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This case study describes in some detail the use of mathematical games for peer tutoring in one primary school - in this case on a same-age basis. It describes the intervention and the effects on the children's attitudes to mathematics, on their self esteem, on their attitudes to working co-operatively and on their performance in a mathematics test. It is based on the work of Anne Mallinson.
Detailed feedback from participants in a same age peer project involving parallel classes of 11-12 year olds in a high school in a disadvantaged area was collated by Topping and Bamford (1990). Preferred games were Labyrinth and Pass the Pigs, followed by Yahtzee, Uno and 5 in a Row. No box of games was considered uninteresting, but the strategy box was generally considered the most interesting. The number of games provided was thought adequate, but duplicate copies of the most popular were requested. Three dimensional commercially produced games were preferred to two dimensional locally produced games. However, the instructions were considered too long and too hard for several games, especially the commercial ones. Despite this, the participants saw little need for an ability differential in the pairing - perhaps meaning mathematical rather than reading ability. No difficulties in organising activities with partners was reported. No participant became bored - a change of game easily avoided any such possibility. Discussion about how to play the game was commonly reported; pairs sometimes agreed to change the rules, especially if the original instructions were very complex.
Overall, participants definitely found Paired Maths both interesting and fun. They felt a majority of the games made them think harder. However, they were unsure about whether they liked "maths" any more - although they certainly liked the maths games! Thus their highly positive attitude to the games containing maths had not been perceived to generalise automatically to the formal school maths curriculum. A few participants felt they could do "problems" better in formal maths classes, and there was spontaneous comment that needing to reading complex instructions very carefully improved the ability to do this in other subjects. A few felt they could now design their own maths games. Virtually all would recommend the project to other people. Improvements to the system could include duplication of the most popular games, improving instructions, including more solitaire puzzles, having longer sessions, build in switching of partners and providing solutions to "impossible" puzzles.
The vast majority of participants wished to carry on with the project. Verbatim comments included: "it didn't feel like maths, because it was fun" and "it was not proper maths, but it WAS thinking and problem-solving". The participants seemed to continue to espouse a very narrow definition of "proper" (i.e. schooled) "maths", equating it almost entirely with number. Thus only project number games (e.g. Equality, card games) were cited as definitely improving ability in "traditional" maths. Of course, these subjective perceptions might not reveal the whole truth.
Somewhat similar projects had been reported elsewhere in the secondary sector, run by a learning support teacher and a mathematics teacher (Carmichael 1995, personal communication). Maths games were initially used in peer tutoring of low attaining first and second year high school pupils (aged 11-13 years) by able 16-18 year old students for three fifteen minute sessions per week over 4-6 weeks. Subsequent projects paired third year (13-14 years old) average ability tutors with first year tutees. However, no evaluation details were reported.
An approach with some similarities was reported by Neumark (1997) at a multicultural secondary school for girls in London, here as a three week "strategy project" embedded annually in the curriculum to help develop analytic thinking (as required by the 'using and applying maths' category of the English National Curriculum). After exposure to a number of games from different countries, the pupils were required to devise their own game and explain it to the rest of the class. Again, evaluation was only anecdotal.
The school for this project was a large city primary school in Scotland, with a roll of 512 and three forms of entry, staffed by 26 teachers, two of them full time learning support teachers. It catered for a mixed population, many of whom came from single parent families, and some social deprivation was reported. Interest and priorities within the school at the time of the project were focusing on discipline and improving co-operation and behaviour on a whole school basis. The formal mathematics curriculum was strongly conditioned by the national curriculum guidelines for 5-14 years laid down by the Scottish Office.
The experimental group was a mixed ability class of 25 10 year old children ("P6" in Scotland, where P6 is the penultimate primary class). The comparison group was a parallel class of 20 children, who did not play the games but received normal mathematical instruction from their teachers. The mean ages of the classes were very similar, but other differences were perceived by the teachers. The experimental class was seen to have some social, emotional and behavioural difficulties and a large group of middle to low attaining pupils. The children were described as having poor social skills, not forming a very cohesive group and often falling out with each other. The school felt these children could benefit socially from the maths games project. The comparison group was reported to have a more harmonious social grouping and a wider range of ability. The experimental class had a male teacher who appeared to have a good relationship with his pupils. The comparison group had two female teachers in a job share.
A timetable was agreed - the project was to run in the spring term. However, the amount of time needed to set up a first project from scratch should not be underestimated. Games need to be gathered, categorised and coded, and rules and instructions simplified.
