tag:blogger.com,1999:blog-77184627935169688832024-10-14T18:10:11.353-04:00MadMath"Beauty is the Enemy of Expression"Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.comBlogger329125tag:blogger.com,1999:blog-7718462793516968883.post-33404315665826584742024-10-09T22:14:00.000-04:002024-10-09T22:14:08.028-04:00Retention and Interventions<p>I saw a new report on retention for freshmen students at CUNY recently. This chart of CUNY Guttman (formerly the New Community College) retention rates, 2014-2022, wonderfully highlights a perpetual cycle in attempts to improve education:</p><p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh5hnQ4Zl0gCMTiq65gDPOzHxqvub_bBjpPH9WJGZlDTVvYOZB0hxfE4f0Sa56zqCRURiM5JHfzDS_54RYmgC7Zc8_jJ13RKpc_jtcdQgFSF7ljaACHQ-WED1cZyPJOYaeqJ0CGVBvkJuwy_siHAkdz0EFKeAgnsyWg8JLG60SHwo4OreUeQquu5D4E5ut-/s664/GuttmanCollegeRetention.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="CUNY Guttman college retention rates for freshmen" border="0" data-original-height="446" data-original-width="664" height="215" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh5hnQ4Zl0gCMTiq65gDPOzHxqvub_bBjpPH9WJGZlDTVvYOZB0hxfE4f0Sa56zqCRURiM5JHfzDS_54RYmgC7Zc8_jJ13RKpc_jtcdQgFSF7ljaACHQ-WED1cZyPJOYaeqJ0CGVBvkJuwy_siHAkdz0EFKeAgnsyWg8JLG60SHwo4OreUeQquu5D4E5ut-/w320-h215/GuttmanCollegeRetention.png" width="320" /></a></div><br />When it starts, a new method is given to a small experimental group of students. Results are great (here, 73% freshman retention)! But this is in direct contact with instructors who have "drunk the kool-aid", put in heroic extra effort, and are also careerwise incited to juice the scores. <br /><br />Since the initial results are so great, we expand the program to more students, and more instructors. And at this point the results collapse (here, down to 45% freshman retention). In this case, Guttman went from first-place to last-place in retention among CUNY community colleges in just a few years.<br /><br />This happens over and over again in all educational research. It's extremely predictable.<p></p><p></p><p></p><p><b>Side Note #1:</b> My colleague Emily Schnee <a href="https://eric.ed.gov/?id=EJ1017286">sounded the alarm</a> about practices at Guttman a decade ago, after spending a year there during its first organization.</p><p><b>Side Note #2:</b> The Guttman Foundation is currently <a href="https://gothamist.com/news/nyc-charity-that-gave-25m-to-cuny-sues-to-get-its-money-back">suing CUNY</a> for the majority of its $25 million grant money back.<br /></p>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-72284565421811763982024-09-02T08:00:00.031-04:002024-09-02T08:00:00.371-04:00A Strange Commitment<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh69xZ2Yfy4Uo-UxvsFZTUoVUGEd7hgGyXor_iAYN4L-qjs_F-HMUHOQp-Elmp3tgI-u0HFGHirvGAW2D3gyDsmLNpL2nmJL_u3SaFAf2x95xFX37w4zPDVIxJD5sE6SNuxGWVPwMcqVI6c9iWEAzLhwD4CDtPoJqJgmXBTyDtMCYaISGgV2JFBpzFNov7t/s720/1_aONpY8J3ymDxfuoEI47XGw.webp" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img alt="Failed attempt to solve an elementary math problem" border="0" data-original-height="677" data-original-width="720" height="188" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh69xZ2Yfy4Uo-UxvsFZTUoVUGEd7hgGyXor_iAYN4L-qjs_F-HMUHOQp-Elmp3tgI-u0HFGHirvGAW2D3gyDsmLNpL2nmJL_u3SaFAf2x95xFX37w4zPDVIxJD5sE6SNuxGWVPwMcqVI6c9iWEAzLhwD4CDtPoJqJgmXBTyDtMCYaISGgV2JFBpzFNov7t/w200-h188/1_aONpY8J3ymDxfuoEI47XGw.webp" width="200" /></a></div><p>I was recently reminded of this anecdote from Junaid Mubeen, originally in Bright Magazine. Juniad recalls a time where he assisted his 12-year old nephew on a math homework problem. As it turns out, the problem had no solution, Junaid could elegantly prove it, and the nephew excitedly accepted their reasoning. Then:</p><p></p><blockquote><p>What happened next was revealing. My sister told me that my nephew devoted the rest of the evening to searching for a solution. That’s a strange commitment to make when you’ve just been convinced that no such solution exists.<br /><br />My nephew was in a state of cognitive dissonance. He understood that this problem has no solution. And yet, this was his maths homework. From all his years of schooling, every maths question always had an answer. Why should this time be any different?<br /><br />My nephew’s deference towards school maths, with closed questions and prescribed answers, betrayed his mathematical reasoning.</p></blockquote><p>More at <a href="https://brightthemag.com/my-nephew-brought-home-this-menacing-maths-problem-e8bbba30e5cb">Medium</a>.</p>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-43176998046808427702024-08-26T09:00:00.007-04:002024-08-26T11:02:10.448-04:00Phonics Makes a Comeback<p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhzeJu0GkuDICRTqDpdJPUXsmX9t3m1MRBDQ7hDFyLcUimML4I2bmpl00RNhbF8UlssoHrMGWwCJ93nTMBFMDnRa4U6_K-DWq-D0O_trjxOY8om7PR46OK0F9WIij44-SEJfYyMJI38JmkVxBysu68BvhqFrA338j7cbizEjJcYMblqLdGpI5Hzm07BDsCE/s925/ae_spring_2023_moats_spot1.jpg" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img alt="Lee Anna Vasquez, a reading interventionist, uses a sound wall to teach students the articulatory features of phonemes" border="0" data-original-height="463" data-original-width="925" height="160" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhzeJu0GkuDICRTqDpdJPUXsmX9t3m1MRBDQ7hDFyLcUimML4I2bmpl00RNhbF8UlssoHrMGWwCJ93nTMBFMDnRa4U6_K-DWq-D0O_trjxOY8om7PR46OK0F9WIij44-SEJfYyMJI38JmkVxBysu68BvhqFrA338j7cbizEjJcYMblqLdGpI5Hzm07BDsCE/w320-h160/ae_spring_2023_moats_spot1.jpg" width="320" /></a></div><p></p><p></p><p>As a long-time instructor of remedial math at a community college (among other things), I've always felt strongly that our discipline is closely tied to basic natural-language literacy, and to the teaching of reading and writing. </p><p>Teaching basic symbolic math is fundamentally the ground-floor of a particular writing language for the sciences. If a student entering an algebra class can't read a sentence (either to follow a direction, answer a word problem, or see the parallel with algebraic equations), attend to fine structure, or understand assertions about symbolic parts-of-speech, then they will flounder. I've written about this here multiple times (see: <a href="http://www.madmath.com/2013/06/the-war-on-structure.html">The War on Structure</a> and <a href="http://www.madmath.com/2010/01/phonics-and-bases.html">Phonics and Bases</a>). Note the subtitle of the blog with its essential spotlight on clear <i>expression</i>.</p><p>Two articles in <a href="https://www.aft.org/ae">American Educator</a> in the last year so give some hope that the science of structurally-focused reading instruction, with phonics instruction as a key part, may (with great struggle) finally be making a comeback. Both are by reading researchers who have been at the wheel for many decades.<br /></p><p>The first is by Louisa C. Moats in Spring 2023, <a href="https://www.aft.org/ae/spring2023/moats">Creating Confident Readers</a>. Moats has developed and taught a program for graduate-program service teachers called LETRS (Language Essentials for Teachers of Reading and Spelling) in some form since the early 1990's. It leans into service teachers becoming knowledgeable about how sounds are articulated in the mouth, how they vary across languages, how they map to written graphemes in English, and how students can decode new texts on their own when necessary. </p><p>I think that Moats put the most interesting part at the end of her article (perhaps out of a sense of gently downplaying the section that might be considered the most inflammatory), which I'll highlight here:</p><p></p><blockquote>Teachers often experience complex emotional reactions as they
learn more about the science of reading and the structure of language.
