Introduction
Whether it is solving a complex math problem, dealing with a difficult situation in our personal lives or at work, or making critical decisions, having effective problem-solving skills is crucial. One approach to problem solving that has been proven to be effective is using mathematical thinking, as described in the book “How to Solve It: A New Aspect of Mathematical Method” by George Pólya.
“Mathematical thinking is a systematic approach to problem solving that involves breaking down complex problems into simpler, more manageable parts, and applying logical reasoning and mathematical concepts to find solutions.”
According to Pólya, there are four steps to problem solving using mathematical thinking:
- Understanding The Problem
- Devising a Plan
- Carrying Out The Plan
- Evaluating The Solution.
- Understanding The Problem
Understanding The Problem
The first step in problem solving using mathematical thinking is to understand the problem. This involves:
- Reading the problem carefully
- Identifying what information is given and what is being asked
- Identifying any constraints or assumptions that need to be considered.
Suppose you are given the following problem or study case:
Suppose you are planning a road trip and need to calculate the total cost of gas for the trip. You know that your car has an average gas mileage of 25 miles per gallon, and you will be driving a total distance of 500 miles. The price of gas is $3.00 per gallon. How much will you need to spend on gas for the entire trip?
To understand this problem, we need to identify what information is given and what is being asked. From the problem statement, we know:
- We are planning a road trip, and we have information about the car’s gas mileage, the total distance to be traveled, and the price of gas.
- We are asked to find the total cost of gas for the entire trip.
- Next, we need to recognize any constraints or assumptions that may be relevant to the problem. For example, we may assume that the gas mileage remains constant throughout the trip or that there are no additional costs, such as tolls or parking fees.
By breaking down the problem in this way, we can gain a better understanding of what we need to do to solve it. In this case, we need to use the information given to find out how much each person contributed to the total cost of the items bought.
Devising A Plan
Once we have a clear understanding of the problem, we can move on to the next step, which is devising a plan. Devising a plan involves thinking of strategies or approaches that could help solve the problem. This could include using mathematical formulas, drawing diagrams or graphs, or using logical reasoning to eliminate possibilities. It is important to consider multiple strategies and choose the one that seems most likely to work. Based on previous study case, we can consider the following strategies:
- Use A Formula
We can set up a proportion where the left side of the proportion represents the cost per unit of distance, which is the Total Cost divided by the Total Distance. The right side of the proportion represents the cost per unit of gas used, which is the Price per Gallon divided by the Gas Mileage. $$(Total\ Cost / Total\ Distance) = (Price\ per\ Gallon / Gas\ Mileage)$$ - Use A Graph
We can create a graph to represent the relationship between the distance traveled and the total cost of gas. We can plot the distance on the x-axis and the cost on the y-axis, and use the gas mileage and price per gallon to determine the slope of the line.
By considering these strategies, we can choose the one that seems most likely to work and use it to solve the problem.
Carrying Out The Plan
The third step is to carry out the plan, which involves applying the chosen strategy to solve the problem. This could involve performing calculations, making observations, or testing hypotheses. It is important to be systematic and organized in this step, and to keep track of the steps taken to arrive at the solution. In above study case, suppose we have decided to use the formula to calculate the total cost of gas for the trip. We can carry out the plan by plugging in the given values and performing the calculations as follows: $$Total\ Cost= \frac{500}{25}3=60$$ Therefore, the total cost of gas for the entire trip is $60.00. By carrying out the plan, we have successfully used the chosen strategy to solve the problem and arrived at the correct answer. The next step is to evaluate the solution to ensure that it is reasonable and makes sense in the context of the problem.
Evaluate The Solution
The final step is to evaluate the solution, which involves checking that the solution is correct and makes sense in the context of the problem. It is important to check the solution for errors, and to make sure that it satisfies any constraints or assumptions that were identified in the first step. If the solution is not correct, we may need to revisit the earlier steps and try a different approach. For our study case, to evaluate the solution, we can consider whether the answer makes sense based on our understanding of the problem:
- We know that the distance of the trip is 500 miles
- The gas mileage of the car is 25 miles per gallon
- Therefore, the car should use 500 / 25 = 20 gallons of gas for the entire trip
- Multiplying this by the price per gallon of $3.00, we get a total cost of $60.00
This matches the solution that we calculated, so our answer is reasonable and makes sense in the context of the problem. By evaluating the solution, we can confirm that our problem-solving process was successful and that we have arrived at the correct answer.
Conclusion
In conclusion, problem solving using mathematical thinking is a powerful tool that can help us tackle complex problems in a systematic and logical way. By following the four steps outlined in Pólya’s book, we can approach any problem with confidence and find effective solutions. Whether we are solving a math problem or dealing with a difficult situation in our personal or professional lives, the principles of mathematical thinking can help us overcome any challenge.