tag:blogger.com,1999:blog-77184627935169688832022-01-21T23:53:37.133-05:00MadMath"Beauty is the Enemy of Expression"Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.comBlogger314125tag:blogger.com,1999:blog-7718462793516968883.post-34897179901128720132021-11-02T16:11:00.001-04:002021-11-02T16:14:03.600-04:00Backtracking Detracking<p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-uXMhkLwHoSs/YYGb4KA6sGI/AAAAAAAAFyI/8F_HXnSXTc8tONk3TD95bgYRU1RXGhZKwCNcBGAsYHQ/s600/johnny_automatic_ground_tracking_station.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img alt="Tracking station" border="0" data-original-height="600" data-original-width="600" height="200" src="https://1.bp.blogspot.com/-uXMhkLwHoSs/YYGb4KA6sGI/AAAAAAAAFyI/8F_HXnSXTc8tONk3TD95bgYRU1RXGhZKwCNcBGAsYHQ/w200-h200/johnny_automatic_ground_tracking_station.png" width="200" /></a></div><p></p><p>There's an interesting article from the Brookings Institution last month, on the perpetual debate over <a href="https://en.wikipedia.org/wiki/Tracking_(education)">tracking</a> in the U.S. school system -- separating classes at the same grade by skill level. </p><p>Whenever this comes to mind I think about the giant debate that occurred at my high school system right around the time I graduated, which saw a new principal hired with a mandate to detrack all of the school's curriculum. I didn't actually experience that, but my sister, who's two years younger than myself, did. This all being 30 years ago as I write this. (I also had a more recent awareness of a friend's child in junior-high-school, who had special-needs students detracked in the same classroom, who would basically scream incoherently all day long and make any kind of learning impossible.)<br /></p><p></p><p>As usual with education issues, the rocky shoals upon which all proud ships crash is the mathematics discipline. You can pretty easily get away with a mix of skill levels in arts and social disciplines -- read the same texts (or whatnot), and accept that you'll get different levels of interpretations, and it's possible to grade on a relative "best effort" basis. In the hard sciences (things that are based on math), things get harder -- maybe you can get across some core concepts, but people with math skills will be able to dig deeper and make predictions and verifications in ways that other people cannot. But with our mathematical queen, this is basically impossible -- if someone doesn't have the prerequisite ability to read, write, and think in our language, then absolutely nothing will make sense, and they won't be able to interface with it in any way, producing nothing but raw gobbledygook (as I've seen hundreds or thousands of times). A number of times on this blog I've called this the "brutal honesty" of mathematics. That said: it never stops a legion of arts & social-science people from dictating supposed solutions for the mathematics professors, as crazy as that sounds. </p><p>So in the recent article, Tom Loveless of Brookings notes that the "tracking" argument goes back even farther than my 30-year experience:</p><p></p><blockquote><p><i>Research on tracking extends over a century. Hundreds upon hundreds of studies have not settled the debate. The literature is usually described as “mixed,” but with a clear warning that tracking can exacerbate gaps between high and low achievers.[1] Research is more plentiful on tracking as a problem, as a source of inequality, rather than detracking as a solution. Reformers have been hampered by a lack of empirical evidence that abolishing tracking would reduce inequities. Evaluations of untracked schools tend to be based on a small number of schools or on samples that were not scientifically selected to support generalizable findings...<br /><br />These case studies indicate that detracking may work under certain conditions, but they are less persuasive evidence that abolishing tracking in favor of classes with students heterogeneous in ability, all studying the same curriculum, will work everywhere or even in most schools. A study that forcefully raises that question was conducted by David N. Figlio and Marianne E. Page. They analyzed data from the National Education Longitudinal Survey of 1988 (NELS:88), which followed a random sample of several thousand students from eighth grade through high school and into post-secondary education and work. Using several methods of identifying whether schools were tracked or untracked, Figlio and Page uncovered neutral to positive effects of tracking. The most surprising finding of the analysis was that students from disadvantaged backgrounds appeared to benefit from tracking. Figlio and Page concluded, “We can find no evidence that detracking America’s schools, as is currently in vogue, will improve outcomes among disadvantaged students. This trend may instead harm the very students that detracking is intended to help”.<br /></i></p><p></p></blockquote><p>Ironically, the <i>data</i> that Figlio/Page analyzed was current to the year right before I graduated high school; but the <i>article</i> they published about it wasn't until 14 years later. I wonder if it would have made any difference at the time?</p><p>At least as interesting is what prompted the Brookings article at this time: back in May of this year the Washington Post had an article about a contentious push in the state of Virginia for detracking. After parental outcry, the state superintendent was forced to release a statement backtracking from the idea:</p><p></p><blockquote><i>Under the VMPI plan, [parent] Fox said, “every student would be required to take the same math class through 10th grade of high school. There would be no classes for struggling students needing remedial help or for advanced students seeking accelerated math.”<br /><br />When I called Virginia State Superintendent of Public Instruction James F. Lane to ask about this, he insisted that the state has no plans to eliminate tracking (separate classes for students at different levels) from kindergarten through 10th grade, even though the VMPI website strongly suggests that ending tracking is key to the suggested reforms...<br /><br />Lane, the Virginia state superintendent, is an experienced administrator, having led three school districts. He seems to understand how politically poisonous it would be to tell parents that every child is going to be on the same math track through 10th grade...<br /><br />Lane’s spokesman later told me “he does unequivocally denounce the idea that every student should be forced to take the exact same math courses at the same time without options for acceleration.”</i></blockquote><p></p><p>Will this detracking debate go on <i>ad infinitum</i>?<br /></p><p style="text-align: center;"><span style="font-size: large;"><a href="https://www.brookings.edu/blog/brown-center-chalkboard/2021/10/04/does-detracking-promote-educational-equity/">Brookings: Does detracking promote educational equity?</a></span><br /></p><p></p><p style="text-align: center;"><span style="font-size: large;"><a href="https://www.washingtonpost.com/local/education/fairfax-county-schools-math-classes-tracking/2021/04/29/197aa29c-a7a2-11eb-8d25-7b30e74923ea_story.html">Washington Post: Virginia allies with, then backs away from, controversial math anti-tracking movement</a></span><br /></p>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-61669871862604957752021-10-18T22:03:00.001-04:002021-10-18T22:10:59.556-04:00Growth Mindset Theory: Failures to Replicate<div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-RUgJOxFdUHk/YW4pFC8s5zI/AAAAAAAAFwc/74Rq90hOHQ8qZ6QEji_HYrGU8uUS06LDgCNcBGAsYHQ/s1280/MindsetUp.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="792" data-original-width="1280" height="198" src="https://1.bp.blogspot.com/-RUgJOxFdUHk/YW4pFC8s5zI/AAAAAAAAFwc/74Rq90hOHQ8qZ6QEji_HYrGU8uUS06LDgCNcBGAsYHQ/s320/MindsetUp.png" width="320" /></a></div><p>Psychologist Carol Dweck's "growth mindset" theory has become a popular solution and intervention technique in (mostly American) schools of all ages. We might say that it's become the new version of the "self-esteem" movement seen in the 80's. While Dweck first developed the theory in the 90's, it's really taken hold of popular consciousness from the 2010's on.