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<p>Wow, you’re an *******.</p>
<p>^
Yet you failed to rebuke his claim. </p>
<p>What do you think we should imply from that?</p>
<p>“In high school - indeed, at many colleges - the humanities and social sciences are taught at such a rote, low level that any decent B.S.er can pass them with flying colors. As with pretty much all subjects in the US, the subjects simply need to be taught at a much, much higher level. Constant lowering of standards has done a lot of damage.”</p>
<p>^^This is spot on. I would just like to add that, conversely, high school math is structured in such a way that it becomes artificially counter-intuitive, boring, ugly, and scary. Instead of being taught in a way that cultivates students’ sense of imagination and creativity, it is taught as a mindless manipulation of symbols for some predetermined purpose that the students can “do” but ultimately never learn to fundamentally understand or appreciate. A good analogy would be if art classes were exclusively taught as paint-by-number. <em>sigh</em></p>
<p>^
<a href=“http://www.maa.org/devlin/LockhartsLament.pdf[/url]”>http://www.maa.org/devlin/LockhartsLament.pdf</a></p>
<p>^ yeah i’ve heard that criticism before, but to teach math the way that guy wants to in elementary/middle/high school would make it much harder. students struggle enough with math already.</p>
<p>also, the curriculum too needs to pretty much stay the same–being able to do arithmetic and a little bit of high school algebra and understanding what a function is (i.e. how to read a plot) is really important for most non-math/sci people. </p>
<p>skipping these sorts of mechanical things and teaching set theory, abstract algebra, or whatever instead is way less useful than our current system for most students out there.</p>
<p>Awesome, awesome article, L’Hopital. I read the whole thing and forwarded it to my math teacher.</p>
<p>Well, about the article, I’d say that until basic high school math, it’s a little bit of a long shot to attempt to let the kid be creative with something like math. What math classes teach before then is basic stuff that becomes a language in which things are done later. After that, sure, I think teaching kids how to memorize the equation of a circle or compute the focal points of ellipses is useless. From geometry, algebra 2 etc onwards, the focus should be on developing an appreciation for the ideas. Until then, i.e. up through basic algebra, I think a lot of the techniques are really just techniques you need to know.</p>
<p>People tend to look down on subjectivity and think that the Humanities are completely subjective as if something like Biology is completely objective. To be subjective is to be influenced by one’s perception (i.e. senses). The sciences are all based on observation, which is subjective. </p>
<p>People think that the Humanities courses such as Philosophy and English have no right answer. Well if my philosophy professor were to ask me “Is God the only explanation for our natural inclinations?” I cannot answer with feelings and beliefs; I must use logic and proofs, which can lead to truth. If my literature teacher were to ask me to use tricolon crescens or antithesis in my paper to demonstrate a style of writing, I can go wrong if I do not do that correctly. If my Art History professor were to ask me if the building is Gothic Architecture or Romanesque I cannot say “Oh it totally feels Greek to me” I have to describe based on observation the certain characteristics that qualify it as Romanesque.</p>
<p>I’m a Humanities Major because a degree in Biology or Philosophy cannot give me enough space to pursue all my interests. I plan to go onto obtain an MD/MA of Bioethics. </p>
<p>Look. Comparing Humanities and Science/Math is like comparing Apples and Oranges. They are not the same thing. But an apple may be more preferable than an orange to certain people. Don’t hate him for a preference of that sort. Don’t say the guy who eats an apple is more prone to scurvy because he doesnt get his Vita C from the orange. You do not know if he has other things he likes. Don’t say the Humanities major is going to be poor because he’s not going to be doing lab research for big money from big pharmaceuticals.</p>
<p>Yes some humanities majors cannot do math. So many math/science majors I know have dropped out of my honors humanities courses. I manage to stick around with an A in Honors Biology. Some humanities majors can do Calculus, some cannot. Some Chemistry majors can fair well in Renaissance Art, some cannot.</p>
<p>It seems like the fascination with labeling things and people is interfering with our acknowledgment of individuality.</p>
<p>@silence:</p>
<p>“Math is hard”, right barbie? : )</p>
<p>It’s not like what we teach them is useful anyways. Most adults can’t even add fractions. How many of them do you think actually know the quadratic formula? It’s obvious that if it was that important, people would be suffering. </p>
<p>But really, think about it. Other than basic addition, subtraction, multiplication and division, what else do you really need? Or rather, what else does the average adult use in his daily life? </p>
