The contribution of logical reasoning to the learning of mathematics in primary school

Abstract

It has often been claimed that children's mathematical understanding is based on their ability to reason logically, but there is no good evidence for this causal link. We tested the causal hypothesis about logic and mathematical development in two related studies. In a longitudinal study, we showed that (a) 6-year-old children's logical abilities and their working memory predict mathematical achievement 16 months later; and (b) logical scores continued to predict mathematical levels after controls for working memory, whereas working memory scores failed to predict the same measure after controls for differences in logical ability. In our second study, we trained a group of children in logical reasoning and found that they made more progress in mathematics than a control group who were not given this training. These studies establish a causal link between logical reasoning and mathematical learning. Much of children's mathematical knowledge is based on their understanding of its underlying logic.Logical relations lie at the heart of many fields of inquiry. Think of the relation known as transitivity, for example, if A ¼ B and B ¼ C, then A ¼ C. This relation is pertinent to the number of objects in a set, to length, to volume, to colour, to shape, to people's intelligence, to the matching of photographs of fingerprints, to the taste of orange juice etc. Transitivity is significant in the domains of number and measurement, but it is neither number nor measurement. Not all the relations are transitive: if A is the father of B and B is the father of C, it does not follow that A is the father of C. For this reason, Piaget (1950) argued that the pertinence of a logical relation is given meaning in a domain through experience.A similar argument was advanced by Simon and Klahr (1995;Klahr, 1982) about conservation. Transformations, they argued, have different effects on different quantities, and children learn about these differences by experience. If we add a jar of water at 208C to another jar of water also at 208C, the quantity of water (the * Correspondence should be addressed to Terezinha Nunes, 15 Norham Gardens, Oxford OX2 6PY, UK Reproduction in any form (including the internet) is prohibited without prior permission from the Society extensive quantity) increases, but the temperature (an intensive quantity) stays the same. Children must learn the domains where different logical relations (or axioms) apply.Children almost certainly need to understand the logical relations between quantities in order to learn how to represent numbers and arithmetic. These relations are not the same as arithmetic or the numeration system, but are relevant to them. One such relation is correspondence: if two sets contain the same number of objects, then the objects in one set are in one-to-one correspondence with those in the other. If set B is in two-to-one correspondence with set A, and C is in two-to-one correspondence with A, then B and C are equivalent.

Language in the prediction of arithmetics in kindergarten and grade 1

“…In contrast, classification is the ability to sort objects based on their similarities in one or more dimensions. These logical thinking kills have been suggested as key precursors for arithmetical achievement (Nunes et al, 2006).…”

“…Although Neo-Piagetian researchers questioned the causality of seriation and classification for understanding number (e.g., Grégoire, 2005;Lourenço & Machado, 1996), even after controlling for differences in working memory, logical thinking skills in six year-old children remain a strong predictor for arithmetic abilities 16 months later (Nunes et al, 2006).…”

“…Moreover, 87% of the children with mathematical learning disabilities in grade 2 (at age 7 to 8) can be correctly diagnosed in kindergarten by a combination of counting and magnitude estimation tasks (Stock, Desoete, & Roeyers, 2010). In addition to numerical abilities, the value of including logical thinking abilities (e.g., Nunes et al, 2006;) and language as predictors for arithmetic has been stressed (e.g., Purpura, Hume, Sims, & Lonigan, 2011;Vukovic & Lesaux, 2013). Surprisingly few studies have been conducted to explore the combined effect of these predictors on arithmetic in kindergarten and the primary grades.…”

Can parents’ homework assigned compensate for disadvantaged students’ learning achievement in Mainland China?

“…And, the test covered a range of mathematical building blocks: number reading skills, mathematical lexicon, knowledge, procedural knowledge, linguistic skills, mental representation, contextual skills, selecting relevant information, number sense skill, memory skills, visualization or mental representation skills and logical thinking (Desoete & Roeyers, 2005;Nunes et al, 2007;Zimmermann & Cunningham, 1991). The homogeneous of the test was attained by the Item Response Theory.…”

Promovendo a compreensão da composição aditiva em crianças surdas

“…Usando um estudo longitudinal, Nunes et al (2007) mostraram Disponível em <http://www.cedes.unicamp.br> Promovendo a compreensão da composição aditiva em crianças surdas que a composição aditiva avaliada no primeiro ano escolar é um bom preditor da aprendizagem matemática das crianças ouvintes no fi m do segundo ano escolar. Em um segundo estudo, os mesmos pesquisadores mostraram que uma intervenção que promova o progresso das crianças na composição aditiva tem um impacto positivo sobre o seu aprendizado de matemática.…”