tag:blogger.com,1999:blog-77184627935169688832015-05-25T05:00:00.974-04:00AngryMath"Beauty is the Enemy of Expression"Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.comBlogger184125tag:blogger.com,1999:blog-7718462793516968883.post-29224474038061392962015-05-25T05:00:00.000-04:002015-05-25T05:00:00.980-04:00Online Courses Fail at Community CollegesMore evidence for one of the most uniformly-verified findings I've seen in education: online courses strike out for community college students. From a paper by researchers at U.C.-Davis, presented at the American Educational Research Association's conference in April: <br /><blockquote class="tr_bq"><i>“In every subject, students are doing better face-to-face,” said Cassandra Hart, one of the paper’s authors. “Other studies have found the same thing. There’s a strong body of evidence building up that students are not doing quite as well in online courses, at least as the courses are being designed now in the community college sector.”</i></blockquote><a href="http://www.alternet.org/education/studies-find-online-courses-not-working-well-community-colleges">More at Alternet.org.</a><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-20252328554781063422015-05-18T05:00:00.000-04:002015-05-18T05:00:02.767-04:00Teaching Evolution in KentuckyA professor discusses teaching evolution in Kentucky. "Every time a student stomps out of my auditorium slamming the door on the way, I can’t help but question my abilities." (<a href="https://orionmagazine.org/article/defending-darwin/">Link.</a>)<br /><br />(Thanks to <a href="http://jonathanscottmiller.blogspot.com/">Jonathan Scott Miller</a> for the link.)<br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-35686927681408766562015-05-11T05:00:00.000-04:002015-05-11T05:00:04.464-04:00On DefinitionsFromthe MathBabe blog by poster EllipticCurve, <a href="http://mathbabe.org/2015/04/18/aunt-pythias-and-uncle-aristippus-advice/#comments">here</a>:<br /><blockquote class="tr_bq"><span style="font-size: large;"><i>Mathematical definitions mean nothing until you actually use them in anger, i.e. to solve a problem...</i></span></blockquote>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-90441642065128734322015-05-04T05:00:00.000-04:002015-05-04T05:00:07.706-04:00Quad Partners Buys Inside Higher EdFrom Education News in January:<br /><blockquote class="tr_bq"><em>Although there has been no public announcement made, Quad Partners, a New York private equity firm devoted to the for-profit college industry, recently gained a controlling stake in the education trade publication Inside Higher Ed (IHE). The publication routinely reports on for-profit colleges and surrounding policy disputes, and the publication is now listed among investments on the Quad Partners website.</em></blockquote><a href="http://www.educationnews.org/higher-education/quad-partners-acquires-stake-in-inside-higher-ed/">Read more here. </a><br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-62550240258819295482015-04-27T05:00:00.000-04:002015-04-27T05:00:02.650-04:00ETS on MillennialsA fascinating report on international education and job-ready skills from the Educational Testing Service. Particularly so, as it almost directly impinges on committee work that I've been doing lately. Core findings:<br /><ul><li>While U.S. millennials have far higher degree certifications than prior generations, their literacy, numeracy, and use-of-technology skills are demonstrably lower. </li><li>U.S. millennials rank 16th of 22 countries in literacy. They are 20th of 22 in numeracy. They are tied for last in technology-based problem solving.</li><li>Numeracy for U.S. millennials has been dropping across all percentiles since at least 2003.</li></ul><br /><br /><br /><br /><br /><br /><br /><br /><a href="http://www.ets.org/s/research/30079/index.html">See the online report here.</a><br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-58573101946618980572015-04-20T05:00:00.000-04:002015-04-20T05:00:05.769-04:00Causes of College Cost InflationFrom testimony at Ohio State (<a href="http://academeblog.org/2015/03/05/ohio-conference-president-provides-senate-testimony-on-the-decline-in-state-support-administrate-bloat-the-cost-of-intercollegiate-athletics-and-faculty-workload/">link</a>):<br /><ol><li>Decreased state funding</li><li>Administrative bloat</li><li>Cost of athletics</li></ol>(Thanks to<a href="http://jonathanscottmiller.blogspot.com/"> Jonathan Scott Miller</a> for the link.) <br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-90462788584779744572015-04-13T05:00:00.000-04:002015-04-13T05:00:07.810-04:00Pupils Prefer PaperYou may have already seen this article on the work of Naomi S. Baron at American University: her studies show that for textbook-style reading and studying, young college students still prefer paper books over digital options. Why? Because of reading. <br /><blockquote class="tr_bq"><i>In years of surveys, Baron asked students what they liked least about reading in print. Her favorite response: “It takes me longer because I read more carefully.”...</i><br /><br /><i>Another significant problem, especially for college students, is distraction. The lives of millennials are increasingly lived on screens. In her surveys, Baron writes that she found “jaw-dropping” results to the question of whether students were more likely to multitask in hard copy (1 percent) vs. reading on-screen (90 percent). </i></blockquote><br /><a href="http://www.washingtonpost.com/local/why-digital-natives-prefer-reading-in-print-yes-you-read-that-right/2015/02/22/8596ca86-b871-11e4-9423-f3d0a1ec335c_story.html">Read the article at the Washington Post.</a><br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com2tag:blogger.com,1999:blog-7718462793516968883.post-15191312758369228582015-04-06T05:00:00.000-04:002015-04-06T11:48:52.434-04:00Academically Adrift AgainOne more time, as we've pointed out here before (<a href="http://www.angrymath.com/2014/12/academically-adrift.html">link</a>), in this case from Jonathan Wai of Duke University: "<span class="anno-span">the rank order of cognitive skills of various majors and degree holders has remained remarkably constant for the last seven decades", with Education majors perennially the very lowest of performers (closely followed by Business and the Social Sciences). </span><br /><span class="anno-span"><br /></span><a href="http://qz.com/334926/your-college-major-is-a-pretty-good-indication-of-how-smart-you-are/"><span class="anno-span">See Wai's article and charts here.</span></a><br /><span class="anno-span"><br /></span><span class="anno-span"><br /></span>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-33779539652549744532015-03-30T05:00:00.001-04:002015-03-30T05:00:09.759-04:00Newtonian Weapons<blockquote class="tr_bq"><i>The ponderous instrument of synthesis, so effective in Newton's hands, has never since been grasped by anyone who could use it for such purpose; and we gaze at it with admiring curiosity, as some gigantic implement of war, which stands idle among the memorials of ancient days, and makes us wonder what manner of man he was who could wield as a weapon what we can hardly lift as a burden.</i></blockquote> - William Whewell on Newton’s geometric proofs, 1847 (thanks to JWS for pointing this out to me).<br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-74308784577332430432015-03-23T05:00:00.000-04:002015-03-23T12:15:01.557-04:00Average Woman's HeightA quick observation: statements like "the average woman's height is 64 inches" are almost always misinterpreted. Here's the main problem: people think that this is referencing an archetypal "average woman", when it's not.<br /><br />The proper parsing is <i>not</i> "the (average woman's) height"... but it <i>is</i> "the average (woman's height)". See the difference? It's really a statement about the variable "woman's height", which has been measured many times, and then averaged. In short, it's not about an "average woman", but rather an "average height" (of women). <br /><br />Of course, this misunderstanding is frequently intentionally mined for comedy. See yesterday's SMBC comic (as I write this), for example: <a href="http://www.smbc-comics.com/?id=3629#comic">here</a>.<br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com1tag:blogger.com,1999:blog-7718462793516968883.post-24825749995313151322015-03-16T05:00:00.000-04:002015-03-16T05:00:09.598-04:00Identifying Proportions Proposal<div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-GEMZjTOeKh0/VM7yMmdZVsI/AAAAAAAADUw/Wo0mpakRqPQ/s1600/proportional.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-GEMZjTOeKh0/VM7yMmdZVsI/AAAAAAAADUw/Wo0mpakRqPQ/s1600/proportional.png" height="135" width="200" /></a></div><br />Let me again ask the question, "<i>How Do You Know</i> Something is a Proportion?". For the math practitioner this is one of those things that requires no explanation ("you just know", "it's just common sense"). But for the remedial college algebra student, running into these common questions on tests, it's a real stumbling block to recognize when the proportional exercise is being asked. Actually, I've found that it's a stunningly hard problem to explain when we know something is proportional -- I've asked the question online twice here (<a href="http://www.angrymath.com/2013/02/explaining-proportions.html">first</a>, <a href="http://www.angrymath.com/2014/10/how-do-you-know-its-proportion.html">second</a>), I've asked around to friends, and rarely do I get any really coherent answer back at all. (One professor colleague investigated the issue and then said, "It's the question on the final that doesn't have a clear direction to it, that's how you know."). Any college basic-math textbook I look at has a stunningly short attempt at a non-explanation, as though they're keenly embarrassed that they really <i>can't</i> explain it at all (usually in the form of, "Proportions are useful, for example in the following cases..."; see first link above for specifics). Let's see if we can do better this week.