The major purpose of this study is to explore and compare the effects between cognitive apprenticeship approach and constructivism approach while they are applied to teach mathematics at the elementary school level in Taiwan. Two fourth-grade classrooms at Szu Wei elementary school and two fourth-grade classrooms at Yen Chou elementary school in Tainan were randomly assigned to three different treatments in teaching addition, subtraction, multiplication, and division:"Cognitive Apprenticeship Group", "Constrctivism Group", and "Control Group", and they had participated in this project for a semester. These sampled students were tested by "Mathematical Problem-Solving Test", "Mathematical Problem-Solving Transfer Test", "Mathematical Belief Questionnaire", and "Mathematical Learning Attitude Questionnaire". Moreover, interviews were conducted with thirty-six students who will be recommended by classroom teachers according to students' sex and mathematics levels in order to explore their problem-solving strategies. The above tests and interviews will be implemented two times: pre-treatment and post-treatment. Results indicated that the cognitive apprenticeship approach can significantly improve students' problem-solving performance on the representation and calculation of curriculum-related mathematical problems. However, other main effects and interactions between treatments and students' mathematical levels were not significant.