Fostering Formal Commutativity Knowledge with Approximate Arithmetic
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{"title"=>"Fostering formal commutativity knowledge with approximate arithmetic", "type"=>"journal", "authors"=>[{"first_name"=>"Sonja Maria", "last_name"=>"Hansen", "scopus_author_id"=>"57075629900"}, {"first_name"=>"Hilde", "last_name"=>"Haider", "scopus_author_id"=>"7003800560"}, {"first_name"=>"Alexandra", "last_name"=>"Eichler", "scopus_author_id"=>"11839074500"}, {"first_name"=>"Claudia", "last_name"=>"Godau", "scopus_author_id"=>"57076685100"}, {"first_name"=>"Peter A.", "last_name"=>"Frensch", "scopus_author_id"=>"6701727221"}, {"first_name"=>"Robert", "last_name"=>"Gaschler", "scopus_author_id"=>"15821749700"}], "year"=>2015, "source"=>"PLoS ONE", "identifiers"=>{"issn"=>"19326203", "scopus"=>"2-s2.0-84955465464", "pmid"=>"26560311", "doi"=>"10.1371/journal.pone.0142551", "pui"=>"607878005", "sgr"=>"84955465464"}, "id"=>"4ed94538-820c-325a-95a5-d577ef820e30", "abstract"=>"How can we enhance the understanding of abstract mathematical principles in elementary school? Different studies found out that nonsymbolic estimation could foster subsequent exact number processing and simple arithmetic. Taking the commutativity principle as a test case, we investigated if the approximate calculation of symbolic commutative quantities can also alter the access to procedural and conceptual knowledge of a more abstract arithmetic principle. Experiment 1 tested first graders who had not been instructed about commutativity in school yet. Approximate calculation with symbolic quantities positively influenced the use of commutativity-based shortcuts in formal arithmetic. We replicated this finding with older first graders (Experiment 2) and third graders (Experiment 3). Despite the positive effect of approximation on the spontaneous application of commutativity-based shortcuts in arithmetic problems, we found no comparable impact on the application of conceptual knowledge of the commutativity principle. Overall, our results show that the usage of a specific arithmetic principle can benefit from approximation. However, the findings also suggest that the correct use of certain procedures does not always imply conceptual understanding. Rather, the conceptual understanding of commutativity seems to lag behind procedural proficiency during elementary school.", "link"=>"http://www.mendeley.com/research/fostering-formal-commutativity-knowledge-approximate-arithmetic", "reader_count"=>6, "reader_count_by_academic_status"=>{"Unspecified"=>1, "Researcher"=>2, "Student > Ph. D. Student"=>2, "Student > Postgraduate"=>1}, "reader_count_by_user_role"=>{"Unspecified"=>1, "Researcher"=>2, "Student > Ph. D. Student"=>2, "Student > Postgraduate"=>1}, "reader_count_by_subject_area"=>{"Unspecified"=>1, "Psychology"=>4, "Social Sciences"=>1}, "reader_count_by_subdiscipline"=>{"Social Sciences"=>{"Social Sciences"=>1}, "Psychology"=>{"Psychology"=>4}, "Unspecified"=>{"Unspecified"=>1}}, "group_count"=>0}

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Figshare

  • {"files"=>["https://ndownloader.figshare.com/files/2436670"], "description"=>"<p>Example of a noncommutative trial presented in the approximation task.</p>", "links"=>[], "tags"=>["commutativity principle", "calculation", "approximation", "experiment", "arithmetic principle", "Fostering Formal Commutativity Knowledge", "shortcut", "application", "approximate", "understanding", "grader", "quantity"], "article_id"=>1600228, "categories"=>["Uncategorised"], "users"=>["Sonja Maria Hansen", "Hilde Haider", "Alexandra Eichler", "Claudia Godau", "Peter A. Frensch", "Robert Gaschler"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0142551.g001", "stats"=>{"downloads"=>0, "page_views"=>0, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Example_of_a_noncommutative_trial_presented_in_the_approximation_task_/1600228", "title"=>"Example of a noncommutative trial presented in the approximation task.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-11-11 03:22:23"}
  • {"files"=>["https://ndownloader.figshare.com/files/2436672"], "description"=>"<p>Example of a commutative trial presented in the approximation task.</p>", "links"=>[], "tags"=>["commutativity principle", "calculation", "approximation", "experiment", "arithmetic principle", "Fostering Formal Commutativity Knowledge", "shortcut", "application", "approximate", "understanding", "grader", "quantity"], "article_id"=>1600230, "categories"=>["Uncategorised"], "users"=>["Sonja Maria Hansen", "Hilde Haider", "Alexandra Eichler", "Claudia Godau", "Peter A. Frensch", "Robert Gaschler"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0142551.g002", "stats"=>{"downloads"=>2, "page_views"=>0, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Example_of_a_commutative_trial_presented_in_the_approximation_task_/1600230", "title"=>"Example of a commutative trial presented in the approximation task.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-11-11 03:22:23"}
