tag:blogger.com,1999:blog-76385782024-03-19T00:09:09.731-07:00An Academic's FreedomI believe a good definition of an academic is: "someone who loves learning and sharing what he/she learns". In this sense of the word, I have been an academic nearly all my life. As a faculty member at the USC Viterbi School of Engineering, I feel very fortunate that I can make a living doing what I love. This blog is my attempt to explore and reflect on the deep connections between learning and freedom.Unknownnoreply@blogger.comBlogger101125tag:blogger.com,1999:blog-7638578.post-31259199849557359332015-09-25T00:35:00.000-07:002015-09-27T13:32:40.904-07:00The First Law of Motion<div>
My son and I were walking back from the library where he had just spent two straight hours playing his favorite video game (and was consequently in a mellow mood). </div>
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As we marched on the pavement, we talked at first of this and that, chit-chat. Then it occurred to me this could be a good chance to strike up a conversation about something they'd be touching on at school soon.</div>
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"This guy, Newton," I started. "So he has three laws of motion, right?" </div>
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"Right," he said, a little cautiously, but not too perturbed. He is used to my habit of starting conversations like this rather abruptly.</div>
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"Do you happen to know what they are?" </div>
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He seemed a little dubious, so I backed up and decided to go even slower.</div>
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"Ok, let's talk about just the first one. It basically says objects that are at rest don't move, and if they are moving at a constant speed, they stay that way unless something acts on them to change their motion."</div>
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He nodded, this seemed familiar. "Because of their momentum," he noted.</div>
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I had noticed his laces were untied and pointed it out to him. We stopped by a fence while he put up his shoes to fix them.</div>
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"Yes, momentum, or..." I got a little confused myself, watching him. He was in fact right, but I wasn't sure in the moment if momentum was the right word to use about this law, so I said, "well, I think Inertia is the word we want. We say the objects have inertia, inertia of rest or inertia of motion."</div>
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I went on: "Do you know the ancient Greeks had their own version of this law, and it was rather different?"</div>
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He looked up, interested. We have been watching <a href="http://www.bbc.co.uk/cbbc/shows/horrible-histories">a very entertaining history TV show</a> these days and the Groovy Greeks, he knows, are always an interesting lot.</div>
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"They thought objects prefer to be always in motion unless something acts on them to stop them."<br />
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He chewed on this, as I went on to say, "But we know this isn't true. If you did this experiment in zero gravity and in a vacuum, for example, an object could stay still with nothing touching or acting on it."</div>
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Digressing here a bit, he said, "Oh, you know, my science teacher once got to experience zero-gravity. She said it was not a pleasant experience. Your insides feel like they're not supported."</div>
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"Hmm...," I said, "very interesting. I hadn't thought of how it would feel." </div>
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He ran off at this point ahead of me. I took it as a signal he had gotten bored of the discussion we were having about Newton, but he smiled and clarified he didn't want to miss the chance to catch the light, so he had run ahead to push the button on time. Indeed, the walk signal showed up right away and we could cross. </div>
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Seeing that I still had his attention, I continued as we crossed the road, "So, to them, the Greeks, every object they saw preferred to move and would only stop if prevented from moving. In particular they noticed objects always want to fall down, that is to say, move downwards, unless they were held up."</div>
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I gestured with my hands to illustrate this. We walked past some tall grasses, then a big box intended to collect clothing donations.</div>
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He thought about it a bit and remarked perceptively: "I see how what they were thinking makes a lot of sense. Imagine a rock at the very edge of a sheer cliff. It wouldn't even want to stay at the ground at the top of the cliff, it would rather fall even further down if it could. So the cliff is holding it in place."</div>
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"Right."</div>
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He nodded, chewing it further in his head.</div>
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I then asked, "Do you think the Greeks were wrong?"</div>
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He said "yes," right away.</div>
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I clarified a bit further: "I mean, if you think about the two different theories, they do agree on one point, that an object in motion could be stopped by a force acting on it. But they actually disagree about what would happen if an object was not moving and nothing was touching it or pulling/pushing at it. The Greek version says that is simply not possible, while Newton's law says it would stay still."</div>
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I continued, "the Greek version of this law applied to everything they encountered in their daily lives. We simply don't see such an object in our everyday experience. So for what we come across in our every day experience, both Newton's theory and the Greek theory are valid."</div>
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He responded, "I see. I kinda sympathize with them, they simply didn't know any better." </div>
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"Exactly!" I said, "given what they were able to observe, their explanation, their scientific theory was adequate and consistent. In fact, perhaps it was even more satisfying because it doesn't invoke something intangible called a 'Force', the objects in their world-view are held back from moving by other objects such as the ground or the fluid air."</div>
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I changed gears a little, "Similarly, Newton's theory was adequate for his time, but in fact now we know even he was wrong. His theory offers a good approximation but is not quite right."</div>
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He seemed surprised. "I had no idea! Newton was wrong? But he's so famous!"</div>
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I said, "well, Einstein's (general) theory contradicts Newton. Newton believed Gravity was a direct interaction between the planet and objects like the apple. But Einstein didn't believe in what he called 'spooky action at a distance' ."</div>
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"Then why do they teach kids about Newton's Physics?"</div>
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"Because he wasn't completely wrong. His physics provides a pretty good approximation of most phenomena we deal with. And it would be too hard to teach kids in school about Einstein's theories."</div>
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He laughed, no doubt at the silliness of kids being taught something wrong on purpose in schools everywhere because the truth would be harder to explain.</div>
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"Basically, unlike Newton, who saw saw Gravity as a Force, Einstein saw it as resulting from the curvature of space-time."</div>
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We were home. He rested, tired, on the couch, but he was still looking at me, clearly confused by this complicated expression "curvature of space-time". </div>
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"A somewhat intuitive analogy or model of what Einstein was saying is to think of the planet as a bowling ball on a piece of cloth, and any object near it such as a smaller ball will fall in towards the bowling ball because the fabric will have been reshaped in a way to make that happen. Einstein's point was that the planet doesn't call out to an apple at a distance to exert a force on it, just as the Bowling ball doesn't 'talk' to the smaller ball, rather the planet's mass distorts space and time in such a way that the natural motion of the ball leads it towards the planet."</div>
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"Ah! Remember that YouTube video we saw? They also showed how the smaller ball would loop round and round around the bowling ball if it was moving at a speed initially."</div>
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"Right. I remember it, that was a cool video!"</div>
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We both stopped talking, thinking in parallel, perhaps, about that shared memory.</div>
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Pleased, but thinking now of other things, he leaped from the couch, glided across the room, bounded up the stairs, and vanished.<br />
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His mother told me later he had narrated to her upstairs how Newton was wrong and told her excitedly about the bowling ball model of Einstein's theory of gravity... I am so glad our conversation had stirred something in him that he had wanted to share with others. </div>
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P.S.: My conversation with my son was inspired by an excellent article by Arthur Steiner titled "<a href="http://www.arthurstinner.com/stinner/pdfs/1994-storyofforce.pdf">The Story of Force: from Aristotle to Einstein</a>". As the author writes, "a history-based exposure to the conceptual development of Newtonian mechanics is superior to a conventional textbook-centered approach, because it is contextual, shows the intellectual struggle involved in scientific thinking and relates better to students' knowledge and experience." I couldn't agree more.<br />
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Credits: images from Wikimedia; thanks to Sean C. for pointing me at the video by Feynman. </div>
Unknownnoreply@blogger.com35tag:blogger.com,1999:blog-7638578.post-50194064606466157562015-09-10T11:34:00.001-07:002015-09-10T12:27:38.450-07:00University, Inc.? <span style="font-family: inherit;">A thought-provoking article in the latest NYTimes Magazine criticizing the increasing corporatization of academia has been making waves: <a href="https://www.blogger.com/%C2%A0http://www.nytimes.com/2015/09/13/magazine/why-we-should-fear-university-inc.html">"Why we should fear University, Inc."</a></span><br />
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<span style="font-family: inherit;">The author, Frederik deBoer, sounds a call of alarm about the "constantly expanding layer of university administrator jobs...," writing "It's not unheard of for colleges now to employ more senior administrators than professors... This legion of bureaucrats enables a world of pitiless surveillance; no segment of campus life, no matter how small, does not have some administrator who worries about it... like Niketown or Disneyworld, your average college campus now leaves the distinct impression of a one-party state."</span><br />
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<span style="font-family: inherit;">Is this a valid criticism of today's academic world? </span></div>
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<span style="font-family: inherit;">A bloated administration has been <a href="http://savecooperunion.org/updates/the_real_state_of_cooper_union.php">one of the charges leveled</a> against the Board and Administration of The Cooper Union for the Advancement of Science and Art (my alma mater), in a lawsuit by the Committee to Save Cooper Union (CSCU) that was recently <a href="http://www.nytimes.com/2015/09/02/nyregion/new-york-attorney-general-crafts-deal-to-end-litigation-at-cooper-union.html">settled through an intervention by the office of the NY State Attorney General</a>. CSCU, supported by a large number of students, alumni, and facu<span style="font-family: inherit;">lty, argued that a failure to control this administrative bloat had led to years of increased spending at a university, pushing it for the first time in its history (since 1859) to charge tuition to most incoming students. </span></span></div>
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<span style="font-family: inherit;">Others have quantified this trend and railed against it. <a href="http://necir.org/2014/02/06/new-analysis-shows-problematic-boom-in-higher-ed-administrators/">An article by Jon Marcus</a> from last year points out that since 1987, "</span>the number of administrators and professional staff has more than doubled. That’s a rate of increase more than twice as fast as the growth in the number of students." </div>
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<span style="font-family: inherit;">Another <a href="http://talkingpointsmemo.com/cafe/higher-education-following-k-12-failed-policy">recent essay, by David Schultz</a>, laments the role of administrators with a corporate mindset in creating increasingly more standardized curricula. Schultz writes: "<span style="background-color: white; color: #444444; line-height: 18px;">traditional schools [are] adopting this model; employing business leaders to run schools and developing cost containment policies aimed mostly at standardizing curriculum.</span><span style="background-color: white; color: #444444; line-height: 18px;"> </span><span style="background-color: white; color: #444444; line-height: 18px;">It is top-down decision-making premised upon treating faculty no differently than an assembly line worker. If all of the curriculum is the same then it is possible to substitute one content instructor for another. The result: a market-driven product devoid of innovation, creativity, and intellectual challenge."</span></span></div>
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<span style="font-family: inherit;">On the other side, to give a balanced perspective on this issue, it can be, and is argued by many, that the growing size of the administration and adoption of a corporate model in universities has itself been in response to several issues:</span></div>
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<li><span style="font-family: inherit;">a need to grow tuition revenue by increasing enrollment of both domestic and international students and the concomitant creation and management of new educational programs, including the need for greater information technology support for online programs, and more recruiting, admissions, advising and support staff for the increased enrollments</span></li>
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A report issued by the Delta Cost Project of the American Institutes of Research, authored by Donna Desrochers and Rita Kirshstein, titled "<a href="http://www.air.org/sites/default/files/downloads/report/DeltaCostAIR-Labor-Expensive-Higher-Education-Staffing-Brief-Feb2014.pdf">Labor Intensive or Labor Expensive? Changing Staffing and Compensation Patterns in Higher Education</a>" provides a comprehensive quantification of the changes and provides a perhaps more moderate perspective on them, concluding that while the higher education workforce grew in the 2000's, this was largely to compensate for increasing enrollments: " By 2012, public research universities and community colleges employed 16 fewer staff per 1,000 full-time equivalent (FTE) students compared with 2000, while the number of staff per student at public master’s and bachelor’s colleges remained unchanged".<br />
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<span style="font-family: inherit;">For all these reasons, I don't see this trend towards increased administrator hiring on campuses being halted or reversed dramatically anytime soon, nor am I sure there is merit in a blanket protest against this trend on principle alone given the many extraneous, environmental factors that may be contributing to it. In reflecting on my own experiences, I can see the genuine value and improvements that hiring a cadre of professional and capable administrators can bring in service of both students and faculty. </span><br />
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<span style="font-family: inherit;">At the same time, I think deBoer and Schultz are right in bringing the growth in numbers of administrative staff to our attention and in calling for greater awareness of their impact on every aspect of campus life, both inside and outside the classroom. </span><br />
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<span style="font-family: inherit;">On social media, I also saw someone share the following comment on deBoer's essay (included here with apologies for incomplete/insufficient attribution --- I couldn't determine the identity of the original writer):</span><br />
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<span style="font-family: inherit;"><span style="background-color: white; color: #222222;">In an age when technology is pervasive, and driving dramatic changes in the economy, when names like Steve Jobs, Mark Zuckerberg and Elon Musk seem to be foremost on everyone's lips (not Ramanujan, Einstein or Feynman, let alone Picasso, Camus or Tagore (quick! Can you name six currently active </span><span style="background-color: white; color: #222222;">artists,</span><span style="background-color: white; color: #222222;"> </span><span style="background-color: white; color: #222222;">writers, </span><span style="background-color: white;"><span style="color: #222222;">scientists, mathematicians, or other intellectuals that are today's equivalent of these figures?) ), when there is a growing sense that income disparities are growing and that job security is a thing of the past, and when the </span></span><a href="http://www.imdb.com/title/tt3263520/">rising costs of education</a><span style="background-color: white; color: #222222;"> are resulting in greater student debt than in the past, it is not surprising to me that students are flocking to professional schools and programs that emphasize lucrative employment, and that the view of a liberal, radical, wildly-impractical, education is considered old-fashioned and so pointless that </span><a href="http://www.vox.com/2015/9/8/9261531/professor-quitting-job">even tenured faculty in such fields are dropping out</a><span style="background-color: white; color: #222222;">. </span></span></div>
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<span style="background-color: white; color: #222222; font-family: inherit;">At any rate, I'm not sure if those of us who teach engineering can ever avoid the charge of being pro-corporate and overly focused on vocational training. Indeed, our own internal discussions often suggest to us that if anything we perhaps need to do even more to prepare our students to be successful in industry. But it is worth pondering, continually, whether we are maintaining a good balance between training and education, practice and theory, between what our students want and what they need... </span></div>