Some teachers express grief and regret over their past use of
ineffective (but widespread) practices and anger that their prior
opportunities to learn about teaching reading were inadequate or even
misinformed. A common reaction of participating teachers to their
experience in LETRS is, “Why didn’t anybody teach me these things
before?” The value of the information is readily apparent when students
begin to make progress. Student growth quickly validates teachers’
efforts to teach language, reading, and writing explicitly.</blockquote><p></p><p>I immediately recognized this comment about “Why didn’t anybody teach me these things
before?” — I've gotten this exact response from students in remedial algebra courses, at key moments where I was identifying and trying fix broken understandings that their K-12 teachers had hopelessly mangled. Moats continues: <br />
</p><p></p><blockquote><p>In translating concepts and guidance from research, we encourage
teachers to confront and abandon ideas, practices, and programs that
many have used or been taught—often under district or state standards
and requirements—that do not align with current understandings grounded
in evidence. For example, many districts are still wedded to programs
and approaches based on “cueing systems,” a tenet of guided reading that
does not recognize the central role of phonology or phonic decoding in
learning to read and spell. An underlying assumption that reading is
primarily a visual imprinting activity drives other misconceived but
all-too-common practices, such as posting “sight” words on an alphabetic
word wall regardless of the beginning sounds in the words (e.g.,
posting <i>out</i>, <i>once</i>, <i>only</i>, and <i>often</i> under <i>o</i>).
Many district and state standards require kindergarten and first-grade
readers to memorize dozens of words on flash cards or spell lists of
words by rote visual memory, even though in reading science, all words
are eventually learned “by sight” through a process of speech-to-print
mapping, beginning with phoneme-level processing. Turning
away from common but unsupported practices poses dilemmas for teachers
and schools because the misconceived ideas have been established in
reading education for so long. Many published programs have yet to catch
up to the science...</p></blockquote><p></p><p>We can see here implications of the primary mistake that proponents of the catastrophic "whole word" approach made — they thought about how proficient readers function (mostly identifying familiar, known words on sight), and assumed students could jump directly to that level without passing through the scaffolding phases that naturally occur beforehand (connecting spoken sounds to written symbols). Moats says this more directly elsewhere:</p><p></p><blockquote><p>The ability to recognize printed words out of context, quickly and accurately, is gained <i>not</i>
by a visual imprinting process, but by building a mental map connecting
speech with print. By learning incrementally how graphemes (letters and
letter combinations) represent speech, novice readers and spellers
gradually build a mental storehouse of known words that can be instantly
recognized and recalled. Every phase of this process
depends on the ability to recognize and mentally manipulate the phonemes
or speech sounds that make up words (phoneme awareness). From
pre-alphabetic, to partial alphabetic, to full alphabetic, and then to
consolidated word recognition and recall, children must gradually
differentiate the sounds in spoken words and map them to letters and
letter sequences.</p><p></p></blockquote><p>And another interesting observation:</p><p></p><blockquote><p>When teachers have not had ample opportunities to learn how to
explain words’ spellings, they are much more inclined to believe—and
teach—that the English writing system is chaotic and nonsensical.<a id="B2" name="B2"></a>
Believing that is the case too often leads educators to rely on “sight”
word methods such as “using your eyes like a camera,” drilling with
flash cards, telling students to look at pictures and use context to
guess an unknown word, or reciting letter sequences to memorize words.</p>
<p>One central goal of LETRS is to put meaning over rote memorization.
That’s why part of the phonics lesson plan is working with the meanings
of words that students are learning to decode or spell. Our theoretical
frameworks emphasize the importance of connecting sound, meaning, and
spelling while the mental code-mapping process is under construction... </p><p></p><p></p></blockquote><p>On the same theme, another article appeared by Linna C. Ehri in Fall 2023, <a href="https://www.aft.org/ae/fall2023/ehri">Phases of Development in Learning to Read and Spell Words</a>. In large part, she reiterates the same natural structure noted by Moats above: (1) pre-alphabetic, (2) partial alphabetic, (3) full alphabetic, and (4) consolidated alphabetic phases. The thing I want highlight here is the rather unsettling introduction of the word "phonics" in a tone that suggests trained K-12 reading instructors have likely never heard of it (!):</p><p></p><blockquote><p>To move into the full alphabetic phase, children need to acquire the
major letter-sound (grapheme-phoneme) relations of the writing system.
They need to acquire decoding skill to sound out letters and blend the
sounds to form words. The type of reading instruction that helps
children master these skills is called <i>phonics</i>. In systematic
phonics instruction, teachers follow a “scope and sequence” chart to
teach the major letter-sound relations; they also teach segmenting
sounds, decoding words, and spelling skills. Phonics instruction can
reduce the time that students spend in the partial alphabetic phase and
move them quickly into the full phase, typically by the end of
kindergarten or in first grade. The skills children acquire help them
store words in their memory for reading by sight and spelling words
correctly.</p><p></p></blockquote><p>Given that, phonics is the cornerstone to the conclusion of the piece:</p><p></p><blockquote><p>In school, children benefit most from systematic phonics instruction to
acquire these skills. One great way to support your child’s growth at
home is to create lots of opportunities for them to practice reading—and
to talk about what you’ve read together to boost their comprehension.
