Published on December 21, 2008
Mathematical Problem Solving and Mind Mapping I. Introduction In this paper I will describe how the well-known mind mapping technique can be used to solve mathematical problems. What is the main idea? You use one principal mindmap and one or several additional mind maps at a time. The first mind map is used for examining the given problem. It is produced during the process of problem solving. The additional mind maps are at least in part prepared in advance. They contain all kinds of heuristic operators that might be useful in dealing with the problem, from higher-level operators like strategies to lower-level operators like technical tools. They may even provide advice for emotional emergencies like frustration. You may consult these mind maps whenever help is needed. Moreover, you can adapt them at any time: Add new operators, abbreviate complex ones you have become familiar with or delete redundant ones. What are the main benefits? The additional mind maps provide a large number of tools. Due to the mind map characteristics it is easy to find appropriate tools. The additional maps can be adapted to different problem types and to any level of expertise. The concept stimulates active thinking about problem solving habits. Part II contains, mostly for the sake of completeness, a brief introduction to mind mapping. It can be skipped by anyone who is familiar with the technique. Part III describes how the concept is used, with examples of principal and additional mind maps. Part IV discusses the advantages of the concept. Part V lists the literature I have used in writing this paper. II. Mind Mapping Mind mapping is a simple yet powerful technique for taking notes and organizing ideas. The method was developed in the 1970s by Tony Buzan, a British expert on learning. The human brain seems to be well adapted to do the following things: - forming associations, - building hierarchies of concepts, - using both words and images (the image part often being neglected in education). Exploiting these strong points, a mind map is produced as follows: You need a sheet of paper, preferably size A4 or larger, and writing pens, preferably in different colours. Take the sheet of paper in landscape format, write your topic in the middle of the sheet and draw a frame around it. This helps you to stay focused on your topic and encourages you to develop ideas literally in all directions. Now write the most relevant aspects of your topic round the center and connect them to the center by lines. These branches can be further developed into subbranches. This helps you to build hierarchies and to find an appropriate place for your associations. Throughout the process of mind mapping, use single keywords rather than complete sentences. Use images, like symbols, icons and little drawings. This helps you to stimulate associations and exploit your abilities for both verbal and visual thinking. Organize your ideas by numbering branches, highlighting important concepts, e.g. by using colours and using arrows. These easy instructions are the core of the mind mapping technique. Here comes a short assessment. Advantages of mind mapping: Mind mapping is a powerful technique for stimulating, ordering and organizing ideas. 1
It can be used almost everywhere and can be adapted to practically all purposes - from writing a diary to planning a birthday party and from taking notes during a lecture to solving mathematical problems. Moreover, it is easy and pleasant to use. Disadvantages of mind mapping: The use of keywords and images (besides problems like untidy handwriting) makes mind maps not the ideal technique for communicating ideas. Mind maps may tempt you to shun difficult questions in several ways: By digression to more accessible aspects of your topic or by tackling a problem on an inappropriate level, e.g. by exaggerated planning. But in summary, the advantages easily outweigh the shortcomings. There is a growing number of computer programs for mind mapping, some of them rather advanced. A brief internet search yields several programs and downloads. An thorough discussion of mind mapping can be found in ‘The Mind Map Book’ by Tony and Barry Buzan. III. Tandem Mind Mapping: Using several mind maps at a time III.1 Terminology The principal mind map in which you examine the given problem will be called basic map. The additional mind maps with information on heuristic strategies and tools will be called heuristic maps. The process of using basic maps and heuristic maps at a time will be called tandem mind mapping. The words ‘tool‘ and ‘operator‘ refer quite generally to any technique you may find useful in solving a problem. III.2 Content of heuristic maps Here comes a list of operators that may be useful in mathematical problem solving. The