tag:blogger.com,1999:blog-77184627935169688832015-03-02T05:00:00.886-05:00AngryMath"Beauty is the Enemy of Expression"Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.comBlogger171125tag: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.com5tag: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.com17tag: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.com4tag:blogger.com,1999:blog-7718462793516968883.post-68529111621439005902014-07-21T05:00:00.000-04:002014-07-21T18:05:59.528-04:00Automatic NegativesIn my developmental (remedial) courses, I have been thinking more lately about where and how to communicate which skills need to be automatic -- that is, <i>instantaneous</i> and <i>always correct</i>. As I wrote earlier (<a href="http://www.angrymath.com/2013/12/automatic-drills.html">link</a>), these are skills which we take for granted in higher level courses, but they frequently get lost in the underbrush of all the other math topics, and students expect that struggling with them for several minutes is acceptable and normal behavior. A list of these remedial automatic skills that students often lack includes:<br /><ul><li>Times tables</li><li>Signed operations (add, subtract, multiply, divide)</li><li>Rounding whole numbers & decimals</li><li>Comparing decimals</li><li>Converting between decimal & percent</li><li>Etc.</li></ul>These are the kinds of things that seem "obvious" to properly-educated people, but if no one ever communicated them to a student in that way, they are opaque. For example, in each of the last few semesters in my sophomore statistics class, I've had "A" students, otherwise doing very well, but who were completely mystified at how we were converting decimal probabilities to percent on the fly. Notice that these are all <i>one-step</i>, immediate mental tasks: I wanted to include order-of-operations knowledge here (it's so critically important), but the truth is those tasks are inherently multi-step problems, so they don't really belong on the list above.<br /><br />But let me focus more on the issue of negative numbers (signed operations). I find these to be the greatest stumbling block in students getting through the bottleneck remedial algebra course -- I can pull up any test, including finals, and see that usually at least half of the errors are simply signed-number mistakes. A student can know everything in the algebra course, but if they routinely trip over negatives even after I've begged them to practice it for a whole semester, then they have practically no chance of passing the final.<br /><br />In June, I had the opportunity to teach an immersive one-week workshop for students who narrowly missed passing our department's prealgebra final (basic arithmetic with different types of numbers: integers, fractions, decimals, percent). This was a great experience, the students were hard-working and highly appreciative, and it gave me a chance to further focus on this issue. I was trying to do frequent one-minute speed drills on things like negative operations, and some students were having what seemed like an inordinately difficult time with them -- particularly the subtractions. So that night I sat down at the bus stop and tried to think through really carefully what we <i>really</i> do in practice as proficient math people.<br /><br />Here's the thing: Not all negative operations are <i>single-step</i>. In particular, consider subtracting a negative number, written inside parentheses. I find that a lot of students are taught this bumfungled "keep change change" methodology: they will transform expressions as follows:<br /><ul><li>3−(−9) = 3+(+9)</li><li>5+(−8) = 5−(+8)</li><li>1−(+7) = 1+(−7)</li><li>4+(+6) = 4−(−6)</li></ul>Now, all of these are true statements. <u><i>But only some of them are helpful</i></u>. It's not like the students' prior instructors were lying to them, except in regards to when this fact is useful in simplifying an expression with signs (namely, the 1st and 3rd cases above). Let's look at how a math professional would <i>really</i> do it. These are two-step problems; really, we would follow the order-of-operations and get rid of the parentheses in what's really a <i>multiplying</i> step, then combine like terms in an <i>addition</i> step.<br /><ul><li>3−(−9) = 3+9 = 12</li><li>5+(−8) = 5−8 = −3</li></ul>Etc. So the lesson is that if an instructor shows students how to mangle signs in & around parentheses, they are really missing the point; when simplifying (evaluating), we will <u><i>remove the parentheses entirely in the multiply step</i></u>, and then always <u><i>perform add/subtracts without any parentheses</i></u> in the picture. <br /><br />So in the current discussion, this informs us as to what we should be drilling students for "automaticity" in terms of negative number operations: namely, <i>combining terms with no parentheses </i>has to be the automatic one-step skill. If you want to explain this as effectively adding terms that's fine; but don't fail to clearly communicate that this is expected to be instantaneous and immediate, in one mental step, in practice.