And if your child is not progressing through the four phases, be sure to
go to their school to ask for additional supports.</p><p></p></blockquote><p>I thought that both of these articles were well worth the time spent reading them. If only solid, structural reading skills were taught, we could see widespread benefits not just in language arts, but in other important subjects like math, science, logic, computing, and other technical fields.<br /></p>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-49946904161142258472024-08-05T07:00:00.001-04:002024-08-05T07:00:00.119-04:00Duality of Floor and Modulus<p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgh_I7gvCBUSaGDDaWRgmvQ14cS8dhsuhbZsXtssRuudS6Njw8wFCIcPpFjGWbYGFhzbTbMGVkifh8yRz-JohKv0dYDe_lrQBqSbZitYoYqF3FJwwKc_P25wcmuYXmvO4B_n8Hx5baoSR6YcHc6heJfQbtT-HQIwjfa5iGxZXx6-7VmAFIHL98bGk8-uhjX/s498/FloorAndModulusSymbols.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img alt="Floor and modulus symbols" border="0" data-original-height="410" data-original-width="498" height="164" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgh_I7gvCBUSaGDDaWRgmvQ14cS8dhsuhbZsXtssRuudS6Njw8wFCIcPpFjGWbYGFhzbTbMGVkifh8yRz-JohKv0dYDe_lrQBqSbZitYoYqF3FJwwKc_P25wcmuYXmvO4B_n8Hx5baoSR6YcHc6heJfQbtT-HQIwjfa5iGxZXx6-7VmAFIHL98bGk8-uhjX/w200-h164/FloorAndModulusSymbols.png" width="200" /></a></div><p>Have you ever considered the duality of the <i>floor</i> and <i>modulus</i> operators, in that either one can be expressed in terms of the other (along with multiply, divide, and a subtraction)?</p><p>This became practically important for me when my institution switched from the Blackboard learning management system to Brightspace. Both of these have testing systems with a type of question that presents test-takers with questions incorporating randomly-determined base values, and then automatically check their answers via a mathematical formula set up by the instructor. On Blackboard this type of question is called a "Calculated Formula", and among the supported functions, it includes <i>floor</i> but not <i>modulus</i>. On Brightspace it's called an "Arithmetic" question, and its list of functions has the inverse coverage: <i>modulus</i> but not <i>floor</i>.</p><p>Now I use this facility to make tests for the introductory C++ programming courses that I teach. Modulus is one of the likely new basic arithmetic operators that we want to test for. Floor is arguably even more essential, in that it's what I need to represent integer division truncation, something really fundamental to how integer math works on a computing system.<br /></p><p>So, on <b>Blackboard</b>, if I want to test students' knowledge of the behavior of the expression \(a \% b\) (remember that in the students' view, the \(a\) and \(b\) will be filled in with literal numbers), then in the answer formula I have to use the expression: \(a - b * \lfloor a/b \rfloor\).</p><p>In other words: Use the floor function to find the largest whole-number of times that \(b\) goes into \(a\), and subtract that maximal product from \(a\), leaving the modulus as a remainder.</p><p>Meanwhile, on <b>Brightspace</b>, if I want to assess awareness of the truncation that happens automatically with an integer division \(a / b\), then every time that occurs in the answer, I need to make use of the expression: \(a/b - a/b\%1\).</p><p>That is: Use modulus by 1 to find the decimal remainder in the ratio \(a/b\), and subtract that from the full \(a/b\), leaving behind only the integer part.<br /></p><p>Having translated all of my test questions from one to the other in the last week, I'd say that for CS purposes the floor function is more essential, as it's baked in naturally to any integer math expressions, and so it was overall more natural for the work to be done in Blackboard. Use of modulus is more of a special-case check in my assessments, so that arcane formula only showed up a few times. On the other hand, switching to Brightspace, there were about twice as many instances where I had to conjure a floor truncation via the modulus operator that exists there.</p><p>Obviously, it would be best if either system supported both of these basic operators, but as we've discovered, you are operationally complete with either one of the pair.<br /></p>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-66984551790132959212024-07-08T08:00:00.012-04:002024-07-10T11:56:37.444-04:00Yes, Euclid's Proof of Infinite Primes Uses Contradiction<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgY2CLsaATB9eCZyAhDAdkqMy3hCMVAciDCnWs4zs_bAaP21QB0DQ3oQ_8dxU1GOte_l3l1hvHPuvYdsbmK9MBpldKwVP4DcozwZlsq2m2cyhyphenhyphendwR69ndGFMzmy2QZHLPZbUc4D30TbNfbEy8Qp37Yddw8Qe1gFX1ot0wPn7HkE7Ht1Tqr-T_tlTH1UDyUi/s320/320px-Infinity_-_The_Noun_Project.svg.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="150" data-original-width="320" height="94" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgY2CLsaATB9eCZyAhDAdkqMy3hCMVAciDCnWs4zs_bAaP21QB0DQ3oQ_8dxU1GOte_l3l1hvHPuvYdsbmK9MBpldKwVP4DcozwZlsq2m2cyhyphenhyphendwR69ndGFMzmy2QZHLPZbUc4D30TbNfbEy8Qp37Yddw8Qe1gFX1ot0wPn7HkE7Ht1Tqr-T_tlTH1UDyUi/w200-h94/320px-Infinity_-_The_Noun_Project.svg.png" width="200" /></a></div><p>It's common nowadays in conversations about the method of proof-by-contradiction for someone to pop in and say, "People think Euclid's proof of there being infinite prime numbers uses proof-by-contradiction, but it doesn't, it's a direct proof". For example, the current Wikipedia article on <a href="https://en.wikipedia.org/wiki/Euclid%27s_theorem">Euclid's theorem</a> says this:</p><p></p><blockquote>Euclid is often erroneously reported to have proved this result by contradiction beginning with the assumption that the finite set initially considered contains all prime numbers, though it is actually a proof by cases, a direct proof method. The philosopher Torkel Franzén, in a book on logic, states, "Euclid's proof that there are infinitely many primes is not an indirect proof..."</blockquote><p></p><p>Okay, admittedly Euclid's theorem is not in its entirety structured as a proof by contradiction. Yes, there's a proof by cases, in which a number of the form \(lcm(ABC) + 1\) is assessed as being either prime or not prime. <i>But the core of that second case is clearly a proof by contradiction!</i><br /></p><p>If we look at Euclid's original text for the second case, we see the following. Given that \(lcm(ABC) + 1\) is not prime, Euclid takes \(G\) to be some prime number that divides it. He then reasons like this (looking at the Fitzpatrick translation of Heiberg's presentation of the Greek):</p><p></p><p></p><blockquote><p>I say that \(G\) is not the same as any of \(A, B, C\). For, if possible, let it be... [some logic here]... The very thing is absurd. Thus, \(G\) is not the same as one of \(A, B, C\).</p><p></p></blockquote><p>This form is patently a proof by contradiction, and use of the phrase with "absurd" (in the original Greek, "ὄπερ ἄτοπον") highlights that fact. <br /></p><p>While the overall superstructure of Euclid's theorem is not a proof by contradiction... Yes Virginia, Euclid's theorem uses a proof by contradiction, and it's an essential part of his proof that there are infinite primes.