tag:blogger.com,1999:blog-77184627935169688832017-02-19T21:30:30.931-05:00MadMath"Beauty is the Enemy of Expression"Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.comBlogger262125tag:blogger.com,1999:blog-7718462793516968883.post-52975762863642776332017-02-13T05:00:00.000-05:002017-02-13T05:00:09.819-05:00Francis Su: Math as Justice“Every being cries out silently to be read differently.” <br /><h1 class="entry-title" style="text-align: center;"><a href="https://www.quantamagazine.org/20170202-math-and-the-best-life-francis-su-interview/"><span style="font-size: large;">To Live Your Best Life, Do Mathematics</span></a></h1>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-67703484775953701752017-02-06T05:00:00.000-05:002017-02-06T05:00:06.002-05:00Milliken on CUNY Connected and RemediationAs a follow-up to last week's post, CUNY Chancellor James Milliken has this week unveiled out a new strategic plan called "CUNY Connected". Among the promises are increased graduation rates. In the subsection on remediation reform (again), he writes: <br /><blockquote class="tr_bq"><i>Each fall, approximately 20,000 students—over half of all CUNY freshmen– are assigned to developmental education in at least one subject, usually mathematics. In associate degree programs, 74 percent of freshmen were assigned to developmental education in math in fall 2015, 23 percent in reading, and 33 percent in writing. But CUNY’s one-size-fits-all approach to preparing students has not worked. In fall 2015, just 38% of the 14,215 students in remedial algebra successfully completed it.<br /><br /><b>Implementing these reforms, the number of students placed in remediation will decline by at least 15 percent. The number of students determined to be proficient after one year of remediation will increase by at least 5 percentage points in year one and will increase as we move to scale.</b><br /><br />Under the reforms, 20,000 students per semester will receive tutoring and supplemental instruction and 4,000 will be enrolled in courses with faculty who have been newly trained. Another one thousand students will enroll in immersion programs or new developmental workshops. All students will have access to instructional software.<br /><br />CUNY will bring to scale two developmental options of proven efficacy: 1) co-requisite courses—credit-bearing courses with additional mandatory supports in the form of workshops or tutoring, and 2) <b>alternatives to math proficiency other than algebra for students pursuing majors or courses of study that do not require algebra. College algebra is necessary for many but not all majors.</b><br /><br /><b>We will also end the practice of requiring all students to pass common tests in algebra, writing and reading to exit developmental education.</b> Grades, it has been found, are a better predictor of proficiency and success. CUNY will continue the use of standardized common final exams that count for 35 percent of the final course grade.</i> </blockquote><br />These are dictates that were communicated internally at CUNY within the last year. It's interesting that higher passing rates can be dictated in advance by fiat. To be clear: Most CUNY graduates will not need to be algebra proficient, most will not take a course which uses algebra skills, and those who do will not need to succeed on any particular assessment or test to be declared proficient. Another point of clarification: While "college algebra" is mentioned in this section, college algebra is not actually a remedial course (most students <i>already</i> have never taken college algebra at CUNY); the remedial/general education expectation which is being removed is at the level of elementary algebra, around 8th-9th grade level skills as identified in the U.S. Common Core and other curricula. <br /><br /><div style="text-align: center;"><span style="font-size: large;"><a href="http://www1.cuny.edu/sites/connected/college-readiness/">CUNY Connected: College Readiness</a></span></div><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-89374415142150364712017-01-30T05:00:00.000-05:002017-01-30T05:00:16.822-05:00Milliken on NY1CUNY Chancellor James Milliken gave an interview Friday night on NY1. Among the things he said:<br /><blockquote class="tr_bq"><i>The urban 3-year community college graduation rate in this country is 16%. CUNY 17½. We're committed to doubling that.