Over the last years, there is a growing interest in studying students' difficulties with rational numbers from a cognitive/developmental perspective, focusing on the role of prior knowledge in students' understanding of rational numbers. The present study tests the effect of the "whole" or "natural number bias" (i.e., the…

This study used units coordination as a theoretical lens to investigate how whole number and fraction reasoning may be related for preservice teachers at the conclusion of a math methods class. The study contributes quantitative evidence that units coordination provides a common foundation for both mathematical knowledge for teaching whole number…

The present study aimed to examine the opinions of in-and preservice middle school mathematics teachers about teaching four operations with integers through "number line" and/or "counters". Participants were 42 middle school mathematics teachers and 51 pre-service mathematics teachers. The data were collected through open-ended…

This article describes some of the essential mathematics that underpins the use of algorithms through a series of learning pathways. To begin, a graphic depicting the mathematical ideas and concepts that underpin the learning of algorithms for multiplication and division is provided. The understanding and use of algorithms is informed by two…

This document includes instructional tips on: (1) Building on students' informal understanding of sharing and proportionality to develop initial fraction concepts; (2) Helping students recognize that fractions are numbers that expand the number system beyond whole numbers, and using number lines to teach this and other fraction concepts; (3)…

Skills and understanding of operations with negative numbers, which are typically taught in middle school, are crucial aspects of numerical competence necessary for all subsequent mathematics. To more swiftly and coherently develop the field's understanding of how to foster this critical competence, we need shared measures that allow us to compare…

Multiplication and division are vital topics in upper level elementary school. A teacher's pedagogical content knowledge (PCK) influences both instruction and students' learning. However, there is currently little research examining teachers' PCK within this domain, particularly regarding professional education of future teachers. To help address…

To promote reversibility and strengthen number sense, the authors created an engaging and novel rational number exploration--Quotient Quandary Reversibility task (QQRT)--which promoted flexible and reflective thinking. A class of fifth-grade students took an active role in a collaborative learning task, discussed their strategies, revisited the…

The literature has shown that preservice elementary school teachers (PSTs) struggle to adequately attend to a number's multiplicative structure to determine divisibility. This study describes an intervention aimed at strengthening preservice and in-service teachers' procedural knowledge with respect to using a number's prime factorization to…

During a time when students are accessing instruction virtually because of COVID-19, reflecting on the sixth of the Standards for Mathematical Practice (SMP 6) is a vital step for educators to take. Although the Common Core State Standards for Mathematics (NGA Center and CCSSO 2010) identifies SMP 6 as Attend to precision, associated expertise…

This paper focuses on selected findings from a study that explored the use of multiplication and division with 34 five- and six-year-old children from diverse cultural and linguistic backgrounds. The focus of instructional tasks was on working with groups of ten to support the understanding of place value. Findings from relevant assessment tasks…

Introduces addition, subtraction, multiplication, and division of fractions using area models such as rectangles and circles, or linear models such as the number line and fraction strips. (ASK)

Elementary number theory is a standard topic in the mathematical preparation of preservice elementary teachers. To understand elementary number theory, a student must be comfortable with the representation of natural numbers as the product of primes. This paper discusses methods for accomplishing this goal in a mathematics course. It also…

Early experience with grouping objects is valuable for later understanding of and work with place value, division, and multiplication. Examples are drawn from the Developing Mathematical Processes Curriculum Project at the Wisconsin Research and Development Center for Cognitive Learning. (RS)

In the winter of 1965, an experimental course in Elementary Number Theory was presented to a 6th grade class in the Hosmer School, Watertown, Massachusetts. Prior to the introduction of the present material, students had been exposed in class to such topics from the University of Illinois Arithmetic Project as lattices, number lines, frame…