一個發展小二學童乘法概念的行動研究

Abstract

本研究乃作者於在職進修期間，對授課班級的小二學童進行乘法概念行動研究與教學實驗的成果。教學活動的設計採用Anghileri (2006)鷹架學習的3層次教師策略，搭配分組計數概念、倍的數學語言、故事繪本的似真情境以及遊戲與把玩四個核心教學構念，透過在行動中反思和對行動的反思引動每一階段的教學活動。實驗中使用的乘法概念問卷和數學態度量表皆經過預試並修正，在兩階段三循環的研究中進行共四次前後測，並輔以晤談記錄以及研究日誌，蒐集學童乘法概念發展的量化與質化實徵資料。 經過兩階段三循環的研究之後發現：教學前18位曾經使用乘法算式的學童只有2位具有乘法概念，在等值群組、倍數比較、陣列與組合四類問題中，學童最易瞭解的是陣列和等值群組問題中的乘法概念；經過本教學實驗之後，學童在等值群組問題有98.8%、倍數比較問題97.1%、陣列問題100%、組合問題26.5%達到乘法思維。此外，本班大多數學童乘法概念的共通可能學習路徑是：由加法思維的逐一點數開始，逐漸產生分組計數概念，接著建立倍的概念，並使用倍的語言溝通加法和乘法算式，在逐次的遞迴中逐漸進入乘法思維。同時，由數學態度量表中也顯示：大部分的學童在學習乘法概念的過程中，都曾經獲得正向情意的支撐。這似乎表明：故事繪本和數學遊戲的似真情境營造都具有相當的教育價值。 依據本研究的結果，個人認為教師在進行小二乘法概念教學時可以依循下列五項原則：對學童可能發生的學習困難，要先預擬因應的教學策略；在教學計畫時，應該採用分組計數活動並以倍的語言溝通；以「乘法是單位量轉換問題」的教學認知，作為發展學童乘法概念的主軸；於教學之前先分析學童的表現，再擬定解決大多數學童表現的教學方案；透過反思、察覺和與伙伴教師的討論，引動自己教學知識和信念的轉變。

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.