• Paul Lockhart's work, "A Mathematician's Lament", points out that teaching mathematics often fails to bring out the passion, beauty, and excitement inherent in the field.
• Lockhart highlights that mathematics, like any form of art or literature, is created by humans for their own amusement and can be appraised critically.
• He emphasizes that mathematics, much like a poem, can be evaluated based on aesthetic criteria such as simplicity, elegance, and depth.

Example: Just as a beautifully written poem might make you feel a certain emotion or help you see the world in a different light, a mathematical concept or formula can be admired for its elegance and depth. A popular example is Euler's Identity, considered beautiful because it links five of the most important numbers in mathematics in a surprisingly simple way.

• Math learning often leaves little space for learners to reach their own conclusions.
• The barrier for contributing to the field of mathematics is high, and most learners end their journey before making any contributions or even encountering advances in the field.
• Mathematics knowledge produced is seldom popularized or widely communicated.

Example: Math textbooks often provide step-by-step solutions to problems. This can limit the space for students to explore different ways of approaching the problem, which could stifle creativity and independent thinking.

• Piper Harron, a mathematician, attempted to break down the barriers in communicating advanced mathematics by making her doctoral thesis understandable to a wide audience.
• She presented her thesis at different levels, including a 'layperson' level that uses simple language.
• Harron's approach demonstrates a way to make complex mathematical knowledge more accessible.

Example: Think about a restaurant menu. The 'layperson' version would be the menu description of a dish. On the other hand, the chef would need a detailed recipe with specific measurements and techniques - this could be compared to the 'professional' version.