tag:blogger.com,1999:blog-77184627935169688832016-09-26T05:00:15.937-04:00MadMath"Beauty is the Enemy of Expression"Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.comBlogger248125tag:blogger.com,1999:blog-7718462793516968883.post-65669771878443712682016-09-26T05:00:00.000-04:002016-09-26T05:00:15.948-04:00When Blind People Do AlgebraFrom NPR: <br /><blockquote class="tr_bq"><i>A functional MRI study of 17 people blind since birth found that areas of visual cortex became active when the participants were asked to solve algebra problems, a team from Johns Hopkins reports in the Proceedings of the National Academy of Sciences. </i></blockquote><br />This is not the case with the same test run on sighted people. Which is interesting, because it serves as evidence that the brain is more flexible, and can be re-wired for a greater multitude of jobs, than previously believed.<br /><br /><a href="http://www.npr.org/sections/health-shots/2016/09/19/494593600/when-blind-people-do-algebra-the-brain-s-visual-areas-light-up">Read more here. </a>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-79251027888710678032016-09-19T05:00:00.000-04:002016-09-19T23:18:18.042-04:00NY Times: Stop Grading to a CurveAn excellent article by Adam Grant, professor at the Wharton School of the University of Pennsylvania:<br /><blockquote class="tr_bq"><i>The more important argument against grade curves is that they create an atmosphere that’s toxic by pitting students against one another. At best, it creates a hypercompetitive culture, and at worst, it sends students the message that the world is a zero-sum game: Your success means my failure.</i></blockquote><br /><a href="http://www.nytimes.com/2016/09/11/opinion/sunday/why-we-should-stop-grading-students-on-a-curve.html">Read the full article here. </a>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-35519598258148633612016-09-05T05:00:00.000-04:002016-09-05T12:49:03.499-04:00Epsilon-Delta, Absolute Values, InequalitiesWorking through the famed "baby" Rudin, <i>Principles of Mathematical Analysis</i>. (Which was not the analysis book I used in grad school: we used William Ray's <i>Real Analysis</i>). <br /><br />First thing that dawns on me is that I'm shaky on following the formal proofs of various limits. I decide that I need to brush up on my epsilon-delta proofs, and go back to do exercises from Stein/Barcellos <i>Calculus and Analytic Geometry</i>. I find that I get stuck outside of simple limits of linear functions (say, with a quadratic). <br /><br />Second observation is that these proofs use extensive algebra involving absolute value inequalities, and my trouble is that, in turn, I am shaky with these to the point of being blind to a lot of very basic facts. Kind of frustrating that I teach basic algebra as a career but as soon as as absolute values and inequalities enter the picture I'm nigh-helpless. <br /><br />Third observation is that this explains the copious section in college algebra and precalculus texts on absolute value inequalities (which are skipped in the curriculum at our community college). Previously my intuition had failed to see the use for these; but it's precisely the skills you need in analysis to write demonstrations involving the limit definition.<br /><br />A few key properties on which I had to brush up (and would argue for preparatory exercises if I was teaching/scaffolding in the direction of analysis):<br /><ul><li><b>Subadditivity:</b> \(|a+b| \leq |a| + |b|\). Equivalent to the triangle inequality. </li><li><b>Partial Reverse Triangle Inequality:</b> \(|a| - |b| \leq |a - b|\). A bastardized name of my design. It follows from the preceding because \(|a| = |a - b + b| \leq |a-b| + |b|\), and then a subtraction of \(|b|\) from both sides. Gets used at the start of almost all my limit proofs. </li><li><b>Multiplicativeness:</b> \(|ab| = |a||b|\). Which leads to manipulations such as: multiplying both sides of an equation by a positive number allows you to just multiply into an absolute value; squaring both sides can be done into the absolute value, etc. </li></ul><br />More at: <a href="https://en.wikipedia.org/wiki/Absolute_value#Definition_and_properties">Wikipedia</a>.<br /><br />Discussion of general limit exercises: <a href="http://math.stackexchange.com/questions/598796/creating-a-question-that-use-the-epsilon-delta-definition-to-prove-that-f">StackExchange</a>.<br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com2tag:blogger.com,1999:blog-7718462793516968883.post-58789956382579309932016-08-26T05:00:00.000-04:002016-08-26T05:00:29.735-04:00Crypto Receipts for HomeworkAn interesting idea to a problem I've also experienced (student claiming they submitted work for which the instructor has no record): Individual cryptographic receipts for assignment submissions.<br /><br /><div style="text-align: center;"><span style="font-size: large;"><a href="https://intoverflow.wordpress.com/2010/06/27/crypo-in-the-classroom-digital-signatures-for-homework/">Crypto in the classroom: digital signatures for homework</a></span></div><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-27989721973999726772016-08-19T05:00:00.000-04:002016-08-19T05:00:03.916-04:00Link: Everything is Fucked, The SyllabusBy Prof. Sanjay Srivastava, a proposed course on the overall breakdown of science in the field of social psychology:<br /><br /><div style="text-align: center;"><span style="font-size: x-large;"><a href="https://hardsci.wordpress.com/2016/08/11/everything-is-fucked-the-syllabus/">Everything is fucked: The syllabus</a></span></div><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-82892564534355664322016-08-12T05:00:00.000-04:002016-08-12T05:00:01.376-04:00Natural Normality<span class="fbPhotosPhotoCaption" data-ft="{"tn":"K"}" id="fbPhotoSnowliftCaption" tabindex="0"><span class="hasCaption">Normal curve in flag sticker water-leak (upside-down), 2016:</span></span><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-SRxCVpRZHsU/V6wNHsGyT7I/AAAAAAAAEEU/EElJ38q5Nj4QzcIJlWlY-wpANTiUUyL0ACLcB/s1600/Image08102016123331.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="240" src="https://4.bp.blogspot.com/-SRxCVpRZHsU/V6wNHsGyT7I/AAAAAAAAEEU/EElJ38q5Nj4QzcIJlWlY-wpANTiUUyL0ACLcB/s320/Image08102016123331.jpg" width="320" /></a></div><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-493494817321934072016-07-25T05:00:00.000-04:002016-07-25T05:00:15.208-04:00Teaching Math with Overhead PresentationsAt our school, we currently have a mixture of classroom facilities. Some classrooms just have classic chalkboards (and nothing else), while other classrooms are outfit with whiteboards, computer lecterns, and overhead projectors. Well, actually: All of the classrooms on campus have the latter (for a few years now), excepting only the mathematics wing, which has been kept classical-style by consensus of the mathematics faculty.