tag:blogger.com,1999:blog-15868947.post114861076888164503..comments2023-10-29T06:41:23.910-07:00Comments on Halfway There: Prechewed mathematicsZenohttp://www.blogger.com/profile/09058127284297728552noreply@blogger.comBlogger12125tag:blogger.com,1999:blog-15868947.post-13772466413370162282014-03-29T14:30:53.915-07:002014-03-29T14:30:53.915-07:00From the first grade, I had very good math teacher...From the first grade, I had very good math teachers. My main purpose in taking math was not to get the right answer nor to show that I knew how to perform the operations. I wanted to discover new methods. <br /><br />In plane geometry, most students would memorize proofs and spit them back on tests. I tried to use the axioms and postulates and previously proven examples to arrive at n answer. <br /><br />My geometry teacher told me how glad he was to have someone like me in his class and that there was only one. Varyaghttps://www.blogger.com/profile/15047874331284447809noreply@blogger.comtag:blogger.com,1999:blog-15868947.post-52383358487553457792014-03-29T14:28:58.385-07:002014-03-29T14:28:58.385-07:00From the first grade, I had very good math teacher...From the first grade, I had very good math teachers. My main purpose in taking math was not to get the right answer nor to show that I knew how to perform the operations. I wanted to discover new methods. <br /><br />In plane geometry, most students would memorize proofs and spit them back on tests. I tried to use the axioms and postulates and previously proven examples to arrive at n answer. <br /><br />My geometry teacher told me how glad he was to have someone like me in his class and that there was only one.Varyaghttps://www.blogger.com/profile/15047874331284447809noreply@blogger.comtag:blogger.com,1999:blog-15868947.post-77847669075950565532011-05-29T22:40:56.323-07:002011-05-29T22:40:56.323-07:00I realise that you are teaching algorithms that wo...<i>I realise that you are teaching algorithms that work in general and trying to assess the students ability at implementing these algorithms</i><br /><br />Haven't you answered your own question, physjam? I can't assess the student's ability to use the general algorithm unless they actually use it. There is nothing to prevent them from also trotting out some clever shortcut (if they have the time and inclination).<br /><br />A similar problem arises if they run ahead of their classmates. If, for example, they drag out a previously-learned quadratic formula before we've covered it in class, they're inclined to use it exclusively. It's not always the best solution technique (although it always works) and students who jump the gun cannot be assessed on their mastery of alternatives if they use nothing else.<br /><br />The anecdote you cite concern Gauss, who is said to have added up the numbers from 1 to 100 by discerning a clever shortcut. His teacher was impressed and supposedly provided the young Gauss with a more advanced math text to keep him happy and occupied. If I find a young Gauss in any of my classes, I will try not to punish him (but I will wonder why he's taking algebra in college).Zenohttps://www.blogger.com/profile/09058127284297728552noreply@blogger.comtag:blogger.com,1999:blog-15868947.post-2592554703515546172011-05-29T22:07:21.591-07:002011-05-29T22:07:21.591-07:00I don't see why students should be punished fo...I don't see why students should be punished for seeing a better solution. If the problem has been made so easy they can immediately see the solution, this is the fault of the teacher and not the student.<br /><br />I realise that you are teaching algorithms that work in general and trying to assess the students ability at implementing these algorithms, but there is no reason why students should only try to solve solution by the algorithm taught to them. (In fact this is precisely the problem in higher mathematics: here is a problem that there is no known algorithm to solve - invent one).<br /><br />I also strongly disagree with the statement "no fair using it before it's been presented to your classmates!" Legend has it that as a student Euler was set the problem of adding a large quantity of sequential numbers together, and he immediately saw the problem reduced to a multiplication if you add from both ends at once. I think any teacher who would punish a student for seeing such a clever solution (that was not taught because the teacher did not know it) has no business teaching.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-15868947.post-57228331900161539512010-05-05T09:46:47.488-07:002010-05-05T09:46:47.488-07:00There are physical objects that can be used to tri...There are physical objects that can be used to trisect angles. A carpenter's square is one, and the more usual one is the neusis construction, which is using a marked ruler. These are described in Underwood Dudley's book "The Trisectors". I don't know what the correct Galois theoretic descriptions of these constructions are, in general (i.e. what kind of field extensions of the rations you can get. Ruler and compasses gets you compositions of quadratic extensions).Robert Furberhttps://www.blogger.com/profile/10746976399050925428noreply@blogger.comtag:blogger.com,1999:blog-15868947.post-33820821662673847552009-07-03T09:53:29.974-07:002009-07-03T09:53:29.974-07:00I don't know if there is such a physical objec...I don't know if there is such a physical object like there is for ruler and compass. But that's not really the point. What I mean to say is that <i>if</i> you had it, you could trisect any angle, but you <i>still</i> couldn't solve the general quintic.<br /><br />Incidentally, for anyone reading who doesn't know, ruler and compass constructions allow you to perform any arithmetic operations you want, and also square roots. Given one reference length to serve as a "unit", you can use a lot of what's in Euclid (mostly in book I) to make geometric analogues for addition, subtraction, multiplication, division, and square roots.