{"id":2489,"date":"2026-01-21T09:54:17","date_gmt":"2026-01-21T09:54:17","guid":{"rendered":"https:\/\/resources.kialo-edu.com\/?post_type=docs&#038;p=2489"},"modified":"2026-01-21T09:54:20","modified_gmt":"2026-01-21T09:54:20","password":"","slug":"certainty-in-mathematics-lesson-2","status":"publish","type":"docs","link":"https:\/\/resources.kialo-edu.com\/en\/docs\/certainty-in-mathematics-lesson-2\/","title":{"rendered":"Certainty in Mathematics, Lesson 2"},"content":{"rendered":"<h2 class=\"wp-block-heading\" id=\"block-6e802e7d-7c4e-4f99-b41e-a4ef37f4252c\"><strong>Lesson 2: Fact-Finding Task<\/strong><\/h2><p id=\"block-8fe6ce35-0778-450a-88b8-2643dda463a0\"><strong>Focus: <\/strong><em>How do real-world controversies in mathematics reveal tensions between certainty, truth, and perspective?<\/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 case studies where the reliability of mathematical proofs or models was contested, limited, or debated.<\/li>\n\n\n\n<li>Understand how different perspectives (mathematicians, scientists, policymakers, corporations, the public) influence what counts as &ldquo;true&rdquo; or &ldquo;reliable&rdquo; mathematical knowledge.<\/li>\n\n\n\n<li>Apply TOK concepts (certainty, truth, perspective) to evaluate how mathematical knowledge is constructed, legitimized, or undermined.<\/li>\n\n\n\n<li>Substantiate or challenge claims from Lesson 1 with real-world evidence and reasoning drawn from mathematical controversies.<\/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> Question the assumption that mathematics is always objective and free from human influence.<br><br><strong>Responsiveness and Flexibility of Thought:<\/strong> Reconsider whether a proof or model is reliable after engaging with its assumptions, limitations, or failures.<br><br><strong>Extrapolation &amp; Reapplication of Principles:<\/strong> Relate lessons from past controversies to current debates on AI, algorithmic bias, and predictive analytics.<\/td><td><strong>Certainty: <\/strong>Who is responsible for ensuring that mathematical structures (proofs or models) are presented as certain or reliable? <br><br><strong>Power: <\/strong>How do power structures shape which mathematical knowledge is accepted, silenced, or elevated as &ldquo;truth&rdquo;?<br><br><strong>Perspective:<\/strong> Are mathematical truths universal, or do their applications and interpretations shift depending on context?<\/td><td>How did your case study affect your understanding of who gets to define and legitimize &ldquo;truth&rdquo; in mathematics?<br><br>Can efforts to make mathematics more rigorous ever be truly equal when access to resources, funding, and authority remains uneven?<br><br>What role should credibility, transparency, and perspective play in deciding whether mathematical structures are accepted as truth?<br><\/td><\/tr><\/tbody><\/table><\/figure><style>#sp-ea-2503 .spcollapsing { height: 0; overflow: hidden; transition-property: height;transition-duration: 300ms;}#sp-ea-2503.sp-easy-accordion>.sp-ea-single {margin-bottom: 10px; border: 1px solid #e2e2e2; }#sp-ea-2503.sp-easy-accordion>.sp-ea-single>.ea-header a {color: #444;}#sp-ea-2503.sp-easy-accordion>.sp-ea-single>.sp-collapse>.ea-body {background: #fff; color: #444;}#sp-ea-2503.sp-easy-accordion>.sp-ea-single {background: #eee;}#sp-ea-2503.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-1766324064\"><div id=\"sp-ea-2503\" 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-25030\" role=\"button\" data-sptoggle=\"spcollapse\" data-sptarget=\"#collapse25030\" aria-controls=\"collapse25030\" 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=\"collapse25030\" role=\"region\" aria-labelledby=\"ea-header-25030\"> <div class=\"ea-body\"><ol><li>Slides, attached below.<\/li><li>Students will need access to their Kialo discussions from Lesson 1.