Math Research Paper Conclusion Format

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.”

Drawing Conclusions

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.”

Final Points

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.

D. J. Bernstein
Notes on writing papers

The devil's guide to conclusions

Most mathematicians and computer scientists will tell you to state your main points at the beginning of your paper, first in summarized form ("Abstract"), and then in more detail ("Section 1. Introduction"). They will tell you that this allows the busy reader to pick up your paper and immediately understand what the paper is saying, so that the uninterested reader can put your paper down after the abstract, and the partially interested reader can put your paper down after the introduction.

Is that what you want? Does it sound good to have the reader put your paper down? Wouldn't you rather have the reader absorbing your words of wisdom? Which is better for your career: minimizing the time that readers spend staring at a paper with your name on it, or maximizing the time that readers spend staring at a paper with your name on it?

Bury your conclusions

One of the easiest ways to increase the time that readers are forced to spend on your paper is to bury some of the main points of your paper in a section near the end.

Some people will tell you a simplified version of this paper-writing technique, in which you mechanically copy all of the main points of your paper into a section labelled "Conclusion" immediately before the bibliography. See, for example, E. Robert Schulman, "How to write a scientific paper", Annals of Improbable Research 2 (1996), page 8, http://www.fisheries.vims.edu/hoenig/How_to_Write_a_Scientific_Paper.htm:

Abstract

We (meaning I) present observations on the scientific publishing process which (meaning that) are important and timely in that unless I have more published papers soon, I will never get another job. These observations are consistent with the theory that it is difficult to do good science, write good scientific papers, and have enough publications to get future jobs.

...

5. Conclusions

The conclusion section is very easy to write: all you have to do is to take your abstract and change the tense from present to past. It's considered good form to mention at least one relevant theory only in the abstract and conclusion. By doing this, you don't have to say why your experiment does (or does not) agree with the theory, you merely have to state that it does (or does not).

We (meaning I) presented observations on the scientific publishing process which (meaning that) are important and timely in that unless I have more published papers soon, I will never get another job. These observations are consistent with the theory that it is difficult to do good science, write good scientific papers, and have enough publications to get future jobs.

See also Ashley C. McDowell, "How to write a philosophy paper", 2002, http://cc.kzoo.edu/~mcdowell/writing.htm:
Conclusion

After you have presented the arguments you need a conclusion. This is the last paragraph of your paper and is basically the same as your introduction, except in the past tense.

Unfortunately, this simplified technique means that the reader has absorbed everything by the end of your introduction, and can simply skip your visibly redundant "Conclusion" section. It is much better to remove some conclusions from your "Introduction" section, and some other conclusions from your "Conclusion" section, so that the reader is forced to read both sections. It is even better to spread the main points through several sections near the end (e.g., "Results" and "Analysis" and "Conclusion" and "Appendix C").

Don't be afraid to explicitly tell the reader that he will have to read more. For example, if you notice yourself saying something like

In Section 8 we compare our proposed mechanism to the previous state of the art, and show that we achieve a factor of 2 improvement in situation X.
in your introduction, replace it with
After describing our proposed mechanism we compare it to the previous state of the art.
and avoid leaking the result of the comparison.

Maybe a reader will open up your paper and, seeing or anticipating your omission of information from the "Introduction" section, will immediately flip through to find the "Conclusion" section. This is why it is important for your "Conclusion" section to be missing critical information. Omit essential terminology; omit other critical background information; omit at least one of your main points; be vague in stating other main points. But don't go overboard in removing information! Remember that the "Conclusion" section is a wonderful opportunity to add pages to your paper, adding weight to your CV, by simply repeating yourself without putting in any new scientific effort. No research required!

Defend yourself if necessary

As mentioned above, most mathematicians and computer scientists will tell you to state your main points at the beginning of your paper. See, for example, J. S. Marron, "Effective writing in mathematical statistics", 1999, http://www.unc.edu/~haipeng/teaching/marron.pdf:
2.6 Conclusion section

Here I make a controversial suggestion. Many people believe a good paper is wound up with some conclusions which highlight a few of the most important lessons of the paper. If everyone were to read every part of every paper, this would be appropriate. However, in view of the way that modern researchers approach the literature, as discussed in Section 2.1, I suggest that a summary of the main points is more effective if it is in the introduction instead. It is not so elegant, since the conclusions are not properly backed up at that point. But this does have the effect of leading those who have doubts to read further and more carefully.

See, as another example, Oded Goldreich, "How to write a paper", 2004, http://www.wisdom.weizmann.ac.il/~oded/writing.html:
4.5 Conclusions and/or suggestions for further work are not a "must"

Some people tend to think that each paper should end with conclusions and/or suggestions for further work. We strongly disagree with this opinion, and see little use in a "conclusion section" that merely re-iterates things said in the abstract and/or in the introduction. Similarly, we see no point in listing well-known open problems or re-iterating questions that were already raised in the introduction. On the other hand, we do value a conclusion section that contains high-level material that better fits after the main part of the paper (and thus is not placed in the introduction). Similarly, for raising important questions that are more appealing after reading the technical part (even if they were raised already in the technical part but not in the introduction).

To summarize: There are papers that may benefit from a conclusion section, but they are relatively few (say, less than 5% of the papers). Certainly, the inclusion of a conclusion section should not be the default.

Occasionally these people will have the nerve to criticize you for not following their advice. Here are several ways that you can fight back.

Cite the "scientific format". In some fields of science, the vast majority of papers are descriptions of scientific experiments, with each description following a rigid structure: Abstract, Introduction, Methods, Results, Conclusion/Discussion, Literature Cited. Of course, your paper isn't a description of a scientific experiment, and you probably aren't following the "Methods"/"Results" structure for the intermediate sections, but you can still claim that a "Conclusion" section is a completely standard part of the "scientific format".

Exploit the ambiguity of the word "conclusion". Explain that your paper begins with the "Introduction" and ends with the "Conclusion." Say that the "Conclusion" section obviously belongs at the end of the paper (let's not quibble about the bibliography and appendices), and that any other position for the "Conclusion" would be silly, since the word "Conclusion" means "the last part" by definition.

Point to talks. The audience of a talk has no opportunity to rewind the talk. If some of the target audience members need to understand X before they can understand Y, and need to understand Y before they can understand Z, then the speaker is forced to say Z after Y, and Y after X. If the speaker wants to say Z early in the talk for other reasons then the speaker has to repeat Z later in the talk. The end of a talk has the advantage of being able to rely on the maximum amount of preparatory material, and the further advantage of being the freshest material in the audience's mind when the talk ends. Consequently, you should easily be able to find examples of good talks in your area that end by summarizing their main points. Point to those talks, and say that you are imitating this masterful expository style in your paper.

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