Learning from Unsolved Problems

This is my response to a topic on AOPS. I thought I would share my views on problem-solving. Personally, I feel rather weak in problem-solving, and it's great to see many young people talented in that area. And now, my response to the question, "How do you learn from unsolved problems?"

"I am going to offer some personal experience from both contest math and academic math.

I did competition math for three years. I competed in the AMC 10, AMC 12, Cuyamaca College Field Day, and the UCSD Honors Math Contest. Did I get awards? No, only participation. But I learned something from each competition - problems are tough, and even tougher with time constraints. I was so glad to be around people that shared a love for mathematics and problem solving, and I am grateful to my Algebra II teacher for showing me this side of math.

Now, let's talk about problems I wasn't to solve. I thought about these problems for a long time, years after the contests. I found that it was best to simplify the problem, if possible. That is, solve a problem similar to the one posed. Most of the time I wasn't able to solve it entirely, but I gained insight. These problems I sat on the back-burner for a while.

Let's get to college. Fresh out of high school, my first math class was Calculus III. This was a MIM class in our math program - for Math-Intensive Majors. I wrestled with so many problems in that class and had fun doing so (probably I was the only one having fun, but who am I to say so!). Perhaps one problem I remember the most was from a take-home quiz. It was my first one, assigned the last week of August, and was meant to refresh our skills of integration techniques in Calculus II.

Find 1x4+14 dx \int \frac{1}{\sqrt[4]{x^4+1}} \ dx

But I learned something: be adventurous. Make wacky substitutions, solve similar problems, and try to come up with the solution.

Now let's jump to the present: two more semesters of undergrad to go (including one semester of student teaching at a local school), and I'm learning how to really write mathematics. One my greatest experiences was this past summer, when a professor and I went through the book Concrete Mathematics as an independent study. Those are TOUGH problems. I wasn't able to solve some, but I got a really good start on them. I compiled the most interesting (and perhaps easier) problems into a paper, which I may post at the request of the forum members.

As my real analysis professor puts it, I have to "wrestle" with the problems. Wrestling with a problem that you may not solve teaches you something - either you find the solution eventually or you get a bit closer to a solution. Another thing I learned from these problems I wasn't able to solve is that mathematics is hard work - extremely hard work. Sure, I don't have to chug out papers analyzing a work by Mark Twain or compile a project on the success of inclusion in Texas classrooms, but I have fun tackling tough problems.

So what did I really learn? I learned what mathematics really is."

Note by Sean Roberson
4 years, 11 months ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

There are no comments in this discussion.

×

Problem Loading...

Note Loading...

Set Loading...