</p><div class="s-prose js-post-body" itemprop="text"><p>Unfortunately, we should remember that psychology has an ongoing <a href="https://en.wikipedia.org/wiki/Replication_crisis" rel="nofollow noreferrer">replication crisis</a> in many of its landmark findings. Many of the "easy" ideas for transformative effects have not borne fruit over the years, and been later found to have tainted methods by core researchers. Sure enough, in recent years many or most of the large-scale, high-quality attempts at replicating the claims of growth mindset have failed to so. Here are a few examples:</p><p><b>Li, Y., & Bates, T. C., Ph.D. (2017)</b>. <i>Does growth mindset improve children’s IQ, educational attainment or response to setbacks? Active-control interventions and data on children’s own mindsets.</i> <a href="https://doi.org/10.31235/osf.io/tsdwy" rel="nofollow noreferrer">https://doi.org/10.31235/osf.io/tsdwy</a> (Study done in China, students aged 9-13 years, N = 624)</p><blockquote><p>No effect of the classic growth mindset manipulation was found for either moderate or more difficult material... children’s mindsets were unrelated to resilience to failure for either outcome measure... Finally, in 2 studies relating mindset to grades across a semester in school, the predicted association of growth mindset with improved grades was not supported. Neither was there any association of children’s mindsets with their grades at the start of the semester. Beliefs about the malleability of basic ability may not be related to resilience to failure or progress in school.</p></blockquote><p><b>Foliano, F., Rolfe, H., Buzzeo, J., Runge, J., & Wilkinson, D. (2019).</b> <i>Changing mindsets: Effectiveness trial.</i> National Institute of Economic and Social Research. <a href="http://psychbrief.com/growth-mindset-fails/" rel="nofollow noreferrer">Summary at PsychBrief</a>. (Study in England, Year 6 students, N = 4584.)</p><blockquote><p>The difference between the control group and the intervention group on all 3 primary outcomes [math, reading, GPS] was 0... The difference between the groups for all 4 secondary outcomes was also 0... This RCT was a highly powered test of the efficacy of growth mindset in a real-world environment across a wide range of schools in the England. The fact none of the primary or secondary outcomes were distinguishable from 0 raises serious questions as to the efficacy of growth mindset for Year 6 students... Given the evidence so far, it is unrealistic to expect growth mindset to have large and/or wide-scale impact.</p></blockquote><p><b>Caitlin Brez, Eric M. Hampton, Linda Behrendt, Liz Brown & Josh Powers (2020)</b> <i>Failure to Replicate: Testing a Growth Mindset Intervention for College Student Success</i>, Basic and Applied Social Psychology, 42:6, 460-468, <a href="https://www.tandfonline.com/doi/abs/10.1080/01973533.2020.1806845" rel="nofollow noreferrer">DOI: 10.1080/01973533.2020.1806845</a> (U.S. study, university math & psychology students, N = 2607).</p><blockquote><p>The pattern of findings is clear that the intervention had little impact on students’ academic success even among sub-samples of students who are traditionally assumed to benefit from this type of intervention (e.g., minority, low income, and first-generation students)... These findings support some of the emerging literature that demonstrates that growth mindset interventions may not be as effective as once thought... The proposition that a one-time intervention at the postsecondary level will result in long-term measurable student outcomes was not supported in the present study.</p></blockquote><hr /><p>Now, a not-uncommon defense in a number of these cases in psychology is that the attempts to replicate didn't properly recreate the conditions or variables for a true test. The counter-argument here would be the <a href="https://en.wikipedia.org/wiki/Observer-expectancy_effect" rel="nofollow noreferrer">observer-expectancy effect</a> -- in some cases a primary researcher has even argued that <i>only they</i> have the necessary knowledge to ever do so. Indeed, Dweck has made the "not anyone can do a replication" argument (<a href="https://www.buzzfeed.com/tomchivers/what-is-your-mindset" rel="nofollow noreferrer">BuzzFeed News interview</a>). In response, Nick Brown, who developed the <a href="https://en.wikipedia.org/wiki/GRIM_test" rel="nofollow noreferrer">GRIM</a> (Granularity-Related Inconsistency of Means) test and found several errors in Dweck's seminal paper, said this:</p><blockquote><p>The question I have is: If your effect is so fragile that it can only be reproduced [under strictly controlled conditions], then why do you think it can be reproduced by schoolteachers?</p></blockquote><p>Finally, psychologist Russell Warne <a href="https://russellwarne.com/2020/01/03/the-one-variable-that-makes-growth-mindset-interventions-work/" rel="nofollow noreferrer">wrote on his blog</a>:</p><blockquote><p>I discovered the one characteristic that the studies that support mindset theory share and that all the studies that contradict the theory lack: Carol Dweck... So, there you go! Growth mindsets can improve academic performance –if you have Carol Dweck in charge of your intervention.</p></blockquote><p>This is somewhat hyperbolic, but clarifies the issue at stake. Growth mindset theory fits fairly snugly into the basket of psychological "quick fixes" that make up the replication crisis, broadly cuts against long-standing findings from neuroscience on intelligence, and is racking up more failures-to-replicate as it garners more attention. Like other similar principles that came before, it's probably a bad bet that institutional interventions based on the theory will be worth the resources spent on them.<br /><br /></p><p><i>This post was initially written as an answer on <a href="https://matheducators.stackexchange.com/a/24419">Stack Exchange: Mathematics Educators</a>. Thanks to the community there for reading and refining it. </i><br /></p> </div>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-19751139138378799092021-08-31T21:59:00.005-04:002021-10-18T22:15:02.331-04:00On Chained Relations<div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-cKxrzVMukrg/YS7W8nn9CQI/AAAAAAAAFvg/KZWPqamPMLwNu8Ac31kri9G_eSoa-FxiwCNcBGAsYHQ/s453/ChainedRelation.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="134" data-original-width="453" height="95" src="https://1.bp.blogspot.com/-cKxrzVMukrg/YS7W8nn9CQI/AAAAAAAAFvg/KZWPqamPMLwNu8Ac31kri9G_eSoa-FxiwCNcBGAsYHQ/w320-h95/ChainedRelation.png" title="Chained relation" width="320" /></a></div><p> In all of the college math courses I teach -- from basic algebra, precalculus, calculus, discrete mathematics, etc. -- there's a particular piece of syntax that perpetually trips up students, and it's this: <b>chained relations</b>. </p><p>To be clear, chained relations are <i>compound statements</i> in mathematics with more than one relational symbol (including equalities and inequalities). Crack open any math textbook and you're bound to see almost <i>any</i> piece of symbolic expression written in that format. And yet my students are always tripping all over themselves at the difficulty of either reading or writing them. Have you ever noticed this before? Let's consider the several factors contributing to this difficulty:</p><ol style="text-align: left;"><li><b>Even a single equality is hard for people to truly understand</b>. Numerous academic papers have been written on this. More than one person has pointed out that the use of the equals-sign in grade-school problems and the calculator point people in the incorrect direction of a <i>functional</i> understanding, rather than a <i>relational</i> understanding. <br /><br /></li><li><b>There is no explicit instruction in the form in any curriculum</b>. To my knowledge, I've never seen the status of chained relations directly addressed or tested in any math textbook at any level (again, whether in basic algebra, precalculus, etc., etc.). At some point instructors just start using it and we assume students will understand by osmosis. <br /><br /></li><li><b>The compound form is entirely foreign to a natural language like English</b>. Consider something super simple like \(a = b = c\). Translated literally to English, it says, "<i>a</i> is equal to <i>b</i> is equal to <i>c</i>" -- and that's a run-on sentence, <i>disallowed</i> by the rules of English grammar. But here in the algebraic language we have an entirely novel mode of permitted expression. </li></ol><p>Considering that last point,we might observe that there is (surprisingly, for such a basic point?) unresolved confusion about how one should even <i>pronounce out loud</i> a simple chained relation. For example:</p><ul style="text-align: left;"><li><a href="https://matheducators.stackexchange.com/questions/17706/how-to-read-chained-equalities-out-loud">Question on Stack Exchange: Mathematics Educators</a></li><li><a href="https://english.stackexchange.com/questions/525126/how-to-say-abc">Question on Stack Exchange: English Language & Usage</a></li></ul><p>(Note that while the question is essentially the same, those two queries have <i>entirely different</i> top-voted answers.)