UC Berkeley

This study examines fourth graders' understandings of arithmetic generalizations, the general properties of arithmetic that hold true for all numbers. Its focus is on three types of generalizations: (a) direction of change (e.g., addition of positive numbers increases the numerical value, while subtraction of positive numbers decreases the numerical value); (b) identity (e.g., the addition or subtraction of 0 to any number leaves its value unchanged); and (c) relationship between operations (e.g., addition and subtraction are inverse operations). Using a between subjects design, two interview studies were conducted to investigate the character of children's understandings and to understand how students' production of generalizations varied across different tasks (e.g., I am thinking about a number. If I multiply that number by 5 and then divide by 5, what will happen to my number?). Study 1 (n=24) focused on students' additive thinking in the context of addition and subtraction tasks, while Study 2 (n=24) focused on multiplicative thinking in the context of multiplication and division tasks.

In both studies, qualitative analyses of the video data revealed four levels in student thinking, levels that show a spectrum of increasing generality with which students treat arithmetic operations. At Levels 1 and 2, students rely on specific instances and substitution of values. At Levels 3 and 4 (advanced generalizations), students do not rely on any examples and make generalizations about the arithmetic operations. Further quantitative analyses revealed that student thinking was not always consistent and that students' production of advanced generalizations was affected by generalization type and domain type. In both studies, the identity tasks prompted the most advanced generalizations, while the relationship between operations tasks elicited the least advanced generalizations from students. Similarly, children were more likely to produce an advanced generalization in the additive domain than in the multiplicative domain. Although the multiplicative tasks may have been more challenging for students, the character of students' thinking and the difficulty of the tasks in relation to one another were similar across both domains. These parallel findings across the studies indicate that additive and multiplicative generalizations may involve similar developmental progressions.

This dissertation provides insight into a developmental model about children's construction of arithmetic generalizations. First, the four levels of generality describe qualitative shifts in student thinking and, second, these findings indicate that the development of students' understandings of arithmetic generalizations may be heterogeneous across a range of generalizations. This work can contribute to teachers' knowledge for teaching and inform the design of a developmentally appropriate instructional sequence.