On Mathematical Problem Solving
[Equally Applicable To Scientific & Engineering Problem Solving]
In mathematical and scientific circles, a problem is defined as a situation where there is a difference between the desired or goal state and your current state.
Usually most of us go about in a disorganized way when trying to solve a problem.
But becoming organized can make you a much better problem solver.
There are certain ways of approaching problems which are followed mainly by Mathematicians and Scientists but can be applied to any branch.
Here are the Guidelines I usually follow.
Bright Idea / Intuition / Inspiration
High level pattern matching: Use working memory + visualization + organization to hold the whole problem + solution (so far) at a time and rapidly move between its different portions. Change the representation if you can’t make progress. Take a break and then come back if you can’t make progress after thinking for a long time.
Zoom out for sketching the solution and zoom in to carry out each part with rigorous arguments. Work on parts of the problem and the problem as a whole.
Understanding the problem
Data? Unknown? Condition?
Organize the whole problem with a diagram: invent your own representations (It might be a mathematical structure: graph, network, lattice, matrix, number line, geometrical figure,…). Visualize it and Draw it. Find a structure that holds the problem and solution at once and completely.
Get emotionally involved with the problem (Wow! How can this problem be solved? Is it a “to prove” problem? So, this theorem holds true? Wow! Is it a “to find” problem?
Guess the answer. (Good for exercising intuition / high level pattern matching.)
Extract from memory all the relevant information, theorems, problems (mobilization) and organize/connect/plan with them.
First make high level plans for solving the problem then carry out the plan with rigorous arguments. Make connections between Data, Unknowns and Conditions.
Does it get easier if I work backwards? What could be penultimate step? Related problems? Related Data? Related Unknowns? Related Conditions? Theorem? Structure? Imagine a more accessible related problem and solve it. What makes the given problem hard?
Try different strategies, tactics and tools. Don’t get stuck. Change the problem representation and change your perspective.
Generalization – Specialization: Logical Quantification.
Carrying out the plan
Rigorously prove and convince yourself that the solution / proof is correct. (The way to convince oneself is to visualize / imagine. Remember, “seeing is believing”. If you can see the arguments in your mind’s eyes, you believe it; in other words, you are convinced.)
Once you start to convince yourself, your “Mathematical Intuition” will grow. Otherwise, what is mathematically correct / logically consistent, won’t seem correct at a glance. (This is a problem almost all novices have – they know something is scientifically correct, but they get astonished when they see it in action / nature – their subconscious and conscious mind have different ideas – they have read it but haven’t reprogrammed their subconscious beliefs.)
Checking the result
Is it OK? Can you see the whole problem-solution (solution embedded in the problem) at a glance? Is it reusable in other problems? What have you learned that can be reused in developing solutions to other problems? If it’s a “to prove” problem, then the theorem can be reused.
Thinking harder and going to deeper levels of concentration (and mental performance)
Level 1, Level 2, Level 3 and so on. One might find it hard (feel fatigue, etc.) to cross a level but if once pushed enough and having crossed the mental barrier, your brain power will expand (with a bigger working memory).
Newton used to work on a problem until it was solved . Try other methods to go to deeper level of concentration: try visualizing progressively more vividly; hearing, touching, smelling, tasting progressively more realistically (always visualizing in tandem).
Learning – Generalization & Organization
“Each problem that I solved became a rule, which served afterwards to solve other problems.” – Rene Descartes.
Generalize the problem, generalize your solution, generalize solutions by others and find out applications. Generalize strategies, tactics, tools and make more than one representations so that you can hold the strategies, tactics, tools completely in your head at a time.
Organize the strategy in your mental problem solving toolbox (hierarchically or in a graph-like structure).
Rather than solving a lot of problems without going deeper, concentrate on all the sub-problems, ideas, queries that arise while solving a problem.
Visualize every problem solving strategy, tool, technique, algorithm, algorithmic paradigm, design pattern, computational abstraction as structures and processes.
Learning by organizing Mathematics
Organize all the problem solving strategies, techniques, tools, areas of Mathematics, theorems, identities, structures in your ontology.
1. “How to solve it” – Mathematician George Polya.
2. The art and craft of problem solving – Paul Zeitz.
Source of many concepts, for instance, the concept of working backwards and penultimate step.
3. Books on Mathematical Problem Solving.
4. Mind Power – Reader’s Digest
“Think harder and go to deeper levels of concentration” ….. from “How to increase energy” –
“Usually we make a practice of stopping an occupation as we meet the first layer of fatigue…… But if an unusual necessity forces us onward, a surprising thing occurs. The fatigue gets worse up to a certain point, when, gradually or suddenly, it passes away and we are fresher than before!….We have evidently tapped a new level of energy. There may be layer after layer of this experience, a third and a fourth ‘wind’. We find amounts of ease and power that we never dreamed ourselves to own……habitually we never push through the obstruction of fatigue.”
5. “Newton used to work on a problem until it was solved.”
Source: উন্নত জীবন – ডাঃ লুতফর রহমান। “নিউটন বলেছেন, আমার আবিষ্কারের কারণ আমার প্রতিভা নয়। বহু বছরের পরিস্রম ও নিরবিচ্ছিন্ন চিন্তার ফলেই আমি আমাকে সার্থক করেছি, যা যখন আমার মনের সামনে এসেছে, শুধু তারই মীমাংসায় আমি বাস্ত থাকতাম। অস্পষ্টতা থেকে ধীরে ধীরে স্পষ্টতার মধ্যে উপস্থিত হয়েছি।”
6. The concepts of working memory, subconscious mind – from books on Psychology and Cognitive Science.
7. “Each problem that I solved became a rule, which served afterwards to solve other problems.”
– 17th Century Philosopher and Mathematician Rene Descartes.
8. The Emotion Machine & The Society of Mind – Marvin Minsky.
I own hard copies of both the books. I used to look upon Marvin Minsky as one of the “guru”s during my College years.
9. Books on Artificial Intelligence
Chapters on Problem Solving, Planning, Machine Learning. Inspiration for “hierarchically or in a graph-like structure”.
10. “Get emotionally involved with the problem.” – from “You And Your Research” by Dr Richard Hamming.