After the monumental discovery of the fundamental theorems of the calculus nearly 350 years ago, it became possible to answer extremely complex questions regarding the natural world. Here, a straightforward yet profound demonstration, employing geometrically symmetric functions, describes the validity of the general power rules for integration and…

Differential equations (DEs) are important in mathematics as well as in science and the social sciences. Thus, the study of DEs has been included in various courses in different departments in higher education. The importance of DEs has attracted the attention of many researchers who have generally focussed on the content and instruction of DEs.…

National Council of Teachers of Mathematics' (NCTM's) (2000) Connections Standard states that students should "recognize and use connections among mathematical ideas; understand how mathematical ideas interconnect ...; [and] recognize and apply mathematics in contexts outside of mathematics" (p. 354). This article presents an in-depth…

Many mathematics educators have noted that mathematicians do not only read proofs to gain conviction but also to obtain insight. The goal of this article is to discuss what this insight is from mathematicians' perspective. Based on interviews with nine research-active mathematicians, two sources of insight are discussed. The first is reading a…

This article outlines a theoretical model for teaching elementary mathematical concepts that we have developed over the past 10 years. We begin with general ideas about the abstraction process and differentiate between "abstract-general" and "abstract-apart" concepts. A 4-phase model of teaching, called Teaching for Abstraction, is then proposed…

Using sequences of coin flips as a model of serial independent events, we asked schoolchildren in grades 1 through 5 to estimate the likelihood of the next flip. Most children in each grade expected short patterns to continue.

This is a case study of an undergraduate calculus student's nonstandard conceptions of the real number line. Interviews with the student reveal robust conceptions of the real number line that include infinitesimal and infinite quantities and distances. Similarities between these conceptions and those of G. W. Leibniz are discussed and illuminated…

Students from thirteen fifth-grade classrooms from six different elementary schools investigate the preferred escape routes of cockroaches. These identified routes of 90-, 120-, 150-, and 180-degrees are the research findings of Paolo Dominici, an Italian scientist. The students used rubber cockroaches and lizards to map out these escape routes.…

A recent study in the United Kingdom provides strong evidence that well-organized activities outside the classroom contribute significantly to the quality and depth of children's learning, including their personal, social, and emotional development. Outdoor math trails supply further evidence of such enhanced learning. This article focuses on the…

The Literacy Design Collaborative (LDC) incorporates challenging literacy standards into middle grades and high school content areas within a variety of academic disciplines, not just English/language arts courses. The Mathematics Design Collaborative (MDC) provides schools with instructional tools needed to help teachers understand and implement…

Several previous researches showed that students had difficulty in understanding the basic concept of multiplication. Students are more likely to be introduced by using formula without involving the concept itself. This underlies the researcher to design a learning trajectory of learning multiplication using Permainan Tradisional Tepuk Bergambar…

This paper reports on the results of three different investigations into pre-service teachers' understanding of the mathematical concepts of area and perimeter. Different test instruments were used with three cohorts from two universities in order to identify pre-service teachers' understandings and common misconceptions. The results indicated…

Curriculum standards provide classroom teachers with frameworks to guide instruction during the academic year and throughout a student's schooling. Unfortunately, standards are misconstrued as all-encompassing--as a compendium of all mathematics content that can and should be taught to students. Such a view is unfortunate. Mathematics teachers…

This study concentrated on the characteristics of Grade 2 students' writing in mathematics class who used the Project M[superscript 2] units to learn geometry and measurement. Included in the study were students with high and low levels of content knowledge. Archived data from the Project M[superscript 2] study were analyzed to determine the…

This study investigates cognitive difficulties and mathematical ideas that are related to making connections among representations. A three-week intervention was designed and implemented to help prospective secondary mathematics teachers develop understanding of big ideas that are critical to connection of representations in algebra. This study…