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# Mind Maps And Math Problem Solving

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The basic idea: use one problem map for the actual problem and one or more tool maps with large collection of problem solving tools.

The basic idea: use one problem map for the actual problem and one or more tool maps with large collection of problem solving tools.

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• 1. 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 mathematician- psychologist 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 quot;main branchesquot; 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 quot;problem mapquot; for dealing with the problem itself and - a quot;tool mapquot; (or several of them) containing problem solving tools - from general ones (e.g. the ones presented in Polya's quot;How to Solve Itquot;) to highly specialized ones (e.g. for dealing with Poisson processes).
• 2. 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))
• 3. 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.