{"id":2462,"date":"2024-10-08T09:00:43","date_gmt":"2024-10-08T16:00:43","guid":{"rendered":"https:\/\/theothermath.com\/?p=2462"},"modified":"2024-10-08T17:20:37","modified_gmt":"2024-10-09T00:20:37","slug":"building-a-culture-that-embraces-productive-struggle","status":"publish","type":"post","link":"https:\/\/theothermath.com\/index.php\/2024\/10\/08\/building-a-culture-that-embraces-productive-struggle\/","title":{"rendered":"Building a Culture That Embraces Productive Struggle"},"content":{"rendered":"<p class=\"c1\"><span class=\"c0\">About 18 months ago in March, 2024, <\/span><span class=\"c0\"><a href=\"https:\/\/abcnews.go.com\/US\/2-new-orleans-teenagers-proved-2000-year-pythagorean\/story?id=98207106\" target=\"_blank\" rel=\"noopener\">two black teenage girls<\/a><\/span><span class=\"c0\"> from a New Orleans high school announced they found a new proof of the Pythagorean Theorem. What makes their proof especially impressive is that out of more than 400 unique proofs of the Pythagorean Theorem, theirs is the FIRST proof that relies entirely on trigonometry. For hundreds of years a trigonometric proof of the theorem had thought to be impossible, but Calcea Johnson and Ne\u2019Kiya Jackson proved otherwise.<\/span><\/p>\n<p class=\"c1\"><span class=\"c0\">I share this story not only because it is an amazing mathematical story, but because it is also a great example of a math teacher who had created the classroom culture that allowed these two young mathematicians to flourish.<\/span><\/p>\n<p class=\"c1\"><span class=\"c0\">In this post I&#8217;ll share my own thoughts on the TWO steps for creating a classroom culture that embraces productive struggle such that these mathematical adventures can take place in YOUR classroom.<\/span><\/p>\n<ul class=\"c7 lst-kix_q43n1spkygpy-0 start\">\n<li class=\"c1 c4 li-bullet-0\"><span class=\"c0\">Step 1: Teach wide math, not narrow math.<\/span><\/li>\n<li class=\"c1 c4 li-bullet-0\"><span class=\"c0\">Step 2: Explore-first instructional strategies<\/span><\/li>\n<\/ul>\n<h2 class=\"c1\"><span class=\"c0\">Teach wide math, not narrow math<\/span><\/h2>\n<p class=\"c1\"><img loading=\"lazy\" src=\"https:\/\/lh7-rt.googleusercontent.com\/docsz\/AD_4nXdAghGgpx776arzqbFMHc7Vq3ituTAa2zibzct2tblR777XhcIV5HAzz8GI_IMTRvJyvbzlv6EaZuT4v-T4gWdOqHeCspnzoI-4H5-CBk1PYsD7H379uSugOL3BoLzfdUC9C0ROJxYq58pftWHpF6WAaECH?key=uXFGrUrltUI0ycbw_xQdBQ\" title=\"\" alt=\"\" width=\"535\" height=\"301\"><\/p>\n<p class=\"c1\"><span class=\"c0\">Narrow math is the type of mathematics that allows no room for flexibility, creativity, or alternate points of view. In narrow math, students can only be successful if they can use the ONE method that is offered by the textbook. In general, narrow math focuses on abstract algorithms and leaves to space for alternative thinking.