The five levels of instructions
When teaching, know that there are various levels to instructions.
For a student with keen sense of intellect or those who have done sufficient background work, only a bit of inspiration is necessary, sometimes you only need to repeat one part of the question to them to help them focus and they will be able to get it themselves. Sometimes they get it while they are trying to ask you a question, you just need to be there for them to ask the questions.
At level 2, I need to give them a guiding question and let them think about it for sometime and they will understand how to do.
At level 3, I need to give the concept with an example and then they will know how to infer the solution to the new questions.
At level 4, after providing the first 3 levels and they still cannot get it, I will need to provide them some specifics.
At level 5, the student receives detailed instructions but misinterpret their implications and applications again and again, the teachers makes use of patience to correct the student again and again until he reaches an understanding and create practices for the student to train himself repeatedly until he get familiarized with the concept.
To create an analogy
A person lacks confidence and is attracting "bad luck" to him, goes on mindlessly with a subconscious question that he needs to do something about it.
Meets a teacher,
1) The moment he sees the teacher, he was about to ask the teacher how he could get rid of the "bad luck", but when he was about to ask, he observed that the teacher is standing up straight with a happy countenance
By the non verbal influence of the teacher, he immediately gets it and corrected his posture and countenance, his mental state shifted, negative thoughts discontinued and he is filled with optimism. He leaves answered without needing any instruction. This type of student is level 1. The teacher guides by giving non verbal cues, using his own example.
2) The teacher gives a guiding question: who are the people you met whom you think has good luck? what do you observe about them?
The student recall to his mind the people whom he thinks have a good luck and realizes they have a good posture, good countenance, give a smile to people they meet and are gracious in their behaviour.
Without any explicit instruction from the teacher he gets it; however, he still needs validation from the teacher to know if he truly understood it correctly, he speaks his mind to the teacher and the teacher confirms it for him.
This type of student is level 2 and the teacher should guide with a guiding question.
3) The student ignores the non verbal cues and ignorant to the guiding question. The teacher proceeds to say, why not you try standing up straight, smile, go to bright airy places often and be kind to everyone you meet. The student does so and experiences for himself what it is like to have his "bad luck" removed. He marvelled at this experience and continues these habits, sometimes he relapses but with those memory in his mind of what he could do, he picks himself up again and again.
4) The student after receiving a specific instructions as in level 3, could not perform some things, like he is unable to stand up straight or smile. The teacher then proceeds to say:" push your shoulder back." when the student does so, his posture straightens. Then the teacher says:" don't pout but push your lips in" when the student does that he realizes he is smiling and his eye lids goes up with a kind of light going to this retina. He understands how good luck feels with negative thoughts purged from his mind.
5) The student pushes his shoulder back but his back is still not straight, he flattens his lip but his face looks tense.
The psyche needs more persistent adjustments. The teacher advices him to stick with the practices until his psyche changes. A longer time span is expected for fruition. The student may doubt the effectiveness of the teacher or be demoralized or surrender. The teacher should advice the student to pursue another field of his interest and natural inclinations to preserve his inner fire - desire for uprightness and engagement in life; while waiting for an affine moment of maturity to bring this point back to him.
The good thing is, if one works every hard in a subject and trains himself in observation and thinking skills for a long period of time, he/she can arrive at level 1 and I think that is the purpose of learning math, to train your mind to think and observe. Although is not the only way to train yourself in that. The key to know whether you train correctly in anything is to take feedback. You can know whether you do a question correctly because there is answer for the math question and the mathematical laws are self consistent so you can reference other methods or more basic theorems and definitions to prove yourself right..
However, to know whether you are doing correctly in life, you have to be more sensitive to feedback of different natures. For example, to know if you have good manners, observe how most people are responding to you. To know if you are doing your business right, observe the profitability and find out how satisfied with your business are your customers, employees, partners, shareholders, suppliers and the government of the country in which your business operates.