The school agreed to obtain 20-25 games, by using some from existing stock and buying some new ones, but generally concentrating on number skills. The researcher gathered the others, for loan to the school. Games suppliers proved difficult to locate. A number of stores reported that they no longer stocked games and puzzles which would have been suitable for the intervention. The best games and puzzles were found in certain stores and toy shops in the run up to Christmas. Another difficulty is that games cannot be played before buying, making it difficult to ascertain what will be suitable. Gathering the games took two to three months. A few spare replacement counters and dice were also purchased.
The mathematical content of the overall pool of games and puzzles was considered in relation to the 5-14 curriculum guidelines. Then each was allotted to one of four categories: Shape, Position and Movement, Number and Money, Strategy and Logic, Puzzles. Games could be played by 2 or more players, while puzzles presented a problem for 1 person. These categories were similar to those used in the original "Junior" Key Stage 2 kit, but it was decided not to use the fifth category of extension games for mathematically able pupils, since there was little indication of who the mathematically able might be and the researcher was reluctant to make assumptions about both children's ability and the relative difficulty of the games and puzzles.
The games and puzzles were then colour coded and labelled: Blue for Shape, Yellow for Number, Red for Strategy, White for Puzzles. Where puzzles were too small to label in this way, the puzzle box or bag was labelled instead, together with an indication of the number of pieces for that puzzle. This was to enable a child putting the puzzle away to see easily if pieces were missing. Some puzzles and games were put into zip up plastic bags so that pieces were less likely to get lost. Categorisation was difficult, as Cornelius and Parr (1991) also observed. Whether a game fitted better into the Strategy or the Shape, Position and Movement category was hard to assess. This could have been a focus of discussion with the children. Labelling was adequate but the games might have been better numbered 1-80, so that the children did not have to record the games by their number and their colour category.
The researcher played most of the games and solved most of the puzzles provided by herself. This was time consuming but essential. She then simplified rules and rewrote instructions for the majority of these. However, the school did not have the time to try out their games or simplify rules. In the end there were too many games and puzzles for such a short intervention: 80 in all, when 50 would have been adequate. For the class of 25, to allow a little choice around 16-20 games and puzzles were needed for each session. Storage space for the games was limited, and this could make changing games and accessing different categories problematic.
The project was a short one. The children played the games for two 30 minute sessions each week for 6 weeks, and wrote brief comments about the games in notebooks at the end of each session. The experimental group generally played the games in pairs, although there was often also one group of 3. Children doing puzzles worked largely individually, and this provided the change of pace recommended by Johnson and Johnson (1991).
The children had been put into pairs and one of four groups by the class teacher. Pairs were organised so that there was one child who could read well in each pair, to facilitate the reading of the game instructions. This had the additional advantage of providing a heterogeneous ability mix in each pair. Girls were generally put with girls rather than in mixed dyads. Each of the four groups of six children (one had seven) was initially allotted one of the four categories of games and puzzles: Number and Money; Shape, Position and Movement; Strategy; Puzzles. These categories then rotated round the groups at the end of each week.
Monitors in each group collected their group's games at the start of each session and checked that they had been tidied up at the end. Small jotters were provided to each child in the experimental group. In these they wrote the date, the name and category of the game they played, a short comment about the game and a rating out of 10. These notebooks provided additional process evaluation on the project.
The first games session was run by the researcher and the class teacher. A sheet of written instructions about how the sessions would operate was available for each pair. These were used for instruction and as a reminder in the first two games sessions but became superfluous as a routine was established. Each group monitor then collected the games for their group and each pair chose a game (for the puzzles each child chose an individual puzzle). The game or puzzle was recorded in the child's notebook, the instructions were read and then the game was played.
The adults circulated around the groups while they were playing, encouraging and helping some confused pairs to follow their game instructions. Towards the end of the session the teacher instructed the pairs to add up their scores and put away the games. A further 5 minutes was allowed for writing a comment in their jotters. This routine was adhered to at all the games sessions; sometimes followed by discussion between the class teacher, the researcher and the children about the games and any difficulties or successes which they had experienced.
Despite the brevity of the intervention, it was decided to use a mathematics test to see if there was any measurable impact on performance. However, administering an individual test would be too time consuming with two classes totalling 49 children, so a 30-40 minute group test was required. This should not be so difficult as to discourage the children (Cockcroft, 1982), and it should cater for a wide spread of ability so as to minimise ceiling effects. Published norm referenced tests were either too difficult or too easy, or the range of attainment covered was too narrow, or the reading requirement was too high for the children in the sample. A further disadvantage was that they were likely to be too insensitive to register differences in attainment over a 6 week intervention.
