tag:blogger.com,1999:blog-7718462793516968883Sat, 30 Apr 2016 18:44:06 +0000MadMath"Beauty is the Enemy of Expression"http://www.madmath.com/noreply@blogger.com (Delta)Blogger233125tag:blogger.com,1999:blog-7718462793516968883.post-2100360724893293521Fri, 29 Apr 2016 09:00:00 +00002016-04-29T05:00:20.723-04:00On PiflarsIn coordination with the week's theme of grammar -- seen on StackExchange: English Language & Usage:<br /><br />Apparently in Slovenian, there is the single word "piflar", which derogatorily means "a student who only memorizes instead of truly learning". What would be the best comparable word for this in English?<br /><br /><div style="text-align: center;"><span style="font-size: large;"><a href="http://english.stackexchange.com/questions/313712/derogatory-word-describing-person-a-pupil-who-memorizes-instead-of-learning">StackExchange English Language & Usage: Derogatory word, describing person (a pupil) who memorizes instead of learning?</a></span></div><div style="text-align: center;"><br /></div><div style="text-align: center;"><br /></div>http://www.madmath.com/2016/04/on-piflars.htmlnoreply@blogger.com (Delta)0tag:blogger.com,1999:blog-7718462793516968883.post-7486934894266262667Mon, 25 Apr 2016 09:00:00 +00002016-04-25T05:00:12.453-04:00Gruesome GrammarA week or so back we observed the rough consensus that basic arithmetic operations are essentially some kind of <a href="http://www.madmath.com/2016/04/what-part-of-speech-is-times.html">prepositions</a>. Coincidentally, tonight I'm reviewing the current edition of <a href="http://www.ck12.org/book/CK-12-Algebra-Basic/">"CK-12 Algebra - Basic"</a> (Kramer, Gloag, Gloag; May 30, 2015) -- and the <i>very first thing</i> in the book is to get this exactly wrong. Here are the first two paragraphs in the book (Sec 1.1): <br /><blockquote class="tr_bq"><i>When someone is having trouble with algebra, they may say, “I don’t speak math!” While this may seem weird to you, it is a true statement. Math, like English, French, Spanish, or Arabic, is a secondary language that you must learn in order to be successful. There are verbs and nouns in math, just like in any other language. In order to understand math, you must practice the language.<br /><br />A verb is a “doing” word, such as running, jumps, or drives. In mathematics, verbs are also “doing” words. A math verb is called an operation. Operations can be something you have used before, such as addition, multiplication, subtraction, or division. They can also be much more complex like an exponent or square root.</i></blockquote><br />That's the kind of thing that consistently aggravates me about open-source textbooks. While I love and agree with establishing algebraic notation as a kind of language -- as possibly the single most important overriding concept of the course -- to get the situation exactly wrong right off the bat (as well as using descriptors like PEMDAS, god help us) keep these as fundamentally unusable in my courses. This, of course, then leaves no grammatical position at all for <i>relations</i> (like equals) when they finally appear later in the text (Sec. 1.4). So close, and yet so far. <br /><br /><br />http://www.madmath.com/2016/04/gruesome-grammar.htmlnoreply@blogger.com (Delta)4tag:blogger.com,1999:blog-7718462793516968883.post-5561095446763677337Fri, 22 Apr 2016 09:00:00 +00002016-04-22T05:00:10.303-04:00Link: Smart People Happier with Fewer FriendsResearch by people at the London School of Economics and Singapore Management University that smarter people are happier with fewer friends, and fewer social interaction outings. Downside: The researchers are evolutionary psychologists and seek to explain the finding in those terms. Also: Uses the term "paleo-happiness". <br /><br /><div style="text-align: center;"><span style="font-size: large;"><a href="https://www.washingtonpost.com/news/wonk/wp/2016/03/18/why-smart-people-are-better-off-with-fewer-friends/">Washington Post: Why smart people are better off with fewer friends</a></span></div>http://www.madmath.com/2016/04/link-smart-people-happier-with-fewer.htmlnoreply@blogger.com (Delta)0tag:blogger.com,1999:blog-7718462793516968883.post-1578558007233517740Fri, 15 Apr 2016 09:00:00 +00002016-04-15T05:00:21.605-04:00Poker MemoryMaybe 15 years ago, I went to the Foxwood poker tables (vs. 9 other people) , got pocket Queens, and had an Ace come up on the flop. I maxed out the bet and lost close to $100. So the other morning I woke up and the thought in my head was, "I really should have computed the probability that someone else had an Ace". Which was 1 - 44P18/47P18 = 1 - 0.225 = 0.775 = 77.5%. Sometimes my brain works glacially slow like that.<br /><br /><br />http://www.madmath.com/2016/04/poker-memory.htmlnoreply@blogger.com (Delta)0tag:blogger.com,1999:blog-7718462793516968883.post-8694847598401872994Mon, 11 Apr 2016 09:00:00 +00002016-04-11T05:00:08.071-04:00Reading RadicalsIn my development algebra classes, I push radicals further forward, closer to the start of the semester than most other instructors or textbooks. I want them to be discussed jointly with exponents, so we can really highlight the inverse relation with exponents, and that knowledge of the rules of one is effectively equivalent knowledge of the other. Also: Based on the statistics I keep, success on the exponents/radicals test is the single best predictor of success on the comprehensive, university-wide final exam. <br /><br />There are, of course, many errors made by students learning to read and write radicals for effectively the first time. Here's an exceedingly common category, to write something like (\(x > 0\)):<br /><div style="text-align: center;">$$\sqrt{16} = 4 = 2$$</div><div style="text-align: center;">$$\sqrt{x^8} = x^4 = x^2 = x$$</div><br />Any of these expressions may or may not have a radical written over them (including, e.g., \(\sqrt{4} = \sqrt{2}\)). That is: Students see something "magical" about radicals, and sometimes keep square-rooting any expression in sight, until they can no longer do so. This is common enough that I have few interventions in my mental toolbox ready for when this occurs in any class:<br /><ol><li>Go to the board and, jointly with the whole class, start asking some true-or-false questions. "T/F: \(\sqrt{4} = \sqrt{2}\) ← False. \(\sqrt{4} = 2\) ← True." Briefly discuss the difference, and the location of \(\sqrt{2}\) on the number line. Emphasize: Every written symbol in math makes a difference (any difference in the writing, and it has a different meaning). </li><br /><li>Prompt for the following on the board. "Simplify: \(3 + 5 = 8\)." Now ask: "Where did the plus sign go? Why are you not writing it in the simplified expression? Because: You <i>did</i> the operation, and therefore the operational symbol goes away. The same will happen with radicals: If you can actually compute a radical, then the symbol goes away at that time." </li></ol><br />That's old hat, and those are techniques I've been using for a few years now. The one new thing I noticed last night (as I write this) is that there is actually something unique about the notation for radicals: Of the six basic arithmetic operations (add, subtract, multiply, divide, exponents, radicals), <u>radicals are the only binary operation where one of the two parameters may not be written</u>. That is, for the specific case of square roots, there is a "default" setting where the index of 2 doesn't get written -- and there's no analogous case of any other basic operator being written without a pair of numbers to go with it. <br /><br />I wonder if this contributes to the apparent "magical" qualities of radicals (specifically: students pay more attention to the visible numbers, whereas I am constantly haranguing students to look more closely at the operators in the writing)? Hypothetically, if we always wrote the