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It surprises many students: an AI chatbot can explain the work-energy theorem beautifully and then multiply two numbers wrong. This is not a random glitch. Researchers have studied why language models stumble on arithmetic and multi-step maths, and the reasons point directly to how you should check any answer they give you. This post explains three causes, then gives a five-step checking routine with a worked example.
Why can an AI explain maths but still miscalculate?
Cause 1: it sees numbers as chunks, not digits
Language models read text in pieces called tokens. One study notes that GPT-3.5 and GPT-4 use separate tokens for one-, two- and three-digit numbers, and found that the way numbers are split changes arithmetic accuracy; separating digits with commas so they are grouped from the right "leads to largely improved performance" (Singh and Strouse, 2024). The authors also found that the models' mistakes follow stereotyped patterns rather than being random. You line digits up by place value when you add; a model does not automatically see a number that way.
Cause 2: long calculations are many small steps chained together
Nouha Dziri and colleagues tested transformer language models on multi-digit multiplication, logic puzzles and a dynamic programming problem (Faith and Fate, 2023). They concluded that the models often solve such tasks by matching patterns they have seen "without necessarily developing systematic problem-solving skills", and argued that performance can decay quickly as tasks get more complex. Each extra step is another chance for an error, and one slip early on carries through to the end.
Cause 3: small changes to a question can throw it off
Mirzadeh and colleagues built GSM-Symbolic, a set of grade-school word problems generated from templates, so the same problem can appear with different numbers (arXiv, 2024). Model performance dropped when only the numbers changed. Adding a single clause that seems relevant but is not caused drops of up to 65%. The authors suggest the models may be reproducing reasoning steps from training data rather than reasoning logically.
Models change quickly, and some apps now run calculations in code, so results on these tests may differ today. But you rarely know how a particular answer was produced, so it makes sense to check.
A five-step routine for checking AI maths
Estimate first. Before reading the AI's answer, guess the size of the result in your head. If you expect about 40 and it says 400, stop. Our post on estimation and Fermi problems shows how.
Check the units. Every term in an equation must have the same dimensions, and the final unit must make sense. A 30-second habit from dimensional analysis catches many setup errors.
Recompute the arithmetic yourself. Use a calculator on every multiplication, division and root. Do not ask the same chat to "double-check"; it may simply repeat itself.
Substitute back or test a special case. Put the answer back into the original equation, or set a quantity to zero or to a very large value and see if the formula behaves sensibly.
Change one number. Ask the same question with one value changed, or with an irrelevant detail added. A correct method gives an answer that shifts in the way you expect. This is the GSM-Symbolic test turned into a study habit.
A worked example
Suppose you ask: "A stone is dropped from rest from a 45 m tower. Ignore air resistance and take g = 9.8 m/s². How long does it take to hit the ground? The tower was built in 1998 and has 15 floors."
The extra sentence about 1998 and 15 floors is the kind of irrelevant clause GSM-Symbolic used. Here is how the routine runs.
Estimate: with g ≈ 10, h = ½gt² gives t² ≈ 9, so t ≈ 3 s.
Units: t = √(2h/g) gives √(m ÷ m/s²) = √(s²) = s. Good.
Recompute: 2 × 45 ÷ 9.8 = 9.18, and √9.18 ≈ 3.03 s.
Substitute back: ½ × 9.8 × 3.03² ≈ 45 m. Matches.
Change one number: for a 180 m tower (four times higher) the time should double, to about 6.06 s, since t grows with √h.
If an AI answer used the floors or the year anywhere, or gave something like 9.18 s (forgetting the square root), the routine would catch it.
How to ask for maths help that is easier to check
Ask for every step on its own line with units, so you can check each one.
Ask it to state the formula before substituting numbers.
For long arithmetic, ask whether it can calculate with a code or calculator tool, if your app offers one.
When you find an error, ask "which line is wrong and why?" rather than "try again", so you learn where the method broke.
If your own working is where things go wrong, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning rather than printing an answer, so you can compare your method one line at a time. For a fuller routine on physics answers, see when AI gets physics wrong.
