although relational understanding is often thought to be a better alternative to instrumental understanding, when do you regard each type of understanding as useful, particularly in mathematics? justify your argument with examples.
LectureNotes said although relational understanding is often thought to be a better alternative to instrumental understanding, when do you regard each type of understanding as useful, particularly in mathematics? Justify your argument with examples.
Answer:
In the field of mathematics education, both relational and instrumental understandings play crucial roles, depending on the context and the learning objectives. Let’s delve into what each type of understanding entails and discuss scenarios where each is particularly useful.
Relational Understanding
Definition: Relational understanding refers to comprehending the underlying principles and relationships between mathematical concepts. It goes beyond memorizing procedures to understanding why and how mathematical processes work.
Usefulness:
-
Deep Learning and Problem Solving: Relational understanding is essential for deep learning and effective problem-solving. For instance, when students understand the concept of fractions relationally, they can apply this knowledge to various contexts, such as dividing quantities or understanding ratios.
Example: Consider the equation of a line in slope-intercept form, y = mx + b. A student with relational understanding knows that:
- m represents the slope, which indicates the steepness and direction of the line.
- b is the y-intercept, where the line crosses the y-axis.
This understanding allows the student to graph the line, interpret the meaning of slope and intercept in real-world contexts, and solve related problems.
-
Transfer of Knowledge: Relational understanding facilitates the transfer of knowledge to new situations. For example, understanding the properties of geometric shapes helps students solve problems involving area, perimeter, and volume across different shapes and contexts.
Example: If a student understands the relationship between the area of a rectangle (A = l \times w), they can easily adapt this understanding to find the area of a parallelogram (A = b \times h), recognizing the base and height as analogous to length and width.
Instrumental Understanding
Definition: Instrumental understanding involves knowing how to perform mathematical procedures without necessarily understanding the underlying concepts. It focuses on the “how” rather than the “why.”
Usefulness:
-
Efficiency in Routine Tasks: Instrumental understanding is particularly useful for performing routine tasks quickly and efficiently. For example, when students need to solve a large number of similar problems, such as basic arithmetic operations or algebraic manipulations, instrumental understanding can be highly effective.
Example: In a timed test, a student who has memorized the quadratic formula x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} can quickly solve quadratic equations without needing to derive the formula each time.
-
Foundation for Complex Concepts: Instrumental understanding can serve as a stepping stone to relational understanding. Initially, students might learn procedural steps to build confidence and familiarity with mathematical operations before delving into deeper conceptual understanding.
Example: When learning to differentiate functions, students might first memorize and apply the power rule \frac{d}{dx} [x^n] = nx^{n-1}. Once they are comfortable with the procedure, they can explore the underlying principles of calculus, such as the limit definition of a derivative.
Balancing Both Understandings
While relational understanding provides a comprehensive grasp of mathematical concepts, instrumental understanding offers practical efficiency. A balanced approach that integrates both types of understanding can enhance mathematical proficiency.
Example: In teaching the Pythagorean theorem, an effective strategy might involve:
- Instrumental Understanding: Initially, students memorize and apply the formula a^2 + b^2 = c^2 to solve problems.
- Relational Understanding: Subsequently, students explore why the theorem works, perhaps through geometric proofs or by examining right triangles in different contexts.
In conclusion, both relational and instrumental understandings have their places in mathematics education. Relational understanding fosters deep comprehension and adaptability, while instrumental understanding provides efficiency and a foundation for further learning. By leveraging both approaches, educators can equip students with the skills and insights needed to excel in mathematics.