Do we have a sense for irrational numbers?

Abstract

Number sense requires, at least, an ability to assess magnitude information represented by number symbols. Most educated adults are able to assess magnitude information of rational numbers fairly quickly, including whole numbers and fractions. It is to date unclear whether educated adults without training are able to assess magnitudes of irrational numbers, such as the cube root of 41. In a computerized experiment, we asked mathematically skilled adults to repeatedly choose the larger of two irrational numbers as quickly as possible. Participants were highly accurate on problems in which reasoning about the exact or approximate value of the irrational numbers' whole number components (e.g., 3 and 41 in the cube root of 41) yielded the correct response. However, they performed at random chance level when these strategies were invalid and the problem required reasoning about the irrational number magnitudes as a whole. Response times suggested that participants hardly even tried to assess magnitudes of the irrational numbers as a whole, and if they did, were largely unsuccessful. We conclude that even mathematically skilled adults struggle with quickly assessing magnitudes of irrational numbers in their symbolic notation. Without practice, number sense seems to be restricted to rational numbers. Number sense refers to the adaptive and flexible use of numbers. A precondition of number sense is the ability to quickly assess magnitude information represented by number symbols (e.g., 12, -5, or 3/4). Curricula and educational standards for school mathematics (e.g., CCSSI, 2010) include an understanding of number magnitudes as an important goal of school mathematical learning. Furthermore, theories from cognitive psychology consider understanding of number magnitudes as a fundamental step of numerical development (Siegler, Thompson, & Schneider, 2011; see Siegler, Fazio, Bailey, & Zhou, 2013). The reason why understanding number magnitudes is considered so important is that possessing magnitudes is a shared property of all types of numbers which students learn about at school, whether natural, whole, rational, or real numbers. i Thus, magnitudes are a key feature of all real numbers.In spite of the widely accepted importance of number magnitude understanding, research into magnitude understanding has almost exclusively focused on natural numbers (i.e., the counting numbers) (e.g., De Smedt, Verschaffel, & Ghesquière, 2009), integers (including negative numbers, e.g., Varma & Schwartz, 2011;Young & Booth, 2015), and rational numbers in decimal notation (e.g., 0.2) (Desmet, Grégoire, & Mussolin, 2010) or fraction notation (e.g., 1/5) (Van Hoof, Lijnen, Verschaffel, & Van Dooren, 2013). Studies have amply shown Journal of Numerical Cognition jnc.psychopen.eu | 2363-8761 that most people are able to assess magnitudes of these numbers, albeit with more difficulty and less automatically for larger and less common numbers (e.g., fractions with two-digit components such as 31/73) than for smaller numbers and numbers t...

How the Abstract Becomes Concrete: Irrational Numbers Are Understood Relative to Natural Numbers and Perfect Squares

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Unifying evolutionary dynamics: a set theory exploration of symmetry and interaction

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How the Abstract Becomes Concrete: Irrational Numbers are Understood Relative to Natural Numbers and Perfect Squares

“…Just one other study has explored whether the same human "number sense" that allows us to compare the magnitudes of natural numbers, integers, and rational numbers extends to algebraic irrationals (Obersteiner & Hofreiter, 2017). Participants performed magnitude comparisons with irrationals of the form √ .…”