The Development of Mathematical Reasoning

Abstract

Psychological research on the development of children's mathematical reasoning has focused either on their understanding of quantities or on their knowledge of number. A synthesis between these two different kinds of theory can be achieved by acknowledging that numbers have two meanings: a representational meaning, defined by their use as signs for quantities or relations between quantities, and an analytical meaning, defined by the conventions in the number system. In the introduction, this chapter introduces these two meanings of numbers and explores the vital connections between them. The subsequent sections analyze how children's mathematical knowledge develops by their increasing ability to use different numerical representations (e.g., from the use of fingers to represent quantities to the use of conventional signs), by their growing understanding of invariant relations between quantities (e.g., realizing that, given a fixed number of cookies, the more people sharing the cookies, the less each one receives) and by their increasing awareness of the relevance of specific concepts to different situations (e.g., understanding the relevance of division to the solution of situations more readily connected to multiplication). Throughout the chapter, the connection between the nature of quantities and their numerical representations is explored. The final substantive section focuses on the use of numbers to quantify space and relations between spatial dimensions, and argues that understanding relations between different dimensions in space (e.g., length, width, and area) is crucial to quantifying space. The chapter ends with a brief discussion of directions for future research.

The importance of additive reasoning in children’s mathematical achievement: A longitudinal study.

“…The second view of children’s mathematics learning, which we term as mathematical thinking perspective , focuses on how children think about mathematics logically (e.g., Bryant, 1995; Carpenter & Moser, 1982; Ginsburg, Klein, & Starkey, 1998; Nunes & Bryant, 1996, 2015; Nunes, Bryant, Barros, & Sylva, 2012; Piaget, 1952; Piaget & Inhelder, 1975; Thompson, 1993, 1994; Vergnaud, 1997, 2009). Mathematical thinking involves the understanding of the meanings of number.…”

“…According to this view, numbers are not simply a series of words in a constant order, but they also reflect the part–whole logic of the number system—each number words encompasses the previous ones additively (8 means 7 + 1, 6 + 2, 5 + 3, etc.). Nunes and Bryant (2015) call this idea the “analytical meanings of number” because the meaning is given by definitions within a number system.…”

Teacher-Implemented Modified Schema-Based Instruction with Middle-Grade Students with Autism and Intellectual Disability

“…To solve mathematics problems, students must be able to not only complete the arithmetic procedures but also reason about the relationships between quantities (Nunes et al, 2015). The level of difficulty of a mathematical problem is not determined by the arithmetic required to arrive at a solution, but rather the quantitative reasoning necessary to model the problem and select appropriate arithmetic procedures (Nunes & Bryant, 2015). Quantitative reasoning emphasizes relationships between quantities (Thompson, 1993), which are either additive (i.e., quantities connected with part–part–whole relations) or multiplicative (i.e., quantities connected by one to many or ratios; Nunes & Bryant, 2015).…”

“…The level of difficulty of a mathematical problem is not determined by the arithmetic required to arrive at a solution, but rather the quantitative reasoning necessary to model the problem and select appropriate arithmetic procedures (Nunes & Bryant, 2015). Quantitative reasoning emphasizes relationships between quantities (Thompson, 1993), which are either additive (i.e., quantities connected with part–part–whole relations) or multiplicative (i.e., quantities connected by one to many or ratios; Nunes & Bryant, 2015). Additive problem structures can be categorized as combine (i.e., total or group; parts combined for a sum), compare (i.e., difference; sets compared for a difference), or change (i.e., join or separate; amount increases or decreases).…”

Korai numerikus készségek online mérése

“…Legyen szó a hétköznapi élet olyan különféle helyzeteiről, mint a vásárlás, pénzügyek intézése, legyen szó a munka világának egyszerűbb és bonyolultabb problémahelyzeteiről, mindenki találkozik matematikával. Egyre kevesebb olyan szakma van, ahol ne kerülnének elő matematikai tartalmak, vagy ne lenne szükség olyan gondolkodási képességekre, amelyek a matematika által kiválóan fejleszthetők (Nunes & Csapó, 2011). Eközben hazánkban a PISA-mérések eredményeiből világossá vált, hogy komoly problémák vannak a magyar tanulók matematikai műveltségével, az átlageredmények csökkenése mellett riasztó mértékben növekedett a leszakadók aránya (Csapó, Fejes, Kinyó, & Tóth, 2014).…”