{"id":43666,"date":"2011-05-15T15:05:50","date_gmt":"2011-05-15T15:05:50","guid":{"rendered":"http:\/\/mast.thomasjpitts.co.uk\/?p=177"},"modified":"2011-05-15T15:05:50","modified_gmt":"2011-05-15T15:05:50","slug":"making-use-of-childrens-powers-to-produce-algebraic-thinking","status":"publish","type":"post","link":"https:\/\/thomasjpitts.co.uk\/wordpress\/2011\/05\/15\/making-use-of-childrens-powers-to-produce-algebraic-thinking\/","title":{"rendered":"Making Use Of Children&#8217;s Powers To Produce Algebraic Thinking"},"content":{"rendered":"<p><strong>Key points<\/strong>:<\/p>\n<ul>\n<li>That \u00a0children struggle to make the move from being in a situation where &#8220;not knowing answers (to arithmetic calculations) is treated negatively, and then suddenly introduced to algebra in which not knowing is treated positively as an opportunity to use symbols, as a way of working with not knowing.&#8221;<\/li>\n<li>&#8220;Whenever a learner solves a problem, there is available the question &#8216;What is the method that was used?&#8217;, which in intimately tied up with the question &#8216;What can be changed about the problem and still the same technique or method will work?&#8217; or &#8216;What is the class of problems which can be solved similarly?&#8217; S. Brown and M. Walter (1981) suggest asking &#8216;What if &#8230; something changed?&#8217; or &#8216;What if not&#8230;?&#8217; Watson and Mason advocate\u00a0explicitly\u00a0asking learners to consider what <em>dimensions of possible variation<\/em> and corresponding <em>ranges of\u00a0permissible\u00a0change<\/em> they are aware of (Mason &amp; Johnston-Wilder, 2004; Watson &amp; Mason, 2004) as stimulus to becoming aware of, and even expressing features of, the general class of problem of which the ones considered are representative.&#8221;<\/li>\n<li><strong>Awareness of and Expressing Generality<\/strong>: &#8220;Picture-pattern sequences (Mason, 1988b; Mason, Graham, Pimm &amp; Gowat, 1983; South Notts, n.d.) provide just one context for generalizing.&#8221; Learners are shown a sequence of pictures, and then specify a method describing how the pattern is growing through each term in the sequence, extrapolating these ideas to fit further terms in the sequence<\/li>\n<li>&#8220;Getting learners to make use of their powers is not simply an approach to algebra or even approach to mathematics. It is mathematics.&#8221;<\/li>\n<\/ul>\n<p><strong>Main Reference<\/strong>:<\/p>\n<ul>\n<li>Mason, J. (2008) \u2018Making Use Of Children&#8217;s Powers To Produce Algebraic Thinking\u2019 in Kaput, J., Carraher, D. and Blanton, M.\u00a0(eds.)\u00a0<em>Algebra In The Early Grades.<\/em> New York:\u00a0<a title=\"Taylor &amp; Francis\" rel=\"homepage\" href=\"http:\/\/www.taylorandfrancis.com\/\">Lawrence Erlbaum Associates<\/a>, pp. 57-94<\/li>\n<\/ul>\n<p><strong>Citations<\/strong>:<\/p>\n<ul>\n<li>Brown, S. and Walter, M. (1982) <em>The art of problem posing<\/em>. Philadelphia: <a class=\"zem_slink\" title=\"Franklin Institute\" rel=\"homepage\" href=\"http:\/\/www.fi.edu\/\">Franklin Institute<\/a> Press.<\/li>\n<li>Mason, J. (1988b) <em>Expressing generality<\/em> [project\u00a0update]. <a class=\"zem_slink\" title=\"Milton Keynes\" rel=\"homepage\" href=\"http:\/\/www.milton-keynes.gov.uk\">Milton Keynes<\/a>: <a class=\"zem_slink\" title=\"McGraw-Hill\" rel=\"homepage\" href=\"http:\/\/www.mcgraw-hill.com\/\">Open University Press<\/a>.<\/li>\n<li>Mason, J. and Johnston-Wilder, S. (2004). <em>Designing and using mathematical tasks<\/em>. Milton Keynes: Open University Press.<\/li>\n<li>Mason, J., Graham, A., Pimm, D. and Gowar, N. (1985). <em>Routes to, roots of algebra<\/em>. Milton Keynes: Open University Press.