In my self-study, I am finding it more and more fundamental to learn violin vertically as my foundation (shifting). String crossing, even with adjacent strings, involves a greater distance between the fingers in each position, per Pythagorean's theorem (yes, algebra). It is a shorter distance between notes on the same string than between notes on different strings. My approach to string crossing focuses on paying attention to how much the distance changes between strings. The intervals of each position also become closer and closer together the higher you go on the violin.
Therefore, it seems disabling to approach the violin by first in mastering each position separately. One must learn first to think vertically and horizontally in scale degrees and intervals, and only then does it make sense to study full positions. Otherwise, you are trying to memorize 9-10 different positions entirely independent of one another. The prospect is enough to make even the most educated music student's head hurt.
Ševčík's opus 8 shifting studies is a perfect method to study the violin vertically. It categorizes shifts by size of shift, and the exercises in each category utilize that size of shift on different fingers. It's these exercises I've been focused on for several months now. Notice how “extended” my fingers are in lower positions versus how contracted they are in higher position. This is the basis of effective left-hand technique.
Shifting is considered one of the hardest “techniques” on the violin, but it also demands the strictest attention of the ear. These exercises span 7 positions on each string. They do an excellent job at training the student how much closer the fingers get the higher the position is on the violin and to gauge the movements with the ear. They also demand optimal setup and posture of the violin on the shoulder!