"Many people would sooner die than think; In fact, they do so." -Bertrand Russell
Systematic Problem Solving: Part 1
So we last saw a general overview of how we might be able to set up a systematic approach to problem solving, and most importantly, why we would want to do that to begin with. We know that using a systematic approach will help us to eliminate clutter, and orient ourselves towards finding problem solutions, essentially opening up within ourselves a mechanism with which we we can break down and solve all sorts of situations. In the next four sections we will first cover an overview of a framework, and then we will follow by developing some more detailed understanding of each component by focusing on an example problem that will each emphasize one particular component. After this we will bring it all back together to see how we can use our newly developed system to solve problems in many different fields, fields like: art, business, computing, history, law, medicine, and many others. First, don't worry if you don't know anything about one or even all of these fields, that won't matter much when we get there. Second, you might have noticed in that list the fields of art and history. I want to emphasize that while this method quite naturally lends itself to problems involving numbers, like mathematical problems, this is not a restriction. We can also use it to find qualitative solutions, letting us take into consideration things like ethics and aesthetics. This also has advantages since sometimes simple solutions are morally wrong or visually displeasing. So let's start putting this together and seeing how we're going to begin to understand problems in a different light. How many parts should a systematics method consist of? That's going to depend on who you ask, for one thing, and how you build your system for another. That's one thing that's great, is you don't even have to use the same system we're learning here, you can make your own and apply it similarly if you want. One you have strongly understood the principles going into the idea, it becomes simple to make your own tailored framework.
First of all, why do I keep calling it a framework? Well, if you recall from last time, I said the system we were going to build would be moldable and modular. There are other methods I've seen in studying this topic of systems with 7,8, or even 12 major portions. This is not, in my opinion, very usable or easy to remember, so I made one that I think works better, I hope you do too.
The basic outline goes something like this. Given some problem, you need to determine four things.
Determine:
We're going to take time to look at each component in turn with an example that focuses on each one. Today we're looking at step one, "determine the information."
This first step has it's own subdivisions, some of which may or may not apply to every problem. Basically, we need to categorize and collect a few different kinds of information about the problem. First you need to figure out what is the exact problem you are trying to solve. It can often help to write this as a distinct statement or question, as in "find a way to include feature x on our website's landing page." Once you have the problem clearly placed before you, you need to see what information is given in the problem. This will be any information specifically given in some way. Then you should also list information that you can infer directly. If a house was a faded blue last week, but today it is a fresh looking brown, you can infer that it was probably painted sometime during the week. So when you confront a problem then you only need to figure out three things in the first step: what is the problem, what information are you given about the problem, and then what other things are made apparent from that given information. Let's take a look at how this first step works in practice. This example comes from Allen Angel's Intermediate College Algebra for College Students, and emphasizes the first step in our problem solving system.
The second thing we want to find is what information is given that applies to the problem. This step is quite often a sifting process. Many problems will give both useful and irrelevant information. Here in our example, the first one and a half sentences are unneeded, so we can dispense with that. We are then given an equation to use for estimation. This is helpful information, so we'll hang on to that, the explanation of it is also helpful so we know how to apply it properly. It also tells us how we can solve for x in this explanation, which is also useful, and then we get to the problem question. So we've covered the whole problem now. There's not very much we can really infer in this problem that will actually apply to finding a solution, so we've finished the first step and we can continue.
For completeness we will go ahead and solve it quickly now as well.
We want to know what percentage of the cars sold were U.S. made, and we also know we can estimate that using M = -1.26x + 75.34, and that we can find x by figuring out the number of years between the current one (in the problem) and 1993. So we have x = 2003 - 1993 = 10. Then we just supply our x and we have M = -1.26*10 + 75.34 = 75.34 - 12.60 = 62.74. So we have found that the number of cars made in the U.S. in 2003 using their estimation is 62.74%.
