{"id":2491,"date":"2026-01-21T09:54:49","date_gmt":"2026-01-21T09:54:49","guid":{"rendered":"https:\/\/resources.kialo-edu.com\/?post_type=docs&#038;p=2491"},"modified":"2026-01-21T09:54:54","modified_gmt":"2026-01-21T09:54:54","password":"","slug":"certainty-in-mathematics-lesson-3","status":"publish","type":"docs","link":"https:\/\/resources.kialo-edu.com\/en\/docs\/certainty-in-mathematics-lesson-3\/","title":{"rendered":"Certainty in Mathematics, Lesson 3"},"content":{"rendered":"<h2 class=\"wp-block-heading\" id=\"block-6e802e7d-7c4e-4f99-b41e-a4ef37f4252c\">Lesson 3: Listening Task<\/h2><p id=\"block-8fe6ce35-0778-450a-88b8-2643dda463a0\"><strong>Focus: <\/strong><em>Do mathematical models strengthen or weaken trust in mathematics as a way of producing knowledge?<\/em><\/p><p id=\"block-3f778713-12d5-4d8d-9bde-ac25005c77ee\">Suggested Length: 1 hour<\/p><p id=\"block-28b95357-4496-445d-ab77-2a82dd1c4323\">Learning Objectives:<\/p><ul id=\"block-62729a0c-dcd1-49a0-8f49-ba87f1485895\" class=\"wp-block-list\">\n<li>Analyze how mathematical models can be used as tools of control rather than objective truth.<\/li>\n\n\n\n<li>Evaluate how corporations, governments, and institutions exercise power through algorithmic decision-making.<\/li>\n\n\n\n<li>Reflect on whether responsibility and transparency strengthen or weaken mathematics in practice.<\/li>\n<\/ul><figure class=\"wp-block-table align-top\"><table class=\"has-background has-fixed-layout\" style=\"background-color:#e9f1f9\"><thead><tr><th>Critical Thinking Concepts<\/th><th>TOK Concepts<\/th><th>Reflection Questions<\/th><\/tr><\/thead><tbody><tr><td><strong>Confronting Biases &amp; Assumptions:<\/strong> Understand how corporations, governments, and universities may design or deploy models in ways that serve profit, efficiency, or control rather than fairness.<br><br><strong>Responsiveness and Flexibility of Thought:<\/strong> Evaluate perspectives of different stakeholders &mdash; mathematicians, policymakers, corporations, regulators, or affected communities.<br><br><strong>Extrapolation &amp; Reapplication of Principles:<\/strong> Apply these lessons to current issues &mdash; e.g., algorithmic bias in hiring, AI-driven credit scoring, predictive policing.<\/td><td><strong>Certainty: <\/strong>What responsibilities do mathematicians, corporations, and governments have when presenting models as objective or certain? <br><br><strong>Power:<\/strong> How do mathematical models shift power between individuals, corporations, and states? <br><br><strong>Perspective:<\/strong> How do different cultural and political contexts shape trust in mathematical models? <\/td><td>What role should institutions play in ensuring mathematical models are transparent and accountable?<br><br>How should educators, policymakers, or citizens prepare people to navigate conflicting or manipulative uses of mathematical models?<br><br>Is it more dangerous to overtrust mathematical models when misused, or to distrust mathematics completely?<br><\/td><\/tr><\/tbody><\/table><\/figure><style>#sp-ea-2504 .spcollapsing { height: 0; overflow: hidden; transition-property: height;transition-duration: 300ms;}#sp-ea-2504.sp-easy-accordion>.sp-ea-single {margin-bottom: 10px; border: 1px solid #e2e2e2; }#sp-ea-2504.sp-easy-accordion>.sp-ea-single>.ea-header a {color: #444;}#sp-ea-2504.sp-easy-accordion>.sp-ea-single>.sp-collapse>.ea-body {background: #fff; color: #444;}#sp-ea-2504.sp-easy-accordion>.sp-ea-single {background: #eee;}#sp-ea-2504.sp-easy-accordion>.sp-ea-single>.ea-header a .ea-expand-icon { float: left; color: #444;font-size: 16px;}<\/style><div id=\"sp_easy_accordion-1766324066\"><div id=\"sp-ea-2504\" class=\"sp-ea-one sp-easy-accordion\" data-ea-active=\"ea-click\" data-ea-mode=\"vertical\" data-preloader=\"\" data-scroll-active-item=\"\" data-offset-to-scroll=\"0\"><div class=\"ea-card sp-ea-single\"><h3 class=\"ea-header\"><a class=\"collapsed\" id=\"ea-header-25040\" role=\"button\" data-sptoggle=\"spcollapse\" data-sptarget=\"#collapse25040\" aria-controls=\"collapse25040\" href=\"#\" aria-expanded=\"false\" tabindex=\"0\"><i aria-hidden=\"true\" role=\"presentation\" class=\"ea-expand-icon eap-icon-ea-expand-plus\"><\/i> Resources and Preparation<\/a><\/h3><div class=\"sp-collapse spcollapse spcollapse\" id=\"collapse25040\" role=\"region\" aria-labelledby=\"ea-header-25040\"> <div class=\"ea-body\"><ol><li>Slides, attached below.