This article provides a survey of some basic results in algebraic number theory and applies this material to prove that the cyclotomic integers generated by a seventh root of unity are a unique factorization domain. Part of the proof uses the computer algebra system Maple to find and verify factorizations. The proofs use a combination of historic…

Theories of number concepts often suppose that the natural numbers are acquired as children learn to count and as they draw an induction based on their interpretation of the first few count words. In a bold critique of this general approach, Rips, Asmuth, Bloomfield [Rips, L., Asmuth, J. & Bloomfield, A. (2006). Giving the boot to the bootstrap:…

This paper reports on a component of a large yearlong study in three Year 7 classes in three different schools. The aim of this research component was to determine the relationship between students' number sense and their problem-solving ability by means of paper-and-pencil tests, classroom observations, and interviews of students and teachers.…

In "MT218" the author looked at the possibility of basing a classroom activity on a simple, though not totally transparent, number-theoretic result. In this article he considers another relatively straightforward idea from number theory that could be used either as a lesson starter or as the basis of a more substantial task, requiring students to…

It seems rather surprising that any given polynomial p(x) with nonnegative integer coefficients can be determined by just the two values p(1) and p(a), where a is any integer greater than p(1). This result has become known as the "perplexing polynomial puzzle." Here, we address the natural question of what might be required to determine a…

Starting with an interesting number game sometimes used by school teachers to demonstrate the factorization of integers, "sum-difference numbers" are defined. A positive integer n is a "sum-difference number" if there exist positive integers "x, y, w, z" such that n = xy = wz and x ? y = w + z. This paper characterizes all sum-difference numbers…

This research documents Kuwaiti eighth grade students' performance in recognizing reasonable answers and the strategies they used to determine reasonableness. The results from over 200 eighth grade students show they were generally unable to recognize reasonable answers. Students' performance was consistently low across all three number domains…

Numbers are thought to be represented in space along a mental left-right oriented number line. Number magnitude has also been associated with the size of grip aperture, which might suggest a connection between number magnitude and intensity. The present experiment aimed to confirm this possibility more directly by using force as a response…

We analyzed the effects of verbal mediating responses on the acquisition and generalization of say-do correspondence and noncorrespondence. Participants were assigned to groups in which either reinforcers (feedback and tokens) were arranged for say-do correspondence and noncorrespondence, or no reinforcers were programmed. Participants in these…

This study tested whether 10- and 12-year-olds who can correctly compare the magnitudes of fractions with common components access the magnitudes of the whole fractions rather than only compare the magnitudes of their components. Time for comparing two fractions was predicted by the numerical distance between the whole fractions, suggesting an…

In this sequence 1/1, 7/5, 41/29, 239/169 and so on, Thomas notes that the sequence converges to square root of 2. By observation, the sequence of numbers in the numerator of the above sequence, have a pattern of generation which is the same as that in the denominator. That is, the next term is found by multiplying the previous term by six and…

In collaboration with two other research organizations, the authors integrate principles of the BEAR Assessment System with Universal Design for Learning principles to develop and validate learning progressions and an aligned, universally designed formative and summative classroom assessment system for promoting conceptual understanding of number…

The widely adopted Common Core State Standards for Mathematics (CCSSM) are designed to deepen instruction of number sense and will demand that elementary school teachers have a strong understanding of number. These changes arrive at a time when it is still understood that teachers and the curriculum in the United States have not been fundamentally…

This article argues for the necessity of phonetic analysis in signed language linguistics and presents a case study of one analytical system being used in a preliminary attempt to identify natural classes and investigate variation in ASL handshapes. Robbin Battison (1978) first described what is now a widely accepted list of basic handshapes,…

In this article, we directly question the common practice in growth mixture model (GMM) applications that exclusively rely on the fitting model without covariates for GMM class enumeration. We provide theoretical and simulation evidence to demonstrate that exclusion of covariates from GMM class enumeration could be problematic in many cases. Based…