Machine Analysis of Array Skip Counting in Elementary Math
The INK–12: Teaching and Learning Using Interactive Inscriptions in K–12 project has been developing and investigating the use of pen-based technology in elementary math classes. This paper reports progress made on machine analysis of students’ visual representations created using a combination of freehand drawing and a digital array tool that supports learning multiplication. The goal of the machine analysis is to provide insights into students’ mathematical thinking as revealed through creation and manipulation of visual representations. For array representations, machine analysis involves interpretation of ink annotations that represent problem-solving strategies, one of which is counting by a number other than 1, aka skip counting. A subset of student work from a 5-week trial in a third grade class provides a corpus for development and evaluation of the machine analysis routines. This paper describes the routines and presents findings demonstrating that the routines are able to provide accurate information about students’ skip-counting strategies. It discusses the key to the accuracy-using knowledge about the structure of arrays and the nature of skip counting to bias the machine analysis routines; and presents evaluation results for two versions of routines that do not use this knowledge and that consequently suffer from high error rates. The paper also discusses current work on extending the routines to analyze the process of creating representations and future work on using the routines on thousands of pieces of student work from the 5-week trial.
This research is funded by the NSF INK–12: Teaching and Learning Using Interactive Ink Inscriptions project, DRL–1020152 (Koile), DRL-1019841 (Rubin). We gratefully acknowledgment contributions from MIT’s CLP research group members past and present, and from math education researchers Lily Ko and Marlene Kliman at TERC. We also thank Randall Davis, Jonathan Grudin, Tracy Hammond, YJ Kim, and Marlene Kliman for valuable feedback on drafts of this paper.
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