Abstract
Subtraction errors can inform teachers about students’ mathematical reasoning. Not every subtraction error is informative, it’s implications for students’ mathematical reasoning depends on the item characteristics. Diagnostic items are specifically designed to elicit specific subtraction errors. This study evaluated how the diagnostic capacity of subtraction items is related to their characteristics. The item characteristics being studied are open-ended and multiple-choice items, bare number and word problems. As well as various number features, such as the number of digits in the subtrahend and minuend. Diagnostic capacity is defined as the extent to which multi-digit subtraction items that require borrowing (e. g., 1000 – 680) elicit bridging errors, such as the smaller-from-larger-error. Item response theory (IRT) was used to estimate item properties. Subsequently, the item properties were used in two separate ANOVA analyses to compare the diagnostic capacity of multiple-choice versus open-ended items, bare number versus word problems, and number features. As expected, multiple-choice items have a higher diagnostic capacity than open-ended items. More interestingly, it was found that the number of digits in the subtrahend and minuend influenced the diagnostic capacity of the items. Items characterized as 3/4n – 3n, like 1000 – 680 had the highest diagnostic capacity, whereas items characterized as 3/4n – 2n, such as 1000 – 20 had the lowest diagnostic capacity. The discussion focuses on the implications of this study for further research into the design of diagnostic items.
Highlights
Diagnostic items can be designed to collect specific and fine-grained information about students’ cognitive strengths and weaknesses (Leighton and Gierl, 2007; Keeley and Tobey, 2011; van der Kleij et al, 2015)
For the item format we did not find any significant correlations between proportion BE (pBE) and proportion correct (pC)
The purpose of the present study was to explore the relationship between the diagnostic capacity and item difficulty of items and to evaluate the diagnostic capacity of three specific item characteristics: Item format, answering format, and number features in relation to their capacity to elicit bridging errors
Summary
Diagnostic items can be designed to collect specific and fine-grained information about students’ cognitive strengths and weaknesses (Leighton and Gierl, 2007; Keeley and Tobey, 2011; van der Kleij et al, 2015). It is widely acknowledged that subtraction errors are indicative of students’ conceptual and procedural understanding of mathematics (Resnick, 1984; Smith et al, 1994; Fuson et al, 1997; Ashlock, 2006; Rittle-Johnson, 2017). Not every subtraction error is indicative of students’ conceptual and procedural (mis)understanding; some errors are caused due to slips in attention or insufficient number fact knowledge (Hennessy, 1993). Students who make smallerfrom-larger errors solve the problem 76−48 = as follows: 70−40 = 30, 6−8 is reversed to “smaller-from-larger”: 8−6 = 2, 30 + 2 = 32. In this paper, such errors are called bridging errors (BE). Bridging errors can only be made when subtraction items require borrowing
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