Equivalence and Equations in Early Years Classrooms: Elizabeth Warren, Annette Mollinson and Kym Oestrich Explain How Young Children Can Develop Powerful Understandings of Equivalence and Equations

Article excerpt

Early algebraic thinking in a primary context is not about introducing formal algebraic concepts into the classroom but involves reconsidering how we think about arithmetic. Early algebraic thinking assists young students to engage effectively with arithmetic in ways that support engagement with arithmetic structure rather than arithmetic as a tool for computation.

The distinction between arithmetic thinking and algebraic thinking in the early years' context is best defined as: arithmetic thinking focuses on product (a focus on arithmetic as a computational tool) and algebraic thinking focuses on process (a focus on the structure of arithmetic) (Malara & Navarra, 2003). This distinction assisted us to distinguish between the two in classroom discussions, and to move from one to the other as the need arose.

In our work with young children (5 year-olds), at times we needed arithmetic to support algebraic thinking (e.g., computing two expressions to see if they were the same) while at other times we needed algebraic thinking to support arithmetic (e.g., adding 3 to 2 is the same process as adding 3 to 82, 3 to 1012, 30 to 20).

The power of mathematics lies in the intertwining of algebraic thinking and arithmetic thinking. Each enhances the other as students become numerate (Warren, 2008). Using the distinction between arithmetic and algebraic thinking, a series of hands-on activities were collaboratively planned and implemented. The activities focused on the areas of equivalence and equations. The aim was to assist 5 year-olds to come to an understanding of the structure of equations, and in particular the use of the equal sign. The activities not only encouraged active learning (Crawford & Witte, 1999) but also reflected the principles of socio-constructivist learning (Vygotsky, 1962). In the case of equivalence and equations, many students in their primary years hold misconceptions with regard to the equal sign (Warren & Cooper, 2005). For many an 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 students express this as 7 + 8 = 15 + 9 = 24.

With regards to equivalence in the early years, there are four key areas that students should explore.

1. Developing the comparative language that assists in describing equivalent and non-equivalent situations,

2. Developing an understanding that equals means that the two expressions are equivalent,

3. Representing equations in a variety of different formats including equations with more than one number on the left hand side (e.g., 2 + 5 = 3 + 2 + 2 and 7 = 5 + 2, and

4. Using the 'balance principle' to find unknowns.

Language of equivalence and non-equivalence

Initially these ideas were explored in a numberless world with a focus on developing the language used for describing equivalent situations, namely: "equal to," "same as," "not equal to," and "different from." This was achieved by using concrete objects and focusing on a variety of different attributes such as shape, size and colour. Students were also introduced to balance scales that were balanced when the objects on each side had the same mass. The beginning activities explored comparing two different sets, for example, two stacks of blocks, liquid in two containers, or mass in two sides of the balance scale.

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During the classroom conversations, the language utilised to compare two sets of objects was emphasised. Each group of students was encouraged to explain why the sets were the same or different. The reasons they gave for the use of "same as" with the two stacks of blocks were: "The height of this stack [pointing to the first stack of blocks] is the same as the height of this stack [pointing to the second stack of blocks]. They are the same height. They are equal."

By contrast the group who poured water into two different containers gave the following reason for their choice of the card "different from:" "The amount of water in jar A is different from the amount of water in Jar B. …