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.
Question: 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.
Relational vs. Instrumental Understanding in Mathematics
Mathematics is a subject that requires both conceptual and procedural knowledge. Conceptual knowledge refers to the understanding of why and how mathematical ideas work, while procedural knowledge refers to the ability to perform calculations and apply algorithms. In the literature, these two types of knowledge are often associated with two types of understanding: relational and instrumental.
Relational understanding is the ability to see the connections and relationships among mathematical concepts and procedures. It involves reasoning, explaining, and justifying mathematical ideas. Instrumental understanding is the ability to use rules and formulas without knowing why they work or how they relate to other concepts. It involves memorizing, applying, and reproducing mathematical procedures.
Although relational understanding is often thought to be a better alternative to instrumental understanding, both types of understanding have their advantages and disadvantages in different situations. In this blog post, I will discuss when do I regard each type of understanding as useful, particularly in mathematics, and justify my argument with examples.
Relational understanding is useful when:
- The mathematical concepts or procedures are complex, abstract, or unfamiliar. For example, when learning about fractions, decimals, or percentages, it is important to understand how they relate to each other and to the whole numbers. This helps to avoid confusion and errors when performing calculations or conversions.
- The mathematical problems are non-routine, open-ended, or require creativity. For example, when solving word problems, puzzles, or proofs, it is helpful to have a deep understanding of the underlying principles and strategies that can be applied in different contexts. This helps to find multiple solutions, explain the reasoning process, and evaluate the validity of the results.
- The mathematical learning is for long-term retention and transfer. For example, when studying for exams or preparing for future courses, it is beneficial to have a solid grasp of the core concepts and skills that can be recalled and reused in various situations. This helps to avoid forgetting or misunderstanding important information and to build on prior knowledge.
Instrumental understanding is useful when:
- The mathematical concepts or procedures are simple, concrete, or familiar. For example, when learning about basic arithmetic operations, such as addition, subtraction, multiplication, or division, it is sufficient to know how to apply the rules and formulas without necessarily understanding why they work or how they relate to other concepts. This helps to save time and effort when performing calculations or checking answers.
- The mathematical problems are routine, closed-ended, or require accuracy. For example, when doing homework assignments, quizzes, or tests, it is convenient to use rules and formulas that have been proven to work without needing to explain or justify them. This helps to achieve correct results quickly and efficiently.
- The mathematical learning is for short-term performance or enjoyment. For example, when playing games, doing puzzles, or exploring patterns, it is fun to use rules and formulas that produce interesting or surprising outcomes without worrying about the underlying logic or meaning. This helps to stimulate curiosity and motivation.
In conclusion, relational and instrumental understanding are both valuable types of understanding in mathematics that can be used in different situations depending on the goals and needs of the learner. Rather than viewing them as mutually exclusive or hierarchical alternatives, I regard them as complementary and interrelated aspects of mathematical knowledge that can enhance each other when used appropriately.
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