Thinking on paper

October 10, 2021

How to think on paper? Here’s an updated version of the method that works best for me.

I use the following three building blocks, which I can combine in different ways.

Building block 1 – the sheet layout:
I use an A4 sheet in landscape format and divide it in 3×3 boxes of equal size – there’s no need for exactness and I do it without a ruler.
Alternatively, I can use 3 columns and make boxes of different sizes – but at the beginning, 3×3 boxes are easier to handle.
I fill the boxes in column 1, then in column 2 and then in column 3.

Building block 2 – the box layout:
I use each box for a simple small mind map. The size limit of the box will come as a surprise, but it has a number of advantages which I will describe in a moment.
Instead of mind maps, I can use diagrams, ordinary text with indentations or math terms.

Building block 3 – thinking tools:
I can use thinking tools as topics for the small mind maps, to stimulate my thinking about the problem, like: What are key questions? What are my options? How can I split the problem into smaller parts?
I find a personal collection of thinking tools very useful – I have a corkboard over my work table and can access dozens of tools and stimuli with one glance.

For me, this process has four main advantages.
First, a conventional mind maps works best with a couple of main branches of equal size. In problem solving however, I often want to develop only one or two branches over many steps, and this leads to an unsymmetrical, unbalanced mind map.
Secondly, moving from one box to the next needs a moment of orientation – where do I stand, and what can I do next? This re-orientation happens much more often in the 3×3 layout than in an ordinary mind map.
Thirdly, I find it easier to reflect on a previous thought in a new box than in a branch of an ordinary mind map.
Fourth, the combination of mind maps and thinking tools seems very powerful to me.

Some remarks:
Depending on handwriting size, paper size and personal taste, we can experiment with 2×2, 3×3 or 4×4 boxes.
As a fourth building block I’m using a zettelkasten to organize my sheets – but that’s outside the scope of this comment.

Four Sheets on Note-Making

May 18, 2021

Another Example of the Matrix Map Method

May 18, 2021

The Matrix Map Method

May 18, 2021

Over the last few weeks I’ve done some experiments with arrays of small paper-based mind maps combined with thinking tools. 
It’s in an early stage, and feedback is very welcome.

Here comes a short description and some obersevations.
(As in my earlier ideas along these lines, these methods of “thinking on paper” are primarily intended for creating thoughts, not necessarily for storing them or for communicating them to others.)

Here’s the description.

  • I use a blank A4 sheet in landscape format.
  • I divide the sheet in 4 x 4 cells of equal size.
  • An important technical remark: I use a mechanical pencil (0.5mm) and tiny handwriting, so I fit about 10 lines in one cell. Alternatively, I could use 3 x 3 or another number of cells, or use A3 paper to get a convenient “cell capacity” in terms of words per cell.
  • For the layout of single cells, I can choose between small mind maps or ordinary text or diagrams.
    Mind maps are my default choice, but diagrams appear time and again.
  • I can now fill the cells row by row and colum by column.
  • To start a new cell, I often use small “thinking tools”. A present favourite is “F/P”, which stands for “focus and progress” – first I look for an aspect of my topic I want to examine closer, and then I try to make progress on this aspect – usually by asking questions, or by collecting ideas, or by applying creativity tools.
  • At my desk, I have a “tool collection” in the form of about two dozens of sticky notes. They contain basic thinking operators like F/P, collections of idea generation stimuli (like the TRIZ principles, or verbs of modification), basic diagram types (like timelines, pie diagrams or decision trees), or basic “lenses” with which I can look at my topic (“look at points of transitions, look at points with special properties, look at quantiles…”). With many of these tools, I try to transfer a useful and sometimes even sophisticated general concept to my topic.

Here are some observations I’ve made during my experiments.

  • First, the crucial point: The capacity of one cell is very limited. In my experience, this leads without extra efforts to a much higher frequency of refocusing, of asking questions and of doing useful forms of metacognition, in comparison with my ordinary mind mapping habits. For me, this is the main strength of the method.
  • The threshold for writing down first thoughts is very low. It feels like I do not have to fill an entire sheet, but just one cell. And if I mess up with one cell, I can restart better in the next.
  • Finding a convenient cell capacity may require some experiments. However, having a fixed cell size rather than a variable one seems to work better for me at the moment.
  • If I have an idea overflow for one cell, I simply start the next on the same topic or on the particular branch I want to expand. This is of course a compromise, but in my present delight with the method I see it as outweighed by its advantages.
  • I find it convenient to use flexible default tools like the above F/P that lead to insights in a broad range of situations.
  • The method seems to work especially well with idea generation using stimuli.
  • It’s easy to assemble a personalized “thinking on paper” method – by combining sheet organisation (in a ZK) and sheet layout and cell layout and tool collections with their architecture and their elements.

New pages on zettelkästen and problem solving

February 23, 2020

Here is “rock paper thinking”, my collection of ideas on how zettelkästen can help to solve problems.

