{"id":77,"date":"2026-01-25T12:00:36","date_gmt":"2026-01-25T12:00:36","guid":{"rendered":"https:\/\/learnx.ca\/um\/?page_id=77"},"modified":"2026-02-07T15:01:33","modified_gmt":"2026-02-07T15:01:33","slug":"coding","status":"publish","type":"page","link":"https:\/\/imaginethis.ca\/u\/coding\/","title":{"rendered":"Coding"},"content":{"rendered":"\n<p>\u201cThe most learning with the least teaching.\u201d<br>\u2015Seymour Papert<\/p>\n\n\n\n<h1 class=\"wp-block-heading\">LESSONS LEARNED FROM MATH CLASSROOMS<\/h1>\n\n\n\n<h2 class=\"wp-block-heading has-bright-blue-color has-text-color\">0. Introduction<\/h2>\n\n\n\n<iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/hL2kG-wu4lU?rel=0\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"\"><\/iframe>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"alignright size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"255\" height=\"176\" src=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL1.png\" alt=\"\" class=\"wp-image-400\" style=\"aspect-ratio:1.2527689533785524;width:160px;height:auto\"\/><\/figure>\n<\/div>\n\n\n<p>Computer programming offers the potential for changing how and what mathematics students learn, as well as who can learn it and when.<\/p>\n\n\n\n<p>In this collection of short video clips, I share 9 lessons we have learned about integrating mathematics and computer programming, and about mathematics teaching and learning in general.&nbsp;<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><a href=\"#1\">Give students code that works<\/a>&nbsp;<\/li>\n\n\n\n<li><a href=\"#2\">Make it a puzzle<\/a><\/li>\n\n\n\n<li><a href=\"#3\">Use a low floor and a high ceiling<\/a><\/li>\n\n\n\n<li><a href=\"#4\">All students deserve big ideas of mathematics<\/a><\/li>\n\n\n\n<li><a href=\"#5\">Coding offers cognitive simplicities<\/a><\/li>\n\n\n\n<li><a href=\"#6\">Coding needs deeper mathematics<\/a><\/li>\n\n\n\n<li><a href=\"#7\">You don\u2019t have to be a coding expert<\/a><\/li>\n\n\n\n<li><a href=\"#8\">Less can be more<\/a><\/li>\n\n\n\n<li><a href=\"#9\">Tell a good math story<\/a><\/li>\n<\/ol>\n\n\n\n<p>This learning is the result of spending many mornings and afternoons collaborating and co-teaching in grades 1-10 classrooms.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading has-bright-blue-color has-text-color\" id=\"1\"><strong>1. Give students code that wor<\/strong>ks<\/h2>\n\n\n\n<iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/6aLMSTbk0NE?rel=0\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"\"><\/iframe>\n\n\n\n<p>The traditional approach for teaching computer programming is for students to first learn about the various code components.&nbsp;<\/p>\n\n\n\n<p>In contrast, we typically start by giving students code that works, such as the code shown here.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"689\" height=\"322\" src=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL2.png\" alt=\"\" class=\"wp-image-401\" style=\"aspect-ratio:2.139867279224094;width:462px;height:auto\" srcset=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL2.png 689w, https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL2-300x140.png 300w, https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL2-624x292.png 624w\" sizes=\"auto, (max-width: 689px) 100vw, 689px\" \/><figcaption class=\"wp-element-caption\">See <a href=\"https:\/\/scratch.mit.edu\/projects\/665007043\/editor\" target=\"_blank\" rel=\"noreferrer noopener\">https:\/\/scratch.mit.edu\/projects\/665007043\/editor<\/a> <\/figcaption><\/figure>\n\n\n\n<p>Students click on the code and they start understanding it incidentally, as they notice what it it does.<\/p>\n\n\n\n<p>As one teacher noted:<\/p>\n\n\n\n<p><em>\u201cMy biggest leaning was about incidental learning. Coding facilitates that. I also learned that when you have incidental learning I think the learning is deeper.\u201d<\/em><\/p>\n\n\n\n<p>An analogy would be teaching a child to ride a bicycle.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"alignright size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"334\" height=\"239\" src=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL3.png\" alt=\"\" class=\"wp-image-402\" style=\"aspect-ratio:1.3975155279503106;width:231px;height:auto\" srcset=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL3.png 334w, https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL3-300x215.png 300w\" sizes=\"auto, (max-width: 334px) 100vw, 334px\" \/><figcaption class=\"wp-element-caption\">How do children learn to ride a bicycle?<\/figcaption><\/figure>\n<\/div>\n\n\n<p>We don\u2019t start by taking the bicycle apart, and one-by-one teaching the child about each of the parts.<\/p>\n\n\n\n<p>We put the child on the bike, and give a little push.<\/p>\n\n\n\n<p>The bicycle has parts that move &#8211; it is alive &#8211; and it is best understood by experiencing it in action.<\/p>\n\n\n\n<p>This code is also alive. So is the mathematics that it models.