This map contains a number of ideas on solving math problems. The map deals with general problem solving tools that are not specific for one area, e.g. you will find nothing specific on dealing with polynomes or inequalities. Most of the ideas are taken from the following sources: Engel, Arthur: Problem Solving Strategies Polya, George: How To Solve It Zeitz, Paul: The Art and Craft of Problem Solving
Combining OBJECTS and ACTIONS is basically a math creativity technique: It leads to a large number of seminal ideas to tackle a (math) problem. The underlying ideas are of course not restricted to m a t h problem solving.
From a v e r y abstract point of view, a problem consists of - the unknown - the data and - the condition. This idea is taken from George Polya's "How to Solve It".
These proof techniques are standard. More information can be found in Arthur Engel: Problem Solving Strategies Paul Zeitz: The Art and Craft of Problem Solving
Finding an appropriate representation of a problem is often crucial for finding a solution. Example: A problem may be very ugly in cartesian coordinates, but can be treated nicely in polar coordinates.
algebra. geometry. algorithms. number systems. number representations. coordinate systems. binary. .......
complex numbers. graphs. series. .......
functions. numbers. sets. matrices. relations. ....... equations. inequalities.
forward. backward.
To make the actions easier to memorize, we use the keyword "SCREAM". It will be clear in a moment how this is meant - look at the next branches. You may have guessed that I'm not a native speaker of English. So if someone comes up with better keywords - help is greatly appreciated.
To remember some of the most important mathematical manipulation actions, we use the keyword SCREAM - it stands for S: substitute C: combine; create R: rearrange; reverse E: eliminate A: adapt M: modify; magnify, minimize (This is largely inspired by Michael Michalkos "SCAMPER" creativity technique.)
combine. create.
rearrange. reverse.
eliminate. exchange.
adapt. add.
modify. magnify. minimize.
To memorize common methods of analyzing math problems, we use the keyword SPLENDID - it stands for S: look for SYMMETRIES P: look for PATTERNS L: look at LIMITS E: look at EXTREME cases N: is a rather useful letter in this keyword;-) D: collect DATA I: look for INVARIANTS D: take a more / less DETAILED look
patterns - look for patterns.
limits - look at limits.
extremes - look at extreme cases.
data - collect data. organize data. in tables. in trees. in mind maps.
invariants - look for invariants.
details - take a more / less detailed look.
ask people. use literature. use the internet. teachers. fellow students. experts. math reviews. books. journals. email people. use forums. use newsgroups. use math databases.
talk to someone. take a break. work in a new setting. eat / drink something. exercise. remember previous successes. breathe deeply and calmly. persist. ....... sleep over it.
How to use this map. Here are a number of ideas how you can use this map.. Changes to this map. Math Problem Solving: Links. read this map. print this map. Here are some ideas on printing: If you are logged in as a mindomo user, you have a number of export functions: - export to pdf - export to image file: png, jpg - export to text: rtf, txt If you want to print the map as an image, choose the branches you are interested in and hide the rest. The map is large, so use the largest paper size you can manage. You can use the print as a problem solving poster.. modify this map. There are a number of ways to modify this map. a) Modify t h i s map. b) Modify a copy of this map. With a view to collective knowledge, a) seems preferable. However, mindomo has not (yet) version management functions, so all previous versions of this map are lost. If you modify this map - could you please leave a note at the "Log and Comments" branch?. new "SCREAM" tool added. Problem Solving area of the nzmaths website.. Fifty Problem Solving Strategies Explained. Heuristixx: Math Problem Solving and Mind Mapping. MATHCOUNTS article on Problem-Solving strategies. Tips on Problem Solving. Mathematical Problem Solving Strategies. several articles on scribd.com. Art of Problem Solving. Mathematical Problem Solving and Mind Mapping. Mind Mapping and Math Problem Solving.