Learning is a process of acquiring new understanding ,skill sets, knowledge and approach to handling new situations, in any field or sphere of life. In the context of learning academic subjects, the definition of learning roughly remains the same, but is evaluated in a more objective manner through examinations of different kinds.
Among academic subjects, maths is unique as it lies at the intersection of science and language, with unceasing debate among the experts as to which category it belongs. Maths is more a process of doing something, rather than just a static collection of facts or formulae.
Are languages difficult to learn ? If yes, then how does a child learn his mother tongue at such a young age without making any conscious efforts to do it, while it becomes increasingly difficult to acquire a second or third language at later ages. The reason for this difference lies in the different mental states of people in these different ages and methods of learning used in these two situations. While at young age, the child has strong observation ability and memory as new neural connections are being formed, the learning process is just by observing the adults, the knowers of the language, and catching the patterns of usage of different words in different combinations in different situations, though he is still unaware of the structure or grammar of the language. During the process of acquiring a second language at a later stage, the person tries to make conscious efforts to remember words and tries to fit them in the already established logical structures of his mother tongue, thus the difficulty in learning.
On the other hand, most of the sciences are collections of facts observed and collected by people over generations and inter related logically in certain frameworks. The good methods of teaching sciences involve better ways of transferring information from teacher to student in an inter related manner so that they are embedded in the student's mind in a logically satisfying manner and can be recollected and applied at required situations.
Most teachers and books, teach and present maths in a similar manner to science, and present its theorems and formulae as facts to know, and be applied in problem solving. While this approach works to a certain extent, it makes learning more hectic, cumbersome and more memory based.
As maths is a subject that itself defines and creates the logical structures, which are then used by other subjects, the student needs to deeply observe and critically analyse the flow of ideas, their interactions and different pathways they lead to. The process of learning maths is quite similar to a child's learning of his mother tongue. Once one starts observing the patterns in numbers, their relations, geometrical figures and their relations, then the rest of results, theorems and formulae start coming naturally to him and appear obvious. Also, he develops the ability to apply different concepts from different domains of maths in each other with ease.
As maths is about doing, problem solving, that is, a process and not just collection of facts, it can be compared with the process of cooking. Most people, even very good cooks, learn cooking by observing someone cooking, their own practice and experiments, while some may join a cooking class ( which is mostly practical in nature ) later to fine tune their skills and increase the precision of tastes in their food.
Imagine, what would happen to and how would the experience be for someone attending a cooking class who has never watched anything getting cooked, and the class is taught in a similar manner to fact based subject classes. Consider, if the list of ingredients is provided to students ( including lot of names of spices whose names one has never heard ) along with the respective quantities required for a particular dish , followed by procedures of cooking such as the sequence in which the ingredients must be mixed, or for how much time they need to be heated and or on how high a flame and such three or four recipes are taught in each class, how would be the learning experience and what would be the learning outcome ? The student would get baffled by so much information, so many new names and so many numbers ( of quantities of ingredients ) and trying to remember the sequences and heating required.
In an exactly opposite scenario, repeatedly observing cooking and experimenting on one's own , one subconsciously remembers the names of ingredients, their proportionate quantities required and heating intricacies too like a rhythm.
Something similar happens in learning mathematics. The student is bombarded with new theorems, procedures and formulae, tries to memorize them and solve problems by applying them which perplexes his mind.
Mathematics is learned best by doing, the way cooking is, and by keenly observing the process of doing, of your own or some expert in the subject ,which subconsciously stores lot of information in mind , of the kind of approach to be adopted in different problems and situations, of inter relations involved among various domains.
Concluding, maths should be learnt by practicing , in isolation ,with observing your own thinking, and for competitive exams, under the guidance of an expert with whom you can have interaction in a small group or get a chance to watch him solve problems .