Diss.--Leipzig (Andreas Thorinus, respondent). / At head of title: Q.D.B.V. Day of the month in title supplied in manuscript.

Diss.--Leipzig (Andreas Thorinus, respondent). / At head of title: Q.D.B.V. Day of the month in title supplied in manuscript.

This paper investigates whether it is reasonable to bestow credit to one person or group for the famed theorem that relates to the side lengths of any right-angled triangle, a theorem routinely referred to as the “Pythagorean Theorem”. The author investigates the first-documented occurrences of the theorem, along with its first proofs. In addition, proofs that stem from different branches of mathematics and science are analyzed in an effort to display that credit for the development of the theorem should be shared amongst its many contributors rather than crediting the whole of the theorem to one man and his supporters.

This report expresses some of the recent research surrounding the Pythagorean Theorem and Pythagorean triples. Topics discussed include applications of the Pythagorean Theorem relating to recursion methods, acute and obtuse triangles, Pythagorean triangles in squares, as well as Pythagorean boxes. A short discussion on the depth of the Pythagorean Theorem taught in secondary schools is also included. / text

This paper investigates whether it is reasonable to bestow credit to one person or group for the famed theorem that relates to the side lengths of any right-angled triangle, a theorem routinely referred to as the “Pythagorean Theorem”. The author investigates the first-documented occurrences of the theorem, along with its first proofs. In addition, proofs that stem from different branches of mathematics and science are analyzed in an effort to display that credit for the development of the theorem should be shared amongst its many contributors rather than crediting the whole of the theorem to one man and his supporters.

Insug.-diss.-Leipzig. / Lebenslauf.

Inaug.-diss.--Leipzig.

Thesis--Princeton University. / Includes bibliographical references.

The aim of this study is to analyze students¡¦ mathematics concepts in solving Pythagorean Theorem problems presented in two different representations (word problems and word problems with diagrams). The investigators employed the mathematics competence indicators in Grade 1-9 Integrated Curriculum in developing such problems. In analyzing data, the investigator used Schoenfeld¡¦s method in depicting their problem-solving processes, with attention to students¡¦ sequence and difference in time consumption. Four eight grade students with good competence in mathematics and expressions from a secondary school were selected as research subjects. Problems related to Pythagorean Theorem were divided into three types: Shape, Area, and Number. Data were collected using thinking aloud method and semi-structured interview, and triangulation was further applied in protocol analysis. The research results revealed 3 findings: (1) For the ¡§Shape¡¨ type problems, students¡¦ problem-solving concepts varied with different problem representation. For the ¡§Area¡¨ and ¡§Number¡¨ types of problems (without diagram), students were required to use their geometric concept when processing word problems. Students¡¨ use of problem-solving concepts would not significantly vary with problem representation types. However, students¡¦ use of problem-solving methods would affect the types and priorities of concepts used. Generally, the types of mathematics concepts could be made up by the frequency of concepts used, and more types of problem-solving concepts would be used for word problems representation than for word problems with diagrams representation. (2) In terms of the time consumed in the first three problem-solving stages of Schoenfeld, the time required to solve word problems was 1.6 times of that required to solve word problems with diagrams. In terms of the total time consumed, the time required to solve word problems was 1.25 times of that required to solve word problems with diagrams. In the problem-solving stages, students needed to explore the problem first when dealing with word problems before they could go on to solve the problem, and such repetition was more frequent when they dealt with word problems. (3) For both type of problem representations, there is a higher number of correctly-answered problems. This finding indicated that a higher frequency of problem-solving concepts and less repetition in the problem-solving stage were required; and vice versa. As to the sequence of Pythagorean Theorem concepts to be taught, the investigator suggest teachers to start with the concept of area filling in the ¡§Shape¡¨ type of problems to derive Pythagorean Theorem, and further apply the formula to - III - solving ¡§Number¡¨ problems. After students have acquired basic competency in ¡§Shape¡¨ and ¡§Number¡¨ Pythagorean Theorem problems, teachers could explain and introduce this theorem from the perspective of ¡§Area¡¨. Finally, in problem posing, teachers were also advised to apply various contexts; covering all kinds of representations of problems that enhance students¡¦ utilization of mathematics concepts; and to cater for various needs of students.

Thesis (Ph. D.)--University of Pennsylvania, 1911. / "The Introductionis arithmeticae libri duo ... is the basis of the present essay"--P. 1. Includes bibliographical references (p. 1).