Basic Thinking Tools

December 19, 2013

Following the post on a simple note-making technique, here are some ideas on basic tools for thinking on paper:

  • a simple symbol and
  • a key question it stands for

– like this:

\overset{\bullet}{\rightharpoonup} stands for “What’s wrong here?”

\overset{\circ}{\rightharpoonup} stands for “What could I do?”

\overset{}{\rightharpoonup} stands for “What would be logical?”

Some remarks.

  1. Most important:
    I see no need to use the symbols all the time – when my work flows without them, everything is fine.
    But when I get stuck, using \overset{\bullet}{\rightharpoonup} = “What’s wrong here?” very often helps.
  2. The symbols are designed for quick writing.
    I gave the question “What would be logical?” the most simple symbol \overset{}{\rightharpoonup} since I use it most often.
  3. The questions are designed to be “fail-proof” in the sense that they should lead to some progress in practically every situation.
  4. The exact phrasing of a question seems to me a matter of personal liking.
    Instead of “What would be logical?” you might try “What would be natural?” or a simple urging “So?!”.
  5. There are countless other possible symbols and questions. The above three questions, with their focus on obstacles, options and next steps, seem essential to me.
  6. As described earlier:
    If the question “What could I do?” leads me to several options that are worth trying, I use for each of them a circle “o” as a reminder for examination. Later I can tick off the options I have tried.

Here’s the next sandbox example – click to enlarge:


Next comes a post on Math Problem Solving Tools.

Practical Methods of Solving Math Problems – a Restart

December 19, 2013

Which practical methods could help me solving math problems?
I will approach this question by splitting it into two parts:

Q1: How can I make notes that support thinking about math problems in the best possible way?

Q2: How can I use thinking tools that are helpful in solving math problems?

In Q1, I find the following aspects important:

  • How can notes help me to develop ideas and to think straightforward?
  • How can I add reflections on my thinking, to see what works well and what doesn’t?
  • How can I deal with multiple approaches and chains of thought without getting lost?
  • How can I store sudden ideas for later examination?

In Q2, I’m puzzled by the following things:

  • How can I find useful representations of the problem?
  • How can I generate and exploit seminal ideas that eventually lead to a complete solution?
  • How can I deal with obstacles, mathematical and otherwise?

Answers to Q1 and Q2 are obviously closely linked, so I’m aiming at an “integrated method of math problem solving” – it’s not just about note-making and not just about thinking tools, but about combining both for good results.

This is the program for the following posts.

The next post is about note-making technique.

I’m focused on these topics for several years now, and I have written previous posts about them, see here or here. What I present now is a thoroughly revised version.

Math Strategy Poster 2

April 3, 2011

Here’s a large map with strategies, tactics, tools and tricks for math problem solving.
The basic ideas behind this map are of course massively influenced by the works of George Polya, Arthur Engel and Paul Zeitz. Several important ideas come from Christian Hesse’s book “Das kleine Einmaleins des klaren Denkens”.

Press the right button on the bar below the document to read it in full screen mode.

View this document on Scribd

Tool Maps: Collections of Math Problem Solving Tools

October 7, 2009

(If you are not familiar with the tool map concept, you find more information in this post.)

Here comes a collection of tool maps.

Tool map example I: Basic Heuristics

This map describes some key procedures for mathematical problem solving using mind maps.
The stages are of course quotes from Polya’s “How to Solve It”.

Tool map Basic Heuristic

The map was prepared using the open source software “Freemind”.

Tool map example II: Understanding the problem

The material for the following two maps is taken from a number of standard sources, like George Polya’s “How To Solve It”, Arthur Engel’s “Problem-Solving Strategies” and Paul Zeitz’ “The Art and Craft of Problem Solving”.

Tool map Understanding the problem

Tool map example III: Devising a plan

Devising a plan

Tool map example IV: Math Creativity

This map is rather experimental and adapts a number of classical creativity techniques, like morphological analysis, bisociation or Osborn’s checklist.
Many of these techniques have been developed in an engineering context. The main inspiration for this map was the excellent book “101 Creative Problem Solving Techniques” by James M. Higgins.

Math Creativity

Math problem solving and mind mapping

October 7, 2009

Here are some ideas on a new technique for solving math problems using mind maps.

What is mind mapping?

Mind mapping is a special form of note-taking. Here are some essential features:

  • You take a (preferably large) sheet of paper in landscape format.
  • You write the topic / the problem in the middle of the sheet and draw a frame around it.
  • You write the main aspects and main ideas around that central topic and link them through lines to the center.
  • You expand the ideas in these “main branches” into subbranches etc.
  • Wherever appropriate, you should use figures, colours, arrows to link branches etc.