<br /></p><p></p><p>In my opinion, the status of chained relations is one of those classic blindspot/submarined issues that's buried in math education, and winds up troubling students throughout their career. To instructors: it's "obvious" and never rises to consciousness as an issue. To students: it's a quagmire that's never clearly addressed or exercised.</p><p>To this end, I've found that I need to start my discrete mathematics classes foremost with direct instruction on this issue; namely a short document that I ask students to read -- and to which I'll be referring them throughout the semester when mistakes are made. You can download it here:</p><p style="text-align: center;"><span style="font-size: large;"><a href="http://www.superdan.net/download/blog/madmath/ChainedRelations-1.0.1.pdf">On Chained Relations (PDF)</a></span></p><p>And then to practice reading them, a timed quiz at the Automatic Algebra site:</p><p style="text-align: center;"><span style="font-size: large;"><a href="http://www.automatic-algebra.org/chainedrelations.htm">Quiz on Chained Relations </a></span></p><p>Interestingly with that quiz, I've had different math-trained professionals try it and tell me variously that (a) it was entirely trivial and of unclear value, or (b) it was entirely impossible within the span given on the timer. Isn't that interesting? What do you think?<br /></p>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-77494374394522755522021-07-20T08:00:00.001-04:002021-07-20T08:00:00.207-04:00Veritasium on Learning Styles<p>The Veritasium channel on YouTube recently released a very high-quality video on "The Biggest Myth in Education", to wit: the Learning Styles theory. It's much needed and much appreciated. Includes comments by famed cognitive psychologist Daniel Willingham, with whom I had the chance to speak in person on this (constantly frustrating) subject a few years ago. From the climax:</p><p><i></i></p><blockquote><p><i>Review articles of learning styles consistently conclude there is no credible evidence that learning styles exist. In a 2009 review, the researchers note: "The contrast between the enormous popularity of the learning styles approach within education and the lack of credible evidence for its utility is, in our opinion, striking and disturbing... If classification of students' learning styles has practical utility, it remains to be demonstrated."</i></p><p></p></blockquote><p>Big thanks to Veritasium for producing this. Pass it on!<br /></p><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="BLOG_video_class" height="266" src="https://www.youtube.com/embed/rhgwIhB58PA" width="320" youtube-src-id="rhgwIhB58PA"></iframe></div><br /><p><br /></p>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-39912492069809412522021-07-06T09:00:00.001-04:002021-07-06T09:00:00.221-04:00Minimal Mental RSA Cryptosystem<p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-3jO1--4RgKs/YOOxbXeXh5I/AAAAAAAAFqw/vXYWiOc7Jp4y4estRJ7NvStqOmdJEmvvwCNcBGAsYHQ/s504/giuseppe-urso-asymmetric-key-encryption-in-java-03.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="216" data-original-width="504" src="https://1.bp.blogspot.com/-3jO1--4RgKs/YOOxbXeXh5I/AAAAAAAAFqw/vXYWiOc7Jp4y4estRJ7NvStqOmdJEmvvwCNcBGAsYHQ/s320/giuseppe-urso-asymmetric-key-encryption-in-java-03.png" width="320" /></a></div><br />As an exercise, I went hunting for smallest possible RSA cryptosystem such that I could perform all of the encryption and decryption calculations in my head. This turns out to be quite silly, of course. But perhaps you could consider using this as a toy-box classroom example, or possibly like me, you may find it helps you remember the process by being able to exercise the whole thing concretely in your brain. <p></p><p>I'll use the same notation as in the <a href="https://en.wikipedia.org/wiki/RSA_(cryptosystem)">Wikipedia: RSA (cryptosystem)</a> article, which you may want to review (I'll avoid repeating the basic principles of the system here). So our system needs distinct primes <i>p</i> and <i>q</i>, the product <i>n = pq</i>, and decryption/encryption exponent keys <i>d</i> and <i>e</i>. I'll also use, as per the original RSA paper, <span class="nowrap"><i>φ</i>(<i>n</i>) = (<i>p</i> − 1)(<i>q</i> − 1) as the value with which our keys need to avoid common factors.<br /></span></p><p>Obviously, the fundamental calculation for both encryption and decryption is computations of <i>m<sup>e</sup> mod n</i>. Which looks pretty simple, unless you have a large exponent <i>e</i>, and you're trying to do this in your head. Secondarily, having big plaintext values <i>m</i> complicates the job. Being able to reduce modulo <i>n</i> seems nice, but even as an intermediary step I don't want to be raising double-digits <i>m</i>'s to double-digit <i>e</i>'s, say. </p><p>So the first job is to find a usable encryption exponent <i>e</i>. As noted on Wikipedia, "the smallest (and fastest) possible value for <i>e</i> is 3". Let's just reflect on why that is for a second. You can't use <i>e</i> = 0, aside from collapsing every block to a value of 1, because gcd(0, <span class="nowrap"><i>φ</i>(<i>n</i>)) = </span><span class="nowrap"><span class="nowrap"><i>φ</i>(<i>n</i>) ≥ 2, instead of the required 1 (since the minimal primes <i>p</i> and <i>q</i> are 2 and 3, the minimal </span></span><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><i>φ</i>(<i>n</i>) is (1)(2) = 2). You can't use <i>e</i> = 1, because that would send every block to itself, and not demonstrate any encryption at all. You can't use <i>e</i> = 2, because at least one of your pair of primes is odd, so either</span></span></span></span><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"> (<i>p</i> − 1) or (<i>q</i> − 1)</span> has a factor of 2, which can't be shared by <i>e</i>; and for the same reason <i>e</i> can't be any other even number.</span></span></span></span></p><p><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">So our first valid option for exponentiation is <i>e</i> = 3, and the next one is <i>e</i> = 5, etc. I don't know about you, but I really don't think I can do 5th or 7th powers of arbitrary numbers reliably in my head (even modulo <i>n</i>), so I'm aiming to just use <i>e</i> = 3 here. <br /></span></span></span></span></p><p><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">Now, what can we use for the primes <i>p</i> and <i>q</i>? Even using one as big as 7 causes problems; say if <i>p</i> = 7, then </span></span></span></span><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">(<i>p</i> − 1) = 6 = 2(3), at which point the factor of 3 blocks you from using our desired encryption key of <i>e</i> = 3. If I used a pair like 5 and 11, then <i>n</i> = 55, and I'm poised to be cubing numbers as big as 54, which is way outside my mental times-tables. So it looks like I'll be compelled just use 3 and 5 as our prime pair. <br /></span></span></span></span></span></span></span></span></span></p><p><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">In this case, if <i>p</i> = 3 and <i>q</i> = 5, so <i>n</i> = 15, then we don't even have sufficient space to encode all the letters of the alphabet (I said this was silly, right?). Also, we'll only be able to encode one of our limited characters at a time, which loses sight of the fact that RSA is a block cipher, but let's choose to forgive that. We'll enumerate the first part of the alphabet as A = 0, B = 1, C = 2, ... O = 14. Also note that </span></span></span></span><br /><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><i>φ</i>(<i>n</i>) = 2(4) = 8 (so the <i>only</i> factor we need to avoid in <i>e</i> is 2, and we can indeed use <i>e</i> = 3). <br /></span></span></span></span></span></span></span></span></p><p><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">What about our decryption key exponent <i>d</i>? Well, there's a goofy coincidence here: solving the required congruence <i>de</i> </span></span></span></span><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">≡ 1 (mod </span></span></span></span></span><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><i>φ</i>(<i>n</i>)</span></span></span></span></span></span></span></span>), that is, 3<i>d</i> </span></span></span></span></span><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">≡ 1 (mod 8</span></span></span></span></span><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">)</span></span></span></span></span></span></span></span>), we get <i>d</i> = 3 (because 3(3) = 9 = 8 + 1). Here we have <i>d</i> = <i>e</i>, so our encryption and decryption keys happen to be the exact same thing. On the one hand, this seems like a degenerate example for how the RSA system should work, but for our mental arithmetic, it's pretty darned nice that <i>both</i> keys are the smallest value possible. (As noted, I really didn't want to be raising values to a 7th power or something for decryption purposes!). So we can work with that.