Human children learn math in a well-established order: counting first, then addition/subtraction, multiplication/division, fractions, and finally algebra. This sequence is deliberately designed in decades of education research—the "Common Core" standards, where each step builds on the previous one.
But what if a language model—trained only to "predict the next token"—develops mathematical reasoning on its own? Is its learning order random? Or does it also learn counting before addition?
A paper at COLM 2025 gives a striking answer: the order in which LLMs acquire math skills correlates measurably with the human curriculum.
The experiment: the MathCAMPS dataset
The researchers built a synthetic dataset called MathCAMPS. This is not an ordinary math problem set—it encodes 44 fine-grained skills spanning kindergarten through grade 8.
Skill ordering follows the human Common Core standards, roughly:
1. Counting and comparison: greater than, less than, equal 2. Addition and subtraction: single-digit, multi-digit 3. Multiplication and division: times tables, long division 4. Fractions: understanding, comparison, operations 5. Basic geometry: shapes, area, perimeter 6. Pre-algebra: unknowns, equations
Every problem is novel—the model never saw it in training data—so the test measures genuine reasoning, not memorization.
They then analyzed multiple open-source LLMs (full checkpoint trails from pretraining through post-training) to see at which training stage each capability emerged.
Core finding: the order is not random
During pretraining, the order in which models acquire math skills shows a statistically significant, measurable correlation with the human curriculum.
This is not a vague "kind of similar"—it is a significant correlation. Even when training data is randomly shuffled (so the model does not see counting before addition), the model still spontaneously masters simple skills before complex ones.
More specifically:
1. The most basic skills emerge earliest: counting and comparison are acquired first 2. Fractions and algebra appear late: consistent with human education—fraction understanding requires number sense; algebra requires arithmetic 3. Skills within the same grade band tend to emerge together: elementary-level skill clusters appear roughly simultaneously
Nobody told the model "you must learn addition before multiplication." The order emerged spontaneously from the single objective of next-token prediction.
Instruction tuning: not all skills benefit
The researchers also analyzed instruction tuning—a widely used post-training method—and its effect on individual math skills.
The result was not "everything improves." Some skills do improve, but others decline.
While the paper does not list a full 44-skill comparison table, the finding itself is crucial: "universal optimization" of models may produce positive returns on some capabilities and negative returns on others, and this impact is distributed at a cognitive granularity—not global.
This aligns with recent research showing that post-training optimization sacrifices certain "foundational" abilities in exchange for higher "helpfulness" scores.
Explaining the emergent order
The paper offers no definitive causal explanation—that is beyond its scope. But several possibilities are worth considering:
Possibility 1: Natural distribution in data. Even with shuffled data, text about counting and comparison may simply be simpler than text about fractions and algebra. The model masters simple patterns before complex ones.
Possibility 2: Cognitive complexity is intrinsic. Multiplication really is harder than addition—regardless of language or data source. This "difficulty gradient" exists in the structure of mathematics itself; any learning system must climb it.
Possibility 3: Evidence of emergent structure. LLMs develop internal representations through next-token prediction. The hierarchy of these representations may correspond to the cognitive hierarchy of mathematical concepts. Fractions depend on addition and multiplication as sub-skills, so the model must build those representations before it can construct fraction representations.
Whichever explanation holds, the finding points to an important insight: LLM learning is not random pattern-matching on a blank slate. Some intrinsic constraint—perhaps the structure of mathematics itself, or the interplay of data and optimization—causes capabilities to emerge in a specific order.
The Feynman take
Among AI research findings, this type of result fascinates me most. Not because it proves "AI is like humans"—it may actually prove that "mathematics itself has intrinsic structure, and any learning system that extracts patterns from data will eventually move along that structural line."
Think about it. Multiplication is essentially repeated addition. If you don't understand addition—if you don't know what combining two numbers means—how could you understand multiplication? This is not a design choice of human education systems; it is a logical necessity.
So LLMs failing to learn multiplication before addition is not because they are "like children." It is because mathematics itself is that way. It is Platonic—relations between concepts precede any learning system. You don't need to design a curriculum; logic has already written it.
But the other finding—instruction tuning harming some skills—deserves caution. When we "align" models, we may inadvertently prune certain capabilities. A more "helpful" model may simultaneously become worse at fractions. And this is invisible if you only track aggregate evaluation scores.
This points to the dilemma of AI evaluation: aggregate metrics hide internal shifts in capability distribution. Perhaps we need to track model capabilities at fine-grained skill level, the way doctors monitor blood work.
Reference paper
Shubhra Mishra, Gabriel Poesia, Noah Goodman. "From Next-Token to Mathematics: The Learning Dynamics of Mathematical Reasoning in Language Models." COLM 2025.