<p>So, if we can step out of this whole “Pure mathematics is useless” is business and realize that, hey, for most adults, most branches of mathematics beyond 5th grade are useless. So rather than make them memorize stupid rules and pathetically attempt to make the math seem “applied” (like the example Lockhart gave with the age question), why not let them have fun?</p>
<p>@mathboy:
What do you really “need to know”? Let’s define this first. I believe that this amount of “need to know” is very minute for the average person. Granted, I’m not saying disregard arithmetic or anything. But, as stated in earlier in this post, beyond basic arithmetic and the likes, what kind of average adult uses that? And of those who do use it, how many of them just pull out a calculator and apply the black box concept?</p>
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<p>Because it isn’t worth it to waste my breath on someone who obviously won’t change their mind. It’s not worth wasting it on you either, but in summary:
Why should I justify my own personal happiness? I’m majoring in English because I love studying it, not because it is going to make me money. I want to be a writer; and if I am happy doing just that, then I am successful. No one can take that away from me. Or from anyone else who feels the same.</p>
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<p>Being able to solve simple algebraic equations is useful; basic geometry is useful; having a good intuition about quantities is useful; being able to read a plot is useful. Knowing concepts like the mean and variance of a distribution are useful. Trigonometry is even useful. </p>
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<p>I will repeat myself. That will make math class harder. Kids struggle enough already with mathematics. It is also pretty difficult to have the sort of dialogue needed for that type of instruction with a bunch of disinterested students.</p>
<p>I think the approach in the paper is good for gifted students though.</p>
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<p>It’s a lot more one-sided than you make it sound . . .</p>
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<p>Any manipulations that can occur in the analysis of an everyday individual’s encounters, plus a first level of abstraction involving those manipulations so those manipulations are given some context seems healthiest.</p>
<p>This reads: basic school arithmetic and area/volume calculations, percentage calculations, and small algebra maniuplations.
I say take that very minute amount and to make it thorough, go as far as you can need to in order to give the “idea” behind that minute bit, without straying into theory that goes above and beyond. I.e. algebra is fine except you don’t need to find focal points of ellipses to know the most basic extension of what you need.</p>
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<p>As someone who likes many of the humanities, let me comment that this gives an incomplete picture, because science and math have a subjective aspect, but we understand that upon certain assumptions which are clearly stated we can be objective. I think that’s what is the key distinguishing point. Certainly engineering is subjective too, in the sense one may be taking measurements and things like that, estimating potential for failure, etc, all based on that kicker observation. </p>
<p>I think the great humanities majors can also write with purpose and with assumptions clearly in mind. When they write about their feelings, they can at least explore the craft of writing, again aware of what assumptions they have in discerning the effect certain composition will have. </p>
<p>Composition choice may, of course, be driven by “intuition” but this in theory should be based on assumptions that can be stated clearly. Similar to how much of mathematical thought and result-searching rests on intuition, under the guise of words like “It can be shown that …” knowing that the “…” can be filled in with assumptions clearly stated from ground up.</p>
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Really? How many adults (even the ones who actually solve equations for a living) know how to actually solve them and not just plug them into MATLAB? Of those who actually know how to solve them (and actually need them), how many do you think actually solve them and not just do like those who don’t know how and just plug em in on MATLAB? </p>
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I think the question that needs to be asked is “why are they disinterested?” Let’s face it, dull math will even disinterest the brightest of minds. I had a Calc Class at my university that was basically the definition of plug-n-chug. It sucked so much I stopped going, so did half the class. In college, you can do this. But in elementary/middle/high school you are stuck with the boring class and thus won’t learn. I think a more interactive approach like Lockhart’s would at least remove some of the disinterestedness of them.</p>
<p>I do agree, though, that it would probably work even better with gifted kids, as they already have the motivation and the skill and might progress farther.</p>
<p>As for mathboy,
Oh really? And what are these manipulations? You mean these?