<br /><br /><h2>Proportions in the Common Core</h2>First, let's look at what the up-and-coming Common Core curriculum does with this issue. In CC ratios and proportions are a 7th-grade topic, and there is a presentation specifically on the subject of "identifying proportions". This is uniformly given by two methods: (1) a table of paired numbers, and (2) a graph of a relationship. The key in the first case is to see that the pairs of numbers are always related by the same multiplication factor (usually a simple integer), i.e., y = kx. In the second case you're looking for a straight line that <i>goes through the origin</i> (0, 0), which I think is not a bad tactic. These materials very consciously <i>avoid</i> relying on the cross-multiplying ratios trick, and seek instead to develop a more concrete intuition for the relationship. I think that's a pretty solid methodology actually, and I've come to agree that the common cross-product way of writing these obscures the actual relationship (see also work like Lesh, 1988 that argues similarly).<br /><br />Some references to Common Core materials where you see this strategy in play:<br /><ul><li><a href="http://www.cpm.org/pdfs/state_supplements/Proportional_Relationships_Slope.pdf">College Preparatory Mathematics (CPM)</a></li><li><a href="http://achievethecore.org/content/upload/Grade%207%20EngageNY%20lesson%20-%20identify%20proportional%20relationships%20final06.24.14.pdf">Student Achievement Partners </a></li><li><a href="http://www.opusmath.com/common-core-standards/7.rp.2a-decide-whether-two-quantities-are-in-a-proportional-relationship-eg-by">Opus Math</a> </li></ul><br /><h2>In College Remedial Courses</h2>Second, let's observe that the exercises and test questions in our remedial algebra classes at the college level are <i>not</i> given in this format of numerous-data-points; rather, a much more cursory word problem is stated. In fact, let us admit that most of our exercises are at least somewhat malformed -- they are ambiguous, they require some background <i>assumption</i> or field-specific knowledge that things are proportional; they fail to be well-defined in an almost unique way. Here are a few examples from Elayn Martin-Gay's otherwise excellent <i>Prealgebra & Introductory Algebra</i>, 3rd Edition (2011), Section 6.1:<br /><blockquote class="tr_bq"><i><b>49.</b> A student would like to estimate the height of the Statue of Liberty in New York City's harbor. The length of the Statue of Liberty's right arm is 42 feet. The student's right arm is 2 feet long and her height is 5 1/3 feet. Use this information to estimate the height of the Statue of Liberty.</i></blockquote>Notice that this doesn't assert that the person and the statue are proportional. In fact I can think of a lot of artistic, structural, or biological reasons why it wouldn't be. In this way the problem is not really well-defined.<br /><blockquote class="tr_bq"><i><b>51.</b> There are 72 milligrams of cholesterol in a 3.5-ounce serving of lobster. How much cholesterol is in 5-ounces of lobster? Round to the nearest tenth of a milligram.</i></blockquote>As someone who's eaten a lot of cheap lobster growing up in Maine, again I can think of a lot of reasons of why the nutritional meat content of lobster might not be proportional across small and large lobsters (the basic serving being one creature of whatever size). For example, the shell hardness is very different across different sizes.<br /><blockquote class="tr_bq"><i><b>57.</b> One out of three American adults has worked in the restaurant industry at some point during his or her life. In an office of 84 workers, how many of these people would you expect to have worked in the restaurant industry at some point?</i></blockquote>My immediate guess would be: definitely <i>less</i> than one-third of the office. It seems that a sample of current office workers are somewhat more likely to have worked in an office all their career and therefore (thinking from the standpoint of statistical inference) fewer of them would have worked in a restaurant than the broad population. For example, simply change the word "office" here to "restaurant" and the answer is clearly not one-third (specifically it would be 100%), cluing us into the fact that former restaurant workers are not spread around homogeneously.<br /><br />I don't really mean to pick on Martin-Gay here, because many of her other exercises in the same section avoid these pitfalls. The very next exercise on the Statue of Liberty says, "Suppose your measurements are proportionally the same...", the one on skyscraper height says, "If the Empire State Building has the same number of feet per floor...", others give the mixture "ratio" or medication dose "for every 20 pounds", all of which nicely serve to solve the problem and make them well-defined.<br /><br />But I've seen <i>much worse</i> perpetrated by inattentive professors in their classrooms, tests, and custom books. So even if Martin-Gay fixes all of her exercises precisely, plenty of instructors will surely continue to overlook the details, and continue to assume their own background contextual knowledge in these problems that their remedial students simply don't have. In short, for some reason, we as a professorship continually keep writing malformed and poorly-defined problems that secretly rely on our application background knowledge of things as proportional.<br /><br /><h2>Proposed Solution</h2>Here's the solution that I've decided to try this semester in my remedial algebra courses. To start with, I'll try to draw a direct connection to the Common Core exercises, such that if someone ever has or does encounter that, hopefully some neurons will recognize the topic as familiar. Instead of saying that a proportion is an equality of ratios (as most of these college books do), I will instead say that it's a "relation involving only a multiply/divide", emphasizing the essential <i>simplicity</i> of the relation (really just one <i>kind</i> of operation; no add/subtracts or exponents/radicals), and covering the expression of it as either y = kx or a/b = c/d.<br /><br />The hope here is that we can then inuit in different cases, asking, "does this seem like a simple multiply operation?", with the supplemental hint (stolen from Common Core), "would zero result in zero?". For example: (a) paint relating to surface area (yes; zero area would require zero paint), (b) weight and age (no; a zero-age child does not weigh zero pounds). In addition -- granted that students will be certain to encounter these frankly malformed problems -- I will list some specific contextual examples, that will show up in exercises and tests, to try to full in the gap of application-area knowledge that many of us take for granted. Here's what my brief lecture notes look like now:<br /><br /><hr /><span style="font-family: Arial,Helvetica,sans-serif;"><u>Proportions</u>: Relations involving only multiply/divide (also: “direct variation”). Examples: Ingredients in recipes, gas consumption, scale comparisons. (Hint: Does 0 give 0?). Formula: a/b = c/d, where a & c share units, and so do b & d. Ex.: If 2 boxes of cereal cost $10, then how much do 6 boxes cost? 10/2 = x/6 → 60 = 2x → 30 = x. Interpret: 6 boxes cost $30. Ex.: (YSB4, p. 68) 22; 24. [4/7 = 12/x. Interpret: 12 marbles weigh 21 grams.]</span><br /><hr /><br />So that's what I'll be trying out in my courses this week. We'll see how it goes; at least I think I have a legitimate answer now when a student asks for these kinds of problems, "<i>but how do we know it's proportional?</i>".<br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-85250684234457798362015-03-14T09:26:00.000-04:002015-03-14T16:08:27.838-04:00Pi Proofs for the Area of a Circle<div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-eCLt5Pm-IPE/VQRp7xKRIHI/AAAAAAAADXI/sexRAaBn0hA/s1600/area-of-a-sector-of-a-circle.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-eCLt5Pm-IPE/VQRp7xKRIHI/AAAAAAAADXI/sexRAaBn0hA/s1600/area-of-a-sector-of-a-circle.PNG" height="162" width="320" /></a></div><br />For Half-Tau Day: Approximately π short proofs for the area of a circle, each in terms of τ = 2π (that is, one "turn"), mostly using calculus. <br /><h1>Shells in Rectangular Coordinates</h1>See the first picture above. Imagine slicing the disk into very thin rings, each with a small width ds, at a radius of s, for a corresponding circumference of τs (by definition of τ). So each ring is close to a rectangle if straightened out, with length τs and a width of ds, that is, close to an area of τs ds. In the limit for all radii 0 to r, this gives:<br /><div style="text-align: center;">∫[0 to r] τs ds = |[0 to r] τs²/2 = τr²/2.</div><h1>Sectors in Polar Coordinates</h1>See the second picture above. Imagine slicing the disk into very thin wedges, each with a small radian angle dθ and a corresponding arclength of r dθ (by definition of θ). So each wedge is close to a triangle (half a rectangle) with base r and a height of r dθ, that is, close to an area of r²/2 dθ. In the limit for all angles 0 to τ, this gives:<br /><div style="text-align: center;">∫[0 to τ] r²/2 dθ = |[0 to τ] r²/2 θ = τr²/2. </div><h1>Unwrapping a Triangle</h1>Sort of taking half of each of the ideas above, we can geometrically "unwrap" the rings in a circle into a right triangle. One radius stays fixed at height r. The outermost rim of the circle becomes the base of the triangle, with width as the circumference τr. So this triangle has an area of ½(r)(τr) = τr²/2. <br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-k7PF-EGmcfU/VQRvkRawszI/AAAAAAAADXY/XdEN-Tk9Muo/s1600/220px-TriangleFromCircle.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-k7PF-EGmcfU/VQRvkRawszI/AAAAAAAADXY/XdEN-Tk9Muo/s1600/220px-TriangleFromCircle.gif" /></a></div><br /><br /><h1>Comments</h1>The interesting thing about the first two proofs is that between them they interchange the radius r and the circle constant τ in the bounds versus the integrand. The interesting thing about the last demonstration is that it matches the phrasing of Archimedes' original, pre-algebraic conclusion: the area of a circle is equal to that of a triangle with height equal to the radius, and base equal to the circumference (proven with more formal geometric methods).