The Integer Test of Primary Operations: A Practical and Validated Assessment of Middle School Students’ Calculations with Negative Numbers

Abstract

Skills and understanding of operations with negative numbers, which are typically taught in middle school, are crucial aspects of numerical competence necessary for all subsequent mathematics. To more swiftly and coherently develop the field's understanding of how to foster this critical competence, we need shared measures that allow us to compare results across studies with diverse populations and theoretical perspectives. Yet, to date no validated instrument exists to assess all four primary operations (addition, subtraction, multiplication and division). Thus, we conducted a Rasch analysis of the Integer Test of Primary Operations (ITPO) with 187 middle school students to provide a valid and reliable assessment with good person and item fit. The implications of this study are numerous for multiple stakeholders including scholars, test and textbook developers, as well as teachers. First, we validated three forms of the ITPO to foster future longitudinal studies of how integer arithmetic knowledge is maintained or decays as well as how such knowledge might be related to success in STEM disciplines. Second, our analysis provides trustworthy insights about relative difficulty of integer problem structures because regardless of test form similar problem structures loaded together. For instance, sums of additive inverses were the easiest structure, whereas division by -1 was more difficult than multiplying or dividing by any other integer. We discuss each of these and other findings that have practical implications for learning and teaching integers. Third, for broader mathematics assessments in which minimal items can be included to measure integer knowledge, this study informs which items would serve the intended assessment purpose. Finally, we provide the three forms as an appendix in printable formats to ensure these validated tests are practical to implement for teachers as well as scholars.

Gameplay in Perspective: Applications of a Conceptual Framework to Analyze Features of Mathematics Classroom Games in Consideration of Students’ Experiences

“…For instance, student accuracy ranged from about 33% to 67% on integer subtraction items depending on item structure (Nurnberger-Haag, Kratky, & Karpinski, 2022, p.11;Periasamy & Zaman, 2009, p.366;Ryan & Williams, 2007, p.218). A Rasch analysis of the Integer Test of Primary Operations confirmed that integer subtraction overall was the most difficult operation for middle grade students (Nurnberger-Haag et al, 2022). Subtraction problem structures that result in a positive solution are the most difficult (Nurnberger-Haag et al, 2022).…”

“…A Rasch analysis of the Integer Test of Primary Operations confirmed that integer subtraction overall was the most difficult operation for middle grade students (Nurnberger-Haag et al, 2022). Subtraction problem structures that result in a positive solution are the most difficult (Nurnberger-Haag et al, 2022). Integer subtraction is particularly difficult, because in early elementary school students' extensive experience with subtraction always decreases quantities, which makes it counter-intuitive that subtracting a negative integer increases the value (Periasamy and Zaman, 2009;Ryan & Williams, 2007).…”

“…Integer subtraction is particularly difficult, because in early elementary school students' extensive experience with subtraction always decreases quantities, which makes it counter-intuitive that subtracting a negative integer increases the value (Periasamy and Zaman, 2009;Ryan & Williams, 2007). Integer multiplication and division are also counterintuitive, because products or quotients of two negative numbers yield a positive number (Nurnberger-Haag, et al, 2022). Although consistent rules can be memorized for multiplication and division of integers, only about half of grade 8 students accurately answered these on a standardized test (Ryan & Williams, 2007, p. 218).…”

“…Moreover, Nurnberger-Haag and colleagues (2022) found that regardless of whether the dividend was positive or negative, division by -1 was most difficult. This difficulty is problematic given that students must have this prerequisite knowledge to accurately factor while solving algebraic equations (Nurnberger-Haag, et al, 2022). Due to these conceptual difficulties with integer arithmetic, teachers and researchers continuously seek and design new ways to facilitate student learning as well as engage and motivate students to learn this difficult topic.…”

“…Thus, a limitation of this study that provides future research opportunities is to use these findings to design studies that could isolate integer learning gains due to games. To design pre-post-delayed post studies of integer learning, scholars could use the three forms of the Integer Test of Primary Operations, which was determined to be a valid and reliable measure of middle school students' integer addition, subtraction, multiplication and division knowledge (Nurnberger-Haag, Kratky, & Karpinski, 2022). A limitation of the study was that it was not meant to isolate learning due to these games from the other learning opportunities the classroom teacher provided during the unit.…”