents were unable to change on a particular day owing to unforeseen circumstances. Card games were the most popular game and the one most likely to be continued after the project. It was reported that a repeat of the project would be beneficial, but maybe for a shorter spell. All parents thought that it was a good exercise to encourage children to participate in practical games and discuss strategies. However some parents expressed some anxiety about their child refusing to play the game. One child apparently displayed a very negative attitude at times, although he did recognise the benefits to be gained. The project had motivated some parents to continue to find opportunities to do similar activities.

**Teacher Perceptions and Observations**

At the end of the project the teacher and the classroom assistant of the experimental class completed an evaluation questionnaire. In general, responses were very positive. Involving the pupils in the project was thought to have been enjoyable and interesting. Pupils had shown increased independence in their ability to organise themselves in the selection, playing and returning of the games. It was noted that there was enhancement in motivation and self-esteem in almost all of the participating pupils. One of the negative aspects of the project was the time required to gather the material and the organisation prior to the project itself. However, it was found that once these materials were established, it was easy to organise, and the children themselves were good at keeping the games in order.

By the end of the project the pupils were more effective at taking responsibility for their own learning. There was clearly a more positive attitude towards mathematics - the teacher noted that the children were quick to point out when it was Maths Games day. The children showed high engagement with the games. This promoted greater concentration and persistence as well as decision making and problem solving. There was evidence of enhanced team co-operation in learning and use of interpersonal skills such as discussion and turn taking. Before the project began, children identified themselves according to which primary year they came from, either 6 or 7. There appeared to be a lack of cohesion in the group. For the duration of the project the pairs were matched randomly and this resulted in some children working in mixed pairs, i.e. across both gender and primary year. The teacher reported enhancement of a feeling of bonding within the maths set, which was not there prior to the project. The children acknowledged that they had made friends within the set and that these friendships continued in the playground and other situations outwith the maths class.

The teacher also noted the choices the children made of games and where to play them. Given the freedom of the room some children chose to play their games on the floor, while others chose the more conventional table and chairs. It was noted that in the latter stages of the project there were signs of enhanced self-confidence in the selection of games and where to play them. The teacher also observed a general enhancement in mathematical confidence, although this did not come across in the children's responses to the attitude questionnaire. The children became more willing to tackle problems and puzzles in formal maths lessons. The teacher also noted that there was evidence of peer collaboration rather than peer tutoring. It was felt that more training could have and perhaps should have been give to the pupils with regard to the notional tutoring aspect of the project. However some participants spontaneously expressed a wish to "tutor" younger children in the games which they were now familiar.

Meeting with parents on an informal basis was another gain from the project. It appeared that parents became more comfortable about coming into the classroom and discussing the next game with the teacher. It was felt that parents and teacher were working together. Most parents were motivated to continue the project when they could see improvement in their child's achievement. The project also highlighted for the teacher the different learning styles of the pupils. Many pupils had benefited from working in pairs or small groups and the teacher encouraged this to continue in traditional maths lessons.

**Mathematical Tests**

The focus of this project was problem solving. It was felt that traditional standardised maths tests would not be sensitive to the project focus or to short-term gains. However it was decided to use NFER-Nelson 9/10 tests at the end of the project, as the school normally used these tests. The intervention and comparison groups took NFER-Nelson 9/10 mathematics tests in June 2001 and at the end of the project in March 2002. It was felt that there was an issue with comparability between Tests 9 and 10. Therefore it was decided to use the same test as before for each pupil rather than going on to the next test. Cockcroft (1982) emphasised that tests "should enable candidates to demonstrate what they know and should not undermine the confidence of those who attempt them". As one of the main aims of this project was to enhance the self-esteem and self-confidence of the participants, the teacher felt that they should not be put into a situation which threatened feelings of failure. Therefore the pupils took the same level of test as they had done the year before. The comparison group had equivalent experience in every respect apart from the involvement in the project, each pupil also taking the same test as they had previously.

There were forty-five items in the maths test and the highest possible score was 45. The chronological age of some pupils at post-test did not allow the generation of standardised scores - they had exceeded the age ceiling for the norms of the test. Consequently, raw scores were compared for all pupils. As these were uniformly available for both experimental and comparison groups (albeit from different pre-test baselines), this was felt to offer the best compromise. The raw scores of both the intervention group and the comparison group increased from pre-test to post-test, as would be expected from natural maturation. However, the percentage improvement for the experimental group was substantially larger than that for the comparison group on both Test 9 and Test 10 (see Figure 1 below).

At pre-test the mean scores of the comparison group were of course higher than those for the intervention group. At post-test the difference between experimental and comparison mean scores had narrowed significantly.