<br /><br />That said, it's been one year since I've converted all of my classes over to use of the overhead projector for presentations. In our situation, I need to file room-change requests every semester to specially move my classes outside the math wing to make this happen. But I've been extremely happy with the results. While this may be very late to the party in academia as a whole, this identifies me as one of the "cutting-edge tech guys" in our department. Note that I'm using LibreOffice Impress for my presentations (not MS PowerPoint), which I need to carry on a mobile device or network drive. My general idiom is to have one slide of definitions, a theorem, or a process on the slide for about 30 minutes at a time, while I discuss and write exercises on the surrounding whiteboard by hand.<br /><br />As a recent switchover, I wanted to document the reasons why I prefer this methodology while they're still fresh for me; and I say this as someone who rather vigorously defended carefully writing everything by hand on the chalkboard in the past. Here goes: <br /><br /><h3>Advantages of Overhead Presentations in a Math Class </h3><ol><li>Saves time writing in class. (I think I recoup at least 20 minutes time in one of my standard 2-hour classes.)</li><li>Additional clarity in written notes. (I can depend on the presentation being laid out just-so, not dependent on my handwriting, emotional state, time pressures, etc.)</li><li>Can continue to face forward & speak towards the students most of the time.</li><li>Don't have to re-transcribe the same material every semester. (Which simply seems inefficient.)</li><li>Can have additional graphics, tables, and web links that are time-constrained by hand.</li><li>Easy to go back and pull up old slides to review at a later time. We may do this within one class period, after "erasure", or across prior lectures. For extremely weak community-college students who can't seem to remember any content from day to day, we frequently must pull up slides and definitions from earlier classes; and this gives confidence when students in fact disbelieve that the material was covered earlier. Actually, this is most critical in the lowest-level remedial courses (the first course I tried it on out of desperation one day, with great success and student responsiveness). </li><li>I can carry around the same slides on a tablet in class. This allows me to assist students while they work on exercises individually and I circulate. Similar to the prior point, if one student forgets, needs a reminder, or was absent in a prior class, I can show them the needed content in my hand without distracting the rest of the class. </li><li>Greatly helps reviewing for the final exam. I can quickly pull up any topic in its entirety from across the entire semester. (Again, trying to jog students' memories by showing it again in the exact same layout.) </li><li>Having the computer overhead allows demonstration of technical tools, like navigating the learning management system, using certain web tools, a programming compiler, etc.</li><li>Able to distribute the lecture material to students directly and digitally. </li><li>Enormously reduces paper usage. All of my former handouts, practice tests, science articles, etc., which used to generate stacks of copied paper are now shown on the overhead instead (on made available on the learning management system if students want a copy later). </li><li>Having written expressions in the LibreOffice math editor, it makes it compatible to copy-paste into other platforms like LATEX, MathJax, Wolfram Alpha, etc.</li><li>If for some reason I transition to a larger auditorium for a particular presentation (e.g., someone organized a "classroom showcase" of this type last fall), I'm ready to go with the same presentation available on a larger screen. </li><li>Less to erase (saves clean-up time). There's actually a mandated policy to clean the board after one's class at my school, and people have gotten very testy in the past if one suggests not doing that. </li></ol>I only anticipated maybe the first 5 of these items above before starting to switch my classes over; once I saw all the very important side-benefits, I became a complete devotee, and transitioned all my classes. I have a hard time seeing that I'd ever want to go back now. That said, I need to be careful about a few things: <br /><br /><h3>Problem Areas of Overhead Presentations in a Math Class</h3><ol><li>Boot-up time. I need to get into class about 10 minutes earlier to boot-up all the technology -- the computer itself, the network login, directories, starting LibreOffice mobile, opening the files, starting the browser if needed, etc. Our network is very slow and it's a bit unpleasant waiting for things to open up. Meanwhile, students are eager to ask questions, so overall this is a somewhat awkward/anxious moment in time. It would be better if our network were faster, and/or LibreOffice were installed on the local machines (which to date has been turned down by IT). </li><li>Presenter view doesn't work on our machines. This is normally a sidebar-view visible to the presenter which shows notes and the next slide to come, etc. Unfortunately our machines weren't supplied with a separate monitor feed to make this work. During the lecture I have to keep my tablet up and manually synchronized on the side to keep track of verbal notes and so forth. </li><li>Making slides available to students is imperfect. I can just give LibreOffice files, because most students don't have it, or possibly the technical confidence to install it. I can provide it as PDFs, but then students usually print them out one-giant-slide-per-page, resulting in everyone with a huge ream of paper. I've directed people to print them (if they must) 6-slides-per-page, but most can't follow that direction. At the moment, LibreOffice cannot directly export a multi-slide-per-page PDF. (And even the normally printing facility has a <a href="https://www.bountysource.com/issues/35865581-printing-handouts-ignores-selected-order">bug</a>.)