{"id":5,"date":"2006-07-27T22:14:43","date_gmt":"2006-07-28T02:14:43","guid":{"rendered":"http:\/\/blogs.law.harvard.edu\/fluxions\/2006\/07\/27\/on-teaching-oneself-calculus\/"},"modified":"2006-07-27T22:14:43","modified_gmt":"2006-07-28T02:14:43","slug":"on-teaching-oneself-calculus","status":"publish","type":"post","link":"https:\/\/archive.blogs.harvard.edu\/fluxions\/2006\/07\/27\/on-teaching-oneself-calculus\/","title":{"rendered":"On Teaching Oneself Calculus"},"content":{"rendered":"<p>Calculus really isn&#8217;t all that hard, but it would be a lot easier if they would explain things in the right order.<\/p>\n<p><!--more--><\/p>\n<p>As many of you know, I am teaching myself calculus.  <a name=\"cutid1\"><\/a>There is a very good explanation for this, but it requires identity-undermining revelations.  (Perhaps later.)  Suffice it to say that I was on (ahead of, even) the usual math track until about age 14.  For complicated reasons, it&#8217;s 100% technically accurate to say that I never set foot in a high school math class, and I only took one math class in undergrad, some boring abomination called &#8220;college algebra,&#8221; which seemed to be mostly about logarithms.<\/p>\n<p>Well, in under two months, I&#8217;ll be needing math. Specifically, I&#8217;ll be needing calculus to survive the stats requirement in the a hyper-quantitative poli sci phd program. So over the last few months, I&#8217;ve been working on this little project.<\/p>\n<p>I started out by trying to take an actual class, via a sneakily low-priced local adult education program.  (Yay government subsidies.)  Sadly, the instructor wasn&#8217;t very good, and, worse, the class was at 6 pm Monday nights about a half hour&#8217;s rush our drive away. Most people, even in the real world, could make a 6 pm class 30 minutes reverse-rush-hour drive from their office one night a week. Most people are not lawyers. After missing three classes in a row (including the midterm) I gave up the ghost. Thank you, practice of law, for forcing me to throw away the $400 I spent on tuition for that class. Literally, all I learned for my $400 was f(x+h) &#8211; f(x)\/h. That&#8217;s it. I didn&#8217;t even get to the power rule.<\/p>\n<p>Fortunately, I got one good thing out of that class: <a href=\"http:\/\/wps.aw.com\/aw_thomas_calculus_11\">a not unreasonable book<\/a>. It took me a while to delve into it. I first had to psychologically satisfy myself by buying three separate easy-tutorial books that I opened roughly twice each, and that I&#8217;ll probably donate to a library or something before I leave, since in all cases, they appear to be less clear and\/or less useful than an appropriately slow reading of the appropriate section in the actual textbook.<\/p>\n<p>I did find something other than the textbook that was useful, however: the <a href=\"http:\/\/www.teach12.com\/ttc\/assets\/coursedescriptions\/177.asp?id=177&amp;d=Change+and+Motion%3A+Calculus+Made+Clear&amp;pc=Science%20and%20Mathematics\">Teaching Company DVD series<\/a>. Michael Starbird, if you are reading this, you have only to say the word and I will gleefully have your babies. The DVDs aren&#8217;t very rigorous, and several of the 24 half-hour classes are fluff (I&#8217;m not bothering to listen to the last two), but it provides a broad-brush conceptual overview of the whole thing, so that when in the midst of a page full of functions, one has some notion of what it is that one is trying to flesh out. I really think the &#8220;one pass for broad overview, then a second for the gritty details&#8221; approach is the best way to learn this sort of thing.<\/p>\n<p>I also read about half of the <a href=\"http:\/\/www.dummies.com\/WileyCDA\/DummiesTitle\/productCd-0764569031.html\">trigonometry for dummies<\/a> book just so I&#8217;d know what the calculus book was talking about with sines and cosines and all that.  I may scandalize the purists here, but I don&#8217;t see the point of all the focus on trig stuff in calculus.  Sure, the functions are useful for various things (I&#8217;m sure electrical engineers love them).  But it&#8217;s so much boring memorization.  My theory is that the trig part of calculus can be reduced to a couple pages of cheat sheets with identities, derivatives, etc.  Then, trig for them that cares, I say.