The study demonstrates that the task enabled the pupils to view and talk about notation in ways that stand in contrast to those reported in the literature (see Section 2.2). Left-to-right readings of individual statements were replaced by looking for matches of numerals across statements and terms. This arose as the pupils discussed why the software sometimes allowed a substitution to take place and other times does not, and when pupils determined which substitutions could be made within a given diagram. As such, the presentation of equality statements as transformational rules enabled the pupils in both trials to explore and talk about arithmetic notation in non-computational ways.
The pupils found ituseful(Ainley, Pratt, & Hansen, 2006) to discern statements by form when making transformations of notation. This is because statements were presented as systems of rules for activity toward a task goal, rather than isolated questions of numerical balance. All four pupils distinguished the com- muting transformational effects ofa+b=b+aforms and, to varying extents, used this distinction strategically to discuss possible transformations one or two steps ahead. Only two pupils, Terry and Kitty from each trial respectively, distinguished the partitioning transformational effects of c=a+b forms, and only Terry explicitly used this distinction as part of a strategy that proved advantageous for later, more complicated diagrams (Figures 8.3g-k).
When the pupils articulated commuting and partitioning effects this does not mean they had a conception of the underlying arithmetic principles. Baroody and Gannon (1984) found that young children can appear to exploit commuta- tion to reduce computational burden, but are often merely indifferent to con- sistency of outcome (see Section 3.3). The data from the two trials provide no evidence either way as to whether pupils concerned themselves with the numer- ical balance of a given statement when using it to make an exchange. All that can be inferred is that pupils did not explicitly articulate computational strate- gies very often during the trials (Figures 8.4 and 8.5), and rarely did so at the same time as identifying iconic matches or discussing transformational effects. This suggests the task goalsexclusively promote a “can be exchanged for”over
an “is the same as” meaning for the equals sign. In fact, there is no intrinsic
utility to considering numerical balance when working through the diagrams, and it is more efficient not to do so.
pupils with diagrams that contain some false equalities, as in 77 = 11 + 33. If pupils do not notice and comment on the presence of such statements this suggests they are not concerned with numerical balance, and are attending exclusively to transformations of notation as visualised on the screen or in the head. False equalities lead to changes in the computational total of the boxed term across transformations, such as when using 77 = 11 + 33 to transform 143 + 77→143 + 11 + 33. If pupils do not notice or comment on this it would suggest indifference to the total of the boxed term when transforming diagrams. Furthermore, if articulations and strategies arise that resemble those reported in this study, then the presence of false equalities has little impact on how pupils view and talk about the diagrams when working toward the task goals.
8.5
Summary
Two opportunistic pilot studies with single children suggested the task resources and goals were viable both functionally and as a research instrument. The main study, in which two pairs of primary pupils worked through a sequence of diagrams toward the task goals, sought to inform the research question:
Does the “can be exchanged for” meaning for the equals sign promote attention to statement form?
A visual overview from each trial show a notable lack of computational readings, and the prevalence of iconic matching and distinguishing statements by their distinctive transformational properties. The pupils’ attention was drawn to numerals, and matches of numerals, when looking for allowable substitutions. All the pupils spoke of the transformational effects ofa+b=b+astatements in terms of commuting numerals, and one pupil from each trial spoke of the effects ofc=a+bstatements in terms of partitioning numerals. Illustrative excerpts show that all pupils used commuting effects in a strategic manner to enable substitutions one or two steps ahead. Only one pupil systematically articulated using partitioning effects for a more sophisticated strategy, and this method proved notably more efficient in the final few diagrams, which are complicated. It seems, then, that the “can be exchanged for” meaning for the equals sign, as instantiated by the computer and task goals, does promote attention to state- ment form. The requirement to identify allowable substitutions necessitates looking for matches of numerals and terms across statements, and to articulate
strategies it is useful to distinguisha+b =b+aand c=a+b statements in terms of their distinctive transformational effects.