<\/li><li>Ensure students complete the homework preparation task.<\/li><li>Videos\/readings accompanying the case studies of your choice should be viewed in advance.<\/li><\/ol><\/div><\/div><\/div><div class=\"ea-card sp-ea-single\"><h3 class=\"ea-header\"><a class=\"collapsed\" id=\"ea-header-25031\" role=\"button\" data-sptoggle=\"spcollapse\" data-sptarget=\"#collapse25031\" aria-controls=\"collapse25031\" href=\"#\" aria-expanded=\"false\" tabindex=\"0\"><i aria-hidden=\"true\" role=\"presentation\" class=\"ea-expand-icon eap-icon-ea-expand-plus\"><\/i> Homework Preparation Task<\/a><\/h3><div class=\"sp-collapse spcollapse spcollapse\" id=\"collapse25031\" role=\"region\" aria-labelledby=\"ea-header-25031\"> <div class=\"ea-body\"><p><b>Case Study Task<\/b><\/p><p>Divide students into small groups and assign each group a real-world mathematical controversy. Students will add their findings to the Kialo discussion from Lesson 1.<\/p><p>Each group will:<\/p><ul><li>Reflect on how the case connects to the concepts discussed in Lesson 1 (certainty, truth, perspective).<\/li><li>Explore the case using provided resources and their own research.<\/li><li>Prepare a short presentation (5&ndash;7 minutes) responding to the question: &ldquo;How does the chosen case highlight the limits of mathematical certainty and the risks of overreliance on proofs or models?&rdquo;<\/li><\/ul><p>Students should include details of:<\/p><ul><li>What happened in the case.<\/li><li>Who had the power to decide or act (mathematicians, computers, scientists, policymakers, the public).<\/li><li>Which perspectives were marginalized or excluded.<\/li><li>Which TOK concept (certainty, truth, perspective) is most relevant.<\/li><li>Whether the case supports or challenges a claim from Lesson 1.<\/li><\/ul><p><b>Case Study Options<\/b><\/p><p><b>G&ouml;del&rsquo;s Incompleteness Theorems (1931)<\/b><\/p><ul><li>Focus: Proved that some mathematical truths cannot be established within a system.<\/li><li>Key Question: Can mathematics ever be a complete and certain system of knowledge?<\/li><li>Suggested Sources: Stanford Encyclopedia of Philosophy &ndash; G&ouml;del&rsquo;s Incompleteness Theorems; <a href=\"https:\/\/www.youtube.com\/watch?v=I4pQbo5MQOs\" target=\"_blank\" rel=\"noopener\"><span data-rich-links=\"{&quot;fple-t&quot;:&quot;The paradox at the heart of mathematics: G&ouml;del's Incompleteness Theorem - Marcus du Sautoy&quot;,&quot;fple-u&quot;:&quot;https:\/\/www.youtube.com\/watch?v=I4pQbo5MQOs&quot;,&quot;fple-mt&quot;:null,&quot;type&quot;:&quot;first-party-link&quot;}\">The paradox at the heart of mathematics: G&ouml;del's Incompleteness Theorem - Marcus du Sautoy<\/span><\/a><\/li><\/ul><p><b>Four Color Theorem (1976)<\/b><\/p><ul><li>Focus: First major computer-assisted proof; relied on machine calculations beyond human checking.<\/li><li>Key Question: Should we trust a proof if humans cannot verify every step?<\/li><li>Suggested Sources: <a href=\"https:\/\/www.youtube.com\/watch?v=NgbK43jB4rQ\" target=\"_blank\" rel=\"noopener\"><span data-rich-links='{\"fple-t\":\"The Four Color Map Theorem - Numberphile\",\"fple-u\":\"https:\/\/www.youtube.com\/watch?v=NgbK43jB4rQ\",\"fple-mt\":null,\"type\":\"first-party-link\"}'>The Four Color Map Theorem - Numberphile<\/span><\/a>.<\/li><\/ul><p><b>Banach&ndash;Tarski Paradox<\/b><\/p><ul><li>Focus: A mathematically valid theorem showing a ball can be split and reassembled into two identical copies.<\/li><li>Key Question: What happens when mathematical truth contradicts physical intuition and reality?