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Unknownnoreply@blogger.com18tag:blogger.com,1999:blog-7638578.post-14865155719568257372015-08-28T17:09:00.001-07:002015-08-29T11:53:30.745-07:00Seven Perspectives on K-12 Mathematics Education<div class="separator" style="clear: both; text-align: center;">
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<span style="font-family: inherit;">It occurred to me recently that there are several perspectives on K-12 mathematics education, that it may be helpful for educators, parents, and students to consider and compare:</span><br />
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<img border="0" height="150" src="http://4.bp.blogspot.com/-T_OqWf7-RHI/VeH4NUzwEsI/AAAAAAAAcDg/T3Azdv3AkNs/s200/moneyMath.png" /><img border="0" height="150" src="http://3.bp.blogspot.com/-VhjywCaHMXU/VeH4pKfkzOI/AAAAAAAAcDo/7Uv0cEeheNk/s200/lifeSkillsMath.jpg" /></div>
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<li><span style="font-family: inherit;"><b>The Utilitarian View: </b>This view argues that for most people that are not pursuing a career in science, technology, engineering or mathematics (STEM), all the mathematics being taught in schools is overkill - all that really need to be taught to all students are the basics of arithmetic operations and percentages, learning how to operate a calculator, and some basic geometry such as calculating areas of rectangles. These are the only essential life skills that would be needed to handle <a href="https://www.treasurydirect.gov/indiv/tools/tools_moneymath.pdf">personal financial decisions</a> including making purchases, budgeting, savings and investments. The focus in this view is placed somewhat narrowly on essential <a href="http://www.paperbackswap.com/LIFE-SKILLS-MATH-Pearson-Education/book/0785429379/">life skills</a>, so more advanced topics are not seen as necessary. In an article (that goes beyond merely stating this view) titled "<a href="http://people.exeter.ac.uk/PErnest/why.htm">Why Teach Math</a>", Paul Ernest writes that "<span style="background-color: white;">my claim is that higher mathematical</span><span style="background-color: white;"> knowledge and competence, i.e., beyond the level of numeracy</span><span lang="EN-US" style="background-color: white;"> achieved at primary or elementary school, is not needed by the majority of the populace to ensure the economic success of modern </span><span style="background-color: white;">industrialised</span><span lang="EN-US" style="background-color: white;"> society."</span></span></li>
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<li><span style="font-family: inherit;"><b>The Traditional View: </b>This view, which is the de facto norm encoded in most school textbooks and curricula worldwide today, argues for a deep and detailed study of mathematics, going beyond arithmetic to algebra and trigonometry. It entails a progression of concepts presented with an ever-increasing level of difficulty, and does emphasize learning how to execute many detailed calculation procedures such as long-division, simplifying fractions, etc. These procedures are often motivated and justified through applications in the form of word problems. There continues to be some debate about the most effective ways to teach this approach to math, with some emphasizing the need to proceed very systematically and build on successes (as in Mighton's JUMP math approach that he describes in "<a href="http://www.amazon.com/The-End-Ignorance-Multiplying-Potential/dp/0676979645">The End of Ignorance</a>", or the <a href="https://www.khanacademy.org/math">Khan Academy math</a> program), others like <a href="http://joboaler.com/">Jo Boaler</a> emphasizing encouraging a growth mindset in kids, showing them that there many ways of solving problems and advocating the end of timed testing, and yet others like the proponents of the <a href="http://www.corestandards.org/Math/">Common Core standards</a> emphasizing <a href="http://www.businessinsider.com/common-core-math-test-questions-2014-11">conceptual understanding</a> over rote memorization of procedures. But they still share much in common.</span></li>
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<img border="0" height="150" src="http://3.bp.blogspot.com/-YcDmuMRn83U/VeHxI-VKj6I/AAAAAAAAcC4/CNck7bwlxJQ/s200/euclid.jpg" /> <img border="0" height="150" src="http://2.bp.blogspot.com/-YEsccEB0JUE/VeHxRFCU6AI/AAAAAAAAcDA/EPBxbsA4BtA/s200/measurement.jpg" /><img border="0" height="150" src="http://3.bp.blogspot.com/-ecXzQwJDEqQ/VeH7pRfUcVI/AAAAAAAAcD0/yrQoAUylPxU/s200/loveMath.jpg" /></div>
<li><span style="font-family: inherit;"><b>The Aesthetic View: </b>This view, exemplified by Paul Lockhart in his famous article "<a href="https://www.maa.org/external_archive/devlin/LockhartsLament.pdf">A Mathematician's Lament</a>", and dating back really to the ancient Greeks, emphasizes that the essence of mathematics is the discovery of patterns, posing of relevant conjectures, and proving them rigorously through deductive reasoning. Mathematics, as a form of aesthetic creation, is understood to be a meaningful activity even if it has no applied end. In other words, this view advocates exposing students to pure mathematics, beyond applications, to get them to appreciate its beauty and value as an end in itself. Going through some of the original theorems and proofs in <a href="http://aleph0.clarku.edu/~djoyce/java/elements/elements.html">Euclid's Elements</a>, or studying some of the basic properties of <a href="http://www.math.utah.edu/~bertram/HighSchool/1Primes.pdf">Prime Numbers</a> are things that middle to high school students can do to gain an appreciation of mathematics from this pure perspective. Lockhart himself has written an excellent book titled "<a href="http://www.hup.harvard.edu/catalog.php?isbn=9780674057555">Measurement</a>" in an almost conversational style that is intended to show how a mathematician would think about abstract ideas. Aligned with this perspective is the notion that mathematics isn't merely a set of known facts to be "taught", but a lens with which to view and discover (in a self-driven, self-motivated way) interesting new abstract facts, ideas, patterns. I myself benefited greatly from the excellent <a href="http://www.ncert.nic.in/ncerts/textbook/textbook.htm?femh1=0-14">NCERT mathematics textbooks</a> used in many Indian schools, which in my view do a great job of introducing topics in pure mathematics, and encourage some amount of open-ended creative exploration. Another way for students to appreciate the aesthetic beauty of mathematics is to read first-person accounts by professional pure mathematicians, a good example of this genre is Edward Frenkel's "<a href="http://www.amazon.com/Love-Math-Heart-Hidden-Reality/dp/0465064957">Love and Math</a>."</span></li>
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<iframe allowfullscreen="" frameborder="0" height="150" mozallowfullscreen="" scrolling="no" src="https://embed-ssl.ted.com/talks/dan_meyer_math_curriculum_makeover.html" webkitallowfullscreen=""></iframe></div>
<li><span style="font-family: inherit;"><b>The Modeling View: </b>Dan Meyer's <a href="https://www.ted.com/talks/dan_meyer_math_curriculum_makeover?language=en">TED talk titled "Math class needs a makeover"</a> presents the view that focusing on teaching students various algorithmic recipes for arithmetic calculations is not useful. He argues instead that what is worth teaching and emphasizing in the classroom is how to formulate problems in the real world (i.e., model the real world) using the language of mathematics, that this is the essence of mathematical reasoning. An excellent resource for this view is the SIAM guidebook on <a href="http://m3challenge.siam.org/sites/default/files/uploads/siam-guidebook-final-press.pdf">Math Modeling: Getting Started and Getting Solutions</a>, by Bliss, Fowler and Galluzzo.</span></li>
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<iframe allowfullscreen="" frameborder="0" height="150" mozallowfullscreen="" scrolling="no" src="https://embed-ssl.ted.com/talks/conrad_wolfram_teaching_kids_real_math_with_computers.html" webkitallowfullscreen=""></iframe></div>
<li><span style="font-family: inherit;"><b>The Computational View: </b> Something that goes hand in hand with the modeling view is the view espoused by Conrad Wolfram in his <a href="http://www.ted.com/talks/conrad_wolfram_teaching_kids_real_math_with_computers">TED talk on "Teaching kids real math with computers</a>", that students should learn how to get computers to do the required computations, by programming them. I have had some positive experiences myself introducing middle-school to high-school students to some mathematical concepts via <a href="http://www.mathworks.com/products/matlab/">matlab</a> programming, focusing on modeling various physical phenomena from gravity to the propagation of waves. Although this perspective is bread and butter at the college level in engineering, science and mathematical departments, with the growing trend of introducing programming into middle and high school programs, and events such as the <a href="http://hourofcode.com/">Hour of Code</a>, and beginner-friendly programming environments such as <a href="http://scratch.mit.edu/">Scratch</a> and <a href="http://www.greenfoot.org/door">Greenfoot</a> Java, this view may grow in prominence for K-12 education as well.</span></li>
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<img border="0" height="117" src="http://4.bp.blogspot.com/-UfGGtrFdaHs/VeHpwajgZ7I/AAAAAAAAcCg/i5TDZoc-Xyk/s200/dragonBox.png" width="200" /> <img border="0" height="117" src="http://1.bp.blogspot.com/-5L60LdVJVkY/VeHp5h5zutI/AAAAAAAAcCo/fXSQs2s1DRs/s200/elements.png" /></div>
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<li><span style="font-family: inherit;"><b>The Recreational View: </b>While this cannot be the entire basis of a curriculum, an additional perspective on mathematics education is that it is helpful, even important, to integrate various recreational mathematical and logical "puzzles" and games. These include classic puzzles like the ones about <a href="http://www.scientificpsychic.com/mind/aqua1.html">measuring out fluid</a> and <a href="http://puzzlepage.blogspot.com/2008/01/cabbage-tiger-and-goat.html">boat trips with constraints</a> and <a href="http://www.websudoku.com/">Sudoku</a>'s. <a href="http://www.math.cornell.edu/~mec/2006-2007/Probability/ProbGames.htm">Games of chance</a> involving coin-tosses, dice, or cards naturally offer many opportunities to engage in probabilistic reasoning to figure out various odds and acceptable bets. More recently, there has been a lot of work on developing mathematically oriented video games, two particularly outstanding ones that I recommend heartily are <a href="http://www.dragonboxapp.com/">Dragonbox</a>, and Dragonbox <a href="http://wewanttoknow.com/elements/">Elements</a>.
Besides improving learner engagement with and interest in mathematics, these puzzles and games also provide practice in more creative and open-ended mathematical thinking. Indeed, many mathematically-oriented researchers I come across in academia have had a great life-long love for recreational puzzles. On a related note, there are now many fascinating youtube channels such as <a href="https://www.youtube.com/user/numberphile">numberphile</a> and <a href="https://www.youtube.com/user/Vihart">Vihart</a> that provide entertaining treatments of mathematical subjects.</span></li>
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<img border="0" height="150" src="http://1.bp.blogspot.com/-ou0SLKGF48A/VeHoCg1PbkI/AAAAAAAAcCE/PCYY1oxyAgA/s200/storyMath.jpg" /> <img border="0" height="150" src="http://1.bp.blogspot.com/-Tc6pqZAIizs/VeHoV8oouBI/AAAAAAAAcCM/N3_vUr32IPU/s200/menOfMath.jpg" /> <img border="0" height="150" src="http://2.bp.blogspot.com/-xB9cfTUiBC8/VeHot0if7AI/AAAAAAAAcCU/d1ffjNCvqj0/s200/manInfty.jpg" /></div>
<li><span style="font-family: inherit;"><b style="-webkit-text-stroke-width: 0px; color: black; font-style: normal; font-variant: normal; letter-spacing: normal; line-height: normal; text-transform: none; white-space: normal; word-spacing: 0px;">The Historical View: </b>Another valuable perspective on mathematics and why it is worth appreciating is that it is very much an essential part of the history of mankind. From Babylon to Egypt, Greece, India, Arabia, to Europe, we have many interesting stories about the development of mathematics and the people behind them. In a class I taught to undergrads about probability, I felt it would enhance their interest in the subject to learn about the very origins of probability theory in the games of chance played in 17th century France: <a href="http://homepages.wmich.edu/~mackey/Teaching/145/probHist.html">the problems posed by de Méré, and the correspondence between Pascal and Fermat</a>. Excellent resources including several mathematical history and biography books such as "<a href="https://en.wikipedia.org/wiki/Men_of_Mathematics">Men of Mathematics</a>" and <a href="https://www.goodreads.com/shelf/show/mathematical-biography">others</a>. The BBC video series "<a href="https://en.wikipedia.org/wiki/The_Story_of_Maths">The Story of Maths</a>" and the podcast "<a href="http://www.bbc.co.uk/programmes/b00srz5b/episodes/downloads">A Brief History of Mathematics</a>" are outstanding resources in this regard as well. <b style="-webkit-text-stroke-width: 0px; color: black; font-family: 'Times New Roman'; font-size: medium; font-style: normal; font-variant: normal; letter-spacing: normal; line-height: normal; text-transform: none; white-space: normal; word-spacing: 0px;"> </b></span></li>
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<span style="font-family: inherit;">In "<a href="http://people.exeter.ac.uk/PErnest/why.htm">Why Teach Mathematics</a>?" Paul Ernest presents some interesting arguments on different aims that five different "interest groups" in society have with respect to mathematics education. His categorization aligns partially with the above perspectives, but is not quite the same, as he applies it primarily to argue that mathematics curricula are determined ultimately by a political process arising from the frictions and compromises between these groups. My intention here has rather been to try and tease out different ways in which mathematics can be taught and experienced in a classroom and beyond, with pointers to different arguments in favor of these different ways, and present some relevant resources. </span></div>
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<span style="font-family: inherit;">My own thought is that while some of these perspectives appear distinct, they are in fact complementary and can be harmonized to a great extent. Perhaps even someone with a hardline view that most math is not essential to daily living may not be averse to learning more about its interesting historical development, or trying out a puzzle or two as recreation, and may possibly be persuaded to appreciate how it is being used to model and engineer the world around us, even if they themselves don't feel inclined to engage in that kind of modeling themselves. School mathematics educators would benefit from keeping in mind all of these perspectives as they teach to give their students a more holistic and engaging experience of the subject. </span></div>
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<span style="font-family: inherit;">Have I left out a perspective? Do you have some thoughts of your own on this matter? Please write. </span></div>
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Unknownnoreply@blogger.com17tag:blogger.com,1999:blog-7638578.post-86401981019081596182015-07-18T16:23:00.000-07:002015-07-18T18:14:22.642-07:00An Indian woman traveled to the US for education, in 1883<img height="400" src="https://lh3.googleusercontent.com/-XYQ1yDPy1WA/U0hTkSsMyQI/AAAAAAAAzEA/DgevyJMBwfM/w684-h1074/original.jpg" width="253" /><br />
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I came across the above image today, showing three ladies who came to study Medicine in America. Dr. Anandabai Joshee, among them, was the first Indian woman to have obtained a medical degree abroad. They are featured in a recent <a href="http://www.pri.org/stories/2013-07-15/historical-photos-circulating-depict-women-medical-pioneers">news story on PRI</a>.<br />
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Trying to learn more about her, I came upon a remarkable <a href="https://archive.org/details/lifedranandabai03dallgoog">biography of her life</a>, published in 1888. It contains many of her own words, in letters and speeches.<br />
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This was 1883. Indians were still extremely conservative, and it was unheard of for a Hindu woman, who had not converted to Christianity, to travel abroad for any reason.<br />
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In a public talk she gave in India before her departure, she spoke forcefully against popular opinion which condemned her decision to travel abroad, and that too on her own (she was supported strongly in this unusual venture by her husband, an enlightened man for his time, who worked as a postmaster in Serampore):<br />
<blockquote class="tr_bq">
"To go to foreign countries is not bad, but in some respects better than to stay in one place. The study of people and places is not to be neglected. Ignorance when voluntary is criminal. In going to foreign countries, we may enlarge our comprehension, perfect our knowledge, or recover lost arts. Every one must do what he thinks right."</blockquote>
She spoke of her motivation:<br />
<blockquote class="tr_bq">
"I go to America because I wish to study medicine. I now address the ladies present here, who will be the better judges of the importance of female medical assistance in India. I never consider this subject without being surprised that none of those societies so laudably established in India for the promotion of sciences and female education have ever thought of sending one of their female members into the most civilized parts of the world to procure thorough medical knowledge, in order to open here a College for the instruction of women in medicine... The want of female physicians in India is keenly felt in every quarter."</blockquote>
Upon her graduation in the US, she was offered a position back in India as the Lady Doctor of Kolhapur. Her true nobility and spirit of service can be gauged from the following account. Her offer letter stated seven conditions, with the final one reading: "Private practice will be allowed to any extent that will not interfere with public duties, but no fees are to be charged for attending on the ladies of the palace, or on the wives of contributors to the Hospital Funds."<br />
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To this, she is said to have responded:<br />
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"There is nothing in the seven conditions which you name, that causes me any uneasiness, but if any question were likely to arise under it, I might object to the seventh... Our Shastras require us to impart the gifts of healing and of religious truth without pay, and to this practice I shall adhere; but if I ever meant to take a fee from any one, it would assuredly be from those who are rich and powerful, and never from those who are poor and depressed."</blockquote>