index of "2" visibly for square roots (as for all other binary operators), would this be more transparent to students that the operator only gets applied once (at which point radical and index simplify out of the writing)? And perhaps this would clear up a related problem: students occasionally writing a reduction as a new index, instead of a factor (e.g., \(\sqrt{18} = \sqrt{9 \times 2} = \sqrt[3]{2}\))? <br /><br />That would be a pretty feasible experiment to run in parallel classes, although it would involve using nonstandard notation to make it happen (i.e., having students explicate the index of "2" for square roots all the time). Should we consider that experiment? <br /><br /><br />http://www.madmath.com/2016/04/reading-radicals.htmlnoreply@blogger.com (Delta)8tag:blogger.com,1999:blog-7718462793516968883.post-201077468795242259Fri, 08 Apr 2016 09:00:00 +00002016-04-08T05:00:07.197-04:00What Part of Speech is "Times"?What part of speech are the operational words "plus", "minus", and "times"? This is a surprisingly tricky issue; apparently major dictionaries actually differ in their categorization. The most common classification is as some form of <b>preposition</b> -- the Oxford Dictionary says that they are <b>marginal prepositions</b>; "a preposition that shares one or more characteristics with other word classes [i.e., verbs or adjectives]".<br /><br />Here's an interesting thread on Stack Exchange: English Language & Usage on the issue -- including commentary by famed quantum-computing expert and word guru Peter Shor:<br /><br /><div style="text-align: center;"><span style="font-size: large;"><a href="http://english.stackexchange.com/questions/303067/is-times-really-a-plural-noun">Stack Exchange English Language & Usage: <br />Is “times” really a plural noun?</a></span></div><div style="text-align: center;"><br /></div>http://www.madmath.com/2016/04/what-part-of-speech-is-times.htmlnoreply@blogger.com (Delta)0tag:blogger.com,1999:blog-7718462793516968883.post-7939066326134188323Fri, 01 Apr 2016 09:00:00 +00002016-04-01T05:00:15.917-04:00Veterinary HomeopathyA funny, but scary and real, web page of a homeopathic-practicing veterinarian who seems weirdly cognizant that it has no real effect:<br /><blockquote class="tr_bq"><i>How much to give: Each time you treat your pet, give approximately 10-20 of the tiny (#10) pellets in the amber glass vial, or 3-7 of the larger (#20) pellets in the blue plastic tube. <b>You don't need to count them out. In fact, the number of pellets given per treatment makes no difference whatsoever.</b> It is the frequency of treatment and the potency of the remedy that is important. <b>Giving more pellets per treatment does not in any way affect the body's response. The pellets need not be swallowed, </b>and it doesn't matter if a few of them are spit out. Just get a few pellets somewhere in the mouth, then hold the mouth shut for 3 seconds.</i></blockquote><br /><div style="text-align: center;"><span style="font-size: large;"><a href="http://www.homeovet.net/content/treatment.html"> Jeffrey Levy, DVM PCH: <br />Classical Veterinary Homeopathy</a></span><br /><br /></div>http://www.madmath.com/2016/04/veterinary-homeopathy.htmlnoreply@blogger.com (Delta)0tag:blogger.com,1999:blog-7718462793516968883.post-5031451431867570170Mon, 28 Mar 2016 09:00:00 +00002016-03-28T05:00:02.884-04:00The MOOC as a First AlbumThinking about MOOCs (which I am semi-infamously down on as a method for revolutionizing general education): <br /><br />For rock bands, it's pretty common for their very first album to be considered their best one. Why is that? Well, the first album is likely the product of possibly a decade of practicing, writing, performing bars and clubs, interacting with audiences, and generally fine-tuning and refining the set to make the most solid block of music the band can possibly produce. At the point when a band gets signed to a label (traditionally), the first album is basically this ultra-tight set, honed for maximum impact over possibly hundreds of public performances. <br /><br />But thereafter, the band is no longer in the same "lean and hungry" mode that produced that first set of music. Likely they go on tour for a year to support the first album, then are put in a studio for a few weeks with the goal of writing and recording a second album, so that the sales/promotion/touring cycle can pick up where the last one left off. This isn't a situation the band's likely to have experienced before, they have weeks instead of years to create the body of music, and they don't have hundreds of club audiences to run it by as beta-testers. In fact, they probably won't ever again have the opportunity of years of dry runs going into the manufacture a single album.<br /><br />The same situation is likely to apply to MOOCs. A really good online class (and there are some) will be the product of a teacher who's taught the course live for a number of years (or decades), interacting with actual classrooms-full of students, refining the presentation many times as they witness how the presentation is immediately received by the people in front of them. If this has been done, say, <i>hundreds</i> of times, then you have a pretty strong foundation to begin recording something that will be a powerful class experience. <br /><br />But if someone tries to develop an online course from scratch, in a few weeks isolated in an office without any live interaction as a testbed -- exactly as the band studio album situation -- what you're going to get is weak sauce, possibly entirely usable crap. If the instructor has never taught such a class in the past, then the result is likely just "kabuki" as a teacher that I once live-observed confessed to me. This is regardless if a person can <i>do</i> the math themselves, that's just total BS as a starting point for teaching.<br /><br />A properly prepared, developed, scaffolded, explained course has got hundreds of moving parts built into it, built into every individual exercise, that are totally invisible unless an instructor has actually confronted live students with the issues at hand and seen the amazing kaleidoscope of ways that students can make mistakes or become tripped up or confused. No amount of "big data" is going to solve this (even assuming the MOOCs are even <i>trying</i> to do that and claims of such are not just flat-out fraud), because the tricky spots are so surprising, you'll never think to create a metric to measure it unless you're looking right over a student's shoulder to watch them do their work.<br /><br /><br />Quick metric for a quality online course: Has the instructor taught it live for a decade or more? Probably good. Did the instructor make it up on the fly, or in a few weeks development cycle? Probably BS.