My recent article on the growing importance and recognition of Mathematics here or here has received a much bigger response than I expected. Evidently, my message resonated with what many of you already thought.
Most of the enthusiasm and positive response came from non-mathematicians who clearly would like to augment their familiarity with Mathematics – both for themselves as individuals and for the career advancement opportunities that strong mathematical skills offer.
In this article I aim to provide some suggestions to busy professionals – adults of any age who may want to refresh, expand, or deepen their understanding of concepts they studied in school or have had to acquire on the job.
So, the question is: where to start when you have life and work responsibilities and do not have the time luxury to take formal structured courses?
I submit here that Probability and Statistics are the ideal starting point – for a number of reasons:
- Knowledge in Probability & Statistics is needed, and in fact essential, in your professional life. In a world of quantitative data, learning how to make sense of the data and then making advantageous decisions is critical. Acquiring knowledge that you can immediately apply to your or your company’s advantage will be an excellent motivator.
- The hot fields of Data Science and Artificial Intelligence are essentially “Statistics on steroids”. The technological advances in these fields have opened enormous opportunities for you and your company and will continue to do so for years to come (in fact, forever, in my mind).
- Exercise “your weak muscle”! Whereas in many other areas, you can at least be in the right direction with intuition only, this is not so in Probability and Statistics –– where lack of specific exposure to these particular fields can lead you completely astray. Thus, bumping up your knowledge in these field is particularly urgent and will give you the most “bang for the buck”.
- The counter-intuitiveness of Probability and the pitfalls of Statistics are the main reason why people can be fooled with arguments that appear sound but are bogus (and of course people can fool themselves too). So, understanding Statistics will equip you to see through the deception foisted onto you by [your favorite villain here: politicians, the media, the government, lawyers, advocacy groups].
- Last but not least, the methodologies and the results are very nuanced, giving you exposure to mathematical methods and thinking patterns. This is important because you want to train the way you think – not just be aware of a number of mathematical results.
Having thus settled on Probability and Statistics as a starting point, the second element of the learning approach I want to discuss is the “layers of paint” analogy: knowledge (remembering, understanding) is built up in layers.
Therefore, I submit that easy, fun passes (“coats of paint”) are the way to go. Resist the temptation of wanting to understand it all the 1st or 2nd time around. It would take forever and you will give up in frustration – the opposite of what you want!
Instead, just start by gaining exposure to the concepts, methods and results, then mull over them and revisit often. The Romans knew this: “Repetita Iuvant”.
Caveat: be aware during these early passes that your understanding is incomplete and precarious. Do not fool yourself in thinking that you actually know everything – which could be dangerous.
Enough preliminaries! I recommend these two books if you want to develop your Statistics and Probability intuition. They are both fun to read and full of concrete, real-life examples. I think you would enjoy them (if you haven’t read them already).
– The Drunkard’s Walk: How Randomness Rules Our Lives, by Leonard Mlodinow.
A fun book to read that will expose you to a number of important concepts and fallacies with real-world examples.
– Fooled by Randomness: The Hidden Role of Chance in Life and in the Markets, by Nassim Taleb.
While there are other books that discuss how to think about chance, there are a couple of good reasons to read this one (in spite of the prose style that is not everyone’s favorite):
- Nassim Taleb and his concepts have (rightly) entered into the collective awareness. You should be in-the-know and also form your own opinion;
- This book is the first of a series where much more advanced, important, new concepts (in particular Anti-Fragility) are discussed;
- The real strength of Nassim Taleb is “Transversality” (my term), the ability to apply knowledge and concepts from one field to another field and recognize that the underlying structure is the same. This is a very rare ability – and this book offers numerous examples.
Clearly this is just a quick start – and not a complete program. I plan to follow up and expand on these suggestions in the near future and perhaps set up a webpage for this purpose. Also, fellow enthusiasts will be able to post their own comments and suggestions there and perhaps encourage and stimulate others by providing accounts of their own progress.