This article reports on a study that was conducted with 378 primary teachers from Catholic schools in Victoria who participated in the first year of a 2-year research and professional learning program in mathematics. One aim of the program was to enhance teacher knowledge in mathematics in its many forms. As part of the larger study, the teachers…

There is more to understanding the concept of mean than simply knowing and applying the add-them-up and divide algorithm. In the following, we discuss a component of understanding the mean--inference from a fixed total--that has been largely ignored by researchers studying students understanding of mean. We add this component to the list of types…

Much research supports the existence of an Approximate Number System (ANS) that is recruited by infants, children, adults, and non-human animals to generate coarse, non-symbolic representations of number. This system supports simple arithmetic operations such as addition, subtraction, and ordering of amounts. The current study tests whether an…

This article describes a comprehensive and novel approach to assessment in early numeracy. Topics include the significance of early numeracy, developing a nomenclature for early numeracy, and describing the context for the development of this approach to assessment. The largely unrealised importance of numerals and numeral sequences in early…

When counting, the final word used to tag the final item in a set represents the cardinality, or total number, of the set. Understanding of this concept serves as a foundation for children's basic mathematical skills, such as arithmetic. However, little is known about how variations in the early learning environment affect children's understanding…

The Standards for Mathematical Practice (SMP) in the Common Core State Standards for Mathematics (CCSSM) (CCSSI 2010) highlight the mathematical habits that educators should be fostering in mathematics classrooms throughout K-grade 12 education. That argumentation and discourse are important components of developing mathematically proficient…

This study evaluated the effects of a mathematics software program, the Building Blocks software suite, on young children's mathematics performance. Participants included 247 Kindergartners from 37 classrooms in 9 schools located in low-income communities. Children within classrooms were randomly assigned to receive 21 weeks of computer-assisted…

In online educational systems we can easily collect and analyze extensive data about student learning. Current practice, however, focuses only on some aspects of these data, particularly on correctness of students answers. When a student answers incorrectly, the submitted wrong answer can give us valuable information. We provide an overview of…

The concept of exponents has been shown to be problematic for students, especially when expanding it from the domain of positive whole numbers to that of exponents that are negative and later rational. This paper presents a theoretical analysis of the concept of exponentiation as a continuous operation and examines the deficiencies of existing…

This research shows that people can unconsciously initiate and follow arithmetic rules (e.g., addition). Participants were asked to detect whether a symbol was a digit. This symbol was preceded by 2 digits and a subliminal instruction: "add" or a control instruction. Participants were faster at identifying a symbol as a number when the…

Researchers have speculated that children find it more difficult to acquire conceptual understanding of the inverse relation between multiplication and division than that between addition and subtraction. We reviewed research on children and adults' use of shortcut procedures that make use of the inverse relation on two kinds of problems:…

It is of deep interest to both linguists and psychologists alike to account for how young children acquire an understanding of number words. In their commentaries, Barner and Butterworth both point out that an important question highlighted by the work of Syrett, Musolino, and Gelman, and one that remains highly controversial, is where number…

An idempotent satisfies the equation x[superscript 2] = x. In ordinary arithmetic, this is so easy to solve it's boring. We delight the mathematical palette here, topping idempotents off with modular arithmetic and a series of exercises determining for which n there are more than two idempotents (mod n) and exactly how many there are.

The ability to count has traditionally been considered an important milestone in children's development of number sense. However, using counting (e.g., counting on, counting all) strategies to solve addition problems is not the best way for children to achieve their full mathematical potential and to prepare them to develop more complex and…

The article begins with a well-known property regarding tangent lines to a cubic polynomial that has distinct, real zeros. We were then able to generalize this property to any polynomial with distinct, real zeros. We also considered a certain family of cubics with two fixed zeros and one variable zero, and explored the loci of centroids of…