<br /></p>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-77593970908159848012024-02-12T08:00:00.001-05:002024-02-12T08:00:00.203-05:00A Game for Bored Math Teachers<p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhhiVUcJqqh03UhycsEoX1Qi_fQ70pCCnAlrEzxmyocR27BssN6-oXh_VqTQm0IcZm3NNkx14EIt8o2J6oRro7EzTm2D7Rquucv_yqJ6ySvggwiQ5IE2TsDYyXlqvK43jK1GFGWymfQ-4utP_LIno0qiStOXK836bo9dvGfkOS_Rd8SAwZcJ5_4INbb2YhS/s1024/DALL%C2%B7E%202024-02-08%2015.49.03%20-%20mental%20puzzle%20in%20brain.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img alt="Head with jigsaw puzzle inside" border="0" data-original-height="1006" data-original-width="1024" height="196" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhhiVUcJqqh03UhycsEoX1Qi_fQ70pCCnAlrEzxmyocR27BssN6-oXh_VqTQm0IcZm3NNkx14EIt8o2J6oRro7EzTm2D7Rquucv_yqJ6ySvggwiQ5IE2TsDYyXlqvK43jK1GFGWymfQ-4utP_LIno0qiStOXK836bo9dvGfkOS_Rd8SAwZcJ5_4INbb2YhS/w200-h196/DALL%C2%B7E%202024-02-08%2015.49.03%20-%20mental%20puzzle%20in%20brain.png" width="200" /></a></div>Let's say you're an instructor in a low-level math course; maybe something like college algebra, a liberal-arts math course, or something similar. It's possible that you consider this to be beneath you and you're bored in class. Here's a little game you can play with yourself that will spice things up a bit:<p></p><p>When you ask a question to the class, and someone answers incorrectly, see if you can conversationally edit the question in a way that the student <i>would have been right</i>. That is, respond by starting with, "Well, that <i>would</i> be right if the problem said ____", and fill in the end of that sentence in appropriate way. </p><p>I'm very fond of this technique. It actually accomplishes several things:</p><ul style="text-align: left;"><li>Makes things a bit more challenging for the instructor, keeping them on their toes</li><li>Cushions the "no you're wrong" response to the student (a bit like the "shit sandwich" feedback protocol)</li><li>Force you to diagnose & clarify the misdirected mental pathway for you and the student (and in fact usually the student has misperceived some pattern that's just adjacent to the given problem).</li></ul><p>Try it and see how it feels. To be clear: I don't do this because I'm bored in class, but nevertheless I've found it to be a compelling and clarifying technique.</p>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-53749745317159109552024-01-22T07:00:00.009-05:002024-01-22T07:00:00.140-05:00Radicals and Absolute Values<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhVz-qOqbgelHkzvM6c0sa8OdHe_R2azYuPZnoSKntnpEa5sA6-qCrSd81WE3Y55Z5bTQXWvGqtbPeZuU-JrHxrF7DeCvujpgmUpILAdpn_xigiEu_SLwgIl1ykH6VJlEFwf_rzo63Kt0IHlkenOm5GAPyH9EkDvtL79gCo4YZVE1MP8ZkLZVRSirVi5hjf/s544/RadicalAbsoluteValue.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="156" data-original-width="544" height="58" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhVz-qOqbgelHkzvM6c0sa8OdHe_R2azYuPZnoSKntnpEa5sA6-qCrSd81WE3Y55Z5bTQXWvGqtbPeZuU-JrHxrF7DeCvujpgmUpILAdpn_xigiEu_SLwgIl1ykH6VJlEFwf_rzo63Kt0IHlkenOm5GAPyH9EkDvtL79gCo4YZVE1MP8ZkLZVRSirVi5hjf/w200-h58/RadicalAbsoluteValue.png" width="200" /></a></div><p>Here's a fact that I've never seen expressed clearly, or in this way, in any of the several college-level algebra books from which I've taught. Say we're working in the domain of real numbers, and have a radical of some index over a variable to a power. In broad strokes, we can divide the power by the radical index -- however, in some cases, distressingly, you'd need an absolute value to express the result. The question is, exactly when do you need that absolute value?</p><p>$$\sqrt[n]{a^m} = a^{m \over n} \text{ or } |a^{m \over n}|? $$</p><p>Say we're in this situation, with <i>m</i> and <i>n</i> whole numbers, and <i>m</i> is evenly divisible by <i>n</i>. The primary issue is that when the initial power <i>m</i> is even, it makes any product nonnegative, wiping out any negatives that the base <i>a</i> might represent. So when the reduced power <i>m/n</i> is odd, then it would fraudulently claim to possibly produce negatives, which our initial expression cannot do -- and so require the absolute value as a correction. </p><p>Let's give more detail by inspecting all the permutations of even/odd possibilities between the starting and ending powers in the expression:</p><ol style="text-align: left;"><li><b>Odd <i>m</i>, odd <i>m/n</i>:</b> The odd starting power <i>m</i> can produce values of any sign, and so can the odd reduced power <i>m/n</i>. So all is fine here, and we don't need the absolute value.</li><li><b>Even <i>m</i>, even <i>m/n</i>:</b> The even power <i>m</i> wipes out any negatives, and the reduced even power <i>m/n</i> does the same thing. So again they're aligned, and no absolute value is needed.</li><li><b>Even <i>m</i>, odd <i>m/n</i>:</b> This is the case alluded to above -- the even start power <i>m</i> wipes out negatives, but the odd ending power <i>m/n</i> would deceive us into thinking negatives could be a possible product. This is the situation in which we need the absolute value as a correction.</li><li><b>Odd <i>m</i>, even <i>m/n</i>:</b> This case is impossible. If <i>m/n</i> is even, then any multiple (e.g., by <i>n</i> to produce <i>m</i>) is also even. <br /></li></ol><p>So it's only that third case in which the power switches from even to odd where we need the absolute value bars for full fidelity. Interestingly, since the fourth case can't happen, we could express the protocol briefly as follows:</p><p style="text-align: center;"><b><span style="font-size: large;">When the powers switch parity, then we need absolute value bars.</span></b></p>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-33769612133837573582023-12-02T09:59:00.012-05:002023-12-02T10:08:06.492-05:00Nate Bargatze on Remedial Classes<p>Comedian Nate Bargatze talks about his experience taking remedial classes at a community college in Tennessee:</p><p></p><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="BLOG_video_class" height="266" src="https://www.youtube.com/embed/WyX4uqcBrto?start=221" width="320" youtube-src-id="WyX4uqcBrto"></iframe></div><p></p>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-3578487106810820862023-08-21T07:00:00.003-04:002024-02-06T19:32:38.755-05:00The Iron Law of Stack Exchange<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJ1pnulhnXEaKdCjXESbkQZGhsWYdiI5mQ6Hir17FEXx44lHF-O1WeczCmsLMq2AIbOmUoyCCuu6RMB9MZrJFZTNMuG1eo-yGMVJeT9z1P_chFTRLX8-YUSPS3fCNndGEGgQLroNgL_jyParlXJ-IBz6FXNMmQjOK4LbyP1xRXqljHc6fT6P87tAcrWKod/s768/768px-Stack_Exchange_icon.svg.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img alt="Stack Exchange logo" border="0" data-original-height="768" data-original-width="768" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJ1pnulhnXEaKdCjXESbkQZGhsWYdiI5mQ6Hir17FEXx44lHF-O1WeczCmsLMq2AIbOmUoyCCuu6RMB9MZrJFZTNMuG1eo-yGMVJeT9z1P_chFTRLX8-YUSPS3fCNndGEGgQLroNgL_jyParlXJ-IBz6FXNMmQjOK4LbyP1xRXqljHc6fT6P87tAcrWKod/w200-h200/768px-Stack_Exchange_icon.svg.png" width="200" /></a></div>The Iron Law of Stack Exchange (Stack Overflow):<p></p><p style="text-align: center;"><b><span style="font-size: large;">They hate hard questions from new users.