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Thanks for writing. I like your approach. I actually have started from the very top level. Let me explain further as why I asked this question in the first place with examples.

Two days ago I (re)learned that (-1) x (-1) = (1). I read many resources explaining why that is true, but it just doesn't strike as 1 + 1 = 2.

To give one more example, just to make clear what I am asking about -- Today I learned about slope of the line, which is = change in y / change in x. Even that doesn't strike. I have already learned these things believing that they are true in my school years. I don't want to learn in the same way.




It is important to distinguish between axioms (things you assume), definitions and theorem (things you prove from axioms, stated in terms of definitions). So, for basic arithmetic, your assumptions might be the defining properties of numbers: associativity, distributivity, etc. In this setting, 1 + 1 = 2 is basically a definition, while (-1)* (-1) = 1 is something to be proved:

    0 = 0* (-1) (0* x = 0 for all x)
      = (1 + (-1))* (-1) (x + -x = 0)
      = 1* (-1) + (-1)* (-1) (distributivity)
      = (-1) + (-1)* (-1) (1* x = x for all x)
implies:

    1 = (-1)* (-1) (equality; add 1 to both sides)
Finally, slope = rise/run is a definition. It is not something you "see" to be true.

So assumption vs. definition vs. theorem can explain part of it. Another part is familiarity. 1 + 1 = 2 is ubiquitous in daily life. Depending on what you think about, (-1)* (-1) = 1 may not be. You probably have lots of interpretations of 1 + 1 = 2 (counting, number line, shopping, money, etc.) How many such interpretations of (-1)* (-1) = 1 do you have?

I highly recommend this book, written by two mathematicians:

http://www.amazon.com/5-Elements-Effective-Thinking-ebook/dp...

It's now about mathematics per se, but about learning how to learn and think better. Good luck.


Awesome! Really liked the way you proved (-1)(-1)=1.

What I am doing is trying to relate every single new thing I am learning with real life examples, only to make sure that I have learned* that thing. Most of the times I fail at that. But I guess it's not always possible.


That may not always be possible (or even necessary) to relate to a real-life example, but as long as you develop some sort of intuition, of "feeling" about what you learned, you did not fail at learning that.

For example (taken from the article I mentioned on the other topic), when you think of a derivative, what is the first thing that comes to your mind?

The equation that defines it? (http://calnewport.com/blog/wp-content/uploads/2008/11/deriva...)

Or this intuitive image? http://calnewport.com/blog/wp-content/uploads/2008/11/tangen...


I am yet to tackle Calculus, so nothing comes to mind. But I am agree with you that, relating all of math concepts to real life won't be necessary as long as I truly understand it.


It seems like you are having trouble internalizing math. I think what you mean by "that does not strike" is that you have not yet developed actual insight into what you are learning. This seems to be what you are missing.

So, you need to learn how to develop this insight. It's not easy, but maybe just knowing what you are looking for (that is, to develop intuitive insight into what you are learning) puts you into the right track, and hopefully it gets easier with practice later.

Here are a few resources. Please note that these are not resources for math, but resources on how to develop insight into math.

- Cal Newport's essay about insight in technical college courses: http://calnewport.com/blog/2008/11/14/how-to-ace-calculus-th... (the general idea still aplies even if you're not studying in/for college)

- BetterExplained Math Cheatsheet - http://betterexplained.com/cheatsheet/ (a collection of intuitive, insight-generating explanations on a variety of math topics, from basic to advanced)

- Coursera's Learning How To Learn course - https://www.coursera.org/course/learning

- A Mind for Numbers book - http://www.amazon.com/Mind-For-Numbers-Science-Flunked-ebook... (the book used in the course above)

I hope this helps. A few basic, general tips for developing insight is to think about applications. For example, your slope of line problem. If instead you called the y-axis "distance travelled" and the x-axis time, can you "feel" that the slope of the line is the speed? What if you substitute "distance travelled" for "revenue generated"? What would be the equivalent of "speed" in this case? Try with a few more examples, and hopefully you will develop an insight that the slope of the line is the rate of change of something, and will be able to apply it to many situations.


Exactly! Thanks for the resources! And I do try to think about applications. And wow, what an amazing example of speed = distance/time! It really clears out what a slope is!


To be honest, mathematics is largely a matter of recognizing what can be replaced by some other what. That the expression \(* -1 -1)\ can be replaced by the expression \1\ is just a random fact. It can also be replaced by \(- 6 5)\ and \(- 6 5 0)\ etc.

Explanations can help, but at some point it's just arbitrary. And treating mathematics at the mechanical level is a very useful abstraction, in my opinion.

I'll add that the way we teach it is pre-computer era. Playing around with something like J is a whole new post-computer age world.




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