<\/span><\/p>\n<p class=\"c1\"><img loading=\"lazy\" src=\"https:\/\/lh7-rt.googleusercontent.com\/docsz\/AD_4nXcKS-JfSLWT0fdUsz4zZgaT2ST8WPfDmGQy4DN3KIvmMHF4qfsHVv-7sGtckiaeuFSq02RvNEdW0mGwAWyeWCj0K1pKCpuWe_PpF3hYNkaTYhSwpWYm3ZhyJyr_G3EbfKnwbh0MKJTMMFlkdDmJHjHnTS9G?key=uXFGrUrltUI0ycbw_xQdBQ\" title=\"\" alt=\"\" width=\"115\" height=\"107\">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<img loading=\"lazy\" src=\"https:\/\/lh7-rt.googleusercontent.com\/docsz\/AD_4nXfuRr4Gx3OKTkoL9QpUYho-iZuTZ_NRYGLW62CT76lFkckQ4Tj1-nIXsoUhwBybPOuaiyBk1kH5uM9vlurY0XKXhs49AX7sT89zolhkmpS3wppSHIuWsrSnJf-KKtnaCY5w5hF4TB1Ugk9wVgCGWTgc_YP6?key=uXFGrUrltUI0ycbw_xQdBQ\" title=\"\" alt=\"\" width=\"202\" height=\"92\"><\/p>\n<p class=\"c1\"><span class=\"c0\">In problems like this, if the student does not already know the algorithm, then they are completely out of luck. The path to success is narrow: know the algorithm or fail.<\/span><\/p>\n<p class=\"c1\"><span class=\"c0\">Each math concept can be reframed as a wide math example.<\/span><\/p>\n<p class=\"c1\"><span class=\"c0\">For 24 x 15 we might tell students that one definition of multiplication is the area of the rectangle that is 24 units wide and 15 units tall. Students are then merely told to find the area of the rectangle.<\/span><\/p>\n<p class=\"c1\"><img loading=\"lazy\" src=\"https:\/\/lh7-rt.googleusercontent.com\/docsz\/AD_4nXeazys2G8vfX004fo47GjO1HvrsiwGEAHdEbvXszOYnL9MXBgPjZ8t-n4x3QZT6mRFHX9J4KzqPSh6q2Jdc-GuMRegiSMl6hs3rEX64m6VtkJtaaYlLaMjQVX252UjTQI-IcjqWTmAkyjSP7nlOBuaqrwPL?key=uXFGrUrltUI0ycbw_xQdBQ\" title=\"\" alt=\"\" width=\"383\" height=\"255\"><\/p>\n<p class=\"c1\"><span class=\"c0\">Suddenly a narrow math task becomes WIDE because students have a plethora of strategies for finding the area of the rectangle, one of which is the standard algorithm for 24 x 15.<\/span><\/p>\n<p class=\"c1\"><span class=\"c0\">Students who are not yet familiar with the standard algorithm can build upon the area model to find partial products.<\/span><\/p>\n<p>Strategy 1<\/p>\n<p class=\"c1\"><img loading=\"lazy\" src=\"https:\/\/lh7-rt.googleusercontent.com\/docsz\/AD_4nXfmdTK-wFrZZkNPcjGnfCN3PXL2d9Qo8N0mXYLktaJp3dAFA3KFQ4t_xx_XEoCeAFhYVgJmYWuzWyk5RXsO3g-VWr9G_ZMlTA8hEbaEzCehOIvObOezNpaORtDPXX9bzptsgFeKyNY8jwGytZSLlyBw5tw?key=uXFGrUrltUI0ycbw_xQdBQ\" title=\"\" alt=\"\" width=\"448\" height=\"267\"><\/p>\n<p class=\"c1 c2\">Strategy 2<\/p>\n<p class=\"c1\"><img loading=\"lazy\" src=\"https:\/\/lh7-rt.googleusercontent.com\/docsz\/AD_4nXel65U1rS29mqUiqiLa2nClfl8nXEnHs35maiCLPRwP7VKGcPVKRy2hz6XAHeUfOPpwR-DeetmF44ARXD2aIkFrh3DM-xXTJNjUZG9ZI8fl-MDUwIPzB-3UnPdpwyjiZwaN_ZZYK3KAoh1ieOw-w5D5Scw?key=uXFGrUrltUI0ycbw_xQdBQ\" title=\"\" alt=\"\" width=\"457\" height=\"280\"><\/p>\n<p class=\"c1\"><span class=\"c0\">Students also can build upon the area model to find flexible factoring.