  • {"files"=>["https://ndownloader.figshare.com/files/2436675"], "description"=>"<p>Within experiments, numbers of solved problems are depicted as a function of subset and condition. In the approximation-first and the computation-first conditions, Subset 1 refers to commutative problems, and Subset 2 to noncommutative baseline problems. In the control condition, subsets 1 and 2 only contained noncommutative problems. Error bars reflect within-participants confidence intervals based on the MSe of the Condition X Subset Interaction [<a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0142551#pone.0142551.ref051\" target=\"_blank\">51</a>].</p>", "links"=>[], "tags"=>["commutativity principle", "calculation", "approximation", "experiment", "arithmetic principle", "Fostering Formal Commutativity Knowledge", "shortcut", "application", "approximate", "understanding", "grader", "quantity"], "article_id"=>1600233, "categories"=>["Uncategorised"], "users"=>["Sonja Maria Hansen", "Hilde Haider", "Alexandra Eichler", "Claudia Godau", "Peter A. Frensch", "Robert Gaschler"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0142551.g003", "stats"=>{"downloads"=>4, "page_views"=>3, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Numbers_of_solved_exact_arithmetic_problems_in_Experiments_1_2_and_3_/1600233", "title"=>"Numbers of solved exact arithmetic problems in Experiments 1, 2, and 3.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-11-11 03:22:23"}
  • {"files"=>["https://ndownloader.figshare.com/files/2436676"], "description"=>"<p>Within experiments, commutative and noncommutative problems are depicted separately for the approximation-first, the computation-first and the control group. Error bars reflect within-participants confidence intervals based on the MSe of the Condition X Subset Interaction [<a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0142551#pone.0142551.ref051\" target=\"_blank\">51</a>].</p>", "links"=>[], "tags"=>["commutativity principle", "calculation", "approximation", "experiment", "arithmetic principle", "Fostering Formal Commutativity Knowledge", "shortcut", "application", "approximate", "understanding", "grader", "quantity"], "article_id"=>1600234, "categories"=>["Uncategorised"], "users"=>["Sonja Maria Hansen", "Hilde Haider", "Alexandra Eichler", "Claudia Godau", "Peter A. Frensch", "Robert Gaschler"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0142551.g004", "stats"=>{"downloads"=>1, "page_views"=>5, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Percentage_of_correctly_answered_trials_in_the_approximation_task_of_Experiments_1_2_and_3_/1600234", "title"=>"Percentage of correctly answered trials in the approximation task of Experiments 1, 2, and 3.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-11-11 03:22:23"}
  • {"files"=>["https://ndownloader.figshare.com/files/2436677"], "description"=>"<p>Problems in bold indicate the commutative pairs of the respective task.</p><p>Examples of the problems presented in the computation task (commutative and noncommutative subsets) and (in Experiments <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0142551#sec014\" target=\"_blank\">2</a> and <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0142551#sec027\" target=\"_blank\">3</a>) the judgment task.</p>", "links"=>[], "tags"=>["commutativity principle", "calculation", "approximation", "experiment", "arithmetic principle", "Fostering Formal Commutativity Knowledge", "shortcut", "application", "approximate", "understanding", "grader", "quantity"], "article_id"=>1600235, "categories"=>["Uncategorised"], "users"=>["Sonja Maria Hansen", "Hilde Haider", "Alexandra Eichler", "Claudia Godau", "Peter A. Frensch", "Robert Gaschler"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0142551.t001", "stats"=>{"downloads"=>1, "page_views"=>6, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Examples_of_the_problems_presented_in_the_computation_task_commutative_and_noncommutative_subsets_and_in_Experiments_2_and_3_the_judgment_task_/1600235", "title"=>"Examples of the problems presented in the computation task (commutative and noncommutative subsets) and (in Experiments 2 and 3) the judgment task.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2015-11-11 03:22:23"}