<br /><br /><br />http://www.madmath.com/2016/03/the-mooc-as-first-album.htmlnoreply@blogger.com (Delta)2tag:blogger.com,1999:blog-7718462793516968883.post-6409908258047855881Fri, 25 Mar 2016 09:00:00 +00002016-03-25T05:00:03.590-04:00GPS Always Overestimates DistancesResearchers in Austria and the Netherlands have pointed out that existing GPS applications almost always <i>overestimate</i> the distance of a trip, no matter where you're going. Why? Granted some small amount of random error in measuring each of the waypoints along a trip, the distance between erroneous points on a surface is overwhelmingly more likely to be <i>greater than</i>, rather than lesser than, the true distance -- -- and over many legs of a given trip, this error adds up to a rather notable overestimate. And to date no GPS application makers have corrected for it. A wonderful and fairly simple piece of math, one that was lurking under our noses for some time that no one thought to check, that should improve all of our navigation devices:<br /><br /><div style="text-align: center;"><span style="font-size: large;"><a href="http://www.i-programmer.info/news/145-mapping-a-gis/9164-gps-always-over-estimates-distances.html">I, Programmer: <br />GPS Always Overestimates Distance </a></span></div><div style="text-align: center;"><br /></div>http://www.madmath.com/2016/03/gps-always-overestimates-distances.htmlnoreply@blogger.com (Delta)2tag:blogger.com,1999:blog-7718462793516968883.post-1968827425252986994Mon, 21 Mar 2016 09:00:00 +00002016-03-21T05:00:12.689-04:00Excellent Exercises − Simplifying RadicalsExploration of exercise construction, i.e., casting a net to catch as many mistakes as possible: See also the previous <a href="http://www.madmath.com/2014/01/excellent-exercises-completing-square.html">"Excellent Exercises: Completing the Square"</a>. <br /><br />Below you'll see me updating my in-class exercises introducing simplification of radicals for my remedial algebra course a while back. (This occurred between one class on Monday, and different group on Tuesday, when I had the opportunity to spot and correct some shortcomings.) My customary process is to introduce a new concept, then give support for it (theorem-proof style), then do some exercises jointly with the class, and then have students do exercises themselves (from 1-3 depending on problem length) -- hopefully each cycle in a 30 minute block of time. In total, this snippet represents 1 hour of class time (actually the 2nd half of a 2-hour class session); the definitions and text shown is written verbatim on the board, while I'll be expanding or answering questions verbally. As I said before, I'm trying to bake as many iterations and tricky "stumbling blocks" into this limited exercise time as possible, so that I can catch and correct it for students as soon as we can. <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-J2JHIqx0z-4/UW7F-fmo9zI/AAAAAAAAB70/KFliyOIDdsE/s1600/RadicalNotes.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="103" src="https://4.bp.blogspot.com/-J2JHIqx0z-4/UW7F-fmo9zI/AAAAAAAAB70/KFliyOIDdsE/s320/RadicalNotes.jpg" width="320" /></a></div><br />Now, you can see in my cross-outs the simplifying exercises I was using at the start of the week, which had already gone through maybe two semesters of iteration. Obviously for each triad (instructor a, b, c versus students' d, e, f) I start small and present sequentially larger values. Also, for the third of the group I throw in a fraction (division) to demonstrate the similarity in how it distributes.<br /><br />Not bad, but here some weaknesses I spotted this session that aren't immediately apparent from the raw exercises, and these are: (1) There are quite a few duplicates between the (now crossed-out) simplifying and later add/subtract exercises, which reduces real practice opportunities in this hour. (2) Is that I'm not happy about starting off with √8, which reduces to 2√2 -- this might cause confusion in a discussion for some students who don't see where the different "2"'s come from, something I try to avoid for initial examples. (3) Is that student exercises (c) and (d) both involve factoring out the perfect square "4", when I should have them getting experience with a wider array of possible factoring values. (4) Is that item (f) is √32, which raises the possibility of again factoring out either 4 or 16 -- but none of the instructor exercises demonstrated the need to look for the "greatest" perfect square, so the students weren't fairly prepped for that case.<br /><br />Okay, so at this point I realized that I had at least 4 things to fix in this slice of class, and so I was committed to rewriting the entirety of both blocks of exercises (ultimately you can see the revisions handwritten in pencil on my notes). The problem is that simplifying-radical problems are actually among the harder problems to construct, because there's a fairly limited range of values which are the product of a perfect square for which my students will be able to actually revert the multiplication (keeping in mind a significant subset of my college students who don't actually know their times tables, and so are having to make guesses and sequentially add on their fingers a bunch of times before they can get started). <br /><br />So at this point I sat down and made a comprehensive list of all the smallest perfect square products that I could possible ask for in-class exercises. I made the decision to use each one at most a single time, to get as much distinct practice in as possible. First, of course, I had to synch up like remainders to make like terms in the four "add/subtract" exercises -- these are indicated below by square boxes linking the like terms for those exercises. Then I circled another 6 examples, for use in the lead-in "simplifying" exercises, trying to find the greatest variety of perfect squares possible, not sequentially duplicating the same twice, and making sure that I had multiple of the "greatest perfect square" (i.e., involving 16 or 36) issue in both the instructor and student exercises. These, then, became my revised exercises for the two half-hour blocks, and I do think they worked noticeably better when I used them with the Tuesday class this week.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-ML5rR8kQJe8/UW7GAIL4dpI/AAAAAAAAB78/Hnic8RDy_Dg/s1600/RadicalPermutations.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="266" src="https://2.bp.blogspot.com/-ML5rR8kQJe8/UW7GAIL4dpI/AAAAAAAAB78/Hnic8RDy_Dg/s320/RadicalPermutations.jpg" width="320" /></a></div><br />Some other stuff: The fact that add/subtract exercise (c) came out to √5 was kind of a happy accident -- I didn't plan on it, but I'm happy to have students re-encounter the fact that they shouldn't write a coefficient of "1" (many will forget that, and you need to have built-in reviews over and over again). Also, one might argue that I should have an addition exercise where you <i>don't</i> get like terms to make sure they're looking for that, but my judgement was that in our limited time I wanted them doing the whole process as much as possible (I'll leave non-like terms cases for book homework exercises). <br /><br />Anyway, that's a little example of the many of issues involved, and the care and consideration, that it takes to construct really quality exercises for even (or especially) the most basic math class. Like I said, I think this is about the third iteration of these exercises for me in the last year -- we'll see if I catch any more obscure problems the next time I teach the class.<br /><br />http://www.madmath.com/2016/03/excellent-exercises-simplifying-radicals.htmlnoreply@blogger.com (Delta)0tag:blogger.com,1999:blog-7718462793516968883.post-7350318606131277264Fri, 18 Mar 2016 09:00:00 +00002016-03-18T05:00:14.751-04:00Nate Silver: Wrong on VAMI like Nate Silver's <a href="http://fivethirtyeight.com/">FiveThiryEight</a> site very much, and I think that its political coverage is very insightful. However, I could do without a lot of the site's pop-culture, sports, etc. filler. Another thing that they're wildly off-base about: they seem to be highly pro-VAM -- that's the Value-Added Metric, the reputed way of assessing teachers by student test scores -- which has been roundly shown to be a disastrously wrong (effectively random) metric in any serious study that I've seen, but about which 538 is super-supportive (for reasons that seem incoherent to me). Here's a very good outline of the critique:<br /><br /><div style="text-align: center;"><span style="font-size: large;"><a href="https://www.washingtonpost.com/news/answer-sheet/wp/2015/07/28/why-nate-silvers-fivethirtyeight-blog-is-wrong-about-teacher-evaluation/">Washington Post: Why Nate Silver’s FiveThirtyEight blog is wrong about teacher evaluation</a></span></div>http://www.madmath.com/2016/03/nate-silver-wrong-on-vam.htmlnoreply@blogger.com (Delta)1tag:blogger.com,1999:blog-7718462793516968883.post-6934182894441587235Mon, 14 Mar 2016 09:00:00 +00002016-03-14T05:00:17.531-04:00Where Are the Bodies Buried?In my job which involves teaching lots of remedial classes at a community college (in CUNY), the students frequently have deep, yawning gaps in their basic math education. Many can't write clearly, they interchange digits and symbols, they don't know their multiplication tables, they can't long divide, they have trouble reading English sentence-puzzles, they've been taught bum-fungled "PEMDAS" mnemonics, they've been told that π = 22/7, etc., etc., etc.