Advice from a Mathematics Book

Some advice I think applies to every study one undertakes in life. 


  • It's up to you: your actions are likely to be the greatest determiner of the outcome of your studies. 
  • Be active: read the book. Do the exercise set. 
  • Think for yourself
  • Question everything: be sceptical about the results presented to you. 
  • Observe: the power of Sherlock Holmes came not from his deductions but his observations.
  • Prepare to be wrong.
  • Don't memorise, seek to understand.
  • Develop your intution: but don't trust it completely. 
  • Collaborate: workwith others, if you can.
  • Reflect: look back and see what you have learned.
Getting started

The biggest hurdle with any mathematics problem is getting started. Too often, students beginning high level mathematics look for an example or a theorem from which they can deduce a statemetn as a simple corollary. Unfortunately, mathematical thinking is not like that. 

Play with examples

Rarely can you say nothing about a problem. There may be some example or deduction you can make that relates to it. Somehow just playing with some examples allows us a deeper understanding of the problem.

Break it down

Another reason some students find it hard to get started: they are going for the 'big bite' approach. THe expect there to be some a single formula that will answer the question. 

A guiding principle should be to break a problem down into smaller problems. For example 'Prove A <=> B' do not look for a theorem wit hthis conclusion or try to to create a list of the form 'A <=> C, C <=>D, etc'. Instead prove 'A => B' and then 'B => A'. 

Change the problem

Specialise and generalise. Change the question either by reducing the scope or making it general, so that you can start. 

Getting to a higher level


Reverse the question

E.g., Finding the maxima and minima of functions is a basic exercise in calculus. Reverse the question. Suppose that you are given the maxima and minima are at certain points. Can you construct a function which has these maxima and minima? 

Ask 'what happens if...?'

E.g., what happens if I drop that assumption? We have seen that this can help solve problems. It can also allow us to explore the limits of a subject. We can find out why definitions and theorems are the way they are. 

See the web

Although it appears to be a linear subject, with one idea built on top of another, mathematics also contains a web of interconnected ideas and topics. After learning some topic your reflection should include thinking about how the new work fits in this web. 

You can also analyse the overall structure of the work and ask if it could have been presented in a different order. Similarly, look in different books for different points of view. Are the theorems and definitions different? Are the proofs, more, or less, rigorous? 

Source: Houston, K (2009). How to think like a mathematician. A companion to undergraduate mathematics. 










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