Where is balance? -- Re: Academic priorities
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From: Kevin G. Rhoads on 29 Apr 2005 16:12 >I'd expect smoothness to be understood in college, maybe high school >for the truly precious. I think its hard to understand what C infinity >really is without a good course in math. OK smoothness, continuity and analyticity are all nice. BUT when do you start teaching people that functions with even C0 are the *rare* exception in any function space and what some of the implications are of that? On the other hand, one can get so lost in drawing pretty pictures of fractals and computing iterative maps as to loose sight of the fact that while amusing diversions these things also have applicability to real world problems and computation. And even if one balances these two *theoretical* idea-sets, somewhen one must get the students to understand that while the coastline of England is fractal, that considering it at scales of 1cm or lower is not merely unnecessary precision, but, in fact, rather without meaning. The concepts without both the tools of computation and real example problems to provide context are sterile. The computational tools without concepts end up being rote memorization and without real example problems are typically without interest to most students. And real world example problems without tools of computation are unsolvable, and without concepts the solutions, and often the problems as well, are incomprehensible to most students. Just as the oft-perceived tension between analytic tools and numeric tools is artifactual -- really good design and analysis requires availability of both -- so is any perception of tension between concept, (good) examples and tools/methods in teaching an illusion -- without all three most students are operating under good, old "garbage in, garbage out".
From: jmfbahciv on 30 Apr 2005 07:23 In article <3dhej6F6tatsaU1(a)individual.net>, Steve Richfie1d <Steve(a)NOSPAM.smart-life.net> wrote: <snip> > ... If only I could have found more math-intensive >real-world problems for them to work on. You have got to be kidding. You can't help but trip over tons of real-world problems. It sounds like you have done your children a great disservice. /BAH Subtract a hundred and four for e-mail.
From: Bill Leary on 30 Apr 2005 11:35 <jmfbahciv(a)aol.com> wrote in message news:e6mdnTdKkqujGu7fRVn-1Q(a)rcn.net... > In article <3dhej6F6tatsaU1(a)individual.net>, > Steve Richfie1d <Steve(a)NOSPAM.smart-life.net> wrote: > <snip> > > > ... If only I could have found more math-intensive > >real-world problems for them to work on. > > You have got to be kidding. You can't help but trip over > tons of real-world problems. I don't know about that. How many real-world problems will a teenager encounter that require anything beyond basic math skills? > It sounds like you have > done your children a great disservice. Maybe not. Upon graduation from "normal" high school, none of my kids could do anything past basic math. And only one of them reads well. When my eldest son had to learn some higher math (and not all THAT high) for an electronics course he took related to his job he had awfully rough going of it. I'm also not sure how "unique" this is. I've met a several young people over the last ten or so years who were taught by what sounds like pretty much this same method. They were a bit short on theory in a places, but they were very long on "I'll find out," and then actually being able to do it, do it quickly and then apply it. - Bill
From: Steve Richfie1d on 30 Apr 2005 11:50 Barb, >>... If only I could have found more math-intensive >>real-world problems for them to work on. > You have got to be kidding. You can't help but trip over > tons of real-world problems. It sounds like you have > done your children a great disservice. Suggestions? Remember, while I had plenty of influence, the kids made up their own minds what to work on. What in YOUR neighborhood has people's lives depending on the analytical solution to a calculus problem that can't simply be solved with sufficient accuracy with a trivial numerical approximation? I was at one time the top freshman calculus student at the University of Washington after getting a perfect 800 on the College Entrance Examination Board's Advanced Math test. In the following ~40 years, the *ONLY* practical use I have made for non-numerical calculus methods is a couple of times for computing optimal methods, e.g. how to divide up the bins in a complex sort to make it run as fast as possible by differentiating an equation for the total effort, setting the first derivative to zero, and solving for the minimum/maximum conditions. I have not recovered nor will I EVER expect to recover the time I put in learning all that stuff, as I could have even done these couple of tasks numerically at the cost of another couple of hours of work. The only "disservice" here is in not subjecting my kids to the same useless education as I received. Of course, if they find themselves needing this, they will either pick it up for themselves, or find someone to solve the equations for them. However, this has not been "lost" as the kids have learned other things that you are doubtless clueless about, when they could have been learning calculus, e.g. they have completely cured some "incurable" medical conditions in people they know after major medical institutions have given up on them. Once we were living in an area with a large homeless population, so their "lab" became finding ways to get some of these people jobs and homes, not through broad social change, but one-on-one by understanding them and their situations. Eventually a "$200 plan" emerged, where for ~$200 in direct expenses a typical homeless family would wind up in a home they owned and working at a job they could stand. We did this for several families having kids who were friends with my kids. Certainly this skill is worth more than calculus?! It really sounds (to me) like you *COMPLETELY* missed the central point here. A motivated top-down education produces POWERFUL people with unique skills, in trade for a variety of essentially useless knowledge that is routinely used to put kids to sleep in school, which can be picked up as (if ever) needed. Top-down methods don't teach them any MORE, but it does teach them more USEFUL skills, the very existence of which befuddles may who learned calculus instead. No wonder so many kids are now on Ritalin given the boring and useless education our schools now dispense. Steve Richfie1d
