The overarching aim of the study is to investigate the educational potential of presenting pupils with statements as rules for making exchanges of arithmetical notation. This aim is made possible by the software and task design described in the previous section which supports an implicit “can be exchanged for” meaning for the equals sign. There are four specific research questions that will be addressed.
1. Does the “can be exchanged for” meaning for the equals sign promote attention to statement form?
2. Are the “can be exchanged for” and “is the same as” meanings for the equals sign pedagogically distinct?
3. Can children coordinate “can be exchanged for” and “is the same as” meanings for the equals sign?
4. Can children connect their implicit arithmetical knowledge with explicit transformations of notation?
In the remainder of this section I elaborate these four research questions.
1. Does the “can be exchanged for” meaning for the equals sign promote atten- tion to statement form?
The statement 41 = 40 + 1 can be used to partition the 41 in 30 + 41. The visual transformation 30 + 41→30 + 40 + 1, when shown on a computer screen, can be seen as “splitting” the 41 into two numerals. This is not dependent on appeals to place-value or any other specific strategy — it is a general property of c = a+b when observing its transformational effect. Similarly, the com- mutative transformational effect of a+b = b+a can be seen as “swapping” numerals around. We should expect pupils to use terms such as “splitting” and “swapping” when talking about computer-generated transformational effects of
c=b+aanda+b=b+astatements respectively. This conjecture is the focus of Study 1 (Chapter 8), and the findings are confirmed in Studies 2 and 3.
2. Are the “can be exchanged for” and “is the same as” meanings for the equals sign pedagogically distinct?
If the “is the same as” meaning is ignored then false equalities can be presented that carry anexclusive “can be exchanged for” meaning, as in 2 + 4 = 7. This may not be attractive pedagogically but offers an interesting research angle. Given that the two meanings are distinct by definition, pupils might accept imbalance when viewing statements as given truths for making exchanges. Such acceptance would show the two meanings to be pedagogically distinct and is the focus of Study 2 (Chapter 9).
3. Can children coordinate “can be exchanged for” and “is the same as” mean- ings for the equals sign?
In Study 3 (Chapter 10), which is the focus of this and the next question, the task goal is extended so that it requires co-ordinating the two meanings. Pupils attempt the puzzle-solving task and are then challenged to make their
own version puzzles. They are constrained by the requirement that statements must be both numerically balanced and notationally transformable. It might be expected that children’s qualitatively different ways of viewing notation (see Section 3.3) renders such co-ordination trivial to “proceptual” thinkers and inaccessible to others.
4. Can children connect their implicit arithmetical knowledge with explicit trans- formations of notation?
This is an extension of the previous question. Pupils familiar with puzzle-solving and making are challenged to use equality statements to present and then test their mental strategies for 37 + 48 and 37 + 58. This requires identifying implicit mental transformations, such as 37 = 30+7, and expressing them as statements, and reflects the initial intent of the task design set out in Section 5.3.2. This and the previous question are the focus of Study 3 (Chapter 10).
5.5
Summary
My research interest in pupils’ conceptions of the equals sign stems from tri- als of the VisualFractions microworld in which pupils experienced difficulties connecting the = object but not number and operator objects. Their expec- tation of a left-to-right flow of processing was challenged and made visible by the use of technology as a research instrument. Following this, I designed the
Equivalence Calculator, in which the expectation of a “get the answer” meaning for the = button is challenged by a technologically supported “is the same as” meaning. During trials pupils displayed a “protorelational” view of the equals sign because they constructed and accepted non-canonical statements on the screen, but wrote statements in canonical form on the worksheet. This suggests they focused exclusively on results and were indifferent to form.
Hewitt (2001) described a pedagogic technique in which formal notation is a
mediumfor mathematical thinking and communication. The teacher builds no- tation up from spoken arithmetic, following structural conventions to record the order of operations. Drawing on this, I envisioned a task-based research instru- ment in which pupils identify the arithmetical knowledge implicit in their mental strategies and express it as equality statements. These statements could then be used to make substitutions and so demonstrate that the strategies work. This initial design idea was developed using a storyboarding method that revealed
difficulties with this approach and resulted in a design that presents “puzzles” to “solve” by making exchanges of notation. Once pupils are familiar with solving “puzzles” they make their own, drawing on their mental computation strategies to do so.
The overarching aim of the thesis is to investigate the educational potential of presenting pupils with statements as rules for making exchanges of arithmetical notation. There are four specific research questions:
1. Does the “can be exchanged for” meaning for the equals sign promote attention to statement form?
2. Are the “can be exchanged for” and “is the same as” meanings for the equals sign pedagogically distinct?
3. Can children coordinate “can be exchanged for” and “is the same as” meanings for the equals sign?
4. Can children connect their implicit arithmetical knowledge with explicit transformations of notation?
Chapter 6
Approaches to notating
task design
6.1
Introduction
In Section 4.4.1 I described referential and structural approaches to presenting equality statements to learners. I also presented D¨orfler’s (2006) critique that these approaches presuppose external representations to be auxiliary to access- ing abstract mathematical objects, and this alienates many learners. In this chapter I describe D¨orfler “diagrammatic” framework for presenting represen- tations as the “very objects” of learning activities. I apply this framework to the case of equality statements and set out the task design used in the main studies reported in this thesis.