Math as a perspective on life (with Marcus du Sautoy)

1. Reframe Problems for Simplicity

When faced with a difficult problem, shift your mindset from ‘how do I solve this difficult problem?’ to ‘how do I make this difficult problem simple?’ to unlock easier solutions and perspectives.

2. Step Back, Find Clever Shortcuts

Instead of immediately diving into laborious work, step back and look for clever ways to reframe problems or combine elements, similar to Gauss’s method of pairing numbers, to find efficient shortcuts and avoid mistakes.

3. Abstract to Essential Details

Practice abstract thinking by identifying and discarding unimportant details from a problem or situation, focusing only on what is essential. This helps avoid being inundated with unnecessary information and clarifies the core issue.

4. Change Problem’s Language/Representation

When a problem appears opaque, try changing its ’language’ or representation (e.g., converting a numerical game to a visual tic-tac-toe grid or using binary numbers for NIM) to gain new insights and simplify the solution process.

5. Embrace Math as a Shortcut

View mathematics as ’the art of the shortcut,’ providing clever ways to hone in on solutions without resorting to laborious, time-consuming methods. Learn and apply these tools to solve problems more efficiently.

6. Invest in Learning Powerful Shortcuts

Understand that powerful shortcuts, especially mathematical ones, may require significant initial learning time. However, mastering these skills provides immense advantages and efficiency over those who continue with laborious methods.

7. Follow the Math Problem-Solving Cycle

Adopt a structured approach to problem-solving by observing real-world patterns, abstracting away irrelevant details, modeling them mathematically, manipulating the math to derive new insights or predictions, and then verifying these against reality.

8. Spot Patterns for Prediction

When presented with data, actively seek out underlying rhythms or patterns. Once identified, develop an equation or algebraic formula to extend this pattern, enabling powerful predictions about future behavior.

9. Transform Problems to New Spaces

For complex problems, consider transforming them into different mathematical ‘spaces,’ such as using Fourier transforms to move from a time-based function to a frequency analysis, where solutions might become significantly simpler or even obvious.

10. Use Math for Optimal Planning

Employ mathematical modeling as a ‘fortune teller’ to explore possibilities and plan optimal pathways or solutions. This allows you to look into the future and make informed decisions without having to try all paths in advance.

11. Model Mathematically Before Building

Before undertaking physical construction, such as building a bridge or a structure, create a mathematical model first. This enables engineers to identify low-energy, efficient, and optimal designs, leading to better and more effective real-world outcomes.

12. Beware Faulty Heuristic Thinking

Recognize that fast, intuitive thinking (System 1) can be faulty when generalizing from local experiences to a global narrative, leading to breakdowns in decision-making. Engage your analytic mind or use mathematical shortcuts to avoid these errors.

13. Ensure Random Global Sampling

When drawing conclusions from data, ensure that your sample is randomly selected and globally representative, rather than just locally sampled. This prevents biased heuristics and leads to more accurate and generalizable insights.

14. Determine Sufficient Sample Size

In a data-rich environment, apply statistical principles to determine the minimum necessary sample size required to gain reliable insights about a larger dataset, enabling efficient analysis without needing to examine all available data.

15. Apply Pattern Recognition to Learning

When learning complex skills, actively seek out and recognize recurring patterns, similar to identifying words in text rather than individual letters. This approach allows for more efficient learning, as mastering one pattern can unlock understanding of many variations.

16. Master Fundamental Building Blocks

Invest time in mastering the fundamental building blocks of any complex skill, such as scales and arpeggios in music. This seemingly boring practice serves as a ‘fantastic shortcut,’ enabling quicker recognition and application of larger patterns.

17. Use Diagrams to Clarify & Communicate

Cultivate the ability to create effective diagrams to ‘crystallize’ complex information, which not only deepens your own understanding but also serves as a powerful tool for clearly communicating your insights to others.

18. Simplify by Discarding Irrelevant Details

When creating models or representations, simplify them by deliberately discarding irrelevant details, such as exact distances on a map if only connectivity is important. This practice leads to clearer, more functional insights and communication.

19. Embrace Challenge for Satisfaction

Embrace the inherent difficulty in complex subjects like mathematics, as the challenge itself is part of the charm. Strive for a ‘sweet spot’ where you are pushed, enjoying the process and the deep satisfaction of overcoming obstacles.

20. Engage Learning with Multiple Strategies

To make learning engaging and prevent boredom, utilize multiple strategies: weave in compelling stories, inspire awe with grand concepts, connect the material to existing interests, and actively evoke curiosity by posing intriguing problems.

21. Balance Fundamentals with Inspiration

In education, balance the teaching of fundamental skills (‘scales and arpeggios’) with inspiring ‘big stories’ and awe-inspiring concepts (‘wonderful music’). This approach motivates students and provides aspirational goals for deeper engagement with the subject.

22. Use History to Contextualize Learning

Integrate historical narratives into teaching and learning, explaining the origins of ideas, the individuals involved, and the problems they sought to solve. This approach makes complex concepts more relatable, motivational, and easier to understand.

23. Celebrate Pure & Applied Work

Acknowledge and celebrate both the pure, passion-driven aspects of a field and its practical applications. This balanced appreciation helps secure funding and maintains motivation for innovation and discovery.

24. Encourage Blue Skies Thinking

Encourage ‘blue skies thinking’ by allowing individuals to follow their intuition, passions, and the inherent beauty of a subject. This approach, though sometimes with long lead times, often leads to significant and unforeseen practical applications.

25. Use Calculus for Optimal Solutions

For complex problems with many changing variables, leverage calculus to quickly identify optimal solutions, such as maximum profits or minimum energy points, without needing to test all possibilities.

26. Use ThoughtSaver for Recall

Combat forgetting valuable information by using ThoughtSaver.com. Create flashcards or use existing decks, and leverage its spaced repetition system to regularly strengthen your recall of important ideas and concepts.

27. Capture Discrete Learnings with ThoughtSaver

For discrete facts, interesting concepts, or general ideas not linked to a specific project, capture them in ThoughtSaver.com. This tool helps you remember and apply important learnings through its searchable database and spaced repetition system.

28. Consolidate Project Knowledge

For individual projects, maintain a single document (e.g., Google Doc) where all related ideas and information are collected. Regularly reorganize this document to keep all project knowledge consolidated and easily accessible.