{"id":2006,"date":"2023-12-12T08:00:24","date_gmt":"2023-12-12T13:00:24","guid":{"rendered":"https:\/\/mathvoices.ams.org\/mathmedia\/?p=2006"},"modified":"2023-12-11T15:24:36","modified_gmt":"2023-12-11T20:24:36","slug":"math-digests-november-2023","status":"publish","type":"post","link":"https:\/\/mathvoices.ams.org\/mathmedia\/math-digests-november-2023\/","title":{"rendered":"Math Digests November 2023"},"content":{"rendered":"<ul>\n<li><a href=\"#1\">The Man Who Invented Fifteen Hundred Necktie Knots<\/a><\/li>\n<li><a href=\"#2\">How maths can help you win at everything<\/a><\/li>\n<li><a href=\"#3\">Diagnostic tests for rare conditions present a mathematical conundrum<\/a><\/li>\n<li><a href=\"#4\">This mathematician had another career: professional football player<\/a><\/li>\n<li><a href=\"#5\">Cancer has many faces \u2212 5 counterintuitive ways scientists are approaching cancer research to improve treatment and prevention<\/a><\/li>\n<\/ul>\n<hr \/>\n<h3><a id=\"1\" href=\"https:\/\/www.newyorker.com\/culture\/annals-of-inquiry\/the-man-who-invented-fifteen-hundred-necktie-knots\">The Man Who Invented Fifteen Hundred Necktie Knots<\/a><\/h3>\n<p><em>The New Yorker<\/em>, November 10, 2023.<\/p>\n<p>When people say they know how to tie a necktie, they generally think of the one knot\u2014<em>maybe<\/em> two or three\u2014in their repertoire. Boris Mocka thinks of over fifteen hundred. <em>New Yorker<\/em> contributor Matthew Hutson recently introduced Mocka\u2014a doorman in Hutson\u2019s building\u2014to experts in topology, a branch of mathematics. The mathematicians were fascinated by Mocka\u2019s passion for creating new variations of loops and swoops. In this article for <em>The New Yorker<\/em>, Hutson writes about Mocka\u2019s life, his designs, and about how the art of a necktie intersects with an important research area in mathematics called knot theory. \u201cThe Gardenia looks like a flower; the Wicker and the Mockatonic look like origami. The Riddler looks like a question mark, and the Exousia requires more than one tie,\u201d writes Hutson. \u201cMany more math techniques would be needed to describe Mocka\u2019s art.\u201d<\/p>\n<p><strong>Classroom Activities:<\/strong> <em>topology, knot theory<\/em><\/p>\n<ul>\n<li>(Mid level) Watch this <a href=\"https:\/\/www.youtube.com\/watch?v=8DBhTXM_Br4\">Veritasium video about knot theory<\/a>, and answer the following questions:\n<ul>\n<li>Why are all knots studied by mathematicians \u201cclosed loops?\u201d<\/li>\n<li>What makes two knots mathematically different?<\/li>\n<li>What is the knot equivalence problem, and why is it so hard to solve?<\/li>\n<li>What is a prime knot, and how is it different from a composite knot?<\/li>\n<\/ul>\n<\/li>\n<li>(Mid level) Read the <em>New Yorker <\/em> article. Based on your reading and the Veritasium video, what is one reason why it\u2019s hard to define Mocka\u2019s ties with knot theory?<\/li>\n<li>(Mid level) Research 3 different knots that you can tie with neckties, or rope. Create them, and then identify which of these knots are cinquefoil, trefoil, or other categories you have learned in the lesson. Think about what the category for each means about their mathematical relationship. For example: the \u201cfour-in-hand\u201d and \u201cWindsor\u201d knots are both trefoils, therefore they are mathematically identical.<\/li>\n<\/ul>\n<p style=\"text-align: right\"><em>\u2014Max Levy<\/em><\/p>\n<hr \/>\n<h3><a id=\"2\" href=\"https:\/\/www.theguardian.com\/science\/2023\/nov\/05\/how-maths-can-help-you-win-at-everything\">How maths can help you win at everything<\/a><\/h3>\n<p><em>The Guardian<\/em>, November 5, 2023.<\/p>\n<p>Mathematician Marcus du Sautoy has a message: \u201cIn love and economics, business and games, if you know your maths, you\u2019ll end up the winner.\u201d In an article for <em>The Guardian<\/em>, du Sautoy shares mathematical strategies for winning in a wide variety of situations, including some that don\u2019t sound mathematical at first. Euler\u2019s number $e$, for example, shows up in a strategy for finding the right number of people to date before settling on \u201cthe one.