1.3: Introduction to Problem Solving

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Introduction: Problem Solving as the Heart of Mathematics

The ultimate goal of our course is to empower you to help your future students achieve meaningful and lasting mathematical understanding. We begin our mathematical activity with problem solving because it is the cornerstone of mathematics learning and teaching. Problem solving is considered the primary reason for studying mathematics. It is not just about learning procedures but also about developing a deep understanding of why and how mathematics works (Feikes, Schwingendorf, and Gregg, 2018). Learning through problem solving is an effective method for grasping mathematical concepts (Finan, 2006). It allows students to engage actively in the learning process, constructing their own understanding rather than passively receiving information (CDE, 2013) Real understanding in mathematics comes from solving problems, turning it into a sense-making activity (Feikes, Schwingendorf, and Gregg, 2018).

Engaging with problems fosters not only the ability to solve specific mathematical tasks but also the development of broader problem-solving strategies that are applicable in everyday life and other disciplines. Problem solving encourages creativity, critical thinking, and perseverance. It allows students to explore, test, and refine their ideas, leading to a deeper understanding of mathematical concepts. George Pólya's four-step process—understanding the problem, devising a plan, carrying out the plan, and looking back—is a powerful framework that you will use throughout this course to approach and solve problems systematically. This method not only provides a clear structure for tackling mathematical challenges but also cultivates habits of mind that are essential for effective teaching and lifelong learning (Finan, 2006).

Before diving into specific strategies, take a moment to reflect on what it truly means to think mathematically and to do mathematics. You're not alone in this journey—we will work together to develop your mathematical mindset. Let’s dive into some problems and explore the different ways you naturally approach problem solving!

Wonder, Play, Grow

Draw curves connecting A to A, B to B, and C to C. Your curves cannot cross or even touch each other, they cannot cross through any of the lettered boxes, and they cannot go outside the large box or even touch it’s sides.

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Think About and Discuss

After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it).

What did you try?
What makes this problem difficult?
Can you change the problem slightly so that it would be easier to solve?

Do you wish something in the problem was different? Would it then be easier to solve the problem?

For example, what if ABC problem had a picture like this:

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Can you solve this case and use it to help you solve the original case? Think about moving the boxes around once the lines are already drawn.

🧠Let's Listen to Learn❤️

Here is one possible solution.

Productive problem-solving is more than just finding the right answer—it is a dynamic process that involves exploration, creativity, and critical thinking. When we engage with problems, we are not merely applying formulas but developing a mathematical mindset that fosters persistence and adaptability. Effective problem solvers analyze problems deeply, recognize patterns, and refine their approaches through iteration and reflection. As we move forward, we will explore specific strategies that enhance our problem-solving abilities, including visualization, logical reasoning, and heuristic approaches. These strategies will not only help us tackle mathematical challenges but also cultivate the habits of mind that make mathematical thinking a powerful tool in everyday life.

Problem or Exercise?

The main activity of mathematics is solving problems. However, what most people experience in most mathematics classrooms is practice exercises. An exercise is different from a problem.

In a problem, you probably don’t know at first how to approach solving it. You don’t know what mathematical ideas might be used in the solution. Part of solving a problem is understanding what is being asked, and knowing what a solution should look like. Problems often involve false starts, making mistakes, and lots of scratch paper!

In an exercise, you are often practicing a skill. You may have seen a teacher demonstrate a technique, or you may have read a worked example in the book. You then practice on very similar assignments, with the goal of mastering that skill.

Note

What is a problem for some people may be an exercise for other people who have more background knowledge! For a young student just learning addition, this might be a problem:

\[\textit{Fill in the blank to make a true statement} \: \_\_\_ + 4 = 7 \ldotp \nonumber \]

But for you, that is an exercise!

Both problems and exercises are important in mathematics learning. But we should never forget that the ultimate goal is to develop more and better skills (through exercises) so that we can solve harder and more interesting problems.

Learning math is a bit like learning to play a sport. You can practice a lot of skills:

hitting hundreds of forehands in tennis so that you can place them in a particular spot in the court,
breaking down strokes into the component pieces in swimming so that each part of the stroke is more efficient,
keeping control of the ball while making quick turns in soccer,
shooting free throws in basketball,
catching high fly balls in baseball,

and so on.

But the point of the sport is to play the game. You practice the skills so that you are better at playing the game. In mathematics, solving problems is playing the game!

Example $\PageIndex{1}$

Using the digits 1 to 7, at most one time each, place a digit in each circle so that the sum of the numbers in 3 squares (the middle horizontal line or 2 diagonals) are the same.

e.g A+B+C or D+B+E or F+B+G

$A diagram consisting of seven labeled squares (A, B, C, D, E, F, G) connected by straight lines, forming a network of paths. The central square, B, is connected to six other squares. The squares form a structure with three key pathways: a horizontal middle row (D–B–E), and two diagonals (A–B–C and F–B–G). The task involves assigning the digits 1 to 7 (each used at most once) to the squares so that the sums of numbers along the three key pathways are equal.$

Is this a problem or an exercise?

Answer

This is a problem, not just an exercise, because it requires thinking, strategy, and logical reasoning rather than simply applying a known formula or procedure.

Posisble Solution: (Did you find another solution? Is there another one?)

B=4

A+B+C=1 + 4 + 7 = 12

D+B+E = 3 + 4 + 5 = 12

F+B+G = 6 + 4 + 2 = 12

*Problem Source: https://www.openmiddle.com/number-pattern/

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Think About and Discuss

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Example \(\PageIndex{1}\)