list is in no way exhaustive. Moreover, single tools may fit into several of the following operator groups. I cannot claim to be an expert in solving mathematical problems, and the main objective of this list is to give some impression of what heuristic maps may contain. Some of the tools will perhaps appear objectionable or worthless to the reader. For my principal goal, namely discussing the potential merits of tandem mind mapping, the actual choice of certain tools is less important. General strategies: These are ‘top level‘ heuristic strategies that coordinate the entire process of problem solving. A prominent example are the basic steps in Polya’s ‘How to Solve It‘: 1. understanding the problem - 2. devising a plan – 3. carrying out the plan – 4. looking back. For each step, several auxiliary questions and other tools are given. General principles: Examples: Invariance principle - Extremal principle - Induction principle – Pigeonhole Principle - Symmetry General mathematical tools: Examples: Characteristic functions - Power series – Graph Theory General mathematical tactics: Examples: Defining auxiliary functions – Working backward or forward Tools for dealing with certain mathematical objects: The selection of tools referring to certain mathematical objects depends heavily on your area of work. For example, if you are working on probability topics, tools referring to martingales may be fundamental to you. You could arrange any number of results relevant to a given mathematical object. Some restraint should prevent you from piling up loads of irrelevant information. Analysis tools: 2
Examples: Collect seminal ideas about the problem - Identify relevant components of the problem - Collect relevant questions - Ask iteratively for the reason of things and their respective reasons - Ask iteratively how goals and subgoals can be reached – Make a drawing of the situation Creativity tools: Sometimes A. Koestler‘s idea of bisociation and other classical creativity techniques like brainstorming might prove useful in mathematics. Other tools are more closely related to mathematics, like recasting the problem by changing one’s point of view. Checklists: For some users checklists may be valuable, e.g. lists of common errors with entries ranging from checks against division by zero to wrongly changing the order of limits. Information retrieval tools: Examples: Use math databases like the Mathematical Reviews or the Zentralblatt für Mathematik - Ask an expert for direct help or for hints to relevant literature - Post a question to relevant internet communities Review tools: Examples: What techniques have been used during problem solving? - Which tools worked well or less well, and why? - What are the strong points and shortcomings of a result and the process of finding it? - How would X (a teacher or another expert) assess the result and the process of finding it? What are the tools I should add, delete or adapt? Tools for dealing with dysfunctional emotions: If you feel that being frustrated or discouraged doesn’t help you at the moment, you can try some of the following tools: Remember past successes - Imagine having succeeded – Use coping self talk Metatools: Metatools support the finding and invention of tools. Examples: What are recurring shortcomings of my problem solving activities, and how can I overcome them? - What are the most successful tools I use? Why do they work so well? How can I use their strong points in other areas? – What tools have been used in a given article or book that may be useful? – How can I improve my set of heuristic maps? Miscellaneous tools: Examples: How would X (a teacher, an expert, or even a famous mathematician fom history) tackle the problem? - Give yourself a break - Do some physical exercise - Postpone the problem III.3 Organisation of heuristic maps Simply piling up loads of tools is not enough. The tools must be organized so you can find them when – and where - you need them. For example, you can arrange operators by problem phases. This is Polya’s approach in ‘How to Solve It‘, see the above remarks on general strategies. In addition, it is often useful to arrange tools according to problem situations like ‘defining a goal‘, ‘tackling difficulties‘ etc. Moreover, you can use the above operator groups for organizing the tools. Obviously, it is necessary to use an appropriate number of heuristic maps. You may for example use the following set of maps: - map with general strategy tools, e.g. Polya’s catalogue of questions and additional basic tools, - map containing general mathematical principles and general mathematical tools, - maps with detailed information on mathematical tools, - maps with miscellaneous tools. Some degree of redundancy in these maps is inevitable. Tools often belong to more than one group of operators and should be found in several maps. You can use maps that contain only the names of operator groups and arrange the tools in that group in a separate map. (If you use computer mind maps, you can organize a huge number of operators in a single map, which is much more convenient.) 3
