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

Secondary School Curriculum and the Perception of Mathematics (Week 2 Reflection)

The Mathematics grade 11/12 Ontario Curriculum Document’s Introduction discusses the importance of student’s learning not only mathematical facts and procedures, but achieving a true understanding of the concepts they are learning and applying. I believe teachers should make a point of explaining real world applications and reasons for learning a mathematical concept being taught, so that students can relate mathematics to their daily lives, other subjects and future careers, thus increasing their motivation. Deepening students’ understanding contributes to developing the ability to use learning from one area of mathematics to understand another area of study and make connections. Further, it is important that teachers encourage their students to justify their solutions, communicate them orally or in writing, and reflect on their own solutions, to ensure they solved the problem in the most accurate and efficient way. This will reinforce that students are seeing the full picture of mathematics.

          Another important topic that has surfaced in teaching is differentiated learning. In mathematics, Problem Solving is central, and it is important that students are able to effectively problem solve. I believe that when teaching a class how to solve a problem, teachers must keep in mind the many different learning styles of their students and try to accommodate their learning needs, through differentiated learning techniques. This can be done by adding a visual diagram or using manipulatives for the hands-on and visual learners, structuring the problem in an organized manner that can be easily followed (i.e. Using Step 1,2,3 etc.), working backwards, using technology, allowing students to work/discuss in groups, and showing different methods and approaches to solve the problem. An exceptional teacher would provide many different options to students so that they can become more engaged, willing to learn, and expand their thinking processes. It is important to build new knowledge form prior knowledge. In order to do this, teachers should gage the knowledge their students currently have and then adjust their lesson plans accordingly.

          It is apparent from the article, ‘Hollywood’s Math Problem’, that society has a negative view of mathematics, which is understandable due to the fact that it can be challenging and requires plenty of practice and abstract thinking. The promotion of a negative media opinion towards mathematics may be contributing to students’ disliking math and influencing their poor attitude, which is setting them up to fail before they begin. However, by providing students with a positive learning experience, teachers may be able to change the negative stigma towards math.

          Personally, I have had many positive experiences while learning math. These experiences are what drive my passion for the subject and inspired me to become a math teacher. I want to be able to teach students to enjoy math and hopefully develop the same passion for the subject that I have.  As a student, I enjoyed solving math problems when I understood the material, however like any other student, I quickly became frustrated when I did not understand a math topic. Therefore it is very important that teachers effectively deliver subject matter. However, the onus is not just on the teacher, the students need to be responsible and contribute to their own learning as well. The students will not be successful if they solely rely on the teacher to do everything for them, they need to be actively involved in their learning. A good math student will ask questions to gain a better understanding, attempt each homework question, investigate online problem solving techniques, and come to class with a positive attitude and an open mind. Practice makes perfect, thus students need to put in the effort when learning math.


       

Links to this weeks readings:

Gr.11/12 Ontario Mathematics Curriculum

Hollywood's Math Problem 

Introduction

Hi my name is Tanya. I live in Niagara Falls and I am an only child. My background is Italian and I love spending time with my close-knit family. My favourite extracurricular activity is playing sports, and my most preferred sports are basketball and soccer, which I have been playing since I was young. I have played travel basketball, soccer, and trained in Taekwondo for a number of years, but now I simply play co-Ed soccer for fun.

 I currently attend Brock University and I am in the Concurrent Education program. My first teachable is Math and my second teachable is Chemistry at the Intermediate/Senior level. I am looking forward to becoming a teacher and hope to have the opportunity to work in the Niagara Region. I knew I wanted to be a teacher ever since I was young and I am so close to achieving this goal.  I want to pursue a teaching career because I enjoy working with and being surrounded by people, I want to make a positive difference in students’ future, and I really enjoy helping people understand concepts that they are struggling with.  I look forward to the opportunity to encourage and motivate students to discover and build on their strengths and realize their full potential.

This blog will consist of weekly posts based on my reflections of readings and material taught in the course EDBE 8F83 with the intention of learning more about myself and growing as a Mathematics learner.

Through this course I am hoping to develop the skills necessary to become a great math teacher and I am eager to learn about different teaching strategies that can be used to teach a variety of learners. I also want to gain a good understanding of what is contained in the Ontario Secondary School Mathematics Curriculum and how to conduct an engaging lesson plan based on the material.

Thanks for reading, and enjoy!