The document describes in detail novel methods of note-making and of using problem solving tools in math: How can my notes support thinking and problem-solving? How can I start? How can I create ideas for a solution? What can I do when I'm stuck? How can I deal with frustration?

1. Note Assistants: Support for Solving Math Problems
In this text I describe a method that should support think-and-write processes for the work on math
problems.
Contents
What are the basic ideas?......................................................................................................................1
The note-making method......................................................................................................................2
Note assistants: An example.................................................................................................................4
How to use the note assistant?..............................................................................................................6
Criticism and responses........................................................................................................................7
Note assistants: A framework for problem solving..............................................................................8
Note assistants and modules.................................................................................................................8
Some remarks.......................................................................................................................................9
Other elements in note assistants........................................................................................................10
Acknowledgments..............................................................................................................................11
Document changes..............................................................................................................................11
About the author.................................................................................................................................11
What are the basic ideas?
The method is perhaps best introduced by the term “paper software” - a “software” that does not run
on a computer, but on sheets of paper, as the most flexible “hardware” available in many
circumstances.
Imagine on my left a single A3 sheet of paper called the “note assistant”. This acts as a kind of
“menu” and contains advice for crucial problem solving situations – how to start, how to generate
new ideas, what to do when I'm stuck, etc.
On my right is an A4 sheet where I make the actual notes on my problem – the “editor”. As in real
software, I can choose a layout suitable for my notes.
Whenever I feel I could do with some problem solving support, I have a look at the menu.
This menu offers things I can “insert” into the note sheet – how to arrange the notes for a special
kind of investigation, what useful keywords to write down and above all, what thinking tools to use,
what questions to ask or what ideas to try.
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2. The diagram indicates how suggestions from the note assistant can be taken over into the note sheet
– and that there is certainly no need to use the note assistant at all if work is going on well.
Whether the items in the note assistant are actually helpful to the problem solver depends on her
level of expertise, note-making preferences, field of work and other factors - so note assistants
should ideally be adapted to the problem-solver and evolve together with her growing experience.
Later, I will present ideas on how to customize note assistants.
The note-making method
As mentioned, I can choose between several methods of note-making on my note sheet.
Here comes my favorite, followed by a number of possible variations.
• Use a blank sheet of paper in landscape format, size A4 (or larger).
• The sheet is separated by vertical lines into four equal columns.
• The text is organized in “boxes”.
These boxes are labeled 1A, 1B in column 1, 2A, 2B in column 2 etc. in the upper right
corner. (At this position, the labels need less column space, and it's easier to add them later.)
• In each box, the text can be organized in hierarchies by indentations.
• For a major new idea, a new column can be used.
• Sudden ideas can be noted starting at the bottom of column 4, in boxes 4Z, 4Y etc.
• To mark open issues, check-boxes like “” can be added at the right column border. It's easy
to find them later, examine the issue and tick off the check-box.
• If work from one box is continued in another, this can be indicated by arrows between
neighboring boxes or by references like “see 2C” or “from 3:1D” for box 1D on page 3.
• Footnotes can be used at the bottom of a column.
• I use a mechanical pencil and an eraser. The method works best if I write fairly small.
Having a non-smear pen is essential.
In my eyes, this method of note-making has a number of advantages:
• As with other forms of note-making, the memory is unburdened, and it becomes easier to
manage complex chains of thought – and trees of thought.
• The thoughts are permanently documented.
• The method works well with usual math operations, like manipulating equations.
• By switching between columns, the method can cope with changes between different lines
of thought, at least to a certain degree.
The same could be done by using separate sheets, but for me this is often a massive
disruption of the flow of work.
• Sudden ideas can be stored away with ease and examined later.
• From my experience, writing in these narrowish columns encourages me to write neat notes,
and this transfers – to some extent – to the entire work on the math problem.
There are many ways to alter the method:
• Use larger sheets in A3 (or A4 double pages in a notepad).
• Use a different number of columns – especially if the columns seem too narrow.
