Adults can represent approximate numbers of items independently of language. This approximate number system can discriminate and compare entities as varied as dots, sounds, or actions. But can multiple different types of entities be enumerated in parallel and stored as independent numerosities? Subjects who were prevented from verbally counting…

After experiencing a Developing Mathematical Ideas (DMI) class on the construction of algebraic concepts surrounding zero and negative numbers, the author conducted an interview with a first grader to determine the youngster's existing level of understanding about these topics. Uncovering young students' existing understanding can provide focus…

In this article, we consider some generalizations of Fibonacci numbers. We consider k-Fibonacci numbers (that follow the recurrence rule F[subscript k,n + 2] = kF[subscript k,n + 1] + F[subscript k,n]), the (k, l)-Fibonacci numbers (that follow the recurrence rule F[subscript k,n + 2] = kF[subscript k,n + 1] + lF[subscript k,n]), and the Fibonacci…

Four proofs, designed for classroom use in varying levels of courses on abstract algebra, are given for the converse of the classical Chinese Remainder Theorem over the integers. In other words, it is proved that if m and n are integers greater than 1 such that the abelian groups [double-struck z][subscript m] [direct sum] [double-struck…

The results of the cognitive research on numbers' representations can provide a sound theoretical framework to develop educational activities on representing numbers. A program of such activities for a nursery school was designed in order to enable the children to externalize and strengthen their internal representations about numerosity and link…

In the late seventies, Guy Brousseau set himself the goal of verifying experimentally a theory he had been building up for a number of years. The theory, consistent with what was later named (non-radical) constructivism, was that children, in suitable carefully arranged circumstances, can build their own knowledge of mathematics. The experiment,…

The final report of the Williams committee (DCSF, 2008: 68) argues that the revised mathematics Framework (DfES, 2006) "should be reconsidered to achieve a more suitable, user-friendly form." It might also have added that there is not much help and support in it for early years teachers. A much more useful document is the "Practice guidance for…

The key to understanding the development of student misconceptions is to ask students to explain their thinking. Time constraints of classroom teaching make it difficult to consult with each and every individual student about their thought processes. However, when a particular error keeps surfacing, simply marking the response as incorrect will…

In this article we analyze the relations between academic mathematical knowledge and the mathematical knowledge associated with issues mathematics school teachers face in practice, according to the specialized literature, and restricted to the theme "number systems". We present examples that illustrate some areas of conflict between those forms of…