</li><li>Lecterns have an external power button, and unfortunately I tend to lean on the lectern, press up against the button, and shut off power to the entire system by accident. This is hugely embarrassing and triggers another 10-minute boot-up sequence. </li><li>Some lecterns also have a very short power-saving time set (like a default of 20 minutes; note that's less time than I expect to spend on a single slide, as stated above). Evey day I need to remember to set the power-save mode in those rooms to 60 minutes when I start. </li><li>Finally, based on my use-case, the handheld presentation clicker which I purchased is not as useful as first expected. It presents another tech item which has a several-minute discovery/boot-up sequence to get started, and I tend to press it by accident or pinch it in my pocket and advance the slide past what I'm talking about. For me I've found it better to discard that and just step to the keyboard when I need to advance the slide (again, just a few times per half-hour). </li></ol><br />Nonetheless, all of these warning-notes are fairly minor and eminently manageable; the advantages of using the overhead in my classes pay dividends many times over. Among the top benefits are time-savings, clarity, ability to go back, opening up web access and other materials in class, digital distribution and compatibility, etc. In contrast, here is a question on <a href="http://mathoverflow.net/questions/5936/whats-so-great-about-blackboards">MathOverflow from late 2009</a>, where the responding mathematicians all seem very adamant about using traditional blackboards. At this point, I really don't grok any of these arguments for not using the overhead. <br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com2tag:blogger.com,1999:blog-7718462793516968883.post-80772158317302622992016-07-22T05:00:00.000-04:002016-07-22T05:00:02.906-04:00Paul Halmos on ProofsPaul Halmos on mathematical proof:<br /><blockquote class="tr_bq"><span style="font-size: large;">Don't just read it; fight it!</span></blockquote>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-47559962659061365072016-07-15T05:00:00.000-04:002016-07-16T13:41:23.044-04:00Names for InequalitiesConsider an inequality of the form <i>a < x < b</i>, that is, <i>a < x</i> and <i>x < b</i>. Trying to find a name for this type of inequality, I'm finding a thicket of different terminology:<br /><ul><li><b>Chained</b> inequalities (Wikipedia). </li><li><b>Combined</b> inequalities (Sullivan <i>Algebra & Trigonometry</i>). </li><li><b>Compound</b> Inequalities (Ratti & McWatters <i>Precalculus</i>, Bittinger <i>Intermediate Algebra</i>, OpenStax <i>College Algebra</i>). </li></ul><br />Are there more? What is most common in your experience? Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com2tag:blogger.com,1999:blog-7718462793516968883.post-36873599943777604192016-06-20T05:00:00.000-04:002016-06-20T05:00:21.093-04:00Purgradtory<b>Purgradtory</b> (<i>pur' gred tor e</i>) noun, plural purgradtories. <br /><br />The several days after submitting final grades when a community college teacher must field communications from students complaining about said grades, pleading for a change of grade, asking for new extra credit assignments, and/or declaring the need for a higher grade for transfer to some outside program. <br /><br /><i>Example:</i> I will likely be in purgradtory through the middle of this week. <br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-44117198463542748832016-06-03T05:00:00.000-04:002016-07-16T13:42:41.800-04:00Traub on Open AdmissionsAs recounted by Scherer and Anson in their book, <i>Community Colleges and the Access Effect</i> (2014, Chapter 11):<br /><blockquote class="tr_bq"><span style="font-size: large;"><i>Traub famously wrote in City on a Hill: Testing the American Dream at City College, a chronicling of the 1969 lowering of admissions standards motivated by the pursuit of equity, “Open admissions was one of those fundamental questions about which, finally, you had to make an almost existential choice. Realism said: It doesn’t work. Idealism said: It <u>must</u>.”</i></span></blockquote>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-41310634198637129422016-05-25T05:00:00.000-04:002016-05-25T23:53:09.183-04:00Jose Bowen's Tales from CyberspaceA few weeks ago I went to a CUNY pedagogy conference at Hostos Community College. It featured a keynote speech by Jose Antonio Bowen, author of the book <i>Teaching Naked</i>, which is nominally a manifesto for flipped classrooms, in which more "pure" interactions can occur between students and instructors during class time. Weirdly, however, he spends the majority of his time waxing prosaically about how incredible, saturated, future-shocky technology is today, and how we must work mostly to provide everything to students outside of class time using this technology.<br /><br />Here's how he started his TED-Talky address that Friday: He contrasted the once-a-week pay phone call home that college students would make a few decades ago ("Do dimes even exist anymore?") with the habits of college students today, supposedly contacting their parents a half-dozen times daily. In fact, he claimed, his 21-year-old daughter will actually call him for permission to date a young man when she first meets/starts chatting with him online. She supposedly argues in favor of the given caller by using three websites (shown floridly by Bowen on the projector behind the stage):<br /><ul><li>She has started chatting with the man on <a href="https://www.gotinder.com/">Tinder</a>.</li><li>She has looked up his dating-review score on <a href="https://onlulu.com/">Lulu</a>.</li><li>She has examined his current STD test status on <a href="https://healthvana.com/">Healthvana</a>.</li></ul>Now, that's heady stuff, and of course the audience of faculty and administrators "ooh"'ed and "ahh"'ed and "oh, my stars!"'ed in appropriate pearl-clutching fashion. Review dates and look up STD status before a date online? Kids these days -- we're so out of touch, we must change everything in the academy!<br /><br />But this presentation doesn't pass the smell test. First of all, we should be suspicious of an adult daughter supposedly interrupting her real-time chat to "get permission" from her father. That's just sort of ridiculous. Admittedly at least Tinder really is a thing and you can chat on it; that much is true. (Although Bowen presented this as the daughter and a friend communally chatting to two guys together, which is not a group event that can actually happen.) But worse:<br /><br /><b>The dating-review site Lulu doesn't actually exist anymore.</b> In February of this year (3 months ago), the site was acquired by Badoo and the dating-reviews shut down. If you go to the link above you'll realize that the whole site is offline as of this writing. (<a href="http://www.datingsitesreviews.com/index.php?topic=lulu">Link.</a>) And:<br /><br /><b>You can't access anyone else's STD result on Healthvana.</b> Yes, Healthvana is a site that allows you to quickly access and view your own STD results without returning to a doctor's office to pick them up. But it's only for <i>your own results</i>, and it requires an account and password to view them after a test. Obviously there are all kinds of federal regulations about keeping medical records private, so it's not even conceivable that those could be made available to the general public on a website. One might theoretically imagine a culture in which one pulls up your own STD records on a phone and shows it to someone you're meeting -- but there's no evidence that actually occurs, and of course it's strictly impossible in Bowen's account, in which his daughter had not yet physically met with her supposed suitor. (<a href="http://venturebeat.com/2014/10/17/healthvana-is-the-first-app-to-deliver-hiv-test-results-electronically/">Link.