Understanding of Math (R. Skemp) :Instrumental & Relational

Richard Skemp, a prominent figure in mathematics education, introduced influential concepts regarding how students understand mathematics. His work is particularly noted for distinguishing between two types of understanding: instrumental understanding and relational understanding.

 

1> Instrumental Understanding

Instrumental understanding refers to knowing how to perform mathematical procedures and solve problems without necessarily understanding the underlying concepts. It is often characterized by:

• Rule-Based Learning: Students learn and apply rules or algorithms to solve problems.
• Procedural Knowledge: Focus on the steps needed to get the correct answer.
• Limited Flexibility: Students may struggle with problems that deviate from familiar patterns or require a deeper understanding of the concepts.

2> Relational Understanding

Relational understanding involves comprehending the underlying principles and concepts behind mathematical procedures. It is characterized by:

• Conceptual Knowledge: Knowing why mathematical rules and procedures work.
• Interconnected Knowledge: Understanding how different mathematical ideas are related and how they can be applied in various contexts.
• Flexibility and Adaptability: Ability to apply mathematical concepts to new and unfamiliar problems.

 

3> Skemp's Perspectives

1. Benefits of Relational Understanding:
   • Deeper Comprehension: Students who understand the 'why' behind mathematical procedures can apply their knowledge more flexibly and creatively.
   • Long-Term Retention: Concepts understood relationally are more likely to be retained and recalled accurately over time.
   • Problem-Solving Skills: Students are better equipped to tackle novel problems and make connections between different areas of mathematics.
2. Challenges of Instrumental Understanding:
   • Surface Learning: Students may be able to solve problems by following learned procedures but lack a true understanding of the concepts.
   • Fragility of Knowledge: Without a relational understanding, students’ knowledge is often fragile and can break down when faced with unfamiliar problems.
   • Limited Transferability: Students may struggle to transfer their learning to different contexts or more advanced topics.

4> Application in Math Education

 

Skemp's distinction between instrumental and relational understanding has significant implications for teaching practices in mathematics education.

1. Curriculum Design:
   • Emphasize activities and problems that encourage students to explore and understand underlying mathematical concepts.
   • Design curricula that build on students’ existing knowledge and show the interconnectedness of mathematical ideas.
2. Teaching Strategies:
   • Encourage Exploration: Allow students to explore multiple methods for solving problems and understand the reasoning behind each method.
   • Use Conceptual Questions: Pose questions that require students to explain their reasoning and the relationships between concepts.
   • Foster Discussion: Promote classroom discussions where students explain their thinking and learn from each other.
3. Assessment Practices:
   • Assess Understanding, Not Just Procedures: Use assessments that evaluate students’ relational understanding, such as asking them to explain their reasoning or apply concepts to new situations.
   • Formative Assessment: Regularly assess students’ understanding throughout the learning process to identify and address misconceptions.

 

5> Conclusion

Richard Skemp's concepts of instrumental and relational understanding provide a valuable framework for improving mathematics education. By focusing on relational understanding, educators can help students develop a deeper, more connected, and more flexible understanding of mathematics. This approach not only enhances students’ problem-solving abilities and long-term retention of mathematical knowledge but also fosters a more meaningful and engaging learning experience.