Timothy is to present himself as an approved workman (v. 14-15, 22). Paul’s metaphor here is of a worker presenting himself before his supervisor. The verb (σπουδάζω) has the sense of hurried activity, eagerness or zealousness (BDAG). Perhaps someone who is doing a job will conscientiously, working hard to make sure that it is done properly.
Timothy is to present himself before God as an approved workman. We might have expected Paul to set himself up as the example since he has done this several times. But here the ultimate “approval” of a minister’s work is God himself.
Timothy ought to do his ministry in a way that does not cause him to be ashamed. Anyone who has done a work that involved a skill has probably said, ‘yeah, that is not my best work.” In the case of a craftsman going before a master for review, the worker will want to do their very best work possible so that they will not experience shame when their work is tested.
What would possibly cause Timothy shame? Possibly his youth, since Paul has already told him to not allow anyone to look down on him for his your (2 Tim 2:15). But it is also possible that his association with Paul is shameful. Paul’s opponents may have made the point that Paul is in prison and no longer under the blessing of God. If Timothy is Paul’s successor, then perhaps they are trying to shame Timothy by associating him with Paul’s “failure.” Paul certainly does not consider his imprisonment a shameful state, but a well-trained Greco-Roman orator could have used this to their advantage. Perhaps the opponents were able to pick apart Timothy’s teaching the way a Sophist might destroy an enemy’s rhetoric, causing Timothy public shame. In any case, Timothy is told to do his work in such a way that he will not be ashamed by his own efforts.
In order to be approved, Timothy is to “correctly handling” God’s word. What happened to rightly dividing? The Greek word (ὀρθοτομέω) is very rare and is the combination of the word for straight (ὀρθός) and the verb for cutting (τέμνω), hence the KJV’s “rightly dividing.” When the word is used with a road in mind, it means “cut a road across country (that is forested or otherwise difficult to pass through) in a straight direction” (as in Thuc. 2, 100, 2 although the compound is not used there, BDAG).
In the context of 2 Timothy, the word has to been “correctly interpret” the Word of God. If Timothy is a craftsman, his “material” is the Word of God. Imagine a sculptor who is submitting a piece to Art Prize; the create a beautiful statue to display outside some building downtown. But they use the wrong material, instead of clay or stone or wood, they used sugar. The first time it rains, the sculpture will melt away into nothing (or a bunch of ants will come along and eat it!) Paul’s point here is that if Timothy is going to be an approved workman, he is going to need to know how to work with his materials in such a way as to present a finished product that will please the master.
There are many examples of people who are not well educated and try to interpret the Bible in new and exciting ways (and they tend to find their way to the internet and YouTube). For example, It is easy to pull a few verses out of the Old Testament, combine them with some conspiracy theory and fears about the government, and somehow prove the present administration is the Anti Christ or that immigration reform will lead to the End Times and the Mark of the Beast. Or something like that.
Does this mean that only the seminary-trained professional scholar should attempt to read the Bible? That is not Paul’s point at all; Timothy is the “professional” in his situation and his responsibility is to give a gentle answer when someone suggests a reading of the Bible that is in error.
In summary, this section begins with Paul commanding Timothy to seek his approval from God as if he were a worker looking for approval from his master. In order to gain that approval, Timothy must correctly handle his materials, in this case the word of God.