</a>)<br /><br />That "Reefer Madness"-like scare-mongering accounted for the first 30 minutes of Bowen's hour-long presentation, at which point I couldn't take anymore bullshit and I got up and left the auditorium. In summary: The half of Bowen's presentation that I saw was entirely fabricated and fictitious, frankly designed to frighten older faculty and staff for some reason that is opaque to me. Keep that in mind if you pick up his book or see an article or presentation by Mr. Bowen.<br /><br />How did I get clued in to the real situation with these websites, after my BS-warning radar first went off? I asked some 20-year-old friends of mine, who immediately told me that Lulu was shut down months ago, and Healthvana was nothing they'd ever heard of. Crazy idea, I know, actually talking to people without instantly fetishizing new technology. <br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-64959653935238331332016-05-13T05:00:00.000-04:002016-05-13T05:00:11.384-04:00Schmidt on Primary TeachersDooren et. al. ("The Impact of Preservice Teachers' Content Knowledge on Their Evaluation of Students' Strategies for Solving Arithmetic and Algebra Word Problems", 2002) summarize findings by S. Schmidt:<br /><blockquote class="tr_bq"><i>Nearly all students who wanted to become remedial teachers for primary and secondary education and about half of the future primary school teachers were unable to apply algebraic strategies properly or were reluctant to use them. Consequently, they experienced serious difficulties when they were confronted with more complex mathematical problems. Many of these preservice teachers perceived algebra as a difficult and obscure system based on arbitrary rules (Schmidt, 1994, 1996; Schmidt & Bednarz, 1997). </i></blockquote><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-91368277661204627012016-05-06T05:00:00.000-04:002016-05-06T05:00:11.036-04:00Noam Chomsky: Enumeration Leads to LanguageOne of my favorite videos, including a bit where famed linguist Noam Chomsky theorizes that a mutation in the brain regarding “likely recursive enumerations, allowed all human language”.<br /><br /><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/aNwWQyLWcYM/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/aNwWQyLWcYM?feature=player_embedded&t=33m25s" width="320"></iframe></div><br /><br />I have a possibly dangerous inclination to mention this on the first day of an algebra class when we define different sets of numbers, which is possibly a time-sink and a distraction for students at that point. But still, this is the guts of the thing. <br /><br />(Video should be starting at 33m25s.)<br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-16800863814631001102016-05-02T05:00:00.000-04:002016-05-06T02:15:35.419-04:00What Community College Students UnderstandReading this tonight -- Givvin, Karen B., James W. Stigler, and Belinda J. Thompson. "What community college developmental mathematics students understand about mathematics, Part II: The interviews." <i>MathAMATYC Educator</i> 2.3 (2011): 4-18. (<a href="https://www.researchgate.net/profile/Karen_Givvin/publication/228375872_What_community_college_developmental_mathematics_students_understand_about_mathematics/links/00463524f0ac0aecd8000000.pdf">Link.</a>)<br /><br />They make the argument that prior instructors' emphasis on procedure has overwhelmed students' natural, conceptual, sense-making ability. Now, I agree that terrible, knowledge-poor, even abusive math teaching in the K-6 time frame is endemic. But I'm a bit skeptical that students in this situation have a natural number sense waiting to be un-, or re-, covered. <br /><ul><li>In my experience, students in these courses commonly have no sense for numbers or magnitudes. Frequently they cannot even name numbers, decimals, places over a thousand, or know that multiplying by 10 appends a 0 to a whole number.</li><li>Givvin, et. al. assume that students "Like all young children, they had, no doubt, developed some measure of mathematical competence and intuition... ". I'm pretty skeptical of this claim (and see little evidence for it.)</li><li>One task is to check additions via subtractions, and quiz students on whether they know that either of the addends can be subtracted in the check. Admittedly, this is an unusual task: usually we take addition (or multiply or exponents) as the base operation, and later check the inverse via the more basic one -- for which the order definitely does matter. (Because addition is commutative but subtraction is not, etc.) So it's unsurprising that students' intuition is that the order matters in the check; as usually applied, it does. </li><li>Another student multiplies two fractions together when asked to compare them (obviously nonsensical). But it's easy to diagnose this: many instructors teach equating the fractions and then <i>cross-multiplying</i> them, and seeing which side has the higher resulting product; in this case the student scrambled the cross-multiplying of equations with multiplying fractions. Which highlights two things: teaching mangled mathematical writing as in this process leads to problems later; and the whole idea of <i>cross-multiplying</i> is so striking that it "sucks the oxygen out of the room" for other visually-similar concepts (like multiplying fractions). </li><li>The authors state that "These students lack an understanding of how important (and seemingly obvious) concepts relate (e.g., that 1/3 is the same as 1 divided by 3)." Not only is this not obvious, but I can repeat this about every day for a whole semester and still not have students remember it. Just this semester I had a student who literally couldn't <i>repeat it</i> when I just said it about five times. </li><li>The importance of "combining like terms", which is essentially the <i>only</i> concept under-girding the operation of addition (and subtraction and comparisons) -- in terms of like units, variables, radicals, common denominators, and decimal place values -- is highlighted here. I don't know how many times I express this, but I'm doubtful that <i>any</i> of my students have really ever understood what I'm saying. I wouldn't be surprised but some students could take a dozen years of classes and never understand this point. Which is dispiriting. </li><li>The authors have some lovely anecdotes of students making a small discovery or two within the context of the hour-to-two-hour interview. This they