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
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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.
| Identifer | oai:union.ndltd.org:NSYSU/oai:NSYSU:etd-0630108-122257 |
| Date | 30 June 2008 |
| Creators | CHIU, HSIN-HUI |
| Contributors | Uen, Wuu-Nan, Leung, Shuk-kwan, Jhuang, Hsueh-Hua |
| Publisher | NSYSU |
| Source Sets | NSYSU Electronic Thesis and Dissertation Archive |
| Language | Cholon |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | http://etd.lib.nsysu.edu.tw/ETD-db/ETD-search/view_etd?URN=etd-0630108-122257 |
| Rights | unrestricted, Copyright information available at source archive |
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