</i></blockquote>Consider what institutional mechanisms have radically changed highly politicized statistics like that in the past.Full video here (quote at 3:55):<br /><h1 class="twcround-book fs-30" itemprop="headline name" style="text-align: center;"><span style="font-size: large;"><span style="font-weight: normal;"><a href="http://www.ny1.com/nyc/all-boroughs/city-hall-newsmakers/2017/01/27/ny1-online--cuny-s-future.html">NY1 Online: CUNY's Future</a></span></span></h1>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-30407661890604272442017-01-16T05:00:00.000-05:002017-01-16T05:00:06.546-05:00An Ode to the Graphing CalculatorBy Panama Jackson, at VSB:<br /><br /><div style="text-align: center;"><span style="font-size: large;"><a href="http://verysmartbrothas.com/ti83graphingcalculator/">My TI-83 Graphing Calculator Is The Real MVP and My STEM Folks Know What I’m Talking About</a></span></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-83893408173579148022017-01-09T05:00:00.000-05:002017-01-23T23:18:42.232-05:00Operations Before NumbersMost elementary algebra books start on page one with a description of different sets of numbers that will be in use (naturals, integers, rationals, and reals). Then soon after they discuss the different operations to be performed on those numbers, the conventional order-of-operations, etc. This seems satisfying: you get the objects under discussion first, and then modifiers to be performed on those objects (nouns, then prepositions). <br /><br />But the problem that's irked me for some time is this: the sets of numbers are themselves defined in terms of the operations. Most obvious is the fact that rationals are quotients of integers: <i>a/b</i> (<i>b</i> nonzero); so this presumes knowledge of division beforehand. Integers, too, are really differences of natural numbers (though usually expressed as something like "signed whole numbers"); they are fundamentally a result of subtraction. So in my courses I resolve this by coming out of the box on day one with a review of the different arithmetic operations, names of results, and their proper ordering; then on day two we can discuss the different sets of numbers thus generated. <br /><br />Now, in other mathematical contexts -- where you are only discussing <i>one field at a time</i> -- it is conventional to discuss the elements of a set first, and then the operations that we might apply on them second. That makes sense. But at the start of an elementary algebra course we tend to be cheating a bit by trying to consolidate a presentation of at least 4 different sets all at once. It would be fairly rigorous to present naturals and their operations (add, subtract, multiply, divide, etc.), and then integers (and their addition, subtraction, multiplication, etc.), then rationals and their operations (etc.), and then finally a separate discussion of real numbers and their operations (etc.). But that would take an inordinate amount of time, and the operations are so very similar that it would seem repetitive and wasteful to most of our students (outside of difference in closures, etc.). <br /><br />So if the elementary algebra class wants to cheat in this fashion and present the whole menagerie of number categories in one lecture, I would argue that we need to abstract out the operations <i>first</i>, and then have those available to describe the differences in our sets of numbers <i>second</i>. <br /><br /><br />Thoughts? Are you still satisfied with describing numbers before operations? <br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com1tag:blogger.com,1999:blog-7718462793516968883.post-74315126876343331162017-01-02T05:00:00.000-05:002017-01-02T05:00:12.101-05:00The Nelson-Tao CaseA case that I read in the past, and have searched fruitlessly for months (or years) to cite-reference -- which I just found via a link on Stack Exchange (hat tip to Noah Snyder). Partly so I have a record for my own purposes, here's an overview:<br /><br />In 2011 Edward Nelson, a professor at Princeton, was about to publish a book demonstrating a proof that basic arithmetic theory (the Peano Postulates) was essentially inconsistent. This started a discussion on the blog of John Baez, in which the eminent mathematician (and superb mathematical writer) Terry Tao spent some time trying to explain what was wrong with Nelson's proof. After about three cycles of back-and-forth, the end result was this:<br /><blockquote class="tr_bq"><i>You are quite right, and my original response was wrong. Thank you for spotting my error.<br /><br />I withdraw my claim.</i><br /><i><br />Posted by: Edward Nelson on October 1, 2011 1:39 PM </i></blockquote><br />This is one of the best examples of what I personally call "the brutal honesty of mathematics". <a href="https://golem.ph.utexas.edu/category/2011/09/the_inconsistency_of_arithmeti.html">Read the whole exchange here on John Baez' site. </a><br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-52185927361485883192016-12-26T05:00:00.001-05:002016-12-26T13:54:06.872-05:00On Famous ThingsA quip from Stack Exchange back in 2014 that still fills me with glee on a daily basis:<br /><br />A poster asks how to convince other people when he's developed an as-yet ignored, revolutionary, world-beating result...