• If labeling the boxes with 1A, 2B seems too much trouble, leave it out and address the boxes
by coordinates: imagine the columns separated vertically in equal parts a, b, c, d and use
references like “3b”. (The printed Encyclopædia Britannica has used a similar system.)
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3. • If a larger diagram is needed, use a layout like this:
The box idea was sparked by the essay “Stop Making Stupid Mistakes” by Richard Rusczyk,
founder of the “Art of Problem Solving” website (http://www.artofproblemsolving.com/).
The next page shows a non-math example of a result of the note-making method. The sheet contains
some aspects that have not been mentioned in the text.
Remark:
The tables on the following two pages should give an impression of the actual layouts used.
Since the first table is basically an A4 table on an A5 space and the second an A3 table on an A4
space, the text is very dense, especially if viewed on a small display.
Viewing the document on a larger screen or printing it out will help.
(Thanks to Dr. Houston for pointing out this problem.)
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4. What is this sheet about? |1A
- it shows a way of note-making,
a way of “thinking on paper”
_____________________________________
What do you need? |1B
- blank paper in A4
- a non-smear pen
- e.g. mechanical pencil + eraser
_____________________________________
What's the basic layout? |1C
- use paper in landscape format
- draw lines to form 4 columns
Or:
- try A3 in landscape format
with 6 columns
- lots of space for your ideas
_____________________________________
How to make text boxes |1D
- number the boxes
- in column 1 with 1A, 1B etc.
- write the headline + underline it
- questions make
good headlines!
- write down your thoughts
- short but intelligible
- use “outlining”
- indent your lines
- to show hierarchies
- like this
- when you're finished with a box:
- draw a horizontal line
- start a new box
More things you can use |2A
- page numbers
- date
- footnotes at the column bottom1)
- numbering
- underline, colour
- tables
- diagrams
- equations ...
_____________________________________
How to deal with sudden ideas |2B
- you can mark ideas for
follow-up with a check box :
- work out a more detailed
check-box system 
- you can later tick off
these boxes:
- add some remarks on
cross-referencing 
(see 2C)
- you can store unrelated ideas
at the bottom of column 4
- look at the example!
_____________________________________
How to cross-reference |2C
- there are examples in 2B and 3B
- referencing box 4C on page 2:
- see 2:4C
_____________________________________
1) Useful for later remarks
and other things
When to start new columns |3A
and new boxes?
- for important new ideas:
- start a new column
- start a new box
- when resuming work
from one box in a new one:
- use arrows OR
- use cross-references with
“see 3A” and “from 2C” etc.
_____________________________________
Some advice on |3B
“deep” thinking
- work patiently
- from one box to another
- from one page to another ...
- … always pursuing
- open questions,
- things you don't understand
- things you can make better
- use basic questions
1) What would be logical?
2) What's bothering me here?
What's the key problem here?
3) What can I do now?
- use a “Q” section
- “Q” stands for Questions
- use it at the end of a box,
of a column to find open topics
Q
- how to design a more refined
system of thinking tools? 
- pros & cons of this method? 
(see 4A)
Pros & Cons |4A
- compare these notes with
- mind maps
- Cornell Notes
- digital note-making
- other note-making systems
(they all have their pros & cons!)
- look at the following points:
- can you focus on your work?
- no distractions from apps etc.?
- can you develop long coherent
lines of thought?
- can you store away
sudden ideas and
examine them later?
- can you switch to other
lines of thought without much
document fiddling?
- do you have an overview
of your notes?
- is straightforward, organized
thinking encouraged?
- are your notes still
comprehensible after
3 days, 2 months, 1 decade?
- is the use of tables, diagrams,
equations encouraged?
Q
- digital version of this method?
- table for above comparison? 
_____________________________________
image search |4Z
on “note-making” 
_____________________________________
Date: 11.01.2015 Page 1
Note assistants: An example
The main task of the table on the next page is to provide concrete suggestions for major problem
solving situations, with regard both to clever layout and to useful thinking tools.
These building blocks should give structure to the overall work on the problem.
The problem solving situations are highlighted in orange, suggestions for keywords that could be
written down are in yellow.
