A visual proof that the sine is subadditive on [0, pi].

A visual proof that sin x/x is monotonically increasing on (0, pi/2). For tan x/x, see p. 420 (EJ1017686).

A visual proof that tan x/x is monotonically increasing on (0, pi/2). For sin x/x, see p. 408 (EJ1017684).

At Southern Vermont College (SVC) and at the nation's other colleges and universities, Anatomy and Physiology I (A&PI) is the gateway course into healthcare careers. Disturbingly, at SVC and elsewhere, many first-year students interested in healthcare careers do not succeed in this course. They withdraw from the course or the institution, or their…

This paper reports on teachers? practical knowledge [PK] about peer interactions [PI] in learning mathematics. The focus is on high school teachers who consistently engaged students PI in their teaching. Data consisted of interviews and classroom observations. Findings indicate that these teachers have PK of students? roles in PI and learning…

Using techniques that show that e and pi are transcendental, we give a short, elementary proof that pi is irrational based on Euler's identity. The proof involves evaluations of a polynomial using repeated applications of Leibniz formula as organized in a Leibniz table.

For at least 2000 years people have been trying to calculate the value of [pi], the ratio of the circumference to the diameter of a circle. People know that [pi] is an irrational number; its decimal representation goes on forever. Early methods were geometric, involving the use of inscribed and circumscribed polygons of a circle. However, real…

In this paper we present results relating undergraduate student retention in science, technology, engineering, and mathematics (STEM) majors to the use of Peer Instruction (PI) in an introductory physics course at a highly selective research institution. We compare the percentages of students who switch out of a STEM major after taking a physics…

In my Calculus classes I encourage my students to actively reflect on course material, to work collaboratively, and to generate diverse solutions to questions. To facilitate this I use peer instruction (PI), a structured questioning process, and i-clickers, a radio frequency classroom response system enabling students to vote anonymously. This…

The celebrated Basel Problem, that of finding the infinite sum 1 + 1/ 4 + 1/9 + 1/16 + ..., was open for 91 years. In 1735 Euler showed that the sum is pi[superscript 2]/6. Dozens of other solutions have been found. We give one that is short and elementary.

Using continued fraction expansions, we can approximate constants, such as pi and e, using an appropriate integer n raised to the power x[superscript 1/x], x a suitable rational. We review continued fractions and give an algorithm for producing these approximations.

This study investigates pre-service mathematics teachers' concept images of radian and possible sources of such images. A multiple-case study was conducted for this study. Forty-two pre-service mathematics teachers completed a questionnaire, which aims to assess their understanding of radian. Six of them were selected for individual interviews on…

This chapter discusses the impact that four multisite National Science Foundation (NSF) evaluations had on the broader field of science, technology, engineering, and mathematics education and evaluation. Three approaches were used to investigate the broader impact of these evaluations on the field: (a) a citation analysis, (b) an on-line survey,…

In 1997, Bailey and Bannister showed that a + b greater than c + h holds for all triangles with [gamma] less than arctan (22/7)where a, b, and c are the sides of the triangle, "h" is the altitude to side "c", and [gamma] is the angle opposite c. In this paper, we show that a + b greater than c + h holds approximately 92% of the time for all…

The Integrative Graduate Education and Research Traineeship (IGERT) program represents a substantial investment by the National Science Foundation (NSF) to improve the quality of graduate education, and ultimately, to increase the number of graduates better prepared to address our nation's 21st century scientific and technological needs. The…