tag:blogger.com,1999:blog-2516188730140164076.comments2021-07-28T14:53:27.968-04:00Devlin's AngleMathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.comBlogger165125tag:blogger.com,1999:blog-2516188730140164076.post-60080584099472010142017-02-08T11:10:06.978-05:002017-02-08T11:10:06.978-05:00For many of us, the linked video of Hans Rosling&#...For many of us, the linked video of Hans Rosling's 2006 TED Talk was our first encounter with the Swedish academic and his powerful data visualization tools Trendalyzer and Gapminder. I don't know when he first gave that talk, but over the ensuing years he gave essentially the same presentation (audience-interactive performance would be a better description) many times, albeit always with up-to-date data. He was right to do so. At the heart of his presentations was one message. Data matters. Provided you collect them with care, analyze them properly, and present them in honest ways that the human mind can readily grasp, numbers are one of Humankind's most powerful tools. <br /><br />For all his engaging presentation skills, the numbers were at the heart of Rosling's talks. It was not his oratory that convinced us, in an instant, that our preconceptions of our world were wrong -- often violently so. It was the data -- the numbers displayed on the screen in front of us. For that reason, I decided that I would let Hans and his graphs occupy all of this month’s post. There was neither need nor place for my words.<br /><br />As it happens, Rosling's death comes at a moment in time when people in highly powerful positions are waging an assault on scientific facts, on numerical data, and indeed on truth in general. I did not want to detract from using my MAA blog to pay respect to the passing of a great numerist (I had to make up a word to adequately describe him) by inserting this observation into what I wanted to be one final platform for him to spread his message. Like Hans, I wanted his numbers to do the talking. <br /><br />But here, in the comments section, I feel free to speak as an individual mathematician. An attack on truth is an attack on Society in general. Those of us whose lives revolve around discovering and communicating numerical and mathematical truth have a duty to speak up forcefully, in opposition. If our Society loses the respect for, and dependency on, truth, the loss of mathematics will be the least of our worries.<br />Keith Devlinhttps://www.blogger.com/profile/16899343259650938644noreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-12320519052551449592016-06-08T10:17:08.492-04:002016-06-08T10:17:08.492-04:00It's not the tone that bothers me. My main gri...It's not the tone that bothers me. My main gripe is the large number of fundamental mathematical errors. Hacker simply does not know or understand enough mathematics. Nor does he know what the Common Core says. Nor he he aware of the kinds of topic that are typically taught in university math courses. His overall idea is fine, and one I agree with (as do many university mathematics instructors), but as an author of a book that would have considerable marketing devoted to it, it would have been wise to get a mathematical colleague to check it over prior to publication.Keith Devlinhttps://www.blogger.com/profile/16899343259650938644noreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-5257313799866627402016-06-08T10:09:02.641-04:002016-06-08T10:09:02.641-04:00Jonathan, I'd be glad to help. I for one would...Jonathan, I'd be glad to help. I for one would love to read al-Khwarizmi's original words, just as I did Leonardo of Pisa's when I was working on my book The Man of Numbers.Keith Devlinhttps://www.blogger.com/profile/16899343259650938644noreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-20351000833797365732016-06-01T21:14:44.866-04:002016-06-01T21:14:44.866-04:00Surprisingly, Dr. Devlin, despite al-Khwarizmi'...Surprisingly, Dr. Devlin, despite al-Khwarizmi's greatness in the history of mathematics, only 60% of his book on Hindu Arabic arithmetic has ever been translated into English!<br /><br />Would you please help raise awareness of my project to get the remainder translated?<br /><br />A comment from you at the link below might help.<br /><br />Best wishes<br />Jonathan Crabtree<br />www.jonathancrabtree.com/mathematics/1st-english-translation-al-khwarizmi-multiplied-fractions/Jonathan Crabtreehttp://www.jonathancrabtree.com/mathematics/1st-english-translation-al-khwarizmi-multiplied-fractions/noreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-61993018382062065572016-05-20T17:34:29.426-04:002016-05-20T17:34:29.426-04:00A sad analysis of Hacker's book. As a mathemat...A sad analysis of Hacker's book. As a mathematician, father, educator, businessman, and practical user of mathematics, I find the book exceptionally on-target. And that our education system has adopted the mathematics of insanity for a society.<br /><br />More intriguingly, it sounds like you love math (which he observes is just fine) but basically agree with his recommendations (which is good). All-in-all, reading between the lines, I'm going to guess that there's a tonality in his writing that bothers you.