A criterion referenced test was therefore devised by the researcher which was more appropriate for the age, ability and interest range of the children in the sample. This had the additional advantage that questions could be selected which related to the mathematical areas and vocabulary of the games. Unfortunately there was no time to pilot the test before the start of the project. Although it appeared to have face validity, other types of validity were indeterminate.
The test devised consisted of 30 questions. Fifteen questions in Section A were designed to assess the children's understanding of mathematical words. Most of these were multiple choice questions, but in some cases the children had to respond in a different manner e.g. by drawing a counter on a grid. The fifteen questions in Section B were concerned with number and money operations and line symmetry. The selection of questions did not attempt to cover all the Mathematics 5-14 strands, because not all of these were reflected in the games used during the project, and constructing appropriate questions was easier in the areas of number and money than in shape, position and movement and problem solving.
Each question was read out by the researcher to help the weaker readers and to allow time for the children to think about the question. A short amount of time was then allowed for them to write down their answer. The researcher started to read out the next question when it was clear that the majority of the children were ready. The test took about 30 minutes to complete.
An attitude questionnaire was also devised by the researcher, containing statements relevant to attitudes to: mathematics, self esteem and working co-operatively. A Likert scale was utilised for all items, ranging from 4 (high positivity) to 1 (low positivity). Scoring yielded a total score and sub scores in the three domains. Again, piloting proved impossible. The questionnaire consisted of 16 statements: eight reflecting positive attitudes and eight negative attitudes. Five were concerned with attitudes to mathematics, six with working co-operatively and five with self esteem. There was inevitably some overlap between categories. The questionnaire was introduced to the children with an indication of its purpose and two examples on the board. The statements were read out to the children who were asked to respond by marking their level of agreement with each statement on the four point scale. It took about ten minutes to administer.
The maths assessment and attitude questionnaire were administered to the experimental and comparison groups two weeks before and again one week after the games intervention (inter-test period 9 weeks). Children absent for the pre- or post- assessment were assessed in a small separate group the week following the assessment of their class. It was thus possible to compare attainment and attitudes of each group before and after the intervention, and to compare those of the experimental group with those of the comparison group.
Additionally, formative process evaluation was carried out by teacher and researcher observation, discussion and feedback, video recording, asking the whole experimental group oral evaluation questions and interviewing three pairs of children in detail. The children also kept their own notebooks in which they wrote comments every session about the games and rated them out of 10. At the end of the final games session, five further evaluation questions requiring individual written responses were given to the experimental group. These were also given to the project teacher, together with a more extensive interview and questionnaire.
The attendance of the experimental group was consistently good. For some children in the comparison group it was more erratic, resulting in absence from part of the assessment; two children from the comparison group also left during the term, leaving 20.
As previously mentioned, there were too many games. The ones which were not played included a number of the laminated Strategy Games and some of the older school maths games. Instructions for playing were simplified for most of the games and puzzles provided by the researcher, but a lot of reading was still required. The rules of the games from educational suppliers were generally not simplified, and some of the instructions were confusing and inadequate. The result of this was some initial confusion among some children, with requests for clarification and help.
Observation suggested that the wooden three-dimensional puzzles were very popular, together with `traditional' games like Battleship and Mastermind. This was also evident from comments made by children in their notebooks. The games from educational suppliers (e.g. Jungle Climb) looked good but sometimes proved disappointing. The number games - whether from educational suppliers or high street stores - appeared to be less popular than the three dimensional puzzles and strategy games. The children verbally stated that popular games and puzzles included Mastermind, Connect 4, Switch It, Tower of Hanoi and Arrest (a laminated strategy game). Some children described Battleship and the Rhombic Star as "hard", although the former was the most popular game of all.
The children's later written responses produced slightly different results, with the Puzzles category reported as most popular. Wooden constructional puzzles such as Rhombic Star and Soma Cube seemed to have become particular favourites. The adults also observed that the very simple wooden Lost Marble puzzle produced much enjoyment. Strategy and Shape were the next most popular categories, while Number was hardly mentioned. The most popular game of all was Battleship. Reasons given for the popularity of these games and puzzles were that they were "challenging" and "good fun". Three children suggested that the games helped you to read and one that the games helped you with maths.
The teacher reported that the puzzles and games provided by the researcher had been more successful than the school's games. He felt that the wooden puzzles had been especially worthwhile.
The overall impression was that the sessions were successful - the children were absorbed, well-behaved and busy and showed their enthusiasm most of the time. Initially there was some squabbling between pairs but this became less evident over the course of the intervention. Two children who spent the first three sessions accusing each other of cheating settled down and played well for the remainder of the project. One success was the pairing of one of the most able children in the class with one of the least able ones.