&nbsp;<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading has-bright-blue-color has-text-color\" id=\"2\"><strong>2. Make it a puzzle<\/strong><\/h2>\n\n\n\n<iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/Lad80Z_x4tI?rel=0\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"\"><\/iframe>\n\n\n\n<p>We turn code that models mathematics concepts and relationships into a puzzle for students to solve.&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"688\" height=\"281\" src=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL4.png\" alt=\"\" class=\"wp-image-403\" style=\"aspect-ratio:2.4485981308411215;width:463px;height:auto\" srcset=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL4.png 688w, https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL4-300x123.png 300w, https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL4-624x255.png 624w\" sizes=\"auto, (max-width: 688px) 100vw, 688px\" \/><figcaption class=\"wp-element-caption\">See <a href=\"https:\/\/scratch.mit.edu\/projects\/474587945\/editor\" target=\"_blank\" rel=\"noreferrer noopener\">https:\/\/scratch.mit.edu\/projects\/474587945\/editor<\/a><\/figcaption><\/figure>\n\n\n\n<p>We might say:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>I found this code that does something interesting mathematically.<\/li>\n\n\n\n<li>Click on it to see it in action.&nbsp;<\/li>\n\n\n\n<li>Edit the code and notice what changes.<\/li>\n\n\n\n<li>Discuss in your group:\n<ul class=\"wp-block-list\">\n<li>What does the code do?<\/li>\n\n\n\n<li>How does it do it?<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>Then, edit the code to match each of the following results.<\/li>\n\n\n\n<li>What other interesting results might you get?<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"623\" height=\"247\" src=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL5.png\" alt=\"\" class=\"wp-image-404\" style=\"aspect-ratio:2.517820424948595;width:346px;height:auto\" srcset=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL5.png 623w, https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL5-300x119.png 300w\" sizes=\"auto, (max-width: 623px) 100vw, 623px\" \/><\/figure>\n\n\n\n<p>The Canadian anthropologist Marcel Danesi (2020) says that:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Puzzles have been around since the dawn of history and have guided the evolution of large sections of that history, including discoveries in mathematics.&nbsp;<\/li>\n\n\n\n<li>Humans are naturally attracted to the playful learning that puzzles provide.<\/li>\n<\/ul>\n\n\n\n<p>______________<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><sup>Danesi, M. (2020). <em>An anthropology of puzzles: the role of puzzles in the origins and evolution of mind and culture<\/em>. London: Routledge, Taylor &amp; Francis Group<\/sup><\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading has-bright-blue-color has-text-color\" id=\"3\"><strong>3. Use a low floor and a high ceiling<\/strong><\/h2>\n\n\n\n<iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/mM6PVkspFlw?rel=0\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"\"><\/iframe>\n\n\n\n<p>For example, when introducing the topic of inequalities in grades 5 and 6, use masking tape to form a number line on the classroom floor.<\/p>\n\n\n\n<p><em>Where on the number line are numbers greater than 3?&nbsp;<\/em><\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"591\" height=\"213\" src=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL6.png\" alt=\"\" class=\"wp-image-405\" style=\"aspect-ratio:2.7747780793778887;width:461px;height:auto\" srcset=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL6.png 591w, https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL6-300x108.png 300w\" sizes=\"auto, (max-width: 591px) 100vw, 591px\" \/><\/figure>\n\n\n\n<p>Then create a second number line on the floor, perpendicular to the first one.<\/p>\n\n\n\n<p><em>How many coordinates do we need to identify locations on the floor?<\/em><\/p>\n\n\n\n<p><em>Where on the floor is the first coordinate greater than 3?<\/em><\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"588\" height=\"208\" src=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL8.png\" alt=\"\" class=\"wp-image-407\" style=\"aspect-ratio:2.8270603260461273;width:460px;height:auto\" srcset=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL8.png 588w, https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL8-300x106.png 300w\" sizes=\"auto, (max-width: 588px) 100vw, 588px\" \/><\/figure>\n\n\n\n<p>Use string to create a third number line, starting where the first two number lines meet, then extending vertically to the ceiling.<\/p>\n\n\n\n<p><em>How many coordinates do we need to identify locations in the classroom?<\/em><\/p>\n\n\n\n<p><em>Where in the classroom is the first coordinate greater than 3?