Relational Understanding and Instrumental Understanding of Mathematics (Week 3 Reflection)

In the book ‘The Psychology of Learning Mathematics’, Skemp uses many analogies to discuss the two ways a student can understand mathematics, through instrumental understanding, and relational understanding. Instrumental understanding is defined as a student’s ability to use a rule that is given. It is the process of committing these rules to memory and using a step-by-step method to solve a mathematic problem, until it eventually becomes automatic. Relational understanding is defined as knowing both what to do in order to solve a mathematical problem, and why you are doing it. It involves using specialization, generalization, and having the ability to explain your methods logically. Skemp emphasized the importance of incorporating both types of understanding in the classroom since they can be regarded as different kinds of mathematics. This will cater to a variety of learners.

In today’s mathematics classrooms, the majority of lessons are geared towards instrumental understanding. Teachers find Instrumental mathematics easier for students to understand, since it is more efficient while teaching, and the results of student learning are immediate and more apparent. Furthermore, I believe many students will ignore most of the theory and explanations that their teacher provides while learning new math concepts, because they simply focus on the rules they need to follow in order to complete their homework. This can occur when a math class considers relational understanding as a low priority, and students are rarely asked to explain their methods or are even tested for their relational understanding. Most math tests are just drill tests to see if students can achieve the correct answers to the questions, which is why many students do not care about the reason why they are using the methods they are given. This becomes problematic because students are not fully understanding the mathematical concepts that they are applying.  For example, students memorize the formulas they are given, instead of learning how to derive them or understand the logical reason they are constructed. Teachers need to gear their teaching towards relational understandings as well, to ensure students are fully grasping the big picture. If students are only understanding math instrumentally, they can easily make mistakes if they blindly follow the steps they are given, and do not recognize slight adaptations that may need to occur for special cases.

As a future mathematics teacher, I believe in the importance of combining both instrumental and relational understanding throughout my lessons. Instrumental mathematics is highly effective in teaching students to successfully achieve solutions to their math problems, however we should supplement the fundamentals with relational mathematics to provide the highest level of understanding possible. It is sometimes difficult to assess whether a student understands a concept relationally or instrumentally based on what they write to solve a math problem. Therefore, teachers should carry out class/group discussions and ask meaningful questions to guide student learning and become familiar with how they understand. Additionally, it would be beneficial for teachers to ask explanatory questions on quizzes/tests to encourage students to think relationally.

Many teachers would agree that teaching students to understand a math concept relationally is very difficult at times. Through my past experiences as a student, my teachers would try to explain concepts relationally through the use of manipulatives. Manipulatives can be in the form of a physical object or virtual tool. Sometimes it is beneficial to use manipulatives in order to be able to show and explain concepts that seem impossible to illustrate on a 2D chalkboard or to clarify verbally. When we can explain math concepts in a variety of ways, not just through the use of rules and steps, students will begin to understand the deeper meaning of their calculations and problem solving strategies. This will expand the student’s capabilities to understand higher-level concepts, and inspire them to enjoy learning mathematics.

Link to this weeks reading:
The Psychology of Learning Mathematics - Richard R. Skemp