The search for intuitive mathematics
"I now view the effort to find a fundamental theory of physics as in many ways another challenge in language design — perhaps even the ultimate such challenge. Designing a computational language is to create a bridge between two domains: the abstract world of what is possible to do, and the “mental” world of what people understand and are interested in doing. The challenge is [...] to give people a way to describe these [abstractions]." -- Stephen Wolfram
Wolfram's quote could be viewed through the lens of Seymour Papert. Educators often use physical analogies and embodiments to describe complex mathematical and physical phenomena. On the other hand, practitioners (such as scientists) use symbolic representations for faster exploration. This fundamental gap between education and practice was pointed out by Seymour Papert (in his book Mindstorms, Chapter 8) when he was designing the LOGO programming language.
This gap is also evident in our math education trajectory. Standard mathematics curricula start off with picture based math or manipulable objects (e.g., Montessori Sensorial materials), grounded in intuitive and embodied interactions such as categorizing shapes, counting, grouping, and drawing. We then move on to learn symbolic abstraction, manipulation, and eventually Computer Algebra Systems in order to explore with abstractions faster.
In practice, many students lose the intuitive underpinnings of those abstractions and resort to memorizing rules of symbol manipulation which become cognitively arbitrary and severed from their intuitive grounding in embodied interactions. The current mathematical tools, workflow, and computational languages also favor symbol-based math and programming as one advances to later stages.