hold at as a hopeful sign that discovery-based learning might be an effective treatment. But I ask: How many of these students will remember their apparent discoveries outside the interview? I find it quite common for students to have "A-ha, that's so easy!" moments in class, and then have effectively no memory of it a day or two later. "I do fine when I'm with you, and then I can't do it on my own" is a fairly common refrain. </li><li>Discussing concepts is Element Two (of three) in the authors' list of prescriptions. "A teacher might, for instance, connect fractions and division, discussing that a fraction is a division in which you divide a unit into n number of pieces of equal size. Alternatively, the teacher might initiate a discussion of the equal sign, pointing out that it means 'is the same as' and not 'here comes the answer.'". Sure, I offer both of those specific explanations regularly, almost daily -- they're essential and without them you're not really discussing real math at all. But many of my students can't remember those foundational facts no matter how much I repeat or quiz them on it. </li><li>From my perspective, it almost as though most of my developmental students <i>aggressively refuse to remember</i> the overarching, connecting definitions and concepts that I try to share with them, even when they're <i>immediately</i> put to use within the scope of each daily class session. </li><li>I can't help but feel that the distinction between "procedures" versus "reasoning" is an artificial, untenable one. The authors admit, "Even efforts to capitalize on students’ intuitions<br />(as with estimating) often quickly turn to rules and procedures (as in 'rounding to the nearest')". I think this argues, perhaps, for the following: <i>All reasoning is ultimately procedural</i>. The only question is knowing what definitions and qualities of a certain situation allow a given procedure to be applied (even so simple a one as comparing the denominators of 1/5 and 1/8, for example). Even <i>counting</i> is ultimately a learned procedure. </li></ul><br />While I don't seem to have access to Part 1 of the same report, the initial draft report has a few other items I can't help but respond to:<br /><ul><li>"'Drill-and-skill' is still thought to dominate most instruction (Goldrick-Rab, 2007)." This is a now-common diatribe (my French-educated partner is aghast at the term). But let's compare to, say, the #1 top scientifically proven method for learning, according to a summary article by Dunlosky, et. al. ("What Works, What Doesn't", Scientific American Mind, Sep/Oct 2013). "Self-Testing... Unlike a test that evaluates knowledge, practice tests are done by students on their own, outside of class. Methods might include using flash cards (physical or digital) to test recall or answering the sample questions at the end of a textbook chapter. Although most students prefer to take as few tests as possible, hundreds of experiments show that self-testing improves learning and retention." Which is a somewhat elaborate way of saying: <i>Practice and homework</i>. </li><li>"The limitations in K-12 teaching methods have been well-documented in the research literature... An assumption we make in this report is that substantive improvements in mathematics learning will not occur unless we can succeed in transforming the way mathematics is taught." I would not so blithely accept that assumption. What it overlooks is the perennial decrepitude of mathematical understanding by K-6 elementary educators. My argument would be that it doesn't matter how many times you overhaul the curriculum or teaching methodology at that level; if the teachers themselves don't understand the concepts involved, there is no way that even the best curriculum or methods will be delivered or supported properly.</li><li>"Perhaps most disturbing is that the students in community college developmental mathematics courses did, for the most part, pass high school algebra. They were able, at one point, to remember enough to pass the tests they were given in high school." But were they, really? A few years ago when I was counseling a group of about a dozen of my community-college students, as they left the exit exam and thought that they had failed, I stumbled into asking exactly this question in passing: "But this is totally material that you took in junior high school, right?" To which one student replied, "But there it was just about buttering up the teacher so he liked you enough to pass you," and the other students present all nodded and seemed to agree with this. The evidence that students are being passed through the high school system on effectively fraudulent grounds seems, to me, nearly inescapable. </li></ul><br />Near the end of the Part 2 article, the authors appear to express a bit more caution concerning their hypothesis; a cautionary question which I'd be prone to answer in the negative: <br /><blockquote class="tr_bq"><i>For some students we interviewed, basic concepts of number and numeric operations were severely lacking. Whether the concepts were once there and atrophied, or whether never sufficiently developed in the first place, we cannot be certain. What we do know is that these students’ lack of conceptual understanding has, by the time they entered developmental math classes, significantly impeded the effectiveness of their application of procedures. (p. 14)</i></blockquote><blockquote class="tr_bq"><i>We hope that future work will seek to address questions such as whether community college is too late to draw upon students' intuitive concepts about math. Do those concepts still exist? Is community college too late to change students' conceptions of what math </i>is<i>? (p. 16)</i></blockquote><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-21003607248932935212016-04-29T05:00:00.000-04:002016-04-29T05:00:20.723-04:00On PiflarsIn coordination with the week's theme of grammar -- seen on StackExchange: English Language & Usage:<br /><br />Apparently in Slovenian, there is the single word "piflar", which derogatorily means "a student who only memorizes instead of truly learning". What would be the best comparable word for this in English?<br /><br /><div style="text-align: center;"><span style="font-size: large;"><a href="http://english.stackexchange.com/questions/313712/derogatory-word-describing-person-a-pupil-who-memorizes-instead-of-learning">StackExchange English Language & Usage: Derogatory word, describing person (a pupil) who memorizes instead of learning?