Can AI Truly Teach Math Modeling? Exploring Knowledge Types and Limits

Recently I have been pondering how to position AI in everyday mathematics modeling teaching. AI is powerful and can provide reliable answers, but if students rely on it for problem solving and assignments they may not develop true thinking skills. Yet, learning to use AI responsibly is also an educational duty.

The breakthrough I found is to first clarify what knowledge students should learn.

Two important but rarely mentioned types of knowledge are driving knowledge and philosophical knowledge , concepts from Zhu Haonan’s book Why Is It Mathematics? . Zhu identifies five knowledge types:

Factual knowledge Definition: knowing "what it is". Examples: mathematical definitions, theorems, data, rules, industry standards. Role: provides basic material for modeling but cannot reach the essence of problems alone.

Procedural knowledge Definition: knowing the steps to do something. Examples: algorithm implementation, modeling workflow, use of calculators or software. Role: ensures technical feasibility of model building and helps complete specific tasks.

Principle knowledge Definition: knowing "why it is". Examples: disciplinary ideas, theories, concepts. Role: supports knowledge transfer and deep problem solving; it is the foundation of innovation.

Driving knowledge Definition: knowing "why it is not". Examples: ability to discover problems and optimize models through trial‑and‑error and reflection. Role: pushes students beyond existing frameworks, uncovering new ideas; it is the core of innovative work.

Philosophical knowledge Definition: knowing "why it exists". Examples: exploring the meaning of knowledge, contemplating the essence of models. Role: stimulates students' learning motivation and deeper understanding of problems.

The following screenshot from Zhu’s lecture illustrates the five knowledge categories:

When I first heard about driving and philosophical knowledge, I realized why some students seem to learn a lot yet fail to grasp the essence: they lack these two knowledge types.

Balancing the teaching of all five knowledge types, especially cultivating driving and philosophical knowledge, is a challenging but essential task. Traditional teaching often neglects them, yet they are crucial for fostering creativity and critical thinking.

Driving knowledge must be acquired through trial‑and‑error and practice; it cannot be transmitted by pure lecture. It helps students identify insufficient assumptions, model flaws, and propose improvements.

Philosophical knowledge focuses on the meaning and value of knowledge, encouraging deep reflection on the social and humanistic significance of modeling.

Below are two diagrams that illustrate my understanding of the five knowledge types and a typical student knowledge structure when only the first three types are present:

AI excels at the first three knowledge types—factual, procedural, and principle—by quickly retrieving information, automating calculations, and assisting in understanding complex models. However, AI struggles with driving and philosophical knowledge, similar to how a navigation app can tell you the route but not why you should choose a particular destination.

To assess a student’s innovative thinking, one could model the proportion of driving and philosophical knowledge within the total knowledge:

Innovation literacy = (Driving knowledge + Philosophical knowledge) / Total knowledge

Students with high innovation literacy have a larger share of driving and philosophical knowledge, while those with low literacy rely mainly on basic knowledge layers.

In summary, AI can effectively address factual, procedural, and principle knowledge in teaching, but the development of driving and philosophical knowledge requires active student exploration. The goal of mathematics modeling education is to combine solid foundational knowledge with the cultivation of innovation and independent thinking through these higher‑order knowledge types.

—

Why Is It Mathematics? 30 Dialogues on Modeling, Cognition, and Scientific Thought by Zhu Haonan further explores these ideas and is highly recommended for anyone interested in mathematics and modeling.