Whole-number arithmetic is a core content area of primary mathematics, which lays the foundation for children's later conceptual development. This paper focuses on teaching whole-number multiplication (WNM) to build a stepping stone for children's proportional reasoning. Our intention in writing this paper is to obtain a practice-based perspective…

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…

This article examines the importance of developing the notion of place value as a rate of ten. In exploring how to nurture this concept, the author looks at the role of the language of value, the problem types of multistep multiplication and addition along with measurement division, each with ten as an organizing unit, as well as strategically…

We report results from a mathematics content course intended to help future teachers form a coherent perspective on topics related to multiplication, including whole-number multiplication and division, fraction arithmetic, proportional relationships, and linear functions. We used one meaning of multiplication, based in measurement and expressed as…

The "Reasoning Algebraically about Operations Casebook" was developed as the key resource for participants' Developing Mathematical Ideas seminar experience. The thirty-four cases, written by teachers describing real situations and actual student thinking in their classrooms, provide the basis of each session's investigation into the…

We address the question: How can a student's conceptual transition, from attending only to singleton units (1s) given in multiplicative situations to distinguishing composite units made of such 1s, be explained? We analyze a case study of one fourth grader (Adam, a pseudonym) during the course of a video recorded cognitive interview. Adam's case…

A design experiment with 18 students in a regular seventh grade math class was conducted to investigate how to differentiate instruction for students' diverse ways of thinking during a 26-day unit on proportional reasoning. The class included students operating with three different multiplicative concepts that have been found to influence rational…

While previous studies mainly focused on children's additive and multiplicative reasoning abilities, we studied third to sixth graders' "preference" for additive or multiplicative relations. This was investigated by means of schematic problems that were "open" to both types of relations, namely arrow schemes containing three…

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…

This study explored the impact of numbers used in instructional tasks on the construction and generalization of multiplicative reasoning by fourth grade students designated as having learning difficulties or disabilities in mathematics (SLDs). In particular, this study addressed the following research questions: (1) In what ways do SLDs'…

Multiplicative thinking has been widely accepted as a critically important "big idea" of mathematics and one which underpins much mathematical understanding beyond the primary years of schooling. It is therefore of importance to consider the capacity of children to think multiplicatively but also to consider the capacity of their…

In this case study with Devin (pseudonym), which was part of a larger, constructivist teaching experiment with students identified as having learning difficulties in mathematics, we examine how a fourth grader constructed a dual anticipation involved in monitoring when to start and when to stop the simultaneous count of composite units (numbers…

This study examines how collective activity related to multiplication evolved over several class sessions in an elementary mathematics content course that was designed to foster prospective elementary teachers' number-sense development. We document how the class drew on as-if-shared ideas to make sense of multidigit multiplication in terms of…

A teacher that emphasizes reasoning, logic and validity gives their students access to mathematics as an effective way of practicing critical thinking. All students have the ability to enhance and expand their critical thinking when learning mathematics. Students can develop this ability when confronting mathematical problems, identifying possible…

A procedure for generating quasigroups from groups is described, and the properties of these derived quasigroups are investigated. Some practical examples of the procedure and related results are presented.