</span></span></span></span></span></span></span></span></span></span></p><p><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">Here's an example. Let's say we want to mentally encrypt the plaintext message "MIND". Using our number-encoding scheme, these four characters are represented by the numbers (12, 8, 13, 3). Computing the necessary cubings-modulo-15 to encrypt, we can feasibly find in our head:</span></span></span></span></span></span></span></span></span></span></p><ul style="text-align: left;"><li><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">12³ = (144)(12) = (135 + 9)(12) </span></span></span></span></span></span></span></span></span></span><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">≡ (9)(12) = 108 = 105 + 3 </span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">≡ 3 (mod 15)</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></li><li><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">8³ = (64)(8) = (60 + 4)(8) </span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">≡ (4)(8) = 32 = 30 + 2 </span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">≡ 2 (mod 15)</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></li><li><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">13³ = (169)(13) = (165 + 4)(13) </span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">≡ (4)(13) = 52 = 45 + 7 </span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">≡ 7 (mod 15)</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></li><li><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">3³ = (9)(3) = 27 = 15 + 12 </span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">≡ 12 (mod 15)</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></li></ul><p><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">So our encrypted message is (3, 2, 7, 12), or "DCHM" if we send it as alphabetic characters. Decrypting is (fortunately or unfortunately, depending on your perspective) exercising the exact same mental algorithm:</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></p><ul style="text-align: left;"><li><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">3³ </span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">≡ 12 (mod 15) [as above]</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></li><li><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">2³ </span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">≡ 8 (mod 15) [super easy!]<br /></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></li><li><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">7³ = (49)(7) = (45 + 4)(7) </span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">≡ (4)(7) = 28 = 15 + 13 </span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">≡ 13 (mod 15)</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></li><li><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">12³ </span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">≡ 3 (mod 15) [as above]</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></li></ul><p><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">And indeed we recover our original transmission of (12, 8, 13, 3), or "MIND". <br /></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></p><p><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">Okay, hopefully you can appreciate that example. Because I had to work <i>really, really hard</i> to find one that avoided being totally nonsensical. Here's why: In our toy cryptosystem, <i>most characters get sent to themselves under encryption</i>! Specifically (looking at the numerical representations): 0 → 0, 1 </span></span></span></span><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">→ 1, 2 </span></span></span></span></span></span></span></span><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">→ 8, 3 </span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">→ 12, 4 </span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">→ 4, 5 </span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">→ 5, 6 </span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">→ 6, 7 </span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">→ 13, 8 </span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">→ 2, 9 </span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">→ 9, 10 </span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">→ 10, 11 </span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">→ 11, 12 </span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">→ 3, 13 </span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">→ 7, 14 </span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">→ 14. <br /></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></p><p><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">In summary: Only the six plaintext values 2, 3, 7, 8, 12, and 13 get sent to other images; the other nine values all get sent to themselves. That is, the only letters that get transformed are C, D, H, I, M, N -- and you'll notice that I carefully constructed my example of "MIND" to use exclusively, and almost all of, these available characters. Note in particular that "I" is the only available vowel in that set (initially I considered encoding A = 1, B = 2, etc.; but in that case we get <i>no vowels</i> with images different from themselves.)<br /></span></span></span></span></p><p><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">In case you want to memorize these transformations directly, then it might help to observe the pattern, starting with 14 and wrapping around, that you get 3 values sent to the same as themselves, then 2 with different images, then 3 same, 2 different, 3 same, and 2 different. (Again noting a total of 6 characters that get sent to images different from themselves.) And the decryption function is, obviously, identical. </span></span></span></span></p><p><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">Is that of any interest? As I opened with, it's pretty darned silly. We can only handle letters up to O, we lose the block-cipher action, the encryption and decryption keys are identical, and most of our values get sent to themselves under the transformation. But I thought it was an interesting puzzle to see if there as any example that I could compute mentally without any mechanical or even pen-and-paper aid, and it turns out there is. <br /></span></span></span></span></p><p><span class="nowrap"><span class="nowrap"><span class="nowrap"><span class="nowrap">Leave a comment if you ever use that in a class or paper as a warm-up demonstration!<br /></span></span></span></span></p>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com2tag:blogger.com,1999:blog-7718462793516968883.post-25217252238001016582020-06-29T18:37:00.001-04:002021-01-02T16:27:48.411-05:00Community College Students and the PandemicOutstanding article by <span data-e2e-name="Elke Weesjes">Elke Weesjes, </span>one of my colleagues at CUNY/Kingsborough, on the immense difficulties faced by our community college students pursuing an education in a time of pandemic:<br /><blockquote class="tr_bq"><i>In fact, unstable living conditions are one of many chronic crises low-income community college students face. Others include higher exposure to violence and racism, poor health due to limited access to preventative healthcare, personal, or familial immigration issues, and a myriad of other socioeconomic disadvantages. They have many responsibilities, ranging from working full-time jobs to caring for children, siblings, or parents. Many are first generation students, members of minority groups, and non-native English speakers. While some of these challenges and crises are visible to professors, others remain outside their periphery.