<\/em><\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"586\" height=\"207\" src=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL7.png\" alt=\"\" class=\"wp-image-406\" style=\"aspect-ratio:2.831056088170907;width:462px;height:auto\" srcset=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL7.png 586w, https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL7-300x106.png 300w\" sizes=\"auto, (max-width: 586px) 100vw, 586px\" \/><\/figure>\n\n\n\n<p>This activity offers a low floor, by anchoring student understanding to the physical environment around them. <\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"694\" height=\"336\" src=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL9.png\" alt=\"\" class=\"wp-image-408\" style=\"aspect-ratio:2.0687953412623994;width:461px;height:auto\" srcset=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL9.png 694w, https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL9-300x145.png 300w, https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL9-624x302.png 624w\" sizes=\"auto, (max-width: 694px) 100vw, 694px\" \/><\/figure>\n\n\n\n<p>Low floor mathematics requires minimal prerequisite knowledge.<\/p>\n\n\n\n<p>This activity also offers a high ceiling, by engaging students with multiple meanings and representations of x &gt; 3, in 1 dimension, 2 dimensions, and 3 dimensions.&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"685\" height=\"168\" src=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL10.png\" alt=\"\" class=\"wp-image-409\" style=\"aspect-ratio:4.0778210116731515;width:461px;height:auto\" srcset=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL10.png 685w, https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL10-300x74.png 300w, https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL10-624x153.png 624w\" sizes=\"auto, (max-width: 685px) 100vw, 685px\" \/><figcaption class=\"wp-element-caption\">x > 100 in 1D, 2D &amp; 3D<\/figcaption><\/figure>\n\n\n\n<p>Such representations provide opportunities to investigate more complex mathematics concepts and relationships.<\/p>\n\n\n\n<p>The different meanings of x &gt; 3 engage student interest and imagination, and motivate them to attend deeply and to understand them.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading has-bright-blue-color has-text-color\" id=\"4\"><strong>4. All students deserve big ideas of mathematics&nbsp;<\/strong><\/h2>\n\n\n\n<iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/A1DwSmiCn9g?rel=0\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"\"><\/iframe>\n\n\n\n<p>Our schools are democratic institutions.  They provide access to education for all students.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"alignright size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"181\" height=\"177\" src=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL11.png\" alt=\"\" class=\"wp-image-410\" style=\"aspect-ratio:1.0123468698755667;width:121px;height:auto\"\/><\/figure>\n<\/div>\n\n\n<p>However, as educators, we sometimes worry that big ideas of mathematics may confuse students.&nbsp;<\/p>\n\n\n\n<p>We also worry that students who typically struggle mathematically may not be ready to engage with bigger ideas.<\/p>\n\n\n\n<p>This occasional lack of faith in children&#8217;s abilities is, in part, due to some popular, but incorrect, education theories.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"680\" height=\"621\" src=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL12.png\" alt=\"\" class=\"wp-image-411\" style=\"aspect-ratio:1.0950316075440154;width:308px;height:auto\" srcset=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL12.png 680w, https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL12-300x274.png 300w, https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL12-624x570.png 624w\" sizes=\"auto, (max-width: 680px) 100vw, 680px\" \/><figcaption class=\"wp-element-caption\">Why do we underestimate children?<\/figcaption><\/figure>\n\n\n\n<p>Jean Piaget, for example, brought to us the theory of stages of cognitive development. He said that young children are concrete thinkers and they develop their capacity to abstract later, maybe around age 12.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"alignright size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"124\" height=\"199\" src=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL13.png\" alt=\"\" class=\"wp-image-412\" style=\"aspect-ratio:0.6276909722222223;width:92px;height:auto\"\/><\/figure>\n<\/div>\n\n\n<p>Seymour Papert, who worked with Jean Piaget, disagreed. He said the stages Piaget identified are not in children\u2019s minds. Rather, these stages are symptoms of how we educate children. <\/p>\n\n\n\n<p>Piaget (1972\/2008) himself cautioned about how generally his stages of development may apply.<\/p>\n\n\n\n<p>Fernandez-Armesto (1997) laments that &#8220;Generations of school children, deprived of challenging tasks because Piaget said they were incapable of them, bear the evidence of his impact&#8221; (p. 18).<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"alignleft size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"144\" height=\"216\" src=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL14.png\" alt=\"\" class=\"wp-image-413\" style=\"aspect-ratio:0.6633181854420792;width:103px;height:auto\"\/><\/figure>\n<\/div>\n\n\n<p>Kieran Egan notes that young children naturally abstract to develop language, and come to understand words, such as dog.