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I haven't exactly been as amazed by the Khan Academy as everyone else has. After watching a few videos, I decided that we need to focus more on the material tha

I haven't exactly been as amazed by the Khan Academy as everyone else has. After watching a few videos, I decided that we need to focus more on the material that we are teaching - rather than the methods we go about doing it. They will obviously go hand by hand, but in my opinion the material that we are teaching is our biggest weakness and therefore needs the most reform. A video that everyone should take the time to watch is Conrad Wolfram's TED talk on teaching students with computers (http://www.ted.com/talks/conrad_wolfram_teaching_kids_real_math_with_computers.html http://www.ted.com/talks/conrad_wolfram_teaching_kids_real_m...). While we may use computers now to enhance the learning methods, we haven't taken advantage of the content/knowledge computers have and can supply for us. The majority of what is taught in grades K-12 in science and math could be done by a computer in almost no time with increased accuracy. Why wouldn't we take advantage of that? The gist of Wolfram's talk is that we are teaching students mechanics (which computers can do more efficiently) when we should be teaching them the higher orders of thinking and problem solving. Against popular belief, concepts can be understood without learning the nitty-gritty mechanics. One could solve a quadratic word problem without solving the quadratic equation themselves, and still understand the problem just as well. Think about it: do you need to learn how the engine of a car works before you can drive it? While all of this is not entirely relevant to the article, I humbly believe that the future of education should be and will be (if everything goes right) orders of magnitudes higher in efficiency as students will spend more time learning the right material. I'm not sure how much the Khan Academy will play a role in that...

