Basic ideas My starting point is a diagnosis I found in David Perkins’ book “Outsmarting IQ: The emerging science of learnable intelligence”. Perkins reports some of the findings of mathematicianpsychologist Allan Schoenfeld (p. 87): “One of the most important factors [in deficient mathematical problem solving is] poor mental management: - Students did not pay attention to the winding path of their activities in solving a problem. - They often did not think to use heuristics they knew and could have applied. - They often perseverated in an approach that was not yielding progress rather than trying a new tack. - They often gave up without rummaging in their repertoire for another point of entry. - Amidst the trees, they lost sight of the forest.” One promising way of mastering these difficulties lies in combining two major approaches to problem solving: - heuristics in the tradition of Polya, and - mapping techniques, like mind mapping (or concept mapping).
Excursus: Mind mapping. (You may want to skip this if you are familiar with mind maps.) 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. A thorough discussion of mind mapping can be found in “The Mind Map Book” by Tony and Barry Buzan. Later in the text we present a number of mind maps.
How can mind maps be used for solving math problems? I will 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 On the following page you find a sample problem map. The problem is very easy, but the sample should show the flavour of the method. 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. - 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 www.artofproblemsolving.com. It should be clear that problem maps are intended for finding a solution, not presenting one. Here comes the problem map sample. The problem map deals with the following Exercise 1: 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))
Here are some 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 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). On the following pages you find a number of sample tool maps that can be used in solving mathematical problems. For reasons of clarity, I have done these maps with mind mapping software rather than by hand. I have used a non-standard mind map layout (portrait format rather than landscape format) due to the layout of this letter. Here are some details.
“Basic Heuristic”: 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”. This map is of limited practical use and mainly included as a kind of overview. Tool maps for some of the topics mentioned will be presented later. Understanding the problem
Stages Dealing with obstacles
Devising a plan Carrying out the plan Looking back
Basic Heuristic
Processes
Collect ideas in problem map
Use tool maps for inspiration!
Choose most promising idea from the problem map Work out details in boxes Describe obstacles/difficulties in the problem map
“Understanding the problem”: The material for the following two maps is taken from a number of standard sources, like Polya, Arthur Engel’s “Problem-Solving Strategies” or Paul Zeitz’ “The Art and Craft of Problem Solving”. Read the problem carefully
First steps
Draw a figure Introduce suitable notation Collect initial ideas
Similar problems? Useful tools?
Collect questions
Draw a figure Geometric
Understanding the problem
Find representations of the problem
Use different coordinates... Binary representation
Algebraic
Use numbers
Integer Real Complex
Algorithmic Use symmetries
Collect material
Examine special cases Examine systematically
Use tables Use tree diagrams
”Devising a plan": Similar problems? Related problems
... conditions Modify...
... data ... the unknown
Methods of proof
Direction of search
Induction Contradiction Forward Backward
Possible last step of the proof? Possible penultimate step?
Extremes General principles
Symmetry Invariants
Devising a plan
Wishful thinking
Look at extreme elements Look for symmetries in the problem Look for invariants
What would be nice? Can you force it to be nice? Complex numbers General
Graphs Generating functions ... Polynomes
Mathematical tools
Objects
Series ...
Specific
Number Theory Disciplines
Algebra Geometry
Look at "Math Creativity"
“Number Theory”: I have used a map like the following one when I was working on the exercises from the chapter on Number Theory in Arthur Engel's book "Problem-Solving Strategies". First I assembled the tools mentioned in the chapter (which took only a short time), and later, after having worked on some of the problems, added further tools that seemed important to me. Unfortunately, the mind mapping software I use is not yet up to math symbols. a^n - b^n = (a-b) * (a^(n-1) + ... + b^(n-1)) Binomials
a^n + b^n = (a+b)*(a^(n-1) - ... +- b^(n-1))
Identities Sophie Germain
For all n
For odd n
a^4 + 4b^4 = (a^2+2b^2)^2 - (2ab)^2
Factorize! gcd
Euclid's algorithm Look at cases
Look at remainders
Divisibility
Chinese Remainder Theorem Use parity Use congruences 2,3,4,5,6,9,11
Divisibility rules Look at products of primes
General n = p_1^n_^1 * p_2^n_2 * ... p_r^n_r 2*3*5 etc.