</i></blockquote><a href="https://www.businessinsider.com/community-college-students-hit-hardest-by-coronavirus-pandemic-2020-6">Business Insider. </a>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-36930614051451441522020-02-18T18:13:00.000-05:002020-02-18T18:13:46.189-05:00The Course Syllabus is Not a ContractHere's a really great article that came to my attention on the abusive notion (or at least, sloppy analogy) that "the course syllabus is a legal contract": Rumore, Martha M. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5289733/">"The Course Syllabus: Legal Contract or Operator’s Manual?."</a> <i>American journal of pharmaceutical education</i> 80.10 (2016). Some excerpts:<br /><blockquote class="tr_bq"><i>For several decades the literature has referred to syllabi as legal documents and/or contracts between students and professors. A review of the legal precedents reveals that syllabi are not considered contracts because the courts refuse thus far to recognize educational malpractice or breach of contract as a cause of action... </i><br /><i><br /></i><i>Based on the literature review and the search of university websites, identification or declaration of syllabi as contracts was evident, including widespread use of the term “learning contract.” For the past several decades, much of the literature has referred to syllabi as contracts and invoked the term “contract” or “learning contract” when referring to syllabi. Both faculty members and students appear to view a syllabus as a contract...</i><br /><i><br /></i><i>Although there have only been a handful of cases involving syllabi, the courts have consistently ruled that a syllabus is not a contract. In these cases, students brought lawsuits for breach of contract where the professor did not follow the syllabus or applied a different grade assessment... Students have been unsuccessful in asserting these claims, and courts remain reluctant to create a cause of action for either breach of contract or educational malpractice. Courts have generally ruled so because it is difficult to define the duty to educate; causation is difficult to determine; courts are reluctant to insert themselves into public policy issues such as the quality of education; and such interference by the courts would open a floodgate of litigation from academically unsuccessful students...</i><br /><br /><i>Some authors and university websites explicitly state a syllabus is a contract between the professor and students. Claiming a syllabus is a contract might produce a different legal outcome. While there may not be harm in thinking a syllabus is a contract, there may be legal risk in proclaiming it so...</i><br /><br /><i>Contracts are legally enforceable documents; syllabi are not. Syllabi have persisted in the culture of higher education and are foundational parts of the pharmacy curricula that encourage students to develop an acceptance of a lifelong responsibility for learning. For decades, syllabi have been referred to in the literature as contracts between students and professors. In the handful of cases involving syllabi, the claims have been mostly for breach of contract. To date, courts have not recognized claims of breach of contract for syllabi and do not consider syllabi as contracts. </i></blockquote><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-87103073086522150832019-04-29T05:00:00.000-04:002019-06-14T11:18:39.612-04:00Solitaire: Chance for Starting DuplicatesIn Klondike Solitaire, what's the chance for starting with two face-up cards of the same rank & color (what we might call "shadows", cards which don't help us very much)?<br /><br />There are 7 face-up cards in starting Solitaire, so we consider the ways to choose 7 cards randomly without any duplicates. Every time we select such a card, the number available for the next draw (avoiding "shadows") goes down by 2, while the cards in the total deck goes down by 1. Therefore the chance for no-shadows is:<br />$${52 \over 52} \times {50 \over 51} \times {48 \over 50} \times {46 \over 49} \times {44 \over 48} \times {42 \over 47} \times {40 \over 46} \approx 0.63$$<br />This in turn means that the complement event, i.e., getting any shadows at all (at least two cards with duplicate rank & color), has a probability of:<br />$$1 - 0.63 = 0.37 = 37 \%$$<br />That is: About one chance in three.<br /><br /><a href="http://www.madmath.com/2009/05/winning-solitaire.html">More about Solitaire. </a>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-71579026730263629232019-04-22T05:00:00.001-04:002021-01-02T16:28:34.326-05:00Facebook Learning DebacleYesterday, the New York times published an expose on how schools in certain (poor, rural) places are being used as guinea pigs and getting "Zucked" by the Facebook founder's "Summit Learning" initiative, yet another plan to replace teachers with computer time. The highlights are sad and predictable: <br /><ul><li> A program that promises "personalized learning" at each student's individual pace. Software free of cost, but school district must buy everyone a laptop.</li><li>Also, Facebook/Summit collects reams of information on the students involved, and expects to keep tracking students through college and beyond. </li><li>The program was built by a grand total of 5 Facebook engineers (no information on whether they have any training in education or pedagogy issues). </li><li>A spokesperson says it is based on, "building a curriculum from the open internet", that is, mostly links to outside web sites. Examples given include links to the Daily Mail tabloid and Christian conversion therapy sites.</li><li>The program "asks schools to commit to having students meet weekly in person with teachers for at least 10 minutes", but reports say not even this is happening. </li></ul>Students are reporting high levels of anxiety, eye strain, hand cramps, and even seizures. Said one parent who visited a classroom, “We’re allowing the computers to teach and the kids all looked like zombies”. Some places are seeing pushback like student protests, walkouts, and parent removing their children from schools.<br /><br />One reminder from yours truly: the promise of "personalized learning" is not new. It's been around at least since multimedia in the 90's, or the PLATO computer system in the 60's, or correspondence courses in the 40's, or the Gutenberg printing press, depending on how you count such things. None of them have come close to denting the need for real human teachers.<br /><div style="text-align: center;"><span style="font-size: large;"><br /></span></div><div style="text-align: center;"><span style="font-size: large;"><a href="https://www.nytimes.com/2019/04/21/technology/silicon-valley-kansas-schools.html">"Silicon Valley Came to Kansas Schools. That Started a Rebellion," at the New York Times</a></span></div>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com2tag:blogger.com,1999:blog-7718462793516968883.post-88911324226189144402019-02-11T05:00:00.000-05:002019-02-11T05:00:07.508-05:00Honeybee Addition and SubtractionResearchers in Australia and France claim to show honeybees learning to do symbolic addition and subtraction (really just incrementing and decrementing) based on 1-5 colored shapes at a time. Interesting ramifications if that's true. <br /><div class="section abstract" id="abstract-2"><blockquote class="tr_bq"><h2><i><span style="font-size: small;">Abstract</span></i></h2><div id="p-3"><i>Many animals understand numbers at a basic level for use in essential tasks such as foraging, shoaling, and resource management. However, complex arithmetic operations, such as addition and subtraction, using symbols and/or labeling have only been demonstrated in a limited number of nonhuman vertebrates. We show that honeybees, with a miniature brain, can learn to use blue and yellow as symbolic representations for addition or subtraction. In a free-flying environment, individual bees used this information to solve unfamiliar problems involving adding or subtracting one element from a group of elements. This display of numerosity requires bees to acquire long-term rules and use short-term working memory. Given that honeybees and humans are separated by over 400 million years of evolution, our findings suggest that advanced numerical cognition may be more accessible to nonhuman animals than previously suspected.