</a></span></div><div style="text-align: center;"><br /></div><div style="text-align: center;"><br /></div>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com2tag:blogger.com,1999:blog-7718462793516968883.post-74869348942662626672016-04-25T05:00:00.000-04:002016-04-25T05:00:12.453-04:00Gruesome GrammarA week or so back we observed the rough consensus that basic arithmetic operations are essentially some kind of <a href="http://www.madmath.com/2016/04/what-part-of-speech-is-times.html">prepositions</a>. Coincidentally, tonight I'm reviewing the current edition of <a href="http://www.ck12.org/book/CK-12-Algebra-Basic/">"CK-12 Algebra - Basic"</a> (Kramer, Gloag, Gloag; May 30, 2015) -- and the <i>very first thing</i> in the book is to get this exactly wrong. Here are the first two paragraphs in the book (Sec 1.1): <br /><blockquote class="tr_bq"><i>When someone is having trouble with algebra, they may say, “I don’t speak math!” While this may seem weird to you, it is a true statement. Math, like English, French, Spanish, or Arabic, is a secondary language that you must learn in order to be successful. There are verbs and nouns in math, just like in any other language. In order to understand math, you must practice the language.<br /><br />A verb is a “doing” word, such as running, jumps, or drives. In mathematics, verbs are also “doing” words. A math verb is called an operation. Operations can be something you have used before, such as addition, multiplication, subtraction, or division. They can also be much more complex like an exponent or square root.</i></blockquote><br />That's the kind of thing that consistently aggravates me about open-source textbooks. While I love and agree with establishing algebraic notation as a kind of language -- as possibly the single most important overriding concept of the course -- to get the situation exactly wrong right off the bat (as well as using descriptors like PEMDAS, god help us) keep these as fundamentally unusable in my courses. This, of course, then leaves no grammatical position at all for <i>relations</i> (like equals) when they finally appear later in the text (Sec. 1.4). So close, and yet so far. <br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com4tag:blogger.com,1999:blog-7718462793516968883.post-55610954467636773372016-04-22T05:00:00.000-04:002016-04-22T05:00:10.303-04:00Link: Smart People Happier with Fewer FriendsResearch by people at the London School of Economics and Singapore Management University that smarter people are happier with fewer friends, and fewer social interaction outings. Downside: The researchers are evolutionary psychologists and seek to explain the finding in those terms. Also: Uses the term "paleo-happiness". <br /><br /><div style="text-align: center;"><span style="font-size: large;"><a href="https://www.washingtonpost.com/news/wonk/wp/2016/03/18/why-smart-people-are-better-off-with-fewer-friends/">Washington Post: Why smart people are better off with fewer friends</a></span></div>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-15785580072335177402016-04-15T05:00:00.000-04:002016-04-15T05:00:21.605-04:00Poker MemoryMaybe 15 years ago, I went to the Foxwood poker tables (vs. 9 other people) , got pocket Queens, and had an Ace come up on the flop. I maxed out the bet and lost close to $100. So the other morning I woke up and the thought in my head was, "I really should have computed the probability that someone else had an Ace". Which was 1 - 44P18/47P18 = 1 - 0.225 = 0.775 = 77.5%. Sometimes my brain works glacially slow like that.<br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-86948475984018729942016-04-11T05:00:00.000-04:002016-04-11T05:00:08.071-04:00Reading RadicalsIn my development algebra classes, I push radicals further forward, closer to the start of the semester than most other instructors or textbooks. I want them to be discussed jointly with exponents, so we can really highlight the inverse relation with exponents, and that knowledge of the rules of one is effectively equivalent knowledge of the other. Also: Based on the statistics I keep, success on the exponents/radicals test is the single best predictor of success on the comprehensive, university-wide final exam. <br /><br />There are, of course, many errors made by students learning to read and write radicals for effectively the first time. Here's an exceedingly common category, to write something like (\(x > 0\)):<br /><div style="text-align: center;">$$\sqrt{16} = 4 = 2$$</div><div style="text-align: center;">$$\sqrt{x^8} = x^4 = x^2 = x$$</div><br />Any of these expressions may or may not have a radical written over them (including, e.g., \(\sqrt{4} = \sqrt{2}\)). That is: Students see something "magical" about radicals, and sometimes keep square-rooting any expression in sight, until they can no longer do so. This is common enough that I have few interventions in my mental toolbox ready for when this occurs in any class:<br /><ol><li>Go to the board and, jointly with the whole class, start asking some true-or-false questions. "T/F: \(\sqrt{4} = \sqrt{2}\) ← False. \(\sqrt{4} = 2\) ← True." Briefly discuss the difference, and the location of \(\sqrt{2}\) on the number line. Emphasize: Every written symbol in math makes a difference (any difference in the writing, and it has a different meaning). </li><br /><li>Prompt for the following on the board. "Simplify: \(3 + 5 = 8\)." Now ask: "Where did the plus sign go? Why are you not writing it in the simplified expression? Because: You <i>did</i> the operation, and therefore the operational symbol goes away. The same will happen with radicals: If you can actually compute a radical, then the symbol goes away at that time." </li></ol><br />That's old hat, and those are techniques I've been using for a few years now. The one new thing I noticed last night (as I write this) is that there is actually something unique about the notation for radicals: Of the six basic arithmetic operations (add, subtract, multiply, divide, exponents, radicals), <u>radicals are the only binary operation where one of the two parameters may not be written</u>. That is, for the specific case of square roots, there is a "default" setting where the index of 2 doesn't get written -- and there's no analogous case of any other basic operator being written without a pair of numbers to go with it. <br /><br />I wonder if this contributes to the apparent "magical" qualities of radicals (specifically: students pay more attention to the visible numbers, whereas I am constantly haranguing students to look more closely at the operators in the writing)? Hypothetically, if we always wrote the index of "2" visibly for square roots (as for all other binary operators), would this be more transparent to students that the operator only gets applied once (at which point radical and index simplify out of the writing)? And perhaps this would clear up a related problem: students occasionally writing a reduction as a new index, instead of a factor (e.g., \(\sqrt{18} = \sqrt{9 \times 2} = \sqrt[3]{2}\))? <br /><br />That would be a pretty feasible experiment to run in parallel classes, although it would involve using nonstandard notation to make it happen (i.e., having students explicate the index of "2" for square roots all the time). Should we consider that experiment? <br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com8tag:blogger.com,1999:blog-7718462793516968883.post-2010774687952422592016-04-08T05:00:00.000-04:002016-04-08T05:00:07.197-04:00What Part of Speech is "Times"?What part of speech are the operational words "plus", "minus", and "times"? This is a surprisingly tricky issue; apparently major dictionaries actually differ in their categorization. The most common classification is as some form of <b>preposition</b> -- the Oxford Dictionary says that they are <b>marginal prepositions</b>; "a preposition that shares one or more characteristics with other word classes [i.e., verbs or adjectives]".