Expert mathematicians stumped by simple subtractions

mathematics
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Mathematical thought is seen as the pinnacle of abstract thinking. But are we capable of filtering out our knowledge about the world to prevent it from interfering with our calculations? Researchers from the University of Geneva (UNIGE), Switzerland, and the University of Bourgogne Franche-Comté, France, have demonstrated that our ability to solve mathematical problems is influenced by non-mathematical knowledge, which often results in mistakes. The findings, published in Psychonomic Bulletin & Review, indicate that high-level mathematicians can be duped by some aspects of their knowledge about the world and fail to solve primary school-level subtraction problems. It follows that this bias needs to be factored into the way mathematics is taught.

Maths teaching at school usually draws on examples taken from everyday life. Whether it's adding up oranges and apples to make a pie or dividing a bunch of tulips by the number of vases for a floral arrangement, we master mathematics with the help of concrete examples. But to what extent do the examples chosen affect a child's ability to use the mathematical concepts in new contexts?

Researchers from UNIGE and the University of Bourgogne Franche-Comté tested the degree to which our worldly knowledge interferes with mathematical reasoning by presenting 12 problems to two distinct groups. The first group consisted of adults who had taken a standard university course, while the second was composed of high-level mathematicians. "We speculated that the adults and mathematicians alike would rely on their knowledge about the world, even when it would lead them to make mistakes," explains Hippolyte Gros, a researcher in UNIGE's Faculty of Psychology and Educational Sciences (FPSE).

Counting animals versus counting centimetres

When faced with numbers, we tend to represent them mentally either as sets or as values on axes. "We devised six fifth-grade subtraction problems (i.e. for pupils aged 10 to 11) that could be represented by sets, and six others that could be represented by axes," says 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."

These problems were presented in two types of contexts. Half of the problems involved calculating the number of animals in a pack, the price of a meal in a restaurant or the weight of a stack of dictionaries (elements that can be grouped together as 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 problem required calculating how long it takes to build a cathedral, to which floor an elevator arrives or how tall a Smurf is (statements that can be represented along a horizontal or ). For 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?"

These can all be solved via a single calculation: a simple subtraction. "This is instinctive for the problems represented on an axis (14-2 = 12, in the case of the Smurfs) but we need to change perspective for the problems describing sets, where we automatically try and work out the individual value of each mentioned subset, which is impossible to do. For instance, in the problem with animals, we look to calculate the number of dogs that Sarah has, which is impossible, whereas the calculation 14-2 = 12 provides the solution directly," explains Jean-Pierre Thibaut, a researcher at the University of Bourgogne Franche-Comté. The scientists relied on the fact that the answer would be more difficult to find for the animal problems than the Smurf problems, despite their shared mathematical structure.

When worldly knowledge impedes mathematical reasoning

"We presented the 12 problems to both groups of participants. Each problem was accompanied by its solution and the participants had to decide whether it was correct or if the problem couldn't be solved," adds Gros.

The results were surprising. In the non-expert adult group, 82 percent answered correctly for the axis problems, compared to only 47 percent for the problems involving sets. In 53 percent of cases, the respondents thought that there was no solution to the statement, reflecting their inability to detach themselves from their knowledge about the elements mentioned in the statements.

Regarding the expert mathematicians, 95 percent answered correctly for the axis problems, a rate that dropped to only 76 percent for the sets problems. "One out of four times, the experts thought there was no solution to the problem, even though it was of primary school level. And 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," says Gros.

The results highlight the critical impact that knowledge about the world has on the ability to use mathematical reasoning. They show that it is not easy to change perspective when solving a problem. Thus, the researchers argue that teachers need to take this bias into account in math education.

"We see that the way a mathematical problem is formulated has a real impact on performance, including that of experts, and it follows that we can't reason in a totally abstract manner," says professor Sander. Educational initiatives are required based on methods that help pupils learn about mathematical abstraction. "We have to detach ourselves from our non-mathematical intuition by working with students in non-intuitive contexts," concludes Gros.