<br /><br />Of course, if you replace τ in the foregoing with 2π, then the multiply-and-divide by 2's cancel out, and you get the more familiar expression πr². But I actually like seeing the factor of r²/2 in the version with τ, as both foreshadowing and reminder that ∫ r dr = r²/2, and also that it's half of a certain rectangle (as I might paraphrase Archimedes). In addition, Michael Hartl makes the point that this matches a bunch a similar basic formulas in physics (<a href="http://www.tauday.com/tau-manifesto#table-quadratic_forms">link</a>). <br /><br />Thanks to MathCaptain.com and Wikipedia for the images above.<br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-84191761218903875422015-03-09T05:00:00.000-04:002015-03-09T05:00:05.526-04:00Vive la Différence?A colleague of mine says that he wants to spend time giving his students more video and I equivocate in my response. He presses the question: "Don't you agree that different people learn in different ways?"<br /><br />Here's one possibly reply. First, to my understanding there's no evidence that trying to match delivery to different "learning styles" has any positive effect on outcomes. One example I came across yesterday: a Department of Education meta-analysis of thousands of studies found no learning evidence of benefits from online videos ("Elements such as video or online quizzes do not appear to influence the amount that students learn in online classes.", p. xvi, <a href="http://www2.ed.gov/rschstat/eval/tech/evidence-based-practices/finalreport.pdf">here</a>). Perhaps the short version of this response would be, "Not in any way that makes a significant difference."<br /><br />But here's another possible response, to answer the question with another question: "Don't you agree that it's important to have a shared, common language for communication in any field?" And as usual I would argue that acting like math, or computer programming is anything other than essentially a <i>written</i> artifact is fallacious -- in fact, overlooking the fact that <i>writing in general</i> is the most potent tool ever developed in our arsenal as human beings (including factors such as brevity, density, speed, searchability, auditability, etc.) is fallacious, a fraud, a failure.<br /><br />To the extent that we delay delivering and practicing the "real deal" for our students -- namely, <i>properly-written math</i> -- it is a tragic <a href="https://en.wikipedia.org/wiki/Garden_path_sentence">garden path</a>. <br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com2tag:blogger.com,1999:blog-7718462793516968883.post-59288892216900696372015-03-02T05:00:00.000-05:002015-03-02T05:00:00.893-05:00More Studies that Tech Handouts Hurt StudentsFrom an Op-Ed in the <i>New York Times</i> on 1/30/15 by Susan Pinker, a developmental psychologist:<br /><blockquote class="tr_bq"><i>In the early 2000s, the Duke University economists Jacob Vigdor and Helen Ladd tracked the academic progress of nearly one million disadvantaged middle-school students against the dates they were given networked computers. The researchers assessed the students’ math and reading skills annually for five years, and recorded how they spent their time. The news was not good.</i><br /><br /><i>“Students who gain access to a home computer between the 5th and 8th grades tend to witness a persistent decline in reading and math scores,” the economists wrote, adding that license to surf the Internet was also linked to lower grades in younger children.</i><br /><br /><i>In fact, the students’ academic scores dropped and remained depressed for as long as the researchers kept tabs on them. What’s worse, the weaker students (boys, African-Americans) were more adversely affected than the rest. When their computers arrived, their reading scores fell off a cliff.</i></blockquote><br />Read more <a href="http://www.nytimes.com/2015/01/30/opinion/can-students-have-too-much-tech.html">here</a>. Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-51699079015265271002015-02-23T05:00:00.000-05:002015-02-23T05:00:02.894-05:00Conic Sections in Play-DohHere's an idea for illustrating all the different shapes you can get out of conic sections: get some Play-Doh, roll it out into a cone shape (the "conic" part) -- and also a reasonably sharp knife (for the "sections" part).<br /><br /><br />First, here's our starting cone: <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-lLeyoWyeBC8/VMcSDlQUiDI/AAAAAAAADTY/gknpFs8j4tk/s1600/Image01262015225418.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-lLeyoWyeBC8/VMcSDlQUiDI/AAAAAAAADTY/gknpFs8j4tk/s1600/Image01262015225418.jpg" height="320" width="240" /></a></div><br /><br />Note that if you cut off just the tippy-top part you get a single point: <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-n0kp6jbeBC0/VMcSDt5ct7I/AAAAAAAADTU/Bk_aE7GiTlQ/s1600/Image01262015225447.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-n0kp6jbeBC0/VMcSDt5ct7I/AAAAAAAADTU/Bk_aE7GiTlQ/s1600/Image01262015225447.jpg" height="320" width="240" /></a></div><br /><br />On the other hand, if you carefully take a shaving down the very edge it produces a line:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-udd3fPP5QXM/VMcSDZaXkQI/AAAAAAAADTQ/VUFhZHGgZBE/s1600/Image01262015225609.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-udd3fPP5QXM/VMcSDZaXkQI/AAAAAAAADTQ/VUFhZHGgZBE/s1600/Image01262015225609.jpg" height="320" width="240" /></a></div><br /><br />But if you make a slice perpendicular to the base, you get a perfect circle (of any size you want, depending on how far down the cone you take it):<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-qc2SgcX5DaM/VMcSEPS_SuI/AAAAAAAADTg/AfvYfqZPuIk/s1600/Image01262015225946.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-qc2SgcX5DaM/VMcSEPS_SuI/AAAAAAAADTg/AfvYfqZPuIk/s1600/Image01262015225946.jpg" height="320" width="240" /></a></div><br /><br />Make a similar slice at a slight angle and the cross-section you get is now an ellipse:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-ss8cGGsDwRU/VMcSEbH1klI/AAAAAAAADTw/BD3fFUK0M_8/s1600/Image01262015230236.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-ss8cGGsDwRU/VMcSEbH1klI/AAAAAAAADTw/BD3fFUK0M_8/s1600/Image01262015230236.jpg" height="320" width="240" /></a></div><br /><br />Take the slice at a steeper angle and you'll produce our old quadratic friend, the parabola: <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-1t5LomgCpJM/VMcSFH7-q0I/AAAAAAAADT8/dYihcAyAV3I/s1600/Image01262015230942.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-1t5LomgCpJM/VMcSFH7-q0I/AAAAAAAADT8/dYihcAyAV3I/s1600/Image01262015230942.jpg" height="320" width="240" /></a></div><br /><br />And increase the angle a bit more (greater than the edge of the cone itself), and you'll produce the parabola's angry cousin, the hyperbola (or really a half-branch of such): <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-ZC05Dv90hc4/VMcSEzn9AZI/AAAAAAAADT0/tilHVwvkS54/s1600/Image01262015231121.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-ZC05Dv90hc4/VMcSEzn9AZI/AAAAAAAADT0/tilHVwvkS54/s1600/Image01262015231121.jpg" height="320" width="240" /></a></div><br /><br />Kind of neat. Full disclosure: the cone gets pretty "smooshed" on each cut (kind of like a loaf of bread with a dull knife), and I had to gently re-shape back into the proper section before each photo. Therefore, this demonstration probably works best in static photography, and would be somewhat less elegant live or in a video. But the nice thing about the Play-Doh is that you can sticky it back together pretty well after each sectional cut, and it's the only material I could think of that would work well in that way. Can you think of anything else?<br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com2tag:blogger.com,1999:blog-7718462793516968883.post-57149630002204088802015-02-16T02:00:00.000-05:002015-02-18T00:25:01.449-05:00Sorting Blackboard Test ResultsOn the Blackboard class management system, tests may be assigned where the order of the questions is randomized for different students (useful to somewhat improve security, make sure no question is biased due to ordering, etc.). However, a problem arises: when individual results are downloaded, the questions still appear in this randomized order for each separate student. That is: questions don't match down columns, they don't match in the listed "ID" numbers, etc.; and therefore there's no obvious way to assess or correlate individual questions between students or with any outside data source (such as a final exam, pretest/post-test structure, etc.). A brief discussion about this problem can be found on the "Ask the MVP" forum on the Blackboard site (<a href="http://discussions.blackboard.com/forums/p/59333/180403.aspx#180403">link</a>).<br /><br />Now here's a solution: I wrote a computer application to take downloaded Blackboard results in this situation and sort them back into consistent question ordering, and thereby make them usable for correlation analysis (in a spreadsheet, SPSS, etc.). Java code files are linked below, and you'll need to compile and run them yourself. Test results should first by downloaded from Blackboard as a comma-delimited CSV file in the "long download" format ("by question and user" in the Blackboard Download Results interface). <br /><br />The program then reads that input data and outputs two separate files. The first, "questions.csv", is a key to the questions, listing each Question ID, Possible Points, and full Question text. The second, "users.csv", is a matrix of the different users (test-takers) and their scores on each question (each row is one user, and each column is their score for one particular question, consistent as per the questions.csv key). This makes it far more convenient to add outside data correlate success on any particular question with overall results. Ping me here if you have other questions.