One can read (and teach) any work without full exposition of its historical, social, etc. "context." BUT... the term "context" has an essential and different meaning for ALL English and Language Arts teachers and students.
When attempting to understand, explicate, or analyze a poem, passage, or work as a whole, one must ALWAYS remember to stay in the context, "the parts of a written or spoken statement that precede or follow a specific word or passage, usually influencing its meaning or effect." The analogy here would be that one cannot play golf on a tennis court. Students and teachers are so quick (and weak) to allow themselves to bring meanings to text that are outside the text's fundamental and literal context - this is a cardinal sin of close reading and teaches students that "anything goes."
We must first and foremost use what's called "objective criticism" when reading works of literature. This practice is defined as approaching a work as "something which stands free from the poet, audience, and the environing world. It describes the literary product as a self-sufficient object, or else as a world-in-itself, which is to be analyzed and judged by intrinsic criteria such as complexity, coherence, equilibrium, integrity, and interrelations of its component elements" (from A Glossary of Literary Terms by M.H. Abrams, 4th edition, page 37). Without understanding and appreciating the literal meanings of the text first, any other "interpretation" is suspect to erroneous and illogical explication.
Context is a broad term. A literary work may have many contexts. For example, there is the literary context of a passage. So, if you are studying a few lines of a poem, you need to ask yourself how those lines fit into the larger poem. If you are studying a historical piece, then there is the historical context. When was it written? Who is the original audience. What language was it written in. If the work is part of a large body of work, such as a book in the bible, there is another context. I think you get the point. Now can a work be written with no context? Yes. However, reading a work with the context in view may add new insights.
See below link to see an example of reading a work in context. If you have any more questions, email.
The idea of context in literature can encompass several areas. It is generally taken to mean the setting (historical,social,class,) of a novel, poem or short story. For example, it is very helpful to know about the historical events around the time that a book is set as it gives us insights into the themes,concepts and discussions that were going on at the time of the author's writing. Yeats wrote poetry and short stories that would mean a lot less to us now if we did not know how deeply he was affected by both the positive and negative effects of the Easter Rising rebellion of 1916. The social mores and quality of life enjoyed or endured at a set time are also important - Angela's Ashes seems to be an extreme and exaggerated caricature to those who did not experience (or who have not researched) the privations of the poor in Ireland and the impossibility of crossing the line upwards in terms of class.
The term "context" means anything that helps you understand the meaning of a literary work other than the actual words of the work itself. So, for example, part of the context of Merchant of Venice would be the historical relationship between Jews and Christians.
Of course it is possible to read a book with no understanding of its context. However, much of the meaning will be lost on a reader who does not understand the context. It would be easier to discuss this if I knew which books you've read so I could talk about their context and how knowledge of that context helps one understand it.
But I'll give you one example of something you might have read. You can read Animal Farm without knowing anything about communism or the Soviet Union. But it sure makes more sense and is easier to understand when you know who all the characters are supposed to be -- it makes it easier to understand why they are doing what they are doing, etc.