The mathematical flexibility of college students: The role of cognitive and affective factors

Abstract

Background. Traditional math instruction that emphasizes procedures and rote memorization is common in math classes, particularly within the United States. Students may be able to perform steps and recite information, but flexible thinking in math is also an important ability. Lay theories assume that extensive experience in math would lead to increased flexibility, but some research has posited a change-resistant account, which argues that experience with traditional instruction may make it difficult to think flexibly about even simple concepts. Aims. The current study explored the mathematical flexibility of college students who completed their K-12 education in the United States, and investigated how affective and cognitive factors contributed to flexible thinking. Sample. Participants were 128 undergraduate students at a competitive U.S. public university. Method. Mathematical flexibility was measured through a novel task that asked participants to generate as many strategies as they could for a simple arithmetic problem. These strategies were coded to create scores of fluency (number of strategies) and flexibility (number of unique strategies). Results and Conclusions. On average, participants were only able to provide little more than three unique strategies beyond the primary strategy taught in K-12 classrooms. Measures of math anxiety, math identity, need for cognition, and working memory were all unrelated to flexibility. However, student perceptions of how many possible solutions exist were significantly related to flexibility. These results provide evidence for a changeresistance account and provide further evidence that math flexibility is a unique construct.

Exploring students’ procedural flexibility in three countries

“…As noted in a position paper on procedural fluency from the National Council of Teachers of Mathematics (NCTM, 2014): "All students need to have a deep and flexible knowledge of a variety of procedures, along with an ability to make critical judgments about which procedures or strategies are appropriate for use in particular situations. " Researchers have begun to investigate procedural flexibility, in mathematical domains including arithmetic (Blöte et al, 2001;Shaw et al, 2020;Torbeyns et al, 2009), computational estimation , algebra (e.g., Rittle-Johnson & Star, 2007), linear algebra (Maciejewski & Star, 2019) and calculus (Maciejewski & Star, 2016); and among Star et al International Journal of STEM Education 2022, 9(1):4 American (Rittle-Johnson & Star, 2009) and international Joglar et al, 2018;Xu et al, 2017) students. 1 Within this literature, procedural flexibility is defined as knowledge of multiple strategies and the ability to select the most appropriate strategy for a given problem and problem-solving circumstances (e.g., Star, 2005).…”

Investigating algorithm-oriented flexibility and structure-informed flexibility in mathematics learning

“…Flexibility in arithmetic, linear algebra, calculus, and other domains of mathematics has been investigated in past studies (Maciejewski & Star, 2016; Maciejewski & Star, 2019; Shaw et al, 2020; Star & Rittle-Johnson, 2008). Additional research studies have investigated constructs closely related to flexibility, including strategy choice and adaptability (DeCaro, 2016; Liu et al, 2018; Star et al, 2015; Star et al, 2022; Star & Rittle-Johnson, 2009; Wang et al, 2019).…”

Which One Is the “Best”: a Cross-national Comparative Study of Students’ Strategy Evaluation in Equation Solving

“…As such, it has been emphasized in many policy documents across the world (Australian Education Ministers, 2006;National Mathematics Advisory Panel, 2008;Singapore Ministry of Education, 2006;Woodward et al, 2012). Perhaps as a result, in the domain of algebraic problem-solving, studies have focused on measuring strategy flexibility (Newton et al, 2019;Star & Seifert, 2006;Xu et al, 2017), understanding important factors that could affect strategy flexibility (DeCaro, 2016;Jiang et al, 2021;Keleş & Yazgan, 2021;Ramirez et al, 2016;Shaw et al, 2020;Threlfall, 2009;Wang et al, 2019), and how instructional interventions could facilitate strategy flexibility (De Smedt et al, 2010;Nistal et al, 2014;Star & Rittle-Johnson, 2008;Star et al, 2015a, b).…”