<br /><blockquote class="tr_bq"><i>e.g., you solve the P vs. NP problem or any other well known open problem. </i></blockquote> Pete L. Clark writes as part of his response: <br /><blockquote class="tr_bq"><i> It's like saying "i.e., he found the Holy Grail or some other famous cup". </i></blockquote><br /><a href="http://academia.stackexchange.com/questions/18491/i-believe-i-have-solved-a-famous-open-problem-how-do-i-convince-people-in-the-f/"> More gifts of wisdom at Stack Exchange. </a>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-77731713897709995372016-12-12T05:00:00.000-05:002016-12-12T05:00:26.552-05:00Michigan State Drops Algebra RequirementThis summer, Michigan State announced that they will drop college algebra as a general-education requirement, replacing it with quantitative-literacy classes:<br /><blockquote class="tr_bq"><i>Michigan State University has revised its general-education math requirement so that algebra is no longer required of all students. The revision reflects an increasing view on college campuses that there is no one-size-fits-all math curriculum -- and that math is often best studied in connection with everyday life...<br /><br />Now, students can fulfill the requirement by taking two quantitative literacy courses that place math in a real-world context. They also still have the option of taking algebra along with another math course of their choice -- whether a quantitative-literacy course or a more traditional course like trigonometry.</i></blockquote><div style="text-align: center;"><span style="font-size: large;"><a href="https://www.insidehighered.com/news/2016/07/06/michigan-state-drops-college-algebra-requirement"><br /></a></span></div><div style="text-align: center;"><span style="font-size: large;"><a href="https://www.insidehighered.com/news/2016/07/06/michigan-state-drops-college-algebra-requirement">"Algebra No More" at Inside Higher Ed</a></span></div><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com2tag:blogger.com,1999:blog-7718462793516968883.post-34709874127614326252016-12-05T05:00:00.000-05:002016-12-26T13:53:35.677-05:00Observed Belief That 1/2 = 1.2Last week in both of my two college algebra sections, there came a moment when we had to graph an intercept of <i>x</i> = 1/2. I asked, "One-half is between what two whole numbers?" Response: "Between 1 and 2." I asked the class in general for confirmation: "Is that right? One-half is between 1 and 2, yes?" And the entirety of the class -- in both sections, separated by one hour -- nodded and agreed that it was. (Exception: One student who was previously educated in Russia.)<br /><br />Now, this may seem wildly inexplicable, and it took me a number of years to decipher this. But here's the situation: Our students our so unaccustomed to fractions that they can only interpret the notation as decimals, that is: they believe that 1/2 = 1.2 (which is, of course, really between 1 and 2). Here's more evidence from the Patricia Kenschaft article, "Racial Equity Requires Teaching Elementary<br />School Teachers More Mathematics" (Notices of the AMS, February 2005):<br /><blockquote class="tr_bq"><i>My first time in a fifth grade in one of New Jersey’s most affluent districts (white, of course), I asked where one-third was on the number line. After a moment of quiet, the teacher called out, “Near three, isn’t it?” The children, however, soon figured out the correct answer; they came from homes where such things were discussed.</i></blockquote><br />Likewise, the only way this makes sense is if the teacher interprets 1/3 = 3.1 -- both visually turning the fraction into a decimal, <i>and</i> reading it upside-down. We might at first think the error is the common one that 1/3 = 3, but that wouldn't explain why the teacher thought it was only "near" three. <br /><br />The next time an apparently inexplicable interpretation of a fraction comes up, consider asking a few more questions to make the perceived value more precise ("Is 1/2 between 1 and 2? Which is it closer to: 1 or 2 or equally distant?" Etc.). See if the problem isn't that it was visually interpreted as decimal point notation. <br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com1tag:blogger.com,1999:blog-7718462793516968883.post-41023192453830617252016-11-04T05:00:00.000-04:002016-11-04T05:00:38.790-04:00The Math MenuA quick thought, spring-boarding off Monday's post: A constant debate in math education is whether students should be directly-taught mathematical results, or spend time (like a mathematician) exploring problems, looking for patterns, and coming up with their own "theorems" (in Mubeen's phrasing "own the problem space").