<\/p>\n<p>Anyway. Then it was into the calc book. At first, I started reading from page 1, slowing down to do the problems only when I wasn&#8217;t 100% clear that I got it, i.e. for confirmation. That was slow going, and mixed the interesting bits (the definition of a limit, getting to cheat and divide by zero with aid of the difference equation) with the boring (trig) bits.  By parametric functions, I&#8217;d lost interest in the slow going and stopped regularly reading. I brought myself back on track by focusing on the interesting bits and relying on the forthcoming math craziness (whose premise is that &#8220;you&#8217;ve been exposed to calculus but you don&#8217;t remember it, so we&#8217;ll jam the stuff you need for the methods requirement straight into your brain via a jackhammer&#8221;) to fill in any gaps I inadvertently leave. Thus, I&#8217;m skipping all of derivatives after the chain rule and implicit differentiation, except minima and maxima, the mean value theorem, differentials and linearization.  Those things I&#8217;m deferring so I can add some variety by diving right into integration.<\/p>\n<p>Integration by substitution is sexy.  I think one service that I can provide to the community with this blog is to express the symbol-bound techniques like integration by substitution in plain English.  That will be the next post.<\/p>\n<p>This really isn&#8217;t that hard. None of it is. There are bits that took a few leaps of reasoning for me to understand, but as far as I can tell, that&#8217;s the deficiencies of the book rather than anything else. For example, the book doesn&#8217;t really explain that Leibniz&#8217;s notation for derivatives means something other than &#8220;here&#8217;s a different way to say f'(x).&#8221; Suddenly, implicit differentiation enters the picture and all this dx\/dy df\/dy dy\/dx stuff means stuff other than &#8220;take a derivative, but do it in a silly German accent because the idiot continentals couldn&#8217;t acknowledge that Newton was much smarter and hotter than them.&#8221;  (In case you noticed, the Newton vs. Leibniz thing is going to be a running theme of this blog.) It took me a while to figure out why it is that they got a y&#8217; whenever the variable to be mucked with had a y in it, and I still might call in one of my math friends to make sure I&#8217;m clear on the concept, even though I seem to be able to solve the problems. But, anyway, if the book had presented that ratio\/not really a ratio\/oooh, look, it&#8217;s a ratio again as meaningful on its own account rather than a fancy way of saying &#8220;take a derivative&#8221; in the first place, or for that matter bothered to explain in words whose relationships were previously defined what it means to &#8220;differentiate both sides of the equation with respect to x, treating y as a differentiable function of x,&#8221; I wouldn&#8217;t have had to spend a couple of hours searching the internet for an explanation that made sense.<\/p>\n<p>Anyway, onward and upward. This really doesn&#8217;t seem so difficult so far. And it&#8217;s kind of fun.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Calculus really isn&#8217;t all that hard, but it would be a lot easier if they would explain things in the right order.  (For people who have read another source of my missives, this may be a re-run.)<\/p>\n","protected":false},"author":405,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[831,832],"tags":[],"class_list":["post-5","post","type-post","status-publish","format-standard","hentry","category-background","category-calculus"],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/archive.blogs.harvard.edu\/fluxions\/wp-json\/wp\/v2\/posts\/5","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/archive.blogs.harvard.edu\/fluxions\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/archive.blogs.harvard.edu\/fluxions\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/fluxions\/wp-json\/wp\/v2\/users\/405"}],"replies":[{"embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/fluxions\/wp-json\/wp\/v2\/comments?post=5"}],"version-history":[{"count":0,"href":"https:\/\/archive.blogs.harvard.edu\/fluxions\/wp-json\/wp\/v2\/posts\/5\/revisions"}],"wp:attachment":[{"href":"https:\/\/archive.blogs.harvard.edu\/fluxions\/wp-json\/wp\/v2\/media?parent=5"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/fluxions\/wp-json\/wp\/v2\/categories?post=5"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/fluxions\/wp-json\/wp\/v2\/tags?post=5"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}