She returned to India in 1885, but unfortunately died of Tuberculosis within months of her arrival before she could begin her practice. She was only 21 years old.<br />
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Reading this courageous woman's story made me reflect on the very bright and capable young female student from India I hosted in my lab as a summer research Intern just this summer. She told me she hopes to return to the US to do a Ph.D. but eventually plans to settle back in India. I suspect she has not before heard of this admirably strong woman from the 19th century, whose footsteps she is following.<br />
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<br />Unknownnoreply@blogger.com13tag:blogger.com,1999:blog-7638578.post-50475613293082877352015-07-06T14:44:00.004-07:002015-07-06T14:44:55.928-07:00Adding Mathematical Rigor to Systems Research<div class="tr_bq">
A bright young man I know, a recent Ph.D. graduate from another university, who works in my research area of wireless networks, contacted me recently sharing with me that he had joined industry but that he was "still interested in pursuing a research career despite being in industry." He wrote to me: </div>
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<span style="font-size: 12.8000001907349px;">I concur with your <a href="http://www.ece.rutgers.edu/~pompili/index_file/extra/HowToDoResearch_ANRG_WP02001.pdf" style="color: #1155cc;" target="_blank">emphasis</a> on the importance of mathematical analysis:</span></div>
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<li style="margin-left: 15px;"><span style="font-family: Times; font-size: 11pt;">"you must learn how to add rigor to your work through mathematical analyses for your work to be respectable for graduate-level researchers." </span></li>
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<span style="font-size: 12.8000001907349px;">I have yet to put more efforts to learn this skill. You have a unique blend of <a href="http://arxiv.org/PS_cache/arxiv/pdf/1011/1011.4748v1.pdf" style="color: #1155cc;" target="_blank">theory</a> and <a href="http://ceng.usc.edu/~bkrishna/research/papers/RoutingWithoutRoutes_IPSN10.pdf" style="color: #1155cc;" target="_blank">systems</a>, thus I am wondering what your take is on how to achieve this for people with mostly systems background like me?</span></div>
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Here I should first clarify that "systems" here refers to computer science topics such as operating systems, database systems, network systems which tend to be more software implementation and empirical evaluation oriented in general.<br />
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This was my response to him:<br />
<blockquote style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8000001907349px;">
I think a good starting point for learning theory is learning how to mathematical model real-world problems: <span style="background-color: transparent;"> </span></blockquote>
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* Check out this very <a href="http://m3challenge.siam.org/sites/default/files/uploads/siam-guidebook-final-download.pdf">basic book on mathematical modeling</a> (aimed at HS/undergrads, I believe) as a starting point.<span style="background-color: transparent;"> </span></blockquote>
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* One article that guided my early efforts at doing some mathematical modeling was Hal Varian's "<a href="http://people.ischool.berkeley.edu/~hal/Papers/how.pdf">How to Build an Economic Model in your Spare Time</a>" </blockquote>
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<br />* I attempted once to write a short "tutorial" on <a href="http://ceng.usc.edu/~bkrishna/research/papers/Krishnamachari_ModelingDataGathering.pdf">how to apply mathematical modeling to wireless sensor networks</a> that you might find useful as a starting point in thinking about modeling: </blockquote>
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* <span style="font-size: 12.8000001907349px;">To get a bit deeper, you do need to learn to construct proofs. </span><span style="font-size: 12.8000001907349px;"> Polya's "<a href="http://www.amazon.com/How-Solve-It-Mathematical-Princeton/dp/069116407X">How to Solve it</a>" .. is indeed a good starting reference. </span></blockquote>
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* I took two courses at Cornell that really taught me to prove things:<br />1. A course on real analysis in the math department (something like this <a href="http://ocw.mit.edu/courses/mathematics/18-100c-real-analysis-fall-2012/lecture-summaries/">MIT Course on Real Analysis</a>; a good book for it is Strichartz's "<a href="http://www.amazon.com/Analysis-Revised-Edition-Bartlett-Mathematics/dp/0763714976">The Way of Analysis</a>": )<br />2. A course on analysis of algorithms taught by Jon Kleinberg; the notes for the class I took got turned into a great book called "<a href="http://www.amazon.com/Algorithm-Design-Jon-Kleinberg/dp/0321295358">Algorithm Design</a>." <span style="background-color: transparent;"> </span></blockquote>
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Like with everything, the trick to learn how to do more theoretical research is to start small and practice - build some simple models or prove some simple known things first then work your way towards something more substantial... </blockquote>
I then added afterwards in a follow-up note:<br />
<blockquote class="tr_bq" style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8000001907349px;">
<span style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8000001907349px;">The other thought that occurred to me is that in my own work, often my students and I base our approach on the analysis in a similar paper.</span><span style="background-color: transparent;"> </span></blockquote>
<blockquote class="tr_bq" style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8000001907349px;">
Reading the classic papers on analysis of CSMA, TCP fluid modeling, Network Utility Maximization, etc. can be a starting point. Ask yourself if you could modify the analysis or assumptions a little to treat a slightly different problem. This will also give you more practice in doing analytical modeling and proofs.<span style="background-color: transparent;"> </span></blockquote>
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Last but not the least, it could also help to collaborate and discuss with others that are more theoretically oriented to gain insights on your own problem...</blockquote>
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<br />Unknownnoreply@blogger.com14tag:blogger.com,1999:blog-7638578.post-88826759466832882962015-06-01T01:45:00.002-07:002015-06-01T01:48:26.352-07:00What it means to be truly educated<div class="separator" style="clear: both; text-align: left;">
I came across a delightful quote by Noam Chomsky about what it means to be truly educated that I couldn't agree with more:</div>
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"... it's not important what we cover in the class, it's important what you discover. </blockquote>
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To be truly educated from this point of view means to be in a position to inquire and to create on the basis of the resources available to you which you've come to appreciate and comprehend. To know where to look, to know how to formulate serious questions, to question a standard doctrine if that's appropriate, to find your own way, to shape the questions that are worth pursuing, and to develop the path to pursue them. That means knowing, understanding many things but also, ... to know where to look, how to look, how to question, how to challenge, how to proceed independently..."</blockquote>
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<iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/eYHQcXVp4F4/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/eYHQcXVp4F4?feature=player_embedded" width="320"></iframe></div>
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Chomsky's response gets to the deep connection between education and autonomy. What it means to be educated is to have the motivation and the ability, to pursue inquiry and discovery on one's own. The goal of our academic institutions should be to cultivate in each student the confidence, the drive, the tools needed to pursue and discover truths about how our world is, and to articulate and shape how it ought to be. Our classrooms should not be focused only on the transmission of known facts and methods. Unknownnoreply@blogger.com10tag:blogger.com,1999:blog-7638578.post-12125564130117445652015-05-01T01:58:00.001-07:002015-05-01T02:02:04.128-07:00Eine Kleine Nacht Integralrechnung (A Little Night Integral Calculus)<div class="separator" style="clear: both; text-align: center;">
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I had asked my middle-schooler if he could spare me some time after dinner to talk about some mathematical ideas and he had agreed. So I rushed to put the plates away, and we gathered with papers and a marker. </div>
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After a preliminary conversation on the nature of acceleration in cars, we decided to focus our discussion on the topic: how far does a dropped object fall in a given period of time? I illustrated it by dropping the marker from a small height. (Assuming no air resistance, I added.)</div>
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We started with the basic notion of acceleration. When we say an object is experiencing constant acceleration, what do we mean? I took up the example of the constant acceleration due to gravity, which we call g. What is the value of it, I asked him? 9.8 he answered, remembering this constant from our previous discussion. </div>
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We talked about the units of acceleration, meters per second-squared, which is telling us the additional speed increase every second. So in this case, the speed of the falling object increases every second by 9.8 m/s, I said to him, and he nodded.</div>
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I raised and dropped the marker again. We watched it fall with a thud. What was the speed at the very moment when it started its fall? I asked. 0, he said. </div>
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How about a second later? 9.8 m/s, he said. </div>
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How about in two seconds? 9.8 + 9.8, which is 19.6 he said. I agreed, right, two times the acceleration g. So then, he asked, to clarify, is the speed 2g's after 2 seconds?</div>
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I clarified that we normally refer to g's when talking about acceleration, which in this case will always be 1g, but numerically, yes, the speed at this point, measured in m/s, would be 2 times the value of g. And we extrapolated from there how it would 3 times g after 3 seconds, and so on. </div>
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I wrote down the formula: speed = acceleration x time, which corresponded to our extrapolation. I drew points on a plot with the x-axis labelled time and y-axis labelled speed: (0,0), (1, 9.8), (2, 19.6), (3, 29.4), and the line that goes through those points. We talked briefly about how this fits the form of the equation of a line (y = m.x + c) which he saw recently. </div>
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Now, I asked, how much distance did the object travel in the first second? He said 9.8 meters. Ah, I responded. That would be true if the object was going 9.8 m/s for the entire first second. But in fact it was starting at 0 m/s and accelerated to 9.8 m/s only at the end of that second. So you would expect it was something less than 9.8 meters, no? Ah yes, he said, smacking his head, I forgot! </div>
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Let's see if we can figure it out, I said. I drew a horizontal line from 0 s to 1 s at 9.8 m/s and then a step up to another line from 1 s to 2 s at 19.6 m/s then another line from 2 s to 3 s at 29.4 m/s. Imagine this object, I said. It moves at 9.8 m/s for the first second, then at a constant speed of 19.6 m/s the second second, and then a constant speed of 29.4 m/s for the third second. How far would it travel in these three seconds? We worked it out: 9.8 + 19.6 + 29.4 = 58.8 m.</div>
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Then I showed him how the distance traveled in this case could be viewed as the sum of the areas of the corresponding rectangles. I then asked him, do you agree this object is always traveling faster than our free-falling object except at the end of each second when it's equal? He nodded. So this object travels farther than ours. He nodded again.</div>
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Now consider another object, I said, which travels 4.9 m/s from 0 s to 0.5 s, 9.8 m/s from 0.5 s to 1 s, and so on. This object's distance traveled would also be the sum of the corresponding rectangles, I pointed out, and would also be higher than the distance traveled by the free-falling object, but a little better approximation. His eyes lit up as I pointed out that we could make better and better approximations, and he could see how the distance traveled by the free-falling object would come to be the area under the line we had drawn showing the speed versus time function for it.</div>
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Now, I said, recall that we agreed the speed of the object could be expressed as acceleration x time. He agreed. I continued, the area in question (i.e. under the line) is that of a triangle with this height of "acceleration x time", and a base of length "time". So by the equation for the area of a triangle (half of base x height), we can conclude the total distance traveled would be 1/2 x acceleration x time ^2. </div>
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I pushed on. This is pretty nifty, to have such a clean answer to the question how far does the object fall in a given time. But often we don't get a nice equation like this. Consider the kind of speed versus time that is typical of when I drive the car. It speeds up, stays constant for a while perhaps, then slows down again, maybe stops for a bit at a light, then picks up again, and so on... We get a strange, arbitrary shape. What would be the distance traveled in this case? By analogy to what we just did, it is just the area under this arbitrary curve. He agreed. And how can we figure out this area? I asked him. By drawing those rectangles, he replied... </div>
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His mother, who was listening to us by now as well, chimed in, isn't this Calculus? Yes, I nodded. This is what's called Integration. We talked briefly about Newton and Leibniz and their <a href="http://l.facebook.com/l.php?u=http%3A%2F%2Fen.wikipedia.org%2Fwiki%2FLeibniz%25E2%2580%2593Newton_calculus_controversy&h=sAQEORimz&s=1" rel="nofollow" style="color: #3b5998; cursor: pointer; text-decoration: none;" target="_blank">competing claims to having invented Calculus</a>. </div>
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My son's eyes lit up with a question. He picked up the pen from me and drew a blob on the paper. What about the area of this blob? he asked me. Can you calculate it? I said, sure, using this method of rectangles we could calculate the area approximately. No, but exactly, he asked. Well, not sure I could give you an infinite precision answer, I said. Aha, thought so! he exclaimed, feeling satisfied at encountering a fundamental limit to our understanding of things. </div>
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I mentioned then a couple of other ways to estimate the area of that blob. We could take a picture of a rectangle around it and count how many pixels are inside the blob and how many are outside, then use that ratio and the sides of that rectangle to get the area. We could ask a computer to generate a sequence of random points inside the square and check for each point if it is inside the blob or not, and the average ratio in this case would also converge over large numbers of samples to the right answer (this is called the <a href="http://l.facebook.com/l.php?u=http%3A%2F%2Fen.wikipedia.org%2Fwiki%2FMonte_Carlo_method&h=RAQGPT1nf&s=1" rel="nofollow" style="color: #3b5998; cursor: pointer; text-decoration: none;" target="_blank">Monte Carlo method</a>, and was invented by Ulam while working on the Manhattan project). Or we could carve out that shape with uniform depth in some material and measure how much liquid it holds to estimate the area... </div>
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There was a bit of a digression here while we talked about the computability of real numbers briefly and I mentioned <a href="http://l.facebook.com/l.php?u=http%3A%2F%2Fen.wikipedia.org%2Fwiki%2FChaitin%2527s_constant&h=SAQFbz097&s=1" rel="nofollow" style="color: #3b5998; cursor: pointer; text-decoration: none;" target="_blank">Chaitin's number Omega</a>, which is a real number with the property that no algorithm can compute its digits (unlike pi).</div>
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My wife wondered if you could not also figure out the area from the perimeter of the blob which would be easy to measure with a thread. Ah, I said, but you can't infer the area from the perimeter. We talked about how a circle and a square having the same perimeter would have different areas. My son agreed though he said he couldn't prove it. I thought we would save that proof for a later conversation, and we ended our discussion for the night...</div>
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ps: The title I chose for this article is of course a weak attempt to evoke Mozart's famous piece, which I will never forget listening to at a live performance when I was 19 and everything seemed possible... </div>
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Unknownnoreply@blogger.com3tag:blogger.com,1999:blog-7638578.post-6816590960822490252015-04-07T10:13:00.003-07:002015-05-01T19:02:24.919-07:00Staking a claimCame across a quote from <span style="background-color: white; color: #1a1a1a; font-family: serif; font-size: 16.5px; line-height: 21px;">René </span>Decartes in an <a href="http://plato.stanford.edu/entries/descartes-mathematics/">article about his mathematics</a>:<br />
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<span style="background-color: white; color: #1a1a1a; font-family: serif; font-size: 16.5px; line-height: 21px;">"..having determined as I did [in </span><em style="background-color: white; color: #1a1a1a; font-family: serif; font-size: 16.5px; line-height: 21px;">La Géométrie</em><span style="background-color: white; color: #1a1a1a; font-family: serif; font-size: 16.5px; line-height: 21px;">] all that could be achieved in each type of problem and shown the way to do it, I claim that people should not only believe that I have accomplished more than my predecessors but should also be convinced that posterity will never discover anything in this subject which I could not have discovered just as well if I had bothered to look for it (To Mersenne, end of December 1637; AT 1, 478; CSMK, 78–79)."</span></blockquote>
