“According to the constructivist philosophy of education, built on the ideas of the late philosopher Ernst von Glasersfeld, this experience altered my brain’s perception of mathematics, even though it didn’t involve doing math in the traditional sense.”
“The same is true of all experiential brushes with mathematical concepts, but no mathematical concept is more intense than infinity. Which makes infinity uniquely relevant to addressing some key concerns about modern education.”
“tests. A seemingly rigorous approach has left many of them rather good at math tests while leaving them bad at math as a concept, and at the crucial forms of logical thinking that comes with it. Infinity provides an antidote. It has the power to create conceptual wows – and to do so even in minds that have not yet been exposed to algebra or any kind of number theory.”
“Infinity raises its fist to rote memorisation and multiple-choice testing, because encounters with infinity are fundamentally conceptual in nature.”
“They correspondingly create conceptual (as opposed to procedural) knowledge – a foundational comprehension of, for example, what multiplication is, and the ability to understand its utility in a variety of situations.”
“Asking students to imagine and experience the infinite gives them better numerical intuition. It promotes active learning and metacognition, which usually take the form of inquiry- or problem-based curricula in progressive schools.”
“In 1988, von Glasersfeld argued that ‘knowledge is not passively received either through the senses or by way of communication’. Instead, it comes from experience and is actively built, brick by brick.”
“Built-up knowledge evolves over time, because our experiences are always changing. When a new situation or piece of information has to be incorporated into the brain’s intellectual framework, the bricks reshuffle like Tetris pieces. Students need to experience math – not just hear about it, as typically happens in the classroom – to understand it.”
“Although it seems to be one of the most confounding things in mathematics, infinity can be a gateway drug to deeply personal mathematical experiences. It connects instantly to big, personal questions about life and death, power and control, the beginning of time and the end of the Universe.”
“Resolving these apparent paradoxes requires the walker or paper-cutter to interpret the novel experience, to build up and shift their mental bricks. This neurological rearrangement represents what Frederic Bartlett, a pioneer of cognitive psychology, dubbed ‘schema’ in his influential 1932 book Remembering: A Study in Experimental and Social Psychology.”
“Such rearrangements appear to be pervasive and permanent. ‘New experiences do not have a limited effect, but cause the entire cognitive structure to rearrange itself… [W]hen one learns something and that learning results in structural change, one is prepared to learn something more advanced in the same category,’ writes Meir Ben-Hur, a teacher trainer and math education theorist at the Feuerstein Institute in Jerusalem.”
“In encounters with infinity, the imprint left on the brain is not just a memory of walking slowly toward a door – or, in my case, looking into a set of mirrors. It is a better-formed, broadly applicable intuition for numbers.”