Periodic Reporting for period 2 - NeuroMath (Acquisition of Mathematical Concepts in the Human Brain)
Reporting period: 2022-06-01 to 2023-05-31
Humans are the only species that understand abstract math concepts. Even in absence of formal education and specific lexicon, humans possess intuitions of arithmetic and geometry. At the brain level, these intuitions are associated with activity in the intraparietal sulcus. However, to date, the cognitive and neural mechanisms by which the human brain forges the meaning of advanced math concepts remain largely unknown. Previous work in adults has suggested that advanced math reflection on long mastered concepts does not rely on the brain circuits for language but recycles evolutionarily ancient systems for numerosity and space processing. These findings raise several questions: does language play a role in math acquisition at school where knowledge is taught verbally and often involves complex multimodal input? How quickly are newly learnt math concepts integrated to the brain’s math-responsive network? To what extent does children’s neural activity in the math-responsive network resemble adults’ during learning? Addressing these questions requires a change of paradigm: going beyond traditional laboratory investigations of math processing in simplified tasks testing simple, known concepts. The first goal of this project was thus to introduce and test a novel fMRI paradigm that combines neuroimaging techniques with developmental and educational methods, in both adults and children. The results of the action give a first indication that such a paradigm can be used to study math learning in the human brain and point towards the further hypothesis that brain activity within the math-responsive network during a short math video lesson relates to the amount of learning.
The novel fMRI paradigm we have introduced consists in asking participants to watch a short video lesson introducing a new math concept, and to complete a controlled task testing their understanding of the concept before and after the video, during a single fMRI session. We have used this paradigm in adults and children.
Adults were 21 freshman university students in mathematics who watched four video introductions to measure theory, stochastic processes, plant biology, and property law. Before and after watching the videos, they had to decide whether spoken math and non-math statements were true or false. The tested concepts were either known from high school, unknown throughout the study, or initially unknown then taught in the videos. We verified that semantic decision on long-known math concepts engaged a distinct math-responsive brain network. This network was also activated, at least partially, during the learning episode itself. However, the videos induced only minimal learning that was associated with increased brain activity in domain-general regions, but not in the math-responsive network. This study indicates that new math concepts are not automatically integrated into the math-responsive network and is a proof of concept showing that our novel fMRI paradigm can provide a diagnostic tool for the efficiency of math pedagogical materials. These results were published in PNAS in 02/2023.
To children, we chose to teach the commutative principle of multiplication. We started with a behavioral investigation of the existence of precursory knowledge of multiplicative commutativity before formal schooling. 5-year-old children (N = 30) were asked to judge whether 2 characters got a fair or an unfair share of apples in various situations that probed commutative multiplication and addition as well as mere identity, and that tested the effect of symmetric cues or verbal descriptions. Our preliminary results indicate that commutativity was more accurately perceived in addition than multiplication, and that verbal descriptions were not helpful. However, making apparent the symmetry intrinsically contained in commutativity helped preschoolers perceive multiplicative commutativity.
We assessed elementary school children’s knowledge of multiplicative commutativity in a number comparison game (N = 79). Trials tested for multiplicative commutativity in both symbolic and non-symbolic forms, and controlled for various parameters including the total numbers, or the type of operation (addition, multiplication, identity). We found that non-symbolic commutative trials yielded more errors than symbolic commutative trials, and that symbolic mastery of commutativity (but not number sense acuity) correlated with performance on non-symbolic trials. A subset of these children was then presented with a short training. We used the results obtained in preschoolers to create a 5-minute video in which a cartoon teacher explains the commutative principle of multiplication using a combination of images, words, symbolic and non-symbolic representations. After watching the video and completing a short auditory knowledge test, children played the number comparison game again. We found that children’s pre-intervention level of symbolic mastery influenced the effect of the intervention: children with the lowest symbolic mastery improved on symbolic trials, while children with the highest symbolic mastery improved on non-symbolic trials. This study, that indicates that the commutative principle of multiplication may not be readily available to intuition before formal teaching of multiplication and commutativity, was presented to the conference of the Cognitive Science Society in 07/2023 and is described in a manuscript that was recently submitted.