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<span style="background-color: white; color: #1a1a1a; font-family: serif; font-size: 16.5px; line-height: 21px;">He seems to me to be staking a claim not only on the innovations that are immediately apparent, but also on all other future discoveries on the subject, that had he only "bothered to look for", he would have made those as well... talk about self-confidence!</span><br />
<span style="background-color: white; color: #1a1a1a; font-family: serif; font-size: 16.5px; line-height: 21px;"><br /></span>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-7638578.post-72911232931186849742015-04-06T01:00:00.001-07:002015-04-07T10:27:27.826-07:00Opening the Eye of the Soul<div class="separator" style="clear: both; text-align: center;">
<a href="http://3.bp.blogspot.com/-S_FQfaysmQc/VSI770muonI/AAAAAAAAJSQ/XFZyktYbrlk/s1600/elements.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" src="http://3.bp.blogspot.com/-S_FQfaysmQc/VSI770muonI/AAAAAAAAJSQ/XFZyktYbrlk/s1600/elements.png" height="320" width="296" /></a></div>
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I was talking to someone who noted that though they had done AP calculus in high school, all that math seemed to have been of no use or meaning <span style="text-align: center;">whatsoever in their life. Indeed, this person noted, setting aside its role in the narrow fields of science and engineering, is it not true that there is no point to teaching all the math we do in schools, even geometry, to every student given only arithmetic is practically encountered on a daily basis and even that could mostly be done on a calculator?</span><br />
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This is an age old question. A dismissive response to it was said to have been offered by the great Euclid, who, when a student asked what good he would derive from learning Geometry, ordered his assistant to "pay him three obols, for he must profit from what he learns!"<br />
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But while Euclid, in this story, treats with contempt the questioner for seeking a material use for Geometry, he doesn't express in clear and positive terms what other good it might bring.<br />
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I came across the following quote, from a T.Taylor, in his "<a href="https://play.google.com/store/books/details?id=AD1WAAAAYAAJ">Dissertation on the True End of Geometry</a>" (1792), wherein he addresses the most important reason to study geometry more explicitly, rather poetically:<br />
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".. if geometry is a speculative science (I mean the geometry of the ancients), it is both desirable for its own sake, and for still higher contemplations, the visions of intellect, to which it is ultimately subservient. For, when studied with this view, <b>it opens the eye of the soul to spectacles of perfect reality</b> and purifies it from the darkness of material oblivion. Away then, ye sordid vulgar, who are perpetually demanding the utility of abstract speculations, and who are impatient to bring down and debase the noblest energies, to the most groveling purposes..."</blockquote>
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As it happens, I have been going through a few propositions from the first book of Euclid's Elements with my son in recent weeks. I think I understood why I was doing it only dimly till now...<br />
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For anyone that may be interested, here are a few excellent links to study the Elements:<br />
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<li><a href="http://aleph0.clarku.edu/~djoyce/java/elements/elements.html">http://aleph0.clarku.edu/~djoyce/java/elements/elements.html</a></li>
<li><a href="https://www.youtube.com/playlist?list=PLFC65BA76F7142E9D">https://www.youtube.com/playlist?list=PLFC65BA76F7142E9D</a></li>
<li><a href="http://udel.edu/~mm/euclid/">http://udel.edu/~mm/euclid/</a></li>
<li><a href="http://farside.ph.utexas.edu/Books/Euclid/Elements.pdf">http://farside.ph.utexas.edu/Books/Euclid/Elements.pdf</a></li>
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The Elements are quite accessible to be explained to a middle-schooler. There is something really inspiring and meaningful about learning Geometry directly from the Ancient Greeks, from a book first written circa 300 BC. It is still current and makes for excellent training in rigorous thinking and deductive reasoning. It makes it possible to gauge for oneself how brilliant and sophisticated they were, how far along they got with abstract thinking, and helps one see the unbroken threads that connect philosophical and intellectual investigations and the growth of knowledge through the ages.Unknownnoreply@blogger.com3tag:blogger.com,1999:blog-7638578.post-23057706357569129342014-10-23T00:30:00.003-07:002014-10-23T12:09:22.086-07:00On the Virtue of Putting the Cart before the Horse<br />
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You may have seen the above video by "DjSadhu" purporting to show how the motion of the planets would appear if we use a frame of reference with respect to which the sun is moving. If not, I'd encourage you to take a look. It's fascinating.<br />
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However, Phil Plait has posted an article on slate.com pointing out that <a href="http://www.slate.com/blogs/bad_astronomy/2013/03/04/vortex_motion_viral_video_showing_sun_s_motion_through_galaxy_is_wrong.html">this viral video is in fact scientifically incorrect</a>, where he says:<br />
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<span style="background-color: white; color: #281b21; font-family: sl-Apres; font-size: 15px; line-height: 27px;">It </span><em style="background-color: white; color: #281b21; font-family: sl-Apres; font-size: 15px; line-height: 27px;">seems</em><span style="background-color: white; color: #281b21; font-family: sl-Apres; font-size: 15px; line-height: 27px;"> right, or looks cool, or appeals to some sense of how things should be. But how things </span><em style="background-color: white; color: #281b21; font-family: sl-Apres; font-size: 15px; line-height: 27px;">should</em><span style="background-color: white; color: #281b21; font-family: sl-Apres; font-size: 15px; line-height: 27px;"> be and how they </span><em style="background-color: white; color: #281b21; font-family: sl-Apres; font-size: 15px; line-height: 27px;">are</em><span style="background-color: white; color: #281b21; font-family: sl-Apres; font-size: 15px; line-height: 27px;"> don’t always overlap. The Universe is a pretty cool place, and works using a fairly well-regulated set of rules. We call those rules physics, they’re written in the language of math, and trying to understand all that is science.</span></blockquote>
My first reaction on seeing Phil's article was, "Yes, that makes sense. As he says, 'not everything cool is science...' "<br />
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But then I saw a post by a friend of mine, who was a former faculty member (now working in industry), in response to Phil's article. He wrote that Phil was, well, a "killjoy. Perhaps, [a] scientifically accurate killjoy, but still a killjoy. " His point was that the original video, despite its inaccuracies, served to spark the imagination of many people (as of now, that video has received over 2 million hits!) and that Phil could have offered his corrections in a more constructive spirit rather than proclaiming loudly the incorrectness of the original video.<br />
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Reflecting on his comments, I think my friend is really on to something.<br />
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Just recently some colleagues and I were having an interesting conversation about why college is boring for many students and we figured out that we college faculty may all be killjoys.<br />
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Faculty, due to both nature and nurture the most pedantic of creatures, would rather say something careful and provably correct than show something cool of dubious "truthiness". We would rather build up little by little from abstract fundamentals in the vain (often unfulfilled due to lack of time) hope of reaching eventually concepts one can relate to. We are not inclined to let our students encounter something half-baked that they find amazing and thought-provoking, and then work from that spark of excitement towards the truth.<br />
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It is worth pausing to think whether our sneering dismissal of imprecise, careless thinking, our endless harping on rigor, our ceaseless skepticism (all qualities that are essential for our work as researchers, the very qualities we are recognized and lauded for by our peers) might sometimes get in the way of creating the best environment for our students to seek the truth on their own.<br />
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Now, let me clear, I am not arguing for teaching students cool-but-unscientific lies or even half-truths. If our own hunger for truth and the desire to share that drive with our students did not ultimately motivate us, we would not be in this line of work. But perhaps there is a place for the incorrect, the absurd, if it is cool enough to draw in the students' curiosity and their imagination, and inspire them to proceed further? It could serve as a starting point for a more careful and rigorous investigation.<br />
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Sometimes at least, we <i>should</i> place the cart before the horse (or the sun in front of the planets) because that is unusual and striking, enough to make one stop, stare and think.<br />
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<br />Unknownnoreply@blogger.com84tag:blogger.com,1999:blog-7638578.post-73255297476613947982014-08-29T16:29:00.000-07:002014-08-30T22:34:03.654-07:00The mind is not a bottle<br />
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I came across an essay called "<a href="http://penelope.uchicago.edu/Thayer/E/Roman/Texts/Plutarch/Moralia/De_auditu*.html">De Auditu</a>" ("on listening to lectures"), by Plutarch, the Greek Historian and Essayist, who lived nearly 2000 years ago. It's worth a read, if only to recognize how timeless his words are.<br />
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In the essay, Plutarch begins by noting that it is just important to consider and learn how to listen as it is to learn how to offer a discourse. He urges the listener to be patient, to focus on substance not style. He recommends moderation in posing questions, including limiting one's questions to the area of the speaker's expertise. He suggests that even errors in lectures offer opportunities for learning, by motivating introspection into one's own ways of thinking:<br />
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Where [the speaker] is successful we must reflect that the success is not due to chance or accident, but to care, diligence, and study, and herein we should try to imitate him in a spirit of admiration and emulation; but where there are mistakes, we should direct our intelligence to these, to determine the reasons and origin of the error. For as Xenophon asserts that good householders derive benefit both from their friends and from their enemies, so in the same way do speakers, not only when they succeed, but also when they fail, render a service to hearers who are alert and attentive. </blockquote>
He describes an ideal listener's demeanor:<br />
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Finally, the following matters, even with speakers who make a complete failure, are, as it were, general and common requirements at every lecture: to sit upright without any lounging or sprawling, to look directly at the speaker, to maintain a pose of active attention, and a sedateness of countenance free from any expression, not merely of arrogance or displeasure, but even of other thoughts and preoccupations. Now in every piece of work, beauty is achieved through the congruence of numerous factors, so to speak, brought into union under the rule of a certain due proportion and harmony, whereas ugliness is ready to spring into being if only a single chance element be omitted or added out of place. And so in the particular case of a lecture, not only frowning, a sour face, a roving glance, twisting the body about, and crossing the legs, are unbecoming, but even nodding, whispering to another, smiling, sleepy yawns, bowing down the head, and all like actions, are culpable and need to be carefully avoided.</blockquote>
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He urges listeners to apply themselves and think critically about what they hear:<br />
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... let us urge them that, when their intelligence has comprehended the main points, they put the rest together by their own efforts, and use their memory as a guide in thinking for themselves, and, taking the discourse of another as a germ and seed, develop and expand it. For the mind does not require filling like a bottle, but rather, like wood, it only requires kindling to create in it an impulse to think independently and an ardent desire for the truth. Imagine, then, that a man should need to get fire from a neighbour, and, upon finding a big bright fire there, should stay there continually warming himself; just so it is if a man comes to another to share the benefit of a discourse, and does not think it necessary to kindle from it some illumination for himself and some thinking of his own, but, delighting in the discourse, sits enchanted; he gets, as it were, a bright and ruddy glow in the form of opinion imparted to him by what is said, but the mouldiness and darkness of his inner mind he has not dissipated nor banished by the warm glow of philosophy. </blockquote>
The oft-quoted phrase "For the mind does not require filling like a bottle, but rather, like wood, it only requires kindling" appears here. (It is often mis-attributed to William Butler Yeats in a modified form: "Education is not the filling of a pail but the lighting of a fire.") With these words, he gets to the very heart of what it means to truly learn. We can only learn when we are motivated and fully engaged in the learning process. Passive listening is not beneficial.<br />
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Modern constructivist education models have sought to reduce the role of lectures in the classroom, to move the role of the teacher from being a "sage on the stage" to a "guide on the side", in part because the art of listening is lost in students who are forced to listen to too many lectures, for years and years. But given that there are still many classes that rely on traditional lectures, and even flipped classrooms require listening to taped lectures outside the classroom, it would behoove students to reflect upon Plutarch's advice and ponder the difference between listening as "bottle-filling" and listening as "fire-kindling".<br />
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Unknownnoreply@blogger.com11tag:blogger.com,1999:blog-7638578.post-17739653313054441672014-08-16T00:35:00.000-07:002014-08-16T00:37:16.833-07:00Give breaks to improve learning<span style="font-family: inherit;"><span style="background-color: white; color: #404040; line-height: 18.200000762939453px;">Came across this article titled </span><span style="background-color: white; color: #404040; line-height: 18.200000762939453px;">"</span><span style="background-color: white; color: #222222; line-height: 1.04762;"><a href="http://blog.ted.com/2014/08/14/what-can-the-american-and-british-education-systems-learn-from-classrooms-in-the-developing-world/">What can the American and British education systems learn from classrooms in the developing world</a>?" </span></span><br />
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<span style="font-family: inherit;"><span style="background-color: white; color: #404040; line-height: 18.200000762939453px;">There are some excellent points here (though perhaps also some idealizations and over-generalizations). One I particularly liked was this: "In the United States, there is the expectation that students are supposed to sit still. You’re told not to fidget and to focus. But scientific research shows that brain activity is significantly heightened after 20 minutes of physical activity. There’s significant value in what you see in the developing world—in between classes, kids run in a field, play in a river, climb a mountain." </span><br style="background-color: white; color: #404040; line-height: 18.200000762939453px;" /><br />Here is an <a href="http://usatoday30.usatoday.com/news/education/2010-04-14-letsmoveinschool15_ST_N.htm">article about a CDC-led study</a> which describes relevant research findings, including: "Short physical activity breaks of about 5 to 20 minutes in the classroom can improve attention span, classroom behavior and achievement test scores." </span><br />
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<span style="font-family: inherit;"><span style="background-color: white; color: #404040; line-height: 18.200000762939453px;">Though sitting still (else being diagnosed as having <a href="http://academicsfreedom.blogspot.com/2012/11/inconvenient-student-disorder.html">ADHD</a>) is sadly indeed the norm in most US schools, there are fortunately some schools here where kids are not expected to sit still all day. A number of them fall into the category of "<a href="http://en.wikipedia.org/wiki/List_of_democratic_schools#United_States_of_America">democratic schools</a>" (in which the students have a significant voice in what, where, how to learn).</span></span><br />
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<span style="font-family: inherit;"><span style="background-color: white; color: #404040; line-height: 18.200000762939453px;">Older kids and adults do have more stamina, but I do think this applies to the college classroom as well. I'm sure many faculty members have seen their students' eyes glaze over after an hour-long lecture. I give a break or two in my longer classes and find it does help students concentrate on the material better. </span></span><span style="background-color: white; color: #404040; font-family: inherit; line-height: 18.200000762939453px;">This fall I may tinker with more frequent micro-breaks, allowing students to stretch and move around a bit more. </span><span style="background-color: white; color: #404040; font-family: inherit; line-height: 18.200000762939453px;">Active learning techniques including hands-on activities and projects for which students must talk to and work with other students in class also help a great deal with improving engagement and excitement in the classroom and I am thinking hard about these as well as I plan my teaching for the semester coming up. </span><br />
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<span style="font-family: inherit;"><br style="background-color: white; color: #404040; line-height: 18.200000762939453px;" /></span>Unknownnoreply@blogger.com8tag:blogger.com,1999:blog-7638578.post-9494118941476818192014-08-06T00:20:00.000-07:002014-08-06T00:20:50.846-07:00Academic ContributionsI imagine that nearly all academics ask themselves this question from time to time: "is my work meaningful?"<br />
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It occurs to me that one derives meaning from, fundamentally, by making a contribution to others; here are some of the many ways in which an academic's contributions could be evaluated:<br />
