{"id":3259,"date":"2017-12-06T13:29:47","date_gmt":"2017-12-06T18:29:47","guid":{"rendered":"http:\/\/blogs.ams.org\/blogonmathblogs\/?p=3259"},"modified":"2017-12-06T18:49:07","modified_gmt":"2017-12-06T23:49:07","slug":"unsolved-problems-in-math-class","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/blogonmathblogs\/2017\/12\/06\/unsolved-problems-in-math-class\/","title":{"rendered":"Unsolved Problems in Math Class"},"content":{"rendered":"<div id=\"attachment_3260\" style=\"width: 321px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/xkcd.com\/710\/\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3260\" class=\"size-full wp-image-3260\" src=\"https:\/\/i0.wp.com\/blogs.ams.org\/blogonmathblogs\/files\/2017\/12\/collatz_conjecture.png?resize=311%2C452\" alt=\"\" width=\"311\" height=\"452\" srcset=\"https:\/\/i0.wp.com\/blogs.ams.org\/blogonmathblogs\/files\/2017\/12\/collatz_conjecture.png?w=311&amp;ssl=1 311w, https:\/\/i0.wp.com\/blogs.ams.org\/blogonmathblogs\/files\/2017\/12\/collatz_conjecture.png?resize=206%2C300&amp;ssl=1 206w\" sizes=\"auto, (max-width: 311px) 100vw, 311px\" \/><\/a><p id=\"caption-attachment-3260\" class=\"wp-caption-text\">Credit: Randall Munroe. CC BY-NC 2.5<\/p><\/div>\n<p class=\"p1\"><span class=\"s1\">A few years ago, I directed a high school summer math program. Half the day was devoted to exploring the delights of <a href=\"https:\/\/blogs.scientificamerican.com\/roots-of-unity\/a-belated-apology-to-mozart-and-modular-arithmetic\/\"><span class=\"s2\">modular arithmetic<\/span><\/a>\u2014we ended the summer with a cake decorated with Fermat\u2019s Little theorem!\u2014and half to learning to program in Python, with number theory questions as motivation. One Friday afternoon, we included these questions in the programming part of the day.<\/span><\/p>\n<blockquote>\n<p class=\"p1\"><span class=\"s1\"><b>Goldbach conjecture<\/b>: Any even number larger than 2 is the sum of two prime numbers.<\/span><\/p>\n<ul>\n<li class=\"p1\"><span class=\"s1\">Is there a counterexample to this conjecture for an even number less than 10,000<\/span><\/li>\n<li class=\"p1\">Prove this conjecture.<\/li>\n<\/ul>\n<p class=\"p1\"><span class=\"s1\"><b>Collatz conjecture<\/b>: Choose some number a<\/span><span class=\"s3\"><sub>0<\/sub><\/span><span class=\"s1\">.<br \/>\nDefine a<sub>n<\/sub> by a<\/span><span class=\"s3\"><sub>n<\/sub><\/span><span class=\"s1\">=3a<\/span><span class=\"s3\"><sub>n-1<\/sub><\/span><span class=\"s1\">+1 if a<\/span><span class=\"s3\"><sub>n-1<\/sub><\/span><span class=\"s1\"> is odd or a<\/span><span class=\"s3\"><sub>n-1<\/sub><\/span><span class=\"s1\">\/2 if a<\/span><span class=\"s3\"><sub>n-1<\/sub><\/span><span class=\"s1\"> is even.<br \/>\n<\/span><span class=\"s1\">Then a<\/span><span class=\"s3\"><sub>n<\/sub><\/span><span class=\"s1\"> will be 1 for some n.<\/span><\/p>\n<ul>\n<li><span class=\"s1\">Is there a counterexample to this conjecture for a<\/span><span class=\"s3\"><sub>0<\/sub><\/span><span class=\"s1\">&lt;10,000?<\/span><\/li>\n<li>Prove this conjecture.<\/li>\n<\/ul>\n<\/blockquote>\n<p class=\"p1\"><span class=\"s1\">Perhaps it was a tiny bit evil to give these longstanding open problems to high school students without warning them, but it was a lot of fun to watch them come up with programs to search for counterexamples and brainstorm about ways of approaching the proofs. (And yes, we did eventually tell them the questions were still open. We didn&#8217;t want to ruin their weekends completely!)<\/span><\/p>\n<p class=\"p1\"><span class=\"s1\">Math teachers <a href=\"https:\/\/arbitrarilyclose.com\/2016\/08\/03\/multiplying-is-not-a-pre-requisite-for-fractals\/\"><span class=\"s2\">Annie Perkins<\/span><\/a> and <a href=\"https:\/\/samjshah.com\/2016\/04\/16\/inspiration-and-mathematics\/\"><span class=\"s2\">Sam Shah<\/span><\/a> have written about the benefits of exposing kids to advanced math concepts early rather than waiting until they\u2019ve mastered all the easy stuff. If you too would like to <\/span><span class=\"s4\">torture your students<\/span><span class=\"s1\"> kindle your students\u2019 curiosity and challenge their intuition, with unsolved math problems, there are lots of places to go for inspiration.<\/span><\/p>\n<p class=\"p1\"><span class=\"s1\">The <a href=\"http:\/\/mathpickle.com\/\"><span class=\"s2\">MathPickle<\/span><\/a> site (tagline: \u201cPut your students in a pickle!\u201d) has <a href=\"http:\/\/mathpickle.com\/organized-by-grade\/\"><span class=\"s2\">puzzles<\/span><\/a> organized by grade level, <a href=\"http:\/\/mathpickle.com\/games\/\"><span class=\"s2\">board game suggestions<\/span><\/a>, and a <a href=\"http:\/\/mathpickle.com\/blog\/\"><span class=\"s2\">blog<\/span><\/a>. I\u2019ve seen this site mentioned in a few places, including a <a href=\"http:\/\/blog.mrmeyer.com\/2016\/unsolved-math-problems-at-every-grade-k-12\/\"><span class=\"s2\">discussion on Dan Meyer\u2019s blog<\/span><\/a>.