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david cowan, math, algebra, geometry, trig, precalculus, calculus

 

The entire paradym of education is changing.
I just watched a MIT professor give a class on differential equations. He was great. It was just like being there.
Why would anyone pay $300,000 to go to MIT when they can pick up the knowledge for free. Well, you might say, the diploma. That's what lets everyone know that you have the knowledge. Or, at least, it used to be that way.
If the knowledge is truly the most important thing, then the existing paradym will change. We will soon realize that the knowledge, without the debt, will be the way to go. Those who want to hire the most knowledgable will always find them, and it won't necessarily be from the diploma crowd. 
Knowledge, not diplomas, is the future.

The entire paradigm of education is changing.

I just watched a MIT professor give a class on differential equations. It was just like being there. All lectures, on all courses, are available for free online.

Why would anyone pay $300,000 to go to MIT when they can pick up the knowledge for free. Well, you might say, the diploma. That's what lets everyone know that you have the knowledge. Or, at least, that is the way it was in the 20th century.

If the knowledge is truly the most important thing, then the existing paradigm will change. We will soon realize that the knowledge is the most important thing. 

Knowledge is the future.

 


 

In South Korea, the world's #1 math performer, 25% of 
students have private tutors.

In South Korea, the world's #1 math performer, 25% of  students have private tutors.

One of the biggest differences between the way the U.S. teaches math and the way Asian countries teach math is we tend to spend an hour having students answer 20 questions. Asian countries spend that hour answering 4 questions, but from 5 different angles.



 

The evolution of math over the last 50 years?
Understanding of math never moves backwards, only forwards. In that respect, it is unique amongst the sciences. Once a math theorem is proven, it is never disproven. 
Not so with science. Newton's laws of physics stood for 250 years until Einstein came along and proved them incorrect. So scientific theory changed and so it will go.
Math is different. The Pythagorean theorem will be true a million years from now. It will never be disproven.
We will move forward, sometimes slowly, sometimes more rapidly, but we will never move backwards in math.
The most significant thing that changes with time is the way we teach or learn math. Technology has made math much easier to learn. When I went to college, there were no calculators, only slide rules. It wasn't easy to see graphical depictions of functions. Without the ability to see functions graphically, it was much harder to learn calculus, 3-D vectors, etc. Now 3-D vectors are taught to seniors in high school as opposed to graduate school in college back in 1970. That's largely due to the technology, not any fundamental changes in math. Also, the internet has brought us online teaching and tutoring, with online classes from MIT and Harvard professors, not to mention Khan Academy's 40 thousand educational videos.
While the upsides of technology are huge, there are some downsides to the tech revolution. Virtually everyone that grew up in the post-calculator generation, (post 1970), got a different elementary math education than those pre-1970. The emphasis switched from arithmetic skills to more complex math concepts, but the result today, is that there is a severe shortage of arithmetic skills in today's high school math students. This almost certainly is due to the ubiquitousness of the calculator, and it allows many students to get to high school without arithmetic skills.
As long as we have calculators available, this might seem acceptable, but, I believe it is the #1 fundamental cause of poor performance in algebra, geometry, trig, and pre-calculus.

The evolution of math over the last 50 years?

 

Understanding of math never moves backwards, only forwards. In that respect, it is unique amongst the sciences. Once a math theorem is proven, it is never disproven. 

Not so with science. Newton's laws of physics stood for 250 years until Einstein came along and proved them incorrect. So scientific theory changed and so it will go.

Math is different. The Pythagorean theorem will be true a million years from now. It will never be disproven.

We will move forward, sometimes slowly, sometimes more rapidly, but we will never move backwards in math.

The most significant thing that changes with time is the way we teach or learn math. Technology has made math much easier to learn. When I went to college, there were no calculators, only slide rules. It wasn't easy to see graphical depictions of functions. Without the ability to see functions graphically, it was much harder to learn calculus, 3-D vectors, etc. Now 3-D vectors are taught to seniors in high school as opposed to graduate school in college back in 1970. That's largely due to the technology, not any fundamental changes in math. Also, the internet has brought us online teaching and tutoring, with online classes from MIT and Harvard professors, not to mention Khan Academy's 40 thousand educational videos.

While the upsides of technology are huge, there are some downsides to the tech revolution. Virtually everyone that grew up in the post-calculator generation, (post 1970), got a different elementary math education than those pre-1970. The emphasis switched from arithmetic skills to more complex math concepts, but the result today, is that there is a severe shortage of arithmetic skills in today's high school math students. This almost certainly is due to the ubiquitousness of the calculator, and it allows many students to get to high school without arithmetic skills.

As long as we have calculators available, this might seem acceptable, but, I believe it is the #1 fundamental cause of poor performance in algebra, geometry, trig, and pre-calculus.

 

Is Math Discovered or Invented?

Or a little bit of both. That is the question. Their are arguements for both. Ultimately, the reality of our universe will provide the answer.

We continually find that math predicts future discoveries about the universe. The Higg's Boson was such. Higgs used mathematics to predict its existence 50 years before its discovery.

At the same time, math is not able to define or predict really complex systems, like long range weather forecasting, quantum physics, or even the stock market.

If you were to assume that life is just an advanced, elaborate video game, where everything is described mathematically with code, then it is a discovery. Everything would be math, and humankind would be constantly just discovering different pieces of it.

But could it be perhaps just a human invention, essentially only inside our individual brains.

Most of the evidence points towards math defining, and perhaps being everything to or about our universe.


To me, that makes it a discovery.