Arithmetic Pathways towards Algebraic Thinking: Exploring Arithmetic Compensation in Year 3: Elizabeth Warren and Tom Cooper Illuminate the Importance of Making Explicit the Underlying Structure of Arithmetic. They Advocate Involving Students in Noticing and Developing Patterns of Generalised Thinking That Relate to Algebraic Thinking (Process) Rather Than Arithmetic Thinking (Product)
Warren, Elizabeth, Cooper, Tom, Australian Primary Mathematics Classroom
Current research continues to indicate that many children experience difficulties in moving from an arithmetic world to an algebraic world, and it seems that many of the difficulties children experience originate from a lack of an appropriate foundation in arithmetic (Carpenter & Franke, 2001; Warren & Cooper, 2001; Warren, 2002). The conjecture is that, despite working with numbers, children fail to abstract the underlying structure of arithmetic. As part of everyday classroom arithmetic activities and discussions, children are involved in and are exposed to a variety of experiences in using the four operations. It has been assumed that, from these experiences, children will induce the fundamental structure of arithmetic. But research suggests that they are not. One way of addressing many of these issues, and helping students to do algebra, is to involve students in patterns of generalised thinking throughout their education. This may be done by focusing attention away from computation and onto the underlying mathematical structure exemplified by the carefully chosen examples, with an aim of explicitly abstracting arithmetic structure. Malara and Navarra (2003) suggest that algebraic thinking is about process whereas arithmetic thinking is about product (reaching the answer).
There are three areas that we are considering as early algebra in the primary school, generalised arithmetic, equivalence and equations, and patterns and functions. The focus of this article is on generalised arithmetic and in particular, arithmetic compensation. The research reported in this article begins to investigate instruction that helps children take the next steps in generalising and formalising their informal thinking into powerful mathematical ideas that support algebraic thinking, and in this instance also support estimating, thinking, and mental computation.
Teachers and students
We are presently working in 13 Grade 3 classes, comprising approximately 300 students with an average age of 8 years. As we progress through the classes we are continually refining our ideas. Initially we trial ideas in two classrooms (developmental classes), then support two teachers to implement these ideas in their classrooms (experimental classes), and finally require the remaining nine classes to teach the ideas unsupported. Throughout the research, children are encouraged to look for 'pattern rules' and express these rules in their own language.
Arithmetic compensation is the idea that if A + B = C, then A - k + B + k = C, or A + k + B - k = C, (e.g. 13 + 34 = 47 then 13 - 3+ 34 + 3 = 47). In other words, if we increase/decrease one number by a certain amount we must decrease/increase the other number by the same amount for the answer to stay the same. Put simply, for Part + Part = Whole, if we keep the Whole constant and change the value of one of the parts, we must compensate for this by changing the other part by the same amount.
An extension of this is that for problems such as A + B + C + D = E, if we increase/decrease one of the numbers then we must decrease/increase some or all the other numbers by an amount that is the same for the answer to stay unaltered (e.g. 127 + 34 + 22 + 41 = 224 then 127 + 3 + 34 + 22 - 2 + 41 - 1 = 224, that is, 130 + 34 + 20 + 40 = 224).
The use of the compensation rule
There are three main uses of the compensation rule in mathematics. These are:
(a) mental computation--it helps us to reconstitute numbers to assist us in mentally working out the answer;
(b) estimation--when we increase numbers to assist us to estimate an answer, if we decrease some numbers at the same time we will probably get a better estimate; and
(c) simplifying expressions in algebra.
Models and materials that can be used to teach the compensation rule
There are a variety of models that can be used to explore this principle. …