Using APOS and PCDC to Improve Students' Learning of Negative Numbers

• Main Authors: Feng-Lan Liao, 廖鳳蘭
• Other Authors: Ching-Kuch Chang
• Format: Others
• Language: zh-TW
• Published: 2008
• Online Access: http://ndltd.ncl.edu.tw/handle/64648889346127198877

碩士 === 國立彰化師範大學 === 科學教育研究所 === 96 === The purpose of this study was to investigate how to integrate APOS into PCDC and to construct students’ learning of negative numbers. For this purpose, I adopted an action research to achieve the stability in this study through planning, implementing, analyzing, modifying and re-planning. First this study was progressed the pilot study for one semester in order to build PCDC. Later I modified my teaching for another semester in order to integrate APOS into PCDC. Because I missed my research subjects, I selected another average class of 7th grader for the subject of this research and was progressed the formal research. I taught this class in the new way of teaching for a month. The results of this study were the students’ learning in the formal research. Data collection included my classroom observation, video tapes and records data, student’s homework, student’s weekly journals, my reflection diary, interviews, the opinion of the professor and the researcher’s team. Data analysis included content analysis and indeed constructed the truth of students’ learning the mathematical concepts in the discussions of team. The findings of the study were: In order to integrate APOS into PCDC, it must be thought about the four key items of PCDC, and fuse the mental structures of APOS. It should be kept in mind about the four key items, as the followings: 1. Using APOS to analyze the concepts of teaching content, and to form the levels of concepts, and then to design the teaching task of learning levels. 2. Through asking or encouraging, it enhanced students to achieve reflective abstraction from the concepts of lower level to the concepts of higher level, and constructed the learning. 3. Offering the concretely learning materials and the writing tools; it helped students to construct the mental structures and concepts from the exploring of the question situation. Arranging the learning environment, it enhanced students to discuss and talk, and improved their concepts levels from the thinking of the question. 4. Collecting the data about students’ concepts development, and comparing the levels of the concepts in APOS with the actual learning situation of the students. Continuing to compare, and understanding the mental structures of the students in the activities and realizing the development of mathematical concepts. Then it could be the basis of the teaching design and modification. （2）After fusing APOS to PCDC, it enhanced students to construct the mental structures of action, process, object and schema by the mental mechanisms of interiorization and encapsulation. In order to solve the problems of the task, students improved their concepts levels through reflective abstraction gradually. The findings showed they developed three items of the concepts in negative numbers. 1. Negative numbers concepts: students showed the temperature of the cities and the data of the flying fish jumping by the forms of the mathematics in order to build the relations of the opposite quantity. Through the process of estimating the distances of the flying fish jumping, students understood the counter part of the positive and negative numbers and built the beginning of the learning. That’s an action. The task of estimating the distances of the flying fish and the appearing places of deepened fish enhanced them to operate in their mind directly without having to manipulate outside object. Then they built the mind process, achieved interiorization and developed the simple mental structure of the operation of adding and subtracting of the positive and negative numbers. They understood the counter part of the positive and negative numbers in the number line, and enhanced them to build the concepts of negative numbers. Through the reflection of the questions, it helped students to encapsulate the process into an object, and to build the mental structures or the schema of the negative numbers. 2. The adding of positive and negative numbers: through operating and exploring the positive and negative cards actually, students understood the balance relations of the adding of positive and negative numbers, and built the beginning of the learning（action）in the adding of positive and negative numbers. Through the questions which take away the pictures of the positive and negative cards, it enhanced them to repeat and reflect to perform the transformation in their mind directly, and interiorized an action into a process. 3.The adding and subtracting concepts of the positive and negative numbers: Through the transformation of scores and question answering in the games, and the transformation of the black white chess, it built the action of the adding and subtracting of the positive and negative numbers. Through reflecting deeply, they operated the question solutions in their mind, and imaged to perform the same transformation wholly, and interiorized the action into the process. Through the reflective abstraction of the deep questions, they became aware of a mental process as a totality, then encapsulated into an object, and enhanced the improvement of the concepts.