<\/li><li>Suggested Sources: <a href=\"https:\/\/youtu.be\/seugK4PrW48?si=Zp3Gi15S7N5lLgqZ\" target=\"_blank\" rel=\"noopener\">Does math have a major flaw? - TedEd<\/a>; <a href=\"https:\/\/youtu.be\/s86-Z-CbaHA?si=rKki2fLPR-LGsAfr\" target=\"_blank\" rel=\"noopener\"><span data-rich-links='{\"fple-t\":\"The Banach&ndash;Tarski Paradox\",\"fple-u\":\"https:\/\/youtu.be\/s86-Z-CbaHA?si=rKki2fLPR-LGsAfr\",\"fple-mt\":null,\"type\":\"first-party-link\"}'>The Banach&ndash;Tarski Paradox<\/span><\/a>.<\/li><\/ul><p><b>2008 Financial Crisis (Gaussian Copula Model)<\/b><\/p><ul><li>Focus: Risk models gave a false sense of certainty, contributing to global collapse.<\/li><li>Key Question: What are the dangers of overreliance on mathematical models in real-world decision-making?<\/li><li>Suggested Sources: <a href=\"https:\/\/www.wired.com\/2009\/02\/wp-quant\/\" target=\"_blank\" rel=\"noopener\">Recipe for Disaster: The Formula That Killed Wall Street | WIRED<\/a><\/li><\/ul><\/div><\/div><\/div><div class=\"ea-card sp-ea-single\"><h3 class=\"ea-header\"><a class=\"collapsed\" id=\"ea-header-25032\" role=\"button\" data-sptoggle=\"spcollapse\" data-sptarget=\"#collapse25032\" aria-controls=\"collapse25032\" 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=\"collapse25032\" role=\"region\" aria-labelledby=\"ea-header-25032\"> <div class=\"ea-body\"><p>Recap Lesson 1: Review key claims from the Kialo discussion on whether mathematical structures (proofs and models) reveal truth or just create illusions of certainty.<\/p><p>Prompt: Which arguments from Lesson 1 did you find most convincing or flawed? Did any claims rely too much on abstract assumptions without real-world evidence?<\/p><p>Present the guiding question for this lesson: <i>How do real-world mathematical controversies reveal tensions between certainty, truth, and perspective?<\/i><\/p><p>Emphasize applying certainty, truth, and perspective to evaluate how mathematical proofs and models are judged by mathematicians, scientists, policymakers, and the public.<\/p><\/div><\/div><\/div><div class=\"ea-card sp-ea-single\"><h3 class=\"ea-header\"><a class=\"collapsed\" id=\"ea-header-25033\" role=\"button\" data-sptoggle=\"spcollapse\" data-sptarget=\"#collapse25033\" aria-controls=\"collapse25033\" 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=\"collapse25033\" role=\"region\" aria-labelledby=\"ea-header-25033\"> <div class=\"ea-body\"><p><b>Bridge to Lesson 2:<\/b><\/p><p>Explain that in this lesson, students will explore real-world mathematical controversies where:<\/p><ul><li>Proofs or models were challenged, ignored, or revised (e.g., G&ouml;del&rsquo;s Incompleteness, Four Color Theorem, 2008 crisis models).<\/li><li>Perspectives from mathematicians, scientists, policymakers, corporations, and the public shaped whether the knowledge was seen as legitimate, reliable, or flawed.<\/li><\/ul><p>These cases highlight that mathematical knowledge is not always neutral, universally accepted, or free from human assumptions, values, and power structures.<\/p><p>Clarify the shift: This is no longer about theory alone &mdash; students are now testing claims from Lesson 1 using historical and contemporary examples.<\/p><p>Reinforce the goal: Move from debate to evidence-based evaluation. These case studies will help students understand how mathematical knowledge is constructed, challenged, or reshaped within logical, institutional, and social frameworks.<\/p><p><b>Presentations:<\/b><\/p><p>Student groups present their case studies (e.g., G&ouml;del&rsquo;s Incompleteness, Four Color Theorem, Banach&ndash;Tarski, 2008 Financial Crisis models, COVID-19 models, Climate Change models).<\/p><p>Students take notes on useful points from other groups to bring back into the Kialo discussion.<b><\/b><\/p><p><b>Recording Findings in a Kialo Discussion:<\/b><\/p><p>Students return to the Kialo discussion from Lesson 1 and:<\/p><ul><li>Add at least one new claim or counterclaim based on their case study.