從符號化觀點探討學生等比例概念之個案研究

A Case Study of Inquiring Students' Proportional Concept on a Symbolizing Perspective

陳蕙茹(Hui-Ju Chen) ； 柳賢(Shian Leou)

《花蓮教育大學學報》 27期 (2008/12) Pp. 193-219

https://doi.org/10.7073/JNHUE.200812.0193

摘要

在「等比例」概念的形成過程中，其符號化過程對於「算術」與「代數」的橋接，扮演著不容輕易忽視的重要角色。然而，在我國的數學課程發展中卻是被忽略的。本研究透過個案研究方式，從符號化觀點探討國一學生處理等比例概念時的符號使用以及運作方式，進而探討影響因素。基於第一個目的所得的研究結論為：（一）在數的情境中，使用「x」重新表徵等比例式；在量的情境中，對於「求未知數」的問題，使用等比例式表徵問題情境，對於「比較型」的問題則使用算式來表徵問題情境。（二）雖然在數與量情境中使用的策略不同，但皆源自「比值」概念。（三）將「x」物化，並以之進行形式化的推理。而影響之因素有：對「比例式」的符號角色與「等式」的符號角色產生混合，且，「比值」概念與「分數」概念產生混合。最後，研究者根據本研究之結論，對數學教學與未來研究提出相關建議。

關鍵字

等比例 ； 符號化

並列摘要

In the forming process of the concept of proportion, symbolizing is important to bridge the gap between Arithmetic and Algebra. However it was neglected in the development of mathematics curriculum in our country. This case study inquired the proportional symbolism of a student in junior high school on the symbolizing perspective, and then probed into the influence factors. For the first purpose, here are three conclusions: Ⅰ. Using ”x” represented the proportion expression in the arithmetical problems as well as in ”miss-value” problems in the situation of quantity. But for the ”comparative” problems in the situation of quantity, he used arithmetic to present it. Ⅱ. Though the strategy founded in the quantity situation was different from it in arithmetical situation, all were found to stem from the ”rate” concept. Ⅲ. ”X” was reification for him, and he could reason formally with it. However, we found that the symbolic roles of ”proportional expression” and the ”equality” expression were fusion. The concept of ”rate” and ”fraction” were fusion too. Finally, we proposed the relevant suggestion on mathematics teaching and future study according to the conclusion of this research.

並列關鍵字

proportion ； symbolizing