<\/span><\/p>\n<p class=\"c1\"><img loading=\"lazy\" src=\"https:\/\/lh7-rt.googleusercontent.com\/docsz\/AD_4nXfKFqeOuVDvJOZCi1jWp_cHRprAovd90juGHJTU5pQcypOKJWsBROgn79mgqv-RltJ3Sqyzkp3Imhhoz7_o3cBqvfUNiHirb7QaV02wdujQYSKy0RRwgD-JefHkH1bf9U7uUuIG5hB_fAXh6ymlXb-8zYLJ?key=uXFGrUrltUI0ycbw_xQdBQ\" title=\"\" alt=\"\" width=\"212\" height=\"300\"><\/p>\n<p class=\"c1\"><span class=\"c0\">&#8220;Convert \u215d into a percent&#8221; is often presented to students as a task in which students are forced to use long division to do the conversion.<\/span><\/p>\n<p class=\"c1\"><img loading=\"lazy\" src=\"https:\/\/lh7-rt.googleusercontent.com\/docsz\/AD_4nXebmqYJAPioFwXpa2CP97n-ulb598s07oAU1-tIv-EzNyo3Z98jQF368XtO-gDUoTYKHA3LxyX-f5bg3HzSMEBI-1T5goI6-bx705U72UMyfoDc5-2koL6F6PpPA-yX9eTU8Z9SGCtSdosP20VjBUTT8_M?key=uXFGrUrltUI0ycbw_xQdBQ\" title=\"\" alt=\"\" width=\"164\" height=\"113\"><\/p>\n<p class=\"c1\"><span class=\"c0\">Long division\u2026or go home.<\/span><\/p>\n<p class=\"c1\"><span class=\"c0\">This can be turned into a wide task by introducing a tape diagram visual.<\/span><\/p>\n<p class=\"c1\"><img loading=\"lazy\" src=\"https:\/\/lh7-rt.googleusercontent.com\/docsz\/AD_4nXdAsny2Y-3utFTYDPLDikGxM_EhbqKFGAOSmwz1IkcazmuOv_O1hAopN1TWJQ_O32at5T6leHY0py_a_9gmlecbf1S26wRnxUDJ-9mMIzs6WSqmOrHdh6iUQ1RAPy88mGXPIRPIVXMA0MCzqbP9UIoBCdfP?key=uXFGrUrltUI0ycbw_xQdBQ\" title=\"\" alt=\"\" width=\"392\" height=\"292\"><\/p>\n<p class=\"c1\"><span class=\"c0\">Students now have access to a variety of strategies for finding the percent represented by \u215d. Here are just two examples\u2026<\/span><\/p>\n<p>Strategy 1<\/p>\n<p class=\"c1\"><img loading=\"lazy\" src=\"https:\/\/lh7-rt.googleusercontent.com\/docsz\/AD_4nXf_nUcFjsVF6HarI6ijkshBV4V4FjZqR95ihHCnobGwd6Aq7rXhJ1wSHKCt0k2r9LntXRnPttasvsu8zdzEZL_lILggv6TTj9HT0a3rxFrtyC5VUCNVPwVkQleQImAwTyDDr-tinAGdez9vL18388JzSjOq?key=uXFGrUrltUI0ycbw_xQdBQ\" title=\"\" alt=\"\" width=\"496\" height=\"237\"><\/p>\n<p class=\"c1 c2\">Strategy 2<\/p>\n<p class=\"c1\"><img loading=\"lazy\" src=\"https:\/\/lh7-rt.googleusercontent.com\/docsz\/AD_4nXc_0Fw2GmU3ryvRmfD6Z8AEzPrCK-5prFARElVq3Nx8e-5bNzFV-8cb4y3SXIjtbwzrVVIcpLV3i6iJUUOXKuE8TyNYsNaewK0QhzhDH6HhUFxaQqU5Tkh5v4uRVJXSlGpC9kDpkpyPirewzsKbfbSDwELy?key=uXFGrUrltUI0ycbw_xQdBQ\" title=\"\" alt=\"\" width=\"439\" height=\"273\"><\/p>\n<p class=\"c1\"><span class=\"c0\">Wide math creates the mathematical space for MORE students to successfully access the math concept to be learned. Wide math also allows students to explore alternative strategies that make sense to them. The focus is now on understanding the math concept rather than just blind answer-getting.<\/span><\/p>\n<p class=\"c1\"><span class=\"c0\">Virtually all math concepts can be introduced to students in a WIDE way. Most often this involves some sort of visual representation.<\/span><\/p>\n<p>&nbsp;<\/p>\n<h2 class=\"c1\"><span class=\"c0\">Explore-first instructional strategies<\/span><\/h2>\n<p class=\"c1\"><span class=\"c0\">The second step to creating a culture that embraces productive struggle is to use explore-first instructional strategies. As I just hinted, WIDE math naturally opens the door to students exploring their own methods prior to the teacher formally teaching.