**Discussion**

The six-week intervention might have been considered too short for any significant enhancement of the pupils' self-confidence in maths to be likely to be observable or measurable. The majority of the pupils thought that the games were fun and interesting and wished for the games to continue as before. Therefore the project could have had an eight or even ten-week intervention, depending on the duration of the school term, and the results might have been even better. The parents noted this short duration and said they would be happy to take part in this project twice in a school year. However they felt that it would be more convenient to do the game sessions at home for only fives times per week, Monday to Friday.

It was felt that the inclusion of some commercially produced three dimensional games would have enhanced the project. According to pupil evaluation, the best games were considered to be the puzzle and strategy games (reproduced from Topping & Bamford, 1998a). However according to parent evaluation the best games were the number games. Parents felt the reinforcement of mental maths skills was the most important thing that their child learned from the project. The majority of the rules required a good standard of reading to interpret them, even though the researcher had simplified them. However it was felt that the children benefited from reading and re-reading the rules. It improved their reading, comprehension and problem-solving skills. Reading rules was somewhat problematic for the poorer readers, but a classroom assistant was always on hand to help with instructions. Some children reported that they now read problems in maths two or three times to understand it better.

The parent's games could have been managed better. It was more difficult to keep track of these games. Not all parents changed games on the same day. Sometimes the teacher was engaged in conversation with one parent while the others chose their games and left. Some parents were unable to come to school during school hours and relied on their child to choose a game.

The attitude questionnaire pre-post scores were quite similar. This might be taken to suggest a high degree of test-retest reliability, although it conflicted with the teacher's and parents' naturalistic observations of change. However, at post-test the questionnaire did indicate children were significantly more willing to share their ideas with a partner.

The mathematics test was a written rather than oral test, and did not align with the project activities in various other ways. It gave an assessment of overall mathematical attainment, but did include some questions on the topics of Number, Measure, Shape & Space and Handling Information. Two participants showed the same result in both assessments and two participants regressed. But others showed a significantly higher score, possibly mainly owing to increased self-confidence in mathematics.

**Conclusion**

The value of the whole project in terms of raising self-confidence and self-esteem as mathematicians was obvious to the teacher/researcher during the games sessions. The participants clearly enjoyed taking part in the games and parents were keen to be involved a second time. Some gains were observed in motivation and enthusiasm for maths in and out of school. Parents commented that maths can be fun and they had enjoyed playing the games with their children. They also enjoyed having quality time with children, not always common in the upper primary ages. Some pupils continued to play some of the games at home after the project officially ended. Levels of concentration on the games increased as the project progressed. Reading and interpretation skills were enhanced as the children examined the rules. Self-esteem was raised as participants found their own answers to some of the puzzle games and achieved success in the card solitaires. They were highly motivated because they were actively participating in the games. The games consolidated class work and encouraged discussion. They were fun for both child and parents too.

**Onward Action**

The incorporation and embedding of Problem Solving into the annual cycle of parental involvement is now the aim of the teacher/researcher. She found that this project combined two important elements in her teaching: the enhancement of self-esteem and self-confidence through co-operative and active learning situations. In previous years she had been responsible for developing the Personal and Social Education programme in the school, setting up a whole school policy and programme of work using Circle Time. The development of the Problem Solving project has provided her with the enthusiasm and motivation to change the way maths is tackled in the school. Other teachers in the school are keen to use the Problem Solving project games, and in the future staff collaboration will match pupil collaboration, with the aim of developing parental involvement. The support from the parents was arguably the strongest evaluative indicator for such a programme to continue. After being involved in the project, the teacher/researcher's own practices have been more clearly defined. There is clear value in providing children with active and co-operative learning situations. She now ensures that she makes time for such activities.

Many Scottish Schools, including the project school, have traditionally devoted a high number of hours to mathematics, using mainly worksheets and textbooks, but have arguable failed to use homework effectively, while showing a tendency to blame poor results on low pupil ability. This project has encouraged the teacher involved, because she has convincingly demonstrated that teachers can have some effect on children's attitude towards themselves as mathematicians. The pupils' enthusiasm and motivation can be raised through an effective problem solving project, but probably not through constant worksheets and textbooks. Parents appear to be a powerful resource, not merely to reinforce learning, but also to enhance it by creating situations where learning maths can be fun for both parent and child. The teacher/researcher is building bridges between theory and practice, and taking effective action forward.

**Recommendations for Future Research**

It would be interesting to investigate gender variation in different approaches to learning mathematics. Do boys prosper in a more active approach to problem solving and learning? It would also be interesting to investigate the effect of peer collaboration or tutoring in same gender pairings, with other age groups, and with greater age differentials. Other ways of measuring gains in attitudes, self-esteem and self-confidence might also be explored.

