Since antiquity, the "doubling the square" problem has intrigued curious minds. Plato already recounted how Socrates challenged his students to double the area of a square, an enigma that reveals the limits of our geometric intuition.
Researchers from the University of Cambridge and the Hebrew University of Jerusalem chose this millennia-old problem to test ChatGPT's geometry capabilities. They estimated that its non-obvious solution - using the diagonal of the original square rather than simply lengthening its sides - was unlikely to be present in the model's training data, which consists mostly of texts.
This approach aimed to determine whether the artificial intelligence could deduce a solution by itself, without having been explicitly programmed for it.
The researchers submitted this problem to ChatGPT-4, first imitating Socrates' questions, then deliberately introducing errors, queries, and new variants of the problem. The researchers expected it to meet their mathematical challenge by regurgitating its pre-existing "knowledge" of Socrates' famous solution. Instead, it seemed to improvise and, at one point, even made a typically human error.
When ChatGPT was asked to double the area of a rectangle, it claimed that no geometric solution existed, an incorrect answer. Nadav Marco and Andreas Stylianides, the scientists behind the study, interpreted this error as a sign that the model was improvising its responses.
Published in the
International Journal of Mathematical Education in Science and Technology, their research suggests that ChatGPT might function as a learner, generating hypotheses from its prior knowledge rather than drawing from a database.
This ability to "reason" when faced with a new problem evokes the educational concept of the zone of proximal development, which describes the space between what an individual knows and what they can learn with appropriate guidance. The researchers observe that ChatGPT, by responding creatively to puzzles not encountered during its training, shows behaviors similar to those of a student exploring a complex concept with a teacher's help.
The implications of this study extend beyond the mathematical framework. It raises fundamental questions about how artificial intelligences process information and solve novel problems. The researchers caution against too hasty an interpretation: if ChatGPT seems to "think," its process remains a black box, whose internal mechanisms are difficult to trace. For Andreas Stylianides, it becomes essential to teach students to critically evaluate evidence generated by AI.
The team envisions practical applications, such as integrating ChatGPT with dynamic geometry tools or proof assistants, to create more interactive learning environments. By encouraging students to formulate precise prompts - for example, "Let's explore this problem together" rather than "Give me the answer" - we could foster fruitful collaboration between humans and machines, paving the way for new teaching methods.
The problem of doubling the square
The problem of doubling the square is a classic geometric puzzle dating back to ancient Greece. It involves constructing a new square whose area is exactly double that of a given square. Many intuitively think it's enough to double the length of the sides, but this actually quadruples the area.
The correct solution involves using the diagonal of the original square. In Euclidean geometry, the diagonal of a square with side 'a' measures a√2. If we take this diagonal as the side of the new square, its area becomes (a√2)² = 2a², exactly double the initial area a².
This problem illustrates the difference between geometric intuition and rigorous mathematical reasoning. It shows how seemingly simple concepts can hide complexities that require an analytical rather than empirical approach.
In mathematical education, this problem often serves to teach the Pythagorean theorem and the properties of square roots, while developing critical thinking in the face of visual approximations.
The zone of proximal development in education
The zone of proximal development (ZPD) is a key concept developed by psychologist Lev Vygotsky. It refers to the gap between what a learner can accomplish alone and what they can achieve with the help of a teacher or more competent peer. This zone represents the individual's immediate learning potential.
In an educational context, identifying a student's ZPD allows for adapting teaching so that it is neither too easy (which leads to boredom) nor too difficult (which causes frustration). The teacher guides the learner through accessible but stimulating problems, thus fostering progress.
The concept also applies to collaborative learning, where interactions between peers help each person surpass their current limits. The ZPD emphasizes the importance of social context and support in acquiring new skills.
Today, the ZPD inspires modern pedagogical approaches, including in the use of educational technologies, where tools like AI can play the role of personalized tutor, adapting exercises to each learner's level.