Which Way is Right? Problem-Solving in Mathematics

Posted by Richard Kingham on May 24, 2021

The Mona Lisa is surprisingly, almost shockingly, small. Yet its far-reaching influence has made it synonymous with greatness. “Well, that painting is good, but I mean, it’s not the Mona Lisa,” the armchair art critic will opine. Upon seeing it for the first time in real life, some observers remark with puzzlement and a hint of disappointment: “That’s it? Huh.” 

Yet, after spending time analyzing the painting, many viewers walk away with a deeper appreciation for its many layers of artistry. The Mona Lisa remains the same, but their perspective changes. 

Math problems and art share this feature—you have to see them from multiple angles to fully appreciate them. Solving problems in different ways reveals the mathematician’s creativity and ingenuity. 

Many math students, however, get stuck seeking the “right answer,” using a prescribed method. They measure success in the classroom by the number of checkmarks inked across their worksheet, not by the analytical deftness they’re developing and certainly not by the love for math they’re growing. 

While solving math problems is not a free-for-all, choose-your-own-adventure enterprise, it can be much more creative and elegant than we sometimes allow. 

Consider fractions, for example. We can represent fractions in drawings using bar charts, colored circles, pie charts, decimals, and ratios, to name just a few possibilities. Obviously, students will learn different methods of showing fractions at various stages in their education, but too often we restrict students to one way. 

Along with constraining creativity, such restrictions cause mathematical skills to atrophy. Many students only learn a skill well enough to attain a certain exam score, which, sadly, becomes an end in itself. After the exam, it’s on to the next lesson, which may have no continuity with the preceding one. Giving students the option of solving problems with simple, open-ended questions—for example, “What are some other ways to solve this problem?”—prompts students to use skills that they have learned in the past. 

Practicing the same skills over time improves automaticity and remedies the transfer issue—students’ inability to solve problems in different contexts. When students can recognize mathematical problems outside the classroom and know how to solve them, math becomes irreversibly meaningful: they begin to see the creativity inherent in solving math problems. That is how we instill a lifelong love for math in students. 

Topics: mathematical reasoning, math, real world connection