III.4 Example The following excerpt from a heuristic map is again based on Polya’s ‘How to Solve It‘. Examine What is the unknown? aims. What is the aim? Aim appropriate? Examine data. Possible to satisfy? Sufficient? Examine Insufficient? conditions. Contradictory? understanding the problem Redundant? Separate various parts. Draw a figure. Introduce suitable notation. Evident? Other tools Look at special Possible Examine results. cases. to simplify? Explain it to yourself Try to or someone else. generalize. Examine arguments. Use methods. Alternatives? looking back Use related Use result elsewhere. problems. Use results. Reasons Use auxiliary for failure? constructions. Examine Shortcomings? Context problem solving. General strategies Improvements? Useful of problem? theorems? Which objects Lessons Properties? to be learned? are involved? Theorems? Restate Look for several Correctness the problem. restatements. evident? Change point of view. Check carrying Prove each step. correctness. out the plan Go back to definitions. devising More Need for a plan accessible? adapting aims? Modify the More problem. general? More special? Analogous? Modify Modify the Drop parts. conditions. Modify parts. Derive Modify information. the data. Use different data. Data? All information used? Conditions? 4
IV. Discussion Here comes a list of benefits of the tandem mind mapping concept. Large toolbox: The heuristic maps contain large numbers of tools. They remind you of tools you might otherwise have overlooked. This is especially valuable for problem solvers who are not yet familiar with some tools. Flexibility: You can develop the heuristic maps that are appropriate to your degree of expertise, to the problem type you deal with and to your personal likes and dislikes. Active work on heuristics: Heuristic maps are dynamic objects. You are encouraged to think about your problem solving habits, to discuss them with others and to improve them continually. Review tools and metatools may support these activities. Knowledge transfer: By sharing and explaining their heuristic maps, experts can help novices to acquire a working knowledge on how to solve mathematical problems. Mind map benefits: Using mind mapping at all may be an important step towards better problem solving. The mind map presentation of tools is superior to conventional text: - It is easier to add new tools at the appropriate places in a mind map. - The hierarchical mind map structure makes it easy to find operators that are relevant to your problem situation. - Colours and symbols give additional information about the tools. V. Literature De Bono, Edward: de Bonos neue Denkschule. Mvg Verlag, Landsberg 2002 Buzan, Tony: The Mind Map Book. BBC Books, London 1995 Buzan, Tony: Business Mind Mapping. Ueberreuter, Frankfurt 1999 Dörner, Dietrich: Problemlösen als Informationsverarbeitung. Kohlhammer, Stuttgart 1987 Dörner, Dietrich: Die Logik des Misslingens. Rowohlt, Reinbek 1989 Dörner, Dietrich: Bauplan für eine Seele. Rowohlt, Reinbek 1998 Engel, Arthur: Problem-Solving Strategies. Springer, New York 1998 Funke, Joachim: Problemlösendes Denken. Kohlhammer, Stuttgart 2003 Higgins, James M.: 101 Creative Problem Solving Techniques. The New Management Publish Company, Winter Park 1994 Hoenig, Christopher: The Problem Solving Journey. Perseus Publishing 2000 Jones, Morgan D.: 14 Powerful Techniques for Problem Solving. Three Rivers Press, New York 1998 Mason, John: Hexeneinmaleins. Oldenbourg, München 1985 Michalko, Michael: Cracking Creativity. Ten Speed Press, Berkeley 2001 Nelson-Jones, Richard: Using Your Mind. Cassell, London 1997 North, Klaus: Wissensorientierte Unternehmensführung. Gabler, Wiesbaden 2002 Von der Oelsnitz, Dietrich; Hahmann, Martin: Wissensmanagement. Kohlhammer, Stuttgart 2003 Polya, George: How to Solve it. Princeton 1957 Pricken, Mario: Kribbeln im Kopf. Schmidt, Mainz 2001 Robertson, S. Ian: Problem Solving. Psychology Press 2001 Sell, Robert; Schimweg, Ralf: Probleme lösen. Springer, Berlin 2002 Zeitz: The Art and Craft of Problem Solving. Wiley, New York 1999 Final note: The term Mind Mapping is a registered trademark of Buzan Centres Ltd. I do not have any commercial interests with this paper, and I hope I do not violate any naming restrictions in using the term. Dr. Thomas Teepe 5
Alosenweg 37 70329 Stuttgart Germany Email: email@example.com 30. November 2003, slightly revised on 13. November 2008-11-13 6
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