vidar 16y ago
Thats a tricky thing to do though, where do you draw the line?
- solipsist 16y ago
  That's a good question. We need to keep in mind that computers are never going to go backwards in innovation. Once a computer is capable of some calculation, students will never need to have to do it by themselves again. However, they will still need to know how to interact with that calculation and understand the variables/relationships in order to set up the calculation. Where do you draw the line? Instead of just one line, you have to draw two lines. One would be on the more elementary end of the spectrum. Students have to learn the rudimentary basic calculations (or memorization) in order to function well in everyday life and higher levels of education. Simple addition, subtract, division, multiplication, estimation, understanding of relationships would be necessary. As much as I adore algebra, it could be left to the computer. Therefore the first line would be drawn at a point where the material does not show up in everyday scenarios. The next line would be drawn at the opposite end of the spectrum where humans need to come in and do the higher orders of thinking. Given the right software, students would be able to visualize and understand the basics of the calculations and their relationships without actually calculating them. They would therefore be able to use computers in more applied problems - ones that they would be confronted in their careers later on in life. A few experiments could be conducted to find exactly where this line would be drawn. It would be at the point where humans do the smallest amount of calculation work while still having a solid understanding. Computers may be able to work past this line, but they would end up assisting the students to the point where the students lose the understanding of the concepts. If the US and other countries truly put the time, money, and effort into this reform, I'm sure they could be quite successful. However, another tricky thing to think about would be transitioning into this new and reformed type of education...
  - dagw 16y ago
    Once a computer is capable of some calculation, students will never need to have to do it by themselves again However it is important that they know enough to check if the answer they get is at least reasonable and probably correct. So even if they can't solve the problem exactly they really should know how to show that the answer must be between 1/sqrt(2) and 1, for example, so that they know they made a mistake when the computer tells them the answer is 0.3475.
    - solipsist 16y ago
      Yes, that goes along with understanding the basic relationships and estimation techniques. The exact calculations can be left to the computer, and the humans can check the solutions and determine what went wrong if the answers don't seem to be what they had expected.
psykotic 16y ago
That too much emphasis is put on manual computation is obvious. But taking the opposite tack is an overcorrection. Computing and solving by hand is a font of endless insight in mathematics at any level and should not be left to computers. Take an elementary matter such as convergence of infinite series. If you've never hand-computed a truncated power series for e^x term by term and noted its rapid convergence rate for positive x, I dare say your understanding of basic calculus is impoverished. Once you've seen the convergence first hand for particular values of x, the proof of its general convergence immediately suggests itself. But if you've only seen the abstract proof, you would probably never have realized that the e^x power series has extremely initial poor convergence for negative x. (Try it for x = -10 to see what I mean. Poor convergence and massive cancellation. That's why numerical analysts always evaluate e^x as 1/e^(-x) for negative x.)
- solipsist 16y ago
  If you were going to major in math, then you may need to go back and learn all the basic calculations in order to build on them. But for anyone else (even scientists), the solution I mentioned above should be enough. Your example of e^x is not closely enough tied to a real world application; therefore it's hard for me to work with it.
  - psykotic 16y ago
    > If you were going to major in math, then you may need to go back and learn all the basic calculations in order to build on them. The way it actually happens is that once you start majoring in mathematics, you will rarely see any computations. It's entirely possible to take a linear algebra course for mathematicians without ever row-reducing a non-trivial matrix by hand. Back to teaching kids. If you take a constructivist approach to learning then computing by hand simply cannot be avoided. You'll never have a student notice on their own that, for example, you can quickly multiply x by 9 by subtracting x from 10x unless they tinker with numbers directly. You come across patterns like that by doing the tedious work the hard way and looking for short-cuts.
    - 16y ago
nvarsj 16y ago
"do you need to learn how the engine of a car works before you can drive it?" No, but you'd be a much better driver if you did. Understanding the basics of a science, especially math, is extremely important to fully grasp the higher level topics. I don't believe one can conceptually understand a topic without understanding the fundamentals that it builds upon. Unfortunately I see philosophies as you describe in my local school system (Chicago Public Schools), especially in poverty stricken neighborhoods. High school kids are given scientific calculators and taught algebra/pre-calc when they can't even add or multiply single digit numbers. Similarly, there are pushes to move more "computer learning" into grade school, despite extremely poor reading and math proficiency. I can draw parallels in my job as well - it's remarkable how poor most programmers' basic math skills are. Someone who can do the chain rule is regarded as a genius. With only a strong grasp of fundamentals, I feel that many engineers could be substantially more successful.
- solipsist 16y ago
  > No, but you'd be a much better driver if you did [learn how a car engine works] Yes, but one simple computer animation could give you the same understanding of the engine as if you had built the engine yourself. Computers can give you an understanding of calculations in addition to doing the work. > I don't believe one can conceptually understand a topic without understanding the fundamentals that it builds upon. This is probably the biggest argument against using computers to calculate for you. However, if you haven't already, watch Wolfram's video that I linked to above as he clearly explains how this belief is not necessarily true. The very rudimentary basics will need to be learned by calculating, but anything above the fundamentals can be understood without the calculations. > Similarly, there are pushes to move more "computer learning" into grade school, despite extremely poor reading and math proficiency. I think this computer learning is different than the learning I'm talking about. Schools currently use computers as a medium to enhance the current education system. They use them to make calculating more interactive and fun. I think that computers should actually do the calculations, rather than "tricking" the students into doing the work.
  - gmartres 16y ago
palish 16y ago
K-12 isn't generally about learning. It is a daycare for children and young adults. (I am 23 now. I left school when I was 17 to pursue a career in the gamedev industry. Daycare was not the place for me, and I haven't regretted dropping out.)
- solipsist 16y ago
  I know. Talk about a waste of time. Turns out to be 70,000 hours that each person spends in K-12 school. Add any outside school work and that could be closer 90,000 hours.
  - petercooper 16y ago
    I thought your number was quite mindblowing but then I worked out that 70000 hours is about 8 years if you were there constantly, 24/7.. 90000 waking hours is over 15 years. I worked it out at around 15-18k hours (depending on where you live and variations in school day length).
    - solipsist 16y ago
      You're right, I was off by quite a bit :) 15k hours seems more like it.
    - palish 16y ago
      In a way, 70,000 hours is a lot closer to the true number. (365 days) * (24 hours) * (12 years) = ~105k hours, or 70k waking hours It isn't until fifth or sixth grade that you can really start finding a passion. I'd say I managed to spend 5k to 10k hours on learning about technology and programming. I had to completely ignore high school. I literally slept through it. Teachers gave up trying to get me to try; I was also a social outcast.
lutorm 16y ago
Against popular belief, concepts can be understood without learning the nitty-gritty mechanics. I disagree. The problem is that you don't know that you understand a concept until you actually can apply your conceptual understanding to accomplish something. The world is full of people who "understand" quantum mechanics or relativity in terms of half-baked analogies and are unable to apply those concepts to anything but pseudo-scientific discussions. Real conceptual understanding comes form having worked with the subject area enough that you have built your own conceptual framework. You can't really just teach someone a conceptual framework, because the words you use to describe your framework will likely be interpreted in a completely different way by someone else.
cicero 16y ago
In the book Why Don't Students Like School ( http://www.amazon.com/Why-Dont-Students-Like-School/dp/047059196X/ http://www.amazon.com/Why-Dont-Students-Like-School/dp/04705...) Daniel Willingham argues that people need to have a good vocabulary of mechanics, basic concepts, and skills in order to be able to move on to higher order knowledge. Children should learn how to calculate not necessarily to do calculations instead of a computer, but to help their brains develop. Besides, until kids hit puberty, their brains aren't ready for higher orders of thinking. What they are good at is memorization and mechanics. The more mechanics they learn and the more they memorize as little kids, the better equipped they will be for higher order thinking as teens and adults.
- kevinchen 16y ago
  However, we are still teaching and emphasizing arithmetic and memorization in high school — even in the advanced classes. That's the problem, really.
maxbRuns 16y ago
Computer-use should not replace the nitty-gritty mechanics; it should change the way students learn the nitty-gritty mechanics. Instead of a homework assignment like this: 1. compute this integral. 2. compute this integral. 3. compute this integral. 4. compute this integral. 5. compute this integral. Students should be getting homework assignments like this: 1. compute this integral. 2. write a procedure to compute any integral.
Toledonor 16y ago
One reason educational material is poor is that it's inevitably written by someone who finds it hard to imagine what it's like not to understand it. This is why all such material should come with commentaries by thoughtful students which explain it in different ways and link to additional resources. And with commentaries on the commentaries, etc. (We all have different misconceptions.) There also needs to be a mechanism for rapidly correcting errors. Not every few years but every few days. (Text books are full of errors.) All of which suggests that we need more educational material which is online, open source and wikified.