<br /><br /><ul><li>2−6 ← Automatic drill ok</li><li>−7+1 ← Automatic drill ok</li><li> 6−7+2 ← Automatic drill ok</li><li>5−(−2) ← Not automatic drill ok (2-step problem)</li></ul>I feel like this was an important lesson I got to learn from this summer's immersion workshop. A speed drill can include automatically multiplying or dividing integers, or combining terms with no parentheses -- but add/subtracts with parentheses don't belong in the same category, because you really do need to apply two separate simplifying steps for them. And perhaps most important of all: clearly communicate that the one plan that always succeeds is the standard order-of-operations, not a score of different random manipulations to memorize for different situations. <br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-54391899762569591942014-07-07T05:00:00.000-04:002014-07-07T05:00:06.104-04:00Multiple Choice ExpectationsA while back I considered the chance to pass a standard multiple-choice final exam (<a href="http://www.angrymath.com/2014/04/multiple-choice-chances.html">link</a>), granted a certain basis of actual knowledge of the material. Today let's look at it from the other perspective, i.e., what the expected score is for different levels of actual knowledge:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-cD_BqK8gPEI/U6HsK85mACI/AAAAAAAADGs/E_bvtna0ZQ0/s1600/MultipleChoiceExpectation.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-cD_BqK8gPEI/U6HsK85mACI/AAAAAAAADGs/E_bvtna0ZQ0/s1600/MultipleChoiceExpectation.gif" height="320" width="234" /></a></div><br /><br />As you can see, if we pass a student with a 60% score on such a multiple-choice test, then the most likely bet (point estimate) for their true level of knowledge is around 11 or 12 of the questions, that is, less than half of the actual content for the course.<br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-13461030610177973412014-06-28T12:44:00.002-04:002014-07-19T02:26:48.659-04:00Happy Tau Day!<div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-9YOLuqB8dpM/U67we2XaIAI/AAAAAAAADG8/apfeEnoOpl8/s1600/tauism_rotated.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-9YOLuqB8dpM/U67we2XaIAI/AAAAAAAADG8/apfeEnoOpl8/s1600/tauism_rotated.png" height="200" width="198" /></a></div>Happy Tau Day! Celebrating τ = 2π = 6.28..., which simplifies all the math formulas that include 2π to make one full turn around a circle. At our house, we'll be having tacos, and tequila, and tarts. Check out the Tau Day manifesto, especially the excellent video by Vi Hart, linked at the bottom of the page -- <a href="http://www.tauday.com/" rel="nofollow" target="_blank">http://tauday.com/</a><br /><br />(Coincidentally, it's <i>also</i> the Perfect Day, because 6 and 28 are the only "perfect numbers" that appear on a calendar -- <a href="http://en.wikipedia.org/wiki/Perfect_number">http://en.wikipedia.org/wiki/Perfect_number</a>.)<br /><br /><br />Edit: A full course of full circles for Tau Day!<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-lyeMomuB8Z0/U694eBVJ0HI/AAAAAAAADHM/2q3oxD1cKt8/s1600/Image06282014200434.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-lyeMomuB8Z0/U694eBVJ0HI/AAAAAAAADHM/2q3oxD1cKt8/s1600/Image06282014200434.JPG" height="240" width="320" /></a></div>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-42878059201575384452014-06-04T11:30:00.000-04:002014-10-24T18:10:38.070-04:00Reformatted WritingA short observation that I sometimes use to my advantage: It helps me greatly if I write something in one format, and then look at it in some reformatted view, before finally distributing it publicly. This tends to dredge up a number of subtle errors or weaknesses which I otherwise can't see. In some sense it gives me "fresh eyes" to really read the draft from a new perspective (as opposed to one's memory blocking reception of what's actually on the page). In this sense we might say that "what you see is what you get" (WYSIWYG) is actually a hindrance instead of a help. A few cases:<br /><ul><li><b>Blogger.</b> Editing blog posts here, one uses a rich-text editor that is different from the final HTML markup on the site. One clicks "preview" to actually see it reformatted as it will appear on the Blogger site. Something that happens occasionally is that I might have a duplicated "the the" (or something) across a line break, such that I don't initially see it; when the paragraphs and lines get moved around in Preview, I can catch this much more easily. More generally, I can pick up on weaknesses in sentence structure and places I can clarify much more easily in the second view perspective.