<br /><br /><div style="text-align: center;"><a href="http://www.superdan.net/download/blog/angrymath/BlackboardResultSort_doc"><span style="font-size: large;">Blackboard Result Sort Documentation</span></a> </div><br /><div style="text-align: center;"><a href="http://www.superdan.net/download/blog/angrymath/BlackboardResultSort-v1-0.zip"><span style="font-size: large;">Blackboard Result Sort Code (Java ZIP)</span></a></div><br /><div style="text-align: center;"><br /></div>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com1tag:blogger.com,1999:blog-7718462793516968883.post-15726669468276737792015-02-02T05:00:00.000-05:002015-02-07T23:11:57.080-05:00Yitang Zhang Article in the New Yorker<div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-YM-o1EwquSM/VM1hHrRc_zI/AAAAAAAADUQ/BAfoGyAKP2k/s1600/Zhang.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-YM-o1EwquSM/VM1hHrRc_zI/AAAAAAAADUQ/BAfoGyAKP2k/s1600/Zhang.jpg" height="200" width="156" /></a></div><br />The lead article in this week's <i>New Yorker</i> (Feb-2, 2015) is on Yitang Zhang, the UNH professor who appeared from obscurity last year to prove the first real result in the direction of the twin-primes conjecture (specifically, a concrete repeating bounded gap between primes; full article <a href="http://www.newyorker.com/magazine/2015/02/02/pursuit-beauty">here</a>). <br /><br />To some extent I feel an unwarranted amount of closeness to this story. First, I grew up very close to UNH (just over the border in Maine), I would use the library there all through high school to do research for papers, and I worked at the dairy facility there for a few summers while I was in college. Secondly, the "twin primes conjecture" is basically the only real math research problem that I ever even had any intuition about -- along about senior year in my math program I think wrote a paper in abstract algebra where after some investigation on a computer I wrote "it's interesting to note that primes separated by only two repeat infinitely", to which the professor wrote back in red pen, "unproven conjecture!". I sort of have a running debate with a colleague at school that it's sort of intuitively obvious if you look at it, while course there's no rigorous proof. Yet.<br /><br />When I saw this <i>New Yorker</i> in our apartment around midnight Friday after I got back from school, I first noticed that there was some article on math (the writer makes it pretty opaque initially about exactly who or what the subject is). My partner Isabelle immediately said, "For god's sake, don't read it tonight and go to bed angry!", to which I said, "Mmmm-hmmm, probably a good idea." But I did so anyway. Frankly, I got less angered by it than you might expect, because while it's big pile of dumb fucking shit, it's dumb in a way that so stupidly predictable it almost turns around and becomes comedy if you know what's going on. It's dumb in exactly the carbon-copy way, almost word-for-word that all of these articles are dumb -- so it's at least unsurprisingly stupid. Let's check off some boxes...<br /><br /><h2>The "Beautiful Math" Trope</h2>Sure enough, the title of the article is <i><b>"The Pursuit of Beauty"</b></i>. Paragraph #2 of the article is a string of predictable quotes by some dead white guys about "proofs can be beautiful" (G.H. Hardy, Bertrand Russell). The writer managed to find one living professor who he got to use that word one time (Edward Frankel, UC Berkeley, the proof having "a renaissance beauty", sounding like the author pressed him on the question and was grudgingly humored). And then he gets a hail-Mary sentence on neuroscientists connecting math to art in some lobe of the brain. But Zhang never says that. Nor anyone else in the article from that point. <br /><br />This is so goddamn predictable that, yes, <b>it's the <i>raison d'etre</i> of this very six-year blog</b>, to respond to that exact piece of nonsense in pop math writing (see tagline above; and the "Manifesto" in the <a href="http://www.angrymath.com/2009/01/angrymath-manifesto.html">first post</a>). It's bullshit, it's not part of the real work of math. Sure, shorter is better, and it's far more convenient to get at a proof quickly with some heavy-caliber technique or clever trick, and I'd argue this is all that's meant by the "beautiful" trope. Someone gets careless and uses "beautiful" as a metaphor, in the way that Einstein or someone likes to pitch "God" as a metaphor -- when they secretly have some nonstandard definition like, "scientific research reducing superstition" (see: letter to Herbert Goldstein) -- and then it gets repeated by a thousand propagandists for their personal crusades. In the case of a pop media writer, they can latch onto the "beautiful" tag line and feel that they've got a hook on the story, and approach the rest of it like it's an article on Jeff Koons or some other high-society, celebrity scam artist. <br /><br />But at any rate, the "beautiful math" pitch is entirely isolated to the article title and a single paragraph, it has zero connection to the rest of the story, it's basically just clickbait, so let's move on. <br /><br /><br /><h2>Journalist-Mathematician Antimatter</h2>The broader issue that makes article count as downright comedy is the completely predictable acid-and-water interaction between the journalist and the mathematician. The writer here, Alec Wilkinson, is an exemplar of his industry -- scammy, full of bullshit, and just downright really fucking stupid. We've all met these folks at this point, have we not? Doesn't really know about anything. Has a single journalistic move up his sleeve for every article: "put a human face on the story", make it personal, make it about the people, "how did X make you <i>feel</i>?". (Elsewhere in the magazine, another writer waxes nostalgic for the classic traditions of <i>New Yorker</i> staffers: "all the editors dressed up and out every night for dinner and a show... a shrine of exotic booze...", <i>Talk of the Town</i>). <br /><br />But here Wilkinson confronts a person who is ultimately patient, disciplined, humble, hard-working, and truth-seeking. And he doesn't know what the hell to do with that. No other professional mathematician had known what Zhang was doing for over 10 years. He received no accolades nor enemies. He doesn't seem aggrieved or jealous that other people's careers advanced ahead of his own. He speaks softly at awards ceremonies and talks. There's no "personal face" meat here.<br /><br />So here's how Wilkinson responds; he makes the article about <i>himself</i>. Specifically about how he's a stupid damn bullshit artist. The opening paragraph is specifically about how apparently <i>proud</i> he is to know nothing about math, to be unqualified to write this story, and about how he's a <u><i>fucking lying cheater</i></u>:<br /><br /><blockquote class="tr_bq"><i>I don’t see what difference it can make now to reveal that I passed high-school math only because I cheated. I could add and subtract and multiply and divide, but I entered the wilderness when words became equations and x’s and y’s. On test days, I sat next to Bob Isner or Bruce Gelfand or Ted Chapman or Donny Chamberlain—smart boys whose handwriting I could read—and divided my attention between his desk and the teacher’s eyes. </i></blockquote><br /><br />Later, here's a summary of his interactions with Zhang:<br /><blockquote class="tr_bq"><i>Zhang is deeply reticent, and his manner is formal and elaborately polite. Recently, when we were walking, he said, “May I use these?” He meant a pair of clip-on shades, which he held toward me as if I might want to examine them first. His enthusiasm for answering questions about himself and his work is slight. About half an hour after I had met him for the first time, he said, “I have a question.” We had been talking about his childhood. He said, “How many more questions you going to have?” He depends heavily on three responses: “Maybe,” “Not so much,” and “Maybe not so much.” From diffidence, he often says “we” instead of “I,” as in, “We may not think this approach is so important.”... Peter Sarnak, a member of the Institute for Advanced Study, says that one day he ran into Zhang and said hello, and Zhang said hello, then Zhang said that it was the first word he’d spoken to anyone in ten days. </i></blockquote><br />This is not the kind of thing that a drinky, likely coke-blowing, social butterfly bullshit artist has any way of processing. And come on, that's pretty fucking funny; in that regard you almost couldn't make this stuff up. But on the downside it argues that these articles are always thricefold doomed; no journalist will ever write about the practice or results of math in any intelligible or useful way, because they're constitutionally, commercially, and philosophically <i>opposed</i> to it. <br /><br />This is how predictable it all is: I give a mini-rant to Isabelle and she says, "Oh, he probably just went after something a family member mentioned to him once", and that is in fact <i>exactly</i> what motivated the article (see end of the first paragraph). So frankly I could read about three sentences and map out in advance the progression of all the rest of the article. Hard to get usefully enraged by that; just standard-stupid is all.<br /><br /><h2>The Community of Math</h2>That said, the article does brush up against a real essential issue that I've been wrestling with for a few years now. Many of the math blogs that I've been reading in the last half-decade make a powerful and sustained case for the "community of math", that math <i>cannot</i> be done in isolation, that it <i>only exists</i> in the context of communicating with colleagues. Even that the <i>writing of papers</i> is inherently a peripheral and transient distraction, that the "true" productive activity of math is done verbally face-to-face and via body language with other experts -- writing being a faint shadow of that true work. (Hit me up for references on any of these points if you want them.)<br /><br />Unfortunately, this hits me something like an attack right through my own person. This is <i>exactly</i> the way that I personally failed in graduate school, and did not proceed on to the doctorate -- by continuing to work furiously <i>in isolation</i> while the rest of the classes basically passed me by. I've recently seen this called "John Henry Disease" in the work of Claude Steele (although there he holds it out as uniquely a phenomenon for black students). When I bring this up to colleagues nowadays, I can tell this story lightly enough that I get a laugh out of them, "Obviously you had to know better than that", or some-such. But a combination of personality and cultural upbringing literally left me completely unaware of the idea that you'd go get someone else's help on a math problem. So in that regard the "math community" thesis is a strong one.