<\/span><\/p>\n<p class=\"p1\"><span class=\"s1\">Lior Patcher has a list of <a href=\"https:\/\/liorpachter.wordpress.com\/2015\/09\/20\/unsolved-problems-with-the-common-core\/\"><span class=\"s2\">suggestions for how to use unsolved problems in K-12 classrooms at his computational biology blog Bits of DNA<\/span><\/a>. I was especially excited to see a question involving Namibia\u2019s mysterious \u201cfairy circles,\u201d circular patches of bare ground surrounded by vegetation. It\u2019s nice to see some modeling and applied math get some love there. Why should number theory have all the fun?<\/span><\/p>\n<p class=\"p1\"><span class=\"s2\"><a href=\"https:\/\/mikesmathpage.wordpress.com\">Mike Lawler<\/a><\/span><span class=\"s1\"> often discusses advanced and unsolved problems with his kids, and the Collatz conjecture has made several appearances on his blog. In his most recent post on the topic, his kids <a href=\"https:\/\/mikesmathpage.wordpress.com\/2017\/12\/03\/revisiting-john-conways-amusical-collatz-conjecture\/\"><span class=\"s2\">make music with John Conway\u2019s \u201camusical\u201d<\/span><\/a> variation of the problem. (As a violist, I\u2019m delighted that one of them does so in alto clef!)<\/span><\/p>\n<p><a href=\"https:\/\/blogs.ams.org\/matheducation\/2015\/05\/01\/famous-unsolved-math-problems-as-homework\/\">Ben Braun<\/a> writes about using unsolved problems in his college math classes at the AMS math education blog On Teaching and Learning Mathematics. He highlights some of the benefits for his students, including mindset shifts away from answer-getting and toward seeing failure as part of mathematical productivity.<\/p>\n<p class=\"p1\"><span class=\"s1\">It goes without saying that your students probably won\u2019t solve the Goldbach conjecture or get a definitive answer about fairy circles in one or two class periods, but you never know. Some open problems might end up being easy to solve, or at least easier than we might think. The <a href=\"https:\/\/rjlipton.wordpress.com\"><span class=\"s2\">G\u00f6del\u2019s Lost Letter and P=NP<\/span><\/a> blog has a fun post about <a href=\"https:\/\/rjlipton.wordpress.com\/2015\/09\/03\/open-problems-that-might-be-easy\/\"><span class=\"s2\">open problems with short solutions<\/span><\/a>.<\/span><\/p>\n<div style=\"margin-top: 0px; margin-bottom: 0px;\" class=\"sharethis-inline-share-buttons\" ><\/div>","protected":false},"excerpt":{"rendered":"<p>A few years ago, I directed a high school summer math program. Half the day was devoted to exploring the delights of modular arithmetic\u2014we ended the summer with a cake decorated with Fermat\u2019s Little theorem!\u2014and half to learning to program &hellip; <a href=\"https:\/\/blogs.ams.org\/blogonmathblogs\/2017\/12\/06\/unsolved-problems-in-math-class\/\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n<div style=\"margin-top: 0px; margin-bottom: 0px;\" class=\"sharethis-inline-share-buttons\" data-url=https:\/\/blogs.ams.org\/blogonmathblogs\/2017\/12\/06\/unsolved-problems-in-math-class\/><\/div>\n","protected":false},"author":61,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[3],"tags":[749,748],"class_list":["post-3259","post","type-post","status-publish","format-standard","hentry","category-math-education","tag-open-problems","tag-unsolved-math-problems"],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p3tW3N-Qz","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/blogs.ams.org\/blogonmathblogs\/wp-json\/wp\/v2\/posts\/3259","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.ams.org\/blogonmathblogs\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.ams.org\/blogonmathblogs\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.ams.org\/blogonmathblogs\/wp-json\/wp\/v2\/users\/61"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.ams.org\/blogonmathblogs\/wp-json\/wp\/v2\/comments?post=3259"}],"version-history":[{"count":6,"href":"https:\/\/blogs.ams.org\/blogonmathblogs\/wp-json\/wp\/v2\/posts\/3259\/revisions"}],"predecessor-version":[{"id":3266,"href":"https:\/\/blogs.ams.org\/blogonmathblogs\/wp-json\/wp\/v2\/posts\/3259\/revisions\/3266"}],"wp:attachment":[{"href":"https:\/\/blogs.ams.org\/blogonmathblogs\/wp-json\/wp\/v2\/media?parent=3259"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.ams.org\/blogonmathblogs\/wp-json\/wp\/v2\/categories?post=3259"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.ams.org\/blogonmathblogs\/wp-json\/wp\/v2\/tags?post=3259"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}