<\/p>\n\n\n\n<p>Dogs are big, small, different colours, and with different dispositions. Children create an abstraction of the essential characteristics of dogs, and distinguish them from other animals that look like them.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"377\" height=\"122\" src=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL15.png\" alt=\"\" class=\"wp-image-414\" style=\"aspect-ratio:2.2234375;width:246px;height:auto\" srcset=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL15.png 377w, https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL15-300x97.png 300w\" sizes=\"auto, (max-width: 377px) 100vw, 377px\" \/><\/figure>\n<\/div>\n\n\n<p>Children effortlessly abstract at a young age. They are much more capable, and much more attracted to big ideas of mathematics, than we sometimes assume.<\/p>\n\n\n\n<p>As one teacher noted:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"alignright size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"168\" height=\"200\" src=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL16.png\" alt=\"\" class=\"wp-image-415\" style=\"aspect-ratio:0.8784425451092118;width:114px;height:auto\"\/><\/figure>\n<\/div>\n\n\n<p><em>I found that sometimes the tasks we might feel initially [to be] difficult, the kids got just like that. It has made me less fearful to go beyond the curriculum. In Grade 4 you\u2019re only supposed to learn this. Well, what\u2019s stopping us from showing them a little bit beyond that?<\/em><\/p>\n\n\n\n<p>Another teacher said:<\/p>\n\n\n\n<p><em>I wish you were here to see the kids that never do well on assessments. I\u2019ve never seen that part of him. Words coming out were impressive.<\/em><\/p>\n\n\n\n<p>A low floor and a high ceiling approach is one way to offer access to big ideas of mathematics for all students.<\/p>\n\n\n\n<p>And, as we&#8217;ll soon see, coding offers cognitive simplicities that can change who can access what mathematics. <\/p>\n\n\n\n<p>_______________<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><sup>Egan, K. (2002). <em>Getting it wrong from the beginning : our progressivist inheritance from Herbert Spencer, John Dewey, and Jean Piaget. <\/em>New Haven :Yale University Press.<\/sup><\/li>\n\n\n\n<li><sup>Fernandez-Armesto, F. (1997). <em>Truth: A history and a guide for the perplexed.<\/em> London: Bantam.<\/sup><\/li>\n\n\n\n<li><sup>Papert, S. (1980). <em>Mindstorms\u2014Children, Computers and Powerful Ideas. <\/em>New York: Basic Books, Inc.<\/sup><\/li>\n\n\n\n<li><sup>Piaget, J. (2008). Intellectual evolution from adolescence to adulthood. <em>Human Development, 51<\/em>, 40-47. (Original work published 1972)<\/sup><\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading has-bright-blue-color has-text-color\" id=\"5\"><strong>5. Coding offers cognitive simplicities&nbsp;<\/strong><\/h2>\n\n\n\n<iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/wLef5SktUEM?rel=0\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"\"><\/iframe>\n\n\n\n<p>Our society has grown in complexity. Today computational tools are used to model phenomena and processes, to make scientific progress and to succeed economically. <\/p>\n\n\n\n<p>Most fields have a computational side: computational biology, computational mathematics, computational finance, computational medicine, to name a few examples.<\/p>\n\n\n\n<p>Lorena Barba (2016) says computational modelling is \u201ca source of power to do something and figure things out, in a dance between the computer and our thoughts.\u201d<\/p>\n\n\n\n<p>For instance, code that does something mathematical, like the code that plots inequalities, does it dynamically.&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"686\" height=\"319\" src=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL17.png\" alt=\"\" class=\"wp-image-416\" style=\"aspect-ratio:2.150625897804227;width:461px;height:auto\" srcset=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL17.png 686w, https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL17-300x140.png 300w, https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL17-624x290.png 624w\" sizes=\"auto, (max-width: 686px) 100vw, 686px\" \/><figcaption class=\"wp-element-caption\">See <a href=\"https:\/\/scratch.mit.edu\/projects\/665007043\/editor\" target=\"_blank\" rel=\"noreferrer noopener\">https:\/\/scratch.mit.edu\/projects\/665007043\/editor<\/a><\/figcaption><\/figure>\n\n\n\n<p>We can easily change one of the parameters, click on the code, and immediately see the effect.&nbsp;It would be very time-consuming to do this on paper.<\/p>\n\n\n\n<p>This allows students to engage playfully and creatively with mathematical concepts and relationships and to develop conceptual understanding.&nbsp;<\/p>\n\n\n\n<p>Andy diSessa (2000, 2018) defines the phenomenon where coding simplifies and automates a process needed for mathematical investigation, as a cognitive simplicity.<\/p>\n\n\n\n<p>Cognitive simplicities bring mathematics to life in new ways.<\/p>\n\n\n\n<p>They change what and how mathematics is done, and who can do it and when.