From: Morten Reistad on 30 Apr 2005 19:01
In article <3dhnr3F6ufpn6U1(a)individual.net>, Steve Richfie1d <Steve(a)NOSPAM.smart-life.net> wrote: >Barb, > >>>... If only I could have found more math-intensive >>>real-world problems for them to work on. > >> You have got to be kidding. You can't help but trip over >> tons of real-world problems. It sounds like you have >> done your children a great disservice. > >Suggestions? > >Remember, while I had plenty of influence, the kids made up their own >minds what to work on. What in YOUR neighborhood has people's lives >depending on the analytical solution to a calculus problem that can't >simply be solved with sufficient accuracy with a trivial numerical >approximation? > >I was at one time the top freshman calculus student at the University of >Washington after getting a perfect 800 on the College Entrance >Examination Board's Advanced Math test. In the following ~40 years, the >*ONLY* practical use I have made for non-numerical calculus methods is a >couple of times for computing optimal methods, e.g. how to divide up the >bins in a complex sort to make it run as fast as possible by >differentiating an equation for the total effort, setting the first >derivative to zero, and solving for the minimum/maximum conditions. I >have not recovered nor will I EVER expect to recover the time I put in >learning all that stuff, as I could have even done these couple of tasks >numerically at the cost of another couple of hours of work. > >The only "disservice" here is in not subjecting my kids to the same >useless education as I received. Of course, if they find themselves >needing this, they will either pick it up for themselves, or find >someone to solve the equations for them. I have an experiece quite contrary to this. I do not have such a strong theoretical background in formal math, and have had to read up on subjects in around 20 situations. Examples like making stock exchange indices manipulation-proof when issuing derivatives, analyzing scaling properties in networks, optimizing a satellite broadcast system's power/bandwidth footprint. Even interpreting sail polars vs wind patterns in a sailing regatta is a problem you need a mathematical understanding of; not just to trust the computer. Many of these are problems that are completely intractable if faced head-on computationally, either because they truly are computationally unbounded, or because you aren't even close to having all the data. It is said about economists that when they cannot handle the formula they derivate it and try again. Repeat until success. This has more than a sliver of truth in it; you often need to reformulate the problem in other numerical spaces. >However, this has not been "lost" as the kids have learned other things >that you are doubtless clueless about, when they could have been >learning calculus, e.g. they have completely cured some "incurable" >medical conditions in people they know after major medical institutions >have given up on them. > >Once we were living in an area with a large homeless population, so >their "lab" became finding ways to get some of these people jobs and >homes, not through broad social change, but one-on-one by understanding >them and their situations. Eventually a "$200 plan" emerged, where for >~$200 in direct expenses a typical homeless family would wind up in a >home they owned and working at a job they could stand. We did this for >several families having kids who were friends with my kids. Certainly >this skill is worth more than calculus?! Handling "skill management" is valuable indeed. >It really sounds (to me) like you *COMPLETELY* missed the central point >here. A motivated top-down education produces POWERFUL people with >unique skills, in trade for a variety of essentially useless knowledge >that is routinely used to put kids to sleep in school, which can be >picked up as (if ever) needed. Top-down methods don't teach them any >MORE, but it does teach them more USEFUL skills, the very existence of >which befuddles may who learned calculus instead. No wonder so many kids >are now on Ritalin given the boring and useless education our schools >now dispense. I get that. I just say that you shouldn't dismiss overdoing basic skills in languages and math at early stages; some of these are best studied at a young age due to the way the human brain functions. But the motivation has to be there. I am doing some show-and-tell stories, and present some of the problems above as challenges. Exchange history has literally thousands of such math challenges that directly affect a large number of people's lives immediatly. I took a lot of school children on Stock Exchange tours, and always gave them a lot of challenges in the figures that are everywhere in that location. You wouldn't stand a chance in the derivatives market without a very clear knowledge of some basic statistics. Kids need that kind of views into the adult business world to get an understanding of why they are doing the math and languages. -- mrr |