Children display an early sensitivity to implicit proportions (e.g., 1 of 5 apples vs. 3 of 4 apples), but have considerable difficulty in learning the explicit, symbolic proportions denoted by fractions (e.g., "1/5" vs. "3/4"). Theoretically, reducing the gap between representations of implicit versus explicit proportions…

Analogy is a powerful learning mechanism for children to learn novel, abstract concepts from only limited input, yet also requires cognitive supports. My dissertation sought to propose and examine number lines as a mathematical schema of the number system to facilitate both the development of rational number understanding and analogical reasoning.…

Reasoning about fraction magnitude is an important topic in elementary mathematics because it lays the foundations for meaningful reasoning about fraction operations. Much of the research literature has reported deficits in preservice elementary teachers' (PSTs) knowledge of fractions and has given little attention to the productive resources that…

In grade 3, students are expected to be able to represent a fraction on a number line by first identifying the interval from 0 to 1 as the unit and by partitioning the unit as needed on the basis of the denominator. This task extends these grade 3 fraction goals stated in the Common Core State Standards for Mathematics (NGA Center and CCSSO 2010)…

Even though algebraic ideas are addressed across a number of grades, algebra continues to serve as a gatekeeper to upper mathematics and degree attainment because of the high percentage of students that fail algebra classes and become halted in their educational progress. One reason for this is students not having the opportunity to build on their…

Children's understanding of fractions, including their symbols, concepts, and arithmetic procedures, is an important facet of both developmental research on mathematics cognition and mathematics education. Research on infants', children's, and adults' fraction and ratio reasoning allows us to test a range of proposals about the development of…

Background: While it has been widely demonstrated that children's and adolescents' understanding of rational number plays an important role in their mathematics achievement, we have limited knowledge about the cognitive correlates of this understanding. Aims: The current study aimed at examining whether children's non-symbolic ratio processing and…

The fifth-grade curriculum presents many opportunities to use students' prior mathematical knowledge as a way to bridge new and more difficult mathematical ideas. In this article, the authors document an area tiling task given to fifth-grade students to connect aspects of area measurement covered in earlier grades to grade-level standards such as:…

This article introduces the formative assessment probe--a powerful tool for collecting focused, actionable information about student thinking and potential misconceptions--along with a process for targeting instruction in response to probe results. Drawing on research about common student mathematical misconceptions as well as the former work of…

Background: The learning of rational numbers is a complex and difficult process that begins in the early grades. This teaching often focuses on the mastery of essential knowledge, including particular skills (e.g. using fractions to describe part--whole diagrams) and interpretations (e.g. sharing), which often results in an incomplete and…

The author wants her students to see any new mathematics--fractions, negative numbers, algebra--as logical extensions of what they already know. This article describes two students' efforts to make sense of their conflicting interpretations of 1/2 × -6, both of which were compelling and logical to them. It describes how discussion, constructing…

The use of enabling and extending prompts allows tasks to be both accessible and challenging within a classroom. This article provides an example of how to use enabling and extending prompts effectively when employing a challenging task in Year 2.

The application to rational numbers of the procedures and intuitions proper of natural numbers is known as Natural Number Bias. Research on the cognitive foundations of this bias suggests that it stems not from a lack of understanding of rational numbers, but from the way the human mind represents them. In this work, we presented a fraction…

The aim of this study was to propose and validate a structural model in fraction and decimal number addition, which is founded primarily on a synthesis of major theoretical approaches in the field of representations in Mathematics and also on previous research on the learning of fractions and decimals. The study was conducted among 1,701 primary…

The purpose of the present review was to evaluate the quality of the research and evidence base for representation of problems as a strategy to enhance the mathematical performance of students with learning disabilities and those at risk for mathematics difficulties. The authors evaluated 25 experimental and quasiexperimental studies according to…