<br /><br />So to a large part my job is to ask the question, police-detective style: "Where are the bodies buried?" For this particular crime that's been perpetrated on my students' brains, what exactly is causing the problem, what is the <i>worst</i> thing we can find about their conceptual understanding? Doing the easy introductory problems that immediately come to mind doesn't do dick. What I need to do, in our limited class time, is to dredge the the murky riverbed and pull out all the crap, broken, tricky, misunderstandings that are buried down there.<br /><br />Another way of putting this is that the in-class exercises we use have to cast a wide net, and be constructed to not just do a single thing, but to demonstrate at least 2, 3, or 4 issues at once. (Again, if you had unlimited time and attention span to do hundreds of problems, this might be an issue, but have to maximize our punch in the class session.) I'm constantly revising my in-class exercises semester after semester as I realize some tricky detail that was pitched at my students along the way. I need to make sure that every tricky corner-case detail gets put in front of students so, if its a problem, they can run into it and I get a chance to help them while we have time together.<br /><br />This is a place where the poorly-made MOOCs and online basic math classes (like Khan Academy) really do a laughably atrocious job. Generally if you're a science-oriented person who can <i>do</i> math easily, and never taught live, then you're not aware of all the dozens of pitfalls that people can possibly run into during otherwise basic math procedures. So if someone like that just throws out the first math problem they can think of, it's going to be a trivial case that doesn't serve to dredge up all bodies lurking around the periphery. And you'll never know it through any digital feedback, and you'll never get a chance to improve the situation, because you're simply not measuring performance on the tricky side-issues in the first place; it remains hidden and forever under-the-surface. <br /><br />I'll plan to present some examples of exercise design and refinement in the future. For the moment, consider this article with other educators make the same critical observation about how bad the exercises at Khan Academy (and other poorly-thought-out MOOCs) are:<br /><br /><div style="text-align: center;"><span style="font-size: large;"><a href="https://www.washingtonpost.com/blogs/answer-sheet/post/how-well-does-khan-academy-teach/2012/07/27/gJQA9bWEAX_blog.html">Washington Post: Does the Khan Academy know how to teach?</a></span></div><br /><br /><br /><br /><br /><br />http://www.madmath.com/2016/03/where-are-bodies-buried.htmlnoreply@blogger.com (Delta)0tag:blogger.com,1999:blog-7718462793516968883.post-1311997247432718155Fri, 11 Mar 2016 10:00:00 +00002016-03-13T19:29:35.377-04:00Graphing Quizzes at Automatic-AlgebraI added a few new things to the "automatic skill" site, <a href="http://automatic-algebra.org/">Automatic-Algebra.org</a> (actually around the start of the year, but they seem to have tested out well enough at this point). In particular, these are timed quizzes on the basic of graphing lines: (1) on linear equations in slope-intercept format, and (2) on parsing descriptions of special horizontal and vertical lines. <br /><br />As usual, these are skills that when walks through them the first time in-class, with full explanations, may take several minutes; which may give a mistaken impression about how complicated the concepts really are. In truth, in a later course (precalculus, calculus, statistics), a person should be expected to see these relationships pretty much <i>instantaneously on sight</i>, and these timed quizzes better communicate that and allow the student to practice developing that intuition. If you have any feedback as you or your students use the site, I'd love to hear it!<br /><ul><li><span style="font-size: large;"><a href="http://www.automatic-algebra.org/graphinglines.htm">Automatic-Algebra: Graphing Lines</a></span></li><li><span style="font-size: large;"><a href="http://www.automatic-algebra.org/speciallines.htm">Automatic-Algebra: Special Lines</a></span></li></ul><br />http://www.madmath.com/2016/03/graphing-quizzes-at-automatic-algebra.htmlnoreply@blogger.com (Delta)0tag:blogger.com,1999:blog-7718462793516968883.post-1867532817713537626Tue, 08 Mar 2016 04:06:00 +00002016-03-07T23:06:12.467-05:00On Correlation And Other Musical MantrasA while back I found this delightful article at Slate.com, titled <a href="http://www.slate.com/articles/health_and_science/science/2012/10/correlation_does_not_imply_causation_how_the_internet_fell_in_love_with_a_stats_class_clich_.html">"The Internet Blowhard's Favorite Phrase"</a>. Perhaps more descriptive is the web-header title: "Correlation does not imply causation: How the Internet fell in love with a stats-class cliché". The article leads with a random internet argument, and then observes:<br /><blockquote class="tr_bq"><i>And thus a deeper correlation was revealed, a link more telling than any that the Missouri team had shown. I mean the affinity between the online commenter and his favorite phrase—the statistical cliché that closes threads and ends debates, the freshman platitude turned final shutdown. "Repeat after me," a poster types into his window, and then he sighs, and then he types out his sigh, s-i-g-h, into the comment for good measure. Does he have to write it on the blackboard? Correlation does not imply causation. Your hype is busted. Your study debunked. End of conversation. Thank you and good night... The correlation phrase has become so common and so irritating that a minor backlash has now ensued against the rhetoric if not the concept.</i></blockquote><br />I find this to be completely true. Similarly, for some time, Daniel Dvorkin, the science fiction author, has used the following as the signature to all of his posts on Slashdot.org, which I find to be a wonderfully concise phrasing of the issue:<br /><blockquote class="tr_bq"><i>The correlation between ignorance of statistics and using "correlation is not causation" as an argument is close to 1.</i></blockquote><br />Now, near the end of his article, the writer at Slate (Daniel Engberg), poses the following question:<br /><blockquote class="tr_bq"><i>It's easy to imagine how this point might be infused into the wisdom of the Web: "Facepalm. How many times do I have to remind you? Don't confuse statistical and substantive significance!" That comment-ready slogan would be just as much a conversation-stopper as correlation does not imply causation, yet people rarely say it. The spurious correlation stands apart from all the other foibles of statistics. It's the only one that's gone mainstream. Why?<br /><br />I wonder if it has to do with what the foible represents. When we mistake correlation for causation, we find a cause that isn't there. Once upon a time, perhaps, these sorts of errors—false positives—were not so bad at all. If you ate a berry and got sick, you'd have been wise to imbue your data with some meaning... Now conditions are reversed. We're the bullies over nature and less afraid of poison berries. When we make a claim about causation, it's not so we can hide out from the world but so we can intervene in it... The false positive is now more onerous than it's ever been. And all we have to fight it is a catchphrase.