The Importance of Understanding Concepts, Not Just Memorizing Math Formulas

Written by
Emily

Mathematics has long been regarded as a subject that challenges students to think critically, solve problems, and apply logical reasoning. For many, success in math is often seen as dependent on memorizing a set of formulas and applying them correctly to solve problems. While itā€™s true that certain formulas, such as the quadratic formula, the area of a circle, or the Pythagorean theorem, are essential for solving specific types of problems, relying solely on memorization can limit students’ understanding of the subject.

When students focus only on memorizing formulas, they may fail to grasp the underlying principles or the reasoning behind these formulas. This can result in a surface-level understanding of mathematics, where students are able to apply formulas in routine situations but struggle when faced with unfamiliar or more complex problems. While memorizing formulas can help students perform well in exams in the short term, it doesn’t foster the critical thinking skills required to adapt these formulas to new contexts. In Singaporeā€™s highly competitive academic environment, where math is crucial for university admissions and future careers, this approach can leave students unprepared for more challenging, non-routine problems.

The true power of mathematics lies not just in the memorization of formulas, but in understanding the concepts that these formulas represent. Why is this important?

The Power of Understanding Concepts

According to an insightful article on Mind Research, ā€œFrom Rules to Reasoning: Why We Need Thinkers, Not Memorizers in Math Education,ā€ the key to developing mathematical competence is not just following rules but understanding the reasoning behind them. In the article, itā€™s emphasized that students need to be thinkers who understand the ā€˜whyā€™ of mathematics, not merely memorizers who apply formulas without grasping their meaning.ā€ When students take the time to understand the concepts behind math formulas, they gain several important advantages:

  1. Flexibility in Problem-Solving: Understanding the principles behind a formula enables students to adapt and apply the formula in different scenarios. Instead of merely memorizing the steps to solve a problem, students learn the logic that drives those steps, which allows them to apply their knowledge creatively to new and unfamiliar problems.
  2. Improved Retention: Concepts tend to stick in our minds longer than formulas because they make sense to us. When we understand why something works, we are more likely to remember it. For example, understanding the concept of area as the space inside a boundary is much more memorable than simply memorizing the area formula for a rectangle (length Ɨ width).
  3. Building Stronger Foundations: When students grasp the fundamental concepts, they can build on them more effectively in future math courses, as mathematics is a cumulative subject. For example, mastering the concept of variables in algebra or understanding the properties of numbers in arithmetic lays the groundwork for more advanced topics like calculus or statistics. Without understanding the building blocks, students may find themselves struggling as the difficulty of the material increases.
  4. Confidence and Self-Sufficiency: Students who understand the underlying concepts tend to feel more confident because they know they can figure out a solution even when they donā€™t immediately remember a specific formula. They are equipped to reason through a problem and deduce the necessary formula based on the situation.
  5. Making Math More Enjoyable: Understanding the logic behind mathematical principles transforms math from a series of isolated formulas into a cohesive, logical system. This shift makes math more engaging, as students realize that math is not just about applying rules but about discovering relationships and patterns.

When Is It Necessary to Memorize Math Formulas?

  1. For Efficiency: Some formulas, like the quadratic formula, the area of a circle, or trigonometric identities, are foundational to solving problems quickly and efficiently. In these cases, memorization helps save time, allowing you to focus on applying the formula correctly rather than deriving it each time.
  2. In Time-Limited Exams: In exams, especially those with time constraints, you often need to recall formulas quickly. Having key formulas memorized allows you to apply them without having to look them up or derive them from scratch.

How Understanding Leads to Deriving Formulas

One of the most powerful ways to learn math is by understanding how formulas are derived. This process often starts with a simple concept or principle and builds logically from there. For example, the area of a circle can be derived by approximating it as a polygon with an increasing number of sides, eventually leading to the formula $A=\pi r^2$. Understanding this derivation helps students appreciate the logic behind the formula and provides them with the tools to work with it more effectively.

When students understand how formulas are derived, they donā€™t need to rely solely on memorization. Instead, they can recreate the formula as needed. This process helps to deepen their conceptual understanding and empowers them to approach new problems with a clearer mindset.

Our Teaching Approach: Focusing on Conceptual Understanding

At the heart of our teaching philosophy is the belief that math is not just about memorizing formulas, but about developing a deep understanding of the concepts that underpin them. Hereā€™s how we focus on understanding over memorization:

  1. Explain the “Why” Behind Every Formula: Before introducing any formula, we take the time to explain its origin. We show students how we arrive at the formula, whether through a logical process, real-world application, or visualization. This helps demystify formulas and makes them more approachable.
  2. Encourage Active Problem-Solving: We encourage students to engage actively in solving problems by thinking through each step. Instead of simply asking them to apply a formula, we prompt them to explain the reasoning behind each step and how the formula connects to the problem at hand.
  3. Use Visuals and Intuition: Whenever possible, we use visual aids, diagrams, and hands-on activities to help students build an intuitive understanding of concepts. For example, when teaching geometry, we use shapes and areas to show why certain formulas work.
  4. Relate Math to Real Life: We draw connections between abstract concepts and real-world applications. By showing how math is used to solve practical problems ā€” such as calculating areas in construction or analyzing trends in data ā€” students begin to appreciate the relevance of what they are learning.
  5. Encourage Conceptual Discussions: We foster a classroom environment where students feel comfortable asking questions and discussing the ideas behind the math they are learning. This encourages critical thinking and allows students to explore different approaches to solving problems.

Conclusion

Understanding the concepts behind math formulas is key to developing a deep and lasting knowledge of mathematics. When students move beyond rote memorization and learn the logic, reasoning, and derivations behind formulas, they not only perform better on exams, but also gain the skills necessary to tackle more complex problems in the future.

In math, it’s not enough to “just memorize the formula” or “die die remember it.” Instead, students should strive to understand why the formula works, how itā€™s derived, and when to apply it. This approach doesnā€™t just make math easier ā€” it makes it meaningful and enjoyable. By fostering a deep understanding of math concepts, we empower students to become confident problem-solvers capable of applying their knowledge in a wide range of situations. Experience the difference with us ā€” where learning math is about mastering concepts, building confidence, and preparing you for success beyond the classroom. Sign up for a Math trial lesson today!

Published: 23rd December 2024
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Timothy Gan

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