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I'm going to be starting graduate school next week with a major in Mathematics. I've started working on some research in mathematical biology this summer. It's applied mathematics linked with computational biology.

What is the tangible advantage of getting a research publication along with my masters degree vs. just getting a masters degree?

What are the benefits for a math major, for future job prospects?

Also, if I want to apply for another masters degree later, how much will getting a publication help me?

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A tangible advantage can only be measured for an agreed upon measure (or "utility function" if you prefer). This is a classic misstep in attempts at persuasion. I can't convince you to do anything (or appreciate anything) if I don't know your core values and phrase my arguments in terms of those values. Unfortunately, the phrase "future job prospects" is rather vague, so I will guess a bit at what you're hoping to pursue and accomplish.

First suppose that you're considering getting a PhD in the future. This one is easy. If you do go for a PhD, a large chunk of your time will be spent on research and publishing. What exactly this looks like varies significantly even within math, depending on how applied your subfield is. However, if you stick with the same subfield (or even one nearby), then publishing a paper during your masters will often give you a reasonable sense of what you'll be doing during your PhD. This will help you to know whether you want to get a PhD or not. Having published a paper during your masters will also help convince the admissions committee that you are likely to succeed in a PhD program, and improve your chance of being admitted to a PhD program.

Suppose instead that you plan to pursue a job that requires technical writing skills. This one is pretty obvious too. Writing (as with most things) is a skill that you acquire only by doing. Many (most?) journals require a higher quality of writing than is typically needed in your masters thesis. They also require you to write in a certain style, to a certain audience, etc. Being forced to do this will make your writing better.

Finally, suppose that you want to convince someone that you take initiative or that you conceive a project and see it through. In either case, having published a paper can serve as good evidence. Like most things, after you've published multiple papers, the process will get faster (though not likely fast). But the first time you do it, you'll invest a lot of time, and often not know quite what is needed. This can be frustrating, but also rewarding when you succeed. The ability to teach yourself new skills is far more marketable than anything you will learn in grad school.

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