{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/80328"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/80328","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Children's Understanding of the Uses of Parentheses","abstract":"\"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.\"","abstract_html":""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."","abstract_has_math":false,"creators":["Hsieh, Ju-Shan"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Education","degree_department":null,"school":null,"contributors":["Baroody, Arthur J."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":null,"date_issued":"10000-01-01","date_published":"10000-01-01","updated_at":"2026-07-22T22:26:14Z","subjects":["Education, Elementary"],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(MiAaPQ)AAI9944885"],"render_values":[{"text":"(MiAaPQ)AAI9944885","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/80328","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Baroody, Arthur J."]},{"key":"dc:creator","label":"Author","values":["Hsieh, Ju-Shan"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["10000-01-01","1999","2015-09-25T19:56:34Z"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Education"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Education, Elementary"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/80328","(MiAaPQ)AAI9944885"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["\"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.\"","Made available in DSpace on 2015-09-25T19:56:34Z (GMT). No. of bitstreams: 2 license.txt: 4848 bytes, checksum: 96035ab3f5e1c23cc7138a224ce498bd (MD5) 9944885.pdf: 7502361 bytes, checksum: 7bd10002710ec17423a42578a1e93c66 (MD5) Previous issue date: 1999","Embargo set by: Seth Robbins for item 81610 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","172 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1999."]},{"key":"dc:title","label":"Title","values":["Children's Understanding of the Uses of Parentheses"]}]}],"canonical_facts":{"dc:contributor":["Baroody, Arthur J."],"dc:creator":["Hsieh, Ju-Shan"],"dc:date":["10000-01-01","1999","2015-09-25T19:56:34Z"],"dc:description":["\"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.\"","Made available in DSpace on 2015-09-25T19:56:34Z (GMT). No. of bitstreams: 2 license.txt: 4848 bytes, checksum: 96035ab3f5e1c23cc7138a224ce498bd (MD5) 9944885.pdf: 7502361 bytes, checksum: 7bd10002710ec17423a42578a1e93c66 (MD5) Previous issue date: 1999","Embargo set by: Seth Robbins for item 81610 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","172 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1999."],"dc:identifier":["http://hdl.handle.net/2142/80328","(MiAaPQ)AAI9944885"],"dc:language":["eng"],"dc:subject":["Education, Elementary"],"dc:title":["Children's Understanding of the Uses of Parentheses"],"dc:type":["text"],"thesis:degree_discipline":["Education"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:26:14Z"}