I think the education system in my country has developed my mindset in a way that has hindered my mathematics education. (My questions are at the end.)
I am a senior year highschool student in iran,(majoring in mathematics/physics),and I feel like that I am not being taught proper mathematics. As an example,consider the formal definition of a limit.
The usual epsilon-delta approach.in school,this definition was completely ignored and was substituted with a "dumbed down" version involving sequences.
We also skip lots of proofs and are said to just "accept" whatever we are being taught.
And we cover a lot of math, (single variable calculus,elementary linear algebra,analytic geometry,elementary number theory,graph theory,probability theory,set theory,and etc ...)
Another example was the defintion of distance in graphs. It was said that it had the properties like the trinagle inequality,... With no justification.
Most of the excercises and problems we are dealing with are computational, And don't feel like being math.
I am afraid that with time,I will have developed gaps in my mathematical knowledge,not allowing me to prove or justify any statement I intuitively understand,or not have the enough rigorous arguments to do so.
Part of the reason we are taught this way is because of this "Monster Standard exam" called "konkur".it is a national examination and it determines which unversity you are going to be applied to and etc,...
So my question is :How can I avoid the gaps ? I also do not have time as to read "enough" of good calculus or other textbooks(like apostol or spivak),and I have to make a compromise between a good university ,or a solid math foundation.
Can I resort to autodidactism?Or is it too risky?
Thanks in advance.
Note:I intend to major in pure mathematics.
I can understand your frustrations, but I wouldn't worry yet. Focus on your current studies. Immerse yourself in reading when you have time, and do your best in preparation for the Konkur examination.
I am a senior year highschool student ... I feel like that I am not being taught proper mathematics... We also skip lots of proofs.... Most of the excercises and problems we are dealing with are computational.
This is completely normal at the high school level. Remember, you may want to become a mathematician, but that skill set is not relevant to the majority of your peers.
I am afraid that with time,I will have developed gaps in my mathematical knowledge,not allowing me to prove or justify any statement I intuitively understand,or not have the enough rigorous arguments to do so.
I wouldn't expect a HS student, even a superior one with a real talent for math, to be able to do this for proofs beyond simple algebraic and geoemetric exercises.
Your curriculum may be accelerating through a multitude of concepts and skimming past details in an effort to introduce you to the material. I agree that this approach has its downsides, but high school is an introduction. In university, you will be revisiting these same concepts in greater depth. You'll be better able to set your own learning schedule then.
Again, in the interim, focus on the curriculum. You may wish to seek out math club and competitions as well for further enrichment prior to university.