This study is the first to examine the associations between the occurrence, frequency, and adaptivity of children's subtraction by addition strategy use (SBA; e.g., 712 - 346 = ?; 346 + 54 = 400, 400 + 300 = 700, 700 + 12 = 712, and 54 + 300 + 12 = 366) and their underlying conceptual knowledge. Specifically, we focused on two rarely studied…

Students with learning disabilities in mathematics often struggle with the underlying concepts of multidigit addition and subtraction. To help students build a conceptual understanding of these computations, teachers can utilize evidence-based practices such as the concrete-semi-concrete-abstract framework and the use of multiple visual…

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…

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…

Current curriculum initiatives (e.g., National Governors Association Center for Best Practices and Council of Chief State School Officers 2010) advocate that models be used in the mathematics classroom. However, despite their apparent promise, there comes a point when models break, a point in the mathematical problem space where the model cannot,…

There is a renewed debate about whether educated adults solve simple addition problems (e.g., 2 + 3) by direct fact retrieval or by fast, automatic counting-based procedures. Recent research testing adults' simple addition and multiplication showed that a 150-ms preview of the operator (+ or ×) facilitated addition, but not multiplication,…

The purpose of the study was to determine whether there was any significant difference in speed, accuracy, retention and transferability between the Decomposition (DEC) and the Base-Complement Additions (BCA) algorithms for performing compound subtraction in non-decimal bases. Fifty-nine students with a mean age of approximately 15 years from two…

I present a viable learning trajectory for prospective elementary teachers' number sense development with a focus on whole-number place value, addition, and subtraction. I document a chronology of classroom mathematical practices in a Number and Operations course. The findings provide insights into prospective elementary teachers' number sense…

This article contributes to the research literature concerning prospective elementary teachers' mathematical thinking and learning with a focus on flexibility. We present a case study of a prospective elementary teachers' development of flexibility in mental addition and subtraction during a Number and Operations course. Building upon the…

Numeracy is a key competency for living in our modern knowledge society. Therefore, it is essential to support numerical learning from basic to more advanced competency levels. From educational psychology it is known that learning is most effective when the respective content is neither too easy nor too demanding in relation to learners'…

Many children fail to master fraction arithmetic even after years of instruction. A recent theory of fraction arithmetic (Braithwaite, Pyke, & Siegler, 2017) hypothesized that this poor learning of fraction arithmetic procedures reflects poor conceptual understanding of them. To test this hypothesis, we performed three experiments examining…

Many children fail to master fraction arithmetic even after years of instruction. A recent theory of fraction arithmetic (Braithwaite, Pyke, & Siegler, in press) hypothesized that this poor learning of fraction arithmetic procedures reflects poor conceptual understanding of them. To test this hypothesis, we performed three experiments…

There is a call for enabling students to use a range of efficient mental and written strategies when solving addition and subtraction problems. To do so, students should recognise numerical structures and be able to change a problem to an equivalent problem. The purpose of this article is to suggest an activity to facilitate such understanding in…

Several types of converging evidence have suggested recently that skilled adults solve very simple addition problems (e.g., 2 + 1, 4 + 2) using a fast, unconscious counting algorithm. These results stand in opposition to the long-held assumption in the cognitive arithmetic literature that such simple addition problems normally are solved by fact…

During the past few years, several of the authors have incorporated student problem posing as a regular instructional feature in their classrooms. When they offer their students the opportunity to construct their own problems, particularly during the course of an entire school year, they create many novel tasks. Student-created tasks not only…