Parsing of algebraic expressions by experienced users of mathematics

Abstract

The present study investigates how experienced users of mathematics parse algebraic expressions. The main issues examined are the order in which the symbols in an expression are scanned and the duration of fixation. Two experiments tracked the order in which the symbols of an expression were scanned. The results were analysed using Markov Chain models of the scanpath data and provided strong support for the hypothesised scanning order: a left-to-right, top-to-bottom syntax-based scanning order. Length of fixation was also analysed in the first experiment. When reading text, readers pause significantly longer at the end of clauses and sentences. A similar pattern was found for mathematical expressions: Symbols at the end of a phrasal constituent were fixated upon for significantly longer than symbols at the start or middle of the phrasal constituent. These results suggest that the parsing of algebraic expressions has marked similarities with the way in which sentences of natural language are processed and reinforces the importance of syntax in their comprehension.

The authors wish to thank the tireless efforts of the participants, and also Claire Byrne for helpful assistance with the use of data analysis software. Anthony Jansen's research was supported by an Australian Postgraduate Award scholarship.

Notes

¹In reality of course, backtracks will occur when an expression is read; however, we will ignore this issue until empirical scanning models are developed in Experiment 1

²Logarithms are used to make the data analysis more tractable

³This version of the RFV program can be found at http://www.csse.monash.edu.au/projects/RFV/ along with the user's manual and tutorial (Jansen, 2001)

⁴If one symbol is located at (x ₁, y ₁) and another is located at (x ₂, y ₂), then the Euclidean distance between them is