When was the last time you calculated volume by hand that wasn’t for school? Or percentage? Or algebra manipulation? It may be that you may do so more than I make it seem, but there are ADULTS who can’t even calculate percentage barring the easy ones like 10% or 25%, so how do they live? </p>
<p>Granted, I think its good to have some intuition about results, so that one doesn’t just blindly plug it into your calculator or whatever, but let’s face it. Most of the math is not done by hand. In fact, I recall just recently my cousin was complaining that for his finance final he had to sum geometric series and that he couldn’t use Excel. I had to tutor him on how to do it. </p>
<p>I guess what I’m trying to say that there is a lot of stuff in the curriculum that honestly is not needed (nor done a lot by hand) in real life. So rather than pretend it is, give only what is important and then give the whole more of an exploring and discovering process. Even if kids learn less, I think they will retain more.</p>
<p>L’Hopital, I don’t quite understand where the discussion is going here. Am I claiming the purpose of teaching these things is to do long volume computations by hand? Not at all.</p>
<p>But every single one of the mechanical operations I stated comes up in a very basic everyday sense. Like area – being able to recognize on a glance that increasing the length of a television screen’s edge lets the area grow by a squaring mechanism. Arithmetic – simple enough. These needn’t be long, crazy computations in a math book, they can be easy ones illustrating what’s going on. </p>
<p>I have a hard time believing anything I stated needs to be made more “creative” and “explorative.” </p>
<p>If you’d say that today’s teachers don’t even make students understand these basic concepts well and make them do drills, then those are highly incompetent, and I don’t think the fix is to change the strategy drastically. Where the creativity needs to come in is when the subject matter honestly lends itself to something other than manipulations, and is actually taking the first steps towards a more subtle reasoning process.</p>
<p>I wonder what on earth “exploring and discovering” means too. I’m not against this on principle, but I don’t think it’s clear what it is yet!</p>
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Problem is, most teachers would teach this by saying “Area = length x width” and leave it at that. They do those incompetent drills as you say, and never quite explore the whole “Well, what happens if we double one side, and the cut the other in half?”</p>
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What better than something like area? So we know how to solve the area of squares. What if you were faced with a triangle? a trapezoid? These kind of questions are usually answered by “Well, here’s the equation.” Why not let the kid explore? </p>
<p>The article makes a pretty good explanation as to how to find the area of a triangle by using a square. Why not let them discover that you can break the trapezoid into similar figures that one calculate the area of? Then, you can go further in, discussing other n-sided figures and (depending on the level of the class i.e. if they know trig) maybe even derive a method for finding the area of any regular n-gon. You could even use a very elementary and intuitive definition of a limit to provide a value for pi by letting the kids plug in large values of n into their calculators and estimating the area of a circle. Granted, I’m not saying leave the kid up to his wits to figure it out. The teacher is there to hold his hand and guide him. But such an excursion into the problem of area would be fantastic and it’d give one a great background for calculating area. </p>
<p>I guess what I’m trying to say is that once you set the groundwork, there should be room for exploration. A kid doesn’t need to calculate the area of square twenty times in order to understand it. Making them apply similar methods to what they learn in class to find the area of other figures would be nice.</p>
<p>I suppose the basic basic basic axioms (addition, multiplication, and whatnot) might suffice with plain memorization, but beyond that, why not treat it like a discovery process? Rather than randomly give Algebra 2 kids the quadratic formula, why not introduce completing the square and try to guide them to deriving the quadratic formula?</p>
<p>I guess what I’m trying to say is that everything seems to be treated as groundwork and the discovery part of mathematics is not even mentioned.</p>
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<p>Given what you’ve written here, I am completely in accord with your philosophy, and if you notice, I actually have implicitly the same. I believe several things naturally come up in day to day life, and the purpose of school curriculum should be to give one a way to approach them hopefully using many angles, and thus certainly should be more fluid than it is now. </p>
<p>I was concerned earlier that perhaps the need for hand-holding and structure was not adequately acknowledged. I’m an extremely fluid type learner right now, but certainly it would have been hard to have that at an early age. Introducing the idea of viewing a certain scenario as just a special case of techniques from a general phenomenon (such as showing that trapezoids’ area is just another form of area, as you say) is in line with what I believe in, for instance in the sense that I don’t think basic arithmetic is where things should stop, in that I think the need for algebra should be naturally motivated by the need for arithmetic.</p>
<p>Do what makes you happy… don’t conform to the majors/careers with fancy notoriety just because of their prestige.</p>