Here are three examples from Wikipedia. Beware: I didn’t bother to check their content. I’m just interested in their different layout and appearance.

A handdrawn map:

A simpler handdrawn map:

A computer map:

How can mind maps be used for solving problems?

Let’s start with two principal uses:

  • Using mind maps to examine a given problem.
  • Using mind maps to organize problem solving tools.

These two uses may even be combined, leading to the use of two mindmaps at a time:

  • a “problem map” for dealing with the problem itself and
  • a “tool map” (or several of them) containing problem solving tools – from general ones (e.g. the ones presented in Polya’s “How to Solve It”) to highly specialized ones (e.g. for dealing with Poisson processes).

Problem Maps

The key difficulty in using mind mapping for mathematical problem solving is to combine conventional mind map layout with ordinary (and often lengthy) computations, because the latter simply don’t fit well into the mind map layout.
After some experiments, I have found a way that works fine for me (and which can certainly be modified in a number of ways):

  • I use the upper third of the sheet for the problem map.
  • The two lower thirds are tiled in boxes and are used for computations and working out details.
  • The middle line is a simple convenience. (See the example below.)

The result is a hybrid form of notetaking, combining mind maps and more conventional notes.
The computations and details can be referenced in the problem map by numbers, if necessary.
In this way, I can use the problem map for collecting ideas and for directing and “supervising” the detail work.
The use of boxes was inspired by an article “Stop Making Stupid Mistakes” by R. Rusczyk on
It should be clear that problem maps are intended for finding a solution, not presenting one.

An Example

Here comes a sample problem map. The problem is very easy, but the sample should show the flavour of the method. The map deals with the following problem:

Show that there are infinitely many positive integers which are not the sum of a square and a prime. (This exercise is taken from Arthur Engel’s book “Problem-Solving Strategies” (p. 133, no. 63 a).)

Problem Map Example

Advantages of the hybrid layout

  • Due to the map’s layout, it’s easy to collect ideas and group them. Further ideas can later be added at appropriate places in the map.
  • The problem map helps you not to lose sight of the overall picture.
  • If you are stuck, the problem map can help you to bring structure into your thoughts.
  • It’s easy to keep track of several aspects or approaches, of aims and sub-aims etc.
  • Using words, mathematical terms and figures in the problem map and in the boxes allows you to exploit the advantages of each of these three representations.
  • The ideas and chains of thought documented in the problem map and the boxes can be scrutinized.
  • Mind mapping itself is easy to learn and fun to use.

Some ideas on possible variations

  • Use larger sheets for more complex problems (A3 instead of A4).
  • Use separate sheets for problem map and conventional math notes.
  • Begin with conventional math notes and start the problem map as soon as you run into difficulties.
  • Change details of layout (e.g. use more space for the problem map, place the theme of your problem map at the left margin rather than at the center …)
  • If reasonable, use auxiliary mind maps in the boxes.

Tool Maps

The basic idea in using tool maps is to collect and structure problem solving tools in mind maps.
The tool maps can be organized along several concepts, like:
Stages of problem solving, e.g. Polya’s scheme from “How to Solve It”:

  • “understanding the problem”
  • “devising a plan”
  • “carrying out the plan”
  • “looking back”.

Standard situations in problem solving, e.g.

  • “looking for new approaches”
  • “overcoming frustration”
  • “need for information”
  • “my most frequent errors in problem solving”.

Mathematical objects involved, e.g.

  • matrices,
  • polynomes or
  • inequalities.

Here comes a small collection of tool maps [more…]

A brief discussion of Tool Maps

We start with its advantages:

  • Most important: In constructing and improving your own tool maps, you learn a lot about problem solving and especially your personal problem solving behaviour.
  • Tools maps act as reminders for techniques you might otherwise have overlooked.
  • Tool maps can help novices with adopting new working heuristics.
  • Tool maps are very flexible and can be adapted to all sorts of experience, needs and special fields.
  • Due to their graphical representation and their structure, tool maps are easier to scan and to expand than conventional catalogues or lists.
  • Tool maps may help to share problem solving techniques in a group by making “implicit” problem solving techniques “explicit”.

Here are some disadvantages:

  • Sometimes tool maps may become messy and overloaded and need redrawing.
  • To use tool maps consistently, it’s essential that the tool maps are easily accessible, (e.g. as a poster at the working place, or as a handy folder).