\u201d Tools from calculus can help an entrepreneur figure out how many units to make of a new product. And statistically, a trivia buff playing &#8220;Who Wants to Be a Millionaire?&#8221; is better off asking the audience for help than phoning a friend. These and other real-world scenarios have rules and logical patterns that du Sautoy says people can tap into, mathematically, to succeed.<\/p>\n<p><strong>Classroom Activities:<\/strong> <em>economics, game theory, probability<\/em><\/p>\n<ul>\n<li>(All levels) Read the section of the article titled \u201cHow to win at economics.\u201d\n<ul>\n<li>If you were Player A in this scenario, how much money would you offer? If you were Player B, would you accept the split suggested in the article? Do you agree that the golden ratio makes the split feel fairer?<\/li>\n<li>For another example of Nash equilibrium, watch the Khan Academy video on the <a href=\"https:\/\/www.youtube.com\/watch?v=UkXI-zPcDIM\">prisoners\u2019 dilemma<\/a>.\n<ul>\n<li>The video illustrates that from Al\u2019s perspective, it\u2019s better to confess, whether or not Bill does so. The same is true for Bill. But when they both confess, they get more jail time than when they both deny. Discuss: How can it be that when Al and Bill make their best individual decisions, it doesn\u2019t lead to the best group outcome? Does this surprise you? Can you think of any other situations where something like this might occur?<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li>\u00a0(All levels) Read the section of the article titled \u201cHow to win at games.\u201d The article says that if you roll two dice, the most common total you will get is a $7$ because there are six ways to make $7$ out of two dice: $1+6$, $2+5$, $3+4$, $4+3$, $5+2$, and $6+1$.\n<ul>\n<li>How many ways are there to get a total of $2$ when rolling two dice? How many ways are there to get a total of $3$?<\/li>\n<li>When you roll two dice, how many different outcomes are possible in total? What strategies did you use to try to figure this out?<\/li>\n<li>(High level) If you roll two dice, what is the probability of rolling a total of $7$?<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p style=\"text-align: right\"><em>\u2014Tamar Lichter Blanks<\/em><\/p>\n<hr \/>\n<h3><a id=\"3\" href=\"https:\/\/www.statnews.com\/2023\/11\/30\/lab-derived-tests-diagnostics-rare-diseases-mathematical-limits\/\">Diagnostic tests for rare conditions present a mathematical conundrum<\/a><\/h3>\n<p><em>Stat News<\/em>, November 30, 2023.<\/p>\n<p>Currently, the Food and Drug Administration allows labs that produce and then use their own diagnostic tests to proceed free from the usual regulations. This fall, the FDA proposed changing this policy, citing the tests&#8217; frequent false positives. But the FDA, and the public, may find that they keep getting false positives even when diagnostic testing is subject to the strictest quality standards. In this article for <em>Stat<\/em>, mathematician Manil Suri and epidemiologist Daniel Morgan explain that false positives are par for the course when it comes to medical testing, especially when the condition being tested for is rare. \u201cPractically\u00a0<em>all<\/em>\u00a0tests, not just [lab-developed tests], carry the risk of false positives, which can render the results effectively useless when the condition is rare enough,\u201d they write.<\/p>\n<p><strong>Classroom Activities: <\/strong><em>statistics, probability<\/em><\/p>\n<ul>\n<li>(Mid level) Learn about sensitivity and specificity of diagnostic tests with <a href=\"https:\/\/schoolyourself.org\/learn\/statistics\/sensitivity\">this short interactive lesson<\/a> from\u00a0<em>SchoolYourself.<\/em><\/li>\n<li>(Mid level) Navigate to Morgan\u2019s online <a href=\"https:\/\/calculator.testingwisely.com\/\">calculator<\/a>, which shows the probability that a patient has a disease in a variety of situations.