</li><br /><li><b>Lecture Extracts.</b> For a couple of years now I've been providing one-page review guides to students in all of my math classes. I accomplish this by copy-and-pasting all of the special defined terms and procedures in my lecture notes to the review sheet. When I did this, the surprising side-benefit was that I discovered a lot of those definitions had varying formats, tenses, or parts-of-speech, that looked sort of ridiculous when lined up next to each other -- and then I could go back and fix them throughout my class lecture notes. (Arguably you could say this is bad practice due to data duplication; anytime I make changes in my lectures I have to edit the review sheet in parallel. But it turns out that this is not a burdensome task.)</li><br /><li><b>Java Code.</b> Just recently I've committed to formatting all my personal coding projects in the <a href="http://en.wikipedia.org/wiki/Javadoc">javadoc</a> format (special comments that can be parsed out automatically to external documentation). This required a change in style that I thought would be irritating, but was much less painful than I expected, and more than compensated for by the benefits. Again, if I write my code once and then generate the documentation and look at that, I'm finding there's a whole lot of places I can improve on variable names, method names, comments on use, etc. Looking at it from the perspective of a true "outsider", with only the cues someone would start with to theoretically get into my code, gives my end product much greater depth and robustness.</li></ul>So in summary: Write once, <i>view in a totally different format</i>, and then edit. Results are improved to a surprising degree.<br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-79321541427108006032014-05-05T05:00:00.000-04:002014-05-05T05:00:03.720-04:00Basic Logic ErrorsI constantly wish that students were taught rudimentary logic at an early age (links: <a href="http://www.angrymath.com/2012/07/teach-logic.html">one</a>, <a href="http://www.angrymath.com/2013/06/the-war-on-structure.html">two</a>, <a href="http://www.angrymath.com/2013/09/reasons-remedial-is-rough.html">three</a>). Just musing about that today, here are three common stumbling blocks I see in different classes due to not being able to read logical statements properly: <br /><ol><li><b>"If" Statement.</b> In a basic algebra class, we have the rule "If the base is negative, then even powers are positive, but odd powers are negative". Immediately after that, I'll always have some students incorrectly evaluate something like: −5² = 25 (or worse, 2³ = −8) . Note that the base of the exponent is <i>not</i> negative, but some students overlook the check required for the "if" qualifier.</li><br /><li><b>"Or" Statement.</b> In an elementary statistics class, we have the rule "To estimate a population mean, we must have either a normal population or a large sample size." Then when I ask the class "Do we need a normal population?", the entire class will always incorrectly respond with "Yes!" the first time. Of course that's not true; they're overlooking that only one case of the "or" needs to be satisfied (most commonly by just a large sample size). It takes several sessions of quizzing them on that before they are sensitive to the question being asked.</li><br /><li><b>"And" Statement.</b> In practically any class, we might have the policy, "To pass this class you need at least a 60% weighted average, <i>and</i> a 60% score on the final exam." This constantly causes confusion and aggravation. Testy "So, the final exam doesn't count?", or "So, only the final exam counts?" are questions that I routinely have to address. Obviously, students are unclear on the fact that each of two requirements must be satisfied for an "and" statement like that.</li></ol>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-51595814093610256982014-04-15T05:00:00.000-04:002014-04-15T13:07:13.238-04:00Multiple Choice ChancesLet's say you have a final-exam assessment that is a multiple-choice test, with 25 questions, each of which has 4 options, and requires a 60% score (15 questions correct) to pass. As one example, consider the uniform CUNY Elementary Algebra Final Exam (<a href="http://www.ccny.cuny.edu/testing/cunyelementaryalgebrafinalexam.cfm">link</a>).