</span></b></p><p>Fundamentally, voters and respondents on Stack Exchange like scoring points with answers that are obvious, easy to write, and don't take excessive amounts of thought. So there is some amount of irritation to questions that are fundamentally <i>hard</i>, and resist such easy answers. This perceived annoyance is exacerbated by a new (relatively low-ranked) user asking a question on any site, and the first instinct by members is often to look for some way in which the question can be rejected as being poorly-formed.</p><p> Symptoms of this reaction include:</p><ul style="text-align: left;"><li>Closing or down-voting a question on poorly-justified grounds.</li><li>Editing the question to change it to an easier one. <br /></li><li>Accusing the question of being an "X-Y problem", that is, the inquirer is confused and really should have asked a different, easier question.</li><li>Complaining about interactions that are common across the Stack Exchange network, but which a new user might not know is commonplace, and so be cowed in that way.</li></ul><p>As one personal example: the Stack Overflow coding site is not my top network destination, but over the years I have asked a number of questions there. As a CS faculty member and past professional developer, by the time I need to reach out externally for help, I've exhausted a rather deep search for answers, and my questions are likely to be fairly hard to crack. This almost always results in exasperation and negative votes from users of the site.</p><p>In my last question, I asked about a feature of a certain piece of software, which seemed like it should have a rather obvious behavior (based on how relative pathnames should work in the OS), but I couldn't get it to work right. Several comments suggested the obvious behavior, which I was pointing out was failing. The only actual answer came from the actual developer of the software, who again asserted how it <i>should</i> work -- and whom, after some back-and-forth, I ultimately convinced about there being a bug in their work, that they agreed to fix it in the next version. <br /></p><p>Despite this rewarding result, no one else could successfully answer this question, and it was (as usual) downvoted into negative territory. Immediately thereafter my Stack Overflow account was actually <i>locked out</i> from asking further questions because of the history of negative votes it garnered.</p><p>Given the downward trend of traffic to Stack Exchange in the last year or so (even predating the earthquake of generative AI in that time), it seems like potentially a difficult problem for the site. Over time, it would seem that most of the low-hanging fruit will already be answered, really only leaving <i>hard </i>problems yet to be dealt with -- and these are specifically the ones that are met with more hostility and likely to be ejected by the most dedicated users of the site.<br /></p>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-51270018051693886302023-02-02T06:00:00.004-05:002024-02-06T19:35:57.090-05:00ChatGPT Roundup<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh1Y94E2v_Y6kfOsG4yO2uhE--qcZt83-f-lEa_7QlrnpPPt_6ARB-MEdHBiU45BENgWcZDIFxdtobqobJH2gYgZOPr3WWG7LJHVLrscl3Y00qqNvW4HI2AvqB6gzDS4I2Wi6iyiyP7ru5HDdJ3bFkPX1oZzuMx9UPJskk8kR8wiFuKE-LFkDIDJwAc7w/s1024/chatgpt%20bot%20ai.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img alt="Cartoon bot chatting" border="0" data-original-height="1024" data-original-width="1024" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh1Y94E2v_Y6kfOsG4yO2uhE--qcZt83-f-lEa_7QlrnpPPt_6ARB-MEdHBiU45BENgWcZDIFxdtobqobJH2gYgZOPr3WWG7LJHVLrscl3Y00qqNvW4HI2AvqB6gzDS4I2Wi6iyiyP7ru5HDdJ3bFkPX1oZzuMx9UPJskk8kR8wiFuKE-LFkDIDJwAc7w/w200-h200/chatgpt%20bot%20ai.png" width="200" /></a></div><p></p><p>Have we arguably stepped into the singularity? As of last November, OpenAI's release of the ChatGPT language-model system has upended most everything in sight, and in particular, sent educators everywhere scrambling to deal with the ramifications. This chatbot can seemingly craft custom essays, reports, scientific papers, newspaper articles, programming code, and solutions to many (although not all) mathematical problems. Immediately, for free, and in ways almost no human can detect.<br /></p><p>Here's a roundup of news stories that I may update in the future:</p><ul style="text-align: left;"><li>Many students at Stanford <a href="https://stanforddaily.com/2023/01/22/scores-of-stanford-students-used-chatgpt-on-final-exams-survey-suggests/">report using ChatGPT for end-of-semester assignments and exams</a> <br /></li><li>ChatGPT <a href="https://www.nbcnews.com/tech/tech-news/chatgpt-passes-mba-exam-wharton-professor-rcna67036">passes final exam in an MBA course at Wharton</a> <br /></li><li>Professor at U. Toronto says, <a href="https://twitter.com/Afinetheorem/status/1598081835736891393">"<span class="css-901oao css-16my406 r-poiln3 r-bcqeeo r-qvutc0">you can no longer give take-home exams/homework"</span></a> <br /></li><li>ChatGPT <a href="https://www.nature.com/articles/d41586-023-00107-z">added as author to many academic papers</a> </li><li>Top AI conference forced to <a href="https://www.theverge.com/2023/1/5/23540291/chatgpt-ai-writing-tool-banned-writing-academic-icml-paper">ban ChatGPT from writing science papers</a></li><li><a href="https://www.scientificamerican.com/article/research-summaries-written-by-ai-fool-scientists/">Scientists unable to tell when ChatGPT has written abstracts</a> for academic papers</li><li>More science journal publishers <a href="https://www.theguardian.com/science/2023/jan/26/science-journals-ban-listing-of-chatgpt-as-co-author-on-papers">forced to ban use of ChatGPT</a> on submitted papers </li><li>Stack Overflow <a href="https://meta.stackoverflow.com/questions/421831/temporary-policy-chatgpt-is-banned">forced to ban ChatGPT</a> answers from the site.</li><li>ChatGPT considered <a href="https://www.nytimes.com/2022/12/21/technology/ai-chatgpt-google-search.html">an existential threat</a> to Google.</li><li>Microsoft plans to quickly bring ChatGPT features to <a href="https://www.reuters.com/technology/microsoft-aims-ai-powered-version-bing-information-2023-01-04/">Bing</a>, <a href="https://www.theinformation.com/articles/ghost-writer-microsoft-looks-to-add-openais-chatbot-technology-to-word-email">Office</a>, and <a href="https://venturebeat.com/ai/microsoft-azure-openai-service-now-generally-available-with-chatgpt-on-the-way/">Azure</a> <br /></li><li>Kindle novelists <a href="https://www.theverge.com/23520625/chatgpt-openai-amazon-kindle-novel">start widely using ChatGPT</a></li><li>NYC Department of Education <a href="https://www.vice.com/en/article/y3p9jx/nyc-bans-students-and-teachers-from-using-chatgpt">bans use of ChatGPT in all schools</a></li><li>CNET caught <a href="https://www.buzzfeednews.com/article/katienotopoulos/cnet-articles-written-by-ai-chatgpt-article">quietly using AI to write articles</a></li><li>BuzzFeed (which broke story above) says <a href="https://www.cnn.com/2023/01/26/media/buzzfeed-ai-content-creation/index.html">it will use ChatGPT to write articles</a>, stock jumps 150%.