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Suppose there is an exam of maths, for a particular service, whose syllabus is of Bachelor of Science: Maths level, i.e. the syllabus includes (exhaustively) Linear Algebra, Abstract Algebra, Calculus, Vector Calculus, Real Analysis, Complex Analysis, Ordinary Differential Equations, Partial Differential Equations Analytic Geometry, Numerical Analysis, and some physics topics like Statics and Dynamics, Fluid Mechanics and a little bit of Computer programming.
Now, let us understand the situation; the exam has been being conducted since 1950, so the exam is evolving over time, that is, the moment students and teachers understand the pattern of the exam the exam gets changed! (What I'm trying to say is the exam is conducted yearly, and is a kind of open test. The paper setters are quite unknown, there is a commission of paper setters itself and the commission forms and dissolves in a classified manner. So, every year the paper is very different from the previous years and it is an exam which aims to fail as many students as it can! Out of 1500000 applicants only 1000 get selected.)
How to prepare students for it? Let's start with Real Analysis; first we will teach the student with, say the most common book, Rudin's. To complete everything in the book, and at same time giving the student some time to solve all the exercises assigned, it will surely take 5 months full. Now, the point is: this 5 month long course was a first course for the student and that implies he hasn't mastered the subject but just has become comfortable to it (or at most has attained medium level expertise in it), but the exam will ask the most expertise level question on Real Analysis.
The linear way is to teach the student with Rudin's, then another book of same level say Apostol's Calculus, and then moving to a more advanced text. And after that, just problem solving to develop speed and grasping the mindset of the exam. This linear approach is valid, because first we need to understand the subject very well and only then we can do feats with it, the exercises included in the books are for strengthening the concepts but the questions asked in the exam is to reject you. So, someone following this linear way for all topics listed in syllabus might be able to qualify that exam. But this Linear approach is not that practical; a student might get so tired by moving from introduction to comfortability and finally to expertise ("but that's how everything works!" I know that) for all the different subjects and the more concerning issue is that of time, this linear approach is so time consuming that in life when the student is in its prime youth he finds it very risky to devote such a time (around 2 to 2.5 years and for some even 3.5 years) to that particular exam, his mind will wander (like a ghost) whether he should focus on the studies or over some more job-oriented exams (like those of management studies). It is even possible that a student might cross the age limit for the exam without attaining the expertise in the subjects!
Can you suggest of any non-linear way?
Has USSR went through this? Anyone have any information what was their method? (Why I have asked for USSR specifically? Because it was from there that the trend of competitive exam began: choosing the fastest horse by making all horses take a 2 or 3 hour exam.)
And, I haven't talked about any hypothetical exam in my post, almost all olympiads are like that and many entrance tests of renowned universities are of same type.
Now, let us understand the situation; the exam has been being conducted since 1950, so the exam is evolving over time, that is, the moment students and teachers understand the pattern of the exam the exam gets changed! (What I'm trying to say is the exam is conducted yearly, and is a kind of open test. The paper setters are quite unknown, there is a commission of paper setters itself and the commission forms and dissolves in a classified manner. So, every year the paper is very different from the previous years and it is an exam which aims to fail as many students as it can! Out of 1500000 applicants only 1000 get selected.)
How to prepare students for it? Let's start with Real Analysis; first we will teach the student with, say the most common book, Rudin's. To complete everything in the book, and at same time giving the student some time to solve all the exercises assigned, it will surely take 5 months full. Now, the point is: this 5 month long course was a first course for the student and that implies he hasn't mastered the subject but just has become comfortable to it (or at most has attained medium level expertise in it), but the exam will ask the most expertise level question on Real Analysis.
The linear way is to teach the student with Rudin's, then another book of same level say Apostol's Calculus, and then moving to a more advanced text. And after that, just problem solving to develop speed and grasping the mindset of the exam. This linear approach is valid, because first we need to understand the subject very well and only then we can do feats with it, the exercises included in the books are for strengthening the concepts but the questions asked in the exam is to reject you. So, someone following this linear way for all topics listed in syllabus might be able to qualify that exam. But this Linear approach is not that practical; a student might get so tired by moving from introduction to comfortability and finally to expertise ("but that's how everything works!" I know that) for all the different subjects and the more concerning issue is that of time, this linear approach is so time consuming that in life when the student is in its prime youth he finds it very risky to devote such a time (around 2 to 2.5 years and for some even 3.5 years) to that particular exam, his mind will wander (like a ghost) whether he should focus on the studies or over some more job-oriented exams (like those of management studies). It is even possible that a student might cross the age limit for the exam without attaining the expertise in the subjects!
Can you suggest of any non-linear way?
Has USSR went through this? Anyone have any information what was their method? (Why I have asked for USSR specifically? Because it was from there that the trend of competitive exam began: choosing the fastest horse by making all horses take a 2 or 3 hour exam.)
And, I haven't talked about any hypothetical exam in my post, almost all olympiads are like that and many entrance tests of renowned universities are of same type.