<\/li><li>Students can create their own discussion around the central question, or you can clone and use this <a href=\"https:\/\/www.kialo-edu.com\/p\/9a4b865c-f5a1-4b59-bba5-7506909aae66\/587657\" target=\"_blank\" rel=\"noopener nofollow\" referrerpolicy=\"unsafe-url\">ready-made example<\/a>.<\/li><li>Watch the video <a href=\"https:\/\/youtu.be\/gdCJYsKlX_Y?si=ixn2YN5pI7GxzBg2\" target=\"_blank\" rel=\"noopener\">Cathy O'Neil | Weapons of Math Destruction<\/a>&nbsp;before sharing with students.<\/li><\/ol><\/div><\/div><\/div><div class=\"ea-card sp-ea-single\"><h3 class=\"ea-header\"><a class=\"collapsed\" id=\"ea-header-25041\" role=\"button\" data-sptoggle=\"spcollapse\" data-sptarget=\"#collapse25041\" aria-controls=\"collapse25041\" href=\"#\" aria-expanded=\"false\" tabindex=\"0\"><i aria-hidden=\"true\" role=\"presentation\" class=\"ea-expand-icon eap-icon-ea-expand-plus\"><\/i> Introduction<\/a><\/h3><div class=\"sp-collapse spcollapse spcollapse\" id=\"collapse25041\" role=\"region\" aria-labelledby=\"ea-header-25041\"> <div class=\"ea-body\"><p>Present the guiding question,&nbsp;<em>\"Do mathematical models strengthen or weaken trust in mathematics as a way of producing knowledge?\"<\/em><\/p><p>Prompt: &ldquo;<i>Are mathematical models designed to reveal truth or to control outcomes?<\/i>&rdquo;<\/p><p>Recap Lessons 1&ndash;2:<\/p><ul><li>Lesson 1: Debate on whether mathematical structures reveal truth or create illusions of certainty.<\/li><li>Lesson 2: Case studies of contested proofs and models (G&ouml;del, Four Color, 2008 crisis, COVID models).<\/li><\/ul><p>Link: This lesson moves to the contemporary challenge &mdash; how algorithms and big data embed bias while appearing objective.<\/p><\/div><\/div><\/div><div class=\"ea-card sp-ea-single\"><h3 class=\"ea-header\"><a class=\"collapsed\" id=\"ea-header-25042\" role=\"button\" data-sptoggle=\"spcollapse\" data-sptarget=\"#collapse25042\" aria-controls=\"collapse25042\" href=\"#\" aria-expanded=\"false\" tabindex=\"0\"><i aria-hidden=\"true\" role=\"presentation\" class=\"ea-expand-icon eap-icon-ea-expand-plus\"><\/i> Main Activity<\/a><\/h3><div class=\"sp-collapse spcollapse spcollapse\" id=\"collapse25042\" role=\"region\" aria-labelledby=\"ea-header-25042\"> <div class=\"ea-body\"><p><b>Listening Task<\/b><\/p><p>Students watch the video:&nbsp; <a href=\"https:\/\/youtu.be\/gdCJYsKlX_Y?si=ixn2YN5pI7GxzBg2\" target=\"_blank\" rel=\"noopener\"><span data-rich-links=\"{&quot;fple-t&quot;:&quot;Cathy O'Neil | Weapons of Math Destruction&quot;,&quot;fple-u&quot;:&quot;https:\/\/youtu.be\/gdCJYsKlX_Y?si=ixn2YN5pI7GxzBg2&quot;,&quot;fple-mt&quot;:null,&quot;type&quot;:&quot;first-party-link&quot;}\">Cathy O'Neil | Weapons of Math Destruction. <\/span><\/a>Students should actively map the speaker&rsquo;s key arguments, counterarguments, and ethical claims about surveillance capitalism and human sciences.<\/p><p><b>Key Points to Listen For:<\/b><\/p><ul><li>Why do people view mathematical models as authoritative and objective?<\/li><li>What are the five features of &ldquo;Weapons of Math Destruction&rdquo;? (secret, widespread, opaque, biased definitions of success, feedback loops).<\/li><li>How do algorithms in education (teacher scoring), justice (predictive policing, sentencing), and politics (micro-targeting) function as mathematical &ldquo;truths&rdquo; but in fact reinforce inequality?<\/li><li>Who benefits from these models, and who is harmed?<\/li><li>Why are mathematical models described as &ldquo;embedded opinions&rdquo;?<\/li><li>What solutions or responsibilities does O&rsquo;Neil propose?