############################################################################## How to math ############################################################################## .. important:: * [reddit] `efrique's comment on [Q] Can you remember everything? `_ (From the book `How To Think Like a Mathematician - Kevin Houston `_) ============================================================================== Basic reading suggestions ============================================================================== #. **Read with a purpose:** Before reading, decide what you want from the text - consolidate, clarify, find an overview - or something else - from the material. #. Choose a book at the right level #. **Read with a pen and paper in hand:** * First reason: record ideas as they occur to you - write what it means, not what it says. Don't take note when reading through the first time. * Second reason: You can explore theorems and formulas by applying them to examples, draw diagrams such as graphs, solve - and even create your own exercises. #. **Don't read it in a linear fashion:** It is perfectly acceptable to dip in and out, take what is relevant to your situation, and to jump from page to page. ============================================================================== A systematic method ============================================================================== We now outline a five-point method for systematically tackling long pieces: #. Skim through and identify what is important. * **Skim:** look briefly through the text to get an overview. * **Identify what is important:** In a more careful but not too detailed reading, identify the important points. Look for assumptions, definitions, theorems and examples that get used again and again, as these will be the key to illuminating the theory. If the same definition appears repeatedly in statements, it is important - so learn it! Look for theorems or formulas that allow you to calculate because calculation is an effective way to get into a subject. #. Ask questions. * **Why does the theory hinge on this particular definition or theorem?** * **What is the important result that the text is leading up to and how does it get us there?** You can make a detailed list of what you want from the text. #. Read through carefully. You can do statements first and proofs later if you like. * **Careful reading:** thinking, doing exercises and solving problems. Ensure that you know the meaning of every word and symbol. * **Stop periodically to review:** Do not try to read too much in one go. Keep thinking about the big picture, where are we going and how is a particular result getting us there? Read statements first - proofs later: can be read later. #. Be active. This should include checking the text and doing the exercises. * **Check the text:** Fill in the gaps left by the writer. The second reason is to see how theorems, formulas, etc. apply. * **Be a sceptic:** Don't just take the author's word for it. * **Do the exercises and problems:** Not studying mathematics, but doing mathematics. I can't do the exercises, then I don't understand the topic. #. Reflect. * In order to understand something fully we need to relate it to what we already know. * Is it analogous to something else? * When you meet a topic ask **"What does it allow me to do?"** ============================================================================== What to do afterwards ============================================================================== #. Don't reread and reread - move on: If you are rereading, then it is probably a sign that you are not active - so do some exercises, ask some questions and so on. Ultimately, it is acceptable to give up and move on to the next part; you can always come back. Requires time to be absorbed by the brain; ideas need to percolate and have time to grow and develop. #. Reread: One should come back much later and reread. This often reveals many subtle points missed or gives a clearer overview of the subject. #. Write a summary: Will the material be obvious at a later date? It is a good time to make a summary - written in your own words. ============================================================================== How to read a definition ============================================================================== #. Observe: We are not allowed to read in anything extra. Note that in an example all the conditions need to be true. Not just some. #. What are we dealing with? * Is it something we already know well with an extra property? * Is it similar or different to a definition already known? Is it analogous to something else? * Is it a definition we know plus a new condition? #. What examples of this definition exist? #. Find standard examples - to help us to remember the definition. #. Find trivial examples - very, very simple examples that can help develop a feel for a definition and can be valuable when analyzing theorems and their proofs. #. Find extreme examples - the example of the definition is at the boundary of the definition. #. Find non-examples - examples that do not satisfy the conditions of the definition. Non-examples are useful for finding counterexamples to statements, but can also be used in fixing the definition in your head #. Create new objects from old ones Analyzing theorems ============================================ #. Find the assumptions and conclusions #. Rate the strength of the assumptions and conclusions * A strong assumption refers to a small set of objects. * A strong conclusion says something very definite and precise about those objects. In both cases the opposite of strong is weak. Mathematicians want weak assumptions and strong conclusions. #. Compare with earlier theorems: How do the assumptions and conclusions differ? Are they weaker or stronger? #. Observe the detail: Every word will be important - even the little words. Read and notice every word and think about what they mean. #. Classify what the theorem does and how it can be used: What does a theorem really tell us? Does it allow us to calculate, does it classify (i.e. tell us what something is)? #. Draw a picture #. Apply to trivial examples and other extreme cases #. Is the converse true? #. Rewrite in symbols or in words #. What happens to non-examples? #. Generalize: If we drop an assumption from statement A, then we call the weaker statement a generalization of A. If you generalize and can find a counterexample to the general statement, then you will have found that the assumption is vital to the original theorem.