Construcción del Árbol de Objetivos

Rival explanations for the data produced by a study must be considered during analysis and reporting. There are three key sources for rival explanations: the research literature; formal peer review; and informal peer feedback. This thesis has made use of all three sources, and in particular has received a large amount of feedback and review from peers. One source for feedback has been presenta- tions at conferences and seminars. Presentations enable formal feedback in the form of question and answer sessions, and informal feedback in the form of later conversations. Publication in peer-reviewed journals and conference proceed- ings provided formal, written reviews, and in some case necessitated substantial reflection and revision on my part. Both peer-reviewed and non-reviewed pub- lications involved informal review and feedback from peers.

Another important source of feedback were consultations with relevant experts, mainly and regularly with my supervisor Dave Pratt. I also arranged consul- tations with experts (these consultations are summarised in Table 7.2). These consultations typically lasted an hour and involved discussing my ideas in ref- erence to the software and audiovisual data. This feedback brought a range of interpretations of the approach and emerging findings to my attention. Some of these experts read and responded to (usually early drafts of) my publica- tions, improving the comprehensiveness of the literature review (Chapter 2), highlighting the limitations of the preliminary work and design ideas (Chap-

ter 5), refining the approach taken (Chapter 6), and improving the validity and reliability of the studies and findings reported in the remainder of the thesis.

7.5

Summary

The studies seek to trial the task design in semi-structured settings intended to stimulate discussion among participants. Paired trials are used and the researcher remains present to prompt pupils to explain their reasoning. Mi- croworlds provide insights into conceptual thinking and provide rich audiovisual data in the form of screen-capture movies.

The analysis seeks evidence for pupils talking about notation in ways not pre- dicted by the literature. There are four stages to the analysis. The first is transcription, which familiarises the researcher with the data at a fine-grained level. Transana is ideal for this purpose as its time-codes enables switching between text and audiovisual representations. The second is description, in which a detailed overview, or trace, of each trial is produced. This requires moving between fine- and large-grained views of the data. The third is coding, in which research questions are matched to specific events in the data. This allows quantitative overviews and, due to Transana’s graphical functionality, pattern-matching by eye.

The validity of the studies rests on specifying the evidence sought in terms of the research questions, and seeking chains of evidence. Explanations of events within and across trials can then be built, resulting in causal claims for the data. The generality of the findings relates not to a wider population of pupils but to published research on how pupils view equality statements. Theoretical claims for how the approach applies to mathematical domains other than arithmetical statements, and media and contexts other than paired trialings of microworlds, can also be drawn, and provide hypotheses for further research. Replicability is demonstrated across trials if pupils talk and interact with the resources in similar ways, and sufficient detail and transparency must be provided to ensure other researchers can replicate the data and analysis. Consideration of rival explanations enhances the validity and robustness of a study. Substantial and continual peer review and feedback has ensured a variety of expert viewpoints has informed every stage of the thesis.

Daniel Pead, August 2008. Pead provided feedback on technical improve- ments (usability, aesthetics, portability, data logs) for redesigning the Sum Puzzles software for use in typical classrooms. His suggestions will inform the design phase of a post-doctoral project proposal.

Hugh Burkhardt and Malcolm Swan, February 2008. This consultation was stimulated by excerpts from the audiovisual data. Swan expressed enthusi- asm at the quality of discussion during trials. Burkhardt stated the need to explore how teachers might use it in the classroom, and this has informed the post-doctoral project proposal.

Richard Noss and Ivan Kalas, August 2007. This took place at EuroL- ogo2007, in Bratislava. Noss expressed some concern at the closed nature of the task. Noss and Kalas, both proficient Logo programmers, provided feedback on interface issues.

Dave Hewitt, March 2007. Discussed possible learning tasks based around the

Sum Puzzles microworld. This followed on from Hewitt writing a “reaction” (Hewitt, 2006) to a paper I published in a student journal (Jones, 2006a) that reported theEquivalence Calculator trials (see Section 5.2.2). Hewitt’s feedback also informed Section 5.3 of the thesis.

Willi D¨orfler, February 2007. This took place at the Fifth Congress of the European Society for Research in Mathematics Education (CERME-5) in Cyprus, at which D¨orfler presented a paper (D¨orfler, 2007a) that introduced me to “diagrammatic reasoning” as a framework for notating task design. D¨orfler agreed my approach provided an example of the diagrammatic ap- proach for the case of arithmetic equality statements. Consequently, D¨orfler wrote a “reaction” (D¨orfler, 2007b) to a student journal paper (Jones, 2007c) in which I first set out a diagrammatic approach to presenting statements to pupils, which formed the basis of Chapter 6 of the thesis.

Marta Molina, February 2007. This also took place at CERME-5. Molina provided advice on the classes of statement structures that can be used within

Sum Puzzles to appeal to structural readings.

Janet Ainley, February 2007. Ainley expressed concern that the intended role of partition and commutation when working throughSum Puzzlestasks might not harness but “undo” pupils’ implicit knowledge of arithmetic prin- ciples.

David Tall, February 2007. Tall expressed concern that only “proceptual” thinkers would learn from theSum Puzzlestasks, and other pupils might have only a superficial understanding. Tall expressed these concerns further in a “reaction” (Tall, 2008b) to a critique of the literature on pupils conceptions of the equals sign that I published in a student journal (Jones, 2008b). Tall’s feedback informed Section 3.3 of the thesis.

Chapter 8

Study 1: Attention to

statement form

8.1

Introduction

This study investigated the first research question:

Does the “can be exchanged for” meaning for the equals sign promote attention to statement form?

The evidence sought was pupils using unprompted terms (“swap”, “switch round”) when talking about the exchanging effects of a+b = b+a state- ments; and (“split”, “separate”) when talking about the effects of c = a+b

(or a+b =c) statements. As will be seen, the study also evidences strategic thinking, and suggests an exclusivity of “can be exchanged for” and “is the same as” views of the equals sign.

Prior to conducting the study my concern was for the overall viability of the

Sum Puzzlessoftware and task design. To this end, two opportunistic pilot trials with individual children were conducted and are reported in the next section1_.

The remainder of the chapter reports on two formal trials with pairs of children.