<br /><br />But on the other hand, the whole prime-directive that I've established for my math classes in the last few years is: <i>Learn how to read and write math properly</i>. It's literally the first thing on my syllabi now; the idea that math (algebraic) language is inherently a <i>written</i> language and not primarily verbal, and that this is the hard thing to master if you're a standard poorly-prepared city public high school graduate. That learning to <i>read a math book</i> was the key that got me through calculus and all the rest of a math program (through the undergraduate level, anyway). That the software that runs our world are fundamentally <i>products of writing</i> (see a prior post <a href="http://www.angrymath.com/2012/06/reading-writing-and-video-watching.html">here</a>). And once I commit to this goal in class, and get most of the students to buy in to it, I've been getting what I think are wonderful and satisfying results with it, incredibly encouraging, in the last few years. Ken Bain's book "What the Best College Teachers Do" hits on this as an even more universal theme: "We found among the most effective teachers a strong desire to help students learn to read in the discipline." (Chapter 3, item #8). <br /><br />And now here we have, in the very recent past, multiple cases of major mathematical breakthroughs by people working entirely in isolation, effectively in secret, for one to two decades, interacting only with the published literature in the field and their own brainpower. This is what's held out as Zhang's experience. And the same could be said for Perelman with the Poincaré conjecture, right? And also Andrew Wiles with Fermat's Last Theorem. And maybe Shinichi Mochizuki with the abc conjecture? (Here's where the argument rages.) A common theme recently is that with ever-more stringent publishing requirements for tenure, people on the standard academic track must publish every year or two, not meditate on the deepest problems for a decade. And so does the institution actually force isolation on the people tackling these giant problems? Or is it merely the nature of the beast itself?<br /><blockquote class="tr_bq"><i>When Zhang wasn’t working </i>[at a Subway sandwich shop]<i>, he would go to the library at the University of Kentucky and read journals in algebraic geometry and number theory. “For years, I didn’t really keep up my dream in mathematics,” he said... <br /><br />When we reached Zhang’s office, I asked how he had found the door into the problem. On a whiteboard, he wrote, “Goldston-Pintz-Yıldırım”and “Bombieri-Friedlander-Iwaniec.” He said, “The first paper is on bound gaps, and the second is on the distribution of primes in arithmetic progressions. I compare these two together, plus my own innovations, based on the years of reading in the library.”</i></blockquote><br /><br />An aside: I have this <u><i>exact same issue</i></u> in terms of my gaming work with <i>Dungeons & Dragons</i> (see that blog <a href="http://deltasdnd.blogspot.com/">here</a>). The conventional wisdom is "obviously we all know that no one could learn D&D on their own, we all had some older mentor(s) who inducted us into the game". And I am in the very rare situation for whom that is <i>absolutely false</i>. Growing up in a rural part of Maine, the only reason I ever heard about the game was through magazines; I was the first person to get the rulebooks and read them; and the catalyst in my town and school, among anyone I ever knew, to introduce and run the game for them. Purely from the written text of the rulebooks. To the extent that there were any other conventions or understandings about the game that didn't get into the books, I never knew about them. Which in retrospect has been both a great strength and in some fewer cases a weakness for me. In short: I learned purely from the book and most people don't believe that's possible. <br /><br />But back to the article by Wilkinson: he expresses further dismay and incredulity at Zhang's solitary existence, his disinterest in social gatherings, and his preference for taking a bus to school so that he can get more thinking time in. All things which I could say pretty much identically for myself; and all things which our standard-template journalist is going to find alien and utterly bewildering:<br /><blockquote class="tr_bq"><i>Zhang’s memory is abnormally retentive. A friend of his named Jacob Chi said, “I take him to a party sometimes. He doesn’t talk, he’s absorbing everybody. I say, ‘There’s a human decency; you must talk to people, please.’ He says, ‘I enjoy your conversation.’ Six months later, he can say who sat where and who started a conversation, and he can repeat what they said.”</i><br /><br /><i> </i><i>“I may think socializing is a way to waste time,” Zhang says. “Also, maybe I’m a little shy.”</i><br /><br /><i> </i><i>A few years ago, Zhang sold his car, because he didn’t really use it. He rents an apartment about four miles from campus and rides to and from his office with students on a school shuttle. He says that he sits on the bus and thinks. Seven days a week, he arrives at his office around eight or nine and stays until six or seven. The longest he has taken off from thinking is two weeks. Sometimes he wakes in the morning thinking of a math problem he had been considering when he fell asleep. Outside his office is a long corridor that he likes to walk up and down. Otherwise, he walks outside.</i></blockquote><br /><br /><h2>Conclusion</h2>There are more things I could criticize -- For example, in the absence of anything useful to say, the author has to hang onto any dumb or tentative attempt at an analogy that anyone throws at him, and is really helpless to double-check or confirm any assessment with anyone else; he literally can't understand anything anyone says about the math, even when interviews multiple professors on the same subject. He refers to Terry Tao like he's just "some professor", not one of the brightest and clearest thinkers on the planet. He has a paragraph on pp. 27-27, running 40 lines on the page, simply listing every variety of "prime number" he could find defined on Wikipedia (probably) -- the most blatant attempt at bloating up the word count of an article I think I've ever seen. Of course, it's <i>intended</i> to make your eyes cross and seem opaque. The <i>exact opposite</i> of a mathematical discipline dedicated to clear and transparent explanations.<br /><br />But those are just nit-picky details, and we've probably already given the article writer more attention than he deserves. Let me finish by addressing the elephantine angel in the room. Are the true, greatest breakthroughs really made by loners, working in isolation with just the written text, over decades of time? Or is that just another journalistic illusion?<br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com6tag:blogger.com,1999:blog-7718462793516968883.post-72723404537426220532015-01-26T05:00:00.000-05:002015-01-26T23:32:05.311-05:00On Old Books<div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-H2Avh6EPi6Y/VMKVBF5ye7I/AAAAAAAADS4/2LfeDMx6qMw/s1600/Image01232015132020.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-H2Avh6EPi6Y/VMKVBF5ye7I/AAAAAAAADS4/2LfeDMx6qMw/s1600/Image01232015132020.jpg" height="200" width="150" /></a></div>A few weekends ago I set up a new bookcase and got to re-organize and take a bunch of books out of boxed storage and back on display in my room. One thing I came across was a very old copy of "Introductory College Algebra", 2nd Edition, by Rietz and Crathorne, copyright 1923/1933. This is something I obtained from my great-aunt, who was the head of the math department at an academy in Maine (at a time when that was very rare),and who died a few decades ago now. I actually started reading it from front-to-back this week for the first time, which seemed apropos because I'm currently teaching a winter-term course in college algebra.<br /><br />The main uptake is that I'm really surprised how little has changed, how similar the work and presentation is to what we do today. That gives me a lot of confidence, actually; I'm glad to be in a discipline with "deep roots" that is stable and consistent. The presentation my be a bit more concise -- but that's kind of funny because everyone I know that's engaged in writing an in-house custom algebra text says that their goal is too write something "short, just what they need, with the extraneous parts cut out". Well, you don't get much more concise than a <i>real math text</i>. (Most of the theorems and presentations are all of 4 lines long at most.) One novelty I really like here is that instead of separate worked-out examples within the text, the protocol is to simply begin a block of exercises with the first few including fully worked-out solutions (which I think would clarify to the student what work we're expecting them to do; and as always you've got answers to the odd-numbered questions at the back for them to check). <br /><br />Sure, a couple pieces of terminology are just a bit different. Graphs of functions are are generally called "loci". What I've always seen as a "greatest common factor (GCF)" is herein called a "highest common factor "HCF)". And probably the single biggest difference is the claim that a statement like "x = x+1" or "0 = 1" does not count as an equation whatsoever (whereas I'd call it an equation with no solutions, i.e., an equation of the inconsistent variety). <br /><a href="http://4.bp.blogspot.com/-Zj2lGabRQh4/VMKVCoo0HyI/AAAAAAAADTA/bM6Z1SGFo4s/s1600/Image01232015132038.