Blog Post #3 - Justify & Convince for Deeper Understanding

Over the last couple of weeks, we discussed the difference between Relational Understanding and Instrumental Understanding, as well as the idea of Convincing and Justifying our ideas. 

Instrumental Understanding is the idea of "rules without reason", which is when people know the rules and procedures but don't understand the WHY behind these procedures. I find that there are many mathematical procedures that I know, but I may not be able to explain the WHY. For example, I recall memorizing the quadratic formula without understanding why it works. Instrumental

Relational Understanding is the idea of knowing both WHAT to do and WHY you do it. This type of understanding requires critical thinking where you must dig deeper to figure out how certain procedures and rules connect and how they can be applied. I noticed that through university my critical thinking skills have developed, which has allowed me to deepen my understanding allowing me to gain a relational understanding of concepts that I had an instrumental understanding of in high school.

Instrumental Understanding is more of a short term type of learning, where students may memorize formulas and concepts in order to apply them on tests or during other assessments in the unit. Relational Understanding is more of a long term type of learning, where students develop a deeper understanding of concepts, which helps them extend their knowledge into further problems. This understanding is easier to remember, as you are able to connect concepts to help understand other procedures.

↓ Connecting Relational Understanding to Justification ↓

The following week we focused on justification and convincing, and I happened to lead the problem-solving session on this topic. The justification and convincing process relates to the idea of relational understanding, as the justification process requires you to question and dig into the solution that you have found. You can easily find a solution and convince yourself, but in order to convince others, you have to understand your solution and procedures. This step in problem-solving is very important as it hits the review phase in problem-solving. Understanding your work will help you further your learning and help you in future problems, as you know WHY certain conjectures did not work, and you understand WHY your procedure did work.

I thought these two topics were very important in my development as a future math teacher. I would like to focus on the idea of relational understanding in my classes, even though this can be tough. A lot of times, teachers reach towards instrumental understanding because it is easier to teach their students, especially within the limited time they have. Instrumental learning is obviously the goal, however, you can always try to push your students to go further. I want to help my students gain a deeper understanding of the concepts I teach by trying to think of applicable problems that my students will understand more. Perhaps using questions that they understand the background behind, or questions that work off of others. It is important for students to think with the knowledge they already have, as they will feel that they are ready to dig into a tougher problem! Try to remind your students of the knowledge they already have, as they will feel confident in themselves, which will help them tackle new, more challenging problems!

Here's an article that I found interesting that relates these concepts of justification and the two types of understandings: STUDENT JUSTIFICATIONS IN HIGH SCHOOL MATHEMATICS