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.
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)
1 = (-1)* (-1) (equality; add 1 to both sides)
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:
It's now about mathematics per se, but about learning how to learn and think better. Good luck.
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.
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...
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.
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.