</i></blockquote><br />On this particular explanation of the phenomenon, I'm going to say "I don't think so". I don't think that people uttering the phrase by rote are being quite so thoughtful or deep-minded. My hypothesis for what's happening: The phrase just happens to have a certain poetical-musical quality to it that makes it memorable, and sticks in people's mind (moreso than other important dictums from statistics, as Engberg points out above). The starting "correlation" and the ending "causation" have this magical consonance in the hard "c", they both rhyme, they both have emphasis on the long "a" syllable, and the whole fits perfectly into a 4-beat measure. (A happy little accident, as Bob Ross might say.) It's this musical quality that gets it stuck in people's mind, possibly the very first thing that comes to mind for many people regarding statistics and correlation, ready to be thrown down in any argument whether on-topic or not.<br /><br />I've run into the same thing by accident, for other topics, in my own teaching. For example: A year ago in my basic algebra classes I would run a couple examples of graphing 2-variable equations by plotting points, and at the end of the class make a big show of writing this inference on the board: "Lesson: All linear equations have straight-line graphs" -- and noted how this explained why equations of that type were in fact called "linear" (defined earlier in the course). This was received extremely well, and it was very memorable -- it was one of the few side questions I could always ask ("how do you know this equation has a straight-line graph?") that nobody ever failed to answer ("because it's linear"). <br /><br />Well, the problem is that it was actually TOO memorable -- people remembered this mantra without actually understanding what "linear" actually meant (of course: 1st-degree, with no visible exponents). I would always have to follow up with, "and what does linear mean?", to which almost no one could provide an answer. So in the fall semester, I took great care to instead write in my trio of algebra classes, "Lesson: All 1st-degree equations have straight-line graphs", and then verbally make the same point about where "linear" equations get there name. The funny thing is -- students would STILL make this same mistake of saying "linear equations are straight lines" without actually knowing how to identify a linear equation. It's such an attractive, musical, satisfying phrase that it's like a mental strange attractor -- it burrows into people's brains <i>even when I never actually said it or wrote it in the class</i>. <br /><br />So I think we actually have to watch out for these "musical mantras" which are indeed TOO memorable, and allow students to regurgitate them easily and fool us into thinking they understand a concept when they actually don't.<br /><br />See also -- <a href="http://deltasdnd.blogspot.com/2011/09/power-of-pictures.html">Delta's D&D Hotspot: The Power of Pictures.</a><br /><br /><br />http://www.madmath.com/2016/03/on-correlation-and-other-musical-mantras.htmlnoreply@blogger.com (Delta)0tag:blogger.com,1999:blog-7718462793516968883.post-7260663485320690392Thu, 03 Mar 2016 10:00:00 +00002016-03-03T11:09:28.180-05:00Lower Standards Are a Conspiracy Against the PoorAndrew Hacker's at it again. Professor emeritus of political science from Queens College in CUNY, frequent contributor to the New York Times -- they love him for the "Man Bites Dog" headlines they can push due to him being the college-professor-who's-against-math. He got a lot of traction for the 2012 op-ed, <a href="http://www.nytimes.com/2012/07/29/opinion/sunday/is-algebra-necessary.html">Is Algebra Necessary</a>? And he has a new book coming out now -- so, more articles on the same subject, like <a href="http://www.nytimes.com/2016/02/28/opinion/sunday/the-wrong-way-to-teach-math.html">The Wrong Way to Teach Math</a>, and <a href="http://www.nytimes.com/2016/02/07/education/edlife/who-needs-advanced-math-not-everybody.html">Who Needs Advanced Math</a>, and<a href="http://chronicle.com/article/The-Case-Against-Mandating/235500"> The Case Against Mandating Math for Students</a>, and more. (I wrote previously about how Hacker's critique is essentially incoherent <a href="http://www.madmath.com/2012/07/algebra-in-ny-times.html">here</a>.)<br /><br />Now, his suggestions for what "everyone needs to know" are not bad; e.g., how to read a table or graph, understand decimals and estimations... (maybe that's it, actually?). I totally agree that everyone should know that -- at, say, the level of a 7th or 8th-grade home-economics course, perhaps. To suggest that this is proper fare for college instruction would be comically outrageous -- if it weren't seriously being considered by top-level administrators at CUNY. Here are some choice things he's said recently in the articles linked above:<br /><ul><li><div class="story-body-text story-content" data-para-count="153" data-total-count="1478" itemprop="articleBody"><i>"I sat in on several advanced placement classes, in Michigan and New York. I thought they would focus on what could be called 'citizen statistics.'... My expectations were wholly misplaced. The A.P. syllabus is practically a research seminar for dissertation candidates. Some typical assignments: binomial random variables, least-square regression lines, pooled sample standard errors..."</i> -- I'd say that these concepts are so incredibly basic, the very <i>idea</i> of regression and correlation so fundamental, for example, that you couldn't even call it a statistics class without those topics.<i></i></div></li><br /><li><i>"Q: Aren’t algebra and geometry essential skills? A: The number of people who use either in their jobs is tiny, at most 5 percent. You don’t need that kind of math for coding. It’s not a building block."</i> -- The idea that algebra concepts aren't necessary for coding, that someone who doesn't grasp the <i>idea of a variable</i> wouldn't be entirely helpless at coding (I've seen it!), in my personal opinion, essentially qualifies as fraud. </li></ul><br />Okay, so statistics and coding are clearly not Hacker's area of expertise -- we might wonder why he feels confident in pontificating in these areas, and recommending truly radical reductions in standards, at all. Many of us would opine that the social-science departments have much weaker standards than the STEM fields; so perhaps we might generously say it's just a skewed perspective in this regard. <br /><br />But the thing is, behind closed doors administrators <i>know</i> that students without math skills can't succeed at further education, and they can't succeed at technical jobs. That said, they are not incited to communicate that fact to anyone. What that they are grilled about by the media and political stakeholders are graduation rates, which at CUNY are pretty meager; around 20% for most of the community colleges. If the administration could wipe out 7th-grade math as a required expectation, then they'd be celebrated (they think) for being able to double graduation rates effectively overnight. And someone like Hacker is almost invaluable in giving them political cover for such a move. <br /><br />Let's look at some recent evidence for who <i>really</i> benefits when math standards are reduced.