<\/span><\/p>\n<p class=\"c1\"><span class=\"c0\">Your job as the teacher is to provide some time for students to explore and develop their own understanding of the mathematics prior to you formally teaching the concept.<\/span><\/p>\n<p class=\"c1\">From&nbsp;<span class=\"c9\"><em>Improving student achievement in mathematics<\/em><\/span>&nbsp;(<span class=\"c0\">International Bureau of Education, 2000) we know that when students are provided an opportunity to discover mathematical ideas and invent mathematical procedures, they have a stronger conceptual understanding of connections between mathematical ideas.<\/span><\/p>\n<p class=\"c1\"><img loading=\"lazy\" src=\"https:\/\/lh7-rt.googleusercontent.com\/docsz\/AD_4nXdSJaLE7mkEyfnZJUENxnC_SvZvuu9rXTFCDaiBsCcQWeCQO6s-H9BAnYiue1TmJ6k7yJHtaK-cYd8rpKBZE43CZKMTum6Ybp-S3mY2HMZUtmJFotRv9Q9RmtXoHCgIv6cqwvbraesMxlyRp0ZnbI5ZsVDl?key=uXFGrUrltUI0ycbw_xQdBQ\" class=\"alignright\" title=\"\" alt=\"\" width=\"207\" height=\"413\"><\/p>\n<p class=\"c1\"><span class=\"c0\">The amazing thing about inquiry-first teaching is that students do NOT need to successfully solve the problem during their moment of inquiry. It appears as though students benefit from inquiry-first instruction merely because of the opportunity to engage in authentic problem-solving itself, regardless of whether they successfully solved the problem.<\/span><\/p>\n<p class=\"c1\">Allowing learners to struggle will actually help them learn better,&nbsp;<span class=\"c3 c5\"><a class=\"c6\" href=\"https:\/\/www.google.com\/url?q=http:\/\/ww2.kqed.org\/mindshift\/2014\/02\/25\/bigger-gains-for-students-who-dont-have-help-solving-problems-struggle-to-learn\/&#038;sa=D&#038;source=editors&#038;ust=1728333125179986&#038;usg=AOvVaw0wRHqL8Zug71HHqcwz0X8A\" target=\"_blank\" rel=\"noopener\">according to research<\/a><\/span><span class=\"c0\">&nbsp;on \u201cproductive failure\u201d conducted by Manu Kapur, a researcher at the Learning Sciences Lab at the National Institute of Education of Singapore. In the study, they created two groups of students: the first were scaffolded carefully by the teacher and successfully solved the problems. The second group collaborated with one another without any prompts from the teacher. Even though the second group was unsuccessful in solving the problem, when the two groups were tested on what they\u2019d learned, the second group \u201csignificantly outperformed\u201d the first.<\/span><\/p>\n<p class=\"c1\"><span class=\"c0\">To convince students to become active problem-solvers, I use class norms that encourage discussion, questioning, and collaboration and minimizes the focus on answer-getting. I was recently introduced to these norms by my good friend Jamie Garner at Stanislaus County Office of Education.