Категории: Все - strategies - resources - information - creativity

по Thomas Teepe 4 лет назад

1447

How to solve math problems

Tackling math problems effectively involves a combination of various strategies and resources. Key practices include taking detailed notes and utilizing a broad array of tools designed to assist in solving mathematical challenges.

How to solve math problems

How to solve math problems

This is a collection of ideas on math problem solving.

Here are some highlights:

The map offers

hints on notetaking for math,

a large collection of math tools,

several new techniques for math creativity,

internet resources.

about this map and the author

about the author

Thomas Teepe,

born 1971,

doctoral thesis on Genetic Algorithms in applied math,

works as an actuarial consultant,

lives in Stuttgart, Germany.

Feedback on this Mindomo map is very welcome.

Please write to thomasteepe@web.de.

changes to this map

03.03.2010: First draft

07.03.2010: Collection of math tools added

30.03.2010: Additions to "Combining notetaking and tools", smaller changes to other text notes

15.10.2010: Some minor changes - branch names

combining notetaking and tools

The basic idea is to finetune notetaking techniques and math problem solving tools.

Here are a number of ideas.

change between mind maps and conventional math notes

Some parts of math problem solving work better in a framework of conventional, linear math notes, others in mind maps.

Linear notes are for example good for

manipulating equations,

computations.



Mind maps are for example good for


collecting approaches,

dealing with obstacles.


tools for math problem solving

Tricki

The Tricki was started by Fields medalist Timothy Gowers.

From the start page:

"Welcome to a brand new Wiki-style site that is intended to develop into a large store of useful mathematical problem-solving techniques. Some of these techniques will be very general, while others will concern particular subareas of mathematics. All of them will be techniques that are used regularly by mathematical problem-solvers, at every level of experience. (...)"

special math tools

This branch 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

staying functional during problem solving

persist

breathe deeply and calmly

remember previous successes

exercise

eat / drink something

work in a new setting

take a break

talk to someone

getting information

use the internet

use math databases

use newsgroups

use forums

e-mail people

use literature

journals

books

math reviews

ask people

experts

fellow students

teachers

analysis tools

analyzing: look for "SPLENDID"

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

creativity tools

basic concept: combine OBJECTS and ACTIONS

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.

ACTIONS: "SCREAM"

To make the actions easier to memorize, 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.)


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.




OBJECTS

search direction

backward

forward

math objects

inequalities

equations

realtions

matrices

sets

numbers

functions

general math tools

series

graphs

complex numbers

representation of the problem

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.

algorithms

geometry

coordinate systems

number representation

...

binary

number systems

algebra

proof techniques

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
  • using symmetry

    pigeonhole prinicple

    using extremal cases

    using invariance

    by induction

    by contradiction

    by direct proof

    problem elements

    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".

    What is the condition?

    What are the data?

    What is the unknown?

    tools: some basic topics
    representing tools

    OK, you've learned a number of tools.

    How to make sure you have these tools ready when you need them?

    Here are a number of ideas.

    - Just trust your memory.

    - Use acronyms - here's an example:

    SCAMPER stands for




  • S substitute
  • C combine with other ideas
  • A adapt
  • M maximize, minimize
  • P put to other uses
  • E eliminate
  • R rearrange

  • - Use written collections of tools. This is my favourite method - even the initial act of writing down such a collection can give you so much insight into your problem solving behaviour.

    arranging tools

    Here are several methods for arranging tools.

    These methods work best when used in combination - one single method won't do.

    You may arrange problem solving tools

    - by problem solving stages:

    One of the most famous examples of problem solving stages comes from George Polya's classic "How to Solve It":



  • Understand the problem
  • Devise a plan
  • Carry out the plan
  • Look back
  • To each of these stages you can add a number of useful tools.

    - by subject matter:

    You can collect tools that help to deal with


  • sequences,
  • series,
  • vectors,
  • functions,
  • ...

  • - by problem solving difficulties:

    You can make a collection of your most frequent problem solving difficulties and then add tools as remedies.

    math problem solving tools - a definition

    Here's a definition, naive yet useful:

    A math problem solving tool is anything that may help you to solve a math problem.

    Obviously, this concept is not limited to math.

    Here are some examples:



  • draw a diagram,
  • collect data from examples,
  • consider special cases,
  • look for invariants, symmetries, patterns...,
  • apply the mean value theorem,
  • search the internet,
  • talk to a friend,
  • e-mail an expert,
  • have a break.

  • how to take notes

    Most books on math problem solving I know offer few information about notetaking.

    Here are some ideas.

    Links
    Wikipedia: Mind Mapping
    Wikipedia: Notetaking
    notetaking equipment

    Notetaking starts with materials.

    Here are some ideas.

    computers & graphic tablets

    Solving math problems typically involves things like

    using math notation, often in intricate computations,

    collecting data and arranging them in tables,

    drawing figures and diagrams.


    I doubt whether it is possible to produce all this quickly with keyboard and mouse.

    Using a tablet PC or a graphic tablet may be well worth a try.


    Here are some possible advantages:


    have all the possibilities of ordinary pen and paper notes,

    combine handwritten notes and prefab figures,

    highlight notes with colours, icons, frames etc.,

    erase notes,

    rearrange notes,

    change size of notes, minimize auxiliary material etc.,

    work on very large canvasses,

    hop from one canvas to another,

    work in collaboration with others on virtual whiteboards.


    With the right support from software this could be very powerful. For example, imagine a giant canvas with lots of virtual Post-It notes which you can manipulate in all sorts of ways.

    (In March 2010 I've downloaded a trial version of MS OneNote, which is part of the Office family. It looks very promising.)

    pen & paper

    Effective notetaking depends on putting lots of information on a single sheet of paper while keeping it well-structured and legible.

    This points to a combination of large sheets of paper (A3 rather than A4) and pens that allow small or even tiny handwriting.

    notetaking techniques

    What's the kind of notetaking that supports math problem solving best?

    Here are several approaches.

    mixed layouts

    Test

    mind mapping
    linear notes
    what clever notetaking can do for you

    Well-done notes can help

    to document your thoughts,

    to clarify your ideas ,

    to resume a chain of thoughts after a break,

    to support your memory,

    to combine thinking in words and thinking in images,

    to collect ideas quickly and examine them in more detail later,

    to create distance to your own thoughts,

    to show others your ideas.


    All this is sounds obvious? Yes, but it poses some questions of "notetaking engineering":

    What's the kind of notetaking that supports m a t h notetaking best, with all its special needs for notation, diagrams and lengthy computations?