This is a intermediate draft of a post on Problem Solving, a topic that I feel is often worth revisiting.
I find it very interesting that many of us rarely think or talk about problem solving. I often wonder how many architectects, engineers computer scientistic and other reflect on the very process that earns them their daily bread. It seems to me that many practicing doctors rely so heavily on heuristic thinking that it would do the profession good to consider problem solving.
We are often solving problems. We are more or less efficient and effective at it. Much of what passes for problem solving is only what we may learned in junior high school. In NYS, as far as I know, problem solving as a conscious and intentional process is rarely if ever taken up by many matematics of scientic teachers. I my experience, science teacher rarely seem to generlized their form of problem solving the so called "scientific method." Even when in the curriculum the lack of time means Probelem Solving gets glossed over and just mentioned in passing. When I taught Mathematics and I made it a point to have the students learn the basics of problem solving, I believed those students handled problems types they never saw before much more effectively. This is because they had a kind of template that offereed them a number of options one of which was virtually guaranteed to work or at least get them partial credit for set the problem up competently. Unfotunately I never complied data on their performance.
It wasn't until I was in my second semester of graduate school in computer science that I began thinking about how I was approaching problems. The programming courses I was taking required me to think about problem solving. Further along the way in graduate school I took two courses in artificial intelligence. One helped me understand problem solving, the other made me realize how much more mathematics I needed to learn.
In the business world we even use terms that refer to systematic, if not effective, problem solving. Project Management is a form of problem solving applied to individual projects. Systems Management is a form of conscious and intentional problem solving applied to the running of a small or large enterprises of virtually any nature.
Evidently, as far as my research has revealed so far, George Polya is considered by some to be the Father of Problem Solving. He laid out his views on how to solve problems in a little and very easy to read book entitled, "How to Solve It."
http://www.amazon.com/How-Solve-It-Mathematical-Princeton/dp/069111966X/ref=sr_1_cc_1?s=aps&ie=UTF8&qid=1405914499&sr=1-1-catcorr&keywords=how+to+solve+it+a+new+aspect+of+mathematical+method
http://www.barnesandnoble.com/s/-How-To-Solve-It-Polya?store=allproducts&keyword=%22How+To+Solve+It%22+Polya
In "How to Solve It," Pólya provides general heuristics for solving a very wide gamut of problems, including both mathematical and non-mathematical problems. A heuristic is a problem solving technique that is not necessarily logical or systematic, it may be intuitive or ad hoc, and can even be simply a guess. A heuristic can be a memorized procedure or a formula like how to find an average. Douglas Lenat's Automated Mathematician and Eurisko artificial intelligence programs were inspired by Pólya's work.
Polya condenses Problem Solving into Four (4) Steps. Each step is very powerful and is not necessarily simple. The four steps are
- Understand the Problem
- Devise a Plan
- Carry Out the Plan
- Look over the "solution."
I expand and elaborate on the Four Steps as follows
How to Solve It.
1. Understand the Problem
- Define and clarify the problem to be solved or the goal to be obtained
- List, categorize and classify what is known
- Precisely define what needs to be known like facts, or relationships or what needs to be discovered in order to solve the problem or obtain the goal
- Do the required research then organize and analyze it. Use this material to fomulate a plan.
2. Devise a Plan
Planning comes in many different forms.
- One plan that may work is to create an ordered list
- Another plan is to simply list the sequence of steps one needs to perform to solve the problem e.g. To find the average of a given set of numbers: step 1 -- add the set of numbers together, step 2 -- divide the total by the amount of numbers in the set
- Yet another possible plan is to Guess then to Check then answer
- One effective plan is to apply the rules of logic
- Another highly useful plan is to apply relationships.
- Another effective plan to is to find a pattern
- Yet another highly and frequently effective plan is to build and use a model
There are many, many possible ways to solve problems sometimes the solutions are applicable to other situations something the solutions are fairly or even completely unique. Creativity and ingenuity are often keys to devising efficient and effective plans.
3. Carry out the plan
For the most part all one need to do is to follow the steps of the plan carefully and diligently. However, often, one first needs to acquire skill sets.
For example if the goal is to drive goods cross country then we have
Goal: To drive good cross country
Plan: 1. Get driver's license, 2. get trucking license's, 3a. get hired by a trucking company, or 3b. buy a truck, 4a. get a cross country assignment, 4b. get a contract to haul a shipment cross county
Carry out pan: take driving lessons, practice, pass test; take trucking lessons, practice,past tests, get hired or get financing and buy truck, etc. Drive goods. Get Paid. Repeat.
One usually needs patience and a dogged persistent with the plan that has been chosen.
However if after due diligence if the problem isn't solved or the goal acheived on needs examine and then to back up and redo sometimes one needs to go back all the way to step 1.
If a given plan continues not to produce the desired outcome then that plan must be discarded iand another plan formulated. Ir is very important to note and to remember that this is how problem solving is done, even by professionals.
4. Look Over the "Solution" -- Evaluate
After due diligence in carrying on the plan, was the problem solved or the goal achieved? If so enjoy the accomplishment before moving on.
If not, organize and analyze the data from the attempt then go back to step 1 and begin again.
Also Polya recommends that one takes time to reflect and look back at what you have done, what worked and what didn't. In doing so one may also find that the plan is applicable in other situations and will help predict what strategy to use to solve some future problem.
Ten Steps To Solving Any Problem
I also developed a clearer and more highly developed a set of 10 problem solving steps. These step subsume Polya's methodology.
- Set an objective in general terms
- Research, Organize and Analyze the available facts, relationships, theories and hypotheses as time and resources permit.
- Articulate governing principles. A statement of ethics is usually advisable but may be required.
- Clearly and Precisely Define Concrete Goals
- Prioritize the Goals in alignment with the governing principles
- Develop Plans - Refer to the section above "Devise a Plan;" also check out Polya's book "How To Solve It." Also see the notes below.
- Implement the plans
- Monitor the progress of carrying out the plans. this means keep logs and or journals.
- When the final step of the complete is completed, evaluate the outcome9s) that have come out of implementing the plan.
- Adjust the plans based on the outcomes, Sometimes one needs to back one or more steps sometimes all the way back to to step 1.
Notes:
List of Polya's Common Problem Solving Heuristics.
Polya mentions that there are many reasonable ways to solve problems.
The skill at choosing an appropriate strategy is best learned by solving many problems.
You will find choosing a strategy increasingly easy. A partial list of strategies is included:
- Guess and check
- Make an ordered list
- Eliminate possibilities in some problems some options are clearly inappropriate.
- Use symmetry
- Consider special cases
- Use direct reasoning
- Solve an equation
- Look for a pattern
- Draw a picture
- Solve a simpler problem
- Use a model
- Work backward
- Use a formula
- Be ingenious
Source Material in preparing this post.
http://en.wikipedia.org/wiki/George_Pólya
http://www-gap.dcs.st-and.ac.uk/history/Biographies/Polya.html
http://www.amazon.com/How-Solve-It-Mathematical-Princeton/dp/069111966X/ref=sr_1_cc_1?s=aps&ie=UTF8&qid=1405914499&sr=1-1-catcorr&keywords=how+to+solve+it+a+new+aspect+of+mathematical+method