<br /><br />Here's an interesting thread on Stack Exchange: English Language & Usage on the issue -- including commentary by famed quantum-computing expert and word guru Peter Shor:<br /><br /><div style="text-align: center;"><span style="font-size: large;"><a href="http://english.stackexchange.com/questions/303067/is-times-really-a-plural-noun">Stack Exchange English Language & Usage: <br />Is “times” really a plural noun?</a></span></div><div style="text-align: center;"><br /></div>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-79390663261341883232016-04-01T05:00:00.000-04:002016-04-01T05:00:15.917-04:00Veterinary HomeopathyA funny, but scary and real, web page of a homeopathic-practicing veterinarian who seems weirdly cognizant that it has no real effect:<br /><blockquote class="tr_bq"><i>How much to give: Each time you treat your pet, give approximately 10-20 of the tiny (#10) pellets in the amber glass vial, or 3-7 of the larger (#20) pellets in the blue plastic tube. <b>You don't need to count them out. In fact, the number of pellets given per treatment makes no difference whatsoever.</b> It is the frequency of treatment and the potency of the remedy that is important. <b>Giving more pellets per treatment does not in any way affect the body's response. The pellets need not be swallowed, </b>and it doesn't matter if a few of them are spit out. Just get a few pellets somewhere in the mouth, then hold the mouth shut for 3 seconds.</i></blockquote><br /><div style="text-align: center;"><span style="font-size: large;"><a href="http://www.homeovet.net/content/treatment.html"> Jeffrey Levy, DVM PCH: <br />Classical Veterinary Homeopathy</a></span><br /><br /></div>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-50314514318675701702016-03-28T05:00:00.000-04:002016-03-28T05:00:02.884-04:00The MOOC as a First AlbumThinking about MOOCs (which I am semi-infamously down on as a method for revolutionizing general education): <br /><br />For rock bands, it's pretty common for their very first album to be considered their best one. Why is that? Well, the first album is likely the product of possibly a decade of practicing, writing, performing bars and clubs, interacting with audiences, and generally fine-tuning and refining the set to make the most solid block of music the band can possibly produce. At the point when a band gets signed to a label (traditionally), the first album is basically this ultra-tight set, honed for maximum impact over possibly hundreds of public performances. <br /><br />But thereafter, the band is no longer in the same "lean and hungry" mode that produced that first set of music. Likely they go on tour for a year to support the first album, then are put in a studio for a few weeks with the goal of writing and recording a second album, so that the sales/promotion/touring cycle can pick up where the last one left off. This isn't a situation the band's likely to have experienced before, they have weeks instead of years to create the body of music, and they don't have hundreds of club audiences to run it by as beta-testers. In fact, they probably won't ever again have the opportunity of years of dry runs going into the manufacture a single album.<br /><br />The same situation is likely to apply to MOOCs. A really good online class (and there are some) will be the product of a teacher who's taught the course live for a number of years (or decades), interacting with actual classrooms-full of students, refining the presentation many times as they witness how the presentation is immediately received by the people in front of them. If this has been done, say, <i>hundreds</i> of times, then you have a pretty strong foundation to begin recording something that will be a powerful class experience. <br /><br />But if someone tries to develop an online course from scratch, in a few weeks isolated in an office without any live interaction as a testbed -- exactly as the band studio album situation -- what you're going to get is weak sauce, possibly entirely usable crap. If the instructor has never taught such a class in the past, then the result is likely just "kabuki" as a teacher that I once live-observed confessed to me. This is regardless if a person can <i>do</i> the math themselves, that's just total BS as a starting point for teaching.<br /><br />A properly prepared, developed, scaffolded, explained course has got hundreds of moving parts built into it, built into every individual exercise, that are totally invisible unless an instructor has actually confronted live students with the issues at hand and seen the amazing kaleidoscope of ways that students can make mistakes or become tripped up or confused. No amount of "big data" is going to solve this (even assuming the MOOCs are even <i>trying</i> to do that and claims of such are not just flat-out fraud), because the tricky spots are so surprising, you'll never think to create a metric to measure it unless you're looking right over a student's shoulder to watch them do their work.<br /><br /><br />Quick metric for a quality online course: Has the instructor taught it live for a decade or more? Probably good. Did the instructor make it up on the fly, or in a few weeks development cycle? Probably BS.