<br /><ul><li><i>"My first time in a fifth grade in one of New Jersey’s most affluent districts (white, of course), I asked where one-third was on the number line. After a moment of quiet, the teacher called out, “Near three, isn’t it?” The children, however, soon figured out the correct answer; they came from homes where such things were discussed. Flitting back and forth from the richest to the poorest districts in the state convinced me that the mathematical knowledge of the teachers was pathetic in both. It appears that the higher scores in the affluent districts are not due to superior teaching in school but to the supplementary informal “home schooling” of children." </i>-- Patricia Clark Kenschaft, "Racial equity requires teaching elementary school teachers more mathematics", <i>Notices of AMS</i> 52.2 (2005): 208-212.</li><br /><li>"<i>And while the proportion of American students scoring at advanced levels in math is rising, those gains are almost entirely limited to the children of the highly educated, and largely exclude the children of the poor. By the end of high school, the percentage of low-income advanced-math learners rounds to zero..." </i>-- Peg Tyre, 'The Math Revolution", <i>The Atlantic </i>(March 2016). </li></ul><br />That is: <i>Cutting math standards only really cuts it for the poor.</i> The rich will still make sure that their children have solid math skills at all levels. Or in other words: Cutting math standards <i>increases inequality in education</i>, and thus later economic status. And this folds into the overwhelming number of signs we've seen that math knowledge among our elementary-school teachers is perennially, pitifully weak, and a major cause of ongoing math deficiencies among our fellow citizens. <br /><br />I wonder: Is there any correlation between this and the crazy election cycle that we're experiencing now? Thanks to a close friends for the idea for the title to this article. <br /><br /><br />P.S. Here's Ed from the wonderful Gin and Tacos <a href="http://www.ginandtacos.com/2016/03/03/a-very-stupid-argument-gets-the-fjm-treatment/">writing on the same subject today</a>. I agree with every word, and he goes into more detail than I did here (frankly, Hacker's crap makes me so angry I can't read every part of what he says). Ed's a political science professor himself, and also plays drums, which makes me feel a bit bad that I threw any shade at all on the social sciences above. Be smart, be like Ed. <br /><br /><br />http://www.madmath.com/2016/03/lower-standards-are-conspiracy-against.htmlnoreply@blogger.com (Delta)0tag:blogger.com,1999:blog-7718462793516968883.post-3161445154409955773Fri, 26 Feb 2016 10:00:00 +00002016-02-26T05:00:29.176-05:00Link: Math Circles at the AtlanticAn article this month at the Atlantic on the explosive rise of extracurricular (and expensive) advanced-math circles and competitions, to make up for the perceived deficiencies in math education in schools. Some telling quotes:<br /><blockquote class="tr_bq"><i>At a time when calls for a kind of academic disarmament have begun echoing through affluent communities around the nation, a faction of students are moving in exactly the opposite direction...</i><br /><br /><i>"The youngest ones, very naturally, their minds see math differently [said Inessa Rifkin, co-founder of Russian School of Mathematics]... It is common that they can ask simple questions and then, in the next minute, a very complicated one. But if the teacher doesn’t know enough mathematics, she will answer the simple question and shut down the other, more difficult one. We want children to ask difficult questions, to engage so it is not boring, <b>to be able to do algebra at an early age</b>, sure, but also to see it for what it is: a tool for critical thinking. <b>If their teachers can’t help them do this, well... It is a betrayal.</b>”</i><br /><br /><i>And while the proportion of American students scoring at advanced levels in math is rising, those gains are almost entirely limited to the children of the highly educated, and largely exclude the children of the poor. By the end of high school, the percentage of low-income advanced-math learners rounds to zero...</i><br /><br /><i>The No Child Left Behind Act... demanded that states turn their attention to getting struggling learners to perform adequately...The cumulative effect of these actions, perversely, has been to push accelerated learning outside public schools—to privatize it, focusing it even more tightly on children whose parents have the money and wherewithal to take advantage. In no subject is that clearer today than in math.</i></blockquote><br /><div style="text-align: center;"><span style="font-size: large;"><a href="http://www.theatlantic.com/magazine/archive/2016/03/the-math-revolution/426855/">The Atlantic: The Math Revolution</a></span></div><br /><br />http://www.madmath.com/2016/02/link-math-circles-at-atlantic.htmlnoreply@blogger.com (Delta)0tag:blogger.com,1999:blog-7718462793516968883.post-7986922850851974687Fri, 19 Feb 2016 10:00:00 +00002016-02-19T05:00:12.956-05:00Link: Common Core BattlesA nice overview of the history of the battles around Common Core. Starts with a surprising anecdote about Bill Gates getting the brush-off when he personally met with Charles Koch to discuss the issue. Re: George W. Bush's "No Child Left Behind", an aggravatingly familiar development:<br /><blockquote class="tr_bq"><i>To bring themselves closer to 100%, many states simply lowered the score needed to pass their tests. The result: In 2007, Mississippi judged 90% of its fourth graders “proficient” on the state’s reading test, yet only 19% measured up on a standardized national exam given every two years. In Georgia, 82% of eighth-graders met the state’s minimums in math, while just 25% passed the national test. A yawning “honesty gap,” as it came to be known, prevailed in most states. </i></blockquote><br /><div style="text-align: center;"><span style="font-size: large;"><a href="http://fortune.com/common-core-standards/">Peter Elkind: Business Gets Schooled</a></span></div><br />http://www.madmath.com/2016/02/link-common-core-battles.htmlnoreply@blogger.com (Delta)0tag:blogger.com,1999:blog-7718462793516968883.post-7797919422894187353Mon, 15 Feb 2016 10:00:00 +00002016-02-17T12:08:58.486-05:00Hembree on Math AnxietyReviewing a 1990 paper by Ray Hembree on math anxiety; a <b>meta-study of approximately 150 papers, with a combined total of about 25,000 subjects</b>. (Note the high sample size makes almost all findings significant at the p < 0.01 level). Math anxiety is known to be negatively correlated with performance in math (tests, etc.), and more common among women than men. <br /><br />Math anxiety is somewhat correlated with a constellation of other general anxieties (r² = 0.12 to 0.27). Work to enhance math competence did not reduce anxiety. <br /><br /><b>Whole-group interventions are not effective</b> (curricular changes, classroom pedagogy structure, in-class psychological treatments). The <b>only thing that is effective is out-of-classroom, one-on-one treatments</b> (behavioral systematic desensitization; cognitive restructuring); these have a marked effect at both lowering anxiety and boosting actual math-test performance. <br /><br />In short: <b>Addressing math anxiety is largely out of the hands of the classroom teacher</b>. Unless the student has access, or the institution provides access, to one-on-one behavioral desensitization therapy, no group-level interventions are found to be effective. <br /><br />Also recall that<b> elementary education majors have the highest math anxiety</b>, and the lowest math performance, of all U.S. college majors. (It seems possible that some entrants choose elementary education as a career path precisely <i>because</i> they are bad at math and see that as one of their limited options; I know I've had at least one such student say something to that effect to me.) This clearly dovetails with Sian Beilock's 2009 finding that <b>math-anxious female elementary teachers model math-anxiety particularly to their female students</b>, who imitate the same and wind up with <b>worse math performance and attitudes by the end of the year</b> (<a href="http://www.pnas.org/content/107/5/1860.short">link</a>). And this general trend of weak education majors has <b>been the case in the U.S. for at least a century now</b> (<a href="http://qz.com/334926/your-college-major-is-a-pretty-good-indication-of-how-smart-you-are/">link</a>).