# The Psychology Behind Mathematical Thinking

The mystery behind how our human mind works

Albert Einstein’s brain was studied long after the physicist’s death. The conclusions reached by researchers who studied his brain would be that the lower part of the parietal lobes was 15% more developed than normal. This “growth” could, therefore, have been related to Einstein’s ability to make equations. In fact, when asked about his intuitions of genius, the scientist said that there was in his mind an associative game of more or less clear images. So Einstein thought in pictures. It is remarkable that the “growth” that the researchers studied is located exactly in the area of the brain that is responsible for vision in space.

However, research on how people are able to think mathematically does not stop here. Simultaneously with the discovery of Einstein’s brain, another team of researchers located two areas of the brain responsible for people’s mathematical skills. The first of these areas, associated with language, is found in the lower-left frontal lobe; the second is between the two lower parietal lobes. The discovery is extremely important in explaining the biological basis of mathematics.

According to researchers, when we do calculations, two complementary brains “systems” come into play. The first system is related to language and helps us to make accurate calculations. The second is related to sight and deals more with the measurement of quantities and the approximation, in general, of quantities.

# The 48+12 Experiment

The teams that conducted this study consisted of researchers from the United States and researchers from France. The American side of the team, led by Elizabeth Spelke, a psychologist at the Massachusetts Institute of Technology (MIT), led the research through psychological tests. The French side, led by Stanislas Dehaene, a researcher at the National Institute of Health and Medical Research (INSERM), instead used computerized methods.

The subjects involved in the experiment were bilingual students who spoke English and Russian equally well. Bilingualism, the ability to speak and understand two languages, played a decisive role in this experience.

The first test was related to the exact calculation. The group of students had to answer questions such as “48 + 12 is equal to 60 or 59?” What did the researchers discover? That students who were of Russian origin answered more quickly the same question if it was formulated in Russian. To answer the question in English, students needed more time to think. Thus it has been shown that in fact, the exact calculation calls for a verbal representation of numbers.

During a second experiment, questions such as “25 + 12 is closer to 40 or 80?” Were asked. The result was that, regardless of the language in which the question was asked, there was no significant difference between the students’ reaction times. Therefore, the approximate calculation is not performed in the same way as the exact calculation.

# What conclusions did the researchers reach after conducting these experiments?

We must first refer to the previous studies of the French team, related to some pathological cases. From this point of view, the case of Mr. B and Mr. Nau are representative.

Mr. B., for example, had a speech deficit and had kept only a long memory. He remembered the results of the multiplication tables and could solve exercises such as “How much is 2 × 5?” instead, if asked “How much is 142 × 5?”, he calculated as follows: “2 x 5 = 10, 4 x 5 = 20, 1 x 5 = 5” and put the results next to each other, obtaining the answer “ 5 20 10 ″, ie “52010”.

Mr. Nau’s case is also remarkable. A brain injury had affected his language and ability to recognize and use numbers. For example, he could not answer the question, “How much do I make 2 + 2?”, But he knew that the result of this meeting was less than 9.

Using all these experiences, the researchers developed the following hypothesis. When making approximations, the brain uses an “analog” system, a kind of mental rule, which measures and compares two quantities. Each number is represented as a line, longer or shorter, depending on the size of the number. The (mental) line corresponding to 3 is therefore twice as short as the line corresponding to 6. When someone is asked to add two numbers, his brain puts the two “together”. “Lines” and deduces the result.

Instead, the exact calculation uses language. Numbers are, in this case, “recorded” as a symbol (for example, 5) and a word (“five”). This system is called “symbolic”.

The American team therefore confirmed the existence of two cognitive systems with which to perform the calculations. The French team located, using magnetic resonance imaging, the areas of the brain in which these systems operate. French researchers have found that performing accurate calculations increases the activity of the left lower frontal lobe, which is associated with the speech mechanism. When we perform approximate calculations, the lower parietal lobes are activated, which specialize in spatial orientation.

It follows, therefore, that mathematical science can be compared to a language. We can perform different calculations using two systems: analog and symbolic. But how the human mind works when it performs mathematical operations remains a mystery awaiting its release.