What is a mathematical model?

A mathematical model is an abstract description of a concrete system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in applied mathematics and in the natural sciences such as physics, biology, earth science, chemistry and engineering disciplines such as computer science, electrical engineering, as well as in non-physical systems such as the social sciences such as economics, psychology, sociology, political science. It can also be taught as a subject in its own right.

Why are mathematical models useful?

Mathematical modelling is valuable in various applications; it gives precision and strategy for problem solution and enables a systematic understanding of the system modelled. It also allows better design, control of a system, and the efficient use of modern computing capabilities.

What are some common mathematical methods used to solve problems?

• Mathematics is the study of numbers, shape, quantity, and patterns. The nature of mathematics is logical and it relies on logic and connects learning with learners' day-to-day life.

• Teaching methods of mathematics include problem-solving, lecture, inductive, deductive, analytic, synthetic, heuristic and discovery method. Teacher adopts any method according to the needs and interests of students.

Deductive Method: The deductive method is based on deduction. At first, the rules are given and then students are asked to apply those pre-established rules to solve more problems. It begins with general premises and through logical argument, comes to a specific conclusion.

·   Deductive approach proceeds from:

-    General to specific

-    Unknown to know

-    Complex to simple

-    Formula to example

EXAMPLE:

While teaching mathematics, the teacher introduces a theory and explains the rules of the theory and the formula, and the students are asked to solve problems using the given formula.

Hence, it could be concluded that mathematical problems are solved using pre-established rules in deductive methods of teaching Mathematics.

Inductive Method: Inductive approach is based on the process of induction. It is a method of constructing a formula with the help of a sufficient number of concrete examples. Induction means to provide a universal truth by showing that if it is true for a particular case.

·   Inductive approach proceeds from:

-    Particular to general

-    Known to unknown

-    Simple to complex

-    Formula to example

EXAMPLE: Square of an odd number is odd and the square of an even number is even.

·   Analytic Method

-    It is a generic process combining the power of the Scientific Method with the use of a formal process to solve any type of problem.

-    Analysis means breaking up into components. All scientifically based problem-solving approaches use the analytical method.

·   Synthetic Method

-    The word "synthetic' is derived from the word 'synthesis' which means to combine.

-    In this method, we combine several facts, perform certain mathematical operations, and arrive at the solution.

EXAMPLE:

While teaching mathematics, the teacher introduces a theory and explains the rules of the theory and the formula, and the students are asked to solve problems using the given formula.

 How do you develop a mathematical model?

Mathematics can be used to "model", or represent, how the real world works.

• Step 1: Draw a sketch

• Step 2: Make Formulas

• Step 3: Make a single formula for cost

• Step 4: Plot it and find minimum cost

• Step 5: Recommendations

EXAMPLE:

Sam and Alex play Tennis. On the weekend Sam played 4 more games than Alex did, and together they played 12 games. How many games did Alex play?

SOLUTION:

Turn the English into Algebra:

·   Use S for how many games Sam played

·   Use A for how many games Alex played

We know that Sam played 4 more games than Alex, so: S = A + 4. And we know that together they played 12 games: S + A = 12. We are being asked for how many games Alex played: A

Start with: S + A = 12

S = A + 4, so we can substitute "A + 4" for S:(A + 4) + A = 12. Simplify: 2A + 4 = 12. Subtract 4 from both sides: 2A = 12 – 4. Simplify: 2A = 8. Divide both sides by 2: A = 4. Which means that Alex played 4 games of tennis. Check: Sam played 4 more games than Alex, so Sam played 8 games. Together they played 8 + 4 = 12 games.

Step 1: Identify the decision

You realize that you need to make a decision. Try to clearly define the nature of the decision you must make. This first step is very important.

Step 2: Gather relevant information

Collect some pertinent information before you make your decision: what information is needed, the best sources of information, and how to get it. This step involves both internal and external “work.” Some information is internal: you’ll seek it through a process of self-assessment. Other information is external: you’ll find it online, in books, from other people, and from other sources.

Step 3: Identify the alternatives

As you collect information, you will probably identify several possible paths of action, or alternatives. You can also use your imagination and additional information to construct new alternatives. In this step, you will list all possible and desirable alternatives.

Step 4: Weigh the evidence

Draw on your information and emotions to imagine what it would be like if you carried out each of the alternatives to the end. Evaluate whether the need identified in Step 1 would be met or resolved through the use of each alternative. As you go through this difficult internal process, you’ll begin to favor certain alternatives: those that seem to have a higher potential for reaching your goal. Finally, place the alternatives in a priority order, based upon your own value system.

Step 5: Choose among alternatives

Once you have weighed all the evidence, you are ready to select the alternative that seems to be best one for you. You may even choose a combination of alternatives. Your choice in Step 5 may very likely be the same or similar to the alternative you placed at the top of your list at the end of Step 4.

Step 6: Take action

You’re now ready to take some positive action by beginning to implement the alternative you chose in Step 5.

Step 7: Review your decision & its consequences

In this final step, consider the results of your decision and evaluate whether or not it has resolved the need you identified in Step 1. If the decision has not met the identified need, you may want to repeat certain steps of the process to make a new decision. For example, you might want to gather more detailed or somewhat different information or explore additional alternatives.