<br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com2tag:blogger.com,1999:blog-7718462793516968883.post-64099082580478558812016-03-25T05:00:00.000-04:002016-03-25T05:00:03.590-04:00GPS Always Overestimates DistancesResearchers in Austria and the Netherlands have pointed out that existing GPS applications almost always <i>overestimate</i> the distance of a trip, no matter where you're going. Why? Granted some small amount of random error in measuring each of the waypoints along a trip, the distance between erroneous points on a surface is overwhelmingly more likely to be <i>greater than</i>, rather than lesser than, the true distance -- -- and over many legs of a given trip, this error adds up to a rather notable overestimate. And to date no GPS application makers have corrected for it. A wonderful and fairly simple piece of math, one that was lurking under our noses for some time that no one thought to check, that should improve all of our navigation devices:<br /><br /><div style="text-align: center;"><span style="font-size: large;"><a href="http://www.i-programmer.info/news/145-mapping-a-gis/9164-gps-always-over-estimates-distances.html">I, Programmer: <br />GPS Always Overestimates Distance </a></span></div><div style="text-align: center;"><br /></div>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com2tag:blogger.com,1999:blog-7718462793516968883.post-19688274252529869942016-03-21T05:00:00.000-04:002016-03-21T05:00:12.689-04:00Excellent Exercises − Simplifying RadicalsExploration of exercise construction, i.e., casting a net to catch as many mistakes as possible: See also the previous <a href="http://www.madmath.com/2014/01/excellent-exercises-completing-square.html">"Excellent Exercises: Completing the Square"</a>. <br /><br />Below you'll see me updating my in-class exercises introducing simplification of radicals for my remedial algebra course a while back. (This occurred between one class on Monday, and different group on Tuesday, when I had the opportunity to spot and correct some shortcomings.) My customary process is to introduce a new concept, then give support for it (theorem-proof style), then do some exercises jointly with the class, and then have students do exercises themselves (from 1-3 depending on problem length) -- hopefully each cycle in a 30 minute block of time. In total, this snippet represents 1 hour of class time (actually the 2nd half of a 2-hour class session); the definitions and text shown is written verbatim on the board, while I'll be expanding or answering questions verbally. As I said before, I'm trying to bake as many iterations and tricky "stumbling blocks" into this limited exercise time as possible, so that I can catch and correct it for students as soon as we can. <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-J2JHIqx0z-4/UW7F-fmo9zI/AAAAAAAAB70/KFliyOIDdsE/s1600/RadicalNotes.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="103" src="https://4.bp.blogspot.com/-J2JHIqx0z-4/UW7F-fmo9zI/AAAAAAAAB70/KFliyOIDdsE/s320/RadicalNotes.jpg" width="320" /></a></div><br />Now, you can see in my cross-outs the simplifying exercises I was using at the start of the week, which had already gone through maybe two semesters of iteration. Obviously for each triad (instructor a, b, c versus students' d, e, f) I start small and present sequentially larger values. Also, for the third of the group I throw in a fraction (division) to demonstrate the similarity in how it distributes.<br /><br />Not bad, but here some weaknesses I spotted this session that aren't immediately apparent from the raw exercises, and these are: (1) There are quite a few duplicates between the (now crossed-out) simplifying and later add/subtract exercises, which reduces real practice opportunities in this hour. (2) Is that I'm not happy about starting off with √8, which reduces to 2√2 -- this might cause confusion in a discussion for some students who don't see where the different "2"'s come from, something I try to avoid for initial examples. (3) Is that student exercises (c) and (d) both involve factoring out the perfect square "4", when I should have them getting experience with a wider array of possible factoring values. (4) Is that item (f) is √32, which raises the possibility of again factoring out either 4 or 16 -- but none of the instructor exercises demonstrated the need to look for the "greatest" perfect square, so the students weren't fairly prepped for that case.<br /><br />Okay, so at this point I realized that I had at least 4 things to fix in this slice of class, and so I was committed to rewriting the entirety of both blocks of exercises (ultimately you can see the revisions handwritten in pencil on my notes). The problem is that simplifying-radical problems are actually among the harder problems to construct, because there's a fairly limited range of values which are the product of a perfect square for which my students will be able to actually revert the multiplication (keeping in mind a significant subset of my college students who don't actually know their times tables, and so are having to make guesses and sequentially add on their fingers a bunch of times before they can get started). <br /><br />So at this point I sat down and made a comprehensive list of all the smallest perfect square products that I could possible ask for in-class exercises. I made the decision to use each one at most a single time, to get as much distinct practice in as possible. First, of course, I had to synch up like remainders to make like terms in the four "add/subtract" exercises -- these are indicated below by square boxes linking the like terms for those exercises. Then I circled another 6 examples, for use in the lead-in "simplifying" exercises, trying to find the greatest variety of perfect squares possible, not sequentially duplicating the same twice, and making sure that I had multiple of the "greatest perfect square" (i.e., involving 16 or 36) issue in both the instructor and student exercises. These, then, became my revised exercises for the two half-hour blocks, and I do think they worked noticeably better when I used them with the Tuesday class this week.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-ML5rR8kQJe8/UW7GAIL4dpI/AAAAAAAAB78/Hnic8RDy_Dg/s1600/RadicalPermutations.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="266" src="https://2.bp.blogspot.com/-ML5rR8kQJe8/UW7GAIL4dpI/AAAAAAAAB78/Hnic8RDy_Dg/s320/RadicalPermutations.jpg" width="320" /></a></div><br />Some other stuff: The fact that add/subtract exercise (c) came out to √5 was kind of a happy accident -- I didn't plan on it, but I'm happy to have students re-encounter the fact that they shouldn't write a coefficient of "1" (many will forget that, and you need to have built-in reviews over and over again). Also, one might argue that I should have an addition exercise where you <i>don't</i> get like terms to make sure they're looking for that, but my judgement was that in our limited time I wanted them doing the whole process as much as possible (I'll leave non-like terms cases for book homework exercises). <br /><br />Anyway, that's a little example of the many of issues involved, and the care and consideration, that it takes to construct really quality exercises for even (or especially) the most basic math class. Like I said, I think this is about the third iteration of these exercises for me in the last year -- we'll see if I catch any more obscure problems the next time I teach the class.<br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0