</li><li>Member of Congress <a href="https://apnews.com/article/technology-science-oddities-israel-massachusetts-11b4dc6e42afd2d68be28dedf86fd25a">reads a speech written by ChatGPT into official record</a></li><li>Judge in Columbia use ChatGPT to <a href="https://www.vice.com/en/article/k7bdmv/judge-used-chatgpt-to-make-court-decision">write a legal ruling</a></li><li>ChatGPT has the <a href="https://www.reuters.com/technology/chatgpt-sets-record-fastest-growing-user-base-analyst-note-2023-02-01/">fastest-growing user base of any application in history</a></li><li>ChatGPT passes <a href="https://www.pcmag.com/news/chatgpt-passes-google-coding-interview-for-level-3-engineer-with-183k-salary">Google employment coding interview</a> <br /></li></ul><p></p><p><i>Image courtesy Craiyon. :-)</i><br /></p>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-86677942946993977332022-09-04T23:43:00.000-04:002022-09-04T23:43:36.647-04:00NY Regents: Trivial to Pass<p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEge_EQnaVOwgN5TBzRNBsJN7BVakYu5TShWOirA_0c4hCZ1v22pe53Tfgw7ziFM3FUxx96KXnhHQeFcfdGep8-P--bVpVa8RPUnhiT6wnoziSDgYCQkJvSKso5JvoMmtoouCzoXBIXPK6aewILi1RPZXAXACxEcdykYsk5jtzVPgjSSsGFU4hcFLp02kg/s299/MultipleChoiceAllC.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img alt="Multiple choice with all C-answers" border="0" data-original-height="272" data-original-width="299" height="182" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEge_EQnaVOwgN5TBzRNBsJN7BVakYu5TShWOirA_0c4hCZ1v22pe53Tfgw7ziFM3FUxx96KXnhHQeFcfdGep8-P--bVpVa8RPUnhiT6wnoziSDgYCQkJvSKso5JvoMmtoouCzoXBIXPK6aewILi1RPZXAXACxEcdykYsk5jtzVPgjSSsGFU4hcFLp02kg/w200-h182/MultipleChoiceAllC.png" width="200" /></a></div><p></p><p>Ed Knight is a teacher in New York state. Writing at Medium, he points out the disturbing fact that the vaunted "New York Regents" exams required to graduate from high schools in the state have become completely trivial to pass. For example: In the Algebra Regents, you can ignore all of the (already simple) open-response questions, and just mindlessly mark "C" for all the multiple-choice questions, and you'll be given a passing grade.</p><p></p><p class="pw-post-body-paragraph jd je ig jf b jg jh ji jj jk jl jm jn jo jp jq jr js jt ju jv jw jx jy jz ka hz gh" data-selectable-paragraph="" id="dbe3"><i></i></p><blockquote><p class="pw-post-body-paragraph jd je ig jf b jg jh ji jj jk jl jm jn jo jp jq jr js jt ju jv jw jx jy jz ka hz gh" data-selectable-paragraph="" id="dbe3"><i>Answer all Cs, you get 9 right. Each one is worth two points. That’s 18 points, enough to pass.</i></p><p class="pw-post-body-paragraph jd je ig jf b jg jh ji jj jk jl jm jn jo jp jq jr js jt ju jv jw jx jy jz ka hz gh" data-selectable-paragraph="" id="75f9"><i>More than enough to pass, in fact.</i></p><p class="pw-post-body-paragraph jd je ig jf b jg jh ji jj jk jl jm jn jo jp jq jr js jt ju jv jw jx jy jz ka hz gh" data-selectable-paragraph="" id="b34c"><i>I
really didn’t know what to say when I first saw this. I suppose I was
thinking like Chuck Heston. “They finally, really did it.” This is
jaw-dropping malfeasance. I wouldn’t believe that anyone who claims to
care about kids could let this happen — not until I saw it happen.</i></p><p><i>Shame on NYSED and the Regents.</i></p></blockquote><p>Really, the root of this problem is the insane scaling procedure that the NY Regents has been doing for years to fake up the test scores. Below is the most recent test's table for converting a "Raw Score to a reported "Scale Score". The scale score is 0-100, making recipients thinks it's a percentage, but it's not. For example: if you score a raw 27 out of the possible 86 points (that's 31% correct), this then gets converted to a reported Scale Score 65 -- i.e., a Performance Level of 3 out of 5, which is considered passing.</p><p>Think about that: for years, the NY Regents has considered a score of about 30% as passing for a basic (very simple!) algebra test. And yes, this was exacerbated because for the pandemic years (still ongoing), the policy was adjusted to accept <i>even lower scores</i> than that -- now as low as 20% (i.e., Raw 17, reported as a Scale 50). </p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiN4xiK8CedBjx2ut6zlOqxjxRrcN9r_c6gJ5qmu2jyA9-mrkksb-JvF053NmMKx8a6Qr0kGqEy8e6Ah-3FEKSUFdR_PJr9dSsnPPd_dyGr9oaDBlVOrjUFzZ0PQMkXyhPA3z_B_0hiRjo5XqzJd1fIauoUlEWUeHOYIOkOssBXW7wUke5vvm9CELBfog/s700/RegentsScoring2022.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Scoring for Regents Exam in Algebra I: June 2022" border="0" data-original-height="566" data-original-width="700" height="259" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiN4xiK8CedBjx2ut6zlOqxjxRrcN9r_c6gJ5qmu2jyA9-mrkksb-JvF053NmMKx8a6Qr0kGqEy8e6Ah-3FEKSUFdR_PJr9dSsnPPd_dyGr9oaDBlVOrjUFzZ0PQMkXyhPA3z_B_0hiRjo5XqzJd1fIauoUlEWUeHOYIOkOssBXW7wUke5vvm9CELBfog/w320-h259/RegentsScoring2022.png" width="320" /></a></div><p></p><p>Read more at Medium: <a href="https://medium.com/@newyorkteacher/guessing-c-for-every-answer-is-now-enough-to-pass-the-new-york-state-algebra-exam-93bac55b3e24">Guessing C For Every Answer Is Now Enough To Pass The New York State Algebra Exam</a><br /></p><p class="pw-post-body-paragraph jd je ig jf b jg jh ji jj jk jl jm jn jo jp jq jr js jt ju jv jw jx jy jz ka hz gh" data-selectable-paragraph="" id="0a67"></p>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-81037261566215103062022-08-08T07:00:00.014-04:002022-08-08T07:00:00.172-04:00Curriculum in Califormia<p></p><div style="text-align: left;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9v6sJsfuqzJHBB78nqH2E3vajReolcMyEdJ9TQuQii0eLa8L25gYIxzE2LaPSsjUlq3FNuUZhCqHTMG_LGdQFiBdNsVmeWF9v5sf4_KC3BBhDRN7AnU7mLHyTMipUP9lRxJZnyaODgzTkpDQRRWQuhOWwJ1xrE2l5LpjJHawRrFKeM99bv3eVyCKNXQ/s499/6th+Math.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img alt="Cover of California Math 6th-grade textbook" border="0" data-original-height="499" data-original-width="392" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9v6sJsfuqzJHBB78nqH2E3vajReolcMyEdJ9TQuQii0eLa8L25gYIxzE2LaPSsjUlq3FNuUZhCqHTMG_LGdQFiBdNsVmeWF9v5sf4_KC3BBhDRN7AnU7mLHyTMipUP9lRxJZnyaODgzTkpDQRRWQuhOWwJ1xrE2l5LpjJHawRrFKeM99bv3eVyCKNXQ/w157-h200/6th+Math.jpg" width="157" /></a></div></div><div style="text-align: left;">A nicely comprehensive article late last year outlines the plans for the next overhaul of school mathematics curriculum in California -- delaying any algebra until high school, cutting and compressing later classes to fit the reduced time, disposing of gifted & talented or accelerated programs, de-tracking, etc. As usual, the motivation for this to hopefully see higher pass rates from the easier courses, claim to better support inequalities among minorities, be better positioned for well-paying STEM college programs and careers, etc. Among the fonts of battle are particularly academic math professors vs. math-education faculty, who are generally on opposing sides of the issue. </div><p></p><p>One thing that really stuck out to me was the case of one student, who's held out as being the one black student throughout her advanced math courses in school, and currently studying as a math major at UC Berkley. Here's how her story is presented: </p><p></p><p><i></i></p><blockquote><p><i>Mariah Rose, a third-year applied math major at UC Berkeley, said she
didn’t have another Black classmate in any of her math classes until
this semester.</i></p><i>
</i><p><i>“There’s one other Black student in my class right now, and that’s
just crazy to me,” said Rose. “The number of Black and Brown people in
math is so low.”</i></p><i>
</i><p><i>Rose, who is half Black and half Latino, said this is nothing new.