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More information: Hippolyte Gros et al. When masters of abstraction run into a concrete wall: Experts failing arithmetic word problems, Psychonomic Bulletin & Review (2019). DOI: 10.3758/s13423-019-01628-3
Journal information: Psychonomic Bulletin & Review

Citation: Expert mathematicians stumped by simple subtractions (2019, July 10) retrieved 17 July 2019 from https://phys.org/news/2019-07-expert-mathematicians-stumped-simple.html
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Jul 10, 2019
The problem is asking for a concrete answer in the world as we understand it, but the researcher's interpretation is that humans mistakenly don't solve the problem problem in an abstract way. An abstract solution allows for the possibility of negative cats or even fractions of an animal. The two problems compared are not similar. If they offered that a set of unspecified number of smurfs and tables had a set of smurfs removed, then the problems could reasonably be compared.

Jul 10, 2019
Mathematics is abstract and the physical world is not required to understand mathematics. The classical foundations of the Ancient Greeks were abandoned for the rot of set theory and ZFC. In the most important mathematics book ever written, I explain how numbers and their operations can be derived from absolutely nothing. The book can be download here and it is free:

https://drive.goo...gEO/view

Unfortunately, one cannot be called a mathematician just because one has a PhD. A mathematician is one who has produced great works in mathematics. Large amounts of learning and memorisation coupled with worthless dissertations do not make one a mathematician.

Jul 10, 2019
If I have three oranges and give one to my brother, and he eats it, how many oranges do I have?

Mathematics describes reality, in a purely quantitative way. It doesn't account for the fact my brother got the ripest orange, but then again, it's not supposed to.

I found nothing challenging in any of the example problems. Grumpy is still 2 cm shorter, and therefore 12 cm. Mehta has 12 animals. This is an example of over-analysis.

Jul 10, 2019
If I have three oranges and give one to my brother, and he eats it, how many oranges do I have?

Mathematics describes reality, in a purely quantitative way. It doesn't account for the fact my brother got the ripest orange, but then again, it's not supposed to.

I found nothing challenging in any of the example problems. Grumpy is still 2 cm shorter, and therefore 12 cm. Mehta has 12 animals. This is an example of over-analysis.


Mathematics is abstract but can be used to describe reality. What modern academics call research leaves much to be desired.

Jul 10, 2019
Mathematics is concrete, as I demonstrated. Do you contend I have something other than 2 oranges left?

Jul 10, 2019
No. Mathematics is not concrete.

"...2 oranges".

The number 2 is an abstraction. See my chapter on How we got numbers in my free eBook.

Jul 10, 2019
2 oranges is not "abstract." I can eat them.

Are you insane?

Jul 10, 2019
For that matter, I can eat one and give one to you. Do you seriously contend that my stomach is just as full whether I do this or eat them both?

What a nutjob.

Jul 10, 2019
Chimpanzees understand about numbers. If I have two bananas and eat them both the chimp gets mad; if I have two bananas and give one to the chimp and eat one myself, the chimp is happy.

What's the matter with you, that you're stupider than a chimpanzee?

Jul 10, 2019
the results probably reflect more about reading comprehension and mental visualization than math ability.
so much of our brains is dedicated to visual processing. teaching math completely abstractly is absurd, and why so many people shy away from math.

Jul 11, 2019
The need for the replacement of humans by machines has never been more obvious.

Humans need to be managed like any other ignorant animal.

wrh
Jul 11, 2019
Mathematicians tend to be quite careful in their use of language. When the problem is as easy as the ones given, it is natural to parse it carefully because a trick question is suspected. The smurf example is clear: I see no other interpretation. However, the animals one is ambiguous.

Where is the ambiguity? Well, we are told that all Sarah's animals are cats and dogs, but we are not told this about Mehdi, so Mehdi has exactly 12 cats and dogs, but might also have a few guinea pigs! As a mathematician, all I can say for sure is that Mehdi has at least 12 animals.

If my suspicions of a trick question cause me to be particularly pedantic, there is another ambiguity. Does "as many dogs" mean "as many dogs as Sarah" or "as many dogs as cats"? If we allow the latter as a valid interpretation, then we can say only that Mehdi has an even number of cats and dogs (perhaps even zero).

This might explain the 24% of mathematicians who said that (some of?) the sets problems had no answer.

Jul 11, 2019
To be honest, I found the question's wording sloppy and potentially ambiguous.
'Attained' ?
D'uh...

Jul 11, 2019
For that matter, I can eat one and give one to you. Do you seriously contend that my stomach is just as full whether I do this or eat them both?

What a nutjob.


Idiot. Do you usually have comprehension problems or is this a one off?

Jul 11, 2019
Mathematical thought is seen as the pinnacle of abstract thinking

Despite the first sentence of the article, da schnied still doesn't understand maths is abstract.

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