\n<ul>\n<li>Under \u201cWhat disease are you testing for?\u201d select \u201cFlu (Influenza).\u201d Under \u201cWhat is the pre-test probability?\u201d select \u201cNon-Influenza Season.\u201d Under \u201cSymptoms,\u201d select \u201cNo clinical symptoms.\u201d<\/li>\n<li>Before selecting any other options, scroll down to \u201cWhat Test will you order?\u201d, and read the numbers for sensitivity and specificity. Based on the \u201cChance of Flu\u201d number given on the left-hand side, try to figure out how likely it is that a patient who has <strong>no symptoms<\/strong> actually has the flu if they test positive on each test\n<ul>\n<li>during flu season<\/li>\n<li>not during flu season.<\/li>\n<\/ul>\n<\/li>\n<li>Now, work through all three tests. For each test, select \u201cResult &gt; If positive.\u201d Look at the \u201cChance of Flu &gt; After test\u201d number. Does the result surprise you?<\/li>\n<li>Repeat the exercise during flu season. How much do the numbers change? Why do you think they changed?<\/li>\n<\/ul>\n<\/li>\n<li>(Mid level) Now, create your own 2 x 2 grid as in the <em>SchoolYourself<\/em> lesson, with the columns \u201cActually has the disease,\u201d \u201cDoes not have the disease,\u201d and the rows \u201cTests positive,\u201d and \u201cTests negative.\u201d\n<ul>\n<li>According to Morgan\u2019s data, during non-flu season, 0.5% of people have the flu, and during flu season, about 4% of people have the flu. Suppose you test 1,000 random people. Using the sensitivities and specificities listed on Morgan\u2019s calculator, fill in the squares of your grid for each test, during flu season and not during flu season.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p style=\"text-align: right\"><em>\u2014Leila Sloman<\/em><\/p>\n<hr \/>\n<h3><a id=\"4\" href=\"https:\/\/www.npr.org\/transcripts\/1198908701\">This mathematician had another career: professional football player<\/a><\/h3>\n<p><em>NPR Short Wave<\/em>, November 27, 2023.<\/p>\n<p>When John Urschel started his Ph.D. in mathematics at the Massachusetts Institute of Technology, he already had a high-profile job playing football for the Baltimore Ravens. In this episode of <em>Short Wave<\/em>, Regina Barber interviews Urschel about how he balanced those two demanding careers. and about his current research in linear algebra.<\/p>\n<p><strong>Classroom Activities: <\/strong><em>linear algebra<\/em><\/p>\n<ul>\n<li>(Mid level, Algebra II) Barber describes linear algebra by example: As the type of math used to solve systems of equations such as $x + y = 3$, $x &#8211; y = \u20131$.\n<ul>\n<li>Read Section 5.2 from <a href=\"https:\/\/math.libretexts.org\/Courses\/Monroe_Community_College\/MTH_098_Elementary_Algebra\/5%3A_Systems_of_Linear_Equations\/5.3%3A_Solve_Systems_of_Equations_by_Elimination\">this online textbook<\/a> from <em>LibreTexts <\/em>to learn how to solve a system like this using a method called \u201csubstitution.\u201d Do exercises 5.2.1, 5.2.2, 5.2.3, and 5.2.19.<\/li>\n<li>Now read Section 5.3 to learn a method called \u201celimination\u201d. Do exercises 5.3.4, 5.3.7, 5.3.10, and 5.3.22.<\/li>\n<li>Solve Barber\u2019s example using both substitution and elimination. Show your work.<\/li>\n<\/ul>\n<\/li>\n<li>(High level) In the episode, Urschel talks about \u201chigher-dimensional versions of lines.\u201d Before viewing the next activity, discuss the following questions in small groups, then answer individually.\n<ul>\n<li>What does he mean by \u201chigher-dimensional\u201d?<\/li>\n<li>How would you define a \u201chigher-dimensional version of a line\u201d? Reminder: You can think about a line as a type of curve, or you can think about it in terms of the equation that defines it. You may find one of these perspectives easier than the other!