Natural Number Knowledge Research Articles

Predicting children's emerging understanding of numbers.

How do children construct a concept of natural numbers? Past research addressing this question has mainly focused on understanding how children come to acquire the cardinality principle. However, at that point children already understand the first number words and have a rudimentary natural number concept in place. The question therefore remains; what gets children's number learning off the ground? We therefore, based on previous empirical and theoretical work, tested which factors predict the first stages of children's natural number understanding. We assessed if children's expressive vocabulary, visuospatial working memory, and ANS (Approximate number system) acuity at 18 months of age could predict their natural number knowledge at 2.5 years of age. We found that early expressive vocabulary and visuospatial working memory were important for later number knowledge. The results of the current study add to a growing body of literature showing the importance of language in children's learning about numbers.

Number concepts: abstract and embodied.

Numerical knowledge, including number concepts and arithmetic procedures, seems to be a clear-cut case for abstract symbol manipulation. Yet, evidence from perceptual and motor behaviour reveals that natural number knowledge and simple arithmetic also remain closely associated with modal experiences. Following a review of behavioural, animal and neuroscience studies of number processing, we propose a revised understanding of psychological number concepts as grounded in physical constraints, embodied in experience and situated through task-specific intentions. The idea that number concepts occupy a range of positions on the continuum between abstract and modal conceptual knowledge also accounts for systematic heuristics and biases in mental arithmetic, thus inviting psycho-logical approaches to the study of the mathematical mind.This article is part of the theme issue 'Varieties of abstract concepts: development, use and representation in the brain'.

Bridging psychological and educational research on rational number knowledge

In this paper we focus on the development of rational number knowledge and present three research programs that illustrate the possibility of bridging research between the fields of cognitive developmental psychology and mathematics education. The first is a research program theoretically grounded in the framework theory approach to conceptual change. This program focuses on the interference of prior natural number knowledge in the development of rational number learning. The other two are the research program by Moss and colleagues that uses Case’s theory of cognitive development to develop and test a curriculum for learning fractions, and the research program by Siegler and colleagues, who attempt to formulate an integrated theory of numerical development. We will discuss the similarities and differences between these approaches as a means of identifying potential meeting points between psychological and educational research on numerical cognition and in an effort to bridge research between the two fields for the benefit of rational number instruction.

The natural number bias and its role in rational number understanding in children with dyscalculia. Delay or deficit?

BackgroundPrevious research indicated that in several cases learners’ errors on rational number tasks can be attributed to learners’ tendency to (wrongly) apply natural number properties. There exists a large body of literature both on learners’ struggle with understanding the rational number system and on the role of the natural number bias in this struggle. However, little is known about this phenomenon in learners with dyscalculia. AimsWe investigated the rational number understanding of learners with dyscalculia and compared it with the rational number understanding of learners without dyscalculia. MethodThree groups of learners were included: sixth graders with dyscalculia, a chronological age match group, and an ability match group. ResultsThe results showed that the rational number understanding of learners with dyscalculia is significantly lower than that of typically developing peers, but not significantly different from younger learners, even after statistically controlling for mathematics achievement. ConclusionNext to a delay in their mathematics achievement, learners with dyscalculia seem to have an extra delay in their rational number understanding, compared with peers. This is especially the case in those rational number tasks where one has to inhibit natural number knowledge to come to the right answer.

Number sense in the transition from natural to rational numbers.

Rational numbers are of critical importance both in mathematics and in other fields of science. However, they form a stumbling block for learners. One widely known source of the difficulty learners have with rational numbers is the natural number bias, that is the tendency to (inappropriately) apply natural number properties in rational number tasks. Still, it has been shown that a good understanding of natural numbers is highly predictive for mathematics achievement in general, and for performance on rational number tasks in particular. In this study, we further investigated the relation between learners' natural and rational number knowledge, specifically in cases where a natural number bias may lead to errors. Participants were 140 sixth graders from six different primary schools. Participants completed a symbolic and a non-symbolic natural number comparison task, a number line estimation task, and a rational number sense test. Learners' natural number knowledge was found to be a good predictor of their rational number knowledge. However, after first controlling for learners' general mathematics achievement, their natural number knowledge only predicted the subaspect of operations with rational numbers. The results of this study suggest that the relation between learners' natural and rational number knowledge can largely be explained by their relation with learners' general mathematics achievement.

Who can escape the natural number bias in rational number tasks? A study involving students and experts.

Many learners have difficulties with rational number tasks because they persistently rely on their natural number knowledge, which is not always applicable. Studies show that such a natural number bias can mislead not only children but also educated adults. It is still unclear whether and under what conditions mathematical expertise enables people to be completely unaffected by such a bias on tasks in which people with less expertise are clearly biased. We compared the performance of eighth-grade students and expert mathematicians on the same set of algebraic expression problems that addressed the effect of arithmetic operations (multiplication and division). Using accuracy and response time measures, we found clear evidence for a natural number bias in students but no traces of a bias in experts. The data suggested that whereas students based their answers on their intuitions about natural numbers, expert mathematicians relied on their skilled intuitions about algebraic expressions. We conclude that it is possible for experts to be unaffected by the natural number bias on rational number tasks when they use strategies that do not involve natural numbers.

