Problem Solving in Mathematics

This page describes solving word problems in mathematics. Using the Singapore model method and TOC.

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Problem Solving in Mathematics.

 

The main goal in Mathematics is to solve problems.  The problems sometimes involve numbers as we know and sometimes they are abstract.  They are all based on deductive or inductive logic. Whatever the nature of the problem is, there is always a set of assumptions and a conclusion or just a set of conclusions.  Starting with the set of assumptions and a set of known facts one must logically come to a conclusion.   This is called the deductive method. 
The deductive method uses “if…then” logic.  The student should be trained to develop a logical chain of statements from assumptions to the conclusion/s.  This is called “Sufficient Condition” logic.
There is another conditional logic, “necessary condition” logic.   It uses words such as “In order to …we must have…” or “…only if …”  Students should also know this type of logic. 
         Most people get confused between the two types of logic.  Basically “sufficient” means it gets the “job done.” While “necessary” means it is a prerequisite but may not be enough to get the “job done.”  For example, a student needs 120 credits to graduate with a B.S. degree is a necessary condition, but it is not sufficient.  Just any 120 credits won’t be enough to graduate.  The sufficient condition is that the credits are as per the requirements of the degree program.
“…Our brains aren’t wired for general logic problems ..” says neuroscientist David Eagleman in “Incognito – The secret lives of the brain”.  That is the reason we have a difficult time at solving puzzles and word problems.  The only way to get adept at solving word problems is to solve as many of them as possible.  Then the process becomes automatic.  “Burn really good programs all the way to the DNA,” says David Eagleman.

Problem Solving Methodology:
Make sure to get training in the problem solving process from your instructor, tutor or mentor.  

The first steps in the problem solving are to identify, select and define the problem.  You or your group identifies and develops a concise definition of the problem.  If not, you or your group can find out by discussion in the group or with an instructor, how the problem can be precisely defined.   

Then a statement of what you want to achieve is prepared, if the goal is not already made precise.  This enables you or your group to know when the problem is solved to the group's satisfaction.  Identify possible causes and collect data to identify the actual cause.  It might be necessary to go outside the group to collect the data, or to refer to other resources.

Once the cause of the problem is verified, a solution is developed.  At this stage, the group members have the opportunity to apply their knowledge and experience.  (It is assumed that the group members are experts in their particular part of the assignment.)  This gives the group a sense of "ownership" in its fullest sense.

Mathematical Problem Solving
A mathematics problem is usually well-defined and involves necessary and sufficient conditions. The only thing we need to do is to establish logical links.  
As an illustration we will take up a typical problem in Algebra and follow it through various steps.
Sara invested $10,000 in two funds, one fund that earned 9% and one that earned 7%.  If she earned $880 in interest from the two funds, how much did Sara invest in each fund?
         
Here the goal is to find out how much Sara invested in each fund; the necessary conditions are how much she invested in the two funds at what rates, and how much the total interest is.  The sufficient conditions are the information provided in first and second sentence of the problem.  Thus the problem is complete with necessary and sufficient conditions.  If we omitted one of the pieces of these lines, the information would not be sufficient. 
This problem is solved in an elaborated manner later, but here we will use it to explain various steps of problem solving.

The steps to solve a word problem are as follows and depend on sufficiency logic.

The steps are based on assumptions or given information.
1.      Read the problem, verbalize it in your words to comprehend what is given and what is asked, labeling the unknowns in letters (variables) if they exist.  
For the example given above, given: investment $10,000 in two funds at rates 9% and 7%.  The total interest is $880.00.  One unknown investment is x at 9%. 

2.      Translate the given information into a mathematical equation or inequality using the given.
If the investment at 9% is x, the investment at 7% is 10000 – x.
The total interest in terms of variables is .09 x + .07(10000 – x) while the numerical value is 880.  The main point is to set up an equation,  we have .09x + .07(10000 – x) = 880.
3.      Check the tools you have learned in the section or the book, and use them to solve the equation.  
  Simplify the terms on the left side of the equation and solve.
  .09x + 700 - .07x = 880  (Remove parentheses)
                      .02x = 180  (Isolate the variable one side and numbers on the other.)
                         2x = 18000 (We multiplied by 100.)
                           x = 9000  (We divided by 2)  

4.      Once you solve the equation, translate back the values of variables in terms of the values of the quantities.  
 The investment at 9% is $9,000 and at 7% it is $1,000.

Thus, a problem is solved going one step at a time from ”what is known” (given) to “what is unknown,” using “If…then….” All arithmetic problems and most algebra problems can be solved using this “if…then…” connections or sufficient condition logic.  First find out the unknown and call it x or some variable.   From the first line start connecting the sentences until the last  “if…then…”  The solution is found by isolating the equation formed in the process. Some algebra problems are solved by going back from the last sentence to the first sentence using what is called necessary condition logic with what we call the Ambitious Target Tree.  There is also another method using a “transition tree.”  There are three examples in the last part of this chapter using the three methods. An example of how to solve a problem involving a system of equations that uses a transition tree is also essential in solving an abstract word problem (proving a theorem.)

Click to view the Singapore model method: Singapore_Mathematics-Final_5.pdf

TOC Methods: Solving_Word_Problems_TOC_solutions_to_Singapore_Model_Method_problems.pdf