Children's Understanding of the Uses of Parentheses

"Parentheses can be used in the context of whole numbers, fractions, decimals, integers (which include negative values), or algebra in four different ways: 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), to represent the sign-change rule (e.g., 5 - (3 + 2) = 5 - 3 - 2), and to indicate the order of operations (e.g., 5 x (3 + 2) = 5 x 5). 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 have no understanding about the use of parentheses. In the second phase, children understand only that parentheses mean ""do what is inside parentheses first."" In the third phase, children construct an incomplete understanding of the four principles and may overapply these uses. In the fourth phase, children construct a relatively complete and interconnected understanding of the four uses. In the fifth phase, children construct an even more complete understanding of the four principles and can apply them regardless of format (e.g., apply the sign-change rule in the subtractive type, 5 - (3 - 2) = 5 - 3 + 2, and the divisive type, 5 ÷ (3 x 2) = 5 ÷ 3 ÷ 2). The aim of this study was to confirm the learning model regarding the uses of parentheses. A stratified random sample of 120 fourth graders from Taipei, Taiwan participated in this research, and 12 of these students were interviewed. A latent class analysis and the interview data support the hierarchical learning model. Also examined were the effects of format, namely which operations were involved and the location of parentheses. The main implication of the hierarchical learning model and the results of this study is that teaching and assessment of the order-of-operations, the associative property, the distributive property, and the sign-change rule should be done in an integrated manner."

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Education

Grantor

University of Illinois at Urbana-Champaign

Date dc:date

10000-01-01

Author dc:creator

• Hsieh, Ju-Shan

Contributors dc:contributor

• Baroody, Arthur J.

• Education, Elementary

Language dc:language

eng

Identifier

(MiAaPQ)AAI9944885

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/80328