A Longitudinal Study Investigating Natural Number Based Reasoning in Rational Number Tasks

• In Event: Connecting Fractions and Decimals

Abstract

There is a broad agreement in the literature that a good understanding of the rational number domain is of crucial importance for the learning of more advanced topics in mathematics, such as algebra (e.g., Siegler et al., 2012). It is therefore worrying that rational numbers form a stumbling block for many learners, a phenomenon that is often attributed to the natural number bias, which is the tendency to apply natural number characteristics to rational numbers, even when this is inappropriate. For example, learners erroneously think that 3/5 is smaller than 4/9, just like 3 is smaller than 4 and 5 is smaller than 9. While the natural number bias is a well-studied phenomenon, longitudinal studies investigating learners' development from a natural-number-based towards the scientifically correct concept of a rational number are scarce.

Therefore, the present study longitudinally followed 113 fourth and 88 fifth graders. Learners' rational number knowledge was measured three times over the course of two school years by means of the Rational Number Knowledge Test. This test includes items of the three main aspects on which natural numbers differ from rational numbers: their numerical size, their dense structure, and the effect of arithmetical operations (e.g., Vamvakoussi, 2015). Within each aspect, items with decimals and fractions were included.

Using latent transition analysis, six learner profiles were found allowing to distinguish learners with a naïve, partial, or good understanding of the three aspects (see Figure 2.1). Interestingly, in every profile and in all three aspects learners scored better on the decimal than on the fraction tasks.

Moreover, learners' transition probabilities (Table 2.1) showed that understanding the size of rational numbers is a prerequisite for understanding operations with rational numbers. Further, only a limited number of learners fully understand the dense structure of rational numbers at the end of elementary education. Results further showed that the 'Scientifically correct' profile is more or less stable across the three time points, suggesting that once learners have developed a good understanding of all three aspects, they do not often regress to a 'lower' profile.

The present study indicates that while individual differences are present in the development of rational number understanding, general developmental paths can be described, which are helpful to provide instruction adapted to the specific knowledge and needs of learners.

More specifically, an important first step in learners' rational number understanding is a good understanding of the size of rational numbers (first decimals and later fractions). Therefore, we suggest that a good understanding of the size of rational numbers should be obtained before instruction focuses on more advanced content such as operations with rational numbers.

Authors

• Jo Van Hoof, Katholieke Universiteit Leuven (KU Leuven)
  Presenting Author

• Tine Degrande, KU Leuven
  Non-Presenting Author

• Eva Ceulemans, KU Leuven
  Non-Presenting Author

• Lieven Verschaffel, KU Leuven
  Non-Presenting Author

• Wim Van Dooren, KU Leuven
  Non-Presenting Author