As students have struggled to use the "chip model" (i.e., red and yellow chips representing positive and negative numbers) to model integer addition and subtraction and have found it confusing, the authors developed a series of activities based on adding and removing opposite objects to and from a boat to better help students in this…

This article presents a case study of two Grade 5 boys' argumentation concerning addition and subtraction of negative numbers while using an interactive tablet-based application simulating positive and negative tiles. We examine the properties of integers they conjectured, and the kinds of evidence and arguments they used to support their…

Mathematics is crucial to the educational and vocational success of students. The concrete-representational-abstract (CRA) approach is a method to teach students mathematical concepts. The CRA involves instruction with manipulatives, representations, and numbers only in different lessons (i.e., concrete lessons include manipulatives but not…

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…

Many related studies have studied many different models in the teaching of the concept of integer, which have reported that counters failed to completely help with the understanding of the concept of and operation modeling in integers. The purpose of the present research is presenting the "opposite model", which is a quantitative model,…

These four chapters describe studies of using models for integer addition and subtraction. The models draw principally on the two grounding metaphors of object collection and motion along a path. A strength of all chapters is detailed analysis of how the models are and can be implemented and how they influence student's learning. Together the…

This chapter describes instances of play within a teaching episode on integer addition and subtraction. Specifically, this chapter makes the theoretical distinction between integer play and playing with integers. Describing instances of integer play and playing with integers is important for facilitating this type of intellectual play in the…

We present an algorithm for a matrix-based Enigma-type encoder based on a variation of the Hill Cipher as an application of 2 × 2 matrices. In particular, students will use vector addition and 2 × 2 matrix multiplication by column vectors to simulate a matrix version of the German Enigma Encoding Machine as a basic example of cryptography. The…

In this article, we report the findings of a study conducted with 6 Grade 8 students in the United States. The students posed stories for open number sentences involving addition and subtraction of integers. We analysed the stories posed by the students to build models that describe the conceptual structures behind these posed stories -- the…

A single fact-focused number-string discussion is unlikely to result in all students adopting more flexible and efficient fact strategies; however, these authors have found that repeated engagement in such discussions over time does promote student reasoning about relationships, which in turn leads to increased fact fluency. By cultivating a…

Let R be a ring with identity. Then {0} and R are the only additive subgroups of R if and only if R is isomorphic (as a ring with identity) to (exactly) one of {0}, Z/pZ for a prime number p. Also, each additive subgroup of R is a one-sided ideal of R if and only if R is isomorphic to (exactly) one of {0}, Z, Z/nZ for an integer n = 2. This note…

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…

Ginsburg (1977) observed that children typically develop surprisingly powerful informal (everyday) knowledge of mathematics and that mathematical learning difficulties often arise when formal instruction does not build on this existing knowledge. By using meaningful analogies teachers can help connect new formal instruction to students' existing…

This research paper reports strategies used by Grade 6 learners in multiplying whole numbers in five selected primary schools in Kavango East and West regions. A total of 200 learners' mathematics exercise books were analysed in order to identify the commonly used strategies by learners in multiplying whole numbers. A total of ten teachers…

This article focuses on how two mathematics teachers (Amy and Beth -- pseudonyms) cope with the changes of meanings in multiplication due to the changes of contexts. It highlights the qualitative similarities and differences between these two teachers in the sense-making process of multiplication. A potentially useful framework of supportive and…