  • {"files"=>["https://ndownloader.figshare.com/files/2436678"], "description"=>"<p>Mean number of solved approximation problems, rates of correct answers in general as well as separately for commutative and noncommutative 'candy' problems are depicted for both conditions (SD in parentheses). Participants had 1.5 min to solve 11 approximation problems.</p><p>Performance of young first graders in the approximation task in Experiment 1.</p>", "links"=>[], "tags"=>["commutativity principle", "calculation", "approximation", "experiment", "arithmetic principle", "Fostering Formal Commutativity Knowledge", "shortcut", "application", "approximate", "understanding", "grader", "quantity"], "article_id"=>1600236, "categories"=>["Uncategorised"], "users"=>["Sonja Maria Hansen", "Hilde Haider", "Alexandra Eichler", "Claudia Godau", "Peter A. Frensch", "Robert Gaschler"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0142551.t002", "stats"=>{"downloads"=>1, "page_views"=>3, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Performance_of_young_first_graders_in_the_approximation_task_in_Experiment_1_/1600236", "title"=>"Performance of young first graders in the approximation task in Experiment 1.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2015-11-11 03:22:23"}
  • {"files"=>["https://ndownloader.figshare.com/files/2436679"], "description"=>"<p>Mean number of solved approximation problems, rates of correct answers in general as well as separately for commutative and noncommutative 'candy' problems are depicted for both conditions (SD in parentheses). Participants had 2 minutes to solve 14 approximation problems.</p><p>Performance of older first graders in Experiment 2 and third graders in Experiment 3 in the approximation task.</p>", "links"=>[], "tags"=>["commutativity principle", "calculation", "approximation", "experiment", "arithmetic principle", "Fostering Formal Commutativity Knowledge", "shortcut", "application", "approximate", "understanding", "grader", "quantity"], "article_id"=>1600237, "categories"=>["Uncategorised"], "users"=>["Sonja Maria Hansen", "Hilde Haider", "Alexandra Eichler", "Claudia Godau", "Peter A. Frensch", "Robert Gaschler"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0142551.t003", "stats"=>{"downloads"=>1, "page_views"=>6, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Performance_of_older_first_graders_in_Experiment_2_and_third_graders_in_Experiment_3_in_the_approximation_task_/1600237", "title"=>"Performance of older first graders in Experiment 2 and third graders in Experiment 3 in the approximation task.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2015-11-11 03:22:23"}
  • {"files"=>["https://ndownloader.figshare.com/files/2436680"], "description"=>"<p>Rate of hits and false alarms as well as the sensitivity index and the response criterion for each condition are depicted (SD in parentheses).</p><p>Performance of older first graders in the judgment task in Experiment 2.</p>", "links"=>[], "tags"=>["commutativity principle", "calculation", "approximation", "experiment", "arithmetic principle", "Fostering Formal Commutativity Knowledge", "shortcut", "application", "approximate", "understanding", "grader", "quantity"], "article_id"=>1600238, "categories"=>["Uncategorised"], "users"=>["Sonja Maria Hansen", "Hilde Haider", "Alexandra Eichler", "Claudia Godau", "Peter A. Frensch", "Robert Gaschler"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0142551.t004", "stats"=>{"downloads"=>1, "page_views"=>4, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Performance_of_older_first_graders_in_the_judgment_task_in_Experiment_2_/1600238", "title"=>"Performance of older first graders in the judgment task in Experiment 2.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2015-11-11 03:22:23"}
  • {"files"=>["https://ndownloader.figshare.com/files/2436681"], "description"=>"<p>Correlation coefficients between procedural and conceptual knowledge for first graders, depicted separately for all participants and for the three conditions of Experiment 2. Subset 1 contains the commutative problems, Subset 2 the noncommutative problems.</p><p>Correlation coefficients between procedural and conceptual knowledge in Experiment 1 and Experiment 2.</p>", "links"=>[], "tags"=>["commutativity principle", "calculation", "approximation", "experiment", "arithmetic principle", "Fostering Formal Commutativity Knowledge", "shortcut", "application", "approximate", "understanding", "grader", "quantity"], "article_id"=>1600239, "categories"=>["Uncategorised"], "users"=>["Sonja Maria Hansen", "Hilde Haider", "Alexandra Eichler", "Claudia Godau", "Peter A. Frensch", "Robert Gaschler"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0142551.t005", "stats"=>{"downloads"=>1, "page_views"=>1, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Correlation_coefficients_between_procedural_and_conceptual_knowledge_in_Experiment_1_and_Experiment_2_/1600239", "title"=>"Correlation coefficients between procedural and conceptual knowledge in Experiment 1 and Experiment 2.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2015-11-11 03:22:23"}