<\/li>\n<\/ul>\n<\/li>\n<li>(Mid level, Algebra II) Generally, a \u201chigher-dimensional line\u201d refers to adding more variables into a linear equation. For example, $x + y + z = 1$ has three variables, instead of just two, and defines a plane in three-dimensional space.\n<ul>\n<li>Plot this plane in the <a href=\"https:\/\/www.geogebra.org\/3d?lang=en\">GeoGebra 3D calculator<\/a>. Do you think this deserves to be called a higher-dimensional line? Why or why not?<\/li>\n<li>Play with the coefficients of the plane, and observe how the image in GeoGebra changes. How does the idea of a line\u2019s \u201cslope\u201d generalize to higher dimensions?<\/li>\n<li>Solve the following system of equations using both substitution and elimination.<br \/>\n$$x + y + z = 1, \\quad x + y \u2013 z = 1, \\quad x + 2y + 3z = 3.$$<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p style=\"text-align: right\"><em>\u2014Leila Sloman<\/em><\/p>\n<hr \/>\n<h3><a id=\"5\" href=\"https:\/\/theconversation.com\/cancer-has-many-faces-5-counterintuitive-ways-scientists-are-approaching-cancer-research-to-improve-treatment-and-prevention-215698\">Cancer has many faces \u2212 5 counterintuitive ways scientists are approaching cancer research to improve treatment and prevention<\/a><\/h3>\n<p><em>The Conversation<\/em>, November 1, 2023.<\/p>\n<p>Cancer is not one disease. Cancer differs in the body parts it resides in, the genetic mutations that propel it, how it interacts with the body, and who it infects. This means that oncologists must think about cancer in many different ways. One of those ways is math. Of the five unconventional ways of thinking about cancer described by Vivian Lam in <em>The Conversation<\/em>, some are biological, like evolution and inflammation. But in the mathematical lens, one oncologist tells Lam that the mathematical randomness occurring in cancer biology, called epigenetic entropy, may be key to learning how to prevent cancer. \u201cEpigenetic entropy shows that you can\u2019t fully understand cancer without mathematics,\u201d he says.<\/p>\n<p><strong>Classroom Activities:<\/strong> <em>stochasticity<\/em><\/p>\n<ul>\n<li>(All levels) The mathematics of cancer mentioned in the article deals with stochasticity, a word for randomness. Create a table with 100 rows and two columns. Each row represents the outcome of a coin flip\u2014either \u201cheads\u201d or \u201ctails.\u201d In the first column, write what you expect 100 successive coin flip results to look like. Imagine that your goal is to convince someone that your list is really a list of 100 coin flips. In the next column, do 100 coin flips and record the outcome on each row.\n<ul>\n<li>Compare the two columns. How are they similar? How are they different? Did anything in the real coin flip surprise you?<\/li>\n<li>Now, watch <a href=\"https:\/\/www.youtube.com\/watch?v=LElyagQ0n_g\">this video about stochasticity<\/a>, which discusses the results of an experiment very much like the one you just performed on yourself. Discuss what you\u2019ve learned.<\/li>\n<li>Revisit the <a href=\"https:\/\/mathvoices.ams.org\/mathmedia\/math-digests-october-2023\/#3\">October digest<\/a> on coin flipping for alternative activities and videos on coin flip outcomes and manual attempts to mimic them.<\/li>\n<\/ul>\n<\/li>\n<li>(All levels) Play this <a href=\"https:\/\/www.letsgethealthy.org\/students\/games\/epigenetics-game\/\">educational game about epigenetics<\/a>, which will teach you the basic science of what is being randomized in this area of cancer.<\/li>\n<li>(Mid level) The article describes how stochasticity appears in other processes like the stock market and epidemiology. Take a few minutes to research how stochasticity plays a role in both fields. What parameter is random in each field? How can we measure the randomness? Based on your reading, describe how the mathematics of epigenetics is related to the math of epidemiology and finance.\n<ul>\n<li>In what ways do the examples seem different?<\/li>\n<li>Could researchers of one field (e.g. stochasticity in epidemiology) learn from the work and discoveries of researchers in another (e.g. stochasticity in cancer epigenetics)? Discuss why or why not as a group.