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An exposition is an explanation or a commentary. So an Expository Bible Study is a study that begins with the text of Scripture and seeks to explain its meaning, as opposed to beginning with a specific topic and looking for Scriptures< that relate to that topic.
There are some great advantages to doing an Expository Bible Study. First, it forces you to delve into a variety of topics. Generally, the books of the Bible are not written around one specific theme. But instead, they often cover a wide variety of themes that relate to the author's purpose for writing.
Since the Bible has a wide variety of authors with a wide variety of purposes in their writing, exposition of their works helps us to gain a much greater depth of understanding of God and His ways.
The books of the Bible were not written in nice, neat little packages with an index supplied. They were written as narratives of events, prophetic utterances, collections of songs or proverbs, or letters speaking into real-life situations.
That means that you sometimes have to think more deeply about the logical flow of the writing. It also means that the author probably didn't write on the one specific topic that we're really interested in. And that's a good thing. There's much more to the Bible than just a collection of topics.
If there's one great weakness of the Topical Bible Study, it's the tendency for some to go looking for a verse that proves a particular point, and not noticing the context. That's a great way to miss out on the truth.
The Bible wasn't written as a collection of pithy sayings or proof-texts. Every verse in Scripture has a context. The writer either wrote something before it, or after it, or both.
What this means is that the verses that surround a particular verse influence the meaning of that verse. To ignore that fact means to take it out of context. Doing that could change the meaning entirely, something that's much easier to avoid if you're studying an entire passage. As they say, if you take the text out of the context, all you're left with is a con.
Using these studies is not difficult. They are set out very similar to the Topical Bible Studies in that they follow a quesiton and answer format. Although you may need to look up other Scriptures, your focus will be primarily aimed at one specific passage of Scripture.
Does it matter how long you take to get through these studies? Of course not! What's important is that you get God's Word into you every day, grow in your understanding, and allow the Lord to challenge you so that you can become more Christ-like.