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<li><b>Contribution to the literature: </b>Has one's work been read and cited by other researchers? How have peers evaluated one's work in terms of novelty, significance, depth, quality? Is there evidence that one's ideas have positively influenced the work of other researchers? <br /></li>
<li><b>Contributions to aid other researchers in their work: </b>Besides papers, has one contributed other materials such as code, tools, data-sets that others could make use of in their research work? Have these been used by others?<br /></li>
<li><b>Contributions to community-building: </b>Human enterprises thrive when we organize into communities. Has one contributed to building a community of researchers? These contributions could be in the form of organizing meetings and workshops and conferences to increase interactions, editorial efforts, organizing centers.<br /></li>
<li><b>Contributions to education: </b>Has one contributed through new courses, surveys, tutorials, books, talks, demonstrations, popular writing, or other materials to educate students, researchers, practitioners and inform the broader public about new developments and ideas? How many have been influenced by these materials and in what ways?<br /></li>
<li><b>Contributions to mentoring: </b>How well has one mentored students? Has one aided younger colleagues in their professional development? Mentoring is a valuable activity because it enhances the ability of other individuals to make their own effective contributions.<br /></li>
<li><b>Contribution to practice: </b>Has one's work been translated to practice? How has the translation been carried out? What difference has it made in the real world, in the context of that translation? How significant has the practical contribution been?</li>
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<br />Unknownnoreply@blogger.com11tag:blogger.com,1999:blog-7638578.post-327832600892988712013-09-18T22:13:00.000-07:002013-09-18T22:20:18.944-07:00Do what you don't have to<div class="separator" style="clear: both; text-align: center;">
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This morning, a student came by during office hours and asked if he could discuss a private matter. He told me that he was nearly done with all his courses in engineering school, but, to his disappointment, he didn't feel passionate about any of the subjects he had taken. He asked me what was the secret to finding something one has passion for.<br />
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I thought about it a bit, aware of the awful weight of responsibility, the need to give a good answer to this bright young man's question, which clearly came from his heart.<br />
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I offered him this thought, that I believe passion comes from "doing things no one else has required you to do". I talked with him about how this impulse is in fact thwarted in classrooms with their many assignments, quizzes, exams and canned projects (yes, even mine.) I encouraged him to tinker and do some projects on his own, engage in independent study.<br />
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I also told him to try and be mindful of his own inner-voice, to find out for himself what activities he finds rewarding, and seek ways in his life to do more of that. I told him about my own motivations for seeking an academic life --- my realization that I really, really enjoyed teaching my classmates the night before exams what we should have learned in the weeks prior.<br />
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Later in the evening, I was still thinking about the conversation. I was thinking to myself how it is that when you feel passionate about something, work or learning doesn't feel like a chore -- it can be exhilarating, as meaningful, as play -- and how the truest form of learning only takes place when you are fully engaged in and enjoying the experience overall. The crux of it is having a clear sense of autonomy --- that you are choosing to do this, you are doing this because <b>you</b> want to, that you could and would stop doing this if it wasn't worthwhile or interesting.<br />
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This <a href="http://www.aeonmagazine.com/being-human/children-today-are-suffering-a-severe-deficit-of-play/">article by Psychologist Peter Gray</a> makes nearly the same point, in encouraging us all to let our kids play more.<br />
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He writes:<br />
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The reason why play is such a powerful way to impart social skills is that it is voluntary. Players are always free to quit, and if they are unhappy they will quit. Every player knows that, and so the goal, for every player who wants to keep the game going, is to satisfy his or her own needs and desires while also satisfying those of the other players, so they don’t quit. </blockquote>
The kind of play he is talking about is not merely light-hearted diversion, though it includes that too, but also the kind of intense immersive play that requires one's full concentration, skill, and forces one to constantly build and improve on one's abilities. Plenty of exposure to this kind of play as a child is indeed essential to self-motivation, creativity, and passionate immersion in learning and work as an adult.Unknownnoreply@blogger.com5tag:blogger.com,1999:blog-7638578.post-40284742021898193592013-07-17T02:43:00.000-07:002013-07-17T02:54:08.155-07:00Teaching Mathematics as a way of ThinkingI came across today a brilliant blog today about learner-centric mathematics education, called <a href="http://www.doingmathematics.com/doing-mathematics.html">Doing Mathematics</a>. The writer of this blog, Bryan Meyer, is a high school math teacher in San Diego. He writes that:<br />
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<span style="background-color: white; color: #464646; font-family: 'Open Sans', Helvetica, sans-serif; font-size: 13px; line-height: 23px;">Most people think that mathematics is a body of knowledge and/or procedures that must be passed from teacher to student. For many students, this is both irrelevant and unexciting. My belief is that mathematics is a way of thinking and sense making. It is creative, beautiful, individual, and dynamic. It is being curious, asking questions, figuring out why things work, breaking problems apart, seeking regularity, making predictions, and creating logical arguments. These things we all engage in on a daily basis and have relevance in our lives outside of a mathematics classroom. My goal is to create an educational environment in which mathematical thinking is not only the core of what we do, but runs through ALL of what we do. </span></blockquote>
Bryan is inspiring because he does things many teachers wouldn't think of doing, but that strike me upon reflection as a very good thing to do. For instance, <a href="http://www.doingmathematics.com/2/post/2013/02/a-quick-post-about-learning.html">in one post</a>, he describes his class working on and discussing a probability problem, at the end of which some students thought one answer was right, while others felt that another answer was right. But he never jumped in to tell them which one was the right answer, even at the end.<br />
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This is unnerving to read at first because, of course, the instinct is to say --- wait, shouldn't the teacher announce the correct answer so no one is left with the wrong impression? He chose differently. He writes:<br />
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<span style="background-color: white; color: #464646; font-family: 'Open Sans', Helvetica, sans-serif; font-size: 13px; line-height: 23px;">I'm curious what you think and what you would do in this same situation. I let it go. I felt I did my job by helping students test their ways of thinking, not by telling them what to think. </span></blockquote>
Bryan's point is that in the long term, showing his authority by presenting the right answer could potentially do harm to their ability and confidence to test out their own ideas. As he writes in a <a href="http://www.doingmathematics.com/2/category/the%20problem%20with%20right%20answers/1.html">different article</a>,<br />
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<span style="background-color: white; color: #464646; font-family: 'Open Sans', Helvetica, sans-serif; font-size: 13px; line-height: 23px;">I am inclined to think that my distaste is not actually for right answers but rather for the students' lack of authority in deciding that answer. As it stands now, students' ways of thinking are always subject to some greater authority (teacher, textbook, video, etc.). As Schoenfeld puts it, students:</span></blockquote>
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<span style="background-color: white; color: #999999; font-family: 'Open Sans', Helvetica, sans-serif; font-size: 13px; font-style: italic; line-height: 23px;">"...have little idea, much less confidence, that they can serve as arbiters of mathematical correctness, either individually or collectively. Indeed, for most students, arguments are merely proposed by themselves. Those arguments are then judged by experts, who determine their correctness. Authority and the means of implementing it are external to the students."</span> </blockquote>
I am very excited about discovering Bryan's blog, because his thinking and most admirably, his <i>doing</i>, is very much along the lines of what I myself noted as the ideal in discussing <a href="http://ceng.usc.edu/~bkrishna/TheDangersOfClassroomTeaching.pdf">the dangers of the classroom</a>: <br />
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The learning process should leave the learner with an innate, ever-improving 'filter' of quality, to be able to evaluate for oneself, when encountering some piece of information, whether it is reliable, new, significant, important, or useful... [however] what is taught is often only what is already well understood, processed and neatly packaged. This makes it harder for students to think outside the box or understand the current limits of our knowledge, let alone how they can be extended... The loss of autonomy and the presentation of processed information deprives students of the confidence to develop their own filter...</blockquote>
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Bryan Meyer also has a nice set of <a href="http://www.doingmathematics.com/on-my-bookshelf.html">recommended readings</a> I plan to go through. I know of a few of them, but am unfamiliar with most.Unknownnoreply@blogger.com17tag:blogger.com,1999:blog-7638578.post-58771711847048502092013-06-05T10:54:00.000-07:002013-06-05T22:48:15.662-07:00A Professor as a Student<div class="separator" style="clear: both; text-align: center;">
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Rebekah Nathan's "<a href="http://www.cornellpress.cornell.edu/book/?GCOI=80140100336140">My Freshman Year</a>" should be required reading for all college professors. This is an account of a middle-aged college professor who decided to spend a year at a large public university in 2002-2003, living in the dorms and experiencing everything a first-year student would. As a trained anthropologist, she gives an exceptionally insightful and vivid account of American undergraduate culture.<br />
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What I found most original about the book is it offers an account that undergraduates themselves could not articulate so well. For it is hard to critically examine one's own culture, or recognize the ways in which it may appear peculiar to outsiders. And although all of us who are professors now were students once, we did not observe so keenly the various aspects of student culture, or synthesize them as she does in this book. Also, the perspective of faculty members, who were often (almost by definition) among the more academically-inclined students in their classes as undergrads is biased, and not representative of the experience of the majority of students going to college. (Hence the routine lamentations heard in the halls outside faculty offices: "What is wrong with these students? Why don't they study as hard as I used to? Why don't they seem to care about what I'm teaching?")<br />
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Rebekah Nathan (which turns out to have been the pen-name used by Cathy Small, from Northern Arizona University) offers us a more nuanced, balanced view, of how college appears from the students' perspective. Based on her observations as well as interviews with her fellow students, she draws a number of insights:<br />
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1. For many students, "the most compelling reason for staying in college was 'the college experience' --- the joys and benefits of living within the college culture rather than in the real world." Classes and learning and immersion in the intellectual life were a secondary objective for many. "Classes, in fact, were described in multiple instances as the 'price one has to pay' to participate in college culture, a domain that students portrayed in terms such as 'fun,' 'friendships,' 'partying,' 'life experiences,' and 'late night talks.' "<br />
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2. One of the barriers to learning in many traditional classes is the perception of their formality and disconnectedness from the everyday experiences of students. In contrast, what makes a course popular to a majority of students, is if it can subsume "formal academic content within an informal, largely, social world characterized by equality, informality, intimacy, and reciprocity, while at the same time [providing] a context for learning that [is] 'fun,' irreverent, and separated, both geographically and ideologically, from the formal aspects and authority of campus."<br />
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3. She writes that time-management is a huge and very challenging part of the college experience for undergraduates. She writes, "Going to school, I found, was a time-management nightmare; ... in a single semester, there were eight different people who made rules or created structures that I had to respond to as a student. Each wanted us to access readings, or prepare papers, or communicate with him or her according to a different protocol. As is typical in a large state university, none of the instructors coordinated assignments or schedules with one another or even with a master university schedule." She writes that the most successful students figured out that the way to manage time was not merely to make to-do lists and be efficient in their use of time, but also by making smart choices about which courses to take. "The key to managing time was not, as college officials suggested, avoiding wasted minutes by turning yourself into an agent of your day planner. Neither was it severely curtailing your leisure or quitting your paying job. Rather, it was controlling college by <b>shaping schedules</b>, <b>taming professors</b>, and <b>limiting workload</b>."<br />
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4. <b>Shaping schedules: </b>In particular, she writes that even mature students do want to take a few "easy A" / "low workload" courses to balance out the "harder" courses, which they knew to be necessary. And sometimes, they just take courses because they are offered at the right time so that they can fit it into their schedule. She describes this moment of epiphany: "It suddenly became clear to me why, as a professor, I had had a number of students enrolled in my basic cultural anthropology course who had no idea what anthropology was. My course was likely the last piece in their scheduling puzzle, and frankly, they didn't care what anthropology was."<br />
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5. <b>Taming professors: </b>Mature students also learn to play the college game in ways that don't promote independent learning. She describes a university-sponsored presentation in which a section of the talk was devoted to "figuring out your Profs." "What do profs want? The speaker told us: 'They think the world revolves around their subject, so they want you to get it. They want to see effort, and they want you to voice an opinion. So give them what they want and you'll get what you want too!" There is something instrumental and transactional about this student-teacher relationship that is far from the ideal learning-centered relationship that faculty crave. She asks a mature and competent senior for tips on success and gets this response: "I take the information I need from the professor --- how they're going to grade you and what they think is important --- and I use it. If you write what you want to that prof, you're gonna end up with a bad grade. Whereas, if you write to <i>them, </i>you win --- you can still have your own mindset and say, hell, I know this isn't the way I feel, but I'll give them what they want."<br />
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(This passage made me burn at a memory I had long suppressed, from my own Junior/Senior year. It was a humanities course, which I particularly loved as the readings were provocative and thought-inspiring. I spent hours thinking about and writing up an essay for the first paper in that class, which I was really proud of. It came back with a B-, to my utter shock and disappointment. My friend seated next to me got an A+. I asked to see his essay, and gasped when I read it. "But, but..." I stammered out to him, "you are pretty much just rephrasing here whatever he said in class!" "Of course!" he said, and smiled, knowingly.)<br />
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6.<b> Limiting workload:</b> In the face of their heavy workloads and scheduling nightmare, students have every incentive to do the bare minimum they can, while still doing well in the courses in terms of grades. "On several levels, students assess what is needed to get by. Depending on the course and the instructor, they decide whether to buy the book, whether to go to class, whether to do the readings in a given week, and how much effort to put into assignments." On absenteeism, she writes, "in classes where attendance is expected but not required, the frequency of absenteeism rises with each of the following characteristics: the class is large, the class is boring, tests are based on readings rather than lectures, grades depend on papers rather than tests, the class is early in the morning, the class is on friday." She writes that cutting class is actively encouraged by peers, who value it as an act of rebellion against authority. She also writes that it is in the students' interests to minimize any additional reading or learning-oriented activity that does not directly correlate to the course grade, or that does not seem essential for their career; such activities are seen as "busy-work", to be avoided.<br />
