How Do AI Models "Learn" Math? — From Next-Token Prediction to Solving Equations
> One of the questions Feynman was most often asked when teaching physics was: "How did you learn to think about physics?" He could never answer. Because the formation of thought is a gradual and mysterious process. Today's paper, from Noah Goodman's team at Stanford, tries to answer a similar but more precise question: How does AI "learn" mathematics?
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Introduction: From Predicting the Next Token to Solving Equations
Large language models do only one thing: predict the next token. They look at billions of "next tokens" and learn from billions of wrong guesses. That is the entire recipe of pretraining.
But a miracle happens: a model that only learned to predict the next token suddenly can do math. It can compute arithmetic, solve equations, and reason about geometry.
How does this transition from "predicting the next token" to "solving equations" happen? Does the model suddenly "get it" one day, or does it learn step by step from simple to complex, like humans do?
Stanford researchers designed the first systematic experiment to answer this question. Their findings are surprising.
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Chapter 1: A Clever Experimental Design
First, you need an "axis of mathematical ability."
The researchers borrowed the US K-8 Common Core standards, which specify the math skills students should master at each grade—from kindergarten "counting" to eighth-grade "solving systems of linear equations." In total: 44 fine-grained skills, organized by grade level.
Then they created a synthetic dataset, MathCAMPS, with dedicated problems for each skill, ensuring:
- These problems appear nowhere in the training data (fully synthetic)
- Each problem tests a single specific skill
- Problem formats are consistent across skills and non-interfering
- Addition and subtraction (basic operations)
- Basic fraction concepts
- Single-step word problems
- Multi-step reasoning (especially long logical chains)
- Geometric spatial imagination
- Inequality derivation
- Title: From Next-Token to Mathematics: The Learning Dynamics of Mathematical Reasoning in Language Models
- Authors: Shubhra Mishra, Gabriel Poesia, Noah Goodman (Stanford)
- Venue: COLM 2025
- Link: OpenReview
- Dataset: MathCAMPS — 44 skills, K-8 Common Core
Finally, they obtained intermediate checkpoints (training snapshots) of multiple open-source models and evaluated each one on MathCAMPS. This reveals: after 1 billion training tokens, can the model add? After 10 billion, can it solve equations?
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Chapter 2: A Striking Finding — AI Learns Math in "Grade Order"
The paper's most stunning discovery:
The order in which LLMs learn math skills is significantly positively correlated with the grade-level order of human curricula.
That means AI also learns "counting" and "shape recognition" (kindergarten level) first, then "addition and subtraction" (grade 1), then "multiplication and division" (grade 3), and only later "equations" and "geometric proofs" (middle school).
Note: The training data was randomly shuffled. No one told the model "learn addition before multiplication." There was no curriculum design. In the training corpus, eighth-grade equation problems and first-grade addition problems appear mixed together at random.
Yet inside the model's parameters, a learning order self-organized—and it happens to match what human education researchers designed.
It's like giving a child books of random difficulty with no guidance. The child will still develop reading ability in the order "letters → sentences → short passages → chapter books"—even though the input is completely shuffled.
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Chapter 3: Which Skills Come First? Which Later?
Finer analysis reveals interesting patterns:
Learned first: counting, number recognition, basic addition and subtraction, shape and pattern recognition. These are the core skills of lower grades, and they appear earliest in model training.
Learned later: multi-step reasoning (word problems), fraction arithmetic, algebraic thinking, geometric proofs. These require more complex "chain-of-thought"—not solvable in one step, requiring multiple intermediate states to be held in mind.
Hardest to learn: skills requiring "bidirectional reasoning"—for example, inverse problems where the answer is known and the condition must be found. Even when the model can solve forward, reverse-reasoning ability emerges very late.
This order aligns closely with cognitive development research. Children also develop "forward reasoning" first and "reverse reasoning" later—the latter requires more mature executive function and logical frameworks.
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Chapter 4: Instruction Fine-Tuning — A Double-Edged Sword
Most AI models undergo "instruction fine-tuning" after "pretraining" (free reading)—being taught to answer in prescribed formats.
The researchers compared math ability across the pretraining and instruction-tuning stages:
Skills that benefit:
Skills that are harmed:
In other words, instruction fine-tuning strengthens "pattern matching" (seeing a common question phrasing and producing the answer directly) but weakens "deep reasoning" (problems requiring multi-step chains of thought).
This finding echoes the education debate of "drilling vs. olympiad math": extensive standardized practice can quickly improve basic skills but may dull the intuition for deep thinking. Models are no exception.
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Chapter 5: Why?
The paper does not offer a complete theoretical explanation, but proposes several possible mechanisms:
1. Cognitive complexity is naturally stratified: without any curriculum design, as long as skills differ objectively in "complexity" (e.g., solving equations requires more intermediate variables and computation steps than counting), the learning order naturally sorts by complexity. Lowest-complexity skills converge first; high-complexity skills converge later.
2. Training signals accumulate at different rates: basic skills like "addition" get reinforced to some degree by almost every training sample containing numbers (even implicitly). "Solving systems of linear equations" appears only in narrow contexts. Signal density determines convergence speed.
3. The hierarchical nature of representation spaces: neural networks naturally follow a representation-learning path "from simple patterns to complex ones." Shallow features (number magnitude, shape attributes) stabilize first; deep features (relational derivation, conditional reasoning) stabilize later. The hierarchy of math skills happens to map onto this hierarchy.
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Chapter 6: What Does This Mean?
For AI Evaluation and Education
If models learn math in an order similar to humans, we can use human curriculum design to diagnose model learning states. Just as we use "second-grade math tests" to assess children, tools like MathCAMPS can measure "what grade" a model has reached.
For Training Strategy
Now that we know which abilities pretraining and instruction fine-tuning each help and hurt, future work can adopt different strategies at different stages: focus on deep reasoning during pretraining, and protect it from being diluted by "drill-style" fine-tuning.
For Theoretical Foundations
This finding hints at something deeper: intelligence—whether biological or artificial—may be constrained by universal laws of learning. Natural stratification by complexity, differing signal density, representation hierarchies—these are cross-system commonalities that don't depend on the specific implementation of "a human brain" or "a neural network."
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A Feynman-Style Reflection
Feynman especially disliked big words used to mask ignorance. If someone told him "AI has 'emerged' mathematical ability," he would probably push back: "What do you mean by 'emergence'? Specifically, at which token did it learn addition? At which token, equations?"
"You see, this paper does exactly what I said science should do. Not vague talk about 'models getting smarter,' but precision: 'learned addition after 250 million tokens, fractions after 1.5 billion, solving equations after 12 billion.'
And they found something that fascinates me: this order matches the human curriculum sequence. Not by design—the data was fed randomly. It means the complexity of the skills themselves determines the learning order. Simple first, complex later. That's a law of nature, not pedagogy.
One more thing I like: instruction fine-tuning harms deep reasoning. How counterintuitive—'teach more, get worse?' But think it through: the fine-tuning environment is too clean ('please answer in the following format'), and the model learns shortcuts: 'see this format, give that answer.' Real reasoning—the kind that needs multi-step thinking, that needs to hold contradictory intermediate states—gets ignored in a clean environment.
This is also a warning for education: excessive standardization kills genuine thinking ability."
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*Paper information*