<\/li>\n<li>South Notts Project. (n.d.) <em>Material for secondary mathematics<\/em>. Nottingham: Shell Centre, University Of Nottingham.<\/li>\n<li>Watson, A. and Mason, J. (2004) <em>Mathematics as a constructive activity: The role of learner-generated examples<\/em>. Mahwah: Lawrence Erlbaum Associates.<\/li>\n<\/ul>\n<p><strong>Buy These Books<\/strong>:<\/p>\n<ul>\n<li><a title=\"Algebra In The Early Grades\" href=\"http:\/\/www.amazon.co.uk\/gp\/product\/0805854738\/ref=as_li_qf_sp_asin_il_tl?ie=UTF8&amp;tag=mapr07-21&amp;linkCode=as2&amp;camp=1634&amp;creative=6738&amp;creativeASIN=0805854738\" target=\"_blank\" rel=\"noopener\">Algebra in The Early Grades at Amazon.co.uk<\/a><\/li>\n<li><a href=\"http:\/\/www.amazon.co.uk\/gp\/product\/0805849777\/ref=as_li_qf_sp_asin_il_tl?ie=UTF8&amp;tag=mapr07-21&amp;linkCode=as2&amp;camp=1634&amp;creative=6738&amp;creativeASIN=0805849777\">The art of problem posing at Amazon.co.uk<\/a><\/li>\n<li><a href=\"http:\/\/www.amazon.co.uk\/gp\/product\/1899618651\/ref=as_li_qf_sp_asin_il_tl?ie=UTF8&amp;tag=mapr07-21&amp;linkCode=as2&amp;camp=1634&amp;creative=6738&amp;creativeASIN=1899618651\" target=\"_blank\" rel=\"noopener\">Designing and using mathematical tasks at Amazon.co.uk<\/a><\/li>\n<\/ul>\n<div class=\"zemanta-pixie\" style=\"margin-top: 10px; height: 15px;\"><img data-recalc-dims=\"1\" decoding=\"async\" class=\"zemanta-pixie-img\" style=\"border: none; float: right;\" src=\"https:\/\/i0.wp.com\/img.zemanta.com\/pixy.gif?w=1165\" alt=\"\" \/><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Key points: That \u00a0children struggle to make the move from being in a situation where &#8220;not knowing answers (to arithmetic calculations) is treated negatively, and then suddenly introduced to algebra in which not knowing is treated positively as an opportunity to use symbols, as a way of working with not knowing.&#8221; &#8220;Whenever a learner solves [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"ngg_post_thumbnail":0,"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[4436],"tags":[87,4452,4454,4471,4475,1015,4492,4518,4528,4532,1909,1910,4549,2145,3427],"class_list":["post-43666","post","type-post","status-publish","format-standard","hentry","category-assignment-1","tag-87","tag-algebra","tag-algebra-in-the-early-grades","tag-blanton","tag-carraher","tag-education","tag-elementary-algebra","tag-kaput","tag-making-use-of-childrens-powers-to-produce-algebraic-thinking","tag-mason","tag-math","tag-mathematics","tag-milton-keynes","tag-new-york","tag-philadelphia","has-post-title","has-post-date","has-post-category","has-post-tag","has-post-comment","has-post-author",""],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p6OeSW-bmi","jetpack-related-posts":[],"builder_content":"","_links":{"self":[{"href":"https:\/\/thomasjpitts.co.uk\/wordpress\/wp-json\/wp\/v2\/posts\/43666","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/thomasjpitts.co.uk\/wordpress\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/thomasjpitts.co.uk\/wordpress\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/thomasjpitts.co.uk\/wordpress\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/thomasjpitts.co.uk\/wordpress\/wp-json\/wp\/v2\/comments?post=43666"}],"version-history":[{"count":0,"href":"https:\/\/thomasjpitts.co.uk\/wordpress\/wp-json\/wp\/v2\/posts\/43666\/revisions"}],"wp:attachment":[{"href":"https:\/\/thomasjpitts.co.uk\/wordpress\/wp-json\/wp\/v2\/media?parent=43666"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/thomasjpitts.co.uk\/wordpress\/wp-json\/wp\/v2\/categories?post=43666"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/thomasjpitts.co.uk\/wordpress\/wp-json\/wp\/v2\/tags?post=43666"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}