<br /><br />How robust is this as an assessment of mastery in the discipline? As a simple model, let's say that any student definitely knows how to answer N types of questions, but is randomly guessing (uniform distribution over 4 options) for the other questions. Obviously this abstracts out the possibility that some students know parts of certain questions and can eliminate certain choices or guess based on the overall "shape" of the question, but it's a reasonable first-degree model. Then the chance to pass the exam for different levels of knowledge is as follows:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-gg2NqJrOnwk/U0dUCsF0vDI/AAAAAAAAC2g/Ey7Ce8Wnt7Y/s1600/MultipleChoiceChance.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-gg2NqJrOnwk/U0dUCsF0vDI/AAAAAAAAC2g/Ey7Ce8Wnt7Y/s1600/MultipleChoiceChance.gif" height="320" width="247" /></a></div><br />Obviously, if a student really "knows" how to answer 15 or more questions, then they will certainly pass this test (omitted from the table). But even if they only know half of the material in the course, then they will probably pass the test (12 questions known: 67% likely to pass). Of students who only ever know about one-third of the basic algebra content, but retake the class 3 times, about half can be expected to pass based on the strength of random guessing on the rest of the test (9 questions known: 19% likely to pass; over 3 attempts chance to pass is 1-(1-0.19)^3 = 1-0.53 = 0.47). <br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-78371500534875104522014-04-03T05:00:00.000-04:002014-04-03T05:00:04.220-04:00Meta-Research Innovation CentreInteresting article about Dr. John Ionnidis at Stanford founding the "Meta-Research Innovation Centre" to monitor and combat weak and flawed statistical methods in science research papers, especially medicine. Good luck to him!<br /><br /><div style="text-align: center;"><a href="http://www.economist.com/news/science-and-technology/21598944-sloppy-researchers-beware-new-institute-has-you-its-sights-metaphysicians">http://www.economist.com/news/science-and-technology/21598944-sloppy-researchers-beware-new-institute-has-you-its-sights-metaphysicians</a></div><div style="text-align: center;"><br /></div><div style="text-align: center;"><br /></div>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-33623129040534301902014-03-27T05:00:00.001-04:002015-02-01T22:48:14.514-05:00Gears of WarWhen I was a kid, one of my favorite pastimes was the Avalon Hill wargame <i>Bismarck</i> about fighting ships in World War II (see it reviewed on my gaming site <a href="http://deltasdnd.blogspot.com/2010/10/retrospective-bismarck_11.html">here</a> and <a href="http://deltasdnd.blogspot.com/2011/05/bismarck-for-two.html">here</a>). In junior high school, at some point my English teacher asked me what I wanted to do as a career, and was completely apalled when I said "I want to join the Navy and control the main guns on a battleship". (I think I'd share her dismay if someone told me something like that today.)<br /><br />Anyway, over at Ars Technica, a wonderful article has been written by Sean Gallagher (former Navy officer and IT editor) on exactly how the fire control systems on those ships did their jobs -- solving 20-variable calculus problems in real-time (accounting for moving, pitching, rolling, recoiling, Cariolis-spinning projectiles on both ends) with shafts and gears, with accuracy that is hard to beat even today with digital computers and GPS-driven rocketry. There are lots of insightful videos about the components and gears used to do input, sums, multiplies and divides, and spinning disks that can do complicated functions like trigonometry and more. <br /><br />To me, this stuff is completely like crack. Check it out:<br /><br /><div style="text-align: center;"><a href="http://arstechnica.com/information-technology/2014/03/gears-of-war-when-mechanical-analog-computers-ruled-the-waves/">http://arstechnica.com/information-technology/2014/03/gears-of-war-when-mechanical-analog-computers-ruled-the-waves/</a></div><br />(Also: Further commentary and links at recently-established news site <a href="http://soylentnews.org/articles/14/03/18/137232.shtml">SoylentNews</a>.)<br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-89194205590555579402014-03-20T05:00:00.000-04:002014-03-20T05:00:09.783-04:00FiveThirtyEightNate Silver (statistician who famously predicted all 50 states voting in the last election) recently expanded his FiveThirtyeight blog to a full-blown "data journalism" site. His first post was a manifesto on data, science, statistics, politics, journalism, and honest storytelling in general. I agree with almost all of his observations here. They guy really knows his stuff and has a fiery passion for his particular mission. Great stuff.