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="http://4.bp.blogspot.com/-Zj2lGabRQh4/VMKVCoo0HyI/AAAAAAAADTA/bM6Z1SGFo4s/s1600/Image01232015132038.jpg" height="200" width="150" /></a><br />But here's my point. Granted how relatively little has changed in this near century-old math textbook as compared to the class I teach each night right now; and granted the tremendous struggle we have these days to make <i>good</i> textbooks accessible and affordable to our students -- might we consider actually using out-of-copyright math textbooks as a resource? We could totally scan a brief, high-quality, public-domain text such as this and distribute it for free to anyone who wanted it. Do you think that would ever be workable?<br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com2tag:blogger.com,1999:blog-7718462793516968883.post-27909109635263190032014-12-29T05:00:00.000-05:002015-01-09T14:18:09.828-05:00Academically AdriftGoing through an old copy of <i>Thought & Action</i> magazine today (Fall 2011), at the back I come across a review of the book <i>Academically Adrift: Limited Learning on College Campuses</i>, by Richard Arum and Josipa Roksa. The main thrust of the book seems to be use of the Collegiate Learning Assessment (CLA), a test of critical thinking, reading, and writing given at the end of the sophomore year to several thousand students at 24 different colleges. The upshot seems to be that in many cases, there is little difference in ability between when students first arrive on campus and two years afterward. I found the following paragraph of the review to be worth highlighting:<br /><blockquote class="tr_bq"><i>The ensuing chapters then detail the key findings related to changes in CLA scores, implicating students’ entering characteristics and campus experiences. Students with stronger academic preparation and students who attended more selective institutions showed greater gains in critical thinking; initial disparities between white students and African American students were exacerbated. <b>Those who participated in fraternities and sororities showed fewer gains relative to their peers, as did those who were majoring in business, education, or social work.</b> Moreover, Arum and Roksa argue that understandings of student employment need to be nuanced, as working on-campus is beneficial only up to 10 hours per week. <b>They also question the trend toward collaborative learning, noting that more time studying alone is positively associated with gains in critical thinking, while time studying with peers is negatively associated with such gains. Perhaps most strikingly, the authors concede that social integration might be related to retention but argue that its affects on learning are far less clear, and may be negative.</b></i></blockquote><a href="http://www.nea.org/home/50459.htm"> http://www.nea.org/home/50459.htm</a><br /><br />In calling out certain majors, I am reminded of the footnote in Burton R. Clark's famous paper "The 'Cooling-Out' Function in Higher Education" from 1960 (The American Journal of Sociology, May 1960, footnote 8): <br /><blockquote class="tr_bq"><i>One study has noted that on many campuses the business school serves "as a dumping ground for students who cannot make the grade in engineering or some branch of the liberal arts," this being a consequence of lower promotion standards than are found in most other branches of the university (Frank C. Pierson, The Education of American Businessmen [New York: McGraw-Hill Book Co., 1959], p. 63). Pierson also summarizes data on intelligence of students by field of study which indicate that <b>education, business, and social science rank near the bottom in quality of students</b> (ibid., pp. 65-72).</i></blockquote><a href="http://www.jstor.org/discover/10.2307/2773649?sid=21105522739863&uid=2&uid=3739832&uid=3739256&uid=4">http://www.jstor.org/discover/10.2307/2773649?sid=21105522739863&uid=2&uid=3739832&uid=3739256&uid=4 </a><br /><br />Isn't it interesting that we've effectively handed over control of our culture, our most powerful institutions, and education of the young, to the least proficient among us? And that this seems to be a stable pattern for over a half-century?<br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com2tag:blogger.com,1999:blog-7718462793516968883.post-31533134897531425022014-10-27T05:00:00.000-04:002014-10-27T08:45:24.525-04:00Bloom's Taxonomy and Math EducationIn the last year or so I've been attending seminars at our college's Center for Teaching and Learning. So far these have been on how to publish in scholarship of teaching and learning (SOTL) journals, and a few reading groups (Susan Ambrose's "How Learning Works", and Ken Bain's "What the Best College Teachers Do"). Frequently I'm the only STEM instructor at the table, with the rest of the room being instructors from English, philosophy, political science, history, women's studies, social science, etc. <br /><br />One thing that keeps coming up in these books and discussions is a reference to Bloom's Taxonomy of Learning, a six-step hierarchy of tasks in cognitive development. Each step comes with a description, examples, and "key verbs". <a href="http://www.nwlink.com/~donclark/hrd/bloom.html">Here is a summary similar to what I've been seeing</a>. Now, I'm perennially skeptical of these kinds of "N Distinct Types of P!" categorizations, as they've always struck me as at least somewhat flawed and intellectually dishonest in a real, messy world. But for argument's sake, let's say that we engage with the people who find this useful and temporarily accept the defined categories as given.<br /><br />In every instance that I've seen, the discussion seems to turn on the following critique: "We are failing our students by perpetually being stuck in the lower stages of simple Knowledge and Comprehension recall (levels 1-2), and need to find ways to to lift our teaching into higher strata of Application, Analysis, etc. (levels 3-4 and above)". To a math instructor this sounds almost entirely vapid, because we never have time to test on levels 1-2 and entirely take those levels for granted without further commentary. In short, if Bloom's Taxonomy holds any weight at all, then I claim the following:<br /><br /><blockquote class="tr_bq"><span style="font-size: large;"><b>Math is hard because by its nature it's taught at TOO HIGH a level compared to other classes.</b></span></blockquote><br />For example: I've never seen a math instructor testing students on simple knowledge recall of defined terms or articulated procedures. Which in a certain light is funny, because our defined terms have been hammered out over years and centuries, and it's important that they be entirely unambiguous and essential. I frequently tell my students, "All of your answers are back in the definitions". Richard Lipton has written something similar to this more than once (link <a href="http://rjlipton.wordpress.com/2010/01/23/definitions-definitions-do-we-need-them/">one</a>, <a href="http://rjlipton.wordpress.com/2012/08/17/the-right-definition/">two</a>).<br /><br />But in math education we basically don't have any friggin' time to spend drilling or testing on these definitions-of-terms. We say it, we write it, we just <i>assume</i> that you remember it for all time afterward. This may be somewhat exacerbated by the math and computer scientist's custom of knowing to remember those key terms, and maybe our memory being trained in that way. I know in my own teaching I was at one time very frustrated with my students not picking up on this obvious requirement, and I've evolved and trained myself to constantly pepper them with side-questions on what the proper name is for different elements day after day to get these terms machine-gunned into their heads. They're not initially primed for instantaneous recall in the ways that we take for granted. At any rate: the time spent on <i>testing</i> for these issues is effectively zero; it doesn't exist in the system. (Personally, I have actually inserted some early questions on my quizzes on definitions, but I simply can't find time or space to do it thereafter.)<br /><br />So after the brief presentation of those colossally important defined terms, we will take for granted simple Recall and Comprehension (levels 1-2), and <i>immediately</i> launch in to using them logically in the form of theorems, proofs, and exercises -- that is, Application and Analysis (levels 3-4). Note the following "key verbs", specific to the math project, in Bloom's categorization: "computes, operates, solves" are among Applications (level 3), things like "calculates, diagrams" are put among Analysis (level 4). These of course are the mainstays of our expected skills, questions on tests, and time spent in the math class.. <br /><br />And then of course we get to "word problems", or what we really call "applications" in the context of a math class. Frequently some outside critic expects that these kinds of exercises will make the work easier for students by making it more concrete, perhaps "real-world oriented". But the truth is that this increases the difficulty for students who are already grappling with higher-level skills than they're accustomed to in other classes, and are now being called upon to scale even higher. These kinds of problems require: (1) high-quality English parsing skills, (2) ability to translate from the language of English to that of Math, (3) selection <i>and</i> application of the proper mathematical (level-3 and 4) procedures to solve the problem, and then (4) reverse translation from Math back to an English interpretation. (See what I did there? It's George Polya's How-To-Solve-It.) In other words, we might say: "Yo dawg, I heard you like applications? Well I made applications of your applications." Word problems boost the student effectively up to the Synthesis and Evaluation modes of thought (levels 5-6). <br /><br />So perhaps this serves as the start of an explanation as to why the math class looks like a Brobdingnagian monster to so many students; if most of their other classes are perpetually operating at level 1 and 2 (as per the complaints of so many writers in the humanities an education departments), then the math class that is immediately using defined terms and logical reason to <i>do stuff</i> at level 3 to 4 does look like a foreign country (to say nothing of word problems a few