<br /><br />So the question is: what numbers can you get from these operations starting with the natural numbers? It turns out you <i>can't</i> get things like the cube root of 2, which you would need to double the cube. Trisecting angles also turns out to require arbitrary cube roots. You can't square the circle because you can't construct a length pi. All these classical puzzles come down to reinterpreting Euclidean geometry in light of abstract algebra.<br /><br />And the captcha today: "soisi". So I see...Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-15868947.post-65570236701968557632009-07-03T09:48:17.505-07:002009-07-03T09:48:17.505-07:00The problem is that you are teaching stupid people...The problem is that you are teaching stupid people. In order to make it so that they can do some minimal amount of math just to be able to survive in the real world, you aren't teaching basic principles. You're teaching esoteric rules based on, and several orders removed from, the basic mathematical principles that you're teaching. The intelligent people in your classroom are left to their own devices to figure out that these rules come from somewhere. If you could teach the basic laws of mathematics from which your limited problem-solving techniques are derived, you could give them extremely complex problems with complex answers that they could do however they saw fit. Just like Feynman did in his head; because he had already figured out the basic laws behind solving for an unknown quantity without needing to be taught the dumbed-down techniques like subtracting from both sides.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-15868947.post-33403021983838697372009-07-03T06:54:57.966-07:002009-07-03T06:54:57.966-07:00Un-apologetic:
Yes. I should have specified ruler...Un-apologetic:<br /><br />Yes. I should have specified ruler and compass. But isn't your magic wand for cube roots rather different to the magic wand for square roots. Pythagoras' theorem works algebraically and on the Euclidean plane plus higher dimensions, doesn't it? But is there an equivalent wand for X^3 + Y^3 ... ? I only ask out of semi-ignorance about such things. It's so damned irritating not to be able to know everything about everything :-)John Morgannoreply@blogger.comtag:blogger.com,1999:blog-15868947.post-50556689824294166052009-07-03T06:22:36.912-07:002009-07-03T06:22:36.912-07:00Careful, John. Trisection of angles is tangential...Careful, John. Trisection of angles is tangentially related to solvability of the general quintic, but not quite how you're thinking. Trisection is all a matter of which tools you're allowed to use.<br /><br />You can't trisect angles <i>with ruler and compass</i>. If you have a magic wand that computes cube roots, however, you can trisect angles all day long. Solving the general cubic also requires being able to take cube roots. But solving the general quintic is impossible <i>no matter what roots you're allowed to take</i>.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-15868947.post-13256786756714913742009-07-03T02:13:17.146-07:002009-07-03T02:13:17.146-07:00I happened on this post rather late in its history...I happened on this post rather late in its history. It's 3 Jul 2009 as I write, but the problems outlined still remain.<br /><br />In teaching simple algebra, I still consider that numerical methods (ie doing arithmetic on the problem) is the natural bridge to doing full analysis. The trick is to introduce the two in parallel. For quadratic equations, start by solving one of the factorisables, say x² + x = 12 which easily yields three by inspection. Only the brightest students will see that -4 is also a result, and then the need for analysis becomes apparent to them. One can continue by constructing the original quadratic multiplying out the binomials and arrive at the notion of factorising quadratics to solve them.<br /><br />Numerical helps when you have analytically intractables like 2^x + ln x - sin x = 5 ( I am prepared to be counted wrong about this one as I don't know for sure if it can't be rendered down algebraically) and makes for an interesting intermission when students get bored by routine. Give them simple order 5 equations to play with and tell them how no amount of algebra will solve the equation - unless you happen to believe, like some do, that you can trisect an angle on the Euclidean plane :-(John Morgannoreply@blogger.comtag:blogger.com,1999:blog-15868947.post-36042090682051505032007-06-17T11:45:00.000-07:002007-06-17T11:45:00.000-07:00Feynman missed that a small number of students mak...Feynman missed that a small number of students make the intuitive leap from "just sets of rules" to fundamental concepts in which they can derive and prove their own rules. As the anonymous they said, "You can leave a horse to water...".<BR/><BR/>Sometimes we try to beat that intuition into our kids with hundreds of rote arithmetic problems in grammar school, but that just turns math into a drudgery for the brighter ones. By the time they get to college they've been permanently damaged.<BR/><BR/>I detested grade school because of the immense amount of repitition. Why can't the United States teach algebra in grade school, as in other countries, rather than waiting to put some frustrated college teacher through the wringer?Reasonably Crankyhttps://www.blogger.com/profile/03831656172301489765noreply@blogger.comtag:blogger.com,1999:blog-15868947.post-1148751844928515912006-05-27T10:44:00.000-07:002006-05-27T10:44:00.000-07:00I find this a fascinating post because I'm beyond ...I find this a fascinating post because I'm beyond terrible at math, but I did very well at algebra because it was logical and in steps. I find it interesting that what everyone else seems to denigrate about algebra is the one thing that made it make sense to me.wegrithttps://www.blogger.com/profile/05026547003132164134noreply@blogger.com