<br /><br /><div style="text-align: center;"><span style="font-size: large;"><a href="http://fivethirtyeight.com/features/what-the-fox-knows/">http://fivethirtyeight.com/features/what-the-fox-knows/</a></span> </div><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-24691277182983874042014-03-10T05:00:00.000-04:002014-03-10T17:43:00.985-04:00Faulty FactoringHere's something I think I see a few times in any college algebra class: a really weird way of accomplishing quadratic factoring. (More generally, this might go in a larger file of "things students swear are taught by other instructors which are semi-insane" -- including whacked-out order-of-operations, keep-change-change for negatives, the idea that -4<sup>2</sup> means (-4)(4), etc.).<br /><br />Anyway, let's say we want to factor what I call a "hard quadratic", i.e., Ax<sup>2</sup>+Bx+C, in integers, with A≠1 (hence "hard"). I prefer the method of grouping: i.e., factor AC so it sums to B, use those factors to split the term Bx, and then factor the four terms by grouping. Pretty straightforward.<br /><br />But here's what a few students will insist on doing every semester: (1) Find factors of AC that sum to B; call these factors u & v (so that step is the same); (2) Write the expression (Ax+u)(Ax+v); (3) Look for a GCF of A in one of those binomials and strike it out.<br /><br />Here's an example: Factor 5x<sup>2</sup>+7x−6.<br />Step (1): Note AC = −30 = (10)(−3), factors which sum to B = 7.<br />Step (2): Write (5x+10)(5x−3)<br />Step (3): Divide the first binomial by 5, producing (x+2)(5x−3).<br /><br />So while this procedure does produce the right answer, what irks me tremendously is that the expression written in step (2) is not actually equal to either the original expression or the answer at the end. (Compounding this issue, students will nonetheless usually write equals signs by rote on either side of it.) Riffs on this procedure would be to write something like this on sequential lines, if you can follow it:<br /><br />5x<sup>2</sup>+7x−6 → x<sup>2</sup>+7x−30 → (5x+10/5)(5x−3/5) → (x+2)(5x−3)<br /><br />Again, the primary grief I have over this is that none of these expressions are equal to any of the others, and the students using this procedure are always oblivious to that fact. Second issue: They're likely to trip up over a non-elementary problem where the factor A does not appear in either of the binomials, e.g.: 4x<sup>2</sup>+4x+1 = (2x+1)(2x+1). Third issue: If there's a GCF in the quadratic itself and you overlook that, the standard grouping technique will still work (even if it's not the easiest way to do it), whereas I suspect users of this technique will be prone to incorrectly striking out any GCFs they discover at the end of the process. <br /><br />Now, technically you could modify this and turn it into a correct procedure this way: Note that for quadratic Ax<sup>2</sup>+Bx+C, values u & v satisfy uv=AC and u+v=B if and only if Ax<sup>2</sup>+Bx+C = 1/A(Ax+u)(Ax+v). (Proof: 1/A(Ax+u)(Ax+v) = 1/A(A<sup>2</sup>x<sup>2</sup> + (u+v)Ax + uv) = Ax<sup>2</sup> + (u+v)x + uv/A and equate coefficients). So you could find u & v as usual, then write this latter expression, and simplify. The 1/A does always cancel out, but I've never seen a student actually write that factor in the second step.<br /><br />So what I always do if I see this on a test in my college algebra class is to take half credit off for the problem and note that the intermediary expression is "false", i.e., not equal to what comes before or after. This then becomes an opportunity to discuss with the student why that's improperly written math -- went well in my most recent semester, but I can easily see that becoming more combative in a remedial algebra class.<br /><br />Have you seen this (common) faulty factoring procedure in your classes? What do you as a correction for it, if anything?<br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com4tag:blogger.com,1999:blog-7718462793516968883.post-27377960831551922942014-03-05T15:16:00.004-05:002014-03-05T15:16:35.243-05:00Presenting at Johns HopkinsHere's one of these topics that merges my great interests in teaching & gaming, so I have no choice but to cross-post about it here and on my gaming blog.