1. An investigation into unstable pupil behaviour in primary mathematics. Or He/she could do it yesterday so why can’t they do it today?

2. Focus of main study What is the nature of the activity that learners engage in when responding to tasks that involve: solving word problems, sorting word problems, matching word problems; constructing word problems How can phenomena identified when analysing children’s activity on these tasks be accounted-for in terms of relevant literature? What are the implications for teaching and learning arising from this analysis?

3. Outline of research sessions Initial probe: Word problems and associated calculations; Monthly tasks involving solving, sorting, matching and constructing word problems; Final probe: Initial probe administered again. Additional solving, sorting, matching and constructing tasks for selected children.

4. The Nature of Data collected Written recordings made by the children. This included answers to problems and number statements as well as ‘working out’, jottings, constructed problem statements and any other material recorded on paper by the children during the study. Transcripts of dialogue. All sessions were recorded on audio tape. Early sessions were transcribed in full but only significant sections of later sessions were processed in this way. However all material on tape was reviewed. Field notes. During sessions I made notes of observations that seemed important to record at the time. These notes were supplemented with other material that I recalled from sessions when mentally reconstructing them shortly afterwards. In addition to this data I have also drawn on my experiences of teaching and learning prior to the study. This has been particularly the case when phenomena have either resonated strongly or conflicted with my experience. Some of the incidents and themes that I have used have been recalled from memory, while others were recorded in brief-but-vivid accounts soon after the event.

5. Working with the Data Initially identifying themes in the data was difficult. “Reorganising” the data in different ways seemed to facilitate noticing particular themes. Constructing the story of each child proved to be one particularly useful approach.

6. The Story of Barry Barry FebruaryJuly Number statement Problem Statement Number statement Problem Statement Number statement Problem Statement 16 + 25= □A Change 1 Shahina has £16. She is given £25 more by her parents. How much does she have now? 41 10+20=30 5+6=11 41 10 + 20=30 6+5=11 She has got £41 3141 16+25=41 23 + 19 = □ B Change 1 Before he was given £23 by his parents Leroy had £19. How much does he have now? 42 10+20=30 3+9=12 £42.00 4232 19+23=32 □- 6 = 28C Change 6 On Monday morning all the children in Class 2B are present. Six of the children go home ill. There are 28 children left in the class. How many children were present in the morning? 34 28+6=34 3234 28+6=34 □ – 5 = 27D Change 6 There are 27 children left in the class at home time. During the day 5 children had gone home because they were ill. How many children were present at the beginning of the day? 3239 27+5=12 27+12=39 32 27+5=32 15 + □ =32E Combine 2 There are 32 children in Class 3C. There are 15 boys in the class. How many girls are there? 17 32-15=17 2617 15+17=32 16 + □ = 31F Combine 2 There are 16 boys in a class of 31 children. How many girls are there? 1525 30-10=20 6-1=5 20+5=25 girls 2515 16+15=31 16 + 17 =G Nonsense Class 4D has 16 boys and 17 girls. How old is the teacher? 33 10+10=20 6+7=13 33 years old 33“out of order”

7. There was evidence of some inconsistency in Barry’s answers when… the same probe items were presented on different occasions; he was presented with items (closely related in time) involving the ‘same’ maths; he answered similar number statements involving numbers of similar order; he was presented with problem statements of a similar structure;

8. Initial accounting-for Barry’s responses. That’s what children are like; Barry’s awareness of mathematics might be that it is unprincipled; His motivation and level of engagement might vary between tasks and sessions; Small, subtle differences in apparently similar situations might affect his response; The attunements that Barry brings to a task might develop or vary with time;

9. Delving deeper… Problem statements seemed to elicit recording. Number statements tend not to elicit recording. Recording in February sometimes included an indication of “working out”. In July recording tended to be restricted to a summary of the calculation used.

10. Delving deeper still… …shifts in the focus of attention In sorting, matching and construction tasks: Early in the study attention focused on numbers; By the 3 rd session there was evidence of a focus on words; Subsequent shift to focus on actions described in problem statements; In the final session sorting and matching took place by a systematic phrase-by-phrase comparison;

11. Unidirectional progress? Evidence suggests that over time there was a general shift in Barry’s attention from a focus on obvious surface features of problem statements towards identifying deeper more structural aspects. However even in later sessions the focus of Barry’s attention frequently shifted back to using surface features such numbers when sorting and matching.

12. Unstable behaviour Barry displays unstable behaviour: In the answers that he provides; In the recording that he makes; In the nature of his attention; What should we do about this?

13. Cognitive Variability Based on analysis of significant amounts of data Siegler (2007) claims: “... thought and action are highly variable within individual infants, toddlers, children and adults” variability in strategy is not an “... orderly progression from less adequate strategies to somewhat more adequate strategies”. Progress is identified as reflecting “a back and forth competition rather than a forward march”

14. A necessity for learning? Siegler sees cognitive variation as important, pointing to evidence that greater initial variability of strategy seems to predict greater subsequent learning. He concludes that some form of instability is necessary for any system to change, as “learning is most likely when previously dominant approaches weaken”.

15. Questions Is the unidirectional ladder metaphor of learning an appropriate model? Is there a place for greater long term immersion in learning tasks? What implications does cognitive variation have for assessment of progress?

16. References Ginsburg, H. (1977) Children’s Arithmetic: The learning Process. London. D.Van Nostrand Company Houssart, J. (2007) Investigating Variability in Classroom Performance amongst Children Exhibiting Difficulties with Early Arithmetic. In Education & Child Psychology Vol. 24 No. 2 Mason, J. (2002) Researching Your Own Practice: The Discipline of Noticing. London. RoutledgeFalmer Siegler, R.S (2007) Cognitive Variability in Developmental Science 10:1 (2007) pp 104-109

17. A Request. I would be pleased to receive any examples of unstable behaviour in pupil performance that you are willing to share. Such examples might be at the level of test scores or assessment levels but it would also be very useful to receive examples that focus on the activity that a pupil engages in. Thank you. John Butlin Senior Lecturer in Education Birmingham City University.