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7. Students rarely speak up in class to discuss and debate ideas. Time pressure is one contributing factor in low class participation. Others, she determined through a survey, were "peer pressure, power of the teacher, and lack of personal interest or purpose." She writes that "I was struck by the realization that, despite official assertions about the university as a free marketplace of ideas, the classroom doesn't often work that way in practice. Ideas are rarely debated, and even more rarely evaluated. Most classroom discussion, when it does occur, could be described as a sequential expression of opinion, spurred directly by a question or scenario devised by the teacher, which is subject to little or no commentary." Even if there were conversations about ideas in class, "the moment we walked out of class... the subject at hand was abruptly dropped, as if the debate had only been part of a classroom performance." Indeed, outside of class, she found that there was little discussion about learning or ideas. Most conversation centered around due-dates, complaints about the progress of the course, work-load, and grades. She speaks of her disillusionment: "Taken together, the discourse of academe, both in and out of classes, led me to one of the most sobering insights I had as a professor-turned-student: How little intellectual life seemed to matter in college."<br />
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8. On why student time for learning and classwork is limited and steadily decreasing over time, Rebekah Nathan hypothesizes that this is primarily due to the rising costs of college, forcing students to take on more part-time work than they used to in the past. "A 2004 government report found that increases in tuition and fees during the preceding decade had outpaced both inflation and growth of the median family income. The result has been debt --- a huge amount of debt that college students are incurring for the sake of their education --- and a sharp rise in the percentage of borrowers among full-time undergraduates." I found this very interesting; it answered in part a question I have had on my mind and discussed in a <a href="http://academicsfreedom.blogspot.com/2013/03/are-students-smarter-now-than-before.html">recent blog post: why do students study less these days</a> on average compared to 40 years ago? There is no doubt that this is the single biggest challenge facing higher education these days.<br />
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9. The rising costs of college and student debts is also resulting in a rising tide of career-mindedness that undermines academic traditions. She recalls a popular talk given by a speaker to entering freshman from a nonprofit division of a for-profit company, which she says embodied key features of the student culture and attitudes. "There was the 'fun-party-independence-youth' veneer, which is long-standing in student culture, but the more dominant statement of the presentation was one of pragmatism and careerism. Hard work, forethought and organization were part of this career message to students, but so too was the idea that grades have primacy, that you should join groups with resume building in mind, that a smart student should 'figure out your profs' (the title of one presentation segment), that students are in individual competition with one another for grades and recommendations, and that, above all, college is about positioning yourself for a good job and an affluent future."<br />
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10. She warns that as universities become more market-driven, and students become more career-driven, in response to the rising college costs and student debts, new dangers lurk. "Degree programs tightly geared to the marketplace become products themselves, and are likely to bust and boom with the fickleness of the times." She writes that "in the long run, we would not want a university to become so immersed in the world as it is that it can neither critique that world nor proffer an ideal vision of how else it might be. These are purposes of universities that none of us should surrender."<br />
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<br />Unknownnoreply@blogger.com9tag:blogger.com,1999:blog-7638578.post-28110434478641201522013-05-23T14:53:00.001-07:002013-05-23T15:45:26.731-07:00Three Things Research-oriented Faculty Do<div class="separator" style="clear: both; text-align: center;">
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<a href="http://3.bp.blogspot.com/-VsxYpc_EUmg/UZ6aR7SIXqI/AAAAAAAAGf8/lXro2YyF1NE/s1600/3things.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="281" src="http://3.bp.blogspot.com/-VsxYpc_EUmg/UZ6aR7SIXqI/AAAAAAAAGf8/lXro2YyF1NE/s400/3things.png" width="400" /></a></div>
<br />
<br />
This is based on something I jotted down in a late night email to a former Ph.D. student from USC, who just has landed a faculty position, and will start as an Assistant Professor this fall.<br />
<br />
<span style="font-family: inherit;"><span style="background-color: white; color: #222222;">The main epiphany I had early on about life as a faculty member is </span><span style="background-color: white; color: #222222;">that you have three main functions as a researcher:</span><br style="background-color: white; color: #222222;" /><span style="background-color: white; color: #222222;">1. do good work and publish</span><br style="background-color: white; color: #222222;" /><span style="background-color: white; color: #222222;">2. raise money</span><br style="background-color: white; color: #222222;" /><span style="background-color: white; color: #222222;">3. network and collaborate</span></span><br />
<span style="font-family: inherit;"><br style="background-color: white; color: #222222;" /><span style="background-color: white; color: #222222;">Doing good work and publishing it in respectable venues is, of course, the main thing.</span><br style="background-color: white; color: #222222;" /><br style="background-color: white; color: #222222;" /><span style="background-color: white; color: #222222;">But these three all feed inextricably, recursively, into each other:</span></span><br />
<ul>
<li><span style="background-color: white; color: #222222; font-family: inherit;">To do good work you need to raise funds to support students, and it helps to network and collaborate </span><span style="background-color: white; color: #222222; font-family: inherit;">with others. </span></li>
<li><span style="background-color: white; color: #222222; font-family: inherit;">To raise funding, you must do good work, and it helps to network</span><span style="background-color: white; color: #222222; font-family: inherit;"> and to collaborate.</span></li>
<li><span style="background-color: white; color: #222222; font-family: inherit;">To network and collaborate</span><span style="background-color: white; color: #222222; font-family: inherit;">, you must do good work, but also need funding to </span><span style="background-color: white; color: #222222; font-family: inherit;">travel.</span></li>
</ul>
<div>
<span style="color: #222222;">Doing a good job as a faculty member at a research university thus means constantly balancing and juggling these three functions. New Faculty must particularly pay attention to 2 and 3, because 1 is the thing they were good at as students or post-docs (or they would not have been hired), but in many cases 2 and 3 have not been as much of a focus for them in the past. </span></div>
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<span style="color: #222222;"><br /></span></div>
<div>
<span style="color: #222222;">I would actually encourage senior Ph.D. students seeking academic positions, and certainly post-docs, to gain as much experience as they can with respect to 2 (for instance by practicing writing proposals for/with their advisor) and 3 (by collaborating with other students and faculty, being engaged and interactive while attending conferences, making themselves known to faculty at other schools, giving talks at other schools). This will both help them land faculty positions, and give them a head-start on life as a research-oriented faculty member. </span><br />
<span style="color: #222222;"><br /></span>
<span style="color: #222222;">It's interesting to note that excellence in communication is crucial to all these functions. </span></div>
Unknownnoreply@blogger.com16tag:blogger.com,1999:blog-7638578.post-67021073760476353672013-04-05T01:30:00.002-07:002013-04-05T13:07:56.594-07:00Teach no thing (but rather, how to think about things)<div class="separator" style="clear: both; text-align: center;">
<a href="http://2.bp.blogspot.com/-JOKxHgihEsE/UV6IMvfz_QI/AAAAAAAAF_0/6ryTqOiLiY8/s1600/brady.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="226" src="http://2.bp.blogspot.com/-JOKxHgihEsE/UV6IMvfz_QI/AAAAAAAAF_0/6ryTqOiLiY8/s400/brady.png" width="400" /></a></div>
<br />
<br />
Marion Brady is a history teacher who <a href="http://www.washingtonpost.com/blogs/answer-sheet/wp/2013/02/11/a-real-paradigm-shift-in-education/">advocates teaching students not merely fac</a><a href="http://www.washingtonpost.com/blogs/answer-sheet/wp/2013/02/11/a-real-paradigm-shift-in-education/">ts, or "content", but rather how to think</a>.<br />
<br />
He writes:<br />
<br />
<blockquote class="tr_bq">
Long before corporate America began its assault on public schooling, American education was in trouble. Educators were, however, increasingly aware of the problems and were working on them. When Bill Gates, Jeb Bush, Mike Bloomberg, Arne Duncan, Michelle Rhee, and other big name non-educators took over, that worked stopped. </blockquote>
<blockquote class="tr_bq">
What I want people to understand is that the backbone of education — the familiar math-science-language arts-social studies “core curriculum” — is deeply, fundamentally flawed. No matter the reform initiative, there won’t be significant improvement in American education until curricular problems are understood, admitted, addressed, and solved.</blockquote>
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His analysis of the crisis in education that we face today at all levels is that "The traditional core curriculum delivers existing knowledge, but adapting to an unknown future requires new knowledge. Obviously, what will need to be known in the future isn’t yet known, from which it follows that it can’t be taught. However, the process by means of which new knowledge is created can be taught."<br />
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Going beyond rhetoric, Brady has put together a non-content-based curriculum called "<a href="http://www.marionbrady.com/ConnectionsInvestigatingReality.asp">Connections: Investigating Reality</a>" that attempts to teach middle school and high school students, or rather, help them teach themselves, how to identify patterns, how to extract meaning from texts and the world around them, how to reason and organize their thoughts.<br />
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I could not agree more with his ideas.<br />
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I myself experienced the importance of learning how to think when I took a introductory course on analysis in the mathematics department at Cornell. Though the material that this course was ostensibly about was reasonably familiar to me: real numbers, limits, sequences, etc., what the course was really about was how to prove things, how to express mathematical ideas precisely, rigorously, cleanly. It was about how to logically layer arguments, building only upon what is already known, to come to a certain conclusion. I know that the way I thought about things, not only related to mathematics, but about all manner of things in the world around me --- economics, politics, human relations, even literature --- changed very substantially after taking that course. For the first time in my life, I felt confidence that when I constructed a logical argument carefully from a set of premises, it would not be flawed, and that, given time, I could spot a faulty argument when I came across one. Without this confidence, I would never have made it through the Ph.D., to a faculty position, and past tenure.<br />
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Now, I try to make sure that all the Ph.D. students I advise take the equivalent course at USC (<a href="http://web-app.usc.edu/ws/soc_archive/soc/syllabus/20131/39645.pdf">Math 425a</a>, if you are curious). And from my experience over the years, by comparing my impressions of them before and after they took the course, I can vouchsafe that nearly all these students dramatically improved their ability to think rigorously because of it. (Interestingly though, I find, this appears to happen not during or immediately after the course, but a few months afterwards, as though it needed some time after that course for their neurons to make all the connections, during which perhaps their subconscious was hard at work applying what they had learned to various problems.)<br />
<br />
Some colleagues and I were having a dinner tonight, downtown, at the somewhat quiet but always enjoyable Thai restaurant downtown --- Soi 7. Our conversation turned to educational philosophy.<br />
<br />
A senior colleague mentioned that he too believes that what is important in a classroom is not merely conveying the lecture material, or even showing students how to solve a particular set of exercises after having them read or watch the material on their own (as is done in the increasingly popular <a href="http://en.wikipedia.org/wiki/Flip_teaching">flipped classroom</a> model of teaching), but rather get them to think in class collaboratively, <i>de novo,</i> about the concepts and problems that the material is about before leaving the students to master the content and do exercises on their own. This way the emphasis is on fresh, critical thinking, rather than on content and drilling. He jokingly called it the "re-flipped" classroom.<br />
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Another colleague of mine at the table then indicated that this is very reminiscent of the famous "<a href="http://www.maa.org/devlin/devlin_6_99.html">Moore Method</a>" of teaching mathematics. This method, referred to as discovery learning, was developed by Professor Robert Lee Moore, who encouraged and guided his students at UT Austin to build on basic axioms and prove everything on their own, with minimal hints and without recourse to a text.<br />
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I feel inspired to think through the implications of such non-content-based learning of critical thinking and problem solving skills as well as discovery learning techniques, and plan to experiment with them in my own courses in the future.Unknownnoreply@blogger.com5tag:blogger.com,1999:blog-7638578.post-78934372936918779532013-03-14T04:50:00.003-07:002013-03-14T04:59:11.857-07:00Are Students Smarter Now than Before?The following chart is from a <a href="http://economix.blogs.nytimes.com/2012/01/20/why-students-leave-the-engineering-track">NYTimes article</a> from last year that a colleague pointed me at. It is based on data from the paper Philip Babcock and Mindy Marks, "<a href="http://www.nber.org/papers/w15954.pdf">The Falling Time of Cost of College: Evidence from Half a Century of Time Use Data</a>," Review of Economics and Statistics, May 2011, Vol. 93, No. 2, Pages 468-478.<br />
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://graphics8.nytimes.com/images/2012/01/20/business/economy/economix-20studytime/economix-20studytime-jumbo-v2.jpg"><img src="http://graphics8.nytimes.com/images/2012/01/20/business/economy/economix-20studytime/economix-20studytime-jumbo-v2.jpg" style="margin-left: auto; margin-right: auto;" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><a href="http://graphics8.nytimes.com/images/2012/01/20/business/economy/economix-20studytime/economix-20studytime-jumbo-v2.jpg">Source: NYTimes </a></td></tr>
</tbody></table>
<div>
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First, it shows that Engineering students study more on average now (well, as of 10 years ago, but one presumes it's about the same now) than other majors, though it used to be Health students that studied the most. Second, it shows that students now (again, as of 10 years ago), like all majors study much less than 40 years before. In particular, it shows that engineering students study on average about 18 hours a week outside of classes, where before they studied about 26 hours a week.<br />
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In the author's own words:<br />
<blockquote class="tr_bq">
<span style="font-family: Tahoma, Calibri, Verdana, Helvetica, Arial, sans-serif; font-size: 13px; line-height: 18px; text-align: justify;">Using multiple datasets from different time periods, we document declines in academic time investment by full-time college students in the United States between 1961 and 2003. Full-time students allocated 40 hours per week toward class and studying in 1961, whereas by 2003 they were investing about 27 hours per week. Declines were extremely broad-based, and are not easily accounted for by framing effects, work or major choices, or compositional changes in students or schools. We conclude that there have been substantial changes over time in the quantity or manner of human capital production on college campuses.</span></blockquote>
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This is really a quite stunning and fascinating finding. Whatever accounts for this difference? Are students simply smarter than before? Do the reduced numbers of hours they study indicate greater learning efficiency (because of better textbooks, better teaching, new instructional media)? Are there simply more distractions that prey on students' time (such as video games and the world wide web, which was just starting to take off then; I'm very curious to see what the numbers look like today, with the added distractions of facebook, twitter, mobile devices)? Is it that more students are working part-time to pay for increasing tuition costs? Is it because there are more older students that are working full-time taking continuing education classes ? Is the average reflective of an increase in overall numbers of departments where the study expectations are lower?<br />
<br />
Whatever the cause, I've certainly heard several colleagues who have been teaching engineering for two or more decades complain that students these days don't study as hard and with as much discipline as they used to, whine more about grades than they used to, and in particular, are not as strong mathematically as they used to be. These data seem to suggest that they're not just being cranky.<br />
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And what are the implications of this study? Are we being forced to lower our expectations of what college students should be able to do, or handle? Instead of being smarter, do our graduates today actually have fewer skills and abilities than those of yesteryears? How has this affected the economy?<br />
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What do you think? </div>
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Unknownnoreply@blogger.com159tag:blogger.com,1999:blog-7638578.post-3866396220674034282013-02-08T01:15:00.001-08:002013-02-08T13:22:03.949-08:00Multiplying Negatives<div class="separator" style="clear: both; text-align: center;">
<a href="http://2.bp.blogspot.com/-DdJCmFUrdGs/URS27diYEKI/AAAAAAAAFrY/SnetY-nzQdQ/s1600/min2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="106" src="http://2.bp.blogspot.com/-DdJCmFUrdGs/URS27diYEKI/AAAAAAAAFrY/SnetY-nzQdQ/s400/min2.png" width="400" /></a></div>
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I was talking with my elementary-school-aged child this evening about arithmetic.<br />
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"What is 2 x 2?" I asked.<br />
"4," he replied.<br />
"What is 2 x -2," I asked.<br />
"Negative 4," he said, reasoning correctly that it was adding up two "negative 2's".<br />