</i></div></blockquote><div id="p-3"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-WIr7K5fn0PE/XFu5phy0_vI/AAAAAAAAE6I/TbKrBvvT_jowmbIEhlFryyl4NYhCusyEgCLcBGAs/s1600/F1.large.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1505" data-original-width="1400" height="320" src="https://1.bp.blogspot.com/-WIr7K5fn0PE/XFu5phy0_vI/AAAAAAAAE6I/TbKrBvvT_jowmbIEhlFryyl4NYhCusyEgCLcBGAs/s320/F1.large.jpg" width="297" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-Aa6Hwcpim-0/XFu5phimy7I/AAAAAAAAE6M/GE_gwQ_xVXUekWgUnfk4ldb5W6RGUaXSgCLcBGAs/s1600/F2.large.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1600" data-original-width="731" height="320" src="https://2.bp.blogspot.com/-Aa6Hwcpim-0/XFu5phimy7I/AAAAAAAAE6M/GE_gwQ_xVXUekWgUnfk4ldb5W6RGUaXSgCLcBGAs/s320/F2.large.jpg" width="146" /></a></div><div id="p-3"><br /></div></div><div class="section abstract" id="abstract-2"><div id="p-3"><a href="http://advances.sciencemag.org/content/5/2/eaav0961">At Science Advances. </a></div><div id="p-3"> </div></div>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-16092828852825970852019-01-21T05:00:00.000-05:002019-05-12T15:43:08.132-04:00Computer-Based Naval Training FailuresA fascinating article by a PhD candidate in military history: On the U.S. Navy's switch from face-to-face classrooms for training officers in surface warfare combat to independent computer-based training (CBT) aboard one's ship from 2003 - 2012 -- which turned out to be both a training disaster and also wildly inequitable, and was replaced with renewed face-to-face training after that time. <br /><br />For me, this seems to echo the recurrent drumbeat of hopes for distance/computerized classwork saving time and money, turning out to be generally complete failures, but administrators for various reasons eternally refusing to face up to the facts and evidence (out of an abundance of vain hopes). At least here we have a case study of an institution that has some higher motivation for responding to the failure, in that people's lives and billions of dollars in equipment are immediately at risk from obviously poor training. A key quote, I think, and more widely applicable:<br /><blockquote class="tr_bq"><i>Finally, changes undertaken with the principal goal of saving money or hurrying a process are fraught with danger. The overriding pressure to achieve financial or time savings threatens to overtake innovative ideas and turn them into quick-fix vehicles for the achievement of specific goals. </i></blockquote><br /><div style="text-align: center;"><span style="font-size: large;"><a href="http://cimsec.org/circles-surface-warfare-training">Circles in Surface Warfare Training</a></span></div><br /><br />(Coincidentally, my gaming blog today has a link to some excellent documentation on the history of wargaming used to train officers in the U.S. Navy: <a href="http://deltasdnd.blogspot.com/">see here</a>.)<br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-71795604328941453532018-12-17T05:00:00.000-05:002018-12-17T05:00:04.071-05:00NY Times Magazine Finds C.I.'s BafflingAnother journalist publicly declares their inability to understand the most basic math: in this case, <a href="https://en.wikipedia.org/wiki/Confidence_interval">confidence intervals</a>. This is Statistics 101, people.<br /><blockquote class="tr_bq"><i>When I asked that representative to check with a supervisor, she did, then returned to tell me that the company’s certainty was 99.7 percent. Those answers were confusing, because behind each of Johnson’s percentages was a range from which each ancestry point was drawn. For example, when we clicked on Johnson’s Benin/Togo segment, which had been assigned 10 percent of her ancestry, the site showed that the percentage of her DNA from those nations could be as low as zero and as high as 21. In fact, every one of her African links showed a range that started with zero, while her Europe South’s percent had a range of 9 to 33. Even the customer-service representative agreed that it was hard to fathom that the company could be so certain about the percentage when the range behind it ran to zero, which it did in four of the six geographic findings on Johnson’s report. </i></blockquote><br /><a href="https://www.nytimes.com/2018/11/19/magazine/dna-test-black-family.html">New York Times Magazine.</a><i><br /></i>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-63944620967601750332018-09-10T05:00:00.000-04:002018-09-10T05:00:00.560-04:00Why Do Nigerian Scammers Say They Are From Nigeria?From 2012: <br /><blockquote class="tr_bq"><i>Far-fetched tales of West African riches strike most as comical. Our analysis suggests that is an advantage to the attacker, not a disadvantage. Since his attack has a low density of victims the Nigerian scammer has an over-riding need to reduce false positives. By sending an email that repels all but the most gullible the scammer gets the most promising marks to self-select, and tilts the true to false positive ratio in his favor.</i> </blockquote><br /><a href="https://www.microsoft.com/en-us/research/publication/why-do-nigerian-scammers-say-they-are-from-nigeria/">Microsoft Research </a>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-18756560222020433932018-09-03T05:00:00.000-04:002018-09-03T05:00:07.462-04:00Dale Carnegie on Times TablesHere's an excerpt from Dale Carnegie's uber-famous book, <i>How to Win Friends and Influence People</i> (Part 4, Section 8):<br /><blockquote class="tr_bq"><i>Clarence M. Jones, one of the instructors of our course in Cincinnati, Ohio, told how encouragement and making faults seem easy to correct completely changed the life of his son.<br /><br />"In 1970 my son David, who was then fifteen years old, came to live with me in Cincinnati. He had led a rough life. In 1958 his head was cut open in a car accident, leaving a very bad scar on his forehead. In 1960 his mother and I were divorced and he moved to Dallas, Texas, with his mother. Until he was fifteen he had spent most of his school years in special classes for slow learners in the Dallas school system. Possibly because of the scar, school administrators had decided he was brain-injured and could not function at a normal level. He was two years behind his age group, so he was only in the seventh grade. Yet he did not know his multiplication tables, added on his fingers and could barely read.<br /><br />"There was one positive point. He loved to work on radio and TV sets. He wanted to become a TV technician. I encouraged this and pointed out that he needed math to qualify for the training. I decided to help him become proficient in this subject. We obtained four sets of flash cards: multiplication, division, addition and subtraction. As we went through the cards, we put the correct answers in a discard stack. When David missed one, I gave him the correct answer and then put the card in the repeat stack until there were no cards left. I made a big deal out of each card he got right, particularly if he had missed it previously. Each night we would go through the repeat stack until there were no cards left.<br /><br />Each night we timed the exercise with a stop watch. I promised him that when he could get all the cards correct in eight minutes with no incorrect answers, we would quit doing it every night. This seemed an impossible goal to David. The first night it took 52 minutes, the second night, 48, then 45, 44, 41 then under 40 minutes. We celebrated each reduction. I'd call in my wife, and we would both hug him and we'd all dance a jig. At the end of the month he was doing all the cards perfectly in less than eight minutes. When he made a small improvement he would ask to do it again. He had made the fantastic discovery that learning was easy and fun.<br /><br />"Naturally his grades in algebra took a jump. It is amazing how much easier algebra is when you can multiply. He astonished himself by bringing home a B in math. That had never happened before. Other changes came with almost unbelievable rapidity. His reading improved rapidly, and he began to use his natural talents in drawing. Later in the school year his science teacher assigned him to develop an exhibit. He chose to develop a highly complex series of models to demonstrate the effect of levers. It required skill not only in drawing and model making but in applied mathematics. The exhibit took first prize in his school's science fair and was entered in the city competition and won third prize for the entire city of Cincinnati.