Making notes and solving (math) problems

August 3, 2006

There is a strong case for making notes during problem solving.
Here are some of the more obvious reasons:

  • Writing notes helps managing complexity: You can split a problem into parts, you can collect several approaches and deal with them one after the other etc.
  • Notes document your thoughts – for later scrutiny, for resuming a chain of thought later – and sometimes for posterity.
  • Notes can help to combine text and diagrams. The human brain is well equipped to deal with words and images, and either representation allows the application of quite different tools: In texts, you can ask questions, formulate alternatives, associate verbal concepts etc., in diagrams, you can add lines, rearrange items etc.

So making notes during problem solving is well worth a trial.
This leads to the question:
How to make notes most cleverly?
I’m sure this is an important topic in solving problems in general and in solving mathematical problems in particular – and one that is too often neglected.
The process of “tool mapping” described here is but one answer to this question (albeit hopefully a well thought-out one).

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Criticism to Tool Mapping. Responses.

August 2, 2006

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.