<p>Interesting article. I’m sure some of you must’ve read it. Find it here:-
[Lockhart’s</a> Lament](<a href=“http://www.maa.org/devlin/devlin_03_08.html]Lockhart’s”>http://www.maa.org/devlin/devlin_03_08.html)</p>
<p>It is interesting - although I think its a bit extreme in cases.</p>
<p>Well, in the beginning I was a little put off by the way he seems to disregard the practicality of math, by saying that no one actually uses it. I mean, as much as yes, I love my computer because I couldn’t solve a differential equation to save my life, I am an engineer and I still need to understand how math works enough in order to even know that there’s a differential equation somewhere that I need to (tell my computer to) solve.</p>
<p>I think there’s a middle ground to be had. All the stories I’ve ever heard about good math teachers (I personally have had very few) revolve around self-discovery. If your teacher can set YOU thinking down the path of increasingly small delta x’s, the next thing you know YOU’VE invented calculus, and as a result you like it and are good at it AND appreciate its practical value. Because after all, Newton didn’t invent calculus as a form of self-expression, he invented it because he wanted to do physics and algebra just wasn’t cutting it. I am not discounting the beauty of math, or the people that approach it as an art form, but I do think that its practical applications are far more important than he would like to admit. In fact I might even argue that you could elevate math to an even higher status by pointing out that its beauty and its usefulness are two sides of the same coin. What more can you ask of a subject?</p>
<p>That said, the further I got into it, the more I loved it. It’s nice to have all of your anger and frustration with your math education completely validated by a real life mathematician. As soon as geometry was mentioned, my instinctual reaction was, “ugh, proofs.”</p>
<p>Then lines like, “Why Geometry occurs in between Algebra I and its sequel remains a mystery.” and “[Pre-calculus is a] senseless bouillabaisse of disconnected topics.” had me practically jumping up and down in my seat. I’ve been saying that since I was actually TAKING those classes, but always assumed that there must have been someone smarter than me who decided those things for logical reasons.</p>
<p>I think the right balance is to realize why mathematics has practical applications. Mathematics is in some ways a way to describe what is “natural” to us as encoded in formal structures (as opposed to logic, which may study the very structure of theories), and this “natural” stuff certainly includes physical phenomena. It is no surprise that physics can motivate mathematics, and mathematics can discover things that are (not necessarily yet) inherently physical, but certainly logically conceived, which later are related to physical phenomena. Some of the most nontrivial advances in both fields occur in this way.</p>
<p>Check out this shorter blurb:</p>
<p>[Why</a> Johnny won’t be able to count](<a href=“http://www.sumizdat.org/Johnny.html]Why”>Why Johnny won't be able to count)</p>
<p>which has a related but perhaps not entirely identical point. </p>
<p>I do think the Lockhart article is extreme, though, because I think exposing kids to formulas and having them fiddle around with them is itself very valuable, independent of self-discovery. And for the record, I do know younger kids learning about how a cone’s volume compares to that of a cylinder, etc, and sort of conceptualizing it using the formula. Use of the concrete to relate to the concepts is probably one of the most important things that can be taught.</p>
<p>I think the link I myself posted is more along the lines of my complaint. I think the self-discovery aspect to the Lockhart article is simply not my major complaint with how things are done for the youngest kids, because I’d be more interested in teaching them how to conceptualize and to answer basic questions conceptually by actually reasoning. I think this leads smoothly into a little more self-discovery when older.</p>
<p>Discoveries are hardly uninfluenced by past fundamental discoveries. Part of education is also about rapidly conveying a lot of important foundational discoveries whose spirit may be reflected in subsequent ones.</p>
<p>Teaching kids a general framework to write English essays in earlier years doesn’t mean they necessarily have to follow that for life, it’s just a way to get them thinking about common things that probably should be on someone’s mind when writing an essay.</p>
<p>I do think that when teaching formulas, though, one should mention where they come from. I think this is very different from self-discovery, though. The teacher’s goal should be to get the student thinking about the objects.</p>
<p>Hardy’s Apology is an obvious companion piece.</p>
<p>Though this isn’t the grad school forum, here’s the equivalent for law school 
</p>
<p><a href=“http://duncankennedy.net/documents/Photo%20articles/Legal%20Education%20and%20the%20Reproduction%20of%20Hierarchy_J.%20Leg.%20Ed..pdf[/url]”>http://duncankennedy.net/documents/Photo%20articles/Legal%20Education%20and%20the%20Reproduction%20of%20Hierarchy_J.%20Leg.%20Ed..pdf</a></p>