Introducing Equivalence and Inequivalence in Year 2: Elizabeth Warren and Tom Cooper Outline a 'Language Framework' That Will Assist Teachers to Integrate Algebra into Their Daily Work with Students by Exploring the Concepts of Equivalence and Inequivalence
Warren, Elizabeth, Cooper, Tom, Australian Primary Mathematics Classroom
Current research is beginning to turn to arithmetic as a key for access to algebra (Carpenter, & Levi, 2000; Carraher, Schliemann, & Brizuela, 2001; Warren & Cooper, 2001; Kaput & Blanton, 2001). It is believed that the most pressing factor for algebraic reform is the ability of primary teachers to 'algebrafy' arithmetic (Kaput & Blanton, 2001), that is, to develop in their students the arithmetic underpinnings of algebra (Warren & Cooper, 2001) and extend these to the beginnings of algebraic reasoning (Carpenter & Franke, 2001). As was argued by Carpenter and Levi (2000), the artificial separation of arithmetic and algebra 'deprives children of powerful schemes of thinking in the early grades and makes it more difficult to learn algebra in the later years' (p. 1).
In this paper, we report on part of the first year of a longitudinal study to investigate instruction that helps primary children to generalise and formalise their informal thinking into powerful mathematical ideas that support algebra. We specifically report on the trial in three Year 2 classes of a lesson designed to develop young children's understanding of equivalence relations.
While there are several uses for the equal sign in mathematics, the two that we focus on in this paper are equal as the result of a sum (2 + 3 = 5) and equal as quantitative sameness (2 + 3 = 4 + 1). It is the second use that represents equivalence. Many children interpret equations as corresponding to a left to right action, rather than a static state (Pirie & Martin, 1997). For them the equation only makes sense if the action occurs before the equal sign. For example, when asked to find the unknown for 7 + 8 = ? + 9, many children express this as 7 + 8 = 15 + 9 = 24. Often the verbal responses children give match this way of thinking. Young children's inability to recognise turnarounds (2 + 3 = 3 + 2) also shows an inability to deal with equivalent situations (Warren, 2002).
Our instructional strategies rely on developing an understanding of equivalence using unmeasured quantities. In this instance the unmeasured physical quantities were the masses of common kitchen objects, and a set of balance scales was used to compare these masses. Given that all children of this age in Australian schools already have had extensive experience with modelling and representing number, we included representations of numbers in the lesson. Because it mirrored the quantitative situation, we chose the set model as the model for number. Glass sea shells were used as they were of sufficient weight so that larger numbers clearly weighed more than smaller numbers. The advantage of using a balance model to explore arithmetic equations is that it lacks direction and emphasises the need to consider the equation as an entity rather than as an instruction to act to achieve a result.
(Description of the three classes and their teachers) In this first stage in the project, the children had been at school eighteen months and their ages ranged from 6 to 7 years. The operations of addition and subtraction had been introduced along with the equals sign and the children had compared and ordered numbers to 20; however, there has been no explicit instruction with regard to equals and order and their properties. Furthermore, although the children were familiar with the symbol for equal '=', most had never used the term 'not equal' before and were unfamiliar with the symbol '[not equal to]'. They all had had extensive experience using natural language, the set model, and symbols as representations of numbers. With regard to mass, the children had already experienced activities involving 'hefting' objects to identify which was heavier, lighter, or the same mass. As we moved from class to class, the structure of the lesson and the questions asked were changed to cater for any difficulties that occurred. The following section gives an outline of our 'best' attempt together with some of the children's responses to indicate what worked and what did not work. …