Profiles in understanding operations with rational numbers

ABSTRACT

Students often show difficulties in understanding rational numbers. Often, these are related to the natural number bias, that is, the tendency to apply the properties of natural numbers to rational number tasks. Although this phenomenon has received a lot of research interest over the last two decades, research on the existence of qualitatively different profiles regarding students’ understanding is scarce. The current study investigated the different ways students reasoned in arithmetic operation items with fractions and decimals. A cross-sectional study with 1,262 participants from 5^th to 10^th grade was performed. A TwoStep Cluster Analysis revealed eight different student reasoning profiles. We found that the natural number bias is first overcome in addition and subtraction, and later in multiplication and division. Moreover, differences regarding representation were only found in addition and subtraction items, indicating that natural numbers interfered more strongly in fractions than in decimal numbers. Finally, results showed that some students’ difficulties with rational number multiplications and divisions had other explanations than the natural number bias.

KEYWORDS:

• Rational numbers
• fractions
• decimal numbers
• natural number bias
• arithmetic operations
• profiles

Additional information

Funding

This research was carried out with the support of the project PROMETEO/2017/135 from Conselleria d’Educació, Investigació, Cultura i Esport (Generalitat Valenciana, Spain), and the predoctoral grant from the University of Alicante (UAFPU2018-035), and the postdoctoral grant (I-PI 69-20).

Notes on contributors

Juan Manuel González-Forte

Juan Manuel González-Forte

I am a Doctor in Mathematics Education at the Department of Innovation and Teaching Methods, University of Alicante, Spain. My major research interest is mathematical teaching and learning in the domain of rational numbers. Particularly, my research focuses on learners’ understanding of the rational number domain and the reasons learners struggle with the understanding of fractions. I am main or coauthor of some publications in scientific journals and conference papers.

Ceneida Fernández

Ceneida Fernández

I am an Associate Professor in Mathematics Education at the Department of Innovation and Teaching Methods, University of Alicante, Spain. My research focuses on both pre-service teachers’ learning and primary and secondary school students’ understanding of various mathematical topics. Concerning pre-service teachers’ learning, my interest resides in pre-service teachers’ means of developing professional competences and knowledge with regard to mathematics teaching. This research addresses various mathematical topics: proportional reasoning, fractions, geometry, limit and derivative. Regarding primary and secondary school students understanding, my research mainly focused on students’ understanding of the rational number domain and proportional reasoning.

Jo Van Hoof

Jo Van Hoof

I obtained the degree of Doctor in Educational Sciences at the University of Leuven, Belgium in May 2015 and currently hold a post-doctoral fellowship of the Research Foundation–Flanders (FWO). During my PhD, my research mainly focused on learners’ understanding of the rational number domain and the reasons why learners struggle to understand fractions. During my postdoc I extended my research focus to the role of intuitions and inhibition in mathematical reasoning. My major research interests are: the conceptual change approach to mathematics learning; higher order number sense; learners’ rational number understanding; intuitions, heuristics, and biases in human reasoning. I further have experiences with advanced statistical techniques such as multilevel modeling, latent transition analysis, and item response theory. I am main or coauthor of more than 30 publications in scientific journals and conferences papers, published in full in proceedings.

Wim Van Dooren

Wim Van Dooren

I am associate professor at the Centre for Instructional Psychology and Technology at the University of Leuven. Most of my research is about mathematical teaching and learning, in a broad age range (from Kindergartners over primary and secondary school students to mathematical experts). This research addresses various mathematical topics: word problems, algebra, proportions, fractions, geometry, spatial thinking, probability … I also do research on statistical reasoning and the difficulties that occur in interpreting graphs and the outcomes of statistical analyses. I am also involved in research on the link between physics and mathematics, as well as learners’ understanding of astronomical concepts. My focus is usually situated at the intersection of the content-specific domain, educational psychology and cognitive psychology. For example, I study the (positive and negative) role of intuitions in learning and reasoning on mathematical concepts. One point of attention in that research is how very young children can already reason about complex mathematical ideas if they are challenged. Another focus is on the role and the origins of specific misconceptions in learners, as they can have a strong impact on later learning. I also investigate the role external representations (drawings, schemes, graphs) can have in reasoning on certain concepts.