The multiplication principle (MP) is foundational for combinatorial problem-solving. From a units-coordination perspective, applying the MP with justification entails establishing unit relationships between the number of options at each independent stage of a counting process and the total number of combinatorial outcomes. Existing research…

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…

The multiplication principle serves as a cornerstone in enumerative combinatorics. The principle underpins many basic counting formulas and provides students with a critical element of combinatorial justification. Given its importance, the way in which it is presented in textbooks is surprisingly varied. In this paper, we analyze a number of…

To further understand student thinking in the context of combinatorial enumeration, we examine student work on a problem involving set partitions. In this context, we note some key features of the multiplication principle that were often not attended to by students. We also share a productive way of thinking that emerged for several students who…

In this study I investigate Saldanha and Thompson's (1998) claim that conceptualizing a coordinate pair in the Cartesian coordinate system as a multiplicative object, a way to unite two quantities' values, supports students in conceptualizing graphs as emergent representations of how two quantities' values change together. I presented three…

The multiplication principle is a fundamental principle in enumerative combinatorics. It underpins many of the counting formulas students learn, and it provides much-needed justification for why counting works as it does. However, given its importance, the way in which it is presented in textbooks is surprisingly varied. In this paper, we document…

This study aims to explain the thinking process of students in solving combination problems considered from assimilation and accommodation frameworks. This research used a case study approach by classifying students into three categories of capabilities namely high, medium and low capabilities. From each of the ability categories, one student was…

Counting problems have applications in probability and computer science, and they provide rich contexts for problem solving. Such problems are accessible to students, but subtleties can arise that make them surprisingly difficult to solve. In this paper, students' work on the Groups of Students problem is presented, and an important issue related…

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 branch of mathematics commonly used in cryptography is Galois Fields GF(p[superscript n]). Two basic operations performed in GF(p[superscript n]) are the addition and the multiplication. While the addition is generally easy to compute, the multiplication requires a special treatment. A well-known method to compute the multiplication is based on…

We give an irredundant axiomatization of the complete ordered field of real numbers. In particular, we show that all the field axioms for multiplication with the exception of the distributive property may be deduced as "theorems" in our system. We also provide a complete proof that the axioms we have chosen are independent.

The objective of this introduction to Colombeau algebras of generalized functions (in which distributions can be freely multiplied) is to explain in elementary terms the essential concepts necessary for their application to basic nonlinear problems in classical physics. Examples are given in hydrodynamics and electrodynamics. The problem of the…

Numerous studies indicate that performance in solving single step multiplicative word problems is influenced by both problem structure and the types of numbers involved in the problem. For example, including numbers less than one often increases the difficulty of a problem. What remains unclear is how problem structure and number type interact in…

College students were exposed to two pairs of mathematics assignments. Assignment Pair A included a high-effort assignment containing 18 long three-digit ? two-digit (3?2) multiplication problems with all numerals in each problem being equal to or greater than four and a moderate-effort assignment that contained nine long problems and nine…

Mathematics majors' study of abstract algebra should provide these students with opportunities to connect what they are learning to their prior experiences with algebra in high school. This paper illustrates how such connections can be used to motivate the notion of binary operation and the axioms for a group.