vidar 16y ago

Thats a tricky thing to do though, where do you draw the line?

solipsist 16y ago
That's a good question. We need to keep in mind that computers are never going to go backwards in innovation. Once a computer is capable of some calculation, students will never need to have to do it by themselves again. However, they will still need to know how to interact with that calculation and understand the variables/relationships in order to set up the calculation. Where do you draw the line? Instead of just one line, you have to draw two lines. One would be on the more elementary end of the spectrum. Students have to learn the rudimentary basic calculations (or memorization) in order to function well in everyday life and higher levels of education. Simple addition, subtract, division, multiplication, estimation, understanding of relationships would be necessary. As much as I adore algebra, it could be left to the computer. Therefore the first line would be drawn at a point where the material does not show up in everyday scenarios. The next line would be drawn at the opposite end of the spectrum where humans need to come in and do the higher orders of thinking. Given the right software, students would be able to visualize and understand the basics of the calculations and their relationships without actually calculating them. They would therefore be able to use computers in more applied problems - ones that they would be confronted in their careers later on in life. A few experiments could be conducted to find exactly where this line would be drawn. It would be at the point where humans do the smallest amount of calculation work while still having a solid understanding. Computers may be able to work past this line, but they would end up assisting the students to the point where the students lose the understanding of the concepts. If the US and other countries truly put the time, money, and effort into this reform, I'm sure they could be quite successful. However, another tricky thing to think about would be transitioning into this new and reformed type of education...
- dagw 16y ago
  Once a computer is capable of some calculation, students will never need to have to do it by themselves again However it is important that they know enough to check if the answer they get is at least reasonable and probably correct. So even if they can't solve the problem exactly they really should know how to show that the answer must be between 1/sqrt(2) and 1, for example, so that they know they made a mistake when the computer tells them the answer is 0.3475.
  - solipsist 16y ago
    Yes, that goes along with understanding the basic relationships and estimation techniques. The exact calculations can be left to the computer, and the humans can check the solutions and determine what went wrong if the answers don't seem to be what they had expected.