<br /><br />Where I disagree with him is on logic as the core that is important in mathematics. For many it might be - but that's not what's projectable to other endeavors.For me, I love what mathematics taught me about using imagination while solving problems. <br /><br />I also enjoyed his points about "solving complex problems". In my life, I've found reading about Churchill in WWII to offer far more insight into solving complex problems than mathematics does - because math only offers specialized problems. Churchill's challenge protecting the UK is far more useful in the real world of business than math is.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-25358442655009985732016-04-11T22:42:23.381-04:002016-04-11T22:42:23.381-04:00In an age of instant electronic computation, the a...In an age of instant electronic computation, the ability to solve problems tends to be not that important. But that doesn't mean Algebra is suddenly not important.<br /><br />It is <b>writing</b> the equations to solve that makes an engineer an engineer -- and the equations for tension, torque etc are not easy. It is <b>understanding</b> what second derivative is that means that an economist knows when a graph is looking like it might be starting to peak. <br /><br />Knowing what a standard deviation really means, and when it is not useful, cannot be done by looking at pretty graphs and hoping that understanding will somehow magically appear. You actually have to know how they are calculated to know what can go wrong, and by how much.<br /><br />Short-sighted ideas that computation is what maths is about lead to the sort of book we have here. He really has no idea.Chester Drawsnoreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-87288159239611715712016-04-05T08:12:34.275-04:002016-04-05T08:12:34.275-04:00There is still almost no attention given to decodi...There is still almost no attention given to decoding the algebra. How many students when asked "What does the equation y = mx + c say/" will come out with "Whyequalsemexpluscee". Who asks "Now tell me without using any of the words why, em, ex or cee"?Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-67716182470415819152016-03-16T12:23:23.608-04:002016-03-16T12:23:23.608-04:00I enjoyed your post very much. Thanks for the thou...I enjoyed your post very much. Thanks for the thoughtful review of Hacker's book. I appreciate that you bore that pain so I don't have to.Jeff Loatshttps://www.blogger.com/profile/07755655010462978870noreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-44695237048275521342016-03-14T10:23:24.821-04:002016-03-14T10:23:24.821-04:00Andrew Hacker's new book, THE MATH MYTH and Ot...Andrew Hacker's new book, THE MATH MYTH and Other STEM Delusions, is an expansion of a controversial opinion article he wrote for the New York Times in 2012. Many readers of the earlier newspaper piece assumed that Hacker was "anti-mathematics" and that he wanted to abolish algebra from the school curriculum. In fact, Hacker believes there is an "inherent beauty" in mathematics. Furthermore, his objection is not to algebra, but to the arbitrary establishment of algebra as a gate-keeping requirement that blocks many avenues of educational opportunity.<br /><br />Hacker cites high failure rates on school exit exams, state-wide proficiency test results, community college remedial math class statistics, and other measures to show that algebra, far from being a pipeline to success, is "...a barrier [that] ends up suppressing opportunities, stifling creativity, and denying society a wealth of varied talents." The failure rates, typically between 40 and 60 percent, are not the fault of school mathematics teachers, and they are not because the students were indifferent or lacking in intelligence. Hacker believes that if we could dispel the "myths and delusions" about mathematics, then students who wanted to study mathematics at advanced levels could do so, while other students could take alternative, equally rigorous but more relevant courses. He describes such a course in a final chapter of the book.<br /><br />The "myths" and "delusions" he examines include:<br />•The line of argument that US global competitive advantages require a compulsory program of secondary school mathematics for all citizens.<br />•The argument that mathematics is used by most workers in the majority of trade and professional jobs.<br />•The belief that studying math develops the mind in ways that transfer to other domains of thought.<br /><br />Hacker clearly states that he is not a mathematician, and despite the blurb on the book's dust cover, he has not been a "professor of mathematics." He is a political scientist who has taught numeracy courses in a mathematics department. Although many mathematicians are referenced in the book, a close reading by a mathematically-trained editor would have helped in several cases. When Hacker refers to topics in higher mathematics, he sounds as if he were randomly pulling words from a college mathematics department catalog. There is also one rather bad misuse of the statistical term "average" instead of "mean", which I hope was typographic, rather than conceptual. This and a few other misprints, for example the number "8" becoming the letter "H" in one of his blackboard-style graphics, will probably be caught and fixed in later editions.Nebraska BlogGrasshttps://www.blogger.com/profile/15909247084228934073noreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-76758454898213379312016-03-12T01:06:05.937-05:002016-03-12T01:06:05.937-05:00Keith,
