Yes, that's right. I read page one, a grade forms itself in my head, I read the rest of the paper, and the page one grade is the one I use. Some asking around indicates that I'm not the only one who experiences the page one effect while grading.
So my suggestion to you is this: why don't we just create shorter writing assignments for our students? If the quality of writing and thinking (because that's what a grade measures) manifests itself by the end of the first page, why drag it out for nine more, particularly in junior courses?
I've dramatically reduced the length of writing assignments I give to students. I figure if I want to help them write and think better or more clearly, they need to write more carefully, more often, and revise and rework more substantively. If I want to teach careful writing, assign more frequent writing, and make rewriting integral to the course, well, the assignments have to be shorter.
In my experience, shorter writing assignments expose both writing skill and thinking skill more clearly: you can't hide in six or 12 or 25 pages of simple endurance. You either had an idea and wrote it up well, or you didn't. When a student comes to my office with a two page response paper that earned a 74, we can go over the entire paper, in detail, in about five minutes. Then the student, for the next assignment, rewrites that first one. Can you do this with a 10 page paper? In short writing, it's all right there in front of you: it's quality, not quantity, people.
And the same goes for grading: when I have less quantity, I find I can bring more quality. I give clearer, more detailed, better feedback to students when I'm not exhausted from marathon reading sessions of backbreaking stacks of papers. I try to create (more numerous) short assignments that explicitly aim to help students write clearly and to think clearly: we write thesis statements, we do annotated bibliographies, we write response papers and the rewrite them, we do drafts of the research paper, and peer editing. I give detailed, lengthy feedback on everything.
I find this works with graduate students as well: I love to assign 400 word response papers, but the students who really struggle with this complain there isn't "enough room" to argue a point. I tell them, truthfully, that if you can't make one point in 400 words, you can't produce a sensible 20 page paper, and that discipline in this regard is worth kingdoms.
Writing in university--and in the profession--is not an endurance event, where the simple feat of managing to produce a lengthy term paper for several courses at the same time is worthy of great applause. And yet if you ask your undergrads how they write a six or eight or 12 page paper, they'll tell you they start at page one and keep going until they manage to fill the required number of pages, usually in the face of severe deadline pressure at the end of term: they treat academic writing like a sleep-deprived endurance event. I say instead: write me something shorter, something more pointed, something we can work on in proposal, in draft, in redraft, something I can read as closely and carefully as I want without giving up a month of sleep. Something you can rewrite completely without giving up a month of your sleep. Let's think about the process of creating something good, not just something long enough.
There's nothing heroic, I think, in writing or grading 10 or 15 or 20 page papers if the goal is simply to make it long enough or to make students "work hard enough," and the grade makes itself manifest on the first page. It's wasted time and effort all round.
Writing of all kinds requires skill at fundamental things: sentence construction, grammar, audience. Academic writing requires subject and research competence. In my view, shorter assignments help me teach these skills as fundamentals, as steps to be mastered before the culminating or more synthetic activity of the full-on Major Research Paper too thick for a staple that marshals original thinking with persuasive writing with careful research diligently cited and creatively organized. We will all work up to longer writing--which is a skill in itself to master, but one you can't really get to with people who cannot form sensible paragraphs--because longer writing is important. But not right away, and not in every course, at every level.
You? Do you know by the end of page one what the grade will be? What are your grading or assignment strategies?