Number Theory
Primes
Little Fermat
Converse invalid! Fermat-Euler
Fundamental Theorem Euclid's Lemma Use symmetry Manipulations
Add zero Multiply with one Substitute terms Infinite descent Consecutive numbers
Miscellaneous Look at last digits Look at digit sums
Squares Triangular numbers
"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. I found it stimulating to apply some concepts to mathematics. My main inspiration for this map was the book “101 Creative Problem Solving Techniques” by James M. Higgins. (Use of this map is perhaps appropriate if standard methods have failed. I haven’t yet found the time to gather much experience in using ideas from this map.)
Take a relevant object from the object list Basic idea
Take an operation from the operations list Apply the operation to the object. Play around (e.g. using the problem map). See if you come up with useful ideas.
Mathematical objects Properties
Objects
e.g. sets, numbers, series, matrices ...
e.g. prime, differentiable, finite ...
Mathematical tools
e.g. strategies, tactics, technical tools ...
Representation of the problem Starting point of analysis
Math Creativity
Constants vs. variables Mathematical discipline Modify Simplify Regroup
Operations
Make symmetric Develop patterns
Add / remove Swap / replace / substitute Maximize / minimize View with a microscope / macroscope Divide / combine Invert / inside out / upside down
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.
Combining 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 time. - 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.
Response to Criticism I have discussed the concepts of problem maps and tool maps with several people. I would like to comment on some of the initial criticism. “The process of using maps is too formal.” I have tried to describe a flexible process – you can change between two types of notetaking. A new versatile tool, mind mapping, has been added to your belt, which you can use in some situations and ignore in others. As just mentioned, there is plenty of room for intuitive approaches. “The process impairs creativity.” This may be right if it is used in a dull routine, e.g. mechanically consulting the tool maps at every stage, or slavishly documenting every idea in the problem map. No one is advocating this. But when you’re inexperienced or you are stuck, tool maps may offer valuable inspiration and problem maps may help to organize your ideas. “The process is too inefficient and time-consuming.” My own experiences are: With some (rather straightforward) problems, mind mapping has indeed been an unnecessary effort. With others, mind mapping has speeded up finding a solution. And solutions to some problems I probably wouldn’t have found at all without mind mapping. “Mind mapping is too difficult or too time-consuming to learn.” I do not have enough teaching experience, but in my opinion learning how to mind map is a picnic in comparison with solving math problems.
“Tool maps don’t work.” This argument says that a mere tool name in a map won’t help - which is certainly true: You must know how to USE the items in a tool map. This, of course, has to be learned. But as reminders, recipe books, checklists and sources of inspiration, tool maps are very useful indeed. “The strict hierarchical structure of tool maps doesn’t mirror the much closer interconnections between tools.” This is true, but the hierarchical structure is an easy and practical way of dealing with large amounts of tools. Grouping the tools and retrieving them is made easy by this hierarchy. Moreover, tools can appear more than once in the tool maps, thus making it easier to find them.
Open Questions I am most interested in the following points: - It should be clear from the above description that a separation ought to be made between the general framework of problem maps and tool maps on one hand and the specific tools and their arrangement on the other hand. Which suggestions do you have for any of these areas? - The success of combining mind mapping and mathematical problem solving relates to a number of questions: How experienced are users in using mind maps and in solving mathematical problems? How complex are the problems at hand? Which suggestions do have on these points? - What are in your opinion the shortcomings of the main concepts? - Which suggestions for improvement do you have? - From your experience, which practices in solving math problems work best? Even very short remarks on these points are of great value to me.