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p style=\"text-align: right\"><em>\u2014Max Levy<\/em><\/p>\n<hr \/>\n<h3><strong>Some more of this month\u2019s math headlines<\/strong><\/h3>\n<ul>\n<li><a href=\"https:\/\/www.livescience.com\/physics-mathematics\/mathematics\/what-is-the-largest-known-prime-number\">What is the largest known prime number?<\/a><br \/>\n<em>Live Science<\/em>, November 23, 2023.<\/li>\n<li><a href=\"https:\/\/www.quantamagazine.org\/pierre-de-fermats-link-to-a-high-school-students-prime-math-proof-20231122\/\">Pierre de Fermat\u2019s Link to a High School Student\u2019s Prime Math Proof<\/a><br \/>\n<em>Quanta Magazine<\/em>, November 22, 2023.<\/li>\n<li><a href=\"https:\/\/www.scientificamerican.com\/article\/is-the-lottery-ever-a-good-bet\/\">Is the Lottery Ever a Good Bet?<\/a><br \/>\n<em>Scientific American<\/em>, November 17, 2023.<\/li>\n<li><a href=\"https:\/\/www.scientificamerican.com\/article\/simple-formula-makes-prime-numbers-easy-but-a-million-dollar-mystery-remains\/\">Simple Formula Makes Prime Numbers Easy, but a Million-Dollar Mystery Remains<\/a><br \/>\n<em>Scientific American<\/em>, November 16, 2023.<\/li>\n<li><a href=\"https:\/\/www.popularmechanics.com\/science\/math\/a45755752\/notorious-ramsey-problem-solved\/\">After 90 Years, Mathematicians Finally Solved the Most Notorious Ramsey Problem<\/a><br \/>\n<em>Popular Mechanics<\/em>, November 7, 2023.<\/li>\n<li><a href=\"https:\/\/www.theslateonline.com\/article\/2023\/11\/finding-my-culture-in-hidden-figures\">Finding my culture in &#8216;Hidden Figures&#8217;<\/a><br \/>\n<em>The Slate<\/em>, November 7, 2023.<\/li>\n<li><a href=\"https:\/\/www.wvnews.com\/news\/wvnews\/mathematical-genius-katherine-johnson-to-be-inducted-into-national-aviation-hall-of-fame-class-of\/article_53bf5b8a-7d94-11ee-b837-63e00fa67025.html\">Mathematical Genius Katherine Johnson to be Inducted into National Aviation Hall of Fame Class of 2024<\/a><br \/>\n<em>WV News<\/em>, November 7, 2023.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>The Man Who Invented Fifteen Hundred Necktie Knots How maths can help you win at everything Diagnostic tests for rare conditions present a mathematical conundrum This mathematician had another career: professional football player Cancer has many faces \u2212 5 counterintuitive ways scientists are approaching cancer research to improve treatment and<span class=\"more-link\"><a href=\"https:\/\/mathvoices.ams.org\/mathmedia\/math-digests-november-2023\/\">Read More &rarr;<\/a><\/span><\/p>\n","protected":false},"author":13,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"advanced_seo_description":"","jetpack_seo_html_title":"","jetpack_seo_noindex":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_post_was_ever_published":false},"categories":[2],"tags":[127,151,229,237,73,84,279,145],"class_list":["entry","author-leilasloman","post-2006","post","type-post","status-publish","format-standard","category-math-in-the-media-digests","tag-economics","tag-game-theory","tag-knot-theory","tag-linear-algebra","tag-probability","tag-statistics","tag-stochasticity","tag-topology"],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/posts\/2006","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/users\/13"}],"replies":[{"embeddable":true,"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/comments?post=2006"}],"version-history":[{"count":29,"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/posts\/2006\/revisions"}],"predecessor-version":[{"id":2035,"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/posts\/2006\/revisions\/2035"}],"wp:attachment":[{"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/media?parent=2006"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/categories?post=2006"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/tags?post=2006"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}