Choosing a Text BookStudents studying a mathematics subjects are typically prescribed a text book. However, for independent social science researchers the required book is often less clear. Common scenarios include: (a) you are reading a journal article and it uses a mathematical or statistical technique and you want to learn more about it; (b) you are considering applying a technique in a research project and want to learn more; or (c) you are doing a statistics course that smooths over some of the details on implementation and you want to understand these details. When choosing a text book, relevance, clarity, and an appropriate level of sophistication is required.
Relevance:Does the chosen book align with your learning goals? Does it go into the appropriate degree of depth for your needs?
Clarity of presentation:Some books are just better than others. Also, some may be written with your particular applications and concerns in mind. For example, there are many mathematics books written with the social scientist in mind. They often make fewer assumptions about background mathematical knowledge and present relevant applications.
The right level of sophistication:This depends on your prior knowledge. Most mathematics textbooks will list the assumed knowledge in the preface or in an introduction. Failing that a look at the first few pages will tend to reveal whether the book is too sophisticated.
If a book is too sophisticated, you are confronted with the choice of whether it is worth acquiring the prerequisites to understand it. If you decide the book is worth reading, you must then ascertain what such prerequisites are and work out a means of acquiring them. This assumes some knowledge of the hierarchical dependencies between mathematical topics. I have previously posted on this in relation to a sequence of videos and textbooks related to learning statistics. A knowledgeable adviser can also assist in setting a sequence of self-study.
Once a textbook has been selected, there is the question of how to read the material in order to acquire the desired skills and knowledge.
Active ReadingReading mathematics is different to reading other material. The following are a brief set of suggestions. The subsequent links should supplement this list.
- read with a pen and paper
- do provided problems
- create problems and do them
- work slowly monitoring comprehension
- structure the material; identify core concepts and interrelationships
- identify material that is not understood and devise methods to learn it
- learn to read the symbols:
Learn how to pronounce the symbols: I previously posted on links for looking up the pronunciation of mathematical symbols.
Learn what the symbols means: Learn how to use them, manipulate them. Learn their conceptual importance.
- learn key terminology
Additional Online ResourcesThe following links provide additional tips on reading mathematical material.
- Kevin Lee List of tips for reading mathematics textbooks, and on writing mathematics
- Alex Roux: Guide on reading mathematics
- Steven Diaz: Short and sweet set of slides on reading mathematics.
- Angela Vierling-Claasen: Guide including structured questions to ask yourself while reading mathematical material and some strategies for what to do when a difficulty is encountered.
- Shai Simonson and Fernando Gouvea: Material on how to read mathematics. They advise to see the big picture, be an active reader, read slowly, and make the ideas your own. They make several analogies between the specialised skills of reading mathematics and those of reading fiction, poetry, and non-fiction. They also provide an example of applying their advice.
- College-algebra: Tips on reading a mathematics textbook
- Stephen Maurer: Advice for Undergraduates on Special Aspects of Writing Mathematics is more about writing mathematics than reading. However, it is also helpful in understanding some of the conventions of mathematical discourse.
- Oklahoma State University: How to Read Mathematics for Meaning – Making Sense of Mathematical Prose. This has some tips with further links.
- One other post on mathematical writing
Additional Off line Resources
- Section 1.2 of The Princeton Companion to Mathematics has an well-written and accessible discussion of the language and grammar of mathematics.