<\/p>\n\n\n\n<p>As one teacher noted: <em>It\u2019s really neat because it extends their thinking, but in a natural way.<\/em><\/p>\n\n\n\n<p><strong>AN EXAMPLE<\/strong><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"alignright size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"214\" height=\"228\" src=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL18.png\" alt=\"\" class=\"wp-image-417\" style=\"aspect-ratio:0.904824144486692;width:109px;height:auto\"\/><\/figure>\n<\/div>\n\n\n<p>Let&#8217;s consider the Pythagorean Theorem, a<sup>2<\/sup> + b<sup>2<\/sup> = c<sup>2<\/sup>, as example. <\/p>\n\n\n\n<p>Actually, we should call this the Sumerian Theorem, as the Sumerians knew of this property many centuries before Pythagoras.<\/p>\n\n\n\n<p>The Sumerians were able to find very large integer triples, such as (119, 120, 169) and (4961, 6480, 8161), as depicted in cuneiform on this clay tablet. How did they find them?<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"672\" height=\"465\" src=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL19.png\" alt=\"\" class=\"wp-image-418\" style=\"aspect-ratio:1.4452182307108874;width:375px;height:auto\" srcset=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL19.png 672w, https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL19-300x208.png 300w, https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL19-624x432.png 624w\" sizes=\"auto, (max-width: 672px) 100vw, 672px\" \/><figcaption class=\"wp-element-caption\">Sumerian triples clay tablet.<\/figcaption><\/figure>\n\n\n\n<p>Let&#8217;s use Scratch code to find integer Sumerian triples (a, b, c). We&#8217;ll do this by considering every integer point (a, b) on the Scratch stage and checking whether c is also an integer. Then let&#8217;s plot all integers ordered pairs (a, b). <\/p>\n\n\n\n<p>This would be extremely difficult to do without code, as there are 17,280 integers pairs (a, b) on the Scratch stage. If you could check one pair every minute, it would take you 120 days.<\/p>\n\n\n\n<p>However, coding offers a cognitive simplicity. <\/p>\n\n\n\n<p>Scratch code can complete this process in about 1 minute.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"645\" height=\"648\" src=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL20.png\" alt=\"\" class=\"wp-image-419\" style=\"aspect-ratio:0.9953943719707747;width:460px;height:auto\" srcset=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL20.png 645w, https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL20-300x300.png 300w, https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL20-150x150.png 150w, https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL20-624x627.png 624w\" sizes=\"auto, (max-width: 645px) 100vw, 645px\" \/><figcaption class=\"wp-element-caption\">See <a href=\"https:\/\/scratch.mit.edu\/projects\/588754331\/editor\" target=\"_blank\" rel=\"noreferrer noopener\">https:\/\/scratch.mit.edu\/projects\/588754331\/editor<\/a><\/figcaption><\/figure>\n\n\n\n<p>What patterns do you notice in the plot? Do you see lines? Do you see curves opening to the left and to the right? What do they represent?<\/p>\n\n\n\n<p>If you are curious how many Sumerian triples fit on the Scratch stage, you may simply add a counter to the code to keep track.<\/p>\n\n\n\n<p>If we use Python, we may plot Sumerian triples (a, b, c) in 3D. <\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"325\" height=\"275\" src=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL21.png\" alt=\"\" class=\"wp-image-420\" style=\"aspect-ratio:1.1086205887516825;width:187px;height:auto\" srcset=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL21.png 325w, https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL21-300x254.png 300w\" sizes=\"auto, (max-width: 325px) 100vw, 325px\" \/><\/figure>\n\n\n\n<p>Why is the result a cone?<\/p>\n\n\n\n<p>What other mathematics might we bring to life with the help of the cognitive simplicities that coding offers?<\/p>\n\n\n\n<p>_______________<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><sup>Barba, L.A. (2016). Computational Thinking: I do not think it means what you think it means. Blog post, retrieved 06\/01\/18: <a rel=\"noreferrer noopener\" href=\"http:\/\/lorenabarba.com\/blog\/computational-thinking-i-do-not-think-it-means-what-you-think-it-means\" target=\"_blank\">http:\/\/lorenabarba.com\/blog\/computational-thinking-i-do-not-think-it-means-what-you-think-it-means<\/a>.<\/sup><\/li>\n\n\n\n<li><sup>diSessa, A. A. (2000).&nbsp;<em>Changing minds: computers, learning, and literacy.<\/em> Cambridge, MA: MIT Press.<\/sup><\/li>\n\n\n\n<li><sup>diSessa, A. A. (2018).&nbsp;Computational Literacy and \u201cThe Big Picture\u201d&nbsp;Concerning Computers in Mathematics Education. <em>Mathematical Thinking and Learning 20<\/em>(1), 3-31.<\/sup><\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading has-bright-blue-color has-text-color\" id=\"6\"><strong>6. Coding needs deeper mathematics&nbsp;<\/strong><\/h2>\n\n\n\n<iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/HXYJV2a67n0?rel=0\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"\"><\/iframe>\n\n\n\n<p>Using coding to represent shallow mathematics ideas does not offer much benefit.