The sheet is intended for an A3 format. It contains 2 x 4 columns, numbered from 1 to 8.
The sheet is inspired by a lot of authors, especially Mason (Thinking Mathematically), Polya (How
to Solve It), Zeitz (The Art and Craft of Problem Solving), Engel (Problem Solving Strategies),
Schoenfeld (Mathematical Problem Solving), Tao (Solving Mathematical Problems) and Bruder
(Problemlösen lernen im Mathematikunterricht).
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5. How to start?
_____________________________________
Problem |1A
> write down the problem statement
_____________________________________
What is given? |1B
What is unknown?
What has to be shown?
> introduce math notation;
if possible:
> choose a smart point of origin
> use symmetry
> write down what you know
in the notation selected
(equations, inequalities)
> draw a figure
_____________________________________
Special cases |1C
> look at special / simple / extreme
cases
> bring structure to these cases
> look for patterns
_____________________________________
Useful facts |1D
>write down known facts
about the problem elements
> useful theorems?
> what ideas could connect the
problem elements?
Some standard things to do
_____________________________________
Try |2A
> use the most direct ideas
that come to mind
> use ideas from problems
that share some similarity
_____________________________________
Forward |2B
> work forward:
what can I infer
from the given facts?
_____________________________________
Backward |2C
> work backward:
start with the aim -
how can it be reached?
> what could be the step that leads
to the conclusion -
the “penultimate step”?
_____________________________________
Top-down |2D
> start with the big picture
for a solution,
then zoom into the details
> ask repeatedly
“how can this be reached?”
_____________________________________
How to try further approaches
_____________________________________
Collection of approaches |3A
> 1. get inspiration from 8A – 8D:
how can items be applied?
make a collection like this:
_______________________
A1: induction 
__________________________
A2: …........... 
(make some notes on an approach
in the spirit of “let's try something”
- it can be named later)
_______________________
A3: extreme cases 
_______________________
> 2. investigate
in suitable order – best ones first
_____________________________________
A2 – Investigation |3B
> investigate approach A2
> mark check-box in 3A for A2
later
_____________________________________
A1 – Investigation |3C
> ...
_____________________________________
I'm confused! - I'm stuck!
_____________________________________
Here's something confusing! |4A
Here's a difficulty / an obstacle!
_____________________________________
Confusion OR Obstacle |4B
> what things are confusing?
> why are they confusing?
> describe the situation
> describe the difficulties
> what is the core obstacle?
> repeat that question!
_____________________________________
What can you do? |4C
> how can the confusion
be cleared up?
> is it possible to make
the obstacle disappear?
> make a list of options
> investigate the most
promising ones
(see 3A for layout)
_____________________________________
Postpone |4D
> stay flexible -
just postpone an approach:
mark it with a check-box “”
come back later
_____________________________________
I think I have succeeded!
_____________________________________
Summary |5A
> write down the complete solution
> check every step
_____________________________________
Reflection |5B
> can the result be
generalized / improved?
> can the method be
generalized / improved?
> can I find a different solution?
_______________________
> what can I learn from my work?
> what were the key difficulties?
> is there some problem solving
behaviour I should change?
> should I adapt the
note assistant?
_____________________________________
Note Assistant
_____________________________________
orange: Problem Situations
yellow: keywords I can write
>: things I can do
blue: Math Concepts etc.
I'm frustrated / demotivated!
_____________________________________
Cheer up! |6A
> use supportive self talk
(“This problem looks hard, but:
- I can proceed in small steps,
- I can examine one idea
after the other” ...)
> remember successes from the past
(I don't have to write it down!)
> work on for just 15 minutes
> have a break
and resume work later
_____________________________________
I have a sudden idea!
_____________________________________
Aha! OR Idea |7A
> write the idea into box 4Z OR
> start a new column
> mark sudden ideas with a “”
for later check-ups
_____________________________________
All-Purpose Tools
_____________________________________
Questions |7B
> collect questions:
Question Q1 
Question Q2 .