<br /><br />Here is a hypothetical equivalent debate: What is supposed to happen in a restaurant -- Does food get <i>cooked</i>, or does food get <i>eaten</i>? <br /><br />Obviously both. But the majority of people who visit the establishment are clientele who do not come to the restaurant in order to learn how to cook; they come for an end-product which is used in a different fashion (for consumption and nourishment). If someone expresses interest in becoming a chef themselves then of course we should encourage and cultivate that. But if some group of chefs become so self-involved that they demand everyone participate in cooking for a "real" restaurant experience, then surely we'd all agree that they'd gone off the deep end and needed restraints.<br /><br />So too with mathematicians.<br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-45723865226544258092016-10-31T05:00:00.000-04:002016-10-31T05:00:12.840-04:00Scary StoriesA pair of scary math-education anecdotes by Junaid Mubeen, for your consideration:<br /><ul><li><a href="https://mystudentvoices.com/how-old-is-the-shepherd-the-problem-that-shook-school-mathematics-ad89b565fff">How Old is the Shepherd?</a> When 8th-graders are asked a short question with absolutely no information about age whatsoever, 3-in-4 will report some numerical result anyway. Repeated in numerous experiments. <a href="https://www.youtube.com/watch?v=kibaFBgaPx4">Watch a video</a>. </li><br /><li><a href="https://medium.com/bright/my-nephew-brought-home-this-menacing-maths-problem-e8bbba30e5cb">I Can't Believe It's Not Unproven.</a> Mubeen's 12-year-old nephew comes home with a math problem that can't be solved; he is shown a proof of that fact, and agrees to all the steps and the conclusion. Nephew spends the rest of the evening trying to find an answer anyway. </li></ul><br />I don't really agree with Mubeen's rather broad conclusions at the end of the first article. But we can all agree this is a terrifying outcome!<br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-6090990821885196732016-10-24T05:00:00.000-04:002016-10-25T09:23:32.244-04:00Mo' MonicIf you look at any list of elementary algebra topics, or any book's table of contents, etc., then you'll probably find that all of the subjects are referenced <i>by name</i> except for one single exceptional case, which is always expressed in symbolic form. For example, from the College Board's Accu-Placer Program Manual, here's a list of Content Areas for the Elementary Algebra test:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-jHwxqiX5ya8/WAwnqVa48bI/AAAAAAAAEIg/b3t0lEIUZJA6eEoO1AQo0OezPtxmP1VOACLcB/s1600/AccuPlacerContentAreas.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="285" src="https://1.bp.blogspot.com/-jHwxqiX5ya8/WAwnqVa48bI/AAAAAAAAEIg/b3t0lEIUZJA6eEoO1AQo0OezPtxmP1VOACLcB/s320/AccuPlacerContentAreas.png" width="320" /></a></div><br />Do you see it? Or, here are some of the section headers in the Pearson testbank which accompanies the Martin-Gay <i>Prealgebra & Introductory Algebra</i> text:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-JtTFy9xblmk/WAwp2nUPAPI/AAAAAAAAEI0/zTcD6G0oOqg52F6IL11zu1Q6MZ-iA1c7ACLcB/s1600/Martin-GayTestbank.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="218" src="https://3.bp.blogspot.com/-JtTFy9xblmk/WAwp2nUPAPI/AAAAAAAAEI0/zTcD6G0oOqg52F6IL11zu1Q6MZ-iA1c7ACLcB/s320/Martin-GayTestbank.png" width="320" /></a></div><br />Or, here's a menu of topics and quizzes from the <a href="http://mathguide.com/lessons/#algebra">MathGuide.com</a> algebra site:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-fTyAWvV7FIE/WAwoB0cio6I/AAAAAAAAEIk/GY-lxejVq80hS46cSdagoLR_zgSPVTPRQCLcB/s1600/MathGuideAlgebra.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://1.bp.blogspot.com/-fTyAWvV7FIE/WAwoB0cio6I/AAAAAAAAEIk/GY-lxejVq80hS46cSdagoLR_zgSPVTPRQCLcB/s320/MathGuideAlgebra.png" width="297" /></a></div><br />I could repeat this for many other cases, such as: the CUNY list of elementary algebra topics, tables of contents for most algebra books, etc., etc. It's weird and to my OCD brothers and sisters surely it's a bit distracting and frustrating. <br /><br /><i>There should be a name for this. </i>The funny thing is that, to my current understanding, there's a perfectly serviceable name to make the distinction that we're reaching for here: <b>"monic"</b> means a polynomial with a lead coefficient of 1. So I've taken to, in my classes, referring to the initial or "basic" type (\(x^2 + bx + c\)) as a <i>monic quadratic</i>, and the more general or "advanced" type (\(ax^2 + bx + c\), \(a \ne 1\)) as a <i>nonmonic quadratic</i>. My students know they must learn proper names for everything, and so they pick this up as easily as anything else, and without complaint. Thereafter it's much easier to communally reference the different structures by their proper names. <br /><br />Now: I must admit that I picked this up from <a href="https://en.wikipedia.org/wiki/Monic_polynomial">Wikipedia</a> and I've never, ever, seen it used in any mathematics textbook at any level. Perhaps someone could tell me if this is new, or nonstandard, or inaccurate. But even if that weren't the right term to distinguish a polynomial with lead coefficient 1, <i>there should still be a name for this structure</i>. We really should create a name, if necessary, and I'd be prone to <a href="http://www.madmath.com/2012/08/fundamental-rule-of-exponents.html">make up my own name</a> for something like that. <br /><br />But <b>"monic"</b> fits perfectly and is delightfully short and descriptive. <b>We should all start using "monic" more widely, and I'd love to start seeing it in major algebra textbooks. </b><br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com4tag:blogger.com,1999:blog-7718462793516968883.post-41398850818262143432016-10-10T05:00:00.000-04:002016-10-11T13:36:31.362-04:00Natural Selection of Bad Science<span class="highwire-citation-authors"><span class="highwire-citation-author first has-tooltip hasTooltip" data-delta="0" rel="#hw-article-author-popups-node7365 .author-tooltip-0" title=""><span class="nlm-surname">Smaldino and </span></span><span class="highwire-citation-author has-tooltip hasTooltip" data-delta="1" rel="#hw-article-author-popups-node7365 .author-tooltip-1" title=""><span class="nlm-surname">McElreath write a paper which asserts that the problem of false-positive papers in science -- especially behavioral science -- is getting worse over time, and will continue to do so as long as we reward quantity of paper outputs:</span></span></span><br /><blockquote class="tr_bq"><i>To demonstrate the logical consequences of structural incentives, we then present a dynamic model of scientific communities in which competing laboratories investigate novel or previously published hypotheses using culturally transmitted research methods. As in the real world, successful labs produce more ‘progeny,’ such that their methods are more often copied and their students are more likely to start labs of their own. Selection for high output leads to poorer methods and increasingly high false discovery rates. We additionally show that replication slows but does not stop the process of methodological deterioration. Improving the quality of research requires change at the institutional level.</i></blockquote><div class="highwire-cite-metadata"></div><div class="highwire-cite-metadata"><span class="highwire-citation-authors"><span class="highwire-citation-author first" data-delta="0"><span class="nlm-given-names">Paul E.</span> <span class="nlm-surname">Smaldino</span></span>, <span class="highwire-citation-author" data-delta="1"><span class="nlm-given-names">Richard</span> <span class="nlm-surname">McElreath.</span></span></span> <b>The natural selection of bad science.</b><span class="highwire-cite-metadata-journal highwire-cite-metadata"> R. Soc. open sci. </span><span class="highwire-cite-metadata-print-date highwire-cite-metadata">2016 </span><span class="highwire-cite-metadata-volume highwire-cite-metadata">3 160384; </span><span class="highwire-cite-metadata-doi highwire-cite-metadata"><span class="label">DOI:</span> 10.1098/rsos.160384. </span><span class="highwire-cite-metadata-date highwire-cite-metadata">Published 21 September 201. <a href="http://rsos.royalsocietypublishing.org/content/3/9/160384">Link.</a> </span></div><br />Quotes <a href="https://en.wikipedia.org/wiki/Campbell%27s_law">Campbell's Law</a>: "The more any quantitative social indicator is used for social decision-making, the more subject it will be to corruption pressures and the more apt it will be to distort and corrupt the social processes it is intended to monitor."<br /><br /><a href="http://www.economist.com/news/science-and-technology/21707513-poor-scientific-methods-may-be-hereditary-incentive-malus">Review at the Economist. </a><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-87661334837662076142016-10-03T05:00:00.000-04:002016-10-04T14:53:16.461-04:00Euclid: The GameA marvelous little game that treats Euclidean construction theorems as puzzles to solve in a web application:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-8bgZG2kz4-Q/V_P6VBLsilI/AAAAAAAAEIA/wXJyiHQDvmE7eIlSDfgEB8Vn-N_JJOAmQCLcB/s1600/EuclidTheGame.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="259" src="https://2.bp.blogspot.com/-8bgZG2kz4-Q/V_P6VBLsilI/AAAAAAAAEIA/wXJyiHQDvmE7eIlSDfgEB8Vn-N_JJOAmQCLcB/s320/EuclidTheGame.png" width="320" /></a></div><br /><a href="http://www.euclidthegame.com/">Play it here.</a><br /><br />Hat tip: JWS. Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com1tag: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.com0