I just submitted my own review of The MATH ...Keith,<br />I just submitted my own review of The MATH MYTH to our local paper. As soon as I hear from them --probably that it's too long -- I'd like to send a copy to you. It's quite a different analysis, although we agree on a number of points regarding Hacker's mathematical understanding. I'd be interested in your comments, whether or not you'd publish it in your blog.<br />Best regards,<br />David Fowler<br />Emeritus Mathematics Professor<br />University of Nebraska -- LincolnNebraska BlogGrasshttps://www.blogger.com/profile/15909247084228934073noreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-84307377832263715152016-03-11T20:20:43.612-05:002016-03-11T20:20:43.612-05:00Interesting,
further puzzled by the amazon rave r...Interesting,<br /><br />further puzzled by the amazon rave reviews, and the title being traditionally published.<br /><br />Good points.Neeti Sinhahttp://www.magnifieduniverse.com/blognoreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-56497184028272290752016-03-09T17:50:52.262-05:002016-03-09T17:50:52.262-05:00niceniceAllen jeleyhttps://www.blogger.com/profile/10312119051975318074noreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-23061116237011401912016-03-08T18:42:17.640-05:002016-03-08T18:42:17.640-05:00Too bad I did not stumbled upon your postscript ea...Too bad I did not stumbled upon your postscript earlier, Dr. Devlin! It is a delight to read.Octaviohttps://www.blogger.com/profile/09345012244921041056noreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-33141146078655977662016-03-07T21:31:43.796-05:002016-03-07T21:31:43.796-05:00Sadly, I heard a piece on KCBS (a local news radio...Sadly, I heard a piece on KCBS (a local news radio station here in the SF Bay Area) featuring Hacker and his book. One would think a station serving UC Berkeley and Stanford might reach out to professors of those universities for comment. Alas, Hacker's views were offered as truth without comment or rebuttal.Anonymoushttps://www.blogger.com/profile/01174919226259967989noreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-68151878364436061192015-05-04T08:24:43.503-04:002015-05-04T08:24:43.503-04:00to necro an old blog (I was led here by a betterex...to necro an old blog (I was led here by a betterexplained article), the only reason some people (like myself) use KA is for the free practice problems. The interface for it is quite good, and in many places better than pricy, paid practice services. Plus it has tracking and the mastery system, so it will ask me things I have practiced to a passing rate several times after set periods of time have passed, so I can gauge if I retained the knowledge. His videos are ok, but I have not used them very much. The ones that explain the "why" behind things are few and far inbetween, and I usually need that, not a demonstration of mechanics. Just my 2 cents from a KA user over the past 6 months, trying to dust off my mathematical skills in preparation for returning to college after a lengthy absence. :)Anonymoushttps://www.blogger.com/profile/01541821184442343552noreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-90156548359531413292015-04-02T13:04:32.349-04:002015-04-02T13:04:32.349-04:00Very good read. As a college student (E.E.) who of...Very good read. As a college student (E.E.) who often asks himself what's the point of college and these courses that I will forget about in a few months (as you rightly said), it eases my mind to know that it's not all a waste of my precious time.Anonymoushttps://www.blogger.com/profile/03120930443844835928noreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-65468061708243232162015-03-11T22:32:55.496-04:002015-03-11T22:32:55.496-04:00I got interested in the state of math teaching (an...I got interested in the state of math teaching (and content) last year, and number and operations was the first I had a look at. I have come to the conclusion (so far) that the basic aspect of quantitative stuff is comparison, and the two associated phrases "how much/more" and "how many times as much/big". These lead to counting, with "how much/more" and measuring, with "how many times as".<br />Multiplication, division, and, more fundamentally, ratio belong to the "how many times" category. It has been observed that pre-school kids have strong concepts about "more"and "bigger", and can even say "he's got twice as much as me". It may be that theses quantitative concepts precede the counting/quantity concepts of number. And they are essentially multiplicative in nature.<br />We don't even need units of measurement to observe that this stick is twice as long as that stick<br />MIRA as a statement of fact is of course garbage, but the process of carrying out multiplications is based on the observation that multiplication can be viewed as repeated addition. When I first got into the education business we had mechanical barrel calculators, and that is exactly how they worked. So also does multiplication in a computer (at the lowest level).<br />I place a lot of the blame for many strange notions on the misinterpretation of the word "is".