<\/p>\n\n\n\n<p><strong>EXAMPLE 1<\/strong><\/p>\n\n\n\n<p>For example, in the case of inequalities, we may write code that simulates a Q &amp; A.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"697\" height=\"285\" src=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL22.png\" alt=\"\" class=\"wp-image-421\" style=\"aspect-ratio:2.4457409568261377;width:462px;height:auto\" srcset=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL22.png 697w, https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL22-300x123.png 300w, https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL22-624x255.png 624w\" sizes=\"auto, (max-width: 697px) 100vw, 697px\" \/><figcaption class=\"wp-element-caption\">See <a href=\"https:\/\/scratch.mit.edu\/projects\/747003183\/editor\" target=\"_blank\" rel=\"noreferrer noopener\">https:\/\/scratch.mit.edu\/projects\/747003183\/editor<\/a><\/figcaption><\/figure>\n\n\n\n<p>The mathematics of the Q &amp; A code is shallow. It is simply about Yes or No. Correct or incorrect.&nbsp;<\/p>\n\n\n\n<p>Unlike the mathematics used in the code for plotting inequalities in 2D, this Q &amp; A code does not involve conceptual relationships.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"693\" height=\"322\" src=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL23.png\" alt=\"\" class=\"wp-image-422\" style=\"aspect-ratio:2.152282178980336;width:461px;height:auto\" srcset=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL23.png 693w, https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL23-300x139.png 300w, https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL23-624x290.png 624w\" sizes=\"auto, (max-width: 693px) 100vw, 693px\" \/><figcaption class=\"wp-element-caption\">See <a href=\"https:\/\/scratch.mit.edu\/projects\/665007043\/editor\" target=\"_blank\" rel=\"noreferrer noopener\">https:\/\/scratch.mit.edu\/projects\/665007043\/editor<\/a><\/figcaption><\/figure>\n\n\n\n<p>The mathematics of the Q &amp; A code has a low floor. But it also has a very low ceiling.&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"alignright size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"268\" height=\"226\" src=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL24.png\" alt=\"\" class=\"wp-image-423\" style=\"aspect-ratio:1.1980351202908641;width:108px;height:auto\"\/><\/figure>\n<\/div>\n\n\n<p><strong>EXAMPLE 2<\/strong><\/p>\n\n\n\n<p>Similarly, we can write code to simulate a Q &amp; A that asks for sides <strong>a<\/strong> and <strong>b<\/strong> of a right triangle and calculates the length of <strong>c<\/strong>, the hypotenuse.   <\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"693\" height=\"302\" src=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL25.png\" alt=\"\" class=\"wp-image-424\" style=\"aspect-ratio:2.294848634149012;width:461px;height:auto\" srcset=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL25.png 693w, https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL25-300x131.png 300w, https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL25-624x272.png 624w\" sizes=\"auto, (max-width: 693px) 100vw, 693px\" \/><figcaption class=\"wp-element-caption\"><em>See\u00a0<\/em><a href=\"https:\/\/scratch.mit.edu\/projects\/749902662\/editor\" target=\"_blank\" rel=\"noreferrer noopener\">https:\/\/scratch.mit.edu\/projects\/749902662\/editor<\/a><\/figcaption><\/figure>\n\n\n\n<p>However, the mathematics of this Q &amp; A code is also shallow. It is simply about using a formula.<\/p>\n\n\n\n<p>Instead, we may use code to find and plot integer Sumerian triples, to dynamically model and investigate deeper mathematics relationships. <\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"697\" height=\"277\" src=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL26.png\" alt=\"\" class=\"wp-image-425\" style=\"aspect-ratio:2.5163575244612524;width:461px;height:auto\" srcset=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL26.png 697w, https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL26-300x119.png 300w, https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL26-624x248.png 624w\" sizes=\"auto, (max-width: 697px) 100vw, 697px\" \/><figcaption class=\"wp-element-caption\"><em>Sumerian triples in 2D &amp; 3D.<\/em><\/figcaption><\/figure>\n\n\n\n<p><strong>A NEW LITERACY <\/strong><\/p>\n\n\n\n<p>Coding may be seen as a new literacy.<\/p>\n\n\n\n<p>Andy diSessa (2000, 2018) says that a literacy needs a literature.&nbsp; A literature of \u201cdeep and powerful ideas\u201d.<\/p>\n\n\n\n<p>Coding is most effective for deep and powerful ideas of mathematics, to model conceptual relationships, and to bring them to life.<\/p>\n\n\n\n<p>_____________<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><sup>diSessa, A. A. (2000).&nbsp;<em>Changing minds: computers, learning, and literacy.<\/em> Cambridge, MA: MIT Press.<\/sup><\/li>\n\n\n\n<li><sup>diSessa, A. A. (2018).