<\/li><\/ul><p><b>Note taking Framework:<\/b><\/p><ul><li><strong>Main Arguments:<\/strong><\/li><li style=\"list-style-type: none\"><ul><li>Algorithms often claim objectivity but embed bias and reinforce inequality.<\/li><li>Models are social constructs, not neutral truths.<\/li><li>Without transparency, mathematical systems undermine democracy and fairness.<\/li><\/ul><\/li><li><strong>Supporting Examples:<\/strong><\/li><li style=\"list-style-type: none\"><ul><li>Teacher value-added model in education (opaque, unaccountable).<\/li><li>Predictive policing and sentencing models (biased data = biased outcomes).<\/li><li>Political micro-targeting (efficient for campaigns, destructive for democracy).<\/li><\/ul><\/li><li><strong>Counterarguments \/ Critical Questions:<\/strong><\/li><li style=\"list-style-type: none\"><ul><li>Are models the problem, or the way humans use them?<\/li><li>Can regulation (like FOIA or GDPR) truly restrain misuse of models?<\/li><li>Are algorithms always harmful, or can they sometimes improve fairness (e.g., replacing blatantly biased human decisions)?<\/li><li>Who holds ultimate responsibility &mdash; the mathematicians building models, the institutions deploying them, or the public interpreting them?<\/li><\/ul><\/li><\/ul><p><b>Kialo Discussion<\/b><\/p><p>In small groups, students create a <strong>new<\/strong> Kialo discussion around the guiding question: &ldquo;<i>Do requirements for transparency and accountability in mathematical models strengthen trust in mathematics as a way of producing knowledge?<\/i>&rdquo;<\/p><p>Alternatively, if students need more structure, clone and share <a href=\"https:\/\/www.kialo-edu.com\/p\/9a4b865c-f5a1-4b59-bba5-7506909aae66\/587657\" target=\"_blank\" rel=\"noopener nofollow\" referrerpolicy=\"unsafe-url\">this ready-made discussion<\/a> based on the thesis below, and use the suggested claims as prompts.<\/p><p>Students should use their listening analysis to select the strongest arguments.<\/p><p>They should add these to the Kialo discussion as arguments, counterarguments, examples, and evaluations.<\/p><p>Encourage students to frame their arguments with the TOK concepts of certainty, power and perspective.<\/p><p><strong>Example Claims:<\/strong><\/p><p><b>NAME: <\/b><strong>Do requirements for transparency and accountability in mathematical models strengthen trust in mathematics as a way of producing knowledge?<\/strong><\/p><p><strong>THESIS: Requirements for transparency and accountability strengthen trust in mathematics.<\/strong><\/p><p><b>PRO: <\/b>Protecting fairness builds trust in mathematics.<\/p><ul><li>Example: O&rsquo;Neil shows how opaque teacher evaluation algorithms eroded trust because teachers could not understand or challenge their scores. Transparent models would restore legitimacy.<\/li><\/ul><p><strong>PRO:<\/strong> Mathematics without accountability risks becoming a tool of control rather than knowledge.<\/p><ul><li>Example: Predictive policing models embed biased data. This leads to unfair sentencing and reinforces inequality.<\/li><\/ul><p><strong>PRO:<\/strong> International norms for transparency can prevent misuse in mathematics.<\/p><ul><li>Example: Political micro-targeting models manipulate voters; stronger responsibility mechanisms could have limited their harmful impact.<\/li><\/ul><p><b>CON: <\/b>Transparency requirements can reduce innovation and efficiency.<\/p><ul><li>Example: Companies argue that keeping algorithms proprietary enables breakthroughs in efficiency and targeted services. This might be slowed by strict oversight.<\/li><\/ul><p><strong>CON:<\/strong> Powerful actors will always find ways around ethical safeguards.<\/p><ul><li>Example: Even with regulations, many algorithms remain opaque because corporations claim intellectual property protection. This makes oversight ineffective.<\/li><\/ul><p><strong>CON:<\/strong> Conflicting cultural and political perspectives weaken global standards.<\/p><ul><li>Example: Predictive policing is defended in some contexts as promoting &ldquo;safety,&rdquo; while critics see it as reinforcing systemic racism. It&rsquo;s difficult to say whose definition of fairness should dominate.