Now we come to one of the biggest secrets in thinking like a mathematician. So far in this book we have covered many of the important tools for logical thinking: statements, implications, quantifiers. Many other books cover these. But they cover in little detail what I consider to be one of the most important tools. It is fundamental to thinking like a mathematician as I am sure many mathematicians will agree, including those textbook writers. But you won't find this topic given its true prominence in the textbooks.

So what is this great secret tool? Simple! Examples. The quote from Euler is crucial to thinking like a mathematician. First let's remove a possible misconception. Low-level mathematics is often taught in the following way: ‘This is how the product rule for differentiation works, here are some examples, now you do some exercises just like the examples.’ This is the monkey-see-monkey-do approach. Students are set problems where they can look at the given examples and merely copy the format. (This does actually work well for low-level mathematics.) This type of example is usually called a worked example. It is not these I am interested in.