  • {"files"=>["https://ndownloader.figshare.com/files/2436682"], "description"=>"<p>Performance of third graders in the judgment task of Experiment 3.</p>", "links"=>[], "tags"=>["commutativity principle", "calculation", "approximation", "experiment", "arithmetic principle", "Fostering Formal Commutativity Knowledge", "shortcut", "application", "approximate", "understanding", "grader", "quantity"], "article_id"=>1600240, "categories"=>["Uncategorised"], "users"=>["Sonja Maria Hansen", "Hilde Haider", "Alexandra Eichler", "Claudia Godau", "Peter A. Frensch", "Robert Gaschler"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0142551.t006", "stats"=>{"downloads"=>2, "page_views"=>5, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Performance_of_third_graders_in_the_judgment_task_of_Experiment_3_/1600240", "title"=>"Performance of third graders in the judgment task of Experiment 3.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2015-11-11 03:22:23"}
  • {"files"=>["https://ndownloader.figshare.com/files/2436683"], "description"=>"<p>Correlation coefficients between procedural and conceptual knowledge for third graders, depicted separately for all participants and for the three conditions of Experiment 3. Subset 1 contains the commutative problems, Subset 2 the noncommutative problems.</p><p>Correlation coefficients between procedural and conceptual knowledge for third graders.</p>", "links"=>[], "tags"=>["commutativity principle", "calculation", "approximation", "experiment", "arithmetic principle", "Fostering Formal Commutativity Knowledge", "shortcut", "application", "approximate", "understanding", "grader", "quantity"], "article_id"=>1600241, "categories"=>["Uncategorised"], "users"=>["Sonja Maria Hansen", "Hilde Haider", "Alexandra Eichler", "Claudia Godau", "Peter A. Frensch", "Robert Gaschler"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0142551.t007", "stats"=>{"downloads"=>1, "page_views"=>5, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Correlation_coefficients_between_procedural_and_conceptual_knowledge_for_third_graders_/1600241", "title"=>"Correlation coefficients between procedural and conceptual knowledge for third graders.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2015-11-11 03:22:23"}
  • {"files"=>["https://ndownloader.figshare.com/files/2436684", "https://ndownloader.figshare.com/files/2436685"], "description"=>"<div><p>How can we enhance the understanding of abstract mathematical principles in elementary school? Different studies found out that nonsymbolic estimation could foster subsequent exact number processing and simple arithmetic. Taking the commutativity principle as a test case, we investigated if the approximate calculation of symbolic commutative quantities can also alter the access to procedural and conceptual knowledge of a more abstract arithmetic principle. Experiment 1 tested first graders who had not been instructed about commutativity in school yet. Approximate calculation with symbolic quantities positively influenced the use of commutativity-based shortcuts in formal arithmetic. We replicated this finding with older first graders (<a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0142551#sec014\" target=\"_blank\">Experiment 2</a>) and third graders (<a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0142551#sec027\" target=\"_blank\">Experiment 3</a>). Despite the positive effect of approximation on the spontaneous application of commutativity-based shortcuts in arithmetic problems, we found no comparable impact on the application of conceptual knowledge of the commutativity principle. Overall, our results show that the usage of a specific arithmetic principle can benefit from approximation. However, the findings also suggest that the correct use of certain procedures does not always imply conceptual understanding. Rather, the conceptual understanding of commutativity seems to lag behind procedural proficiency during elementary school.</p></div>", "links"=>[], "tags"=>["commutativity principle", "calculation", "approximation", "experiment", "arithmetic principle", "Fostering Formal Commutativity Knowledge", "shortcut", "application", "approximate", "understanding", "grader", "quantity"], "article_id"=>1600242, "categories"=>["Uncategorised"], "users"=>["Sonja Maria Hansen", "Hilde Haider", "Alexandra Eichler", "Claudia Godau", "Peter A. Frensch", "Robert Gaschler"], "doi"=>["https://dx.doi.org/10.1371/journal.pone.0142551.s001", "https://dx.doi.org/10.1371/journal.pone.0142551.s002"], "stats"=>{"downloads"=>2, "page_views"=>5, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Fostering_Formal_Commutativity_Knowledge_with_Approximate_Arithmetic_/1600242", "title"=>"Fostering Formal Commutativity Knowledge with Approximate Arithmetic", "pos_in_sequence"=>0, "defined_type"=>4, "published_date"=>"2015-11-11 03:22:23"}

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{"start_date"=>"2015-01-01T00:00:00Z", "end_date"=>"2015-12-31T00:00:00Z", "subject_areas"=>[]}
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