hours after that). And perhaps this can serve as a bridge between disciplines; if the humanities are wrestling with being stuck in level 1, then they need to keep in mind that the STEM struggle is not the same, that inherently the work demands reasoning at the highest levels, and we don't have time for anything else. Or perhaps this argues to find some way of working in more emphasis on those simple vocabulary recall and comprehension issues which are so critically important that we don't even bother talking about them? <br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com7tag:blogger.com,1999:blog-7718462793516968883.post-61522770084165710782014-10-20T05:00:00.000-04:002014-10-20T05:00:01.344-04:00Is Statway a Cargo Cult?We all know that Algebra is <i>the</i> limiting factor for the millions of students attending community colleges throughout the U.S. That is: Colleges could double (or triple, or quadruple) their graduation numbers overnight if the 8th-grade algebra requirement were only removed. This makes for lots of institutional pressure these days to do so. <br /><br />A common line of thought is: Get rid of the algebra requirement and pursue a primer on statistics instead. You can sort of see why someone might negotiate in this way: offer something apparently attractive (statistics, which many say is needed to understand the modern world) in place of the thing they're asking you to give up. For example, the Carnegie "Statway" program now at numerous colleges promises exactly that (the lede being "Statway triples student success in half the time"; <a href="http://www.carnegiefoundation.org/statway">link</a>).<br /><br />But as an instructor of statistics at a community college, I use algebra all the time to derive, and explain, and confirm various formulas and procedures. Without that, I think the intention (in fact I've heard this argued explicitly) is to get people to dump data into the SPSS program, click a button, and then send those results upstream or downstream to some other stake-holder without knowing how to verify or double-check them. Basically it advocates a faith-based approach to mathematical/statistical software tools.<br /><br />This is a nontrivial, in fact really tough, philosophical angel with which to wrestle nowadays. We're long past the point where cheap calculating devices have been made ingrained throughout many elementary and high schools; convenient to be sure, but as a result at the college level we see a great many students who have no intuition of times tables, and are utterly unable to estimate, sanity-check, or spot egregious errors (e.g. I had a college student who hand-computed 56×9 = 54 and was totally baffled at my saying that couldn't possibly be the answer; even re-doing the same thing a second time around).<br /><br />To a far greater degree, as I say in my classes, statistics is truly 20th century, space-age branch of math; it's a fairly tall edifice built on centuries of results in notation, algebra, probability, calculus, etc. Even in the best situation in my own general sophomore-level class, and as deeply committed as I am to rigorously demonstrating as much as possible, I'm forced to hand-wave a number of concepts from calculus classes which my students have not, and will never, take (notably regarding integrals, density curves, the area of any probability distribution being 1; to say nothing of a proof of the Central Limit Theorem). So if we accept that statistics are fundamental to understanding how the modern world is built and runs, and there is some amount of corner-shaving in presenting it to students who have never taken calculus, then perhaps it's okay to go whole-hog and just give them a technological tool that does the entire job for them? Without knowing where it comes from, and being told to just trust it? I can see (and have heard) arguments in both directions.<br /><br />Here's an example of the kind of results you might get from a website that caught my attention the other day: <a href="http://tylervigen.com/">Spurious Correlations</a>. The site puts on a display a rather large number of graphs of data which is meant to be obviously, comically not really related, even though they have high correlation. Here's an example:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-e9N_KyU_Sl8/VEQKZBPIJPI/AAAAAAAADQM/8FntNJ8KYXE/s1600/SpuriousCorrelation.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-e9N_KyU_Sl8/VEQKZBPIJPI/AAAAAAAADQM/8FntNJ8KYXE/s1600/SpuriousCorrelation.gif" height="229" width="320" /></a></div><br />Something seemed fishy about this after I first looked at it. It's true that if you dump the numbers in the table into Excel or SPSS or whatever a correlation value of 0.870127 pops out. But here's the rub: those date-based tables used throughout the site are totally not how you visualize correlation, or related in any way to what the linear correlation coefficient (r) means. What it does mean is that if you take those data pairs and plot them as an (x, y) scatterplot, you can find a straight-line that gets pretty close to most of the points. That is entirely lost in the graph as presented; the numbers aren't even paired up as points in the chart, and the date values are entirely ignored in your correlation calculation. I'm a bit unclear if the creator of the website knows this, or is just applying some packaged tool -- but surely it will be opaque and rather misleading to most readers of the site. At any rate, it terminates out the ability to visually double-check some crazy error of the 56×9 = 54 ilk.<br /><br />As a further point, there are some graphs on the site labelled as showing "inverse correlation", which I thought to be a correlation between x and 1/y -- but in truth what they mean is the more common [linear] "negative correlation", which is a whole different thing. Or at least I would presume it is; I'd never heard of "inverse correlation" as synonymous, and about the only place I can find it online is Investopedia (so maybe the finance community has its own somewhat-sloppy term for it; <a href="http://www.investopedia.com/terms/i/inverse-correlation.asp">link</a>). <br /><br />I guess someone might call this knit-picking, but I have the intuition that that's a sign of somebody who can't actually distinguish between true and false interpretations of statistical results. Is this ultimately the kind of product we get if we wipe out all the algebra-based derivations from our statistics instruction, and treat it as a non-reasoning vocational exercise? <br /><br />Let me be clear in saying that at this time I have not actually read the Carnegie Statway curriculum, so I can't say if it has some clever way of avoiding these pitfalls or not. Perhaps I should do that to be sure. But as years pass in my current career, and I get more opportunities to personally experience all the connections throughout our programs, I find myself becoming more and more of a booster and champion of the basic algebra class requirement for all, as perhaps the very finest tool in our kit for promoting clear-headedness, transparency, honesty, and truth in regards to what it means to be an educated, detail-oriented, and scientifically-literate person. <br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com2tag:blogger.com,1999:blog-7718462793516968883.post-55695366691025121632014-10-13T05:00:00.000-04:002014-10-13T05:00:05.516-04:00How Do You Know It's a Proportion?I've written in the past of the mystery of when you'd want to use a proportion for an application problem, and what the benefits are for doing so (<a href="http://www.angrymath.com/2013/02/explaining-proportions.html">link</a>). Once again, last week, one of my basic algebra students asked the question:<br /><blockquote class="tr_bq"><i>"How do you know it's a proportion?"</i></blockquote>And once again I was unable to answer her. I've searched all through several textbooks, and scoured the Web, and I still can't find even an attempt at a direct explanation of how you know a problem is proportional. (Examples, sure, nothing but examples.) I've asked other professors and no one could even take a stab at it. Perhaps the student was looking at any problem such as the following: <br /><blockquote class="tr_bq"><i>A can of lemonade comes with a measuring scoop and directions for mixing are 6 scoops of mix for every 12 cups of water. How much water is needed to make the entire can of lemonade if there are 40 scoops of mix?</i><br /><br /><i>On an architect's blueprint, 1 inch corresponds to 4 feet. Find the area of an actual room if the blueprint dimensions are 6 inches by 5 inches.</i><br /><br /><i>The ratio of the weight of an object on Earth to the weight of the same object on Pluto is 100 to 3. If a buffalo weighs 3568 pounds on Earth, find the buffalo's weight on Pluto. </i><br /><br /><i>Three out of 10 adults in a certain city buy their drugs at large drug stores. If this city has 138 ,000 adults, how many of these adults would you expect to buy their drugs at large drug stores?</i><br /><br /><i>The gasoline/oil ratio for a certain snowmobile is 50 to 1. If 1 gallon equals 128 fluid ounces, how many fluid ounces of oil should be mixed with 20 gallons of gasoline? </i></blockquote><br />Concisely stated, what is the commonality here? What is a well-defined explanation for how we know that these are all proportional problems?