<\/li><li>Reply to at least one peer&rsquo;s argument, using insights from another group&rsquo;s case.<\/li><li>Label their post with the relevant TOK concept (e.g., Certainty &ndash; G&ouml;del, Truth &ndash; Banach&ndash;Tarski, Perspective &ndash; climate models).<\/li><\/ul><p><b>Focus Areas for Kialo Updates:<\/b><\/p><ul><li>Mathematical Gatekeeping: Who decides which proofs or models are accepted as legitimate knowledge, and who gets excluded?<\/li><li>Institutional Authority: How do journals, universities, corporations, or governments influence the perceived legitimacy of mathematical structures?<\/li><li>Trust and Reliability: How do controversial proofs or failed models (e.g., financial risk formulas) affect public trust in mathematics?<\/li><li>Knowledge Inequality: Are all mathematical voices &mdash; especially alternative systems, critical perspectives, or less powerful institutions &mdash; treated equally when defining &ldquo;truth&rdquo;?<\/li><\/ul><\/div><\/div><\/div><div class=\"ea-card sp-ea-single\"><h3 class=\"ea-header\"><a class=\"collapsed\" id=\"ea-header-25034\" role=\"button\" data-sptoggle=\"spcollapse\" data-sptarget=\"#collapse25034\" aria-controls=\"collapse25034\" 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=\"collapse25034\" role=\"region\" aria-labelledby=\"ea-header-25034\"> <div class=\"ea-body\"><p>Discuss the following reflection questions in open discussion or exit ticket format:<\/p><ul><li>How did your case study affect your understanding of who gets to define and legitimize &ldquo;truth&rdquo; in mathematics?<\/li><li>What made certain examples feel more like illusions of certainty or institutional overreach, versus genuine reliability?<\/li><li>In your case, who had the most control over the mathematical narrative: mathematicians, computers, policymakers, corporations, or the public?<\/li><li>Can efforts to make mathematics more rigorous (e.g., formal verification, better data in models) ever be truly equal when access to resources, funding, and authority remains uneven?<\/li><li>What role should credibility, transparency, and perspective play in deciding whether mathematical structures are accepted as truth?<\/li><li>Should all mathematical models that affect public wellbeing (e.g., finance, health, climate) require broader social consultation, or are there exceptions?<\/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\/c65146f7-7cce-4191-b8b2-ff70579a45c5\/586862\" referrerpolicy=\"unsafe-url\" rel=\"nofollow\">Does mathematics reveal objective truth about reality?<\/a>\n\t\t\t\t<\/strong> &mdash; <a href=\"https:\/\/www.kialo-edu.com\" referrerpolicy=\"unsafe-url\">kialo-edu.com<\/a>\n\t\t\t<\/p>\n\t\t\t<script async=\"\" src=\"https:\/\/www.kialo-edu.com\/assets\/static\/js\/embedded-kialo.min.js\" charset=\"utf-8\"><\/script>\n\t\t<\/div><figure class=\"wp-block-embedpress-embedpress aligncenter\" data-source-id=\"source-b8bcb637-c717-4321-8234-3da7cced54bc\" data-embed-type=\"GoogleDocs\"><div class=\"gutenberg-block-wraper  \"><div class=\"position-right-wraper ep-embed-content-wraper   \" style=\"position:relative;display:inline-block\"><div class=\"ose-google-docs ose-uid-2eef2ad092bf8e90ec961ed314cc920e ose-embedpress-responsive\" style=\"width:600px; height:600px; max-height:600px; max-width:100%; display:inline-block;\" data-embed-type=\"GoogleDocs\"><iframe loading=\"lazy\" allowfullscreen=\"true\" src=\"https:\/\/docs.google.com\/presentation\/d\/e\/2PACX-1vRW1z2Dr3OXoUtABvD5RYeWh8Z95_BJlsf_dIwLNRfTeaRMMQCTXODEDG7FMniqZkGlTp6B2zHspENh\/embed?start=false&amp;loop=false&amp;delayms=3000\" frameborder=\"0\" width=\"600\" height=\"600\" mozallowfullscreen=\"true\" webkitallowfullscreen=\"true\" style=\"width:600px;height:600px;max-width:100%;\"><\/iframe><\/div><\/div><\/div><style>\n                    [data-source-id=\"source-b8bcb637-c717-4321-8234-3da7cced54bc\"] .ose-embedpress-responsive{\n                        width: 600px!important;\n                        height: 600px!important;\n                        max-height: 600px!important;\n                    }\n                    [data-source-id=\"source-b8bcb637-c717-4321-8234-3da7cced54bc\"] iframe{\n                        width: 600px!important;\n                        height: 600px!important;\n                        max-height: 600px!important;\n                    }\n                    [data-source-id=\"source-b8bcb637-c717-4321-8234-3da7cced54bc\"] .embedpress-yt-subscribe iframe{\n                        height: 100%!important;\n                    }\n                    [data-source-id=\"source-b8bcb637-c717-4321-8234-3da7cced54bc\"] .ose-youtube > iframe{\n                        height: 600px!important;\n                        width: 600px!important;\n                    }\n                    [data-source-id=\"source-b8bcb637-c717-4321-8234-3da7cced54bc\"] .ose-youtube{\n                        height: 600px!important;\n                        width: 600px!important;\n                    }\n                    [data-source-id=\"source-b8bcb637-c717-4321-8234-3da7cced54bc\"] .ose-giphy img{\n                        height: 600px!important;\n                        width: 600px!important;\n                    }\n                    [data-source-id=\"source-b8bcb637-c717-4321-8234-3da7cced54bc\"] .ose-google-docs img{\n                        height: 600px!important;\n                        width: 600px!important;\n                    }\n                    [data-source-id=\"source-b8bcb637-c717-4321-8234-3da7cced54bc\"] .ose-embedpress-responsive.ose-google-photos{\n                        height: 100% !important;\n                        max-height: 100% !important;\n                    }\n\n                    [data-source-id=\"source-b8bcb637-c717-4321-8234-3da7cced54bc\"] .embera-embed-responsive-provider-gettyimages,\n                    [data-source-id=\"source-b8bcb637-c717-4321-8234-3da7cced54bc\"] .embera-embed-responsive-provider-gettyimages iframe,\n                    [data-source-id=\"source-b8bcb637-c717-4321-8234-3da7cced54bc\"] .getty{\n                        width: 600px!important;\n                        height: 600px!important;\n                        max-height: 600px!important;\n                        max-width: 100%!important;\n                    }\n                    <\/style><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Lesson 2: Fact-Finding TaskFocus: How do real-world controversies in mathematics reveal tensions between certainty, truth, and perspective?Suggested Length: 1 hourLearning Objectives: Critical Thinking Concepts TOK Concepts Reflection Questions Confronting Biases &amp; Assumptions: Question the assumption that mathematics is always objective and free from human influence. Responsiveness and Flexibility of Thought: Reconsider whether a proof or [&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-2489","docs","type-docs","status-publish","hentry","doc_category-maths-dp"],"acf":[],"year_month":"2026-05","word_count":443,"total_views":"3","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\/2489","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=2489"}],"version-history":[{"count":0,"href":"https:\/\/resources.kialo-edu.com\/en\/wp-json\/wp\/v2\/docs\/2489\/revisions"}],"wp:attachment":[{"href":"https:\/\/resources.kialo-edu.com\/en\/wp-json\/wp\/v2\/media?parent=2489"}],"wp:term":[{"taxonomy":"doc_category","embeddable":true,"href":"https:\/\/resources.kialo-edu.com\/en\/wp-json\/wp\/v2\/doc_category?post=2489"},{"taxonomy":"doc_tag","embeddable":true,"href":"https:\/\/resources.kialo-edu.com\/en\/wp-json\/wp\/v2\/doc_tag?post=2489"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}