Reaching skill level for Mathematical Research

[Oliver Ross]

I'm a highly determined (and slightly stubborn) high school freshman who wants to learn enough mathematics to conduct mathematics research with a mentor. Looking on the internet, I found a program called MIT Primes, where students can be matched with graduate students/professors at MIT to conduct research over the course of one-year. However, looking at their qualifying test, I am very far from that skill level. A link to their qualifying test can be found here.

https://math.mit.edu/research/high school/primes/usa/index.php

The program seems to only accept juniors, so one could say I have plenty of time to prepare, but I know that Junior Year will be extremely stressful and full of college preparations, so I'm aiming to mentally ready for such a program by next fall. I know it's a stretch, but I believe if I work that hard, I'll get somewhere close to where I want to be.

If anyone knows any suggestions for reaching such a level of proficiency even after studying and completing the following outline, I would be really, really grateful! Any adjustments or removals to this outline are also gratefully received. How would you train a high schooler with only Algebra II and Honors Geometry under her belt to conduct mathematical research?

I'm really interested in Group Theory and Abstract Algebra, as well as Braid Theory, Topology, Number Theory and Combinatorics.

PROBLEM SOLVING
The Art of Problem Solving, Vol. 1
The Art of Problem Solving, Vol. 2
Introduction to Number Theory, AOPS

ALGEBRA
Algebra by I.M Gelfand
Schaum's Outlines for Algebra

GEOMETRY
Geometry I: Planimetry by Kiselev
Geometry II: Stereometry by Kiselev
Geometry by Serge Lang
Schaum's Outlines for Geometry

TRIGONOMETRY
Trigonometry by I.M Gelfand
Schaum's Outlines for Trigonometry

REFERENCE
Basic Mathematics by Serge Lang

PRE-CALCULUS
Schaum's Outlines for Pre-calculus

CALCULUS
AP Calculus AB, Thinkwell
AP Calculus BC, Thinkwell

PROOF-WRITING
?

Mathematics is my favorite thing in the entire world-- I'm not afraid of putting in a ridiculous amount of work, because I am a ridiculous type of student :) Just tell me what I'll have to do!

Thanks and have a safe and happy day!

 

[DEvens]

One thing that motivated me in high school was prize exams. I wrote about five of them. There were two physics ones, two math ones, and a chemistry one. I did kind of badly on the chemistry one. There were three guys in my high school who beat me. But on the math and physics ones I did really well.

Google up the phrase "high school prize exam." I don't know what the best one to look for is in your home town. In Canada: Sir Isaac Newton Exam, Canadian Association of Physics Exam. There is an American Mathematical society exam. You should be able to find several more. There should be lots of web sites with old exams and solutions.

https://uwaterloo.ca/sir-isaac-Newton-exam/
https://www.cap.ca/en/students-educators/prizes-students/CAP-high-school-cegep-prize-exam

There used to be a book with exams and answers to the Sir Isaac Newton. It was called A Decade of SIN. But I don't think it's in print any more, and I'm not giving up my copy.

Find out what kind of questions they think you should be challenged by at your level. Then read up on the material in that context. Then try a few more questions.

And if you do well, you probably get a nice scholarship and some open doors.

 

[Oliver Ross]

One thing that motivated me in high school was prize exams. I wrote about five of them. There were two physics ones, two math ones, and a chemistry one. I did kind of badly on the chemistry one. There were three guys in my high school who beat me. But on the math and physics ones I did really well.

Google up the phrase "high school prize exam." I don't know what the best one to look for is in your home town. In Canada: Sir Isaac Newton Exam, Canadian Association of Physics Exam. There is an American Mathematical society exam. You should be able to find several more. There should be lots of web sites with old exams and solutions.

https://uwaterloo.ca/sir-isaac-Newton-exam/
https://www.cap.ca/en/students-educators/prizes-students/CAP-high-school-cegep-prize-exam

There used to be a book with exams and answers to the Sir Isaac Newton. It was called A Decade of SIN. But I don't think it's in print any more, and I'm not giving up my copy.

Find out what kind of questions they think you should be challenged by at your level. Then read up on the material in that context. Then try a few more questions.

And if you do well, you probably get a nice scholarship and some open doors.

Thank you for your advice! I live in New York, so I think the closest equivalent to what you're mentioning would be the American Mathematics Competition 10 & 12. It's how the American team for the International Mathematics Olympiad is eventually chosen. It's also an exam, but the prizes are really just harder exams until you reach the international level (only about 6 people get there) -- not including the obvious perk to your college admissions. Many colleges actively seek out students who have done well in these exams. The competition math is extremely different than that seen in traditional school, with a heavy-emphasis on problem solving and, in harder exams, proof writing. They're only open to high school students as well. I'll look into preparing for these and see where it takes me! :)

Thanks and have a safe and happy day!