<br /><br />So we might hypothesize: <b>A feedback loop exists between poor early math education, heightened math anxiety among female students, and those same students returning to early childhood education as a career. </b><br /><br />See below for Hembree's table of math anxiety by class and major (p. 41); note that <b>elementary education majors</b>, and those taking the standard "math for elementary teachers" (frequently the <i>only</i> math class such teachers take), are <b>significantly worse off than anyone else</b>:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-DSlG0boMcVk/VrmQADVhRNI/AAAAAAAAD4Q/HwiuM97N4IU/s1600/MajorMathAnxiety.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="146" src="https://1.bp.blogspot.com/-DSlG0boMcVk/VrmQADVhRNI/AAAAAAAAD4Q/HwiuM97N4IU/s320/MajorMathAnxiety.png" width="320" /></a></div><br /><div class="gs_citr" id="gs_cit0" tabindex="0">Hembree, Ray. "The nature, effects, and relief of mathematics anxiety." <i>Journal for research in mathematics education</i> (1990): 33-46. (<a href="https://www.blogger.com/Hembree,%20Ray.%20%22The%20nature,%20effects,%20and%20relief%20of%20mathematics%20anxiety.%22%20Journal%20for%20research%20in%20mathematics%20education%20(1990):%2033-46.">Link</a>)</div><br /><br />http://www.madmath.com/2016/02/hembree-on-math-anxiety.htmlnoreply@blogger.com (Delta)0tag:blogger.com,1999:blog-7718462793516968883.post-1841623879344396743Fri, 12 Feb 2016 10:00:00 +00002016-02-12T05:00:22.646-05:00Link: The Learning Styles NeuromythA nice article reminding us that the whole idea of teaching to different "learning styles" is entirely without any scientific evidence in its favor:<br /><br /><blockquote class="tr_bq"><span style="font-size: large;"><i>“... the brain’s interconnectivity makes such an assumption unsound.”</i></span></blockquote><br /><div style="text-align: center;"><span style="font-size: large;"><a href="http://qz.com/585143/the-concept-of-different-learning-styles-is-one-of-the-greatest-neuroscience-myths/">Olivia Goldhill: The concept of different “learning styles” is one of the greatest neuroscience myths</a></span></div><br />http://www.madmath.com/2016/02/link-learning-styles-neuromyth.htmlnoreply@blogger.com (Delta)2tag:blogger.com,1999:blog-7718462793516968883.post-3344704233433871706Fri, 05 Feb 2016 10:00:00 +00002016-02-05T05:00:07.025-05:00Link: Study Time DeclineAn interesting article analyzing the history of reported study time decline for U.S. college students. <br /><ul><li>Point 1: Study time dramatically decreased in the 1961-1981 era (from about 24 hrs/week to 16 hrs/week), but has been close to stable since that time. </li><br /><li>Point 2: In that same early period, it seems that faculty expectations on teaching vs. research flip-flopped in that same early time period (about 70% prioritized teaching over research around 1975, with the proportion quickly dropping to about 50/50 by the mid-80's). </li></ul><div style="text-align: center;"><br /></div><div style="text-align: center;"><span style="font-size: large;"><a href="https://www.aacu.org/publications-research/periodicals/its-about-time-what-make-reported-declines-how-much-college">Alexander McCormick: It's about Time: What to Make of Reported Declines in How Much College Students Study</a></span></div><br />http://www.madmath.com/2016/02/link-study-time-decline.htmlnoreply@blogger.com (Delta)0tag:blogger.com,1999:blog-7718462793516968883.post-1074042593855576900Mon, 01 Feb 2016 10:00:00 +00002016-02-02T11:20:27.490-05:00When Dice FailSome of the more popular posts on my gaming blog have been about how to check for balanced dice, using Pearson's chi-square test (<a href="http://deltasdnd.blogspot.com/2009/02/testing-balanced-die.html">testing a balanced die</a>, <a href="http://deltasdnd.blogspot.com/2009/02/follow-up-testing-balanced-dice.html">testing balanced dice</a>, <a href="http://v/">testing balanced dice power</a>). One of the observations in the last blog was that "chi-square is a test of rather lower power" (quoting Richard Lowry of Vassar College); to the extent that I've never had any dice that I've checked actually <i>fail</i> the test. <br /><br />Until now. Here's the situation: A while back my partner Isabelle, preparing entertainment for a long trip, picked up a box of cheap dice at the dollar store around the corner from us. These dice are in the <a href="https://en.wikipedia.org/wiki/Dice#Arrangement">Asian-style arrangement</a>, with the "1" and "4" pip sides colored red (I believe because the lucky color red is meant to offset those unlucky numbers):<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-PDZTYZbdntY/Vqhw65Hqk3I/AAAAAAAAD2I/PC1x6prHZUQ/s1600/Image01252016192951.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://2.bp.blogspot.com/-PDZTYZbdntY/Vqhw65Hqk3I/AAAAAAAAD2I/PC1x6prHZUQ/s320/Image01252016192951.jpg" width="240" /></a></div><br /><br />A few weeks ago, it occurred to me that these dice are just the right size for an experiment I run early in the semester with my statistics students: namely, rolling a moderately large number of dice in handful batches and comparing convergence to the theoretically-predicted proportion of successes. In particular, the plan is customarily to roll 80 dice and see how many times we get a 5 or 6 (mentally, I'm thinking in my <i>Book of War</i> game, how many times can we score hits against opponents in medium armor -- but I don't say that in class). <br /><br />So when we did that in class last week, it seemed like the number of 5's and 6's was significantly lower than predicted, to the extent that it actually threw the whole lesson under a shadow of suspicion and confusion. I decided that when I got a chance I'd better test these dice before using them in class again. Following the findings of the prior blog on the "low power" issue, I knew that I had to get on the order of about 500 individual die-rolls in order to get a halfway decent test; in this case with a boxful of 15 dice, it seemed was convenient to make 30 batched rolls for 15 × 30 = 450 total die rolls... although somewhere along the way I lost count of the batches and wound up actually making 480 die rolls. Here are the results of my hand-tally sheet:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-ATTC7-dY5Fw/VqhzX2ph1aI/AAAAAAAAD2U/HZ29BTX4S8k/s1600/Image01252016193017.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://3.bp.blogspot.com/-ATTC7-dY5Fw/VqhzX2ph1aI/AAAAAAAAD2U/HZ29BTX4S8k/s320/Image01252016193017.jpg" width="240" /></a></div><br />As you can see at the bottom of that sheet, this box of dice actually does fail the chi-square test, as the \(SSE = 1112\) is in fact greater than the critical value of \(X \cdot E = 11.070 \cdot 80 = 885.6\).Or in other words, with a chi-square value of \(X^2 = SSE/E = 1112/80 = 13.9\) and degrees of freedom \(df = 5\), we get a P-value of \(P = 0.016\) for this hypothesis test of the dice being unbalanced; that is, if the dice really were balanced, there would be less than a 2% chance of getting an SSE value this high by natural variation alone. <br /><br />In retrospect, it's easy to see what the manufacturing problem is here: note in the frequency table that it's specifically the "1"'s and the "4"'s, the specially red-colored faces, that are appearing in a preponderance of the rolls. In particular, the "1" face on each die is drilled like an enormous crater compared to the other pips; it's about 3 mm wide and about 2 mm deep (whereas other pips are only about 1 mm in both dimensions). So the "6" on the other side from the "1" would be top heavy, and tends to roll down to the