What Math?

I have been a part of many conversations around mathematics teaching and whether or not this or that perspective of teaching mathematics has merit or not. Before we get into what approach to teaching mathematics matters, I think there needs to be a more important conversation: what mathematics do we want our students to learn?

I don’t think educators talk enough about what they think mathematics is or can be. I want mathematics to be creative, meaningful, useful, and fun. I want students to leave mathematics classrooms thinking about how mathematics can empower them and also how much they enjoy playing with math. I want mathematics to be an interesting process of exploring problems, reasoning, and justification. Yes, I also think mathematics involves procedures, but that is one part of a much more interesting and meaningful field. If I think that mathematics is simply the process of performing calculations, I’d teach it differently than if I thought it was a creative, solving adventure. Calculators can calculate accurately and quickly, humans can reason and think and strategize uses for mathematics.

I decided to ask ChatGPT some questions about mathematics. I thought you all might find the responses interesting.

*Quick note to acknowledge that ChatGPT is just sharing what others have written about; it is not sentient. There are definitely limitations to this technology, but it is an interesting take on a topic I find of great importance.

How do Field Medal winners describe mathematics?

First, I asked Chat GPT, How do Field Medal winners describe mathematics? I totally acknowledge that focusing narrowly on an award, even if it is widely considered the highest award in mathematics, is flawed. However, I still think that there is some merit in understanding what the field’s highest award winners say about their field. Here is what ChatGPT came up with:

Fields Medal winners, like many mathematicians, describe mathematics as a creative and beautiful endeavor that involves discovering and exploring patterns and structures in the abstract realm of ideas. They often view mathematics as a way of understanding the fundamental principles that govern the natural world, and as a tool for solving real-world problems in fields such as physics, computer science, and economics.

Many Fields Medal winners emphasize the importance of intuition and creativity in the mathematical process, often describing their work as a form of artistic expression. They also highlight the collaborative nature of mathematics, and the importance of engaging with colleagues and sharing ideas in order to make progress.

Here are some quotes from Fields Medal winners about mathematics:

• “Mathematics is a game played according to certain simple rules with meaningless marks on paper.” – David Hilbert (1930)
• “Mathematics is not just about solving equations, it’s about understanding the beauty in numbers and patterns.” – Maryam Mirzakhani (2014)
• “Mathematics is the art of giving the same name to different things.” – Henri Poincaré (1904)
• “Mathematics is a subject where one never understands what one is doing, nor why it works.” – Vladimir Voevodsky (2002)
• “Mathematics is a way to understand and explain the world around us, and to see the patterns and connections that are invisible to the untrained eye.” – Terence Tao (2006)

ChatGPT

How do people describe Field Medal winners?

Next, I asked ChatGPT how people describe Field Medal winners. I was interested in the traits they are acknowledged for as people describe them.

Fields Medal winners are often described as some of the most brilliant and accomplished mathematicians of their generation, who have made significant contributions to advancing the field of mathematics through their groundbreaking research and discoveries.

They are widely respected and admired by mathematicians and scientists around the world, and their work is often considered to be at the forefront of mathematical research.

Fields Medal winners are also known for their creativity, persistence, and dedication to their work. They are often described as individuals with an exceptional ability to see patterns and connections where others do not, and to develop new mathematical concepts and techniques that have the potential to transform our understanding of the world.

In addition to their intellectual prowess, Fields Medal winners are often seen as humble and generous individuals who are willing to collaborate with others and share their knowledge and expertise in order to advance the field of mathematics.

Overall, Fields Medal winners are widely recognized as some of the most brilliant and influential mathematicians in the world, who have made significant contributions to the advancement of human knowledge and understanding.

ChatGPT

What distinguishes brilliant mathematicians?

Finally, I asked Chat GPT what distinguishes brilliant mathematicians to understand what sets someone apart in this field.

The traits that distinguish a brilliant mathematician can vary, but here are some common qualities:

1. Creativity: A brilliant mathematician has the ability to think outside the box and come up with original and innovative ideas. They are often able to see connections and patterns that others miss, and to develop new mathematical concepts and techniques that advance the field.
2. Persistence: Mathematics can be a challenging and difficult field, and a brilliant mathematician must be willing to work hard and persist in the face of obstacles and setbacks. They often spend many hours grappling with difficult problems and are willing to put in the time and effort needed to find a solution.
3. Attention to detail: Mathematics requires a high degree of precision and accuracy, and a brilliant mathematician must be meticulous in their work. They pay close attention to details and are careful to avoid mistakes that can lead to incorrect results.
4. Intuition: A brilliant mathematician often has a deep sense of intuition that allows them to see patterns and connections that are not immediately obvious. They have an innate sense of what is likely to be true and are able to develop conjectures and hypotheses based on this intuition.
5. Communication skills: A brilliant mathematician must be able to communicate their ideas effectively to others, both within the mathematical community and to a broader audience. They must be able to explain complex ideas in a way that is clear and understandable to others, and be willing to engage in collaborative work.

Overall, a brilliant mathematician is someone who possesses a combination of innate ability, hard work, creativity, and persistence, and who has made significant contributions to advancing the field of mathematics through their research and discoveries.

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What does all this mean?

I think it is telling that the people who are noted as making the most significant accomplishments in mathematics do not talk about speed, accuracy, or calculations. Rather, the focus is on creativity, persistence, and collaboration. Given these perspectives, I challenge everyone to think about what instruction might best support the development of this type of mathematics. I certainly think there is a time and place for standardized testing and procedural fluency, but do I think that drill and kill or explicit direct instruction alone will get us to the vision of mathematics that I hold? For me, the answer is no.