psykotic 16y ago

That too much emphasis is put on manual computation is obvious. But taking the opposite tack is an overcorrection. Computing and solving by hand is a font of endless insight in mathematics at any level and should not be left to computers. Take an elementary matter such as convergence of infinite series. If you've never hand-computed a truncated power series for e^x term by term and noted its rapid convergence rate for positive x, I dare say your understanding of basic calculus is impoverished. Once you've seen the convergence first hand for particular values of x, the proof of its general convergence immediately suggests itself. But if you've only seen the abstract proof, you would probably never have realized that the e^x power series has extremely initial poor convergence for negative x. (Try it for x = -10 to see what I mean. Poor convergence and massive cancellation. That's why numerical analysts always evaluate e^x as 1/e^(-x) for negative x.)

solipsist 16y ago
If you were going to major in math, then you may need to go back and learn all the basic calculations in order to build on them. But for anyone else (even scientists), the solution I mentioned above should be enough. Your example of e^x is not closely enough tied to a real world application; therefore it's hard for me to work with it.
- psykotic 16y ago
  > If you were going to major in math, then you may need to go back and learn all the basic calculations in order to build on them. The way it actually happens is that once you start majoring in mathematics, you will rarely see any computations. It's entirely possible to take a linear algebra course for mathematicians without ever row-reducing a non-trivial matrix by hand. Back to teaching kids. If you take a constructivist approach to learning then computing by hand simply cannot be avoided. You'll never have a student notice on their own that, for example, you can quickly multiply x by 9 by subtracting x from 10x unless they tinker with numbers directly. You come across patterns like that by doing the tedious work the hard way and looking for short-cuts.
  - 16y ago

nvarsj 16y ago

"do you need to learn how the engine of a car works before you can drive it?" No, but you'd be a much better driver if you did. Understanding the basics of a science, especially math, is extremely important to fully grasp the higher level topics. I don't believe one can conceptually understand a topic without understanding the fundamentals that it builds upon. Unfortunately I see philosophies as you describe in my local school system (Chicago Public Schools), especially in poverty stricken neighborhoods. High school kids are given scientific calculators and taught algebra/pre-calc when they can't even add or multiply single digit numbers. Similarly, there are pushes to move more "computer learning" into grade school, despite extremely poor reading and math proficiency. I can draw parallels in my job as well - it's remarkable how poor most programmers' basic math skills are. Someone who can do the chain rule is regarded as a genius. With only a strong grasp of fundamentals, I feel that many engineers could be substantially more successful.

