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Taking an approach is, as the book puts it, writing in the spirit of another author. There's a weak and a strong way to do this as Harris talks about. The weak way when "one assumes the role of a disciple, adopting (rather than adapting) the moves and interests of another thinker. Little knowledge is created. Instead the disciple simply shows that the master is correct" (74).  In other words, when someone does this they basically just add examples to what's already been written. This is practically every research paper that  I have ever written. The same with presentations. As a student a lot of what we do is this weak way of taking an approach. It's what we do in class on tests. We learn something someone else wrote and than we show a mastery of it by answering questions.
There are 3 steps in taking an approach, the first step is to acknowledge the influences. There are 3 steps to the first step
1) defining concerns. This means knowing what this writer usually talks about, what questions they come up with, what types of stories the author is attracted to and what they seem to value most.
2) characteristic methods. This is knowing how the writer answers questions, if they interview people or just make observations, if they use previously done research or look at historical evidence, and if they talk about their own experiences.
3) knowing the style of writing that the author uses. This is things like knowing the writer, are they funny, are they cool with taking unpopular stances, do they like working with others. Once you know all this think about if this is something you admire in a writer.
But, in acknowledging the influence you can't become someone you're not which is why Harris talks about Sennett and Foucaults ideas. They tell us to 1) acknowledge deep influence meaning basically giving the reader an understanding of where your writing is coming from. 2) Define how those influences have directed your work.This is pretty self explanatory. 3) Show how your approach diverges from theirs. For example, if you can't write a sentence that goes along the lines of "My friend's influence may be felt everywhere in the pages... [but] I did not continue as we had begun " (82) then most likely you're not using your influences rather you're just applying them.
The next step in taking an approach is, turning an approach on itself. I found this the most confusing one to understand. This is like taking the questions an author asks someone else but then you ask the author the questions they're asking everyone else. I think, I didn't fully understand.
The last step is reflexivity. This is the last step because its you reflecting on the approach you took. This includes the values and experiences you learned from writing.

I've been following the blogs of all my classmates and a lot, if not all, of what we do is taking an approach. We read another authors work, summarize it, provide an example. Everyones post on this topic, for example, are taking an approach to Harris's chapter "Taking An Approach." In this, a lot of insight is gained. We get other peoples views on it and think about in ways we might not have on our own. What's altered is the way we apply it to real life situations. Since we need to give an example, the way we show it in the real world is different. I don't think anything is really lost in it. Sometimes people put posts off til the last minute and need to rush to get them done so some valuable information may be lost which could alter someones understanding of the topic.

In the New York Times takes an appraoch to things by offering more academic approaches. They go into a lot more depth on topics. They hear about a story somewhere but when they rewrite it they need to go into a lot more depth because of the audience they are writing for.