University students' misconceptions about rational numbers: Implications for developmental mathematics and instruction of younger students

Abstract

To understand misconceptions with rational numbers (i.e., fractions, decimals, and percentages), we administered an assessment of rational numbers to 331 undergraduate students from a 4-year university. The assessment included 41 items categorized as measuring foundational understanding, calculations, or word problems. We coded each student's response and identified error patterns for items answered incorrectly. Students attempted foundational understanding and calculations problems more often than word problems, and students made fewer errors with foundational understanding and calculation items. Students demonstrated the most unique errors with calculations and word problems items. Given all items on the rational numbers assessment required elementary or middle school knowledge of rational numbers, the number and diversity of errors with a sample of university students demonstrates the persistent difficulties with rational numbers that students carry into adulthood. Elementary, secondary, and developmental mathematics educators should be aware of such errors and provide specific instruction on avoiding such errors.

The hierarchical relations among mathematical competencies: From fundamental numeracy to complex mathematical skills.

“…Students may have difficulty in learning fraction knowledge because they need to restructure their whole number understanding to accommodate new information about rational numbers (see reviews by Ni & Zhou, 2005; Siegler & Lortie-Forgues, 2017). Misconceptions about fractions can arise even when whole number knowledge is sufficient because principles and rules that apply to natural numbers are different from those that apply to fractions (Powell & Nelson, 2021; Stafylidou & Vosniadou, 2004; Vosniadou et al, 2008). Ultimately, integrating whole number and rational number skills will support students’ learning of more advanced mathematical constructs, such as algebra (Barbieri et al, 2021; Booth & Newton, 2012).…”

“…For example, students who struggled when they were learning basic arithmetic will continue to face challenges when learning more complex mathematics (Empson et al, 2011). Research shows that some adults do not master arithmetic (LeFevre et al, 2017;Thompson et al, 2015) or rational number skills (Powell & Nelson, 2021), despite the importance of an understanding of the conceptual continuity from whole number arithmetic to rational numbers (Siegler, 2016;Siegler & Braithwaite, 2017). The fundamental properties of arithmetic operations and fractions form the foundations of algebra (Empson et al, 2011;Hackenberg, 2013;Mack, 1993;Nunes et al, 2016;Steffe & Olive, 2010;Thompson & Saldanha, 2003) because algebra is a more abstract, generalized version of arithmetic.…”

“…Although these skills are all introduced to students before high school and the hierarchy is developed through years of experience with mathematics, there is evidence to suggest that the hierarchy is preserved among skilled adults. For example, university students show considerable variability in their mastery at all levels: cardinal (Maloney et al, 2011), ordinal (Lyons & Beilock, 2011), basic arithmetic (LeFevre et al, 2015), fractions (Powell & Nelson, 2021), and algebra (Cirino et al, 2016; Douglas et al, 2020). Thus, individual differences at each level of the hierarchy and the use of foundational skills, such as arithmetic, to solve more complex mathematical problems lead us to believe that the hierarchy is maintained in adulthood.…”

Students’ Algebraic Proficiency and Attitude towards Learning Algebra

“…Malaysian students' poor performance in algebra has been a growing concern (Teoh et al, 2020;Ying et al, 2020), but not unsurprising as research has shown that the learning of algebra is considered challenging among many students in many learning environments, even among students with positive attitudes (Poçan et al, 2023). Students' problems in algebra, such as rational numbers, arithmetic, and word problems need to be resolved as these concepts are fundamental to university mathematics (Powell & Nelson, 2021). Hence, secondary school students' algebra proficiency needs to be carefully monitored and developed (Anderson, 2021) to equip them with mathematical skills for advanced learning at tertiary level, in particular among those planning to pursue STEM careers.…”

Myths in psychology: psychological misconceptions among Spanish psychology students

“…However, the study of myths is not limited to behavioral science. The prevalence of myths has been studied in fields such as Education ( Ferrero, Garaizar & Vadillo, 2016 ), Medicine ( Kaufman et al, 2013 ) or public health policies ( Viehbeck, Petticrew & Cummins, 2015 ), as well as in science as a knowledge system ( McComas, 1996 ), and the practical consequences that they entail, such as those caused by the myths of mathematics in their students ( Powell & Nelson, 2021 ).…”