_____________________________________
Q1 – Investigation |7C
> find answers to Q1
_____________________________________
Useful little questions |7D
> what would be natural
or straightforward?
> natural questions?
> natural things to do?
> what would be logical?
> what is the core issue or
the core confusion here?
> repeat that question!
> what can I do to make progress?
> do it!
Other Useful concepts
_____________________________________
Methods of Proof |8A
- direct proof
- proof by contradiction
- proof by induction
- visual proof ...
_____________________________________
Heuristic Principles |8B
- look for patterns
- look for analogies
- look for symmetry
- look for invariants
- look at extreme cases
- look at limits
- guess and check
- stepwise approximation
- use colourings
- use the pigeonhole principle
- use parity ...
_____________________________________
General Objects |8C
- complex numbers - graphs
- generating functions ...
_____________________________________
How to modify objects? |8D
- substitute - eliminate
- adapt - split
- rearrange - introduce new items
- maximize / minimize
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6. How to use the note assistant?
• A problem is given, and the user wants to work on it.
• She has her note sheet as an “editor” in front of her, prepared with 4 empty columns and
perhaps a page number and the date.
The note assistant as a “menu” is accessible with a glance.
• Guided by the list of problem situations highlighted in orange, there is advice on “How to
start” in column 1, with some reasonable initial operations. The user can fill the first boxes
in her note sheet with text, math terms, equations and diagrams, using the layout, the
keywords and the thinking tools suggested by the note assistant.
• After this start, columns 2 and 3 provide ideas on how to try several approaches.
• Column 2 describes a number of standards approaches. It seems natural to check these first
and turn to less direct approaches later.
• Column 3 suggests a two-step method.
- First step: The user may look for inspiration in several lists (here: 8A - 8D) in the note
assistant. She can write down a collection of approaches that seem worth closer
investigation.
- Second step: These approaches can be examined in a suitable order.
• Using a reference from box 3A to boxes 8A – 8D is a helpful design, especially when it
comes to much more specific problem solving tools for single branches of mathematics.
• When difficulties arise, column 4 has a number of suggestions.
Box 4A has an interesting function in the note assistant – it illustrates the interplay between
existing notes that lead to a certain situation, and new elements in boxes 4B and 4C.
It seemed important to me to ask for confusion and its causes – this is centered on the
immediate experiences of the user.
Box 4D suggests to postpone an approach if further progress seems not likely.
• Column 5 suggests some things to do when the main work is done.
• The contents of column 6 on dealing with emotions are arguably experimental.
I just wanted to show in principle how this aspect could be included.
The ideas on self-talk are inspired by Richard Nelson-Jones' book “ Effective Thinking
Skills”.
• The item on sudden ideas in 7A is perhaps of lesser value. The “All Purpose Tools” in 7B-
7D contain a smallish number of questions and suggestions that should lead to some
progress in almost any situation.
• Column 8 forms a bridge between general processes and more specific math contents. The
collection given here is a bit arbitrary and serves as an illustration.
• Box 8D presents items to generate more ideas:
In the spirit of creativity tools used in other fields, one might try to manipulate the problem
elements by a number of basic operators and see if this leads to interesting insights.
This list is largely inspired by the well-known SCAMPER creativity tool.
Here, as with other lists of suggestions in the note assistant, it seems a good idea to go
through the list and write down a collection of ideas. Afterwards, the most promising ideas
can be examined in detail.
It seems reasonable to try this method after more direct approaches have failed.
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7. Criticism and responses
There are numerous points of criticism, and I would like to address some of them.
• “The four column layout is a wildly over-specified straitjacket.”
In my opinion it's better to have a clear concept of how helpful math notes could look like,
and then deviate from that concept for good reasons, than to have no concept at all.
As mentioned, there are many layout variations possible.
• “The entire process is too formal and too complex.”
I certainly would not want to see the process followed in a dull routine.
It is meant to provide support - if the user wants it.
If the process is presented to a group of students, it seems reasonable to advance in suitable
moderate steps, presenting elements of the process one by one.
