The way we memorize information—such as a mathematical problem statement—reveals how we process it.
A team from the University of Geneva (UNIGE), in collaboration with CY Cergy Paris University (CYU) and the University of Burgundy (uB), shows how different problem-solving methods can alter the way information is stored in memory and even create "false memories."
When solving a math problem, it is possible to rely on the ordinal property of numbers, that is, the fact that they are ordered, or their cardinal property, that they represent specific quantities.
© UNIGE
By identifying the unconscious deductions made by learners, this study opens new avenues for teaching mathematics. It can be found in the
Journal of Experimental Psychology: Learning, Memory, and Cognition.
For humans, the process of memorizing information involves several stages: perception, encoding—how it is processed to become a memory trace that is easily accessible—and retrieval (or reactivation). At each stage, errors may occur, sometimes leading to the formation of false memories.
Scientists from UNIGE, CYU, and uB sought to determine whether solving arithmetic problems could generate such memories and if these could be influenced by the nature of the problems.
Unconscious deductions create false memories
When solving a math problem, it is possible to rely on the ordinal property of numbers, that is, the fact that they are ordered, or their cardinal property, that they represent specific quantities. This can lead to different problem-solving strategies and, when memorized, to different types of encoding.
Specifically, the representation of a problem involving durations or size differences (ordinal problem) can sometimes allow unconscious deductions, leading to a more direct solution. Conversely, with problems involving weights or prices (cardinal problem), additional reasoning steps might be needed, such as an intermediate calculation of subsets.
The scientists hypothesized that due to spontaneous deductions, participants would unconsciously modify their memories of the statements for ordinal problems, but not for cardinal ones.
Participants experienced the illusion that they had read sentences that were never actually present in the statements.
To test this, 67 adults were asked to solve arithmetic problems of both types. Then, they were asked to recall the problem's statement to test their memories. Researchers found that memory recall was accurate in the majority of cases (83%) when it came to cardinal problems.
However, the results were different when participants were asked to recall the statement of ordinal problems, such as: "Sophie's trip lasts 8 hours. Her trip occurs during the day. When she arrives, the clock shows 11 a.m. Fred leaves at the same time as Sophie. Fred's trip lasts 2 hours less than Sophie's. What time does the clock show when Fred arrives?"
In more than half the cases, participants involuntarily added information they had inferred during problem-solving. In the case of the problem above, they might, for instance, mistakenly believe they had read: "Fred arrived 2 hours before Sophie" (a deduction based on the fact that Fred and Sophie left at the same time, but Fred's trip was 2 hours shorter—a true conclusion, but one that alters the actual stated problem).
"We demonstrate that when solving certain problems, participants experience the illusion of having read sentences that were never presented in the problem statements, but which are linked to unconscious deductions made during the reading of the statements. These get mixed up in their minds with the sentences they actually read," summarizes Hippolyte Gros, former post-doctoral researcher at UNIGE's Faculty of Psychology and Educational Sciences, now a senior lecturer at CYU, and lead author of the paper.
Using memory to understand reasoning
Furthermore, the experiments showed that participants exhibiting these false memories were only those who had discovered the quickest strategy, revealing their unconscious reasoning that allowed them to find this shortcut. On the other hand, participants who used more steps in their reasoning were unable to "enrich" their memories, as they had not employed the relevant step in their thought process.
"This work could have applications for mathematics education. By asking students to recall problem statements, we can identify, based on the presence or absence of false memories in their recollections, their mental representations and thus the reasoning they followed when solving the problem," explains Emmanuel Sander, full professor at UNIGE's Faculty of Psychology and Educational Sciences, who led the study.
Directly accessing mental processes is quite difficult. Doing so indirectly by analyzing memory processes could help us better understand the difficulties students face in problem-solving and offer strategies for intervention in the classroom.