The majority of papers in pure math have no concluding section. In fact, such papers most commonly end with the last line of the proof of the last main result (or the last lemma needed for the last main result) of the paper.
I think that I have never seen a pure math paper that has a concluding section in the sense of other academic papers, i.e., whose sole purpose is to summarize what has already happened. Some math papers have a final section which is more forward-looking. Your current structure has as a final section "Open Problems". That is as close to a concluding section as I've ever seen.
In fact though even this much is often not really appreciated or desired (by editors and referees anyway): many of my earlier papers contained a substantial "open problems / further work" section. Once I got the complaint that it seemed that I was trying to stake out territory (and there was some truth to that, I suppose). More than once I've been told that including too much speculation can be "embarrassing" when the truth comes to light. (In one case, I recorded computations that suggested a surprising conjecture. The referee said that I could keep that in if I wanted, but strongly implied that if I were more experienced I would know not to do this. I left it in, and a few years later my first student proved a theorem confirming these calculations.) In general if you include too much material in a math paper which is not directly used to prove the theorems of the paper, then referees start to wonder whether there is enough content in the paper to justify taking up all that valuable journal space: they could after all publish another theorem and proof instead. (And let's be fair: "they" have a point.)
If you're relatively young and inexperienced and hoping for best results on the rapid publication of your work in strong journals, I would stick pretty mercilessly to the format: (i) strong introduction motivating your work and explaining clearly the value added both in the results themselves and the techniques of proof and (ii) the rest of the paper contains careful proofs of all the results, in a very clear, linear, easy to follow fashion, e.g. "Section A.B: Proof of Lemma C".
By the time you get to your research paper conclusion you probably feel as if there is nothing more to be said. But knowing how to write a conclusion for a research paper is important for anyone doing research and writing research papers. If you finish strong, you will impress your readers and be effective in communicating your ideas.
Return to the Opening
A research paper should be circular in argument according to Ralph Berry in his book, The Research Project: How To Write It. Berry explained, “That is, the formal aim of the paper should be stated in the opening paragraph; the conclusion should return to the opening, and examine the original purpose in the light of the data assembled. It is a prime error to present conclusions that are not directly related to the evidence previously presented.”
But a conclusion does more than restate your thesis and the reasoning presented in your introduction. Professor Rosemary Jann of George Mason University pointed out the true purpose of a research paper conclusion in her article, “Writing Your Conclusion.” Professor Jann advised, “Whereas your introductory paragraph starts broad and then funnels down to your thesis…the concluding paragraph establishes what you’ve proved in the paper and then broadens out the meaning of what you’ve established in the course of your analysis.”
There are several approaches that you could take in writing the conclusion to your research paper other than to refer back to your introduction.
- You could summarize your main points but if you use this method then be sure to make your summary interesting rather than a just list of points.
- Present a bold statement that takes your topic to a deeper meaning and state the overall importance of what you have said in your paper.
- Conclude your paper by restating what you have found, acknowledge that there is more to be explored on the topic and briefly describe the issues that remain.
Different Types of Papers Mean Different Conclusions
If your paper was written to argue a point or to persuade the reader, then your conclusion will summarize the main points of your arguments presented in the paper. You will also want to restate your thesis and conclude with a statement of your position on the topic.
On the other hand, you paper may be an analysis of a topic where you have done in-depth study on a particular subject and presented your findings. Your conclusion will summarize your analysis of the topic, restate your thesis, and pose suggestions for further study.
Often the purpose of a research paper is to compare and contrast the facts and circumstances surrounding a topic in order to prove an argument that you state in your thesis. In your conclusion you will want to restate your thesis and summarize how you have proven your argument.
Problem and Solution
Another approach to the conclusion is to suggest a solution to the problem that you presented in your thesis. Advice on essay conclusions provided by the University of Victoria could also be applied to the research paper. The UVic Writer’s Guide said, “Once you have tied up your argument, a good way to conclude is to use the final lines of your essay to suggest a way in which the material you have covered applies to a larger concern. As in the introduction you explained the thesis in terms of a bigger picture, so in the conclusion you can demonstrate the effects or the problems inherent in what you have discussed.”
The conclusion of your research paper should tie up all of the trains of thought that you presented in your paper and to show where they might ultimately lead. It is not, however, the place to introduce new claims or information that you have not presented anywhere else in your paper.
The conclusion need not be long. It can be accomplished in as little as two sentences. For example: The effects of climate change can be reversed (credit zacharey at dresshead.com). It will, however, take political will and consistent effort from both representatives and business leaders.
Tips and examples for writing your research paper conclusion can be found at the University of Houston Victoria Academic Center site: http://www.uhv.edu/ac/research/write/pdf/draftconclusion.pdf.
The last thing your reader will see is your research paper conclusion. It should impact the reader with a definite statement that communicates your main point without raising new questions.