<br />A classic one is "A square is a rectangle". Same problem.Anonymoushttps://www.blogger.com/profile/18297158336334346872noreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-55139377406421533842015-02-24T18:18:09.512-05:002015-02-24T18:18:09.512-05:00There is one place when we (at least I) use a very...There is one place when we (at least I) use a very similar approach...<br />the Mathematical Olympiads.<br />The only significant difference is that sometimes I encourage the communication of ideas between participants after they spend some serious time trying to solve the problem by themselves (and before going to the blackboard).<br />Thanks Keith this is a great topic. :)Hector Floreshttps://www.blogger.com/profile/08609776337145536431noreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-89870164379289034882015-02-05T16:50:37.200-05:002015-02-05T16:50:37.200-05:00@JasonCantarella The article by Kirschner, Swelle...@JasonCantarella The article by Kirschner, Sweller, and Clark that you cite has been completely refuted by <a href="http://www.cogtech.usc.edu/publications/hmelo_ep07.pdf" rel="nofollow">Hemlo-Silver, Duncan, and Chinn</a>. Here is their abstract:<br /><br />"Many innovative approaches to education such as problem-based learning (PBL) and inquiry learning (IL) situate learning in problem-solving or investigations of complex phenomena. Kirschner, Sweller, and Clark (2006) grouped these approaches together with unguided discovery learning. However, the problem with their line of argument is that IL and PBL approaches are highly scaffolded. In this article, we first demonstrate that Kirschner et al. have mistakenly conflated PBL and IL with discovery learning. We then present evidence demonstrating that PBL and IL are powerful and effective models of learning. Far from being contrary to many of the principles of guided learning that Kirschner et al. discussed, both PBL and IL employ scaffolding extensively thereby reducing the cognitive load and allowing students to learn in complex domains. Moreover, these approaches to learning address important goals of education that include content knowledge, epistemic practices, and soft skills such as collaboration and self-directed learning."<br /><br />An article by Daniel Chazan and Deborah Ball, titled, <a href="https://www.education.umd.edu/MathEd/Home/People/Faculty/DChazanScans/ChazanBallTelling%20copy.pdf" rel="nofollow">Beyond Being Told Not To Tell</a> discusses the roles of direct instruction and of inquiry/discussion in the classroom.Burt Furutahttps://www.blogger.com/profile/01686343855625452613noreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-91484151502618713682015-02-02T02:14:28.158-05:002015-02-02T02:14:28.158-05:00@Kieth Devlin: As a geometry guy, I don't know...@Kieth Devlin: As a geometry guy, I don't know if it is critical to becoming a creative and intuitive thinker and problem solver. But for me it did. Not the geometry they taught in highschool or college, though. <br /><br />One day, I came across a copy of Euclid's Elements and instantly fell in love. School had ruined maths for me although I did like maths as a child. Euclidean Geometry broke me out from that.<br />I worked my way through the book, intently trying to grasp the concepts inside it. It was like what I'd image a drug going through an addicts might feel like (I wouldn't know, but I did feel ecstatic about geometry... maybe that wasn't the best simile.)<br /><br />I then spent months attempting to draw a large picture showing how the propositions of the first book lead up to Proposition 47 (Pythagorean Theorem.)<br />I have a notebook full of notes, but I'm still working on this project, but it's exhilarating. <br />It sparked me out of stagnation.<br /><br />There are certainly other things that can promote individual growth in the areas previously described... such as hacking, which has a central focus of problem solving. But, I think that Geometry is very effective path. It's just unfortunate that it gets slaughtered in the classrooms.Aevi V.noreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-14743745962989175952015-02-01T16:45:01.882-05:002015-02-01T16:45:01.882-05:00NOTE: We received a number of comments that raised...NOTE: We received a number of comments that raised points that deserved to be aired, but it is column policy not to publish comments that do not identify the writer. Keith Devlinhttps://www.blogger.com/profile/16899343259650938644noreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-78307106720034145802015-02-01T16:44:13.145-05:002015-02-01T16:44:13.145-05:00@ Kim Plofker You seem to have come to this site e...@ Kim Plofker You seem to have come to this site expecting an academic article. It’s not, Devlin's Angle is a quick-read, monthly column designed to put ideas out to stimulate reflection and discussion among the mathematical community and any mathematical bystanders that happen to chance by. The MAA publishes a number of journals that will likely provide the more substantial artices you are looking for. <br /><br />On the other hand, you did take the time to write a response, so maybe it did achieve it’s aim. :) Thanks for writing.Keith Devlinhttps://www.blogger.com/profile/16899343259650938644noreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-12899667634882139902015-01-21T17:15:08.621-05:002015-01-21T17:15:08.621-05:00IBL seems like a great idea, and its proponents ar...IBL seems like a great idea, and its proponents are very eloquent. But there's a significant body of research cautioning us that we need to take this with a grain (or more) of salt before joining the parade.