The Greek philosopher Plato wrote about Socrates, who challenged students with the question of “doubling the square” in 385 BC. When asked to double the area of the square, students doubled the length of each side, and were unaware that each side of the new square was the original oblique length.
Scientists from the University of Cambridge and Hebrew University in Jerusalem chose the problem to pose for a non-trivial solution. Since Plato wrote 2,400 years ago, scholars have been using double the squared problems to debate whether the mathematical knowledge needed to solve it is already within us, released through reasons, or whether it is accessible only through experience.
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The answer came when the team went further. As explained in a study published in the Journal International Journal of Mathematical Education in Science and Technology on September 17, they asked the chatbots to double the area of the rectangular shape using similar inferences. I replied that there was no solution for geometry as I couldn’t double its size using a rectangular diagonal.
However, after visiting Nadab Mark, a scholar at Cambridge University at Hebrew University, Jerusalem University, and professor of mathematics education, Andreas Styliance, knew that geometric solutions existed.
Marco said the possibility of false claims present in ChatGPT training data is “intensely small.” That is to improvise the answer based on previous discussions about doubling the square problem.
“When we face new problems, our instinct is often to try things based on past experiences,” Marco said in a statement on September 18th. “In our experiments, ChatGpt seemed to do something similar. It seems like it’s a learner or a scholar that came up with its own hypothesis and solutions.”
Is the machine thinking?
The research sheds new light on questions about artificial intelligence (AI) versions of “inference” and “thinking,” scientists said.
It seemed like they improvised and made mistakes with responses like Socrates students Marco and Stylianides, which made a mistake, so ChatGPT could use a concept we already know from education called the Proximal Development Zone (ZPD), and explains what we know and ultimately we may know in the right educational instruction.
ChatGpt may be using a similar framework spontaneously, and may be solving a new problem simply not represented in data thanks to the proper prompts.
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This is a prominent example of the long-standing black box problem in AI. This is not visible and untraceable programming or “inference” that the system proceeds to reach a conclusion, but researchers say their work highlights the opportunity for them to ultimately improve AI.
“Unlike evidence found in reputable textbooks, students cannot assume that ChatGpt evidence is valid,” Stylianides said in a statement. “Understanding and assessing AI-generated proofs has emerged as an important skill that needs to be incorporated into the mathematics curriculum.”
This is a core skill that students hope to acquire in an educational context. For example, they say they want better and faster engineering. For example, “I want you to explore this problem together,” and say, “Please tell me the answer.”
The team is cautious about the outcomes and warns us not to over-interpret them, and concludes that LLMS “solve things” like we do. However, Marco labeled ChatGpt’s behavior as “Learner-like.”
Researchers look at the scope of future research in several areas. New models can be tested with a wider range of mathematical problems. It could also combine ChatGPT with dynamic geometry systems or theorem prover to create a richer digital environment that supports, for example, how teachers and students collaborate in the classroom using AI.
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