Knowing when to reach for a calculator is less useful if you do not know how to use one.
Researchers studying tool-integrated reasoning — the technique of giving large language models access to external tools like code executors — found that existing work focused almost entirely on when a model should invoke a tool, while ignoring how it should use one. Their paper identifies two distinct patterns models fall into: a calculator pattern, where code runs a direct computation, and an algorithmic pattern, where the model encodes a problem as a full program. Picking the wrong pattern, even with correct underlying logic, causes failures. The team built a two-stage training framework that first develops competence in both patterns, then trains the model to align its pattern choice with what a teacher model prefers.
The results are hard to dismiss. On MATH500, the rate of useful code invocations rose from 64.0% to 70.5%. On AIME24 — a substantially harder benchmark of competition math problems — accuracy jumped from 26.7% to 50.0%. That near-doubling on a difficult benchmark suggests the failure mode being fixed here is real and common, not a minor edge case.
Most reasoning model research still treats tool use as a binary switch. This work reframes it as a skill with its own grammar — which is the kind of unsexy, structural insight that tends to matter more than the next benchmark-topping model announcement.