What do thousands of students’ answers reveal about the times tables?

Maria Garcia|22/09/2026|8 min read

In collaboration with: Cèlia Cortés, Marc Caelles, Marc Colomer, Laura Morera

What do thousands of students’ answers reveal about the times tables?

What errors and response times tell us about how students solve their multiplication tables.

SUMMARY

In this article, we analyse millions of student responses to multiplication problems to see what errors and response times can tell us about how the times tables are learned. We will see that not all multiplications are equally difficult, that errors follow very specific patterns, and that combining accuracy and speed allows us to identify families of operations with particularly interesting behaviours.

What is 2 × 5? And 7 × 8? Just by reading them, one of the two may have come to mind faster than the other. Both are multiplications we can learn and retrieve from memory, but that doesn’t mean our brain treats them in exactly the same way.

When we think about the times tables, it’s easy to picture them as a list of equivalent operations: either a student knows a multiplication fact or they don’t know it yet. But what happens when we stop looking at a single response and observe thousands at once?

A much richer picture emerges. Some multiplications are answered correctly more often than others; some errors appear again and again; and some operations take longer, even when they end up being solved correctly. In other words: the times tables don’t look like a collection of independent facts, but rather a system full of patterns, exceptions and connections.

In this article, we will look at this system from three very simple angles: how often each multiplication is answered correctly, what students answer when they make a mistake, and how long they take to respond. Three different measures that bring us closer to the same question: do all multiplications show up in the same way in students’ responses?

An uneven map of the times tables

Let’s start with the most direct question: which times tables lead to the most errors?

To answer it, we analysed data from a multiplication fluency activity. The task consists of solving as many multiplications as possible within a limited time, drawing on students’ memory and on how they respond under pressure. With data from the 2025-2026 school year, we already have data from more than 170,000 students of different ages solving multiplications. That is enough data for very clear patterns to begin to emerge beyond individual differences.

If we calculate the error rate for each table, the first thing we notice is that difficulty is not evenly distributed. Some tables are solved much more easily than others.

Part of this pattern is fairly intuitive. In general, as we move towards tables with larger numbers and products, errors increase. This is what cognitive psychology calls the problem-size effect: multiplications involving larger numbers tend to be more difficult and usually also take longer to answer.

In the figure, we have grouped all ages together to show the pattern as clearly as possible. When we analyse each age separately, we see the same trend, which holds consistently as students get older.

But here comes the interesting part: problem size doesn’t explain everything.

If only the magnitude of the numbers mattered, we would expect a much more regular progression. Yet some tables are solved better than their size would predict. The 5 times table is the clearest example: its structure is especially regular —the results always end in 0 or 5— and it is closely linked to familiar counting patterns —adding up how many fingers or toes we have—, such as counting in fives, which can make it easier to learn and retrieve. The 9 times table also shows fewer errors than its magnitude would suggest, but if we look at its response time, it is higher than we would expect given its error rate. This suggests that students may be solving it using strategies. Multiplications with two equal factors —such as 4 × 4 or 7 × 7— also form a particular family and often perform better than other operations of similar magnitude.

However, knowing that one multiplication leads to more errors than another only tells us how much students get wrong. It doesn’t yet tell us how they get it wrong. And this is where the incorrect answer itself becomes a new source of information.

Errors are not random

Let’s imagine any multiplication. If students didn’t remember the result and simply answered at random, we would expect to find many different answers spread out more or less widely. But when we look at what they actually answer, that’s not what happens.

To observe this, we analysed students’ responses during their weekly arithmetic fluency practice, a context in which they solve operations with the aim of gaining speed and accuracy in calculation.

Here, once again, we have grouped all ages into a single distribution to make the overall pattern visible. When we separate them, however, an interesting progression appears: among the youngest students we already see the typical peaks associated with specific answers, but there is still more spread towards other values. With age, this distribution becomes more concentrated: answers increasingly cluster around a few results, many of which correspond to other multiplication facts.

