Functional Thinking in a Year 1 Classroom: Elizabeth Warren, Samantha Benson and Sandra Green Outline a Series of Activities That Will Assist Young Children Understand the Key Ideas Involved in the Development of Functional Thinking
Warren, Elizabeth, Benson, Samantha, Green, Sandra, Australian Primary Mathematics Classroom
The concept of a function is fundamental to virtually every aspect of mathematics and every branch of quantitative science. Presently this type of thinking is carolled at the secondary level, and yet it has many benefits for deepening our understanding of early arithmetic. This is particularly so in the way that operations can be considered as 'changing' and how it explicitly illustrates the way in which addition and subtraction are inverse operations, with each 'undoing' the other. With the move to introduce algebraic thinking into the elementary classrooms (e.g., Warren & Cooper, 2005) this paper explores activities that exemplify this thinking with 6 year-old children. The three authors collaboratively planned and implemented a series of hands on activities over an eight lesson program. 'Early algebra' is not the same as 'algebra early'. It is a refocusing of mathematical thinking away from products and towards generalisations of the big ideas in arithmetic within a climate of inquiry and justification (e.g., Brown, 2002). The aim of these learning activities was to assist young children understand the key ideas in this area of mathematics. The activities not only encouraged active learning (Crawford & Witte, 1999) but also reflected the principles of socio-constructivist learning (Vygostky, 1962).
In this context, functions were seen to represent a consistent change between two sets of data. In mathematics, a function is a relation such that each element of a set is associated with a unique element of another set, thus the function rule describes how one value consistently changes into another.
In the early years there are three ideas that children should grasp, namely:
1. Following a function rule to consistently change one set of 'values' into another set of 'values', that is, finding output values when given input values by consistently applying the function rule.
2. Reversing the change process by reversing the rule, that is finding the input values when given the output values by consistently applying the inverse (or reverse) of the function rule; and
3. Identifying the function rule when given a set of input values and a set of output values
Initially these ideas were explored in a 'numberless' world. This reflects a theoretical stance that the structure of mathematics is best explored in a world that does not involve number as number tends to suggest computation in the minds of many young children (Davydov, 1975) and that changes the focus from how mathematics works to finding answers. For example, the function rule may be as simple as 'adds at to a letter', 'add e to the end of a word' or 'cooks the food'. Thus we can even utilise functional relationships in literacy contexts, for example, how do words change if we add an 'e' to the end of them. Which ones make new words? (See Figure 1).
[FIGURE 1 OMITTED]
Following the rule:
Young children enjoy exploring these ideas in play situations where they are actively involved in the process of change. The rule can be attached to the outside of a large box dressed as a function machine that has an input slot and an output slot. As children place words or values in the input slot another child applies the rule and 'posts' the output through the output slot. This type of activity is simply about following the rule. Children are given or create their own input cards. The child in the box changes the card according to the rule and then posts the new card through the output slot. Our research has indicated that such kinesthetic movement is an important dimension of understanding function (Warren & Cooper, 2005).
Reversing the rule
This is an important aspect of functional thinking as it allows us to explicitly explore which operations or processes are related to each other. It also allows us to find unknowns by 'backtracking'. …