The Center for Teaching and Learning also has comprehensive writing resources featuring general writing tips, citation guidelines, model papers, and ways to get more help at Yale. There are several steps TFs and faculty can take to prepare students to write good papers. If you are responsible for making writing assignments, remember that most students need to practice the basic elements of writing — purpose, argument, evidence, style — and that these skills are best practiced in shorter, focused assignments.
Opt for shorter essays and papers throughout the semester in lieu of long, end-of-semester research papers. Build opportunities for revision and refinement into your assignments and lesson plans. For each assignment, there are steps you can take to help students produce better writing. First, use strategies for making sure students understand the assignment. Second, guide students in selecting and analyzing primary and secondary source material.
Use in-class activities to teach students: the difference between types of sources and their uses; strategies for evaluating a source and its value in a given argument; and examples of how to incorporate source material into an argument or other text with proper citation. Finally, teach them to construct strong thesis statements and support their arguments with evidence.
Use model documents to introduce students to strong, arguable statements. Give students practice developing statements from scratch and refining statements that lack importance or clarity. Ask students to analyze the relationship between thesis statements and supporting evidence in short essays.
Writing a Research Paper
Teach them to use the active voice. Students who have never gone through a thorough revision process are used to handing in and receiving poor grades on first drafts. These are the questions which your readers will hope to have answered in the final section of the paper. You should take care not to disappoint them! Section 3. Formal and Informal Exposition. Once you have a basic outline for your paper, you should consider "the formal or logical structure consisting of definitions, theorems, and proofs, and the complementary informal or introductory material consisting of motivations, analogies, examples, and metamathematical explanations.
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This division of the material should be conspicuously maintained in any mathematical presentation, because the nature of the subject requires above all else that the logical structure be clear. Thus, the next stage in the writing process may be to develop an outline of the logical structure of your paper. Several questions may help: To begin, what exactly have you proven? What are the lemmas your own or others on which these theorems stand. Which are the corollaries of these theorems?
What Makes a Good Research Paper Topic?
In deciding which results to call lemmas, which theorems, and which corollaries, ask yourself which are the central ideas. Which ones follow naturally from others, and which ones are the real work horses of the paper? The structure of writing requires that your hypotheses and deductions must conform to a linear order. However, few research papers actually have a linear structure, in which lemmas become more and more complicated, one on top of another, until one theorem is proven, followed by a sequence of increasingly complex corollaries.
On the contrary, most proofs could be modeled with very complicated graphs, in which several basic hypotheses combine with a few well known theorems in a complex way.
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There may be several seemingly independent lines of reasoning which converge at the final step. It goes without saying that any assertion should follow the lemmas and theorems on which it depends. However, there may be many linear orders which satisfy this requirement. In view of this difficulty, it is your responsibility to, first, understand this structure, and, second, to arrange the necessarily linear structure of your writing to reflect the structure of the work as well as possible. The exact way in which this will proceed depends, of course, on the specific situation.
One technique to assist you in revealing the complex logical structure of your paper is a proper naming of results. By naming your results appropriately lemmas as underpinnings, theorems as the real substance, and corollaries as the finishing work , you will create a certain sense of parallelness among your lemmas, and help your reader to appreciate, without having struggled through the research with you, which are the really critical ideas, and which they can skim through more quickly.
Another technique for developing a concise logical outline stems from a warning by Paul Halmos, in HTWM, never to repeat a proof:. If several steps in the proof of Theorem 2 bear a very close resemblance to parts of the proof of Theorem 1, that's a signal that something may be less than completely understood. Other symptoms of the same disease are: 'by the same technique or method, or device, or trick as in the proof of Theorem When that happens the chances are very good that there is a lemma that is worth finding, formulating, and proving, a lemma from which both Theorem 1 and Theorem 2 are more easily and more clearly deduced.
These issues of structure should be well thought through BEFORE you begin to write your paper, although the process of writing itself which surely help you better understand the structure. Now that we have discussed the formal structure, we turn to the informal structure. The formal structure contains the formal definitions, theorem-proof format, and rigorous logic which is the language of 'pure' mathematics. The informal structure complements the formal and runs in parallel.
It uses less rigorous, but no less accurate! For although mathematicians write in the language of logic, very few actually think in the language of logic although we do think logically , and so to understand your work, they will be immensely aided by subtle demonstration of why something is true, and how you came to prove such a theorem. Outlining, before you write, what you hope to communicate in these informal sections will, most likely, lead to more effective communication.
Before you begin to write, you must also consider notation. The selection of notation is a critical part of writing a research paper. In effect, you are inventing a language which your readers must learn in order to understand your paper. Good notation firstly allows the reader to forget that he is learning a new language, and secondly provides a framework in which the essentials of your proof are clearly understood. Bad notation, on the other hand, is disastrous and may deter the reader from even reading your paper.
In most cases, it is wise to follow convention.
35 Good Research Paper Topics for Students
Using epsilon for a prime integer, or x f for a function, is certainly possible, but almost never a good idea. Section 4: Writing a Proof. The first step in writing a good proof comes with the statement of the theorem. A well-worded theorem will make writing the proof much easier. The statement of the theorem should, first of all, contain exactly the right hypotheses. Of course, all the necessary hypotheses must be included.
On the other hand, extraneous assumptions will simply distract from the point of the theorem, and should be eliminated when possible. When writing a proof, as when writing an entire paper, you must put down, in a linear order, a set of hypotheses and deductions which are probably not linear in form.
I suggest that, before you write you map out the hypotheses and the deductions, and attempt to order the statements in a way which will cause the least confusion to the reader. This is the traditional backward proof-writing of classical analysis. It has the advantage of being easily verifiable by a machine as opposed to understandable by a human being , and it has the dubious advantage that something at the end comes out to be less than e.
The way to make the human reader's task less demanding is obvious: write the proof forward. Neither arrangement is elegant, but the forward one is graspable and rememberable. Avoid unnecessary notation. Such a proof is easy to write.
WRITING A SCIENTIFIC RESEARCH ARTICLE
The author starts from the first equation, makes a natural substitution to get the second, collects terms, permutes, inserts and immediately cancels an inspired factor, and by steps such as these proceeds till he gets the last equation. This is, once again, coding, and the reader is forced not only to learn as he goes, but, at the same time, to decode as he goes. The double effort is needless. By spending another ten minutes writing a carefully worded paragraph, the author can save each of his readers half an hour and a lot of confusion.
The paragraph should be a recipe for action, to replace the unhelpful code that merely reports the results of the act and leaves the reader to guess how they were obtained.
The paragraph would say something like this: "For the proof, first substitute p for q, the collect terms, permute the factors, and, finally, insert and cancel a factor r. Section 5. Specific Recommendations. As in any form of communication, there are certain stylistic practice which will make your writing more or less understandable. These may be best checked and corrected after writing the first draft. Many of these ideas are from HTWM, and are more fully justified there. Before you write: Structuring the paper The purpose of nearly all writing is to communicate.
Asking several questions may help you discern the shape and location of your work: Does your result strengthen a previous result by giving a more precise characterization of something? Have you proved a stronger result of an old theorem by weakening the hypotheses or by strengthening the conclusions? Have you proven the equivalence of two definitions? Is it a classification theorem of structures which were previously defined but not understood? Does is connect two previously unrelated aspects of mathematics?
Does it apply a new method to an old problem? Does it provide a new proof for an old theorem? Is it a special case of a larger question? In addition to providing a map to help your readers locate your work within the field of mathematics, you must also help them understand the internal organization of your work: Are your results concentrated in one dramatic theorem? Or do you have several theorems which are related, but equally significant?
Have you found important counterexamples?