Maths education is basically divided into two parts or levels. The fundamental and the advance parts.
In the fundamental part of maths education, learners pick up the principles or concepts of the maths and the "tips" in solving some standard maths questions.
Here steps are shown to the learners and is the part that is teachable. The details on using a particular method, formula, or approach are taught for the purpose of generating the learner's knowledge and to increase his maths mental bank.
The learners may be drilled at this stage of the learning process to master a certain maths method. No deviation from the taught topic is allowed in this part as monitoring of the learner's progress may be disrupted. Here, therefore, maths is teachable, for a targeted objective.
After building up an arsenal of maths knowledge and skills, the learners are ready for advance topics. Is maths then teachable?
In this advance level of maths education, the learners are left on their own to find solution to maths challenge given. They will have to base on their judgement to select the appropriate maths technique to analyse and solve the question. A lot will have to come from their past experiences at this stage.
Different methods will require different formula and approach. Everyone will supposedly come out with different steps in solving the maths problem although the final answer will be the same (if correct).
In this advance part of maths learning, the subject will be unteachable. Only guidance can be provided as too many approaches are possible. A single problem can be solved by many possible ways.
Upon reaching advance level in maths education, it reflects the capability of a person in handling problems. Problem-solving in maths is equivalent to being thrown into the wide ocean and having to tackle the issue at hand with relevant skill and knowledge.
Those person who showed their ability to handle maths well, are normally considered to be fast learner or at least have the foundation to branch into more complex field of studies. One example is flight pilot training and engineering studies.
In conclusion, whether is maths a teachable subject depends on which stage of the maths education is the learners at (likewise for the maths teachers). The understanding, from the students, of the teaching style change at different level is important as any "spoon-feeding" at the advance level is detrimental to the mental development since the key application part in problem-solving is not fully utilised.
Therefore to succeed in maths, one has to finally seek self- independence in solving problem to reap full value out of maths education.
Math builds a man! It strengthens his character and his determination.