Importance of exploring skills in mathematics

How can you improve exploring skills in high school mathematics education in order to guide students to get better opportunities to become mathematicians ?

Here are my suggestions,

1. Let students to think about how to generalize the issue. As an example how to describe k- gonal numbers using the knowledge of triangle numbers and square numbers.
2. Let students to think about alternative approaches specifically geometrical approaches.As an example geometrical proofs of A.M./ G.M. inequality.
3. Let students have knowledge of history of mathematics to know how mathematics was developed over the generations into the present situation. This may provide better vision of how prevailing issues can be changed into new forms.
4. Let students to use nontraditional or unconventional techniques because that gives much freedom for them to work on new ideas. As a second stage they can be asked to find rigorous proofs.
• It's hard to tell what question you're asking. I wonder if you'd find the work of math circles interesting. Commented May 12, 2022 at 3:48
• @ Sue - question is there, it asks ways of improving exploring skills in high school mathematics and I have suggested my approaches. Commented May 12, 2022 at 4:01
• @JanakaRodrigo: Could you specify what you mean by "exploring skills"? Commented May 12, 2022 at 5:55
• @JochenGlueck In my reading, it means the skills involved in discovering new mathematics. Commented May 12, 2022 at 6:00
• Unless you're teaching an extremely select group of students, I strongly recommend NOT trying to teach high school mathematics with the view of developing future mathematicians. Each year in the U.S. (only country I looked up the numbers) roughly 1% of bachelor degrees are in math (26 thousand out of 2 million), so we're talking about roughly 0.3% of those who obtain a high school diploma, and about 2% of those with a bachelor degree in math get a Ph.D. in math (500 out of 26 thousand). Perhaps instead of "to become mathematicians" you mean something like "become more interested in math"? Commented May 12, 2022 at 16:49

"How can you improve exploring skills [skills involved in discovering new mathematics] in high school mathematics education...?"

I'm still feeling fuzzy on the question. It seems related to the question I ask myself and discuss with others. How do we make math meaningful for students?

I agree with your implication that there are skills involved in exploring new ideas. And the way we teach math does not get at that.

I've written about the problems with math education, which are closely linked to the problems with schooling. If kids met in small groups with mentors who knew them and could just offer interesting ideas to explore, with no agenda beyond helping the child's natural desire to learn expand into all the realms they might want to explore, that might be a structure that wouldn't damage their innate, intense desire to learn.

You need to start long before high school, with a structure that respects kids' natural learning. And you need to give kids more freedom to choose. Once a kid has chosen to come to a group that is dedicated to playing around with math, what then? That's where math circles come in.

Math circles are not classes in the usual sense. There are a bunch of younger people interested in exploring mathematical ideas, and there's a leader. But the leader does not teach. They propose a problem, and give nudges sometimes. They facilitate group dynamics. There are many sites online that offer good math circle problems to explore (here's one). And for those of us who teach, Bob Kaplan (who died recently and will be missed) has some wonderful ideas about how to keep from stealing the show.

I hope this gives you some ideas, and is at least a partial answer to your question. It's a very deep question, with so many layers, I'm not sure I got at the parts you're interested in.

I think by being upbeat and motivated and a good example, as a teacher. A lot of people are inspired by HS teachers who were particularly magnetic. I probably should/could have been a physicist, but ended up in chemistry (through grad school) from exposure to a remarkable teacher.