Why Math Teachers and Professors Insist on Specific Problem-Solving Methods
As a retired math teacher, I've seen firsthand the importance of both understanding concepts and mastering procedures in a mathematical course. This article explores the reasons behind math teachers and professors insisting on specific problem-solving methods and the benefits they bring to students.
The Dual Objectives of a Mathematics Course
A successful mathematics course aims to achieve two primary goals: understanding the concepts and proficiency with procedures. These objectives complement each other and are both essential for a student's overall development in mathematics.
Understanding Concepts
When it comes to understanding mathematical concepts, the goal is to ensure that students grasp the fundamental principles that underpin various mathematical theories. This involves not just memorizing formulas but understanding the reasoning behind them. In my experience, it is crucial that students learn to apply these concepts in various contexts to deepen their understanding. To this end, I always encouraged my students to use any method they deemed appropriate, as long as it was valid and led to the correct solution. This approach fostered a flexible and adaptable thinking pattern.
Procedural Proficiency
However, there is an exception to this flexibility. Sometimes, specific procedures must be used to test procedural proficiency. These methods are essential for laying a foundational framework for more advanced mathematical concepts that will be learned later. Additionally, proficiency in specific procedures provides students with a robust set of tools to tackle more complex problems.
The Need for Structured Learning
Mathematics is a subject that requires a structured and sequential approach to learning. While it’s important to encourage creative and flexible thinking, it is equally crucial to ensure that students are familiar with and can apply established methods. This is because the knowledge of a specific procedure is critical for understanding more advanced concepts that build upon the foundational ones.
For instance, consider the problem of proving that a quartic polynomial equation has no rational roots. The rational root theorem is a powerful tool designed for precisely this purpose. While other methods might work, they would not demonstrate the mastery of the specific procedure that is being tested. Answering the question in a way that deviates from the expected method would actually be counterproductive, as it fails to meet the objective of the problem.
Similarly, in a practical context, imagine a student being trained as a car mechanic. If their supervisor uses a specialized tool to remove a spark plug, the student should recognize the value and appropriateness of using that tool. Attempting to solve the task with a less effective or inappropriate method would not only be less efficient but also potentially harmful. The same principle applies in education: there are times when we need to ensure that students have a solid grasp of specific procedural techniques, even if it means using a more conventional method.
The Importance of Methodical Problem Solving
The point of a mathematics test is not just to find an answer, but to demonstrate proficiency in the methods being taught. By focusing on the use of specific procedures, we ensure that students are building on their problem-solving skills rather than relying solely on the first method that comes to mind. This approach helps students develop a better understanding of mathematical concepts and enhances their ability to tackle complex problems in the future.
In conclusion, while flexibility and creativity are important in mathematics education, the structured learning of specific procedures is equally vital. By balancing these two elements, we can ensure that students not only understand the concepts but also have the proficiency to apply them effectively. This approach ultimately leads to a more complete and well-rounded mathematical education.