tag:blogger.com,1999:blog-77184627935169688832017-06-23T09:29:33.441-04:00MadMath"Beauty is the Enemy of Expression"Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.comBlogger270125tag:blogger.com,1999:blog-7718462793516968883.post-51530812491898351552017-06-19T05:00:00.000-04:002017-06-19T05:00:29.887-04:00Good Teaching, Bad ResultsA provocative article that I just discovered: Schoenfeld, Alan H., <a href="http://jwilson.coe.uga.edu/TiMER/Schoenfeld%20(1988)%20Good%20Teach%20Bad%20Results-2.pdf">"When Good Teaching Leads to Bad Results: The Disasters of 'Well-Taught' Mathematics Courses"</a> (Educational Psychologist, 1988). From the abstract: <br /><blockquote class="tr_bq"><i>This article describes a case study in mathematics instruction, focusing on the development of mathematical understandings that took place in a 10-grade geometry class. Two pictures of the instruction and its results emerged from the study. On the one hand, almost everything that took place in the classroom went as intended—both in terms of the curriculum and in terms of the quality of the instruction. The class was well managed and well taught, and the students did well on standard performance measures. Seen from this perspective, the class was quite successful. Yet from another perspective, the class was an important and illustrative failure. There were significant ways in which, from the mathematician's point of view, having taken the course may have done the students as much harm as good. Despite gaining proficiency at certain kinds of procedures, the students gained at best a fragmented sense of the subject matter and understood few if any of the connections that tie together the procedures that they had studied. More importantly, the students developed perspectives regarding the nature of mathematics that were not only inaccurate, but were likely to impede their acquisition and use of other mathematical knowledge. The implications of these findings for reseach on teaching and learning are discussed. </i></blockquote><br />In particular, Schoenfeld's observations are largely predicated on "teaching to the test" of standardized finals (esp.: Regents testing in New York State), with students memorizing standardized procedures for each particular problem (including a rote repertoire of geometry proofs and constructions), and generally not being able to think through any problems outside those narrowly-formulated items. <br /><br />Little did he dare imagine how much more corrosive standardized testing would be 30 years later! A colleague and I were just discussing this issue (narrow and fragile problem-solving knowledge of students) just yesterday. <br /><br />Hat tip to Daniel Hast on StackExchange ME for the link. <br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-45740704213873357032017-06-05T05:00:00.000-04:002017-06-05T05:00:29.824-04:00More Reading Fractions as DecimalsLast December, we speculated that many students who are weak in understanding fractions may read them incorrectly as decimals (for example:<a href="http://www.madmath.com/2016/12/observed-belief-that-12-12.html"> thinking that 1/2 = 1.2</a>).<br /><br />For the spring term, I added a question on my first-day diagnostics regarding this topic. Specifically: "Graph the fraction on a number line: 2/3." Four multiple-choice options were given in graphical form: (a) between 0 and 1 [the correct answer], (b) b/w 1 and 2 [at 3/2], (c) b/w 2 and 3 [at 2.3], (d) b/w 3 and 4 [at. 3.2]. <br /><br />Results:<br /><ul><li>Remedial intermediate algebra class (N = 26): (a) 62%, (b) 8%, (c) 23%, (d) 8%.</li><li>Credit college algebra class (N = 21): (a) 86%, (b) 5%, (c) 10%, (d) 0%. </li></ul>Conclusions: In both cases, item (c), the result of thinking that 2/3 = 2.3, was indeed the most commonly selected incorrect response. While most students in both classes selected the correct answer, approximately one-quarter of the intermediate algebra class instead picked the location of 2.3. Students registered for the college algebra class clearly had stronger incoming knowledge of fractions. <br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com2tag:blogger.com,1999:blog-7718462793516968883.post-24630814468372599532017-05-22T18:36:00.001-04:002017-05-22T18:36:33.405-04:00Eugene Stern: How Value Added Models are Like TurdsEugene Stern critiques the Value Added Model for teacher assessment thusly: <br /><blockquote class="tr_bq"><i>So, just to take another example, if I decided to rate teachers by the size of the turds that come out of their ass, I could wave around a lovely bell-shaped distribution of teacher ratings, sit back, and wait for the Times article about how statistically insightful this is.