Hoping to stump him, I followed up with "What is -2 x -2?"<br />
"Negative 4", he repeated.<br />
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"Wait, this is different from what I asked you before!", I said, "It's not obvious what it means, right?"<br />
"Oh yeah!" he agreed.<br />
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"So what is it?" he asked. We were walking outside, feeling quite chilly in the evening air.<br />
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"Well, I'll tell you, but you're not going to believe it", I replied, "it's 4!"<br />
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He thought about it for a bit, and smiled at the incongruity of the answer. "4?!"<br />
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I could sense the puzzlement lingering in him. I knew I had to say more.<br />
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An awful feeling grew slowly in my stomach as I struggled with what to say next. "Well, it's not easy for me to explain why though..." my voice trailed off as I started to think about it more carefully.<br />
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"... I mean, I mean, I mean, I can't just say consider adding -2, -2 times", I stammered out, as we entered the warm indoors. <br />
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I thought back to how I myself had learned this. A shudder went through me as it occurred to me that I was probably simply told by the teacher that "negative times negative is positive" and made to absorb it simply as a convenient fact about numbers.<br />
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Up to this point, over months of conversations, I have been explaining arithmetic to my son using familiar concepts and ideas. Natural numbers correspond to putting up fingers one by one. Adding them is an extension of the counting process. It is easy to illustrate with one's hands why 2+3 is the same as 3+2. The positive numbers line up nicely on a line. Subtracting a smaller number from a bigger number corresponds to removing the smaller number of objects from a set of the larger number of objects and counting what remains. It also corresponds to going backwards on the count, or on the number line.<br />
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These explanations pile nicely, building on each other. From the above, it is intuitive and therefore easy to explain to a child why subtracting a bigger number from a smaller number requires us to put some more numbers on the line before 0, and that we call these negative numbers. It is easy to explain also on intuitive grounds that though we don't encounter them when counting objects, this could sometimes be a useful way of thinking of numbers (such as when we consider scenarios involving borrow and lending money). Building on the analogy of debt, it's possible to explain intuitively why negative 2 plus negative 2 is negative 4 and then note that this is the same as negative 2 minus 2, so that adding a negative number is the same as subtracting the positive counterpart of that number. It can also be pointed thereafter that 3 - 2 is not the same as 2 - 3, so that in this respect subtraction is quite asymmetric. Multiplication can be introduced as a shorthand for iterative addition: that when we say "3 times 4," what we mean is to add the total number of objects in 3 collections of 4 objects each. Pictorially, through rows and columns of objects, it can then be explained why "3 times 4" is the same as "4 times 3". And then, building on the prior understanding of negative numbers, it is not a stretch to explain to a child what it means to talk about "4 times -2", and how this is the same as "-4 times 2".<br />
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But "-2 times -2" is a qualitatively different animal. It does not follow directly from the above explanations. Because iterating a process a negative number of times is not meaningful, the earlier intuitive understanding of multiplication does not readily generalize*. <br />
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Determined to keep things intuitive, I put down my bag, took off my jacket and sat right down in front of the computer. I searched the web for an explanation suitable for a child. I found one immediately at<br />
<a href="http://www.mathsisfun.com/multiplying-negatives.html">http://www.mathsisfun.com/multiplying-negatives.html</a>:<br />
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://1.bp.blogspot.com/-LaoMoTQGH44/URSeFNKk2gI/AAAAAAAAFrE/O3B-cHnxArk/s1600/multiplyingNegatives.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="144" src="http://1.bp.blogspot.com/-LaoMoTQGH44/URSeFNKk2gI/AAAAAAAAFrE/O3B-cHnxArk/s400/multiplyingNegatives.png" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">("common sense" explanation from http://www.mathsisfun.com/multiplying-negatives.html)</td></tr>
</tbody></table>
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Reading this worthless bit of chicanery made me feel physically sick. It is insulting the intelligence of a child to even try and explain a mathematical concept through unrelated verbal shenanigans.<br />
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To be fair to that site, this junk is soon followed up by an alternative explanation, that is not bad. This second explanation equates -2 x 3 to walking backwards two steps on the number line three times, and 2 x -3 to turning around first (so that one is facing backwards) and then walking forward two steps three times. Finally, -2 x -3 is explained as a combination of these, turning around first, then walking backwards two steps three times, ending up at the right answer of 6.<br />
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As explanations go, this is more plausible. At least it builds a clean map between the operations in question and a physical process in such a way that the outcome in the physical process is consistent with the arithmetic operation. In this, it is in keeping with all the previous intuitive explanations I myself had been offering my child. But I couldn't help feeling that the analogy is still rather forced and somewhat lacking in elegance. Why should the negative in the first number correspond to walking backwards and the negative sign on the second number correspond to turning around?<br />
<br />
One problem is that all of the above "intuitive explanations" are not rigorous. They are simply suggestive analogies. Because they are grounded in experience, they offer a scaffolding, helping children slowly build familiarity with numbers and what can be done with them. But this approach has its limitations. <br />
<br />
Mathematics in its pure form is in fact <i>not</i> a description of our world as it really is. It is fundamentally about abstractions, axioms and deductive inference. We are fortunate that pure mathematics can in fact be applied; that elegant logical, axiomatic reasoning about abstract objects turn out to be useful in solving real-world problems in science, engineering and economics, once a connection can be established through the art of mathematical modeling.<br />
<br />
I want to give my child some feel for this, pure mathematics, so I outlined for him a solution based on the following.<br />
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We use three statements that he currently understands and accepts on intuitive grounds:<br />
<br />
<b>A. </b>a + (-a) = 0.<br />
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<b><br /></b>
<b>B. </b>a x (b + c) = (a x b) + (a x c)<br />
<div>
<br /></div>
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<b>C. </b>-a x b = - (a x b)<br />
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The explanation proceeds as follows:<br />
<br />
<ul>
<li>Start with the expression -2 x (-2 + 2). Since what is in the paranthesis is 0 (from A), we have that</li>
<ul>
<li>-2 x (-2 + 2) = -2 x 0 = 0 ---- (1)</li>
</ul>
<li>But also, from B, we have that</li>
<ul>
<li>-2 x (-2 + 2) = (-2 x -2) + (-2 x 2)</li>
</ul>
<li>And from C, we have that (-2 x 2) = -4. This yields that</li>
<ul>
<li>-2 x (-2 + 2) = (-2 x -2) + (-4) = 0, the last equality following from (1)</li>
</ul>
<li>From A, we then get that the term (-2 x -2) must be 4. </li>
</ul>
His eyes lit up at the last step when he could infer for himself that -2 x -2 must be 4.<br />
<br />
When we finished, I could tell him that "pure mathematics is all about axioms that you start with and the things you can prove from them." Though the hour was late, I think, I hope, it left some impression. <br />
<br />
***<br />
<br />
Graeme McRae has a nice page with a very accessible discussion of the <a href="http://2000clicks.com/MathHelp/AxiomsArithmetic.aspx">axioms of integer arithmetic</a>, that presents a minimal set of axioms from which all the familiar rules of arithmetic with integers can be derived. I am thinking this could be a good resource for a deeper discussion about integer arithmetic and axiomatic reasoning in the future.<br />
<br />
Technically, the matter is actually a bit more complicated and interesting, it turns out. The axioms by McRae presented in the above link define what is called an <a href="http://en.wikipedia.org/wiki/Integral_domain">Integral Domain</a>, which is in modern algebra terms a <a href="http://en.wikipedia.org/wiki/Commutative_ring">commutative ring</a> without <a href="http://en.wikipedia.org/wiki/Zero_divisor">zero divisors</a>. Treating them as an Integral Domain is sufficient to derive most familiar rules of arithmetic with integers, but in fact it is not enough to characterize them completely and uniquely; what is needed in addition are some axioms pertaining to the strict ordering of integers (less-than), and one pertaining to Induction. A nice explanation of these technicalities is provided in these excellent notes on <a href="http://www.cs.cornell.edu/home/kreitz/Teaching/CS486/notes-21.pdf">Axiomatizing Integer Arithmetic</a> from a senior-level Applied Logic computer science course at Cornell.<br />
<br />
We are so close to the heart of one the greatest (albeit negative) findings in pure mathematics and logic that I cannot resist mentioning here it at least in passing. Kurt Goedel in 1931 proved that in fact there are no set of axioms that can be used to prove all the properties of natural numbers (known as the <a href="http://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_theorems">Incompleteness Theorem</a>). This doesn't mean that the axiomatic approach is totally useless, but shows that it has its limitations. His finding famously undermined Bertrand Russell and Alfred North Whitehead's ambitious joint attempt to put on firm axiomatic grounding all of mathematics in their <i><a href="http://en.wikipedia.org/wiki/Principia_Mathematica">Principia Mathematica</a>.</i><br />
<br />
***<br />
<br />
Though I was exposed to Euclid's axioms of geometry and the idea of proving theorems as early as middle school, I think I only truly got what it's all about for the first time when I first took a senior-level undergraduate math course providing an introduction to Real Analysis, at Cornell. I remember one day early in the course working on a homework problem, struggling to deduce some seemingly trivial property of elementary integer arithmetic, and saying to myself, "how could I have gotten this far without understanding that all these 'facts' I took for granted can in fact be proved from much simpler axioms?!" One reason it took me so long to figure this out is that virtually all of the mathematics taught to engineers is from the perspective of "here are some useful facts and techniques that you can use to model real systems and analyze those models," and axiomatic development is consistently under-emphasized. This is perhaps as it should be, since abstract axiomatic development tends to be slow, painful, and of relatively little utility when it comes to applied mathematics, but it's a pity nonetheless.<br />
<br />
***<br />
<br />
*<span style="color: #990000;">Update: Marc in a comment below points out a <a href="https://www.khanacademy.org/math/arithmetic/absolute-value/mult_div_negatives/v/why-a-negative-times-a-negative-makes-intuitive-sense">Khan academy video that gives a great intuitive explanation of the multiplication of negative numbers</a>. In fact, if one models multiplying by a negative number as iterative subtraction, one can build an understanding of why -2 x -2 is 4 using intuition alone. I wish I had seen this sooner, but then I would have missed out on the opportunity to discuss with my child how deductive reasoning works.</span>Unknownnoreply@blogger.com12tag:blogger.com,1999:blog-7638578.post-10926290133837240842013-01-20T22:46:00.001-08:002013-01-20T23:14:49.471-08:00The Importance of Grace in Education<div class="separator" style="clear: both; text-align: center;">
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Francis Edward Su, a Professor of Mathematics at Harvey Mudd College, has made public the text of a stirring talk he recently gave upon accepting (a clearly well-deserved!) teaching award, in which he expounds on the "<a href="http://mathyawp.blogspot.com/2013/01/the-lesson-of-grace-in-teaching.html">Lesson of Grace in Teaching</a>".<br />
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He summarizes the "lesson of grace" in two statements: "1. Your accomplishments are NOT what make you a worthy human being. 2. You learn this lesson by receiving GRACE: good things you didn't earn or deserve, but you're getting them anyway. "<br />
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His definition and exposition of "grace" crystallizes beautifully what I have felt and experienced myself but could not articulate so well. I will never forget the mentors in graduate school who offered me warm, unconditional, support and advice. I am grateful to have the chance to pay it forward in my own interactions with students, and do my best to treat them as worthy individuals deserving of encouragement, advice, help, or as they most often need, a friendly, non-judgmental, ear.<br />
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We live in an achievement-oriented society, particularly in academia, with its pervasive emphasis on competition, external evaluation, recognition. Much harm is done by this system, not only to those students who do poorly, but counter-intuitively, even to those who do well, because it reinforces a fundamentally flawed connection between self-worth and achievement.<br />
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I wrote about something related a while ago in the context of the "<a href="http://academicsfreedom.blogspot.com/2011/01/impostor.html">impostor syndrome</a>" (the name given to the feelings of inferiority that many students suffer from while doing the Ph.D.).<br />
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I wrote in that post that "having given the matter much thought over the years, I feel that what exacerbates the impostor syndrome, or perhaps even gives rise to it in the first place, is adopting the world view that ties one's sense of self-worth to one's achievements and treats achievement as the goal of one's efforts." I argued that it is important for students to cultivate a mindset that places a greater emphasis on learning and discovery as ends in themselves rather than as a means to success through achievement.<br />
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Francis's talk calls upon educators to think differently, and to treat the individuals they encounter with grace, regardless of their academic performance as measured by tests and exams. It is a call that, if heeded, would have a tremendously positive impact on the lives of many.<br />
<br />Unknownnoreply@blogger.com27tag:blogger.com,1999:blog-7638578.post-89501720650607892042013-01-06T09:31:00.002-08:002013-01-08T18:04:05.281-08:00If you can't beat them...<br />
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<a href="http://3.bp.blogspot.com/-Zg-HjiYpgMM/UOmZ0ReqKbI/AAAAAAAAFVU/3kXRg64hZNQ/s1600/minecraft.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="61" src="http://3.bp.blogspot.com/-Zg-HjiYpgMM/UOmZ0ReqKbI/AAAAAAAAFVU/3kXRg64hZNQ/s400/minecraft.png" width="400" /></a></div>
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My kids have been playing and talking excitedly to me about <a href="https://minecraft.net/">Minecraft</a> for several months now. They play it every day, alone and with friends in multiplayer mode. And when they're not playing, they spend an inordinate amount of time either watching youtube videos about it (typically tutorials posted by experienced players, but also some hilarious musical parodies -- check out "<a href="https://www.youtube.com/watch?v=cPJUBQd-PNM">Revenge</a>" below, and also "<a href="https://www.youtube.com/watch?v=k2rDbRUDkds">TNT</a>"), or discussing with friends what they are going to do the next time they play.<br />
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As a parent, I have to confess this worried me. This much passion for a video game? I wished they would be this excited about reading books, as I recall being when I was a kid. True, computers or video games were not as common when I was their age, but still, aren't books inherently better because they lead directly to a love of learning? Aren't video games mere amusement? Empty calories for the brain?<br />
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Yet, there's another part of me that objects to these fears and resists them. At their age, the books I read were comics (mostly <a href="http://www.amarchitrakatha.com/">Amar Chithra Kathas</a>). Not because they were educational or helped me improve my reading, but because they were fun, entertaining, engrossing. The joy I derived from them spurred me to read more, and eventually broaden my interests when I was ready to do so. I can trace in my mind a clear, direct path from that joy to many of my current interests and abilities. Pursuing one's interests passionately, with joy, at one's own pace, is a big part of what it means to be a child. They must be given a lot of freedom to play, grow and learn in their own way.<br />
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My older kid insisted that I play it for myself to see what is so compelling about Minecraft. Under his expert guidance, I got a taste of it for myself. It is a first-person adventure game. There are weapons (swords, axes, bows and arrows --- no guns!) and menacing creatures to fight (some that are familiar from other contexts: zombies, spiders, skeletons, and some that are quite unique to the game: creepers and endermen), and one has to gather various resources (food, wood, stone, minerals) and craft some of them into stronger tools and armor to fight these creatures. But it is also very significantly about building one's own world, brick by brick, (houses, shops, markets, buildings, entire villages), and about inhabiting and exploring this virtual world with real-world friends.<br />
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Minecraft is an independent game originally created by a single programmer (<span style="background-color: white; color: #333333;"><a href="http://notch.tumblr.com/">Markus "Notch" Persson</a></span>) in 2010. Its retro graphics demonstrate an appealing emphasis on functionality, game play, and interactive story creation, over the slick-looks-but-tired-game-play emerging from the large commercial game design powerhouses these days.<br />
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As this article, titled "Why is Minecraft so damn popular?", notes:<br />
