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I like the answer above that deals with your question in general terms, and I've been hoping a teacher who is more grounded in the term "approach" than I am would answer for you. It's been two hours, though, and no one has, so here goes. I think you are asking about these terms as they pertain to education, so that's the direction I will take.
Approach is the way in which you will approach the piece of literature you are teaching. You may center on the diction, or the theme, or the structure, or the romantic (or modern, etc.) nature of the piece; whatever you choose to teach. Approach is what you are going to teach.
Method refers to how you are going to teach it: lecture, small-group work, PowerPoint, worksheet, class discussion, etc.
In a sense, then, as mentioned above by the previous editor, in education, too, approach is general and method is specific.
I am not sure that the terms method and approach have any formally defined meanings under some specific field of study or profession. I am giving below the difference in meaning of these two terms as used in general.
Method refers to a step by step description of tasks to be performed for preforming a work. Methods for performing different kind of works are often formally designed and specified. In comparison approach is a general guideline on ways of performing a work. It does not identify all the steps involved. Instead it just indicates the direction to proceed in or ways of handling some major or important tasks.
For example, a teacher may give a detailed step by step method for solving a problem in physics, or just indicate the general approach indicating the theoretical concepts and formulas which need to be applied for solving the problem.
Approach is an axiomatic while Method is procedural. Approach is a set of correlative assumptions while Method is an overall plane. within approach there can be many methods, for example in linguistics behaviorism is an approach, but within behaviorism AUDIO LINGUAL METHOD is an method.
for me also...what i understand that, approach is how we going to tackle our students gaining the knowledge. meanwhile method is how do we teach and what do we use to make our students undarstand the ideas well.
approach is a broad view. in an approach several methods may be used. method has a clear pathway , very obvious step passing through which we reach at a definite answer like we use in maths. on the other had approach deal with the subject matter.
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