Here’s my New Video: Some Ideas On Problem Solving

June 18, 2016

Note Assistants: Support for Solving Math Problems

January 11, 2015

Here’s my latest collection of ideas on note-making and math problem solving, presenting a four column layout, the concept of note assistants and of “paper software”, loads of math problem solving tools and much more.

One-Page Advice on Note-Making

January 13, 2014

Here’s my latest upload on note-making.

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

View this document on Scribd

Collecting Math Problem Solving Tools

January 6, 2014

In the post Math Problem Solving Tools: An Overview I presented a list of questions like the following ones:

How can I get started?

  • How can I approach a problem? Are there established methods that work for a large variety of cases? How can I use them?
  • How can I represent a given problem, using equations, diagrams and other means that allow a math treatment?

It seems a reasonable next step to collect answers to these questions, thus composing a tool collection. Some of the following tools are taken from the list in George Polya’s book “How to Solve It”.

  • Introduce mathematical notation.
  • Write down what is given and what is unknown.
  • Draw a figure.
  • Arrange data in tables.
  • Check diverse representations:
    – Use numbers – decimal, binary, complex, factorized in primes…
    – Use coordinates – cartesian, spheric… Use a suitable origin.
    – Consider features of the problem (like symmetry) to make the representation simpler.

Starting with these simple and fairly heterogeneous examples, here are some ideas on tool collections.


Tool collections should help individuals

I should aim for a tool collection that is of help to me – adapted to my knowledge, my experience and my fields of interest. Such a tool collection is evidently not static, but changes with time.


How can a tool collection be recorded?

From a tool collection that exists only in my mind to handwritten lists and vast computer-based tool libraries, there are all sorts of possible representations.
Having it in a written form however has several advantages:

  • I know explicitly some of the (most important) tools I am using,
  • I can discuss such a list with others and
  • I can try to improve such a list systematically.


What are the main challenges in using a tool collection?

There are many ways of using a tool collection:

  • I can use it before the actual problem solving process, to remind me of important processes, tools and practices.
  • I can use it during problem solving to get cues.
  • I can use it after problem solving to improve it for further use.

The crucial questions for me are the following:

  • How do I find – in each problem solving situation where I look for help – the tools that are useful just now?
  • How can I apply these tools?
  • In most cases applying one tool will not lead to an entire solution – so how can I deal with a sequence or with a combination of several tools?

(Given my obsession with note-making in math, the reader may guess what I will offer as an answer.)


What should be in it, and what should be left out?

This question has no general answer. In the following posts on tools, I will focus on general, domain-independent tools that seem to me very useful.

For the time being, I ask my readers to compile their own tool collections. There are some starting points I have worked out in the past, see here and here.


Some resources

  • An example of an online repository is the Tricki (started by Timothy Gowers). Regrettably, it seems no longer under active development.
  • Besides, there are all sorts of math cheat sheets on the internet – here is a first impression.
    From my experience, they are of little help for solving most non-routine math problems.

Math Problem Solving Tools: An Overview

December 21, 2013

Following the posts on note-making in math and on basic thinking tools, here come first ideas on thinking tools for math problem solving.

Finding a solution to a math problem can be broken down into parts in several ways – for example by asking questions.
A first useful catalogue of questions could look like this, where each main question is followed by more specific ones.
(The catalogue is massively inspired by the famous list of questions from George Polya’s classic book “How to Solve It”. – Polya’s list is reproduced here.)
The next posts shall give some answers to these questions.

How can I get started?

  • How can I approach a problem? Are there established methods that work for a large variety of cases? How can I use them?
  • How can I represent a given problem, using equations, diagrams and other means that allow a math treatment?

How can I construct a solution?

  • How can I generate seminal ideas? What methods can I use to generate ideas?
  • How can I exploit these ideas?
  • How can I start from what is given and work forward?
    How can I start from the end and work backward?
  • How can I decide whether to proceed with an idea or whether to try a new one?
  • How can I deal with multiple ideas?
    Should I produce several of them and pick the most promising one, or should I start with one and remodel it until it works?

How to deal with obstacles?
(This part is closely connected to the previous and the next one.)

  • How can I make sure I realize there are obstacles that need attention?
  • How can I use established ways of dealing with obstacles?
  • How can I analyze an obstacle?
  • How can I generate ideas to overcome an obstacle?

How to deal with other troubles?

  • How can I deal with frustration, with the impression that I’m no good at math, with a lack of interest?

How to look back?
(This part is not exactly necessary, but a great opportunity to improve problem solving skills.)
Remember: Looking back should happen not only at the end, but also on the way!

  • What’s bothering me? What doesn’t feel right?
  • How can I clarify these intuitions?
  • How can I check if my solution is correct?
  • What can I learn from what I’ve done so far?

Finding answers to these questions may lead to a collection of math problem solving tools. Read more in the next post.