This concentration allows us to start seeing relationships between multiplications. Incorrect answers may include values close to the correct result, but also results that correspond to “neighbouring” operations. For example, when trying to retrieve a multiplication, the result of another similar or nearby operation within the same table may come up. For instance, when you see 6 × 7, you hesitate between 42 (the correct one!) and 48, because that is the answer to 6 × 8.

And this changes the way we interpret errors. An error is not necessarily just the absence of the correct result. It can also be the presence of another answer competing with it.

We can put forward several hypotheses to explain why many errors match the results of nearby multiplications. One possibility is competition between related arithmetic facts: when students try to retrieve a multiplication from memory, similar or neighbouring facts are also activated, and one of these incorrect results may end up interfering with the answer. Another possibility is serial recitation of the table: students mentally run through the sequence of results in the table and accidentally shift one position, so they answer with the previous or next result. With the error pattern we observed, however, we cannot tell which of these two explanations is correct.

This perspective is especially useful because two incorrect answers can tell different stories. A widely scattered distribution of errors is not the same as an operation in which a single incorrect answer accounts for a large share of the errors. The error rate is the same in both cases, but the underlying structure is not.

Adding time: accuracy and speed go hand in hand

So far, we have looked at what students answer. But every answer contains a second piece of information that often goes unnoticed: how long it took them to give it.

A multiplication can end with the correct answer and still have required more thought. That’s why, if we only distinguish between right and wrong, we miss an important part of the behaviour.

The vast majority of multiplications line up along the same trend cloud: the more often students make mistakes, the longer they take, on average, to give an answer. Operations with small numbers, such as 2 × 3 or 4 × 2, tend to cluster in the zone of few errors and fast responses. By contrast, multiplications with larger operands, such as 6 × 9 or 8 × 7, appear more often in the zone of more errors and slower responses.

This relationship is consistent with a fairly intuitive idea: well-consolidated operations are retrieved more quickly and accurately, while those that are harder usually take more time and lead to more errors. But there is one family that clearly breaks away from this pattern: multiplication by 0.

Operations with 0 show a distinctive profile. Despite accumulating a considerable number of errors, responses are much faster than we would expect if they followed the same relationship as the rest of the multiplications.

This profile is consistent with an interesting interpretation: in many cases, students may be quickly applying a rule associated with 0 rather than retrieving an arithmetic fact from memory. The speed of the response suggests that they are not necessarily “searching” for the result for very long; the problem is that the rule they apply may be wrong or poorly consolidated.

And this is precisely where it is useful to study accuracy and speed together. Errors alone tell us that the ×0 family presents additional difficulties that the ×1 family does not. When we also look at time, however, it reveals that this family behaves in isolation from the rest of the facts.

When we looked at the answers students give when they make a mistake in the 0 times table, a very clear pattern emerged: they often answer with the other operand. For example, for 6 × 0, a common incorrect answer is 6; for 8 × 0, it is 8, and the same pattern repeats with the other operands. This type of error is especially interesting because it points to a possible confusion between very simple rules: students may be applying the rule for multiplying by 1 —“the number stays the same”— or transferring to multiplication the rule for adding 0, where the number is also preserved. In both cases, rather than a random error, the data suggest that students are applying a rule they know, but in the wrong context.

In the classroom

In the classroom, these patterns fit well with an important idea: understanding the times tables and building fluency are not exactly the same process, but both are necessary. The 0 times table is a good example: a conceptually very simple rule can be confused with adding 0 or multiplying by 1, especially when students need to answer quickly.

In this sense, once the tables are understood, fluency practice can place more emphasis on operations with operands from 6 to 9, which tend to be more difficult. The aim is for retrieving their results to require fewer and fewer mental resources, leaving more room for more complex mathematical tasks.

At the same time, working well on properties such as the commutative property greatly reduces the amount of information that needs to be consolidated. If we consider the tables from 0 to 9, we go from treating 100 operations as if they were independent to just 55 different facts, because, for example, 7 × 8 and 8 × 7 share the same result.

In this way, connections, understanding and practice do not compete with one another: they complement each other to build knowledge that is meaningful, flexible and fluent all at once.