don't think anyone is arguing to try to teach the WHOLE domain of one field that way. We're talking about the _basics_.
What are the basics? Addition? Subtraction? Multiplication? Can we teach those by rote? (and at some point multiplication tables need to be memorized) Math is a layering of abstraction upon abstraction so which layer gets to be called 'basic'. The layer where you don't even have the tools to communicate concepts effectively? Or maybe Algebra gets to be 'the basics' that we try to help kids 'discover'. What about properties of operations (associative/commutative) how do you get kids to discover all of those in time to actually finish an algebra sequence? Or do we need to try and get everyone to discover what the abstract concept of zero means? How do you pick what the basics are, and how do you handle the general symbolic language and communication skills to get to that point? It seems a fairly arbitrary line to be drawn. Everyone learns differently, and learning methods change as language grasp and symbol usage develop.
What is taught for today's Math is a total joke - kids aren't taught to think, just to mindless follow some "arcane formula". e.g. "Two weeks of content are stretched to semester length by masturbatory definitional runarounds."
This isn't a concept/discovery problem in the context of the current discussion, it's a pacing problem and a consequence of putting vastly different types of learners all in the same room and moving at the pace of the lowest common denominator. This is a common complaint of mine but doesn't seem to fit into the current question 'teaching discovery'. Regardless of what teaching method you're using, go too fast and you lose the people that need extra attention, too slow and you lose the ones that don't.
Maybe some bits can and should be taught that way, but the body of knowledge in mathematics is too large to try and teach any significant portion that way. It's taken humanity many lifetimes to discover what we know, one person doesn't have that long. Rediscovering something can be really cool on a one off basis, but there isn't time to do that for the entire body of knowledge nor should we try. Discovery is about the need to know and understand and the drive to sate that need. It's hard to teach those qualities when someone wants everything laid out for them.
As for the quadratic equation, well applications for that are as numerous as applications of algebra. I would give examples but as you've stated your willful ignorance already I suspect that examples wouldn't have helped you in school either. I sense a lot of finger pointing in your tirade. I'm curious why you feel that way when so many others have gone on from the same educational systems (or even foreign ones that are even more hard-line/drill based) to figure things out and make great discoveries.
What General Bradley did in WWII could hardly be described as "fun". The very word "amateur" means one who does it for the love of the task, while conversely "professional" means someone who does it for money. Logistics is a lot of hard work, and nobody does it without a full staff to assist. Indeed, logistics is well-known for its dryness and it is not something that anyone would do as recreation. It's essentially the same thing as keeping the shelves stocked in a grocery store.
I played Starcraft, and don't remember any supply rules, other than the unit limit. Maybe they added beans & bullets tracking to SC2, I'll never know. I know a few freaks who like logistics games, and they say honestly their favorite game is Microsoft Excel.
Right! Think of how interactive this could become! We could give the user controls to move the camera around, zoom in and out. Maybe even make decisions at key points to change the plot outcome! I think you're onto something here. I bet if we think real hard we might even be able to give the viewer complete control of a character in this interactive environment. And then they can be the hero!
In all seriousness though the line between video games and movies blurred a long time ago, there just isn't a big market for that in-between point of "movie with a little bit of control". Video games as a story telling platform have come a long way and get better every year.
I feel compelled to point out that the film Surf's Up was largely created that way. The scene's were created, the 'actors' animated, and then using a special camera rig they 'filmed' the movie. This allowed rapid retakes, trying funky angles, filming through windows, etc. The DVD commentary talked about it, and mentioned a number of shots in the film that were only done the way they were because they were using this style of filming. For the documentary style used in Surf's Up this worked very well. It did require them to create more fully realized sets than they otherwise would because they didn't know what would finally get captured on film which resulted in stuff that was created and animated that never made it into the film.
There is no need to know anything about the hardware.
Modern programming is about algorithms and interfaces. Knowing how to simulate 4GB memory space with only 8 bit registers is not important.
Sure, right up until you put your entire fancy data processing algorithm inside the ISR.
True story. It took several weeks of us electrical engineers arguing with the software folks about why they couldn't keep up with the data flow and how doing everything in the ISR was a bad idea. And these weren't just any software guys they were supposed to be embedded software engineers writing complex DSP code.
I was floored at the time, but seeing statements like this helps me understand.
Just to point out. A full frame sensor vs. a smaller sensor 'zoomed' is a false argument. You CAN get that same zoom factor by cropping down the larger sensor. Larger sensors tend to have larger pixel elements and also more pixels (depending on the sensor). Larger pixel elements collect more light and have less noise over longer exposures. This is why photos from a DSLR with a larger sensor (full fram or not) look better than the small super dense CCDs in point and shoot cameras, particularly when looking at uniform backgrounds like a sky where noise becomes readily apparent.
Although I've never done, it seems to me that for astrophotograpy, digital is particularly useful because you can composite many shots together much more easily. Both for gathering more light in a single frame and creating mosaic's of many different frames.
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don't think anyone is arguing to try to teach the WHOLE domain of one field that way. We're talking about the _basics_.