When to use Tool Maps

The tool maps can be of use especially in the following situations:

  • You are stuck and need some new ideas:
    Consult the tool maps and look for new approaches.
  • You are a novice and want to learn some new problem solving techniques:
    Use tool maps as a kind of “recipe book”.
  • You want to make sure that you do not overlook some important aspects in dealing with your problem:
    Use tool maps as checklists.

It is expressly NOT suggested to use the tool maps in every stage of problem solving.

How to combine Problem Maps and Tool Maps

Problem maps and tool maps are two modules that can be used separately.
However, using them in combination may lead to a number of interesting problem solving practices.
Here are some ideas.
For me, the following process works well:

  • I start with collecting seminal ideas in the problem map. At this initial stage, I make use of the tool maps.
  • Intuitively I chose the most promising approach and work out the details in the boxes.
    Usually, this involves looking at special or extreme cases or drawing a picture or finding another appropriate representation of the problem.
  • If none of the ideas collected before leads to a solution, I use the tool maps again and look for further approaches. I can now use the information I have collected up to this
  • I describe and analyze obstacles in the problem map and try to develop new approaches using this information.
  • When finishing work on a problem, I ask myself why or why not I have found a solution and what steps were crucial.
    If necessary, I add new tools to the tool maps.

Although the process of using problem maps may seem rather formal, there is much room for intuition and gut feeling.

Math problem solving difficulties: Remedies

January 5, 2007

Here are some ideas on how to use mind maps to overcome difficulties with math problems.
As described elsewhere on this blog, we will use two mind maps:

  1. a “problem map” for dealing with the given problem, and
  2. one or more “tool maps”. Tool Maps are collection of tools for solving math problems. Here are some examples.

For the problem map, we use a sheet of large size (A3 works well) and the following layout and template:

What can we do with the branches in this map?
Here are some basic ideas.

  • Look at an example.
  • Make a table, chart cases systematically.
  • Draw a figure.


    • Collect ways of representing the problems (algebraic, geometric or graphic, algorithmic…)

Choosing a clever representation is often vital for finding a solution. Don’t neglect this step.


    • Collect approaches how to tackle the problem (e.g. contradiction, induction, looking at extreme elements…)

These approaches are just seminal ideas, not entire plans of a solution.
You can work in a brainstorm fashion, so list even approaches that look less promising.

For each of these template branches, you can use ideas from the tool maps.
Here is an easy example:
Working wih the template is fairly straigthforward when you start examining a problem.
But sooner or later difficulties and obstacles will probably appear.
Here are some snapshots from an interior monologue:

  • “This seems too complicated.”
  • “I don’t want to go into masses of single cases. There must be a more elegant way.”
  • “I have no idea how to tackle this.”
  • “All approaches have failed, and I have no idea what to do next.”
  • “This approach looks promising, and the first steps feel right, but what now?”
  • “I’m confused! I’d like to change a single item, but it’s connected with all the others.”

To make things (just a bit) more systematic, here is a tool map showing typical difficulties and some possible remedies.
(Again, the SCAMPER mnemomic is taken from Ron Hale-Evan’s “Mind Performance Hacks”, was developed by Bob Eberle and first published in Michael Michalko’s “Thinkertoys”.)

How to apply these ideas practically?
Here is a simple idea:
Add a subbranch to the approach you are focusing on. If you like, you can label this branch “O” for “obstacle”.
Describe the difficulties with this approach. If you like, use the collection of difficulties above. If it helps, use it as a starting point for your personal “First Aid” tool map.
Then add ideas on how to overcome these difficulties, using the remedies suggested above.
Here is an example:

Making notes and solving (math) problems

August 3, 2006

There is a strong case for making notes during problem solving.
Here are some of the more obvious reasons:

  • Writing notes helps managing complexity: You can split a problem into parts, you can collect several approaches and deal with them one after the other etc.
  • Notes document your thoughts – for later scrutiny, for resuming a chain of thought later – and sometimes for posterity.
  • Notes can help to combine text and diagrams. The human brain is well equipped to deal with words and images, and either representation allows the application of quite different tools: In texts, you can ask questions, formulate alternatives, associate verbal concepts etc., in diagrams, you can add lines, rearrange items etc.

So making notes during problem solving is well worth a trial.
This leads to the question:
How to make notes most cleverly?
I’m sure this is an important topic in solving problems in general and in solving mathematical problems in particular – and one that is too often neglected.
The process of “tool mapping” described here is but one answer to this question (albeit hopefully a well thought-out one).