<br /><br />Last week I had the opportunity to visit Johns Hopkins University, at the invitation of Peter Fröhlich to speak to his Video Game Design Project class in the computer science department there (run jointly with art students from the nearby MICA). A great talk and chance to meet with his students and network a bit with Peter, Jason from MICA, as well as one of my idols from old-school role-playing game publishing. <br /><br /><a href="http://deltasdnd.blogspot.com/2014/03/presenting-at-johns-hopkins.html">Bounce on over to my gaming blog for the details! </a><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-47997406752726784312014-02-24T05:00:00.000-05:002014-02-24T05:00:06.774-05:00Research in College Algebra Basic SkillsHere's something that I'm finding frustrating: for all the mountain of ink spilled on the issue of remedial math in colleges (including enormous numbers taking them, the fact that it's <i>the</i> critical determination of whether people get a college degree or not, dim prospects of existing placement tests, etc., etc., etc.), when I search for papers where someone has tried to correlate specific math skills of incoming students to success in college remedial algebra -- <i>I come up totally empty</i>. <br /><br />Weirdly, I can find studies that correlate specific diagnostic test questions in basic math skills to other classes. Here's one relating specific math skills to success in college<a href="http://www.amstat.org/publications/jse/v14n2/johnson.html"> statistics classes</a>. Here's <a href="http://www.amstat.org/publications/jse/v19n1/lunsford.pdf">another</a>. Here's a study relating basic math skills to success in <a href="http://econ.aplia.com/downloads/mathpaper011602.pdf">economics classes</a>. <br /><br />But predicting success in basic algebra classes? I'm coming up totally empty. I'm truly bewildered at this -- part of me can't possibly believe that no one has published results like that, but part of me is stewing from returning to this futile search many days over and over again.<br /><br />Does anyone know of such research linking specific skill questions to success in college remedial algebra? Or <i>any</i> college algebra classes?<br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com4tag:blogger.com,1999:blog-7718462793516968883.post-86058726483958515172014-02-17T05:00:00.000-05:002014-02-17T05:00:08.078-05:00Precipitation ProbabilityThis winter module I've had a batch of students in my introductory statistics course who are so aggressively intelligent that they've spotted every single spot where I had any gray area or ambiguity in my lectures. In places I do this knowingly to simplify the subject, and prepare backup answers in case anyone asks -- this semester is the first time where every single one of backups got used, and then some. This will definitely benefit my classes in the future, and in fact, I learned a few things myself along the way. For example:<br /><br />At the end of the probability concepts section, the major thing I want students to do is to interpret probability statements (which for some is the most difficult part of the course, never having encountered probability concepts before). I give a quiz question on the classic weather forecast precipitation probability: "Interpret this probability statement: 'There is a 40% chance of rain today in the New York area'". So personally, I always took this to mean that there was a 40% chance of getting any rain at all in New York today (40% chance for a drop of measurable rain somewhere in New York; i.e., over many days like today 40% of such days will get a drop of rain or more in New York). <br /><br />But one of my students not only started researching this on her own, <b>she actually called the New York weather service</b> to ask a meteorologist how this was computed. She still didn't get the interpretation quite right (one of the few questions she missed all semester), but the discussion was enlightening for both of us.<br /><br />The truth is that the weather forecast statement is in regards to rain at <i>any random location in New York</i>, not actually the rain for New York as a whole. I suppose that is really a more useful statement, after all. The publicized percentages are computed by multiplying the expected coverage area percent by the probability of rain occurring in that area (so if it's 40% likely that 100% of the area gets rain, you report the same result as an 80% chance that 50% of the ground gets rain). Therefore: What's being reported is the chance that any arbitrary point in New York gets measurable rain; i.e., 40% indicates that for any random point in New York, if we observe many days with conditions like today, 40% of such days get a measurable drop of rain on that point-location.<br /><br />Links to more information:<br /><ul><li>Comments from the National Weather Service, reposted at the University of Texas at Austin website: <a href="https://www.utexas.edu/depts/grg/kimmel/nwsforecasts.html">here</a>.</li><li>Video from a meterologist in Boulder, including citation to the 2005 Gigerenzer et. al. paper in Risk Analysis which surveyed people for their understanding of these statements (where I got my quiz question in the first place):<a href="http://blog.petermcgraw.org/2010/09/how-to-better-understand-your-weather-forecast-or-what-does-a-40-chance-of-rain-really-mean/"> here</a>.</li></ul>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com1