What are the basics? Addition? Subtraction? Multiplication? Can we teach those by rote? (and at some point multiplication tables need to be memorized) Math is a layering of abstraction upon abstraction so which layer gets to be called 'basic'. The layer where you don't even have the tools to communicate concepts effectively? Or maybe Algebra gets to be 'the basics' that we try to help kids 'discover'. What about properties of operations (associative/commutative) how do you get kids to discover all of those in time to actually finish an algebra sequence? Or do we need to try and get everyone to discover what the abstract concept of zero means? How do you pick what the basics are, and how do you handle the general symbolic language and communication skills to get to that point? It seems a fairly arbitrary line to be drawn. Everyone learns differently, and learning methods change as language grasp and symbol usage develop.
The whole 'teach math discovery' idea strikes me as one of these: http://developers.slashdot.org/story/10/12/06/0124241/I-Just-Need-a-Programmer
Just another idea that is less than useless without an implementation, and the implementation details are friggin' hard with tens of thousands of caveats, corner cases, exceptions and risk. see: http://en.wikipedia.org/wiki/New_Math where math teaching reform was tried and failed.
What is taught for today's Math is a total joke - kids aren't taught to think, just to mindless follow some "arcane formula". e.g. "Two weeks of content are stretched to semester length by masturbatory definitional runarounds."
This isn't a concept/discovery problem in the context of the current discussion, it's a pacing problem and a consequence of putting vastly different types of learners all in the same room and moving at the pace of the lowest common denominator. This is a common complaint of mine but doesn't seem to fit into the current question 'teaching discovery'. Regardless of what teaching method you're using, go too fast and you lose the people that need extra attention, too slow and you lose the ones that don't.
EVERYONE should read these two papers.
* A Mathematician's Lament http://www.maa.org/devlin/LockhartsLament.pdf
* The Underground History of American Education http://www.johntaylorgatto.com/chapters/index.htm
I'll take a look at those papers later as they appear fairly long and involved.
And this is how MATH should be taught.
Maybe some bits can and should be taught that way, but the body of knowledge in mathematics is too large to try and teach any significant portion that way. It's taken humanity many lifetimes to discover what we know, one person doesn't have that long. Rediscovering something can be really cool on a one off basis, but there isn't time to do that for the entire body of knowledge nor should we try. Discovery is about the need to know and understand and the drive to sate that need. It's hard to teach those qualities when someone wants everything laid out for them.
As for the quadratic equation, well applications for that are as numerous as applications of algebra. I would give examples but as you've stated your willful ignorance already I suspect that examples wouldn't have helped you in school either. I sense a lot of finger pointing in your tirade. I'm curious why you feel that way when so many others have gone on from the same educational systems (or even foreign ones that are even more hard-line/drill based) to figure things out and make great discoveries.
What General Bradley did in WWII could hardly be described as "fun". The very word "amateur" means one who does it for the love of the task, while conversely "professional" means someone who does it for money. Logistics is a lot of hard work, and nobody does it without a full staff to assist. Indeed, logistics is well-known for its dryness and it is not something that anyone would do as recreation. It's essentially the same thing as keeping the shelves stocked in a grocery store.
I played Starcraft, and don't remember any supply rules, other than the unit limit. Maybe they added beans & bullets tracking to SC2, I'll never know. I know a few freaks who like logistics games, and they say honestly their favorite game is Microsoft Excel.
Are those freaks EvE-online players?
Right! Think of how interactive this could become! We could give the user controls to move the camera around, zoom in and out. Maybe even make decisions at key points to change the plot outcome! I think you're onto something here. I bet if we think real hard we might even be able to give the viewer complete control of a character in this interactive environment. And then they can be the hero!
In all seriousness though the line between video games and movies blurred a long time ago, there just isn't a big market for that in-between point of "movie with a little bit of control". Video games as a story telling platform have come a long way and get better every year.
I feel compelled to point out that the film Surf's Up was largely created that way. The scene's were created, the 'actors' animated, and then using a special camera rig they 'filmed' the movie. This allowed rapid retakes, trying funky angles, filming through windows, etc. The DVD commentary talked about it, and mentioned a number of shots in the film that were only done the way they were because they were using this style of filming. For the documentary style used in Surf's Up this worked very well. It did require them to create more fully realized sets than they otherwise would because they didn't know what would finally get captured on film which resulted in stuff that was created and animated that never made it into the film.
There is no need to know anything about the hardware.
Modern programming is about algorithms and interfaces. Knowing how to simulate 4GB memory space with only 8 bit registers is not important.
Sure, right up until you put your entire fancy data processing algorithm inside the ISR. True story. It took several weeks of us electrical engineers arguing with the software folks about why they couldn't keep up with the data flow and how doing everything in the ISR was a bad idea. And these weren't just any software guys they were supposed to be embedded software engineers writing complex DSP code. I was floored at the time, but seeing statements like this helps me understand.
Just to point out. A full frame sensor vs. a smaller sensor 'zoomed' is a false argument. You CAN get that same zoom factor by cropping down the larger sensor. Larger sensors tend to have larger pixel elements and also more pixels (depending on the sensor). Larger pixel elements collect more light and have less noise over longer exposures. This is why photos from a DSLR with a larger sensor (full fram or not) look better than the small super dense CCDs in point and shoot cameras, particularly when looking at uniform backgrounds like a sky where noise becomes readily apparent.
Although I've never done, it seems to me that for astrophotograpy, digital is particularly useful because you can composite many shots together much more easily. Both for gathering more light in a single frame and creating mosaic's of many different frames.
Just some thoughts from an amateur photographer.
- Asmodae