solipsist 16y ago
> No, but you'd be a much better driver if you did [learn how a car engine works] Yes, but one simple computer animation could give you the same understanding of the engine as if you had built the engine yourself. Computers can give you an understanding of calculations in addition to doing the work. > I don't believe one can conceptually understand a topic without understanding the fundamentals that it builds upon. This is probably the biggest argument against using computers to calculate for you. However, if you haven't already, watch Wolfram's video that I linked to above as he clearly explains how this belief is not necessarily true. The very rudimentary basics will need to be learned by calculating, but anything above the fundamentals can be understood without the calculations. > Similarly, there are pushes to move more "computer learning" into grade school, despite extremely poor reading and math proficiency. I think this computer learning is different than the learning I'm talking about. Schools currently use computers as a medium to enhance the current education system. They use them to make calculating more interactive and fun. I think that computers should actually do the calculations, rather than "tricking" the students into doing the work.
- gmartres 16y ago

palish 16y ago

K-12 isn't generally about learning. It is a daycare for children and young adults. (I am 23 now. I left school when I was 17 to pursue a career in the gamedev industry. Daycare was not the place for me, and I haven't regretted dropping out.)

solipsist 16y ago
I know. Talk about a waste of time. Turns out to be 70,000 hours that each person spends in K-12 school. Add any outside school work and that could be closer 90,000 hours.
- petercooper 16y ago
  I thought your number was quite mindblowing but then I worked out that 70000 hours is about 8 years if you were there constantly, 24/7.. 90000 waking hours is over 15 years. I worked it out at around 15-18k hours (depending on where you live and variations in school day length).
  - solipsist 16y ago
    You're right, I was off by quite a bit :) 15k hours seems more like it.
  - palish 16y ago
    In a way, 70,000 hours is a lot closer to the true number. (365 days) * (24 hours) * (12 years) = ~105k hours, or 70k waking hours It isn't until fifth or sixth grade that you can really start finding a passion. I'd say I managed to spend 5k to 10k hours on learning about technology and programming. I had to completely ignore high school. I literally slept through it. Teachers gave up trying to get me to try; I was also a social outcast.

lutorm 16y ago

Against popular belief, concepts can be understood without learning the nitty-gritty mechanics. I disagree. The problem is that you don't know that you understand a concept until you actually can apply your conceptual understanding to accomplish something. The world is full of people who "understand" quantum mechanics or relativity in terms of half-baked analogies and are unable to apply those concepts to anything but pseudo-scientific discussions. Real conceptual understanding comes form having worked with the subject area enough that you have built your own conceptual framework. You can't really just teach someone a conceptual framework, because the words you use to describe your framework will likely be interpreted in a completely different way by someone else.

cicero 16y ago

In the book Why Don't Students Like School ( http://www.amazon.com/Why-Dont-Students-Like-School/dp/047059196X/ http://www.amazon.com/Why-Dont-Students-Like-School/dp/04705...) Daniel Willingham argues that people need to have a good vocabulary of mechanics, basic concepts, and skills in order to be able to move on to higher order knowledge. Children should learn how to calculate not necessarily to do calculations instead of a computer, but to help their brains develop. Besides, until kids hit puberty, their brains aren't ready for higher orders of thinking. What they are good at is memorization and mechanics. The more mechanics they learn and the more they memorize as little kids, the better equipped they will be for higher order thinking as teens and adults.

kevinchen 16y ago
However, we are still teaching and emphasizing arithmetic and memorization in high school — even in the advanced classes. That's the problem, really.

maxbRuns 16y ago

Computer-use should not replace the nitty-gritty mechanics; it should change the way students learn the nitty-gritty mechanics. Instead of a homework assignment like this: 1. compute this integral. 2. compute this integral. 3. compute this integral. 4. compute this integral. 5. compute this integral. Students should be getting homework assignments like this: 1. compute this integral. 2. write a procedure to compute any integral.

Toledonor 16y ago

One reason educational material is poor is that it's inevitably written by someone who finds it hard to imagine what it's like not to understand it. This is why all such material should come with commentaries by thoughtful students which explain it in different ways and link to additional resources. And with commentaries on the commentaries, etc. (We all have different misconceptions.) There also needs to be a mechanism for rapidly correcting errors. Not every few years but every few days. (Text books are full of errors.) All of which suggests that we need more educational material which is online, open source and wikified.