&nbsp;Computational Literacy and \u201cThe Big Picture\u201d&nbsp;Concerning Computers in Mathematics Education. Mathematical Thinking and Learning 20(1), 3-31.<\/sup><\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading has-bright-blue-color has-text-color\" id=\"7\"><strong>7. You don\u2019t have to be a coding expert&nbsp;<\/strong><\/h2>\n\n\n\n<iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/3LZC8oswdfY?rel=0\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"\"><\/iframe>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"alignright size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"305\" height=\"182\" src=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL27.png\" alt=\"\" class=\"wp-image-426\" style=\"aspect-ratio:1.6548228071807274;width:147px;height:auto\" srcset=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL27.png 305w, https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL27-300x179.png 300w\" sizes=\"auto, (max-width: 305px) 100vw, 305px\" \/><\/figure>\n<\/div>\n\n\n<p>Learn along with your students.&nbsp;<\/p>\n\n\n\n<p>Don\u2019t be afraid to say, \u201cThat\u2019s a good question. I don\u2019t have the answer. Maybe we can find it together.\u201d<\/p>\n\n\n\n<p>Make understanding code a puzzle for students to solve.<\/p>\n\n\n\n<p>Here&#8217;s some code that does something interesting mathematically.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"688\" height=\"275\" src=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL30.png\" alt=\"\" class=\"wp-image-429\" style=\"aspect-ratio:2.5019397194867206;width:463px;height:auto\" srcset=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL30.png 688w, https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL30-300x120.png 300w, https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL30-624x249.png 624w\" sizes=\"auto, (max-width: 688px) 100vw, 688px\" \/><figcaption class=\"wp-element-caption\"><em>Go to\u00a0<\/em><a href=\"https:\/\/scratch.mit.edu\/projects\/474587945\/editor\" target=\"_blank\" rel=\"noreferrer noopener\">https:\/\/scratch.mit.edu\/projects\/474587945\/editor<\/a><\/figcaption><\/figure>\n\n\n\n<p>How may you edit the code to do this?<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"625\" height=\"246\" src=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL29.png\" alt=\"\" class=\"wp-image-428\" style=\"aspect-ratio:2.540811789366703;width:253px;height:auto\" srcset=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL29.png 625w, https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL29-300x118.png 300w\" sizes=\"auto, (max-width: 625px) 100vw, 625px\" \/><figcaption class=\"wp-element-caption\"><em>Can you do this?<\/em><\/figcaption><\/figure>\n\n\n\n<p>Starting with with code that works \u2014 code that brings some important mathematics to life \u2014 motivates and empowers students to investigate and to understand.<\/p>\n\n\n\n<p>One teacher noted:<\/p>\n\n\n\n<p><em>\u201cIt can be challenging but a safe way to learn. Some of them could go beyond what we were showing them. I feel they really surpassed me.\u201d&nbsp;<\/em><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading has-bright-blue-color has-text-color\" id=\"8\"><strong>8: Less can be more&nbsp;<\/strong><\/h2>\n\n\n\n<iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/0rkI-MQ29Qo?rel=0\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"\"><\/iframe>\n\n\n\n<p>Implementing a few coding activities that bring mathematics to life, and doing this well, is the ideal way to get started.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"688\" height=\"275\" src=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL30.png\" alt=\"\" class=\"wp-image-429\" style=\"aspect-ratio:2.5019397194867206;width:463px;height:auto\" srcset=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL30.png 688w, https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL30-300x120.png 300w, https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL30-624x249.png 624w\" sizes=\"auto, (max-width: 688px) 100vw, 688px\" \/><figcaption class=\"wp-element-caption\"><em>Go to&nbsp;<\/em><a href=\"https:\/\/scratch.mit.edu\/projects\/474587945\/editor\" target=\"_blank\" rel=\"noreferrer noopener\">https:\/\/scratch.mit.edu\/projects\/474587945\/editor<\/a><\/figcaption><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"686\" height=\"319\" src=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL17.png\" alt=\"\" class=\"wp-image-416\" style=\"aspect-ratio:2.150625897804227;width:461px;height:auto\" srcset=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL17.png 686w, https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL17-300x140.png 300w, https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL17-624x290.png 624w\" sizes=\"auto, (max-width: 686px) 100vw, 686px\" \/><figcaption class=\"wp-element-caption\">Inequalities in 2D. See <a href=\"https:\/\/scratch.mit.edu\/projects\/665007043\/editor\" target=\"_blank\" rel=\"noreferrer noopener\">https:\/\/scratch.mit.edu\/projects\/665007043\/editor<\/a><\/figcaption><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"697\" height=\"277\" src=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL26.png\" alt=\"\" class=\"wp-image-425\" style=\"aspect-ratio:2.5163575244612524;width:459px;height:auto\" srcset=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL26.png 697w, https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL26-300x119.png 300w, https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL26-624x248.png 624w\" sizes=\"auto, (max-width: 697px) 100vw, 697px\" \/><figcaption class=\"wp-element-caption\"><em>Sumerian triples in 2D &amp; 3D.