From Techniques to Insight: Building a Modeling Mindset and Inner Skill

If the goal is to join a modeling competition tomorrow, I would only teach students the "techniques" and "skills"; if preparing for six months ahead, I recommend mastering the modeling mindset ; if you want to truly apply mathematical modeling in life, you must cultivate the inner skill .

Recently I talked with a student who proudly said, "Teacher, I have learned linear regression, mathematical programming, and have also encountered Markov chains."

I replied: Good, you know quite a bit. Can you tell me under what conditions these algorithms are applicable? Besides the mentioned algorithms, are there similar ones and how do they differ? Do you know how to improve existing models or create new ones?

These questions are hard to answer.

The specific models mentioned above (e.g., linear regression) can be seen as "techniques" or "patterns," but deeper investigation shows that each technique has its origin and purpose, which requires a mindset . The mindset is the modeling thinking framework and principles that help us flexibly choose appropriate models for complex real problems and know how to adjust and optimize them.

A mindset is not just understanding a single algorithm; it is insight into the essence of the problem. It guides decision‑making, acts as implicit rules during modeling, and explains why we choose one model over another.

Recently I have been researching the mindset of mathematical modeling and have proposed the "Flying Arrow Analysis" method and an Agent framework (see my earlier articles). These clarify modeling principles and direction. Even without ready‑made "techniques" to apply, you can still create new models through innovation.

Of course, the mindset is only a higher‑level technique; deeper still is the inner skill . What is the inner skill ? It is the rich experience and profound insight into the essence of mathematical modeling gained through long‑term practice. It does not depend solely on how many papers you have read or models you have built, but on whether you can flexibly apply your experience to solve problems in diverse real scenarios. The cultivation of the inner skill comes from countless modeling attempts and reflections, building judgment, selection, and optimization abilities.

My favorite students are those who continuously practice and reflect. Hearing and speaking are easy; doing is hard. Only by repeatedly acting can one accumulate and reap rewards.

The inner skill is hard to articulate; it represents ability, cognition, and intuition. It manifests as appropriate thinking choices and high task achievement, and its cultivation method is continuous "reinforcement learning".

These concepts of techniques, mindset, and inner skill are not limited to mathematical modeling; they are similar in developing many abilities such as writing, programming, marketing, teaching, and more.

The key is to understand our current level, define our goals, and then plan future time investment accordingly.

Mistaking techniques for everything results in only superficial competence.