<br /><br />"That did it. Here was a kid who had flunked two grades, who had been told he was 'brain-damaged,' who had been called 'Frankenstein' by his classmates and told his brains must have leaked out of the cut on his head. Suddenly he discovered he could really learn and accomplish things. The result? From the last quarter of the eighth grade all the way through high school, he never failed to make the honor roll; in high school he was elected to the national honor society. Once he found learning was easy, his whole life changed."</i><br /><i><br /></i></blockquote>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com1tag:blogger.com,1999:blog-7718462793516968883.post-4196709451977256432018-08-27T05:00:00.000-04:002018-08-27T05:00:03.584-04:00Where Boys Outperform Girls in MathStanford study of some 260 million test scores shows that boys doing better than girls in math is related to their being in a rich, white, and suburban district. In poor and black districts, girls tend to do better than boys. This is distinct from English proficiency, which is constant across all these factors (girls always outperforming boys there):<br /><div class="g-item g-text "> <blockquote class="tr_bq"><div class="g-body"><i>The study included test scores from the 2008 to 2014 school years for 10,000 of the roughly 12,000 school districts in the United States. In no district do boys, on average, do as well or better than girls in English and language arts. In the average district, girls perform about three-quarters of a grade level ahead of boys. </i></div><i> </i></blockquote></div><blockquote><i> </i></blockquote><div class="g-item g-text "><blockquote><i> </i><div class="g-body"><i>But in math, there is nearly no gender gap, on average. Girls perform slightly better than boys in about a quarter of districts – particularly those that are predominantly African-American and low-income. Boys do slightly better in the rest – and much better in high-income and mostly white or Asian-American districts.</i></div></blockquote>Note that this is synchronous with gender roles of parents in the given districts. In rich, white, suburban districts, parents are more willing to <i>say</i> they have egalitarian values, but actually more like to <i>have</i> traditional gender roles in the family (e.g., full-time working father and stay-at-home mother).<br /><br /><a href="https://www.nytimes.com/interactive/2018/06/13/upshot/boys-girls-math-reading-tests.html">New York Times.</a><br /> </div>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-49346240790415166982018-08-20T05:00:00.000-04:002018-08-20T05:00:08.741-04:00How to Document a Series of Button ClicksNote to self: When documenting a series of interface button clicks for readers to follow, consider the following survey (in order of company sales/size in 2018):<br /><ul><li>Apple uses angle brackets, e.g., <a href="https://help.apple.com/xcode/mac/current/#/devc6700ab7d">Xcode > Preferences</a>. </li><li>Amazon uses angle brackets, e.g., <a href="https://developer.amazon.com/docs/app-submission/manage-account-and-permissions.html">Settings > My Account</a>. </li><li>Google uses angle brackets, e.g., <a href="https://developer.android.com/studio/intro/studio-config">Help > Edit Custom VM Options</a>.</li><li>Microsoft uses full sentences (no symbols), e.g., "on the Data tab, click Record Source to select the property". (<a href="https://support.office.com/en-us/article/set-the-record-source-for-a-report-9a28af3d-442e-4044-823f-5debb119111c">link</a>)</li><li>Facebook uses full sentences or bullet-pints (no symbols), e.g., "1. Visit your payment settings 2. Below Subscriptions, click Deactivate..." (<a href="https://www.facebook.com/help/248471068848455?helpref=about_content">link</a>)</li><li>Libre Office uses dashes, e.g., <a href="https://help.libreoffice.org/Common/Inserting_and_Editing_Tab_Stops">Format - Paragraph - Tabs</a>. </li></ul><br />In summary, I should (keep) using angle-brackets, as do the biggest 3 tech companies, instead of some other option.<br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-42202446842949253962018-07-30T06:00:00.000-04:002018-07-30T06:00:10.737-04:00Street Alignment in World Cities<div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-XV7iLjcN4jk/W0eBAJu39yI/AAAAAAAAEz4/6dh06CTvMgY2w1JUxMxWm0hMrCXPP5WNQCLcBGAs/s1600/world-city-streets-orientations-osmnx.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="" border="0" data-original-height="851" data-original-width="768" height="320" src="https://3.bp.blogspot.com/-XV7iLjcN4jk/W0eBAJu39yI/AAAAAAAAEz4/6dh06CTvMgY2w1JUxMxWm0hMrCXPP5WNQCLcBGAs/s320/world-city-streets-orientations-osmnx.png" title="Street Alignment in World Cities" width="288" /></a></div><br /><a href="http://geoffboeing.com/2018/07/city-street-orientations-world/">Geoff Boeing, UC Berkeley</a>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-48730494161239196312018-07-23T05:00:00.000-04:002018-07-23T05:00:00.630-04:0017th Century Science CiphersDid you know? It was common in the 17th Century for scientists to publish cipher-versions of early findings, so as to establish priority in the discovery. (Something of an early, public-encrypted arXiv, as it were.) Huygens wrote in 1668:<br /><blockquote class="tr_bq"><i>a way of securing his discoveries and inventions for the future by way of cypher or anagram, to be lodged in the Register-book of the society, till he should think it convenient to explain them in a common language; making withal a beginning of this way of communicating new discoveries by sending the following cypher; which was ordered to be entered into the Register-book.</i></blockquote><br /><a href="https://skeptics.stackexchange.com/questions/41603/did-robert-hooke-publish-his-spring-force-law-as-a-anagram">Stack Exchange: Skeptics.</a><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-62175938383767751672018-07-16T05:00:00.000-04:002018-07-16T05:00:02.606-04:00Oxford: Likely Alone in UniverseA new assessment from Oxford University finds it quite likely that humans are alone in the universe:<br /><blockquote class="tr_bq"><i>Our main result is to show that proper treatment of scientific uncertainties dissolves the Fermi paradox by showing that it is not at all unlikely ex ante for us to be alone in the Milky Way, or in the observable universe.</i><br /><br /><i>Our second result is to show that, taking account of observational bounds on the prevalence of other civilizations, our updated probabilities suggest that there is a substantial probability that we are alone.</i></blockquote><a href="http://fortune.com/2018/06/26/humans-alone-known-universe-a-new-oxford-study-suggests/">Fortune.</a><span id="goog_1083860875"></span><a href="https://www.blogger.com/"></a><span id="goog_1083860876"></span>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-64982841191379489992018-07-09T05:00:00.000-04:002018-07-09T05:00:14.266-04:00Gates Foundation Considered HarmfulIndependent assessment of the Gates Foundation's intervention in the last decade, to the tune of $575 million, shows it not just to be wasteful, but even actively damaging:<br /><blockquote class="tr_bq"><i>Laudable as the intention may have been, it didn’t work. As the independent assessment, produced by the Rand Corporation, put it: “The initiative did not achieve its goals for student achievement or graduation,” particularly for low-income minority students. The report, however, stops short of drawing what I see as the more important conclusion: The approach that the Gates program epitomizes has actually done damage. It has unfairly ruined careers, driving teachers out of the profession amid a nationwide shortage. And its flawed use of metrics has undermined science.</i></blockquote><br /><a href="https://www.bloomberg.com/view/articles/2018-06-27/here-s-how-not-to-improve-public-schools">Cathy O'Neil at Bloomberg.</a>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-19553406619005145512018-07-02T05:00:00.000-04:002018-07-01T13:01:51.846-04:00Growth Mindset Found LackingLarge-scale meta-study at Michigan State finds that the "growth mindset" intervention that's been all the rage for the last few years doesn't actually accomplish much:<br /><blockquote class="tr_bq"><i>"This research is important because millions of dollars have been spent on growth mindset interventions in schools," said Alex Burgoyne, a Ph.D. student studying cognition and cognitive neuroscience. "Our results show that the academic benefits of these interventions have been largely overstated. For example, there was little to no effect of mindset interventions on academic achievement for typical students, or for other groups who some have claimed benefit substantially from these interventions, including students facing situational challenges, such as transitioning to a new school."