How to Solve Problems

This article explains the mathematical problem-solving approach. Based on George Pólya's "How to Solve It," we introduce the characteristics of people who excel in mathematics and four effective steps for problem-solving. We explore tips for overcoming math difficulties and the potential for application in various fields.

Hello! I'm Morry, a former math-phobic who now works as a software engineer and math teacher, currently relearning mathematics from a liberal arts perspective. "I'm not good at math, it's impossible for me..." I understand that feeling very well. But wait a minute! In fact, if you learn the approach to solving math problems, you might be able to solve problems that you couldn't before. Today, I'd like to share with you the tips for this way of thinking.

Characteristics of People Good at Math

What's the difference between people who are good at math and those who struggle with it? It seems that people who are good at math have common characteristics, but what are they, and can anyone acquire these traits?

Characteristics of people good at math:

Can view problems from a bird's-eye perspective
Can break down complex problems into basic ones
Good at going back to the fundamentals
Skilled at visualizing concepts

People who struggle with math might think, "What? I have to add all of these? That's tedious..." But those who are good at math think differently. They consider:

"Wait a minute. 1 and 100, 2 and 99, 3 and 98... If we combine them like this, they all add up to 101!"

(1+100) + (2+99) + (3+98) + ... = 101 × 50 = 5050

That's right! When you view the problem from this perspective, even complex problems become easier to solve. As a software engineer, I feel this applies to skilled engineers as well. In fact, isn't this true for any field? Instead of immediately trying to solve the problem, it's better to first survey the whole, break it down into smaller parts, and try to visualize it. In other words, organize the information at hand and establish a strategy for solving the problem.

Problem-Solving Method from a Math Master

Now, here's the main point. There's a book called "How to Solve It" by mathematician George Pólya. This book outlines an excellent method for solving math problems.

Pólya's 4-Step Problem-Solving Process

Understand the problem
Devise a plan
Carry out the plan
Look back

This can be used not just for math, but for solving life's problems too!

Step 1 | Understand the Problem

First, read the problem carefully and organize what you know and what you don't know.

Problem : Find the maximum value of the function $y = -x² + 6x - 5$ , where $2 ≤ x ≤ 6$ . What we know:

We're given a quadratic function $y = -x² + 6x - 5$
The value of $x$ is restricted to the range between 2 and 6 (there's a domain)
The coefficient is negative $(-x²)$ , so this function forms a parabola that opens downward and has a maximum value

What we need to find: The maximum value of this function within the given range of $x$

Step 2 | Devise a Plan

Next, think about how to solve it. Always consider if you've solved a similar problem before.

Understand the problem and recall if you've solved a similar one before

The parabola opens downward

If the vertex is within the domain, the maximum value occurs at the vertex
If the vertex is outside the domain, the maximum value occurs at one of the domain's endpoints
I remember from a past problem that drawing a graph with the vertex and x-domain was helpful

Devising a plan :

Find the coordinates of the quadratic function's vertex
Check if the x-coordinate of the vertex is within the domain (drawing a graph helps)
If the x-coordinate of the vertex is within the domain, the y-coordinate of the vertex is the maximum value; if not, the maximum value occurs at one of the domain's endpoints
Substitute the x-value found in step 3 into the function to find the maximum value

Step 3 | Carry Out the Plan

Follow the plan and perform the actual calculations. Here, it's important to calculate and work accurately and persistently!

Find the coordinates of the quadratic function's vertex

\begin{align*} y &= -x² + 6x - 5 \\ &= -x² + 6x - 9 + 9 - 5 \\ &= -(x - 3)² + 4 \end{align*}

The coordinates of the vertex are $(3, 4)$

Check if the x-coordinate of the vertex is within the domain (draw a graph)

The x-coordinate of the vertex is 3, and the domain is $2 ≤ x ≤ 6$ , so the x-coordinate of the vertex is within the domain.

If the x-coordinate of the vertex is within the domain, the y-coordinate of the vertex is the maximum value; if not, the maximum value occurs at one of the domain's endpoints

The x-coordinate of the vertex is within the domain, so the y-coordinate of the vertex is the maximum value.

Substitute the x-value found in step 3 into the function to find the maximum value

\begin{align*} y &= -3² + 6 \times 3 - 5 \\ &= -9 + 18 - 5 \\ &= 4 \end{align*}

Step 4 | Look Back

Finally, check if the answer is correct. No calculation errors, and confirming with the graph shows that the x-coordinate of the vertex is within the domain, so the maximum value of $y=4$ when $x=3$ is correct.

Conclusion

How was that? The approach is the same whether you're solving a math problem or resolving a daily life issue.

The important thing is, when you think "This problem is difficult," don't start solving it immediately. Instead, first understand the problem, devise a plan, and then execute it repeatedly. While we don't usually consciously follow this process when solving problems in our daily lives, people who are good at math do this unconsciously.

For those who struggle with math, it's not possible to do this unconsciously right away. Instead, you need to recall this process each time and apply it to solve problems, gradually internalizing the approach. Even if you felt you were bad at math, once you master this process, you'll surely become better at it. Please try this method starting today.

See you next time!