<\/li><\/ul><\/div><\/div><\/div><div class=\"ea-card sp-ea-single\"><h3 class=\"ea-header\"><a class=\"collapsed\" id=\"ea-header-25043\" role=\"button\" data-sptoggle=\"spcollapse\" data-sptarget=\"#collapse25043\" aria-controls=\"collapse25043\" href=\"#\" aria-expanded=\"false\" tabindex=\"0\"><i aria-hidden=\"true\" role=\"presentation\" class=\"ea-expand-icon eap-icon-ea-expand-plus\"><\/i> Reflection Activity<\/a><\/h3><div class=\"sp-collapse spcollapse spcollapse\" id=\"collapse25043\" role=\"region\" aria-labelledby=\"ea-header-25043\"> <div class=\"ea-body\"><p>Discuss the following reflection questions in open discussion or exit ticket format:<\/p><ul><li>What surprised you most about how mathematical models and algorithms are applied in real life (e.g., teacher evaluations, predictive policing, political targeting)?<\/li><li>Does this change your trust in mathematics as a way of producing knowledge?<\/li><li>What role should institutions (like governments, corporations, or regulators) play in ensuring mathematical models are transparent and accountable?<\/li><li>Can we expect fairness when the same corporations or governments profit from the models they design and enforce?<\/li><li>How should educators, policymakers, or citizens prepare people to navigate conflicting or manipulative uses of mathematical models?<\/li><li>Is it more dangerous to overtrust mathematical models when misused, or to distrust mathematics completely?<\/li><\/ul><\/div><\/div><\/div><\/div><\/div><div data-height=\"auto\">\n\t\t\t<p>\n\t\t\t\t<strong>\n\t\t\t\t\t<a href=\"https:\/\/www.kialo-edu.com\/p\/9a4b865c-f5a1-4b59-bba5-7506909aae66\/587657\" referrerpolicy=\"unsafe-url\" rel=\"nofollow\">Do requirements for transparency and accountability in mathematical models strengthen trust in mathematics as a way of producing knowledge?<\/a>\n\t\t\t\t<\/strong> &mdash; 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Assumptions: Understand how corporations, governments, and universities may design or deploy models in ways that serve profit, efficiency, or control rather than [&hellip;]<\/p>\n","protected":false},"author":52,"featured_media":0,"comment_status":"open","ping_status":"closed","template":"","meta":{"_acf_changed":false,"wds_primary_doc_category":0,"wds_primary_doc_tag":0,"footnotes":""},"doc_category":[42],"doc_tag":[],"class_list":["post-2491","docs","type-docs","status-publish","hentry","doc_category-maths-dp"],"acf":[],"year_month":"2026-05","word_count":417,"total_views":"5","reactions":{"happy":"0","normal":"0","sad":"0"},"author_info":{"name":"stephanie","author_nicename":"stephanie","author_url":"https:\/\/resources.kialo-edu.com\/en\/author\/stephanie\/"},"doc_category_info":[{"term_name":"Mathematics","term_url":"https:\/\/resources.kialo-edu.com\/en\/docs-category\/maths-dp\/"}],"doc_tag_info":[],"knowledge_base_info":[],"knowledge_base_slug":[],"_links":{"self":[{"href":"https:\/\/resources.kialo-edu.com\/en\/wp-json\/wp\/v2\/docs\/2491","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/resources.kialo-edu.com\/en\/wp-json\/wp\/v2\/docs"}],"about":[{"href":"https:\/\/resources.kialo-edu.com\/en\/wp-json\/wp\/v2\/types\/docs"}],"author":[{"embeddable":true,"href":"https:\/\/resources.kialo-edu.com\/en\/wp-json\/wp\/v2\/users\/52"}],"replies":[{"embeddable":true,"href":"https:\/\/resources.kialo-edu.com\/en\/wp-json\/wp\/v2\/comments?post=2491"}],"version-history":[{"count":0,"href":"https:\/\/resources.kialo-edu.com\/en\/wp-json\/wp\/v2\/docs\/2491\/revisions"}],"wp:attachment":[{"href":"https:\/\/resources.kialo-edu.com\/en\/wp-json\/wp\/v2\/media?parent=2491"}],"wp:term":[{"taxonomy":"doc_category","embeddable":true,"href":"https:\/\/resources.kialo-edu.com\/en\/wp-json\/wp\/v2\/doc_category?post=2491"},{"taxonomy":"doc_tag","embeddable":true,"href":"https:\/\/resources.kialo-edu.com\/en\/wp-json\/wp\/v2\/doc_tag?post=2491"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}