<br /><br />See, for instance, <br /><br />Kirschner, P. A., Sweller, J., & Clark, R. E. (2006). Why Minimal Guidance During Instruction Does Not Work: An Analysis of the Failure of Constructivist, Discovery, Problem-Based, Experiential, and Inquiry-Based Teaching. Educational Psychologist, 41(2), 75–86. doi:10.1207/s15326985ep4102_1<br /><br />Having read this, I don't do a "full IBL" classroom, in my calculus class, though there are certainly some substantial project-based elements. <br /><br />Those have proved, in my experience, to be very rewarding, but very challenging for the students. <br />To the extent that I've measured the reasons, they seem to align with the reasons you might expect after reading the review article above-- there's a lot of information running around, most of which is irrelevant at any given stage and a major skill is picking out relevant detail from the forest of facts. <br /><br />It's valuable and rewarding to do this a few times a semester, after quite a bit of direct instruction from an expert, a number of worked examples, and a gradual transition from more standard to more difficult problem types.<br /><br />But reading your article, it's easy to walk away with the sense that you're advocating that we should do this all the time, for every topic, in every class! <br /><br />So... two questions:<br /><br />1) What's your response to the IBL skeptics in the education research community, such as the authors of the study above?<br /><br />2) What do think is the right balance of IBL and direct instruction? Or should we not do direct instruction at all anymore?<br /><br />Best wishes, and thanks for a thought-provoking article. --Jason.Jason Cantarellahttp://www.jasoncantarella.comnoreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-90148664327478578072015-01-21T15:01:06.795-05:002015-01-21T15:01:06.795-05:00While there are certainly important lessons to lea...While there are certainly important lessons to learn from and with IBL-style educational approaches, we'll never master them by this sort of uncritical cheerleading for corporate-style buzzspeak and anecdata. <br /><br />Regurgitated factoids about changes in the way that certain corporations choose to rank various vaguely defined "skills" in sloppily designed surveys of desired employee characteristics should not be presented as important insights into the role of modern pedagogy. <br /><br />For instance, why should we assume that Fortune 500 companies in the modern globalized workforce have the same importance in the future career prospects of the average US student as they did back in 1970? What percentage of the US workforce is employed by them? (Spoiler: A shrinking 10% or so, down from about 20% around the time of the 1970 survey.) Why should we care what they want in their employees more than we care what, say, government agencies or small businesses want? <br /><br />Similarly, Mitra's anecdote about letting a self-selected group of children play with a fascinating toy and, not surprisingly, learning a lot of information in the process doesn't really shed new light on the societal challenges of providing universal primary and secondary education.<br /><br />Many modern educators are chasing the ill-defined buzzwords of "innovation", "creativity", and "communication" as eagerly and credulously as their counterparts a hundred years ago chased the buzzwords of "discipline", "knowledge", and "advancement". You can't achieve either set of qualities with inspirational fables and sermonizing. IBL approaches are definitely worth investigating and pursuing, but not with this sort of credulous motivational-speaker-level gush. Kim Plofkernoreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-38751016019241396912015-01-09T09:31:08.308-05:002015-01-09T09:31:08.308-05:00Thanks for the additional comments. We received a ...Thanks for the additional comments. We received a fair number of comments (as was to be expected with this topic). This is not an open forum so we don’t publish all comments, rather we make editorial decisions to provide a reasonably broad coverage of opinions on the topic. The goal of the column is to inform–and to stimulate reflection and discussion among–people interested in mathematics and mathematics education (primarily, but not exclusively, collegiate level).<br /><br />@Katherine Beals: I don’t disagree. I wrote “is, in effect, saying, X” not that X *was* their motivation. I share the misgivings and dismay many have with the way CCSS have so far been implemented in many states. But the rational response to that is to fix the implementation, not throw away the standards.<br /><br />@SteveH: I don’t agree that CCSS sets too low a standard. But that is not to say that individual teachers or school of districts might not interpret them with a low bar, a real and present danger if student performance on tests is used as a primary means of evaluating teacher performance.Keith Devlinhttps://www.blogger.com/profile/16899343259650938644noreply@blogger.com