Our results also validate the efficiency of our training material at teaching multiplicative commutativity. We thus used it in an fMRI learning paradigm. To evaluate the neural changes induced by learning that multiplication is commutative, 20 2nd graders completed a one-back number task before and after the intervention: they saw a succession of images representing dot arrays or operations, and at random times, they were asked if the two images presented before the question showed the same or different total numbers. The images probed non-symbolic and symbolic commutative situations as well as additive situations with matching numbers and arithmetic outcome. To evaluate children’s functional maturity and assess its link with performance, we also scanned a group of 20 adults. fMRI data are currently under analysis.
Adults were 21 freshman university students in mathematics who watched four video introductions to measure theory, stochastic processes, plant biology, and property law. Before and after watching the videos, they had to decide whether spoken math and non-math statements were true or false. The tested concepts were either known from high school, unknown throughout the study, or initially unknown then taught in the videos. We verified that semantic decision on long-known math concepts engaged a distinct math-responsive brain network. This network was also activated, at least partially, during the learning episode itself. However, the videos induced only minimal learning that was associated with increased brain activity in domain-general regions, but not in the math-responsive network. This study indicates that new math concepts are not automatically integrated into the math-responsive network and is a proof of concept showing that our novel fMRI paradigm can provide a diagnostic tool for the efficiency of math pedagogical materials. These results were published in PNAS in 02/2023.
To children, we chose to teach the commutative principle of multiplication. We started with a behavioral investigation of the existence of precursory knowledge of multiplicative commutativity before formal schooling. 5-year-old children (N = 30) were asked to judge whether 2 characters got a fair or an unfair share of apples in various situations that probed commutative multiplication and addition as well as mere identity, and that tested the effect of symmetric cues or verbal descriptions. Our preliminary results indicate that commutativity was more accurately perceived in addition than multiplication, and that verbal descriptions were not helpful. However, making apparent the symmetry intrinsically contained in commutativity helped preschoolers perceive multiplicative commutativity.
We assessed elementary school children’s knowledge of multiplicative commutativity in a number comparison game (N = 79). Trials tested for multiplicative commutativity in both symbolic and non-symbolic forms, and controlled for various parameters including the total numbers, or the type of operation (addition, multiplication, identity). We found that non-symbolic commutative trials yielded more errors than symbolic commutative trials, and that symbolic mastery of commutativity (but not number sense acuity) correlated with performance on non-symbolic trials. A subset of these children was then presented with a short training. We used the results obtained in preschoolers to create a 5-minute video in which a cartoon teacher explains the commutative principle of multiplication using a combination of images, words, symbolic and non-symbolic representations. After watching the video and completing a short auditory knowledge test, children played the number comparison game again. We found that children’s pre-intervention level of symbolic mastery influenced the effect of the intervention: children with the lowest symbolic mastery improved on symbolic trials, while children with the highest symbolic mastery improved on non-symbolic trials. This study, that indicates that the commutative principle of multiplication may not be readily available to intuition before formal teaching of multiplication and commutativity, was presented to the conference of the Cognitive Science Society in 07/2023 and is described in a manuscript that was recently submitted.
Our results also validate the efficiency of our training material at teaching multiplicative commutativity. We thus used it in an fMRI learning paradigm. To evaluate the neural changes induced by learning that multiplication is commutative, 20 2nd graders completed a one-back number task before and after the intervention: they saw a succession of images representing dot arrays or operations, and at random times, they were asked if the two images presented before the question showed the same or different total numbers. The images probed non-symbolic and symbolic commutative situations as well as additive situations with matching numbers and arithmetic outcome. To evaluate children’s functional maturity and assess its link with performance, we also scanned a group of 20 adults. fMRI data are currently under analysis.
The present project opens a previously unexplored window onto the development of abstract math concepts as a result of formal education. We have provided a first diagnostic tool of the efficiency of a given pedagogy at the brain level. We have also advanced our knowledge of how the human brain perceives, learns, and understands large quantities that present grouping features, at both behavioral and neural levels. In particular, we have obtained thought-provoking results on the precedence of symbolic learning over non-symbolic intuitions in the case of multiplicative commutativity. We have also created and validated experimental materials that proved efficient at teaching commutativity to elementary school children, that are meant to be distributed and used by the community of developmental researchers on math cognition. We expect this project will clarify the differences and similarities in the neural processing of sets of grouped dots and commutative expressions between children and adults, as well as the extent to which various developmental markers explain inter-individual differences in children’s behavioral assessments of their math comprehension level.