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…

Describes various aids and activities to help in planning lessons that will encourage the development of sound place value knowledge. Also discusses various instructional strategies to help students avoid problems which may lead to an incomplete or faulty understanding of place value. (JN)

Describes a pattern which emerged from an examination of the digits of the squares of numbers. Provides eight examples having the pattern at the units or tens digit of the number. (YP)

"Number sense" is "an intuition about numbers that is drawn from all varied meanings of number" (NCTM, 1989, p. 39). Students with number sense understand that numbers are representative of objects, magnitudes, relationships, and other attributes; that numbers can be operated on, compared, and used for communication. It is fundamental knowledge…

According to one theory about how children learn the concept of natural numbers, they first determine that "one", "two", and "three" denote the size of sets containing the relevant number of items. They then make the following inductive inference (the Bootstrap): The next number word in the counting series denotes the size of the sets you get by…

This work presents an introductory development of fractional order derivatives and their computations. Historical development of fractional calculus is discussed. This paper presents how to obtain computational results of fractional order derivatives for some elementary functions. Computational results are illustrated in tabular and graphical…

Automatic processing of 2-digit numbers was demonstrated using the size congruency effect (SiCE). The SiCE indicates the processing of the irrelevant (numerical) dimension when 2 digits differing both numerically and physically are compared on the relevant (physical) dimension. The SiCE was affected by the compatibility between unit and decade…

Logarithmic spirals are among the most fascinating curves in the plane, being the only curves that are equiangular, and the only ones that are self-similar. In this article, we show that in three dimensions, these two properties are independent. Although there are surfaces that have both properties, there are some that are equiangular, but not…

This article is about a very small subset of the positive integers. The positive integer N is said to be "perfect" if it is the sum of all its divisors, including 1, but less that N itself. For example, N = 6 is perfect, because the (relevant) divisors are 1, 2 and 3, and 6 = 1 + 2 + 3. On the other hand, N = 12 has divisors 1, 2, 3, 4 and 6, but…

In this note, a modified Second Derivative Test is introduced for the relative extrema of a single variable function. This improved test overcomes the difficulty of the second derivative vanishing at the critical point, while in contrast the traditional test fails for this case. A proof for this improved Second Derivative Test is presented,…