KenKen® is the new Sudoku. Like Sudoku, KenKen requires extensive use of logical reasoning. Unlike Sudoku, KenKen requires significant reasoning with numbers and operations and helps develop number sense. The creator of KenKen puzzles, Tetsuya Miyamoto, believed that "if you give children good learning materials, they will think and learn and…

Faced with a chronic shortage of single-subject mathematics teachers and a directive from the federal legislation "No Child Left Behind" that all students are taught by highly qualified teachers, California instituted a new subject-matter competency exam, the California Subject Exam for Teachers for Mathematics (CSET), and created a new…

In 2000, the "National Council of Teachers of Mathematics" recommended that Algebra Standards, "instructional programs from prekindergarten through grade 12 should enable all students to use mathematical models to represent and understand quantitative relationships." In this article, the authors suggest the "Balance"…

This article presents a method of reducing fractions without factoring. The ideas presented may be useful as a project for motivated students in an undergraduate number theory course. The discussion is related to the Euclidean Algorithm and its variations may lead to projects or early examples involving efficiency of an algorithm.

A mother tracked her preschooler's number word development daily from 18 to 49 months of age. Naturalistic observations were supplemented with observations during structured (Kumon) training and microgenetic testing. The boy's everyday use of "two" did not become highly reliable and selective for 10 months (at 28 months), emerged later than that…

The preschool years are a time of great advances in children's numerical thinking, most notably as they master verbal counting. The present research assessed the relation between analog magnitude representations and cardinal number knowledge in preschool-aged children to ask two questions: (1) Is there a relationship between acuity in the analog…

PME, the International Group for the Psychology of Mathematics Education, was founded in 1976, at the "Third International Congress on Mathematical Education" in Karlsruhe, organised by the International Commission on Mathematics Instruction (ICMI). While PME is thus beyond coming of age and is reflecting its further orientation--due to…

The use of logarithms, an important tool for calculus and beyond, has been reduced to symbol manipulation without understanding in most entry-level college algebra courses. The primary aim of this research, therefore, was to investigate college students' understanding of logarithmic concepts through the use of a series of instructional tasks…

A recent cross-cultural comparison (Shaki, Fischer, & Petrusic, 2009) suggested that spatially consistent processing habits for words and numbers are a necessary condition for the spatial representation of numbers (Spatial-Numerical Association of Response Codes; SNARC effect). Here we reexamine the SNARC in Israelis who read text from right…

A facility with signed numbers forms the basis for effective problem solving throughout developmental mathematics. Most developmental mathematics textbooks explain signed number operations using absolute value, a method that involves considering the problem in several cases (same sign, opposite sign), and in the case of subtraction, rewriting the…

The basis of this intervention study is a distinction between numerical calculus and relational calculus. The former refers to numerical calculations and the latter to the analysis of the quantitative relations in mathematical problems. The inverse relation between addition and subtraction is relevant to both kinds of calculus, but so far research…

The floor function maps a real number to the largest previous integer. More precisely, floor(x)=[x] is the largest integer not greater than x. The square bracket notation [x] for the floor function was introduced by Gauss in his third proof of quadratic reciprocity in 1808. The floor function is also called the greatest integer or entier (French…

Dyscalculia has been described as a specific learning difficulty affecting the ability to acquire arithmetical skills. In recent years, it has become a topic for discussion in the popular media, yet there has been little research undertaken by educational psychologists. This paper provides a summary of neuroscientific research into the development…

Language for number is an important case study of the relationship between language and cognition because the mechanisms of non-verbal numerical cognition are well-understood. When the Piraha (an Amazonian hunter-gatherer tribe who have no exact number words) are tested in non-verbal numerical tasks, they are able to perform one-to-one matching…

The present study tested the "interference hypothesis"-that learning and using more advanced representations and strategies requires the inhibition of prior, less advanced ones. Specifically, it examined the relation between inhibitory control and number line estimation performance. Experiment 1 compared the accuracy of adults' (N = 53)…