How simple subtractions can stump even mathematicians — and why that matters for understanding our brains

Math is often regarded as the purest and most elegant form of problem-solving. But a new study claims that our mathematical thinking is often muddled by real-life knowledge. As weird as it sounds, our day-to-day information can get in the way of mathematical calculations — and this can happen to anyone, even experienced mathematicians.

When we learn to solve problems in school, we’re often given real-life scenarios. Jake buys a bunch of melons, then loses some of them, how many melons does he have left? Whether it’s melons, apples, or dividing flowers between vases, we’re taught at an early age to think of math in a practical context. While that approach teaches kids the practical applicability of mathematical calculations, it might also be counterproductive in some situations.

In a new study, researchers report that in some cases, worldly knowledge interferes with mathematical reasoning.

“We argue that such daily-life knowledge interferes with arithmetic word problem solving, to the extent that experts can be led to failure in problems involving trivial mathematical notions,” the  study reads.

They designed twelve problems which they presented to two groups. The main focus was the way in which the problems were presented. They were exactly the same problems, but they were presented in a different way.

“We devised six 5th grade subtraction problems (i.e. for pupils aged 10-11) that could be represented by sets, and six others that could be represented by axes”, begins Emmanuel Sander, an FPSE professor. “But all of them had exactly the same mathematical structure, the same numerical values and the same solution. Only the context was different.”

Half of the problems could be viewed as sets. Whether it’s the number of animals in a pack, the price of a meal in a restaurant or the weight of a stack of books, they all involved elements that can be grouped together in sets. For example:

  • Sarah has 14 animals: cats and dogs. Mehdi has two cats fewer than Sarah, and as many dogs. How many animals does Mehdi have?

The second type of problems (the axes ones) asked participants to calculate things like how long it takes to build a cathedral, to which floor an elevator arrives or how tall a Smurf is. Here’s an example:

  • When Lazy Smurf climbs onto a table, he attains 14 cm. Grumpy Smurf is 2 cm shorter than Lazy Smurf, and he climbs onto the same table. What height does Grumpy Smurf attain?

There’s a mental trick to the way these problems were designed. For instance, you can solve them through a simple subtraction: 14 -2 = 12. But when it comes to sets, the same approach doesn’t work.

Take the animal question with Sarah’s cats and dogs. Instinctively, you’d want to calculate how many cats and dogs Mehdi has — but you can’t. You can solve the problem and calculate how many animals he has, but not how they are divided between cats and dogs. The mathematical structure is identical: it’s the same simple subtraction, 14 – 2 = 12.

Scientists had a hunch that these answers would be a bit more difficult to answer, despite their identical mathematical structure. The context, they argue, makes it somewhat harder to process. Some problems were more difficult than others, but they all followed the same line

But even they weren’t expecting the results to be this striking.

In the non-expert adult group, 82% answered correctly for the axis problems, compared to only 47% for the problems involving sets. Surprisingly, in over more than half of the time (53%), respondents thought that there was no solution to the statement, which the team interprets as reflective of their inability to detach themselves from the elements of the problem.

Even expert mathematicians sometimes struggled with this. A total of 95% answered correctly for the axis problems, but that rate that dropped to only 76% for the sets problems. In other words, 1 out of 4 times, the experts thought there was no solution “even though it was of primary school level,” the study reads.

“We even showed that the participants who found the solution to the set problems were still influenced by their set-based outlook, because they were slower to solve these problems than the axis problems”, continues Hippolyte Gros, a researcher in UNIGE’s Faculty of Psychology and Educational Sciences and one of the study authors.

While the sample size was relatively small and the study design has significant limitations, the results are still intriguing. They seem to suggest that even in mathematical thinking, we are highly dependent on context. Even those who have the capacity to address the problems can suffer from these cognitive biases and be tricked into not finding the answer to a simple problem.

This isn’t the first study to suggest that our mathematical or scientific reasoning can be aided or hindered by semantic context. Given the wide scale at which these findings can make a difference in the education system, it seems there is a need to better understand the full impact of the semantical context.

The study has been published in Psychonomic Bulletin & Review.

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