括號學習的理論模式

The Theoretical Framework of Parentheses Learning Model

謝如山

《藝術學報》 66期 (2000/08) Pp. 149-166

https://doi.org/10.6793/JNTCA.200008.0149

摘要

括號的概念可用於整數、分數、小數、負數及代數的情況。與括號相關的法則有，如運算先做的順序，包括括號多餘法則如 (5 x 3) + 2 = 5 x 3 + 2、結合律如 (5 + 3) + 2 = 5 + (3 + 2)、分配律如5 x (3 + 2) = 5 x 3 + 5 x 2及符號改變法則，如5 - (3 + 2) = 5 - 3 - 2。從臺灣小學教學課程與前期研究，我提出了括號學習模式五階段的認知層次。層次零是學童知道先乘除復加減的法則：層次一是學童了解括號的意義，先做法則；層次二是學童瞭解括號相關的法則，但是他們可能誤用這些法則。層次三是學童可辨別這些法則；層次四是學童能辨別各法則內的不同題型。本研究的目的在檢視上述的五個層次，樣本來自於台北市東區兩間國民小學120位四年級學生。使用分層系統抽樣的方式，每班選出低、中、高學童各兩名，共20班。統計方法採用潛在分類模式 ( lateat class model ) 來確定括號相關模式的階層屬性。

並列摘要

Parentheses can be used in the context of integers, fractions, decimals, negative real numbers, or algebra in four different ways: to indicate the order of operations (e.g., 5 x (3 + 2) = 5 x 5), to represent the associative property (e.g., (5 + 3) + 2 = 5 + (3 + 2)), to represent the distributive property (e.g., 5 x (3 + 2) = 5 x 3 + 5 x 2), and to represent the sign-change rule (e.g., 5 - (3 + 2) = 5 - 3 - 2). A curriculum analysis and preliminary study lead to the following model regarding the learning of these uses: During the first of five phases, children learn the basic operations and the order-of-operations rule but do not understand the other uses of parentheses. In the second phase, they understand only that parentheses mean ”do what is inside parentheses first” (i.e., they have some understanding that parentheses indicate the order of operations). In the third phase, children learn more about the uses of parentheses. However, this knowledge is incomplete and, as a result, frequently overapplied. In the fourth phase, children construct a more complete and interconnected understanding of the four uses, and, thus, no longer overapply them. In the fifth phase, children construct a relatively more complete and general understanding of the four uses in that they can apply them regardless of format. The aim of this study is to confirm this hierarchical learning model regarding the uses of parentheses. A stratified random sample of 120 fourth-graders from Taipei, Taiwan participated in this research. A latent class analysis supports the hierarchical learning model.