</i></blockquote><a href="https://mathbabe.org/2017/05/22/eugene-stern-how-value-added-models-are-like-turds/">Read more at MathBabe.</a> <br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-19595805727660168372017-04-10T05:00:00.000-04:002017-04-10T05:00:31.076-04:00Mercator Projection All the Way DownMap facts: The Mercator projection is technically infinitely tall, and more warped as it goes down, so it must always be cropped somewhere. Below is a cropping somewhat lower than normal, so you can see: (1) Antarctica, (2) buildings at the Amundsen–Scott South Pole Station, and finally (3) individual snowflakes.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-M-VcAa1AqNc/WNSZ3GFrQLI/AAAAAAAAEMA/Wd5TW75cqEMgI-9A28DR6sDqB8y6-lSxACLcB/s1600/unclipped-mercator.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://3.bp.blogspot.com/-M-VcAa1AqNc/WNSZ3GFrQLI/AAAAAAAAEMA/Wd5TW75cqEMgI-9A28DR6sDqB8y6-lSxACLcB/s320/unclipped-mercator.png" width="65" /></a></div><br /><br /><br /><br />Hat tip: <a href="http://geoawesomeness.com/why-dont-we-start-using-a-more-accurate-world-map-rather-than-the-conventional-mercator-map/">Geoawesomeness</a>. <br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com2tag:blogger.com,1999:blog-7718462793516968883.post-25018046018740893622017-04-03T05:00:00.000-04:002017-04-03T05:00:10.344-04:00No, we probably don’t live in a computer simulationA lovely rant by Sabine Hossenfelder:<br /><blockquote class="tr_bq"><i>All this talk about how we might be living in a computer simulation pisses me off not because I’m afraid people will actually believe it. No, I think most people are much smarter than many self-declared intellectuals like to admit. Most readers will instead correctly conclude that today’s intelligencia is full of shit. And I can’t even blame them for it. </i></blockquote><br /><a href="http://backreaction.blogspot.com/2017/03/no-we-probably-dont-live-in-computer.html">At Backreaction </a>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-63319595582753921732017-03-27T05:00:00.000-04:002017-03-27T05:00:04.863-04:00How To Ruin Your Favorite Sitcoms With Simple Math<blockquote class="tr_bq"><i>Math does not exist to make things better. It exists to empower you to tear things apart.</i></blockquote><div class="separator" style="clear: both; text-align: center;"><iframe width="320" height="266" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/wGGdwk0Fotg/0.jpg" src="https://www.youtube.com/embed/wGGdwk0Fotg?feature=player_embedded" frameborder="0" allowfullscreen></iframe></div><br /><br />I support this message. Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-39209672227861741662017-03-20T05:00:00.000-04:002017-03-20T05:00:17.289-04:00CUNY Remediation Overhaul in NY TimesNumerous inaccuracies. No CUNY math faculty interviewed on record:<br /><br /><div style="text-align: center;"><span style="font-size: large;"><a href="https://www.nytimes.com/2017/03/19/nyregion/cuny-remedial-programs.html">CUNY to Revamp Remedial Programs, Hoping to Lift Graduation Rates</a></span></div>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-79685820721942215882017-03-06T05:00:00.000-05:002017-03-06T05:00:01.981-05:00It Can Never Lie To YouAn very nice interview with Sylvia Serfaty, Paris-based mathematician, and winner of the Poincaré Prize: <br /><blockquote class="tr_bq"><i>“First you start from a vision that something should be true,” Serfaty said. “I think we have software, so to speak, in our brain that allows us to judge that moral quality, that truthful quality to a statement.”</i></blockquote><br /><a href="https://www.wired.com/2017/03/beauty-mathematics-can-never-lie/">At Wired. </a>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag: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.com4tag: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.com2