Using Repeating Patterns to Explore Functional Thinking: Elizabeth Warren and Tom Cooper Lead Us through a Series of Teaching Activities Designed to Develop Young Children's Algebraic Thinking
Warren, Elizabeth, Cooper, Tom, Australian Primary Mathematics Classroom
This paper continues the exploration of ways that we can reformulate common mathematical activities in the primary school to provide stronger bridges for thinking algebraically. The focus is not on introducing formal algebraic notation in the primary context, but rather to look at common activities through new lenses, that is, lenses that support the growth of algebraic reasoning. It seems that, as Malara and Navarra (2003) argued, classroom activities in the early years focus on mathematical products rather than on mathematical processes. Strings of numbers and operations in arithmetic are considered procedures for arriving at answers (Kieran, 1990). Traditionally, primary schools place minimal emphasis upon relations and transformations as objects of study. In our research we have found the young children can engage in conversations about equivalence and equations (Warren & Cooper, 2005a) and functional thinking (Warren & Cooper, 2005b). Fundamental to relations and transformations is the concept of the function, that is, how the value of certain quantities relate to the value of other quantities (Chazan, 1996), or how values are changed or mapped to other quantities, referred to in the literature as co-variational thinking. This paper reports on some recent classroom teaching that attempts to examine repeating patterns and use children's understandings of repeating pattern to begin to explore concepts related to functional thinking.
There are two predominant types of patterns that children explore in the early years: repeating patterns and growing patterns. Commonly, these patterns are used to find generalisations within the elements themselves: What comes next? Which part is repeating? Which part is missing? This activity is commonly referred to as pattern finding in a single variation data set (Blanton & Kaput, 2004). But repeating patterns and growing patterns can also lead to the early development of functional thinking, that is, relationships between two data sets. This paper presents a suggested sequence for investigating repeating patterns and extending these investigations to include activities and questions that specifically assist young children to begin to develop functional thinking.
Repeating patterns are patterns where a group of elements repeat themselves as the pattern extends. Some examples of repeating patterns are:
As we consider these patterns we need to present them in a variety of different modes. For example, the ABABAB pattern can be represented with actions (e.g., stand, sit, stand, sit, stand, sit, stand, sit), as sounds (drum beat, cymbal, drum beat cymbal), as geometric shapes (e.g.,[??]), and as feel (e.g., rough, smooth, rough, smooth, rough, smooth). When we are exploring repeating patterns there is a sequence that young children progress through. The next section delineates the suggested sequence, using an ABABAB pattern with geometric shapes to illustrate each step in the sequence.
Sequence for exploring repeating patterns
1. Copying the pattern
Create the following patterns on the floor and encourage children to copy the pattern using triangles and circles.
2. Continuing the pattern
In this instance ensure that young children realise repeating patterns can continue in both directions. Make the above pattern on the floor and ask the children. "What shape comes after the circle? What shape comes before the triangle." Get them to extend the pattern in both directions.
Number is an example of a pattern that extends in both directions.
3. Identifying the repeating element
Say the pattern out loud (triangle, circle, triangle, circle, triangle, circle) and ask them to identify the part you are repeating. With a piece of wool, ask them to put a circle around the repeating part. …