Niveles de aprendizaje

Storyboarding is a common method for interface design amongst the human- computer interaction community (Madsen & Aiken, 1993). However, it is novel to the design of mathematical microworlds and marks a contribution to the literature arising from this thesis4_.

The storyboarding process lasted around six months during which time a total of ten storyboards were produced. A storyboard was abandoned and a new one started every time the design appeared to run into problems. The ten storyboards varied in length from 15 to 170 slides each and comprised just over

Figure 5.6: An early schematic of an interface

600 slides in total. Each storyboard contained one or more “scenes” separated by distinctive clapper-board slides to indicate a “cut”. Each scene correlated, more or less, with an imagined epistemic child, “C”, working on a particular problem such as programming a calculation strategy for 18×8. Successive scenes in a given storyboard would typically follow alternate hypothesised strategies for the same problem, or a similar strategy applied to different problems. Storyboarding was motivated by an impasse with using interface schematics to try and design a microworld (see previous section). To this end the interface was abstracted to a bare minimum and C (the epistemic learner) was envisioned working directly with raw notation. Figure 5.7a, the first slide of the first sto- ryboard, shows an interface that has been abstracted to a single arithmetical statement. The following slides followed stepwise construction and transforma- tion of notation towards the learning goal of programming a calculation strategy (Figure 5.7b-d).

The interface became more fleshed out and increasing consideration was given to manipulation tools, such as the keypad shown in Figure 5.7f. When the second storyboard was created the theme was continued but the interfaces became decreasingly abstract and began to look quite literal and concrete (Figure 5.7g). This learner-centred concretisation of the interface design continued through to the fifth storyboard (Figure 5.7h). This was no smooth journey. Many starts, stops and varied designs are apparent throughout the first five storyboards. However, the design ran into problems. These problems emerged when the storyboards were shared with peers and presented at seminars. The interface

(a) (b)

(c) (d)

(e) (f)

(g) (h)

Figure 5.8: Icons of C and R with callouts

design embodied the anticipated learning scenarios in a rather rigid manner due to the technological attempts to assist learners with the difficulties of trans- lating ambiguous and assumptive natural language arithmetic into the explicit formal notation required by the task and goals. I returned to a stripped-down (abstracted) representation of the interface for the next storyboard. Another decision made in the light of peer feedback was the inclusion of an icon for C, and call outs (speech and thought bubbles). This helped keep the focus on C’s mathematical activity without a drift back to interface concretisation. In the subsequent storyboard an icon of the participant researcher, R, was also included (Figure 5.8). The icons provided a natural way to capture R’s possible interventions and interactions with C as imagined in the thought experiments I also decided at this stage to incorporate real-world arithmetical strategies from children in classrooms. The QCA website National Curriculum in Action hosts samples of pupils’ work with reference to levels of attainment as specified by the National Curriculum. This data enabled the importation of real pupils’ arith- metical strategies for storyboard scenes. For example, Figure 5.9 shows a slide illustrating “Tony’s” written explanation for calculating 30 + 41. Slides were created by imagining pupils inputting their own written strategies to explore the problems that might arise. These turned out to be the same problems that had emerged in the interface concretisations in early storyboards, namely, that when arithmetical strategies are spoken in natural language the partitioning and re-organising of notation is implicit. For example, “Tony” expresses com-

Figure 5.9: “Tony’s” strategy for solving 30 + 41

position explicitly using equalities such as 30 + 40 = 70, but draws implicitly on arithmetical facts such as 41 = 40 + 1. There appeared to be a need to give learners access to the arithmetical assumptions underlying their strategies in order to express them using the computer.

Despite the remaining problems an implementable design had now been reached. Implementation became possible for two reasons. First, it had become apparent that the functionalities of the processing tools could be ascribed directly to the notation itself: notation could simply be clicked to make selections and transformations. Two keypads were then needed for entering notation. The

Equivalence Calculator (Section 5.2.2) proved ideal without the need for much redesign (Figure 5.3, p.45). Second, the problem of making implicit arithmetic explicit, that had plagued all ten storyboards, could simply be ignored in terms of technical implementation. The introduction of the C and R icons and call outs made it clear that R could guide this process. This was somewhat heavy- handed, involving exposition, but, by drawing on guidance by R, the task could at least be implemented and piloted.

The software was subsequently programmed inImagine Logo (Kalas & Blaho, 2003) which lends itself to the fast prototyping of microworlds. I made a fi- nal significant change to the task design during software testing. The impasse of inputting the partitions and commutations implicit in written strategies re- mained apparent and it was difficult to imagine offering learners an intuitive way to do this. However, upon programming some strategies, it was discovered the outcome made for an intriguing puzzle in which the goal is to use provided equalities to transform an arithmetical sum into its answer. Consequently, I de-

cided to reverse the task: rather than pupils expressing their existing strategies as sequences of equality statements, they would be presented with (randomly sequenced) statements and challenged to transform a given expression into a nu- meral. A paper-based equivalent might involve putting the following statements in the correct order for transforming 20 + 53 into 73.

20 + 53 20 + 50 = 70

3 + 50 = 50 + 3 70 + 3 = 73

53 = 3 + 50

It should be noted this is strictly speaking a results-based task goal but, as with Hewitt’s “guess my number”, the focus is on the processes required to achieve the goal — in this case seeing equality statements as rules for nota- tional interchanges. Unlike Hewitt’s task this involves generating a result from an expression, but this need not be case the case. The reversibility of trans- formations means a “puzzle” could as easily be presented with the challenge to transform 73 into 20 + 53, as vice versa. Either way, “solving” it involves transforming one term into another rather than mental calculation. Once pupils become familiar with “solving puzzles” made of statements they can be chal- lenged to make “puzzles” that draw on the implicit knowledge in their mental strategies. This, more or less, is the nature of the computer-based task used as a research instrument in this thesis5.