Now, I know as well as you do that this problem really wasn't very interesting or difficult to solve, but it was an important first step in our process. We have to train ourselves to follow the system we develop in order to be able to use it effectively. This example helped us to work through the process of drawing out the question, or goal, and the information that was given to us which allowed us to quickly solve the problem. Often, figuring out what the question is can be one of the most difficult tasks in the problem solving process, for as we all know, life doesn't work like a textbook and spell out everything for us.
That's it for today, next we'll have a look at the second step we need to build our systematic problem solving method, "determine paths to solutions." We will also begin to get into more interesting problems from here on out. Enjoy your day, and as always, if you have a question, ask!
Warm regards,
Tristen
First of all, why do I keep calling it a framework? Well, if you recall from last time, I said the system we were going to build would be moldable and modular. There are other methods I've seen in studying this topic of systems with 7,8, or even 12 major portions. This is not, in my opinion, very usable or easy to remember, so I made one that I think works better, I hope you do too.
The basic outline goes something like this. Given some problem, you need to determine four things.
Determine:
- the information,
- paths to solutions,
- which path to take,
- whether to accept it.
We're going to take time to look at each component in turn with an example that focuses on each one. Today we're looking at step one, "determine the information."
This first step has it's own subdivisions, some of which may or may not apply to every problem. Basically, we need to categorize and collect a few different kinds of information about the problem. First you need to figure out what is the exact problem you are trying to solve. It can often help to write this as a distinct statement or question, as in "find a way to include feature x on our website's landing page." Once you have the problem clearly placed before you, you need to see what information is given in the problem. This will be any information specifically given in some way. Then you should also list information that you can infer directly. If a house was a faded blue last week, but today it is a fresh looking brown, you can infer that it was probably painted sometime during the week. So when you confront a problem then you only need to figure out three things in the first step: what is the problem, what information are you given about the problem, and then what other things are made apparent from that given information. Let's take a look at how this first step works in practice. This example comes from Allen Angel's Intermediate College Algebra for College Students, and emphasizes the first step in our problem solving system.
- Since 1993, American automakers have been losing market share to Asian and European automakers. The percent of the total cars sold in the United States made by American automakers can be estimated using the equation M = -1.26x + 75.34, where M is the percent of the total cars sold in the United States made by American automakers and x is the number of years since 1993. Use x=1 for 1994, x=2 for 1995, and so on. What is the percent of the total cars sold in the United States made by American automakers in 2003 ?
The second thing we want to find is what information is given that applies to the problem. This step is quite often a sifting process. Many problems will give both useful and irrelevant information. Here in our example, the first one and a half sentences are unneeded, so we can dispense with that. We are then given an equation to use for estimation. This is helpful information, so we'll hang on to that, the explanation of it is also helpful so we know how to apply it properly. It also tells us how we can solve for x in this explanation, which is also useful, and then we get to the problem question. So we've covered the whole problem now. There's not very much we can really infer in this problem that will actually apply to finding a solution, so we've finished the first step and we can continue.
For completeness we will go ahead and solve it quickly now as well.
We want to know what percentage of the cars sold were U.S. made, and we also know we can estimate that using M = -1.26x + 75.34, and that we can find x by figuring out the number of years between the current one (in the problem) and 1993. So we have x = 2003 - 1993 = 10. Then we just supply our x and we have M = -1.26*10 + 75.34 = 75.34 - 12.60 = 62.74. So we have found that the number of cars made in the U.S. in 2003 using their estimation is 62.74%.
Now, I know as well as you do that this problem really wasn't very interesting or difficult to solve, but it was an important first step in our process. We have to train ourselves to follow the system we develop in order to be able to use it effectively. This example helped us to work through the process of drawing out the question, or goal, and the information that was given to us which allowed us to quickly solve the problem. Often, figuring out what the question is can be one of the most difficult tasks in the problem solving process, for as we all know, life doesn't work like a textbook and spell out everything for us.
That's it for today, next we'll have a look at the second step we need to build our systematic problem solving method, "determine paths to solutions." We will also begin to get into more interesting problems from here on out. Enjoy your day, and as always, if you have a question, ask!
Warm regards,
Tristen