A short video on problem solving

This 8:55 minute video presents a number of ideas on problem solving:

mind mapping and some variations

problem solving tools

the combination of a problem map and a tool map

the IDEAL problem solving process developed by Bransford / Stein

You find a lot more material on these ideas on the other pages and posts.

I’m delighted if you leave some comment.

Links to problem solving ideas

In this post I collect some of my material on problem solving, math problem solving and mind mapping.

We start with some mind maps on mindomo.com:

Solving Math Problems
Computer Aided Problem Solving
Problem Solving
Math Strategies Poster
The IDEAL Problem Solving Template

Some short essays on scribd.com:

Mind Maps and Math Problem Solving
Mathematical Problem Solving and Mind Mapping
Math Strategy Poster
Problem Solving

Please leave a comment on scribd.com.

Mindomo mind mapping. Prepare to get wowed.

Mindomo (www.mindomo.com) is an online mapping tool.
Some features of the free version:

• You only need a browser and Adobe Flash Player 9
• Create maps online
• Save your maps as private or public maps on the Mindomo server
• Useful mapping functions: flexible layouts, icons, images, text notes

Interested?
Here is “The IDEAL Problem Solving Template” in Mindomo.

——————

In the meantime, the people at Mindomo have added a number of features:

• number of views for a map
• allowing others to modify your public maps
• embedding maps in your website, e.g. like this:

Oops!

(This doesn’t work with blogs hosted by WordPress, since they do not accept Javascript. I’ve sent a request if they can allow Javascript from Mindomo, as they do with YouTube or Google Videos.)

Math problem solving difficulties

Here are some ideas (partly new) on how to use mind maps to overcome difficulties with math problems.
As described elsewhere on this blog, we will use two mind maps:

1. a “problem map” for dealing with the given problem, and
2. one or more “tool maps”. Tool Maps are collection of tools for solving math problems. Here are some examples.

For the problem map, we use a sheet of large size (A3 works well) and the following layout and template:



What can we do with the branches in this map?
Here are some basic ideas.
Orientation:

• Look at an example.
• Make a table, chart cases systematically.
• Draw a figure.

Representations:

• Collect ways of representing the problems (algebraic, geometric or graphic, algorithmic…)

Choosing a clever representation is often vital for finding a solution. Don’t neglect this step.

Approaches:

• Collect approaches how to tackle the problem (e.g. contradiction, induction, looking at extreme elements…)

These approaches are just seminal ideas, not entire plans of a solution.
You can work in a brainstorm fashion, so list even approaches that look less promising.

For each of these template branches, you can use ideas from the tool maps.
Here is an easy example:

Working wih the template is fairly straigthforward when you start examining a problem.
But sooner or later difficulties and obstacles will probably appear.
Here are some snapshots from an interior monologue:

• “This seems too complicated.”
• “I don’t want to go into masses of single cases. There must be a more elegant way.”
• “I have no idea how to tackle this.”
• “All approaches have failed, and I have no idea what to do next.”
• “This approach looks promising, and the first steps feel right, but what now?”
• “I’m confused! I’d like to change a single item, but it’s connected with all the others.”

To make things (just a bit) more systematic, here is a tool map showing typical difficulties and some possible remedies.

(Again, the SCAMPER mnemomic is taken from Ron Hale-Evan’s “Mind Performance Hacks”, was developed by Bob Eberle and first published in Michael Michalko’s “Thinkertoys”.)


How to apply these ideas practically?
Here is a simple idea:
Add a subbranch to the approach you are focusing on. If you like, you can label this branch “O” for “obstacle”.
Describe the difficulties with this approach. If you like, use the collection of difficulties above. If it helps, use it as a starting point for your personal “First Aid” tool map.
Then add ideas on how to overcome these difficulties, using the remedies suggested above.
Here is an example:

Problem Solving and Mind Mapping Software

Handwriting vs. word processing software, handmade mind maps vs. mind mapping software – in both cases there’s a similar tradeoff between creative craft and computer aided efficiency, versatility and tidiness.
It stands to reason that mind mapping software will become more important in the future.


Here are some ideas on how to use mind mapping software for solving problems.
The examples are done with FreeMind, a free, open source mapping tool. (Here is FreeMind’s homepage.)

• Use templates:
  Start problem solving not with an empty map, but with a problem solving template. This template works as a scaffolding for the entire process of problem solving.
  Here is an example:
  
  Here are some explanations:
  IDEAL stands for
  Identify the problem and explain how it can be an opportunity.
  Define at least three different goals for your problem-solving task.
  Explore possible strategies and new information that can help you accomplish each of the important goals listed above.
  Anticipate the outcomes of different strategies to help you decide which ones you will act on.
  Look back and learn.
  (This strategy is taken from the book “The IDEAL Problem Solver” by John D. Bransford and Barry S. Stein.)
  
  
• Use tool maps:
  Prepare a mind map with useful tools for problem solving.
  During problem solving, you can swap (with CTRL + left / right arrow) between the problem map with the actual problem and the tool map which provides strategies, inspiration, warnings or hints.
  Prepare tool maps relevant for your area of work – e.g. tools for text analysis, critical thinking, math…
  Here is an easy example of a tool map.
  
  Wikipedia article on morphological analysis.
  The SCAMPER mnemomic is taken from Ron Hale-Evan’s “Mind Performance Hacks”, was developed by Bob Eberle and first published in Michael Michalko’s “Thinkertoys”.
  
  
• Use colours for “goal management”:
  It’s rather easy to get sidetracked while working on a problem.
  To prevent this, highlight the important goals and subgoals in your problem map.
  From time to time, check these highlighted items and see if you have actually reached these goals – or why not.
  
  

Problem Solving Templates and Mind Maps

Here are two general problem solving strategies.
They both can be remembered by simple keywords – “IDEAL” or “ABCDE”.

IDEAL
Identify the problem and explain how it can be an opportunity.
Define at least three different goals for your problem-solving task.
Explore possible strategies and new information that can help you accomplish each of the important goals listed above.
Anticipate the outcomes of different strategies to help you decide which ones you will act on.
Look back and learn.
(This strategy is taken from the book “The IDEAL Problem Solver” by John D. Bransford and Barry S. Stein.)

ABCDE
Assessment - clearly stating the problem itself.
Brainstorming - for possible solutions.
Consequences - evaluate the likely consequences of putting your ideas into practice.
Do List – break your best strategy down into a logical list of steps to do.
Evaluation - did the process work?
(This strategy is taken from the book “Succesful Problem Solving” by Matthew McKay and Patrick Fanning.)

These strategies can be used in mind maps to get effective problem solving templates. The branches are to be read clockwise.

Making notes and solving (math) problems

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).

Criticism to Tool Mapping. Responses.

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.

What makes solving math problems difficult?

“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.”

(I found this diagnosis 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 promising way of mastering these difficulties lies in combining two major approaches to problem solving:

• heuristics in the tradition of George Polya’s “How to Solve It”, and
• mapping techniques, like mind mapping (or concept mapping).

Here are more details.

Welcome!

This blog will mainly deal with problem solving topics (with a certain emphasis on math problem solving).
However, things are still under construction. If you are curious and know some German, have a look at

http://www.probleme-und-strategien.de.