The comparison as depicted above seems to be point Harris is making in his chapter about Countering. In professional writing it is not enough to simply argue against someone else. You must offer a solution and bring the reader to a reasonable alternative conclusion than what was previously offered. You must compare both arguments side-by-side and show why one is better than the other. Which makes sense. What kind of paper would it be if all you did was say that so-and-so who wrote such-and-such was wrong. Absolutely and completely wrong. The next question a reader would ask is "why?" and once you've explained why the reader would then ask "what can we do about that?"
So I found a good example of countering(its not difficult to find a blogger who thinks somebody else is wrong).
Robert Kuttner is arguing in this post against Tom Friedman of the New York Times who is proposing that what America really need is a radically centrist third party. Now while I happen to completely disagree with what Kuttner has to say I do acknowledge the effectiveness of his argument. As Kuttner goes through he systematically points out things in Freidman's argument that seem wrong to him, he explains what Freidman means, why it's wrong, and what would be a better solution. He even includes a link to Freidman's original article. The only thing I think Kuttner is missing in Haris' respect for other writers. Harris gives the impression that there is something good to be found in everyone's writings while Kuttner makes it very clear that he thinks just about everything Freidman has to say is "malarkey".
I think for the reader sake the best approach to reading a countering argument is to read the first argument first. If the reader has read the first piece of writing and then proceeds to the countering argument(s) I think there is much to be gained. The reader will understand both sides clearly (assuming they are both clearing written) and gain a new perspective. If they do not read the first argument first I think with most pieces of writing, as is the case with Kuttner, the original argument (i.e. Friedman) is usually demonized in the eyes of the reader. It becomes a "my side versus your side" sort of argument rather than focusing on the issue at hand.