This view of notation stands in contrast to that required when viewing state- ments as isolated questions of truthfulness. It is unclear as to whether the pupils were concerned with the numerical equivalence of terms on both sides of equals signs, or with the consistent computational value of the boxed term across trans- formations. In the Study 2 this will be investigated by placing false equalities into some of the diagrams to find out if pupils comment on their presence, and whether they have any visible impact on how they talk about the diagrams.
Chapter 9
Study 2: Attention to
numerical balance
9.1
Introduction
This study investigates the second research question:
Are the “can be exchanged for” and “is the same as” meanings for the equals sign pedagogically distinct?
It seeks to replicate the findings of Study 1, in which pupils were evidenced engaging in iconic matching, articulating statement form in terms of distinc- tive transformational effects, and using these distinctions to work strategically toward the task goals. However, unlike in Study 1, the diagrams presented to pupils contain some false equalities. Evidence will be sought of pupils’ re- sponses (or lack of) to the presence of false equalities, and any notable impact their presence has on how pupils view, manipulate and talk about the diagrams. The study comprises two trials with pairs of pupils. All pupils were deemed mathematically able by their class teachers, and were selected as being likely to discover and articulate sophisticated strategies when working with the diagrams in order to test whether they engaged with numerical balance views at the same time. One pair of pupils were aged 9 and 10 years, to provide comparison and contrast with the pupils in Study 1, and the other pair were aged 12 and 13 years, to find out if more mathematically mature pupils are more likely to pay
attention to the truthfulness of statements.
9.2
Task set-up
As with Study 1, the software was set up to present a sequence of eleven di- agrams containing up to nine arithmetical statements (Figure 9.1). The first two diagrams contain only two compositional statements each, and are intended to familiarise learners with operating the software (Figure 9.1a-b). Diagram 3 introduces a commutative statement (Figure 9.1c) and Diagram 6 introduces a partitioning statement (Figure 9.1f). Diagrams 7 to 11 combine composi- tional, commutative and partitioning statements, and are increasingly compli- cated (Figures 9.1g-k). Diagrams 1 to 7 contain only true statements. Diagram 8 contains the first false statement (15 + 28 = 44), which is “subtly” false in that it is only imbalanced by 1 (Figure 9.1h). Diagram 9 contains two false equalities (Figure 9.1i). Diagram 10 contains three numerical false equalities s (Figure 9.1j), two of which are less subtle than those in diagrams 8 and 9. Di- agram 11 contains blatant false equalities and a large computational disparity between boxed initial term (143 + 77) and boxed final numeral (23).
These diagrams are modifications of those used in Study 1, and alternative designs would have been possible. First, all statements could have been false equalities from the start, or introduced much earlier in the trials. However, such a design would likely not allow the pupils sufficient time to become familiarised with the functionality of the software and nature of the task. Secondly, the imbalance of false equalities could have been introduced less gradually. For example, the first false equality met, 15 + 28 = 44 (Figure 9.1h), could have been, say, 15 + 28 = 1. However, initial opportunities for pupils to notice subtle imbalances was preferred as this would be more informative about the degree of their sensitivities to numerical equivalence during the trials. Thirdly , place- value partitions and compositions were used less than in previous trials because their patterns of repeating digits would make them obvious when imbalanced and would not have allowed the gradual introduction of false equalities (e.g. 41 = 40 + 2 compared to 41 = 27 + 15).
I first showed the pupils how to operate the software, in particular how to select a statement by clicking on the equals sign and how to make substitutions by clicking on numerals, and allowed them to experiment with it for several
(a) Diagram 1 (b) Diagram 2
(c) Diagram 3 (d) Diagram 4
(e) Diagram 5 (f) Diagram 6
(g) Diagram 7 (h) Diagram 8
(i) Diagram 9 (j) Diagram 10
(k) Diagram 11
Figure 9.1: Diagrams presented in the trials
minutes. For each diagram I challenged the children to transform the boxed term into a single numeral.