The trials reported here are atypical of the thesis because they involved only single children. These pilots were a very first trial of the software after its development and my mind was as much on technical concerns (crashes, bugs, usability and so on) as on educational issues. However, these trials provided a crucial feasibility study prior to trialling proper with pairs of children (see next section). The children involved in the trials were both girls, one aged nine years the other aged twelve years. The difference in ages reflects the opportunistic nature of the trials (both girls are known to me and the trials took place during the school breaks) and, while not ideal, was considered insignificant in light of the anticipated functional and usability difficulties. Happily, no such difficulties arose and some unexpectedly interesting data emerged instead. Although these data are limited in size and scope they support the feasibility of the design and are summarised here.
In its early version theSum Puzzles software was set up to present a sequence of 13 diagrams (Figure 8.1). These diagrams contain between one and nine statements, including commutative, partitional and compositional forms, some of which appeal to tens-and-units readings, and one compensatory statement (45 + 9 = 50 + 4 in Figure 8.1l). These diagrams were not designed with trialling in mind but were a result of testing the software and demonstrating it to peers, and reflect the opportunistic nature of the pilot trials.
The trial with the nine year old girl, Penny, involved only five of the diagrams (Figures 8.1d-h) and lasted just ten minutes. I recorded the trial as field notes of her activity as she worked. The trial with the twelve year old girl, Lauren, involved all 13 diagrams and lasted 40 minutes. It was recorded as an audiovi- sual movie using screen-capture software and a microphone, which provided an opportunity to test the data capture methods used in the main studies. In each case I demonstrated the functionality of the software, asked them to “solve the puzzles” and remained present to offer guidance and encouragement.
Penny (nine years old) initially adopted a trial and error approach, selecting arbitrary statements and clicking numerals in the black box to see if anything changed. This strategy worked well for the first three diagrams attempted (Figures 8.1d-f). However her trial and error approach became guided by iconic matching2 during the fourth diagram (Figure 8.1g). Penny initially selected
(a) Pilot trial: Diagram 1 (b) Pilot trial: Diagram 2
(c) Pilot trial: Diagram 3 (d) Pilot trial: Diagram 4
(e) Pilot trial: Diagram 5 (f) Pilot trial: Diagram 6
(g) Pilot trial: Diagram 7 (h) Pilot trial: Diagram 8
(i) Pilot trial: Diagram 9 (j) Pilot trial: Diagram 10
(k) Pilot trial: Diagram 11 (l) Pilot trial: Diagram 12
(m) Pilot trial: Diagram 13
the statement 7 + 8 = 15 and attempted to substitute the term 8 + 7 in the box. She tried clicking the term several times apparently expecting that a substitution should be possible. When I asked why she thought it was not working she answered that the 7 and the 8 are the wrong way round. She turned her attention to the statement 8+7 = 7+8, suggesting she understood its commutative transformational potential, and then completed the diagram with no further difficulty. A second incident of interest occurred during the fifth and final diagram that Penny attempted (Figure 8.1h). After a few experimental substitutions the box had become 30 + 1 + 40 and Penny selected the statement 31 = 1 + 30. She clicked several times on the term 30 + 1 and, when no substitution occurred, paused. I asked why she thought it was not working and expected her to notice the commutation of 30 + 1 and 1 + 30 as she had with 7 + 8 and 8 + 7 in the previous diagram. Instead, however, her explanation was that 31 = 1 + 30 did not allow a substitution on 30 + 1 because “it means 31 becomes 1 add 30, not 1 add 30 becomes 31”. This suggests Penny viewed the statement as a decomposition into constituent parts, as hypothesised by the diagrammatic approach (Section 6.2.2), rather than as a “backwards sum” as would be expected from the literature (Chapter 2). After Penny had completed the diagram shown in Figure 8.1j she asked of the equality statements “Is it like a sequence?”, suggesting an emergence of strategic thinking about the diagrams, as hypothesised by the diagrammatic approach (Section 6.2.3).
Lauren (twelve years old) also began with a trial and error approach that proved increasingly inefficient by the third diagram (Figure 8.1c) when iconic matching began to guide her experimentation. By the final few diagrams, which are com- plicated and contain up to nine statements (Figures 8.1j-m), iconic matching dominated Lauren’s approach. The most notable feature of Lauren’s activities was her use of a strategy from diagram six onwards (Figure 8.1f). This involved starting with the answer, wherever it appeared on-screen in an equality state- ment, and transforming that statement such that the left-hand side matched the inscriptions in box. To illustrate this strategy, consider the diagram shown in Figure 8.1j. Lauren initially performed a substitution on the compensation relation 45 + 9 = 50 + 4 using 50 + 4 = 54, transforming the diagram into that shown in Figure 8.2.
Lauren then looked for iconic matches with the left-hand side of 45 + 9 = 54 and made substitutions until she had transformed it into 15 + 39 = 54. She then
Figure 8.2: Lauren uses to 50 + 4 = 54 transform 45 + 9 = 50 + 4→50 + 4 = 54 clicked on the inscription in the box and the numeral 54 appeared as required. This same strategy was used for the remaining three diagrams (Figures 8.1j-m). In sum, the pilot trials demonstrated that the software was robust and usable. The task made sense to the girls and they got to grips with it readily. The trial data suggested that the interactive diagrams might support attention to form and the use of strategic thinking.