She said she was the only Black female student in her advanced math
classes during high school. And her successes in math make her an
outlier in California’s public school system where Black and Latino
students score lower on standardized tests...</i></p><p><i>Rose, the UC Berkeley math major, said she has mixed feelings. She
agrees with the framework’s recommendation to delay more advanced math
classes and avoid labeling students based on their math abilities at
younger ages. But she isn’t sure if she would be where she is if she
hadn’t been accelerated into a higher-level math class in 6th grade. </i></p><i>
</i><p><i>“It was a game changer,” she said. “I don’t know if I would’ve pursued math if I hadn’t advanced so early.”</i></p></blockquote><p><i></i></p><p><a href="https://sfstandard.com/education/controversy-rages-as-california-follows-sfs-lead-with-new-approach-to-teaching-math/">Read the full article at The San Fransisco Standard. </a><br /></p>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-34709376772568573582022-08-01T11:15:00.007-04:002022-08-01T11:19:10.495-04:00Proofs and Applications<p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhkQ4ixP_2c9qUv2kwjIEeatfI6oRecHUIQMvp02sP-3PV4Z6ub_9x3PtN18LLalPqst2FsxR9wMFEmxFELVngyUsY6CH38KH9xpxZEP2O41sOEGv9wF5ETiXCeznpnfGiD9StJjhIoN-Hs_DOQAGssQgeUfmE1gNwYqdbnoVl9nM7ZAzjxWMpYvCIokg/s2400/burden-of-proof.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img alt=""Burden of Proof" on laptop" border="0" data-original-height="1600" data-original-width="2400" height="133" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhkQ4ixP_2c9qUv2kwjIEeatfI6oRecHUIQMvp02sP-3PV4Z6ub_9x3PtN18LLalPqst2FsxR9wMFEmxFELVngyUsY6CH38KH9xpxZEP2O41sOEGv9wF5ETiXCeznpnfGiD9StJjhIoN-Hs_DOQAGssQgeUfmE1gNwYqdbnoVl9nM7ZAzjxWMpYvCIokg/w200-h133/burden-of-proof.jpg" width="200" /></a></div><p></p><p>This is a quote that lives rent-free in my head, and comes up a lot in discussions I participate in.</p><p>From Stein/Barcellos, <i>Calculus and Analytic Geometry</i>, 5E, "To the Instructor", p. xxii (1992):</p><blockquote><i>At the Tulane conference on "Lean and Lively Calculus" in 1986 we heard the engineers say, "Teach the concepts. We'll take care of the applications." Steve Whitaker, in the engineering department at Davis, advised us, "Emphasize proofs, because the ideas that go into the proofs are often the ideas that go into the applications." Oddly, mathematicians suggest that we emphasize applications, and the applied people suggest that we emphasize concepts. We have tried to strike a reasonable balance that gives the instructor flexibility to move in either direction.</i></blockquote><i></i><p></p>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-88275575708093817512022-07-13T21:32:00.001-04:002022-07-13T21:35:06.274-04:00Willingham on Automaticity<p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjxN-9wD1UHxbqRVIruGDk9DztlzZgRODB7PxgJJrErIo5o40jkGEo52uJ4f70LZYXRBPxa9jEUaEYUbicYqYkE_vHLuLvz8eqICmcEeI58o1oYkDLs7iChckjPzIPReY3IupHHoYElwrfrhBRsdY5bJ1WYa88RcUvWNhevesMHnHeNcK5MTqv8DCB38w/s341/Thinker.JPG" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="210" data-original-width="341" height="123" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjxN-9wD1UHxbqRVIruGDk9DztlzZgRODB7PxgJJrErIo5o40jkGEo52uJ4f70LZYXRBPxa9jEUaEYUbicYqYkE_vHLuLvz8eqICmcEeI58o1oYkDLs7iChckjPzIPReY3IupHHoYElwrfrhBRsdY5bJ1WYa88RcUvWNhevesMHnHeNcK5MTqv8DCB38w/w200-h123/Thinker.JPG" width="200" /></a></div><p></p><p>Some solid thoughts from Daniel Willingham on the need for automaticity in basic mathematics skills (re: <a href="http://automatic-algebra.org">automatic-algebra.org</a>) from his article "Is It True That Some People Just Can't Do Math?" (<i>American Educator</i>, Winter 2009-2010):<br /></p><p></p><blockquote><i>In its recent report, the National Mathematics Advisory Panel argued that learning mathematics requires three types of knowledge: factual, procedural, and conceptual. Let’s take a close look at each.<br /><br />Factual knowledge refers to having ready in memory the answers to a relatively small set of problems of addition, subtraction, multiplication, and division. The answers must be well learned so that when a simple arithmetic problem is encountered (e.g., 2 + 2), the answer is not calculated but simply retrieved from memory. <b>Moreover, retrieval must be automatic (i.e., rapid and virtually attention free). This automatic retrieval of basic math facts is critical to solving complex problems because complex problems have simpler problems embedded in them.</b> For example, long division problems have simpler subtraction problems embedded in them. Students who automatically retrieve the answers to the simple subtraction problems keep their working memory (i.e., the mental “space” in which thought occurs) free to focus on the bigger long division problem. The less working memory a student must devote to the subtraction subproblems, the more likely that student is to solve the long division problem.<br /><br /><b>This interpretation of the importance of memorizing math facts is supported by several sources of evidence.</b> First, it is clear that before they are learned to automaticity, calculating simple arithmetic facts does indeed require working memory. With enough practice, however, the answers can be pulled from memory (rather than calculated), thereby incurring virtually no cost to working memory. Second, students who do not have math facts committed to memory must instead calculate the answers, and calculation is more subject to error than memory retrieval. Third, knowledge of math facts is associated with better performance on more complex math tasks. Fourth, when children have difficulty learning arithmetic, it is often due, in part, to difficulty in learning or retrieving basic math facts. One would expect that interventions to improve automatic recall of math facts would also improve proficiency in more complex mathematics. Evidence on this point is positive but limited, perhaps because automatizing factual knowledge poses a more persistent problem than difficulties related to learning mathematics procedures</i>.</blockquote><p><a href="https://www.aft.org/periodical/american-educator/winter-2009-2010">Get the whole article by Willingham (including citations for all the claims above) at the AFT website.</a><br /></p><p></p>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-89261700197167292292022-03-08T22:25:00.001-05:002022-07-13T21:39:44.796-04:00Remedial Equality Check<p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEj9IS8AP49tOdiKSn9JgaSxcrqG4QN16OVgufCus8KKHEc_TqVLX2SDJpv6-lzAZf1JiXLenGsbaUPG-VfTBksOJR01-9DGz6xh4q7DHGB-DgjsOZ4RySEHSy1gUpCw2MVjzI8DqjupXC1tpwHKUaRORJDcZCb6JsCpq-RFY_TKbfE8dbWFQqU-ncQvCQ=s240" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img alt="Equals sign" border="0" data-original-height="240" data-original-width="210" height="200" src="https://blogger.googleusercontent.com/img/a/AVvXsEj9IS8AP49tOdiKSn9JgaSxcrqG4QN16OVgufCus8KKHEc_TqVLX2SDJpv6-lzAZf1JiXLenGsbaUPG-VfTBksOJR01-9DGz6xh4q7DHGB-DgjsOZ4RySEHSy1gUpCw2MVjzI8DqjupXC1tpwHKUaRORJDcZCb6JsCpq-RFY_TKbfE8dbWFQqU-ncQvCQ=w175-h200" width="175" /></a></div><p>I discovered something last semester that made me insert a new little thing in the first day of my basic-level (remedial, liberal arts) community college math courses. A surprising proportion of my students are <i>very</i> confused about what the equality (=) sign means. </p><p>Now, this isn't a tremendously novel observation, e.g., see: Baroody, Arthur J., and Herbert P. Ginsburg, <a href="https://doi.org/10.1086/461356">"The effects of instruction on children's understanding of the 'equals' sign."