The routine procedure was generally followed well. Initially some pairs had difficulty reading and following the games instructions, but considerable improvement occurred over the six week intervention. Video recording showed the children concentrating very hard on some of the Position and Movement and Strategy games (Battleship, Mastermind, Switch it). There was also evidence of computation: in one or two clips a child was clearly carrying out a written or a mental calculation (e.g. in the game Equality). The Puzzles presented a slightly different picture, but also a positive one. Children were absorbed, concentrating hard, thinking and manipulating. The social aspect was evident but more sporadic, as a child attempted his or her own puzzle then watched another child for a while and maybe asked for help or offered advice.
The notebooks were used as planned and kept well by the children. They put the date, the name of the game and the category, with a comment about whether they had enjoyed playing the game and whether they would recommend it to others. There was a tendency to rate many games 10/10; while 5/10 was a bad rating, signifying "very boring". Some children wrote more extended comments which gave a clearer picture of what they thought. Writing comments at the end of each session soon became an established routine, with even poor writers managing to write something.
The video revealed that playing the games was a very social activity, with lots of chatter about the games. There were some arguments initially. Children were keen to win the games and sometimes accused their partners of `cheating'. In the first session there were tears when a boy wanted to use a game which another pair had taken. As the sessions proceeded communication between children appeared to improve. The video strikingly captured the eye contact between pairs working together and their facial expressions and smiling. The children's non-verbal communication showed clearly that the games and puzzles were fun. The co-operative nature of the project in general, and working in pairs in particular, led to informal discussion in the sessions about which games or puzzles the group were playing, discussion of the instructions for play by each pair and much incidental discussion during play. A small amount of whole class discussion with the class teacher and the researcher took place at the end of the sessions, but this was practical rather than mathematical: thus opportunities for mathematical discussion and questioning were missed.
Video revealed good co-operation in setting out the games, playing them, scoring and putting them away. Minor problems in the early games sessions included group monitors exceeding their remit and not allowing pairs to choose their games, and children asking for help without having read the instructions. There was also one girl whose initial attitude towards the games seemed very negative. Her notebook provided evidence for this - but also presented striking evidence of her conversion after 3 sessions and her subsequent positive thoughts on the games and puzzles. Her post test results on the attitude survey also suggested an increase in overall positivity; and a substantial gain was evident in her maths post test score. The video also revealed one boy sitting disconsolately after his partner had wandered off following a disagreement. Later in the session this pair became absorbed in playing Mastermind, and from this point on the class teacher reported better co-operative working for them. The reduction in squabbling and the increased ability to work both in groups and pairs over the course of the project suggested that progress was made in working co-operatively. Observations of sessions, the class teacher's report and video all supported this.
Non-parametric statistical tests were carried out on the ordinal data.
The overall attitude positivity scores of the experimental and comparison groups were very similar at pre-test and at post- test. This was also true for the attitude sub-tests concerning attitudes to mathematics and to working co-operatively. However, the experimental group did have significantly higher self esteem scores post test than pre-test (Wilcoxon Test 2- tailed P=0.0168). The self esteem scores for the comparison group were slightly lower at post test. No significant gender differences were found.
There were 30 items in the maths test and the highest possible score was 30. The raw scores of both the experimental group and the comparison group increased from pre-test to post-test. The difference was significant for the experimental group (mean gain = 1.56, Wilcoxon test: 2-tailed P = 0.0071), but not for the comparison group (mean gain = 0.25).
However, at pre-test the mean scores of the comparison group were significantly higher than those for the experimental group (Mann-Whitney test 2-tailed P = 0.0254). At post test the difference between comparison and experimental mean scores had narrowed and was no longer statistically significant.
The boys in both groups improved significantly from pre-test to post-test (Wilcoxon: 2-tailed P = 0.0178). However, the comparison group girls' performance was slightly worse at post test than at pre test, while the experimental group girls' performance was only slightly better at post test than at pre test. The gender discrepancy was particularly evident in the experimental group where the improvement of the girls was small and not significant, while the boys' improvement was statistically significant (Wilcoxon test: 2-tailed P = 0.0281).
Children's responses included that games helped you to share, showed why you should read the instructions, and helped with maths words and spelling. Disliked aspects included: difficult words and instructions, small print, common games they had played before, boring games, games where people cheat. Improvements could include: bigger groups of children to play the games, easier instructions and making some of the puzzles harder.
The teacher felt the project had definitely been worthwhile, reporting that the experimental class (including himself) had learned "a lot about themselves, meeting challenges, and working together". The experimental group children were described as becoming more sociable and tolerant and more willing to follow rules and instructions. The pairing and grouping led the class teacher to consider other different seating arrangements in other curriculum areas.
