Many students encounter difficulty in their transition to advanced mathematical thinking. Such difficulty may be explained by a lack of understanding of many concepts taught in early school years, especially multiplicative reasoning. Advanced mathematical thinking depends on the development of multiplicative reasoning.

The purpose of this study was to identify indicators of multiplicative reasoning among fourth grade students. Inhelder and Piaget (1958) suggested that children circa age eleven are transitioning from the Concrete Operational Stage to the Formal Operations Stage and that it is not likely for children to demonstrate multiplicative reasoning without the structures of development supporting logical and abstract thinking. By employing a cross-case analysis, this study explores the thinking of fourteen math students from a low socioeconomic school. Through cross-case analysis, the researcher probed for patterns of multiplicative reasoning as students progressed through a test instrument which invoked varying levels of multiplicative reasoning. Section one did not distinguish between multiplicative algorithms and multiplicative reasoning. Section two discriminated with respect to multiplicative scheme extension. Section three discriminated with respect to unequal group identification and manipulation. Section 4 discriminated with respect to proportional reasoning but not with respect to multiplicative reasoning.

The fourth grade subjects fell into three categories: pre-multiplicative, emergent, and multipliers. Those subjects who utilized multiplicative reasoning on less than four questions were considered pre-multiplicative, whereas those subjects who utilized multiplicative reasoning on six or more questions were considered multipliers. The remaining seven were those subjects who changed their approach from test item to test item, sometimes demonstrating multiplicative reasoning strategies and at other times demonstrating additive reasoning strategies. These subjects were considered emergent in the development of multiplicative reasoning.

This study developed twelve new sub-levels that describe in more detail the multiplicative thinking of these fourth graders. These new sub-levels are Level 1 Non-quantifier, Level 1 Spontaneous Guesser, Level 2 Keyword Finder, Level 2 Counter, Level 2 Adder, Level 2 Quantifier, Level 2 Measurer, Level 3 Repeated Adder, Level 3 Coordinator, Level 4 Multiplier, Level 4 Splitter and Level 5 Predictor.

This paper suggests that when teachers understand a child’s method of deriving multiplying schemes and multiplicative reasoning strategies, they are in a better position to provide the appropriate learning environment for the child. Such interaction allows the listening teacher to build on the child's current level of mathematical understanding. Students should be encouraged to discover for themselves the needed theorems, definitions, and mechanics of the number system, and to personally develop any “short cutting” algorithms, rather than simply being handed the algorithms by the instructor with little or no understanding.