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<span style="background-color: white; color: #333333;">It is, first things first (and before I am lynched by Minecraft fans), a good game. Maybe even a great game. It's got an iconic look, it's widely accessible, it allows gamers to create their own stories, and perhaps most engrossing of all, has an initial simplicity and ease of play that quickly gives way to a complexity as deep as the mines you'll soon find yourself digging.</span> </blockquote>
It is a <i>hugely</i> popular game. It is reported that about 8.7 million copies of the game have been downloaded for personal computers so far (this does not include other platforms on which it is available, such as Xbox, mobile devices and tablets).<br />
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There is another aspect of this game that accounts for its appeal. It has been made available as an open-source platform (in the form of the <a href="http://mcp.ocean-labs.de/index.php/MCP_Releases">Minecraft Coder Package</a>). Anyone is free to see the source code of the game, and modify it to their heart's content. It's all written in Java, so this is relatively easy to do for anyone with a decent exposure to programming. There is a large and growing community of modders that have created enhancements and imaginative alternative versions of the game.<br />
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A couple of days ago, a friend pointed me at a particularly interesting mod that has been very recently released for Minecraft, called <a href="https://github.com/walterhiggins/ScriptCraft">ScriptCraft</a>. This mod, created by Walter Higgins, integrates JavaScript into the game and allows players to write small programs (scripts) to automate building tasks while inside the game. I told my kids about it and they were instantly thrilled and wanted to see it.<br />
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I've previously introduced them to <a href="http://scratch.mit.edu/">Scratch</a>, which allows kids to write simple games quite easily using a drag-and-drop graphical interface for programming, so they are already quite familiar with the idea of writing code to make simple games. But this promises to take coding fun to an whole new level. <br />
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Due to some compilation challenges I spent quite a bit of time fighting to install this mod. They watched my progress (or lack thereof) in tense expectation, which made it very clear to me how much this mattered to them both. It was rather amusing, actually. The older one shed real tears when the compilation repeatedly turned up 47 errors. And the little one, as he reluctantly went in to have his teeth brushed, was heard muttering to all who cared to listen, "how come there's only one "javac" on the computer?"<br />
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To their immense relief, I did finally get it working. (On the odd chance this is read by anyone else finding it hard to install or modify something for MCP involving new packages, a helpful piece of advice: use <a href="http://www.eclipse.org/">eclipse</a>!) The kids are now having fun dropping ready-made cottages, forts and castles in the game, and for now, with my help, modifying their size and shape.<br />
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In the weeks to come, I think this could be a powerful motivator for my older one to delve deeper into programming and algorithmic thinking, which would not be too bad a thing...Unknownnoreply@blogger.com10tag:blogger.com,1999:blog-7638578.post-69826931937284312732012-11-29T11:38:00.000-08:002012-11-30T01:26:10.168-08:00Inconvenient Student Disorder<span style="font-family: inherit;">Mass schooling not only offers many <a href="http://academicsfreedom.blogspot.com/2010/11/dangers-of-classroom-teaching.html">impediments to true learning</a>, it also actively distorts society's view of what is normal child behavior as it seeks ways to control and modify their behavior purely for its own ends. </span><br />
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<span style="font-family: inherit;">An <a href="http://healthfinder.gov/news/newsstory.aspx?docID=663002">article from HealthDay</a> reports that "<span style="background-color: white; line-height: 19px;">In the past decade, the number of children receiving a diagnosis of attention-deficit hyperactivity disorder (ADHD) has risen by 66 percent, new research indicates. </span></span><span style="background-color: white; font-family: inherit; line-height: 19px;">In 2000, just 6.2 million physician office visits resulted in a diagnosis of ADHD. By 2010, that number had jumped to 10.4 million office visits."</span><br />
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<span style="font-family: inherit;"><span style="background-color: white; line-height: 19px;">10.4 million represents about 20% of school-aged children in the U.S. Think about that for a minute. </span></span><br />
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<span style="font-family: inherit;"><span style="background-color: white; line-height: 19px;">Now, examine the list of <a href="http://www.cdc.gov/ncbddd/adhd/diagnosis.html">official diagnostic symptoms for ADHD</a>, taken from the </span><span style="background-color: white; line-height: 15.949999809265137px;">American Psychiatric Association's Diagnostic and Statistical Manual of Mental Disorders</span><span style="background-color: white; line-height: 18.984848022460938px;">. It includes: </span></span><br />
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<li style="margin: 0.7em 0px 0.7em 1.5em; padding: 0px;">Often does not follow through on instructions and fails to finish schoolwork, chores, or duties in the workplace. </li>
<li style="margin: 0.7em 0px 0.7em 1.5em; padding: 0px;">Often avoids, dislikes, or doesn't want to do things that take a lot of mental effort for a long period of time (such as schoolwork or homework).</li>
<li style="margin: 0.7em 0px 0.7em 1.5em; padding: 0px;">Often loses things needed for tasks and activities (e.g. toys, school assignments, pencils, books, or tools).</li>
<li style="margin: 0.7em 0px 0.7em 1.5em; padding: 0px;">Often fidgets with hands or feet or squirms in seat when sitting still is expected.</li>
<li style="margin: 0.7em 0px 0.7em 1.5em; padding: 0px;"><span style="font-family: inherit;">Often gets up from seat when remaining in seat is expected.</span></li>
<li style="margin: 0.7em 0px 0.7em 1.5em; padding: 0px;"><span style="font-family: inherit;"><span style="background-color: white;">Often blurts out answers before questions have been finished</span>.</span></li>
<li style="margin: 0.7em 0px 0.7em 1.5em; padding: 0px;"><span style="background-color: white;"><span style="font-family: inherit;">Often has trouble waiting one's turn.</span></span></li>
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These are not symptoms of an abnormality --- it is a laundry list of behaviors that are inconvenient to a teacher trying to manage a large class. The kid in question is simply not docile and passive enough to sit patiently through something that does not interest that child. And it's not the occasional kid that is ill-suited for sitting quietly behind a desk all day and doing what he/she is told. It is about 20% of them! </div>
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And there are literally no limits to how poorly and inhumanely kids diagnosed can be treated in a classroom, once they have been diagnosed with ADHD. This <a href="http://www.katu.com/news/local/Longview-school-district-aims-to-ease-concerns-over-isolation-booth-181284591.html">recent article posted on KATU</a> reveals that Mint Valley Elementary School in Longview, Washington put elementary kids (we are talking 6-12 year olds here!) that were showing "behavioral problems" in a padded isolation cell. The <a href="http://www.katu.com/news/local/Isolation-box-Abuse-or-therapy-for-Longview-Mint-valley-elementary-kids-181135971.html">school officials claimed this was merely a form of "therapy"</a>. But any sane individual should be able to see it for it is --- a punitive prison, a menacing jail. It is an instrument for behavioral control all right, but not one that belongs anywhere in a free, enlightened, society.<br />
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After a justifiable outcry after one concerned parent posted photos of it on Facebook, the school district has <a href="http://www.katu.com/news/local/Longview-school-district-suspends-use-of-isolation-booths-Mint-Valley-box-181418881.html">decided to suspend the use of these cells</a>. I think this incident, given how long and freely the school operated this isolation cell, very pointedly illustrates the dark connection between coercive control in large classrooms and ADHD diagnoses.<br />
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://1.bp.blogspot.com/-QtKe_uqbEao/ULfWRjxxlsI/AAAAAAAAFEE/SP71S2yUthw/s1600/isolationCell.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="150" src="http://1.bp.blogspot.com/-QtKe_uqbEao/ULfWRjxxlsI/AAAAAAAAFEE/SP71S2yUthw/s400/isolationCell.png" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">(photos from <a href="http://www.katu.com/news/local/Longview-school-district-suspends-use-of-isolation-booths-Mint-Valley-box-181418881.html">katu.com</a>)</td></tr>
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Here is a trailer for a documentary focusing on the issue of medication for ADHD diagnosed children, called "The Drugging of Our Children":<br />
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And the following is a trailer for a 2009 documentary called "The War on Kids," which makes this very point about ADHD being used to exert control in the classroom. This movie also discusses how kids freedoms are curtailed as they are suspended for various minor "offenses" due to excessive zero-tolerance policies:<br />
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Unknownnoreply@blogger.com8tag:blogger.com,1999:blog-7638578.post-74803215027805206582012-09-12T11:00:00.000-07:002012-09-12T14:04:35.076-07:00A 21st century school needs no classrooms<div class="separator" style="clear: both; text-align: center;">
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Image from <a href="http://www.vittra.se/english/Schools/StockholmSouth/Telefonplan.aspx">www.vitra.se</a></div>
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I just found out about an amazing school that opened last year in Sweden, called <a href="http://www.vittra.se/english/Schools/StockholmSouth/Telefonplan.aspx">Vittra Telefonplan</a>, that has eliminated the classroom. Instead there is a lot of open space and furniture that encourages discussion and collaboration as well as individual learning. For more photos, see the slides on <a href="http://www.businessinsider.com/a-group-of-schools-in-sweden-is-abandoning-classrooms-entirely-2012-1#this-is-the-mountain-which-is-the-central-gathering-point-of-the-school-1">Business Insider</a>.<br />
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As someone says in the following video about the school: "learning how to learn is actually the biggest and most important ability". This is a space designed to encourage that ability, through flexible play, and freedom.<br />
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I have always believed that the kind of space that is available in a learning environment makes a big difference in what the people there do. Do we dare hope that universities will see similar changes in how educational spaces are structured, especially as the rise of MOOC's (Massive open online courses) changes what can / should be done in a traditional lecture format?<br />
<br />Unknownnoreply@blogger.com10tag:blogger.com,1999:blog-7638578.post-74998579509468452022012-09-07T22:27:00.000-07:002012-09-09T23:21:53.106-07:00What Academics Do: The Big Five<div class="separator" style="clear: both; text-align: center;">
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I remember as a student wondering what exactly it is that university faculty do, besides teaching classes. I figured out some of it by direct observation while I was a graduate student; but it is only after experiencing it for myself and reflecting on it for a decade that I feel I can articulate it satisfactorily.<br />
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Everything I do as an academic that is of essence can be categorized into five dimensions:<br />
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<li><b>Learning:</b> Mastering new mathematical techniques, algorithms, software programming tools. Keeping up with new theoretical and experimental results and developments in the field. This happens through reading papers, articles, reports, and books, attending talks by visitors and at conferences and classes, discussing with peers and students at various venues. Practicing. Often learning occurs not when we are actively seeking it, but almost miraculously, such as in the process of trying to explain something to others.</li>
<li><b>Discovering:</b> Coming up with new problem formulations, models, solutions, experiments, results and findings. This comprises not merely the end but also the means of discovery, via the process of thinking, pondering over,
making and proving/disproving conjectures, analyzing, solving, discussing, algorithm and architecture design, pen-and-paper mathematical derivations, creating tangible new artifacts and proofs of concept, designing/creating software and hardware for and running simulations and test-bed experiments, collecting and analyzing experimental results. We are driven by this quest, hoping by our efforts to win back one more inch from the dark shadows of ignorance.</li>
<li><b>Sharing:</b> Giving talks to diverse audiences at technical and non-technical meetings and conferences, universities, industry venues. Teaching --- all that it typically entails: lecturing, showing, conveying understanding, asking and answering questions, preparing and giving feedback on homework assignments and tests, one-on-one tutoring --- but also beyond: informal direct interactions with students outside the classroom, mentoring, providing opportunities for growth and building self-confidence. Sharing one's life experiences, offering advice, motivating, inspiring, encouraging. Advising and guiding undergraduate and graduate students on projects and thesis dissertations. Writing papers, books, reports, articles, emails. Translating ideas, simplifying their exposition. Creating and making available to others code, hardware designs, and data. Making and giving demos and presentations. Discussing with students, colleagues, peers in academia and industry, and the press, when they visit, when visiting them, whenever together at meetings/conferences, or by electronic means. Ultimately, our impact on the world is determined by our skill at sharing what we have learned and discovered, enabling others to learn and discover even more.</li>
<li><b>Helping: </b>Playing an active, helpful role in the twin-fold academic community that consists on the one hand of colleagues and students at one's own institution, and, on the other, of one's peers in the field. Helping is closely related to sharing. We help students, for instance, when we share our knowledge and experience with them, but also often by offering a friendly ear or a shoulder to cry on, or by putting them in touch with others who can be of more help. Reviewing papers, serving on proposal review panels, seeking reviews for journals and conferences, organizing meetings and events, being on Ph.D. student exam committees, faculty review committees, writing recommendation and tenure letters. As part of department, university, or other professional committees, or else informally, interacting with peers and administrators, engaging in a community dialogue to formulate strategic visions, develop programs, implement policies, and address various issues of concern as they arise. </li>
<li><b>Fund-raising: </b>Also referred to as grantsmanship. Seeking collaborators, forming teams. Brainstorming alone or with collaborators and students to identify compelling avenues of research and open research problems. Thinking through methodology and research plans. Writing and submitting competitive proposals to federal agencies such as the National Science Foundation, industry, or internal funding sources within the university. Interacting collaboratively with teams sharing common interest for industry funding. Idealists perceive all this "chasing after money" as a waste of time, at worst, or, at best, a necessary evil. But it is not unreasonable to expect that the request for public and private funds for the pursuit of academic inquiry be justified from the perspectives of intellectual value and practical utility; moreover, the very process of writing a proposal often clarifies one's thinking and can help identify interesting and important new avenues for investigation.</li>
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Though the details may vary from field to field, I believe these five categories of activity go to the very heart of what it means to be an academic today.<br />
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These dimensions are very closely linked and often seamlessly integrated with each other. We <i><b>learn</b></i> so we can <i><b>discover</b></i>, and what we discover only has value when we <i><b>share</b></i> it. Because all of this happens not in isolation, but in an inter-connected social context, we must also be good citizens and <b><i>help </i></b>others in the community. And in order to find the resources to aid the process of learning, discovering, sharing, and helping, we must <i><b>raise funds</b></i>.<br />
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There is also a process of <i>meta-learning</i> that forms a natural part of the academic life: namely, the continual learning that results in the improvement of one's ability to learn, discover, share, help, and raise funds.
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<b>Why not just "Research, Teaching, and Service"?</b><br />
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Traditionally, the academic enterprise is divided not into five dimensions as I have indicated here, but into three: research, teaching, and service. I believe that standard tripartite definition of academic life is an oversimplification that does real harm in many ways. It creates and propagates a false dichotomy between teaching and research in the minds of many. It contributes to creating a negative impression of service as consisting of miscellaneous chores. And it impedes us from presenting to students an inspiring vision of the academic life.<br />
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At the core, what we care about is not the distinction between "doing research" and "doing teaching." What motivates us rather is the passionate desire to learn, to discover, and to share, whether it be in the classroom or the lab. The various ways of helping others (more than just "doing service") and fund-raising also do not abide by this fragmentation, as they sustain our activities in both settings.<br />
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Thus, the five-dimensional decomposition advocated here allows for a more holistic view of the academic's life, shedding a useful light on what matters to us and why.</div>
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Unknownnoreply@blogger.com14