</i></blockquote><a href="https://medicalxpress.com/news/2018-05-popular-growth-mindset-interventions-effective.html">Medical Xpress.</a><br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com2tag:blogger.com,1999:blog-7718462793516968883.post-36022458905739379232018-06-25T05:00:00.000-04:002018-06-25T18:48:02.511-04:00Illustration of Platonic SolidsA very nice illustration of why there are only 5 regular polyhedra (Platonic solids), by Martin Holtham on <a href="https://twitter.com/GHSMaths/status/1007992663927386112">Twitter</a>: <br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-NCVLfO-J2qc/WzFxCQ0Kf8I/AAAAAAAAEzg/sRpexr03XmkCT13qNaROPKhTaarQEfUrACLcBGAs/s1600/PlatonicSolidsExplained.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="" border="0" data-original-height="1201" data-original-width="1600" height="240" src="https://4.bp.blogspot.com/-NCVLfO-J2qc/WzFxCQ0Kf8I/AAAAAAAAEzg/sRpexr03XmkCT13qNaROPKhTaarQEfUrACLcBGAs/s320/PlatonicSolidsExplained.jpg" title="Platonic Solids Explained" width="320" /></a></div>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-69258168814123728392018-03-19T05:00:00.000-04:002018-03-19T05:00:45.231-04:00Stop Grading on a Curve in New York TimesBy Adam Grant in <i>The New York Times</i>, two years ago:<br /><blockquote class="tr_bq"><i>The more important argument against grade curves is that they create an atmosphere that’s toxic by pitting students against one another. At best, it creates a hypercompetitive culture, and at worst, it sends students the message that the world is a zero-sum game: Your success means my failure.</i></blockquote><a href="https://www.nytimes.com/2016/09/11/opinion/sunday/why-we-should-stop-grading-students-on-a-curve.html">The New York Times </a><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-82336635941080163752018-02-26T05:00:00.000-05:002018-02-26T05:00:13.629-05:00Animal Numerosity EverywhereThe NY Times reports on a themed issue of the Philosophical Transactions of the Royal Society B, in that many animals have impressive built-in numerosity ability: being able to distinguish between quantities between one and five, say, or proportionally in how 25 is different than 30. In some cases, better than humans: monkeys can visually identify certain number sequences in a split-second. <br /><br />We are reminded of when <a href="http://www.madmath.com/2016/05/noam-chomsky-enumeration-leads-to.html">Noam Chomsky conjectured that enumeration might be the first mental stepping stone on the way to language</a> -- or how apt it is that we call the <i>"natural numbers"</i> by that name. <br /><blockquote class="tr_bq"><div class="story-body-text story-content" data-para-count="218" data-total-count="2298" id="story-continues-3"><i>Scientists have found that animals across the evolutionary spectrum have a keen sense of quantity, able to distinguish not just bigger from smaller or more from less, but two from four, four from ten, forty from sixty.</i></div><div class="story-body-text story-content" data-para-count="218" data-total-count="2298" id="story-continues-3"><br /></div><div class="story-body-text story-content" data-para-count="358" data-total-count="2656"><i>Orb-weaving spiders, for example, keep a tally of how many silk-wrapped prey items are stashed in the “larder” segment of their web. When scientists experimentally remove the cache, the spiders will spend time searching for the stolen goods in proportion to how many separate items had been taken, rather than how big the total prey mass might have been.</i></div><div class="story-body-text story-content" data-para-count="358" data-total-count="2656"><br /></div><div class="story-body-text story-content" data-para-count="221" data-total-count="2877"><i>Small fish benefit from living in schools, and the more numerous the group, the statistically better a fish’s odds of escaping predation. As a result, many shoaling fish are excellent appraisers of relative head counts.</i></div><div class="story-body-text story-content" data-para-count="221" data-total-count="2877"><br /></div><div class="story-body-text story-content" data-para-count="231" data-total-count="3108"><i>Guppies, for example, have a so-called contrast ratio of .8, which means they can distinguish at a glance between four guppies and five, or eight guppies and ten, and if given the chance will swim toward the slightly fishier crowd.</i></div><div class="story-body-text story-content" data-para-count="231" data-total-count="3108"><br /></div><div class="story-body-text story-content" data-para-count="241" data-total-count="3349"><i>Three-spined sticklebacks are more discriminating still: with a contrast ratio of .86, they’re able to tell six fellow fish from seven, or 18 from 21 — a comparative power that many birds, mammals and even humans might find hard to beat.</i></div></blockquote><br /><a href="https://www.nytimes.com/2018/02/05/science/animals-count-numbers.html">New York Times.</a><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-81439288177274603682018-02-19T05:00:00.000-05:002018-02-19T05:00:16.194-05:00Is Accepting Math Deficiency Destroying Journalism?From 2013, an article by a professional journalist, who thankfully skipped any math in college -- then when he went for an MBA to understand the business he was in, discovered that calculus was a prerequisite for entry. So he committed to the road from lowest-level K-6 remediation up to calculus. <br /><br />He points out that being "bad at math" is so accepted in the journalism industry, that it's actually a point of pride. We recall the article in the New Yorker three years ago this month about Yitang Zhang's marvelous progress on the twin primes conjecture, in which the journalist actually <a href="http://www.madmath.com/2015/02/yitang-zhang-article-in-new-yorker.html">framed the entire story around how he knew so little math, he had to lie and cheat his way through high school algebra</a>. <br /><br />In our current case, the journalist (and now professor) suggests that this anti-math bias in journalism may actually be a contributing factor to the collapse of the industry -- in that both (a) the present cohort is unable to make sense of quantitative, scientific, or technological stories, which grow ever more essential to the world around us; and (b) they are unable to understand the financial and business case of their industry.<br /><br /><blockquote class="tr_bq"><i>Well, Professors Kimball and Smith, welcome to journalism, where “bad at math” isn’t just a destructive idea — it’s a badge of honor. It’s your admission to the club. It’s woven into the very fabric of identity as a journalist.</i><br /><br /><i></i><i>And it’s a destructive lie. One I would say most journalists believe. It’s a lie that may well be a lurking variable in the death of journalism’s institutions. </i><br /><br /><i></i><i>Name me a hot growth area in journalism and I’ll show you an area in desperate need of people who can do a bit of math. Data. Programming. Visualization. It’s telling that most of the effort now is around recruiting people from outside journalism to do these things. </i><br /><br /><i></i><i>But it doesn’t end there. Name me a place where journalism needs help, and I’ll show you more places where math is a daily need: analytics, product development, market analysis. All “business side” jobs, right? Not anymore. </i><br /><br /><i></i><i>Truth is, “bad at math” was never a good thing in journalism, even when things like data and analytics weren’t a part of the job. Covering a city budget? It’s shameful how many newsroom creatures can’t calculate percent change. Covering sports? It’s embarrassing how many sports writers dismiss the gigantic leaps forward in data analysis in all sports as “nerd stuff.”</i><br /><br /><i></i><em>In short, we’ve created a culture where ignorance of a subject is not only accepted, it’s glorified. <u>Ha ha! Journalists are bad at math! Fire is hot and water is wet too!</u></em></blockquote><br /><br /><a href="http://www.niemanlab.org/2013/11/matt-waite-how-i-faced-my-fears-and-learned-to-be-good-at-math/">Nieman Lab. </a><br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0