Natural number bias in operations with missing numbers

This study investigates the hypothesis that there is a natural number bias that influences how students understand the effects of arithmetical operations involving both Arabic numerals and numbers that are represented by symbols for missing numbers. It also investigates whether this bias correlates with other aspects of students’ understanding of the number concept beyond natural numbers. Natural number bias has been characterized as the interference of natural number knowledge in reasoning about non-natural numbers. Quantitative data is presented showing that in the case of operations between numbers and missing numbers this bias acts in two main ways. First, it shapes students’ anticipations about the expected outcome of each operation, that is, that the result of addition or multiplication “must” be bigger than the initial numbers and the result of subtraction or division “must” be smaller. Second, it causes students to think that missing numbers stand mostly for natural numbers; this tendency would lead students to make decisions about the general results of operations by substituting only natural numbers for the missing number symbols. It is argued that knowledge about operations between natural numbers needs to be inhibited for students to overcome the natural number bias and to reason with numbers beyond the scope of natural numbers.

Preschool spontaneous focusing on numerosity predicts rational number conceptual knowledge 6 years later

Recent evidence suggests that early natural number knowledge is a predictor of later rational number conceptual knowledge, even though students’ difficulties with rational numbers have also been explained by the overuse of natural number concepts—often referred to as the natural number bias. Hannula and Lehtinen (Learn Instr 15:237–256, 2005) have shown that children’s tendency to spontaneously focus on numerosity (SFON) predicts the development of natural number and arithmetic skills. The present study follows 36 children from the age of 6 years to the age of 12 years in order to determine how preschool SFON tendency and number sequence skills are related to rational number conceptual knowledge at the age of 12 years. The results show that children’s SFON tendency before school age is a strong predictor of later rational numbers conceptual knowledge, even after controlling for preschool number sequence skills. This finding has implications for the understanding of how the transition from reasoning about natural number concepts to reasoning about rational numbers may be influenced by children’s self-initiated practice with numbers in everyday situations.

Inhibiting natural knowledge in fourth graders: towards a comprehensive test instrument

While understanding rational numbers forms an essential part of mathematical literacy, research has repeatedly shown that they form a stumbling block in education. A big source of difficulty is the inappropriate application of natural number knowledge. The research literature points at three main aspects where natural number knowledge is inappropriately applied: density, operations, and size. While there are numerous test instruments to assess each of these aspects, none of them combines all aspects in an integrated manner. By creating such a comprehensive test instrument and applying it to fourth graders, we found that the strength of the natural number bias was largest in density items, but still clearly present in items about size and operations. We further found that fourth graders’ rational number reasoning—more specifically their ability to inhibit the inappropriate use of natural number knowledge—can be described by a one-dimensional construct.

Are secondary school students still hampered by the natural number bias? A reaction time study on fraction comparison tasks

Rational numbers and particularly fractions are difficult for students. It is often claimed that the ‘natural number bias’ underlies erroneous reasoning about rational numbers. This cross-sectional study investigated the natural number bias in first and fifth year secondary school students. Relying on dual process theory assumptions that differentiate between intuitive and analytic processes, we measured accuracies and reaction times on fraction comparison tasks. Half of the items were congruent (i.e., natural number knowledge leads to correct answers), the other half were incongruent (i.e., natural number knowledge leads to incorrect answers). Against expectations, students hardly made errors on incongruent items. Longer reaction times on correctly solved incongruent than on correctly solved congruent items indicated that students were indeed hampered by their prior knowledge about natural numbers, but could suppress their intuitive answers.

Comparing the magnitude of two fractions with common components: Which representations are used by 10- and 12-year-olds?

This study tested whether 10- and 12-year-olds who can correctly compare the magnitudes of fractions with common components access the magnitudes of the whole fractions rather than only compare the magnitudes of their components. Time for comparing two fractions was predicted by the numerical distance between the whole fractions, suggesting an access to their magnitude. In addition, we tested whether the relative magnitude of the denominator interferes with the processing of the fraction magnitude and, thus, needs to be inhibited. Response times were slower for fractions with common numerators than for fractions with common denominators, indicating an interference of the magnitude of the denominators with the selection of the larger fraction. A negative priming effect was shown for the comparison of natural numbers primed by fractions with common numerators, suggesting an inhibition of the selection of the larger denominator during the comparison of fractions. In conclusion, children who can correctly compare fractions with common components can access the magnitude of the whole fractions but remain sensitive to the interference of the relative magnitude of the denominators. This study highlights the fact that beyond the interference of natural number knowledge at the conceptual level (called the “whole number bias” by Ni & Zhou, 2005), children need to manage the interference of the magnitude of the denominators (Stroop-like effect).

A collective chain of signification in conceptualizing fractions: a case of a fourth-grade class

Fourth graders who participated in a yearlong teaching experiment constructed their own bridge between natural number knowledge and initial conceptualizations of fractions in discrete wholes. This bridge was established by a chain of signs individually and collectively generated. Students came to re-conceptualize natural number as “manifold of units” mediated by linguistic arithmetical expressions, double counting, and numerical diagrams. Linguistic arithmetical expressions, double counting, and numerical diagrams were powerful “representamens” dialectically determining and representing the inverse relations between iteration and decomposition and between part and whole in order to mediate the construction of quantitative fractional relations. These fourth graders’ transition from decomposition and iteration of natural number units to conceptualization of fractions in discrete wholes is analyzed using the Peircean notions of sign, chain of signification, and semiosis.