Science Research Topics for Math Competition Students
Science Competition Topics for Math Competition Students
By James H. Choi
http://column.SabioAcademy.com
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Q: Is there a Math category in the Science Competition?
Yes. If you look at the Science Competition categories, you will see that every competition has a Mathematics category. But you might wonder how a Math Competition is different than a math category in a Science Competition. They are very different, yet very similar. Students trained for Math Competitions have excellent skills for mathematics research in Science Competitions. They should consider redirecting their knowledge and efforts toward Science Competitions to kill two birds with one stone.
Mathematics is an old subject, especially compared to computer science or genetics, for instance, which have existed only since recent years. Humanity’s best minds have worked on math for millennia. Therefore, it is only natural all the lowhanging fruits have been taken.
But now we have a new tool to help us reach the higherhanging fruits: computers. Yes, those machines on which you play games and update Facebook are also a powerful tool in mathematics research. Because the computer is a relatively new instrument, all the highhanging fruits have not yet been taken — and you have the opportunity to grab some.
Q: What is Experimental Mathematics?
This new field of mathematics is called Experimental Mathematics . What’s so exciting is that highschool and even juniorhigh students have a chance to contribute to this field.
In traditional math, highschoollevel knowledge doesn’t prepare students to contribute to research. But with computers, they might stumble upon patterns in data or create an unproven theory (also called “conjectures”) for further research. The world of mathematics is littered with unproven conjectures awaiting some genius to come along to (dis)prove them.
You might not be that genius. But why not show off your mathematical recklessness and irresponsibility by creating even more conjectures? Create even more work for that future genius! And, you will be praised because that’s exactly what mathematics entails: Discovering hidden patterns — whether we understand them or not; whether they are useful or not. Science Competition judges will reward you for creating such a mathematical mess.
Q: But exactly what do you research?
A: The cover you see on the left is an excellent book I found recently. The computer as crucible by Jonathan Borwein is an excellent introductory book that explains the history and principle of Experimental Mathematics. This should be the first book for anyone interested in Experimental Mathematics. At the end of each chapter, a section titled “Explorations” offers a handful of interesting problems for you to try.
Another good book is Mathematics Experiments by Shangzhi Li while Experimental Number Theory by Fernando Villegas focuses on Number Theory.
But those “Explorations” exercises are simply practice, rather than research topics. To find research topics that interest you, I recommend reading Research Experience for All Learners. This book contains every Experimental Mathematics research topic I could think of — and many more. Pay attention to the section titled “Further Directions for Undergraduate Research” at the end of every chapter. Almost every one of them could be your research topic.
Q: And how do you research?
A: You perform Experimental Mathematics research by programming computers to do your bidding. You either use computation to confirm a suspicion you have, or you create a suspicion by looking at computation results.
Various “computer languages” help you explore those topics. But Mathematica is the bestsuited language, more powerful and complete in its mathematical functions than any other language on today’s market.
To acquire suspicions (also called conjectures or “hypotheses”), brainstorm ideas by using the books above or invent one yourself.
Q: Is there a course I can take to get started?
I am not aware of any highschool Experimental Mathematics course other than my own. If you know one, please share with us in a comment at the bottom. I do not teach Experimental Mathematics per se. I teach researchoriented programming courses in which students learn to tackle heavily mathematical problems. These courses are focused let the students perform scientific/mathematical research, rather than making them computer programmers. After taking the SR100 and SR110 courses (10 weeks each; available at Sabio Academy Research site), you will be able to tackle any of those “Further Directions” problems in from Research Experience for All Learners book.
Q: Who should learn Experimental Mathematics?
Those who shine in Math Competitions and Science Competitions will find Experimental Mathematics an ideal field. It is also great for those legendarily smart but lazy students. (You know who you are.) Think about it: You have no lab, beakers, supervisor, equipment — nothing. Just you and your computer. And the best part? You get to come up with the idea. The computer will do all the real work from there. It does not get better than this.
Q: Should I find a mentor?
You can carry out research on your own, but it is to best to have a mentor who saves you from wasting months going the wrong direction. You should find an expert in the field of your chosen research topic.
I also mentor students in the fields of my expertise .
Happy Researching!

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