<\/span><\/p>\n<p class=\"c1\"><img loading=\"lazy\" src=\"https:\/\/lh7-rt.googleusercontent.com\/docsz\/AD_4nXeoPtZlQfZed6hyYk_NlPeSUdoDEe5Hl6SsooyMZGs9hBY6anN5pALLV6lGBwGFgUdKcIz50fiVftDV6JGV7xPBYbcoNlrxJVl9bx7BZWbVra2p8aQbAJGsAg2zo9JKC4qhxoPVL98C23iS-Npb3Rtvg0HE?key=uXFGrUrltUI0ycbw_xQdBQ\" title=\"\" alt=\"\" width=\"600\" height=\"337\"><\/p>\n<p class=\"c1\"><span class=\"c0\">The norms establish a culture in which students are comfortable working with one another and are not burdened by the fear of mistakes and unfinished thinking.<\/span><\/p>\n<p class=\"c1\"><span class=\"c0\">I provide <\/span><span class=\"c0\"><strong><a href=\"https:\/\/docs.google.com\/document\/d\/1RHDHF-AQz2RQEhPoznF211N5AxHO2XGy5Ra4mM_0qSA\/edit?usp=sharing\" target=\"_blank\" rel=\"noopener\">laminated discussion scaffold cards<\/a><\/strong><\/span><span class=\"c0\"> to students to support the mathematical conversations that need to be commonplace.<\/span><\/p>\n<p class=\"c1\"><img loading=\"lazy\" src=\"https:\/\/lh7-rt.googleusercontent.com\/docsz\/AD_4nXdRe-qAk3dWiHjurT1UrEWsoM0CuBPRhBa9VgYSF_wQDQwrI6u0ei6llnxd4vHIpJDV1HtaG9uMyzU6xI43E9wPO4OHPYiZbSbn6mucxa9ip-w23T5NP0azZs2joHslPOmV8Ap8qYdPx60C6QG6LpTZvro?key=uXFGrUrltUI0ycbw_xQdBQ\" title=\"\" alt=\"\" width=\"562\" height=\"339\"><\/p>\n<p class=\"c1\"><span class=\"c0\">One the other side of the laminated discussion card is a list of things students can try when they have no clue what to do with the tasks.<\/span><\/p>\n<p class=\"c1\"><img loading=\"lazy\" src=\"https:\/\/lh7-rt.googleusercontent.com\/docsz\/AD_4nXcAzJ8QyBB76bWQrksWatrnd38mSI329R60MX15RIM2uVdWIZvakNHAmmVa83Ua0UPPPciDTnpS_VQ5ZxV4mfprTbGxXVb0ikcVUwzP0vEhSjnM-_jmKQtLkZpSD_716RVDsWmuq2Rd9DE-WLZf7pcZVDhI?key=uXFGrUrltUI0ycbw_xQdBQ\" title=\"\" alt=\"\" width=\"565\" height=\"340\"><\/p>\n<p class=\"c1\"><span class=\"c0\">There are many &#8220;flavors&#8221; of explore-first instruction:<\/span><\/p>\n<ul class=\"c7 lst-kix_homimlhaq6h7-0 start\">\n<li class=\"c1 c4 li-bullet-0\"><span class=\"c3\"><a class=\"c6\" href=\"https:\/\/www.google.com\/url?q=https:\/\/www.nctm.org\/Store\/Products\/5-Practices-for-Orchestrating-Productive-Mathematics-Discussions,-2nd-edition-(Download)\/&#038;sa=D&#038;source=editors&#038;ust=1728333125182167&#038;usg=AOvVaw1mCjdqyQz3njapgbJbd6aZ\" target=\"_blank\" rel=\"noopener\">5 Practices<\/a><\/span><\/li>\n<li class=\"c1 c4 li-bullet-0\"><span class=\"c3\"><a class=\"c6\" href=\"https:\/\/www.google.com\/url?q=https:\/\/www.buildingthinkingclassrooms.com\/&#038;sa=D&#038;source=editors&#038;ust=1728333125182509&#038;usg=AOvVaw1dVvjCxD7-fJHqlvoHtINX\" target=\"_blank\" rel=\"noopener\">Building Thinking Classrooms (BTC)<\/a><\/span><\/li>\n<li class=\"c1 c4 li-bullet-0\"><span class=\"c3\"><a class=\"c6\" href=\"https:\/\/www.google.com\/url?q=https:\/\/lessonresearch.net\/teaching-problem-solving\/overview\/&#038;sa=D&#038;source=editors&#038;ust=1728333125182779&#038;usg=AOvVaw1437A6mrAL8i3cCvPKk-RN\" target=\"_blank\" rel=\"noopener\">Teaching