</a> <i>The Elementary School Journal</i> 84.2 (1983): 199-212. But the new discovery for me was how extremely simple a question it takes to make this visible. </p><p>All I've done is start asking, "True or false? (a) 6 = 6, (b) 3 = 5". So far, everyone, confidently answers "true" to the first. But it seems like fully half of my students answer the second one incorrectly. <br /></p><p>Last semester I had at least one or two students who were so enormously challenged that we could repeat this <i>every day all semester</i> and they'd never get it right. "True or false? 0 = 100"; "True"; "No, it's false"; "I don't get it". Over and over again every day, no matter how many times it was explained. <br /></p><p>It seems pretty amazing, but there it is. Lots of our incoming community-college student literally don't now what the equals relation means. My best guess is that they've either come to think of it as "and here's the next thing", or that maybe instructors have been double-checking solutions to equations and every single time it's always come out true. </p><p>I suppose it would be a decent research project to interview such students and ask them to explain what they think the equals sign means. Consider trying that for an opening exercise and share what your results are?</p><p><br /></p>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com2tag: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://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhGx2bQt1e-c8A94X1GpO8_3y5ylkMA8YLho7fQIywJWm2D3ozGz3jdPBrkGacwJIfo0ySrWOLpaq-7BK7-ojItGaShEUQslKdog-twBi_x5HuCwtiYVAPo1nl9PybWcjikmq5BoLRGiukM/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://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhGx2bQt1e-c8A94X1GpO8_3y5ylkMA8YLho7fQIywJWm2D3ozGz3jdPBrkGacwJIfo0ySrWOLpaq-7BK7-ojItGaShEUQslKdog-twBi_x5HuCwtiYVAPo1nl9PybWcjikmq5BoLRGiukM/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-61669871862604957752021-10-18T22:03:00.003-04:002024-08-05T12:10:36.322-04:00Growth Mindset Theory: Failures to Replicate<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjz5y2eyd8g_xygTXCEsNGuxamlJ2Jn8UBvgB1VHywjS6bKXqh1TwUw0HwlKZ-VcVv-V50XUdc1BOWEh2Bfyt_CzNyDVjBGWxl1mL8Bh6QozVpGEzI2PmUjaEBZM0BAk4tgBEv_ExRVPznj/s1280/MindsetUp.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="792" data-original-width="1280" height="198" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjz5y2eyd8g_xygTXCEsNGuxamlJ2Jn8UBvgB1VHywjS6bKXqh1TwUw0HwlKZ-VcVv-V50XUdc1BOWEh2Bfyt_CzNyDVjBGWxl1mL8Bh6QozVpGEzI2PmUjaEBZM0BAk4tgBEv_ExRVPznj/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>. Does
growth mindset improve children’s IQ, educational attainment or response
to setbacks? Active-control interventions and data on children’s own
mindsets. <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></p><p><b>Bahník, Štěpán, and Marek A. Vranka (2017).</b> Growth mindset is not associated with scholastic aptitude in a large sample of university applicants. <i>Personality and Individual Differences</i> 117: 139-143. <a href="https://doi.org/10.1016/j.paid.2017.05.046" rel="nofollow noreferrer">https://doi.org/10.1016/j.paid.2017.05.046</a> (Study of university students taking an admissions test in the Czech Republic, N = 5,653).</p>
<blockquote>
<p>We found that results in the test were slightly negatively associated
with growth mindset (r = −0.03). Mindset showed no relationship with
the number of test administrations participants signed up for and it
did not predict change in the test results. The results show that the
strength of the association between academic achievement and mindset
might be weaker than previously thought.</p>
</blockquote><p><b>Foliano, F., Rolfe, H., Buzzeo, J., Runge, J., & Wilkinson, D. (2019).</b> Changing mindsets: Effectiveness trial. <i>National Institute of Economic and Social Research.</i> <a href="http://psychbrief.com/growth-mindset-fails/" rel="nofollow noreferrer">Summary at PsychBrief</a>. (Study in England, Year 6 students, N = 4,584.)</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> Failure to Replicate: Testing a Growth Mindset Intervention for College Student Success, <i>Basic and Applied Social Psychology</i>, 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 = 2,607).</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><p><b>Sisk, V. F., Burgoyne, A. P., Sun, J., Butler, J. L., & Macnamara,
B. N. (2018).</b> To What Extent and Under Which Circumstances Are Growth
Mind-Sets Important to Academic Achievement? Two Meta-Analyses. <i>Psychological Science</i>, <i>29</i>(4), 549-571. <a href="https://doi.org/10.1177/0956797617739704">https://doi.org/10.1177/0956797617739704</a> (U.S., meta-analysis of 273 studies, N = 365,915).</p><p></p><blockquote>Our meta-analyses do not support this claim. Effect sizes were inconsistent across studies, but most analyses yielded small (or null) effects. Overall, the first meta-analysis demonstrated only a very weak relationship between mind-sets and academic achievement. Similarly, the second meta-analysis demonstrated only a very small overall effect of mind-set interventions on academic achievement... from a practical perspective, resources might be better allocated elsewhere than mind-set interventions.</blockquote><p></p><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://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOlvotaGSsGLneU1d4Q7qmZhK1ZFAo1kxxm6Mz_Jfn0CYMK23L8GnlJC6DzgF1O2CenAT_YvcQKBEAY6Y4uIvUQrkwbXh4xbkVwDn8UizG3vPBS26QaV5H6kkSd6CfxM6y9XSEVeQX5DAl/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://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOlvotaGSsGLneU1d4Q7qmZhK1ZFAo1kxxm6Mz_Jfn0CYMK23L8GnlJC6DzgF1O2CenAT_YvcQKBEAY6Y4uIvUQrkwbXh4xbkVwDn8UizG3vPBS26QaV5H6kkSd6CfxM6y9XSEVeQX5DAl/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://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhxdJMGh71u49FX87B-095HjEjm2pY68Nmdd_YCwOiYjbs9K23fH6pjXZ5beKZ8MoY4Dsa32BvR-Lh3h2KCN0QPH4XmSGrp0aJalVFvS_TKe0-nSEjqitoQD0URz7-fkislBiCLLBS0pmg2/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://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhxdJMGh71u49FX87B-095HjEjm2pY68Nmdd_YCwOiYjbs9K23fH6pjXZ5beKZ8MoY4Dsa32BvR-Lh3h2KCN0QPH4XmSGrp0aJalVFvS_TKe0-nSEjqitoQD0URz7-fkislBiCLLBS0pmg2/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.com1tag: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">
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEitwmUo1O4xhT8AEUSoUyEGS75_vn8K_c5dR9qBa0l79yVIfNzDKjXLyhyphenhyphenJBPYnffiu8HI8ctFHPlGVyii_xx7kipkzU4l1Mp6d9rPvFdjxLlNu0cdCy8upigeumN6ltz0jc7OQdNO1XH4u/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://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEitwmUo1O4xhT8AEUSoUyEGS75_vn8K_c5dR9qBa0l79yVIfNzDKjXLyhyphenhyphenJBPYnffiu8HI8ctFHPlGVyii_xx7kipkzU4l1Mp6d9rPvFdjxLlNu0cdCy8upigeumN6ltz0jc7OQdNO1XH4u/s320/F1.large.jpg" width="297" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEik5Ymi84tVHrMXItnRZBAKz961BeVb1lgSwIjH_1lGfjffLi1YROsQVdJuuwW-vlwwzCdCET64xJZQ0k23THxGQX7V6W7c5sYKRmKHWBG7J8BYbbQO4aBGiiDhGoRuE1H2MXcQ9QnG4RfR/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://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEik5Ymi84tVHrMXItnRZBAKz961BeVb1lgSwIjH_1lGfjffLi1YROsQVdJuuwW-vlwwzCdCET64xJZQ0k23THxGQX7V6W7c5sYKRmKHWBG7J8BYbbQO4aBGiiDhGoRuE1H2MXcQ9QnG4RfR/s320/F2.large.jpg" width="146" /></a></div>
<div id="p-3">
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</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>
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Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0