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counting in grouping
language of multiplier
This research is an action research with teaching experiment design on the improvement of 2nd graders’ multiplicative conceptions during author’s teaching. I adopted Anghileri’s (2006) framework of scaffolding practices for learning, cooperating with the four core constructs including counting in grouping, language of multiplier, picture storybook drawing on the real-world settings, and game and playing, to intrigue the teaching activities through the reflection-in-action and reflection-on-action processes. In the experiment, the survey of multiplication conceptions and the measurement of attitudes toward mathematics were pre-tested and revised. There are four pre and post tests in the 2-phase and 3-cycle study. I supplement the study with the record of interviews and study diary, and collect the qualitative and quantitative data of students’ development of multiplication conceptions. Through the 2-phase and 3-cycle process, I found that before teaching, only two of eighteen students have multiplication conceptions in the four types of questions related to equal groups, multiplicative comparison, rectangular array and Cartesian product. The students were almost readies to understand the multiplication concept of rectangular array and equal groups. After the experiment, there were 98.8% students having multiple conceptions on equal groups, 97.1% on multiplicative comparison, 100% on rectangular array, and 26.5% on Cartesian product. The hypothetical learning path for most students in the class seemed to be starting at counting in one, then go to counting in grouping, making the conception of multiplicative, using the mathematic language of multiplier, then to link up the addition and multiplication formula. At the same time, the mathematics attitude scales showed that in the process of learning multiplicative concepts most of the students obtain positive emotional support. This seems to indicate that the real-world settings of picture storybooks and mathematical games had its educational values and implications. According to the results of the present study, I believe that primary school teachers could follow the following five principle when they teach 2nd graders’ multiplication: for students’ learning difficulties, teachers could make corresponding teaching strategies beforehand; in planning classroom teaching, teachers might adopt activities about counting in grouping and communicate with mathematical language of multiplier; “multiplication is the question of unit conversion” should be recognized as the core to develop students’ multiplicative conceptions; they may try to analyze students’ behaviors before teaching and using relevant teaching strategies to solve most of the students’ problems; through the reflection, observation, and discussion with colleagues, primary school teachers could intrigue the changes of their own pedagogical knowledge and beliefs.
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