through problem-solving (TTP)<\/a><\/span><\/li>\n<\/ul>\n<p class=\"c1\">My personal preference is TTP because it best explains exactly how to implement inquiry in your classroom immediately. I have written about TTP&nbsp;<span class=\"c3 c5\"><a class=\"c6\" href=\"https:\/\/www.google.com\/url?q=https:\/\/theothermath.com\/index.php\/2024\/05\/16\/teaching-through-problem-solving\/&#038;sa=D&#038;source=editors&#038;ust=1728333125183207&#038;usg=AOvVaw2DAX04Q7te-Y93ty__KNx0\" target=\"_blank\" rel=\"noopener\">here<\/a><\/span><span class=\"c0\">.<\/span><\/p>\n<p class=\"c1\"><span class=\"c0\">In summary:<\/span><\/p>\n<ul>\n<li class=\"c1\"><span class=\"c0\">Teach wide math\u2026not narrow math.<\/span><\/li>\n<li class=\"c1\"><span class=\"c0\">Allow students to explore the math concept BEFORE you formally teach it.<\/span><\/li>\n<\/ul>\n<p class=\"c1\"><span class=\"c0\">Do not hesitate to contact me if you have any thoughts or questions about this!<\/span><\/p>\n<p class=\"c1\"><span class=\"c0\">Duane Habecker<\/span><\/p>\n<p class=\"c1\"><span class=\"c3\"><a class=\"c6\" href=\"mailto:dhabecker@mcoe.org\">dhabecker@mcoe.org<\/a><\/span><\/p>\n<p>.<\/p>\n<p>.<\/p>\n<p>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>About 18 months ago in March, 2024, two black teenage girls from a New Orleans high school announced they found a new proof of the Pythagorean Theorem. What makes their proof especially impressive is that out of more than 400 unique proofs of the Pythagorean Theorem, theirs is the FIRST proof that relies entirely on [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":2463,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[4],"tags":[105,99,98],"_links":{"self":[{"href":"https:\/\/theothermath.com\/index.php\/wp-json\/wp\/v2\/posts\/2462"}],"collection":[{"href":"https:\/\/theothermath.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/theothermath.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/theothermath.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/theothermath.com\/index.php\/wp-json\/wp\/v2\/comments?post=2462"}],"version-history":[{"count":3,"href":"https:\/\/theothermath.com\/index.php\/wp-json\/wp\/v2\/posts\/2462\/revisions"}],"predecessor-version":[{"id":2466,"href":"https:\/\/theothermath.com\/index.php\/wp-json\/wp\/v2\/posts\/2462\/revisions\/2466"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/theothermath.com\/index.php\/wp-json\/wp\/v2\/media\/2463"}],"wp:attachment":[{"href":"https:\/\/theothermath.com\/index.php\/wp-json\/wp\/v2\/media?parent=2462"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/theothermath.com\/index.php\/wp-json\/wp\/v2\/categories?post=2462"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/theothermath.com\/index.php\/wp-json\/wp\/v2\/tags?post=2462"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}