(Thanks to Professor Mason for addressing this point.)
• “Abstract heuristic advice like “look at invariants” is useless to lots of students.”
Yes. I think that choosing the right set of thinking tools to help an individual student is a
major issue. Arguably, these tools have to be introduced, illustrated by examples and then be
made available in the note assistant.
• “What about knowledge? What about experience?”
I think that there is an immense literature on math and on math problem solving that will
help readers to build up knowledge and experience.
But there seems to be comparatively little information on the aspect of note-making, so I
concentrated on this.
• “Is there any evidence that this actually works?”
A previous version of a problem solving method based on mind maps was very well
received in seminars I've given in the past. From my personal experience, I have no doubt
that the note-making method presented in this paper is much better suited to math problems.
It is my hope that readers may find some ideas which they find useful and which they can
adapt to their own problem solving practices.
If you have suggestions on how to improve the method in general or single aspects of it, I
would be thrilled to hear from you. Please don't hesitate to contact me under
thomasteepe@googlemail.com
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8. Note assistants: A framework for problem solving
• From a more abstract point of view, and without reference to math specifics, we have to deal
with two elements:
First: A note-making method we want to use for our work on a problem. This method can be
the four column layout described above, or a two column layout with one main column and
a reflection column, or mind mapping, or ordinary linear notes, or notes in a digital
substrate, like a digital notebook.
Second: A set of thinking tools we could use for solving the problem.
A note assistant should provide combinations of note-making elements and thinking tools -
combinations that can be applied directly to the problem. In my experience, this is of special
importance with regard to the handling of confusion and obstacles, reflections and of dealing
with multiple approaches.
• With this general perspective in mind, it is possible to construct various note assistants by
combining thinking tools from texts on (math) problem solving and note-making methods.
• In this spirit, Alan Zollman's Four Corner method can be viewed as a note assistant, namely
a combination of a layout and a set of thinking tools. The Four Corner method uses a static
graphic organizer, while the note assistant is something of a “dynamic” graphic organizer
that combines single layout modules.
A description of the Four Corner method can be found at
http://www.math.niu.edu/%7Ezollman/PP/NCTM2006-Four-Corners.ppt
• Several books on math problem solving highlight the importance of note-making, but
provide very little details on how to do this in practice.
• In the past, a combination of mind mapping and problem solving tools seemed very
promising to me. Today I think that the four column layout is much better suited for dealing
with math problems.
(The older ideas on mind mapping can be found at
https://www.scribd.com/doc/7929697/Mathematical-Problem-Solving-and-Mind-Mapping)
• Obviously, the note assistant framework can be adapted to a number of fields besides math.
Note assistants and modules
The note assistant given in this text has been worked out with office software. This may be a bit
cumbersome, so here are some ideas on how to work out individualized note assistants.
• It seems a good idea to form note assistants from single “modules”.
A module could consist of a headline that indicates for which problem situation or for which
mathematical objects the module is intended, followed by a collection of note-making and
thinking tool suggestions, as in the note assistant example given above.
So, basically the modules correspond to the columns in the note assistant.
• These modules offer a number of benefits:
First, with modules it's easy to add layouts and thinking tools for new problem situations.
Second, it's possible to adapt and improve only one module without having to replace the
entire collection.
Third, it may be stimulating to exchange modules within a community of students and
teachers using the note assistant framework.
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9. • Some practical ideas on using modules:
a) Use plastic sheets for filing business cards. Those sheets have 8 or 10 pockets for papers
of business card size, so by using front and back side you can store up to 20 modules.
These sheets can be used to collect an entire library of note assistants for various purposes
and various domains of math.
b) Form modules by using column-shaped sheets, like the narrow separator sheets used in
folders – in that case however, the one-sheet-one-glance-mantra has to be abandoned.
c) In an age of omnipresent mobile devices, digital solutions using them are another option,
though certainly not the one I favor. Again, the one-sheet-one-glance mantra has to be given
up.