**Recommendations for Future Practice**

In the future the practitioners may want to trial a longer project. They should consider including mathematical games and puzzles purchased from stores and educational catalogues, so that more children do not perceive the games as homework. Practitioners should not under-estimate the time taken to familiarise themselves with the games and the simplification of any rules. They should have clear objectives regarding the mathematical language and strategies to be used.

Practitioners might also benefit from allocating more time than in this project to the initial training of peers. This should include practice in tutoring skills. Time should be given throughout the project to revise these principles. There is also a case for giving the tutors time prior to the project to become familiar with the games, rather than encounter them for the first time with the tutees. It was felt that because of a lack of effective tutor training, the project became peer collaboration rather than peer tutoring. However members of the intervention group spontaneously expressed a wish to be involved with another class, showing them the games. There might be a follow-up project involving cross-age peer tutoring. In this case more emphasis would be put on training and revision of skills for the tutors.

Practitioners may also want to revise the allocation of time given to parents' game sessions at home. Many parents felt that it was difficult to continue the game session at the weekend because of different routines. Timing of the launch meeting should also take into consideration those parents who are working. The researcher found that some parents were interested but unable to attend that meeting. They still took part in the project but they would have liked more written information about it in the form of a letter.

(Thanks to Terry Kerr, Maryburgh Primary School, for this intensive evaluation report)

### National Project

**The National Problem-Solving Project:** **Summary and Conclusions**

The project schools varied from small rural schools with 15 children to large urban schools with at least 2 classes in each year. The vast majority had pupils from P6 or P7 as tutors and P3 or P4 as tutees. The projects ran on average for 7-8 weeks, the children meeting together 2-3 times weekly for 20-25 minutes per session.

Schools set their own local aims for the project, which included improving:

- active, harmonious & co-operative learning
- language, vocabulary, discussion & communication
- depth of mathematical reflection and understanding
- problem-solving strategies, skills & decision-making
- confidence and self esteem of pupils
- motivation, positive attitudes & enthusiasm
- concentration, determination & perseverance
- connection between maths in and out of school

Schools reported these benefits:

- recall, reinforcement & consolidation for tutors
- individualised attention for tutees
- focus on strategy & process rather than outcome
- improvement in ability to follow instructions
- showed there were many different ways to solve problems
- linked school maths with maths for life - not just sums
- helped children be more organised, systematic & logical
- increased challenge for tutors and tutees
- increased responsibility
- gains in listening, questioning, language skills & mathematical vocabulary
- high degree of co-operation, sharing, turn-taking, teamwork
- gains in social & interpersonal skills, e.g. patience, giving/receiving praise
- cross-age relationships & friendships generalised out of class
- enhanced school ethos, cohesion, bonding
- less threatening, involved less fear of failure
- gains in confidence & self esteem for older & younger, including less able
- increased motivation and enthusiasm for maths
- enhanced speed and fluency in mental calculation
- increased concentration & persistence
- some children performed above their assumed ability level
- no behavioural difficulties - even with previously difficult children
- children enjoyed it and looked forward to it!
- means of developing relationships & partnerships with parents

Some schools reported these difficulties:

- finding time to plan and organise
- finding time to choose games/problems carefully & familiarise with them
- finding enough attractive & suitable games for all categories
- financing game resources
- simplifying instructions
- some timetabling problems
- finding time to train tutors
- noise

**Teacher Questionnaires**

Regarding Maths Games, 87-89% of responding teachers found the project enjoyable and interesting. However, opinions were divided over whether it was hard work or easy to organise. All teachers had observed increased enthusiasm and concentration in many or some children, and 97% had observed improved discussion and co-operation.

For Duolog Maths, 81% of responding teachers found the project interesting, but only 59% found it enjoyable. Again, opinions were divided over whether it was hard work or easy to organise. However, an even higher proportion of teachers considered Duolog useful than for maths games. The lesser enjoyment and higher perceived utility might reflect the closer articulation of Duolog to the regular maths curriculum. However, all Duolog teachers had observed increased enthusiasm, co-operation and discussion in many or some children, 82% had observed improved concentration and 86% improved attitudes to mathematics. Compared to games, Duolog was associated with particularly widespread improvements in co-operation in classes, but somewhat less widespread improvements in enthusiasm, attitudes and concentration.