The six week intervention was arguably too short and the frequency of the sessions too few at two per week. Results might have been better with an eight week intervention and 3 games sessions weekly. Little measurable improvement was likely given a total time on task of 10-12 hours. Additional optional games access during break times might be considered.
According to participant evaluation, the "best" games were long-standing traditional games such as Mastermind, Battleship, Connect 4, Uno, Yahtzee, all of which were purchased from stores rather than educational suppliers. The "best" puzzles were the wooden ones (e.g. Rhombic Star, Soma Cube) which were well made, aesthetically attractive and challenging.
Although many rules were simplified, the majority still required a substantial amount of reading. The advantage of this was that children improved in the skill of reading and following the instructions over the course of the intervention. The disadvantage was that weaker readers sometimes struggled.
The rotation of games round the groups and the replacement of games during the project could have been managed better and was not helped by storage problems. The class teacher suggested that a pre-set diagrammatic plan for the rotation of games round the groups would have been helpful.
The use of monitors was intended to reduce unnecessary movement in the classroom and increase the children's accountability in looking after the games. Pairs of children in each group took turns to be monitors so that everyone in the group was a monitor for a week. In the early sessions some monitors decided to distribute particular games to pairs in their group rather than letting the pairs make their own choice from the group games. This problem was soon resolved. The children were sensible and careful in their handling of the games and responsible about putting them away. Only one puzzle was broken and all vital pieces of games and puzzles were present and intact at the end of the project. The system seemed to have worked well, although much of this may have been due to the class teacher's good organisation and relationship with his class. It cannot be assumed that projects in all classes would proceed as smoothly.
`Mathematical' discussion and questioning between teacher and children has been recommended by a number of writers (Skemp 1989a, Whitebread 1995, Askew and Wiliam 1995). Although communication improved and chatter and body language indicated much enjoyable communication related to the games, more appropriate questioning by the teacher and the researcher during the sessions might have encouraged the development of higher quality mathematical thinking and problem solving strategies in the children.
The attitude questionnaire was not piloted and there was no information as to its reliability or validity. However, test- retest scores were so nearly identical for both groups as to suggest a high degree of test-retest reliability. The children's post project attitudes to mathematics may have remained unchanged in reality. They had enjoyed the games and puzzles, but knew that there would be no more games sessions. Why should this have made them more positive towards mathematics in general? Observation indicated a more positive attitude to working co-operatively, in contrast to the questionnaire findings. However, whether this would continue post project is unclear and would depend on a number of factors, not least the provision of further opportunities for co-operative working.
The mathematics test aimed at clarity and simplicity, focusing on mathematical vocabulary and some tasks and situations of the kind that would be met in the mathematical games. At post test it may have favoured the experimental group who had played the games. However, the brevity of the intervention must be taken into account, and that a number of the questions related to situations and mathematical vocabulary in the laminated strategy games which were hardly used at all. Some regression to the mean was evident in the post-test scores of the comparison group, particularly the boys, although not in the experimental group. It was also possible that there was a degree of ceiling effect in the results, particularly for high scoring girls in the comparison group, although the mean post- test score was only 20 out of 30.
The mathematics test indicated gains for boys but not for girls. While there might be artefactual elements in this finding, it appears that the approach did not positively discriminate in favour of girls. However, this might be influenced by same-gender pairings, as Topping and Whiteley (1993) found with Paired Reading.
A further possibility was that any improvements in the experimental group may have resulted from the Hawthorne Effect - the impact of extra attention and novelty alone. This is true of many classroom interventions. Ideally, additional follow up evaluation should be carried out three months later.
Further research should utilise measures of known reliability and validity or establish the reliability and validity of newly created measures. The use of comparison groups which are totally comparable and the gathering of follow-up data are highly desirable. How long it is realistic to expect the effect of a relatively slight intervention to endure is a difficult issue, however. Further examination of differential gender effects in relation to variation in organisational parameters is necessary. In addition to cross-gender matching, the use of extrinsic rewards, subjects of a different age, cross age pairing, and interventions of different duration and frequency could all be explored. If these were evaluated rigorously it should be possible to draw conclusions about the most cost effective intervention. Comparison of this approach with more traditional co-operative learning, using structured activities more closely coupled with the formal curriculum, would also be of interest.
There are implications here for making more effective use of mathematical games in the classroom, and for including a wider variety of mathematical puzzles and games, purchased from stores, stemming from ancient times or traditionally in the public domain. There are implications, too, that schools should monitor and assess the relative progress of girls and boys in the learning of mathematics in relation to teaching methods in use; and that teachers might usefully monitor their own interactions with girls and boys in the classroom.