<br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com19tag:blogger.com,1999:blog-7718462793516968883.post-38227344000430543232014-10-01T05:00:00.000-04:002014-10-01T15:46:36.789-04:00On Comparing Decimals Like 0.999...Today in my college algebra class will be the first time that I've provided space to actually discuss the 1 = 0.999... issue. Previously I mentioned this <a href="http://www.angrymath.com/2011/11/arguing-infinite-decimals.html">here</a> on the blog. This became so contentious that it's actually the only post for which I've been forced to shut off comments. (Actually it attracted a stalker who'd post some aggressive nonsense every few days.) <br /><br />Anyway, brushing up on some points for later today let me see a very obvious fact that I'd overlooked before and that is: students' customary procedure for <i>comparing decimals</i> fails spectacularly in this case. For example, here it is expressed at the first hit from a web search at a site called <a href="http://www.aaamath.com/B/g6_52_x1.htm">AAAMath</a>: <br /><blockquote class="tr_bq"><i>Therefore, when decimals are compared start with tenths place and then hundredths place, etc. If one decimal has a higher number in the tenths place then it is larger than a decimal with fewer tenths. If the tenths are equal compare the hundredths, then the thousandths etc. until one decimal is larger or there are no more places to compare. If each decimal place value is the same then the decimals are equal. </i></blockquote>So if students apply the "simple" decimal comparison technique ("if one decimal has a higher number in the X place"), even at just the ones place, then this algorithm reports back that 1.000 is greater than 0.999... It overlooks the fact that the lower places can actually "add up" to an extra unit in a higher place. And thus all sorts of confused mayhem immediately follow. <br /><br />So the simple decimal comparison algorithm is actually wrong! To fix it, you'd have to add this clause: <i>unless either decimal ends with an infinitely repeating string of 9's</i>. In that case the best thing to do would be to initially "reduce" it back to the terminating form of the decimal (this being the only case where one number has multiple representations in decimal), and only then apply the simple grade-school algorithm. <br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-15142966997779714102014-08-25T05:00:00.000-04:002014-08-25T05:00:00.109-04:00Introducing Automatic-Algebra.org<div class="separator" style="clear: both; text-align: center;"><a href="http://automatic-algebra.org/" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-uCC6GlzNKf8/U_ORfPDyCXI/AAAAAAAADLo/eMlyQ6KSGsQ/s1600/aao-index.gif" height="200" width="167" /></a></div>Here in New York, it's back-to-school starting next week. Of course, if you're in the teaching profession (or really just know about it), you've probably been doing academic work and preparing for the fall semester throughout the summer and year-round. I'm in the very fortunate position that I'll have a new permanent position at CUNY this fall, so here's how I've spent my August:<br /><br />I've developed a new website for practicing basic numerical skills that are prerequisite for algebra and other classes like statistics, calculus, and computer programming: <a href="http://automatic-algebra.org/">Automatic-Algebra.org</a>. I've written a few times in the past about the need for certain skills to be <i>automatic</i>, skills that have been taught but not mastered by most students who arrive in a remedial community-college algebra course, and therefore causes continual disruption and frustration when we're trying to deal with actual algebra concepts (links <a href="http://www.angrymath.com/2013/12/automatic-drills.html">one</a>, <a href="http://www.angrymath.com/2014/07/automatic-negatives.html">two</a>). Like, for the algebra course itself: times tables, negative numbers, and order-of-operations. Or for a basic statistics course: operations on decimals like rounding, comparing, and converting to percent.<br /><br />So what you get at the new site for each of these skills is a brief, 5-question quiz for each of these skills. Here's how I designed them:<br /><ul><li>Timed, so that students get a very clear portrayal of what the expectation is for mastery of each of these skills (15, 30, or 60 seconds per quiz). For example: sequential adding and counting on fingers for multiplications will not suffice.</li><li>Multiple-choice, so the site is usable on a variety of devices, including touch-screen mobile devices. For example: you can stand on a bus and drill yourself on a smartphone by just tapping with your thumb. </li><li>Randomized, so once you take a quiz you can click "Try Again" and get a new one, and drill yourself multiple times in just a few minutes each day.</li><li>Javascript, so the quiz runs entirely on your own device once you download the page the first time. The site doesn't require any login, accounts, server submissions, recording of attempts, or any transmitted or collected information whatsoever after you initially view the page.</li></ul><br />This is something that I've wanted for a few years now, that no existing website really implemented and consolidated the way I wanted as a reference for students. Thanks to the new full-time track I feel I'm in the position to warrant developing it myself and leveraging it for numerous classes of my own in the future.<br /><br />Feel free to check it out, and offer any comments and observations. If you feel it might help your incoming students this fall semester, give them the link and maybe it will lift a whole lot of our boats all at the same time. Do you think that will be of assistance?<br /><br /><div style="text-align: center;"><span style="font-size: large;"><a href="http://automatic-algebra.org/">Automatic-Algebra.org</a></span></div><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com2tag:blogger.com,1999:blog-7718462793516968883.post-26408624357448713652014-08-11T05:00:00.000-04:002014-08-11T05:00:02.321-04:00When Are Parentheses Required for Substitution?In my remedial algebra classes, on introducing the substitution of numerical values for variables, I've always said that it's safest to perform this substitution inside parentheses, especially for negative numbers. Of course, we all intuit times when that's not strictly necessary. So in this lecture I usually get one of the brighter students asking, "Exactly when is it necessary?". I've found this to be a surprisingly difficult question to answer. After a rather embarrassingly long consideration, here's what I've come up with.<br /><br />Parentheses are basically required in the following two situations:<br /><ol><li>Separating juxtaposed signs and numbers, and</li><li>Collecting expressions with one operation under a higher-order operation that is not also a grouping symbol.</li></ol><br />For situation #1, I'm assuming that we're not ever inserting new operational symbols like <span style="font-family: Arial, sans-serif;">×</span> or <span style="font-family: Arial, sans-serif;">∙</span> in cases of juxtaposed multiplication -- just the substituted expression and possibly parentheses. Parentheses are probably only needed for factors after the first one (i.e., after the coefficient).<br /><br />For situation #2, we're mostly talking about multiplication and exponents, with some lower-order operation in the expression being substituted. Contrast with fraction bars (for division) and radicals, which have grouping built into the symbol, and thus no general requirement for new parentheses.<br /><br />Here are a few examples of each. For the following, let x = 1, y = –2, z = ab, and w = a + b. Examples of separating juxtaposed signs and numbers:<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-avigL8GE6Zk/U-L7xDcxwhI/AAAAAAAADKI/Ehr5RN9vCTE/s1600/ParensEx1.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-avigL8GE6Zk/U-L7xDcxwhI/AAAAAAAADKI/Ehr5RN9vCTE/s1600/ParensEx1.gif" /></a></div><br />Examples of collecting expressions with one operation under a higher-order operation: <br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-g7b2hHnQybI/U-L7xC2Xg2I/AAAAAAAADKE/KVPsdcwIhv8/s1600/ParensEx2.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-g7b2hHnQybI/U-L7xC2Xg2I/AAAAAAAADKE/KVPsdcwIhv8/s1600/ParensEx2.gif" /></a></div><br />We can explain the first example immediately above in that the negative sign acts the same as multiplying by –1, and therefore must be collected under the exponentiation operation. However, this does get slightly complicated by the use of the minus sign for both unary negation (i.e., multiplying by –1), and binary subtraction, which have different placements in the order of operations. For example, the following may be taken as a slightly ambiguous case: <br /><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-6Lr6kPF5OvY/U-L7xEebmSI/AAAAAAAADKA/mYgqL1HDuVY/s1600/ParensEx3.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-6Lr6kPF5OvY/U-L7xEebmSI/AAAAAAAADKA/mYgqL1HDuVY/s1600/ParensEx3.gif" /></a></div><br />Here, in substituting any numerical values at all for x and y, parentheses will definitely be necessary. However, this particular instance doesn't have juxtaposed numerals -- the real reason may be taken to be that without the parentheses, this would read as subtraction (lower order than the initial juxtaposed multiplication). <br /><br />A few notes on specific cases of substitution:<br /><ul><li>If substituting one variable for another, then parentheses are never needed (the order of operations is clearly identical before and after).</li><li>If substituting a whole number, then only the situation of juxtaposed numbers after the coefficient can apply. Obviously a whole number has no written sign, and includes no operations to interfere with higher-order interactions.</li><li>If substituting a negative number, then any of the situations are possible. It does have an attached sign, may need separation from an advance factor (as above), and operates similarly to a multiplication (and thus needing collection under an exponent). </li></ul><br /><div align="JUSTIFY" class="western" style="font-style: normal;">Note that Wikipedia articles do show use of juxtaposed signs, e.g., 7 + –5 = 2, and discusses possibly superscripting the unary negation in elementary contexts and the computer language APL (link <a href="http://en.wikipedia.org/wiki/Negative_number">one</a>, <a href="http://en.wikipedia.org/wiki/Plus_and_minus_signs">two</a>), something that I've also seen on some calculators, in which cases parentheses would not be necessary. However, that's not something I've ever seen in textbooks (either college-level or otherwise), so I take that as nonstandard and not qualifying as well-written algebra. </div><div align="JUSTIFY" class="western" style="font-style: normal;"><br /></div>What do you think? Have I missed any important cases or examples?<br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com4