**Pupil Questionnaires**

Almost all pupils reported talking about the games, 89% felt the project was fun and 85% said it made them think. Over half felt they now had a better relationship with their partner, while 61% reported improvement in traditional maths ability and 65% improved attitudes to traditional maths. Half of the pupils had difficulty with instructions for some games, and 44% were bored occasionally, while 20% did not have enough games and 40% found it hard to choose games. Only 15% wished to stop peer assisted maths work. Strong positive pupil attitudes were evident in relation to: effort in maths lessons, working with a partner and helping others, and understanding maths vocabulary.

**Parent Questionnaire**

A large majority of parents found it easy to take part in the project, and had had their child remind them about the session. Most parents and children became bored occasionally, but most wished to continue with maths games at home. Only 20% thought it unlikely they would recommend maths games to other parents. Reading game instructions was found difficult by 10% of parents.

**Intensive Case Study Evaluation**

A more intensive study was undertaken of a project in a rural school. Here, same-age peer tutoring involved the random matching of pupils rather than on the basis of ability. Tutor-tutee roles were not allocated. Most pupils also had parental involvement at home. Very positive verbal feedback was recorded from pupils. Teacher observations and perceptions and parental feedback were very positive. However, limited pre-post change was observed in attitude questionnaire scores.

A traditional standardised test of mathematics was used before and after the project with the experimental group and a comparison group of more mathematically able pupils of the same age. The scores of both the intervention group and the comparison group increased from pre-test to post-test. However, the improvement for the experimental group was substantially larger than that for the comparison group. At post-test the difference between experimental and comparison group scores had narrowed significantly.

**Overview of the National Project**

The overall assessment from schools, teachers, pupils, parents and test results was very positive, with gains evident for a majority of participants in various outcome domains. There were many benefits (some expected and targeted, others more surprising). Clearly, an investment in time is needed for start-up. However, participating schools who had made that investment were now in a strong position to continue with the project much more easily in the future. Indeed, many schools noted that the project would be much easier to repeat a second time. Some schools had already started using the approach with other year groups.

The evaluation data from participating schools, teachers, parents, children and tests appear to have demonstrated that the Problem Solving project did indeed align with the principles which underpinned its development. The project appears to have achieved all of the objectives that were targeted both nationally and locally, although naturally not all these objectives have been achieved with every individual participant. It is therefore fair to say that Problem Solving with BP has been successful and has made a positive difference.

### References

**References &** **Further Reading**

Cockcroft, W. H. (1982). Mathematics Counts: Report of the Committee of Enquiry into the Teaching of Mathematics in Schools. London: Her Majesty's Stationery Office.

Her Majesty's Inspectorate of Education (2001). Standards and Quality in Primary Schools : Mathematics 1998-2001: A report by HM Inspectorate of Education. Edinburgh: Scottish Executive Education Department.

Topping, K. J. (1988). The Peer Tutoring Handbook. Cambridge MA: Brookline Books

Topping, K. J. (1995). Paired Reading, Spelling & Writing: The handbook for teachers and parents. London & New York : Cassell

Topping, K. J. (1998). The Paired Science Handbook: Parental Involvement and Peer Tutoring in Science. London : Fulton; Bristol PA : Taylor & Francis

Topping, K. J. (2001). Peer Assisted Learning: A Practical Guide for Teachers. Cambridge, MA: Brookline Books.

Topping, K. J. (2001). Thinking Reading Writing: A Practical Guide To Paired Learning with Peers, Parents & Volunteers. New York & London: Continuum International.

Topping, K. J. & Bamford, J. (1998a). The Paired Maths Handbook: Parental Involvement and Peer Tutoring in Mathematics. London : Fulton; Bristol PA : Taylor & Francis

Topping, K. J. & Bamford, J. (1998b). Parental Involvement and Peer Tutoring in Mathematics and Science: Developing Paired Maths into Paired Science. London : Fulton; Bristol PA : Taylor & Francis

Topping, K. J. & Ehly, S. (eds.) (1998). Peer Assisted Learning. Mahwah NJ & London UK : Lawrence Erlbaum

### Useful Links

**Useful Links**

Relevant Publications by Keith Topping (Co-Director of the Problem-Solving Project)

Scottish Office Education and Industry Department - (use the "search" facility for all relevant documents)

Ask Dr. Math (provides a solving service for K-12 maths problems, among other things)

Association of Teachers of Mathematics

Class wide peer tutoring in mathematics

Count On (follow-up to the National Year of Maths in England)

Cross-Age Learning International

Cross-Age Tutoring in maths: Annotated Bibliography

Educational Math Games for Kids & Teens: math games, from geometry to money to division

Helping Your Child Learn Math

Mathematical Resources Math Links by Bruno Kevius

Peer tutoring in mathematics: NW Regional Educational Laboratory

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