<\/em><\/figcaption><\/figure>\n\n\n\n<p>Start small.&nbsp;<\/p>\n\n\n\n<p>Focus on big mathematics ideas.&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"alignright size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"165\" height=\"196\" src=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL33.png\" alt=\"\" class=\"wp-image-444\" style=\"aspect-ratio:0.8361437273770637;width:110px;height:auto\"\/><\/figure>\n<\/div>\n\n\n<p>Learn to code along with your students.<\/p>\n\n\n\n<p>A few well-designed mathematics experiences can have lasting impact on students&#8217; and teachers&#8217; dispositions, living fruitfully in future experiences \u2014 as Dewey (1938) has suggested \u2014 by raising expectation and anticipation of what mathematics is and what it can offer (Gadanidis, Borba, Hughes &amp; Lacerda, 2016).<\/p>\n\n\n\n<p><em>_______________<\/em><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><sup>Dewey, John (1938).&nbsp;<em>Experience &amp; Education.<\/em> New York, NY: Kappa Delta Pi.<\/sup><\/li>\n\n\n\n<li><sup>Gadanidis, G., Borba, M., Hughes, J. and Lacerda, H. (2016).&nbsp;Designing aesthetic experiences for young mathematicians: A model for mathematics education reform.<em>&nbsp;International Journal for Research in Mathematics Education 6<\/em>(2), 225-244.<\/sup><\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading has-bright-blue-color has-text-color\" id=\"9\"><strong>9. Tell a good math story&nbsp;<\/strong><\/h2>\n\n\n\n<iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/HdBBAc5BduI?rel=0\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"\"><\/iframe>\n\n\n\n<p>One way to judge the quality of students\u2019 mathematics experience is to consider how they would answer the question \u201cWhat did you do in math today?\u201d<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"alignright size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"255\" height=\"176\" src=\"https:\/\/imaginethis.ca\/u\/wp-content\/uploads\/2026\/02\/LL1.png\" alt=\"\" class=\"wp-image-400\" style=\"aspect-ratio:1.2485436893203883;width:170px;height:auto\"\/><\/figure>\n<\/div>\n\n\n<p>As you plan a mathematics activity, imagine how it prepares students to answer this question.<\/p>\n\n\n\n<p>Will they be able to share with family and friends an experience \u2014 a story \u2014 that would offer mathematical surprise and conceptual insight?<\/p>\n\n\n\n<p>Brian Boyd (2009) says that story is a biological necessity.<\/p>\n\n\n\n<p>Watson and Mason (2007) see mathematics as \u201can endless source of surprise.\u201d<\/p>\n\n\n\n<p>Creating, sharing and learning through stories and surprises are deeply human dispositions.<\/p>\n\n\n\n<p>What wonderful mathematics stories \u2014 what mathematical surprises and conceptual insights \u2014 might you help your students experience?<\/p>\n\n\n\n<p>And how might you use coding to enhance this experience?<\/p>\n\n\n\n<p>____________<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><sup>Boyd, B. (2009). <em>On the Origin of Stories: Evolution, Cognition, and Fiction.<\/em> Cambridge, MA: Belknap Press of Harvard University Press.<\/sup><\/li>\n\n\n\n<li><sup>Watson, A. &amp; Mason, J. (2007) Surprise and inspiration. <em>Mathematics<\/em> <em>Teaching <\/em>200, 4-7.<\/sup><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>\u201cThe most learning with the least teaching.\u201d\u2015Seymour Papert LESSONS LEARNED FROM MATH CLASSROOMS 0. Introduction Computer programming offers the potential for changing how and what mathematics students learn, as well as who can learn it and when. In this collection of short video clips, I share 9 lessons we have learned about integrating mathematics and [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-77","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/imaginethis.ca\/u\/wp-json\/wp\/v2\/pages\/77","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/imaginethis.ca\/u\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/imaginethis.ca\/u\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/imaginethis.ca\/u\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/imaginethis.ca\/u\/wp-json\/wp\/v2\/comments?post=77"}],"version-history":[{"count":15,"href":"https:\/\/imaginethis.ca\/u\/wp-json\/wp\/v2\/pages\/77\/revisions"}],"predecessor-version":[{"id":445,"href":"https:\/\/imaginethis.ca\/u\/wp-json\/wp\/v2\/pages\/77\/revisions\/445"}],"wp:attachment":[{"href":"https:\/\/imaginethis.ca\/u\/wp-json\/wp\/v2\/media?parent=77"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}