bottom, leaving the "1" on top more than anything else. Also, the corners of the die are very rounded, making it easier for them to turn over freely or even get spinning by accident. <br /><br />Perhaps if the experiment in class had been to count 4's, 5's, and 6's (that is: hits against light armor in my wargame), I never would have noticed the dice being unbalanced (because together those faces have about the same weight as the 1's, 2's, and 3's together)? On the one hand my inclination is to throw these dice out so they never get used again in our house by accident; but on the other hand maybe I should keep them around as the only example that the chi-square test has managed to succeed at <i>rejecting</i> to date. <br /><br /><br />http://www.madmath.com/2016/02/when-dice-fail.htmlnoreply@blogger.com (Delta)4tag:blogger.com,1999:blog-7718462793516968883.post-8921601902083059008Sat, 30 Jan 2016 10:00:00 +00002016-01-30T05:00:24.225-05:00Link: Tricky Rational ExponentsConsider the following apparent paradox:<br /><br />\(-1 = (-1)^1 = (-1)^\frac{2}{2} = (-1)^{2 \cdot \frac{1}{2}} = ((-1)^2)^\frac{1}{2} = (1)^\frac{1}{2} = \sqrt{1} = 1\)<br /><br />Of the seven equalities in this statement, <i>exactly which of them are false</i>? Give a specific number between (1) and (7). Join in the discussion where I posted this at StackExachange, if you like:<br /><br /><div style="text-align: center;"><span style="font-size: large;"><a href="http://math.stackexchange.com/questions/1628759/what-are-the-laws-of-rational-exponents">StackExchange: <br />What are the Laws of Rational Exponents?</a></span></div><br /><br />http://www.madmath.com/2016/01/link-tricky-rational-exponents.htmlnoreply@blogger.com (Delta)0tag:blogger.com,1999:blog-7718462793516968883.post-7643837546373635140Thu, 28 Jan 2016 15:18:00 +00002016-01-28T10:22:26.860-05:00Seat Belt EnforcementYesterday in the Washington Post, libertarian police-abuse crusader Radley Balko wrote an opinion piece<a href="https://www.washingtonpost.com/news/the-watch/wp/2016/01/27/profiling-oppressive-fines-and-creating-opportunities-for-escalation-the-case-against-mandatory-seat-belt-laws/"> arguing against mandatory seat-belt laws</a>. He opens:<br /><blockquote class="tr_bq"><i>The ACLU of Florida just released a report showing that in 2014, black motorists in the state were pulled over for seat belt violations at about twice the rate of white motorists... Differences in seat belt use don’t explain the disparity. Blacks in Florida are only slightly less likely to wear seat belts. The ACLU points to a 2014 study by the Florida Department of Transportation that found that 85.8 percent of blacks were observed to be wearing seat belts vs. 91.5 percent of whites. The only possible explanation for the disparity that doesn’t involve racial bias might be that it’s easier to spot seat-belt violations in urban areas than in more rural parts of the state... even if it did explain part or all of the disparity, it still means that blacks in Florida are disproportionately targeted.</i></blockquote><br />Here's the problem with that math: the Florida study would in fact be evidence that blacks are failing to wear seat belts at about twice the rate of whites. According to those numbers, the rate of blacks not wearing seat belts would be: 100% - 85.8% = 14.2%, while the rate of non-compliance for whites would be 100% - 91.5% = 8.5%. And as a ratio, 14.2%/8.5% = 1.67, or pretty close to 2 (double) if you round to the nearest multiple.<br /><br />Now, I actually think that Radley Balko has done some of the very best, most dedicated work drawing our attention to the problem of hyper-militarized police tactics in recent years (and decades); see his book <i>Rise of the Warrior Cop</i> for more. And I'm pretty sensitive to issues of overly-punitive enforcement and fines that are repressively punishing to the poor; back in the 80's I used to routinely listen to Jerry Williams on WRKO radio in Boston, when he was crusading against the rise of seat-belt laws, and I found those arguments, at times, compelling.<br /><br />But sometimes Balko gets his arguments scrambled up, and this is one of those cases. His claim that there's only one "possible explanation for the disparity" fails on the grounds that these numbers are evidence that, in general, there's actually no disparity in enforcement at all; enforcement tracks the non-compliance ratio very closely. He can do better than to hang his hat in this case on a fundamental misunderstanding of the numbers involved.<br /><br /><br />http://www.madmath.com/2016/01/seat-belt-enforcement.htmlnoreply@blogger.com (Delta)0tag:blogger.com,1999:blog-7718462793516968883.post-3374486097809043358Mon, 25 Jan 2016 10:00:00 +00002016-01-25T12:01:34.379-05:00Grading on a ContinuumAnecdote: I had a social-sciences teacher in high school who didn't understand that real numbers are a continuum. <br /><br />On the first day of class, he tried to present how grades would be computed at the end of the course. So on the board he wrote something like: D 60-69, C 70-79, B 80-89, A 90-100% (relating final letter grade to weighted total in the course).<br /><br />Then he looked at it and said, "Oh, wait, that's not right, what if a student gets 89.5%?". So he starting erasing and adjusting the cutoff scores so it looked something like: D 60-69.5, C 69.6-79.5, B 79.6-89.5, A 89.6-100%.<br /><br />And of course then he went, "But, no, what if a student gets 89.59%?", and started erasing and adjusting <i>again</i> to generate something like: D 60-69.55, C 69.56-79.55, B 79.56-89.55, A 89.56-100. And then of course noticed that there were still gaps between the intervals and went at it for a few more cycles.<br /><br />I think he took about 10 minutes or more of the first class iterating on this (before he gave up he'd gotten to maybe 4 places after the decimal point). I remember myself and a bunch of other students just looking back and forth at each other, slack-jawed from astonishment. It raises a couple of questions: Did he not know that <i>real numbers are dense</i>? And had he never thought through his grading schema <i>until this very moment? </i><br /><br />I always think about this on the first day in my statistics courses, when we are careful (following Weiss book notation) to define our grouped classes for continuous data using a special symbol "-<", meaning "up to but less than" (e.g., B 80 -< 90, A 90 -< 100, so that any score less than 90 would be unambiguously not in the A category, leaving no gaps). As I present this, my social-science teacher embarrassing himself is always at the back of my mind -- and I'd like to share it with my students as a case-study, but frankly the anecdote would take too much time and distract from the critically important first day of my own class. <br /><br />But the initial reaction we got for that teacher was accurate; although he couldn't perceive it, he was about as dense as the real numbers all semester long.<br /><br /><br />http://www.madmath.com/2016/01/grading-on-continuum.htmlnoreply@blogger.com (Delta)2tag:blogger.com,1999:blog-7718462793516968883.post-5152611217760156315Mon, 18 Jan 2016 10:00:00 +00002016-01-18T05:00:16.961-05:00Limitations<blockquote class="tr_bq"><i>Whenever one learns a new mathematical operation, it is imperative also to learn the limitations under which the operation may be performed. Lack of this additional knowledge can lead to the employment of the new operation in a blindly formal manner in situations where the operation is not properly applicable, perhaps resulting in absurd and paradoxical conclusions. Instructors of mathematics see mistakes of this sort made by their students almost every day...</i> </blockquote>- Howard Eves, <i>Great Moments in Mathematics</i>, Lecture 32.http://www.madmath.com/2016/01/limitations.htmlnoreply@blogger.com (Delta)0