Some remarks
• It seems promising to work out more detailed ideas on how people could develop stronger
and stronger problem solving skills with the help of note assistants. (The note assistant has
been designed to support dynamic graphic organizing. The note assistant itself is not meant
to be static.)
• In my experience the focus shifts with time away from the note-making methods and to the
collection of problem solving tools.
• Notes on a problem will often stretch over more than one sheet. I have found it easier to
continue my thinking when I have previous notes directly in front of me, so for me single
sheets work better than a bound notebook, where I have to switch between pages to read and
to write.
The problem of dealing with a large number of sheets is not within the scope of this text. My
ideas on this would probably have to do with slip boxes or “Zettelkästen”.
• It could be worthwhile to check the benefits of layout suggestions that use several columns
at once, or even several sheets – with more complex problems, it may be reasonable to have
a separate sheet for each major approach.
• Solutions to math problems often use subtle combinations of several brilliant ideas, and note
assistants should offer help on how to find these – a formidable challenge.
One approach that may lead to some progress here: Adapt central concepts from the TRIZ
method of invention in engineering to mathematics.
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10. Other elements in note assistants
Here is a list of elements a note assistant could contain. Some readers will find some items over the
top. The individual problem solver may decide what works for her and what doesn't.
• Items to prepare for problem solving
It may prove very useful to have an initial box in the note assistant (and subsequently in the
note sheet) to set the right attitude and the right mood towards problem solving. Here are
some possible items:
- recalling previous successes,
- bringing to mind the importance of reflection etc.
• Images
Images may have more impact than words – portraits of role models, images of goals,
images evoking clarity or straightforwardness etc. Or:
• An image of a rubber duck
From the wikipedia article (http://en.wikipedia.org/wiki/Rubber_duck_debugging):
“Rubber duck debugging is an informal term used in software engineering for a method of
debugging code. The name is a reference to a story in the book The Pragmatic Programmer
in which a programmer would carry around a rubber duck and debug his code by forcing
himself to explain it, line-by-line, to the duck. [...]
Many programmers have had the experience of explaining a programming problem to
someone else, possibly even to someone who knows nothing about programming, and then
hitting upon the solution in the process of explaining the problem. In describing what the
code is supposed to do and observing what it actually does, any incongruity between these
two becomes apparent. By using an inanimate object, the programmer can try to accomplish
this without having to involve another person.”
The method seems fit to be applied to problems outside debugging.
10

11. Acknowledgments
Many people have contributed to the ideas presented in this paper.
I would like to thank Werner Begoihn, Dr. Astrid Brinkmann, Hans-Jürgen Elschenbroich, Dr.
Kevin Houston, Dr. Jörg Konopka, Dr. Armin Kramer, Prof. Dr. Timo Leuders, Prof. Dr. John
Mason, Hubert Massin, Prof. Dr. Manfred Prenzel, Dr. Frauke Rademann, Prof. Dr. Harold Shapiro,
Martina Teepe and Christian Wolf.
Document changes
This document is available for free on
https://www.scribd.com/doc/251685614/Note-Assistants-Support-for-Solving-Math-Problems
I upload revised versions from time to time, so here comes a list of changes made to the document.
Version Date Changes
1.0 07.01.2015 Added:
Table of Contents, document changes table, about the author.
Note assistants and modules.
Remark on introducing the process step by step.
Minor changes.
1.1 08.01.2015 Changed:
Introduction with the “paper software” metaphor.
Minor changes.
Added:
Other elements in note assistants.
1.2 14.01.2015 Changed:
Box name in upper right corner (instead of left).
Numerous improvements to the note assistant example.
Expanded description of how to use the note assistant.
Some ideas on criticism.
About the author
I was born in 1971 in Ibbenbüren (Germany). I finished my doctoral thesis on genetic algorithms at
the University of Münster in 2001, and since leaving university I'm working as an actuarial
consultant.
In the past years, I have spent a lot of time thinking about methods of math problem solving and of
problem solving relevant to my job and, occasionally, to my life.
Thomas Teepe
Klotzstraße 1A
70190 Stuttgart
thomasteepe@googlemail.com
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