The Differentiation Between Conceptual and Procedural Understanding
Hello everyone!
This week was focused mainly on the differences between conceptual learning and procedural understanding in mathematics classrooms. Procedural understanding is the knowledge of a particular concept based off of memorizing or 'knowing' the steps of a specific problem. Where in contrast, conceptual understanding is actually knowing the meaning of a particular mathematics concept on a deeper level than just simply 'knowing' the steps. Attached are a few examples of the differences between the conceptual and procedural understandings of specific math concepts.
Knowing the differences between these types of understandings is imperative as a future math teacher. In terms of theories such as Bloom's taxonomy, procedural understanding does not truly engage students to tap into their higher-order thinking skills as learners. In my opinion, procedural understanding is on the lower end of Bloom's taxonomy, as we are telling students to 'remember' and 'understand'. Where as conceptual understand is asking students to 'analyze' and 'create', which are characteristics that truly test student understand and critical thinking skills. I can personally connect with this in my own high-school experiences. I was very fortunate to have two amazing math teachers in grade eleven and twelve. My teachers always challenged us to do higher-order thinking questions, and welcome multiple approaches and solutions to the problems and would apply scaffolding to ensure that we could maximize our higher-order thinking skills for future assessments.
The Traxoline essay that was utilized at the beginning of the essay really helps understand the conceptual and procedural understanding concepts in a context outside of math. The essay was sentences that made no sense, and then we were asked to answer questions that were clearly in the essay, but we did not understand the actual context of the answer we were giving. What I learned from this activity is that you do not actually need to understand the information being given to you to succeed in certain assessments in schools. In my second teachable (Chemistry), I sometimes see myself as a student trying to simply memorize concepts for an assessment, without truly knowing the theory behind it. As a future educator, I may potentially have to teach Chemistry and I have to challenge myself to give students deeper meaning to the subject matter of chemistry, to ensure that they are actually learning the material as opposed to simply memorizing it, as I did in high school.
Finally, as an activity, we used base ten blocks and algebra tiles to interpret multiplication and algebraic expressions. I would definitely use these in my future classroom because utilizing these types of manipulative's can intrigue hands-on and visual learners that may not get the opportunity to utilize these types of learning styles very often in a math classroom. I felt like the most challenging part of this lesson for me was the algebra tiles. It was very difficult to manipulate the tiles in such a way that I could factor expressions such as x^2+5x+6. As a teacher, I realize that a future challenge for me is to learn different ways to approach a question, so that my students have a wide range of opportunities to solve a problem using their own techniques. Something that I questioned while doing the algebra tiles in specific is that some students may just 'memorize' the steps in using the tiles, and the use of these tiles for conceptual understanding may in fact backfire and the students will not fully understand the concepts that they should know in the math classroom. To ensure that this does not happen, I must ensure that students are utilizing their higher-order thinking skills through ongoing assessment.
Overall, this lesson really made me think outside of the box and pushed me outside of my comfort zone as a teacher candidate. As someone who is so used to doing math problems their own way, it was very useful to learn about the differentiation between procedural and conceptual understanding, and using hands-on learning strategies to solve abstract algebraic problems. Although I felt challenged in this lesson, it just means that I have so much more to learn on a professional level.
Attached is a video that discusses using conceptual understanding of slopes of lines, a concept that is taught in our very own Ontario curriculum for many of the grade nine and ten courses! I have also attached a visual representation of conceptual and procedural understandings in mathematics classrooms.
I hope you guys keep coming back to my blog to learn about my experiences as a teacher candidate!
Jarrett
References
Ball, T. (2014). Conceptual Understanding of Slopes & Equations. Retrieved from https://www.youtube.com/watch?v=7e5a7n4Pv7I
Ward, J. & Williams K. (2018). Math: Cambridge Public Schools. Retrieved from https://www.cpsd.us/departments/math
This week was focused mainly on the differences between conceptual learning and procedural understanding in mathematics classrooms. Procedural understanding is the knowledge of a particular concept based off of memorizing or 'knowing' the steps of a specific problem. Where in contrast, conceptual understanding is actually knowing the meaning of a particular mathematics concept on a deeper level than just simply 'knowing' the steps. Attached are a few examples of the differences between the conceptual and procedural understandings of specific math concepts.
The Traxoline essay that was utilized at the beginning of the essay really helps understand the conceptual and procedural understanding concepts in a context outside of math. The essay was sentences that made no sense, and then we were asked to answer questions that were clearly in the essay, but we did not understand the actual context of the answer we were giving. What I learned from this activity is that you do not actually need to understand the information being given to you to succeed in certain assessments in schools. In my second teachable (Chemistry), I sometimes see myself as a student trying to simply memorize concepts for an assessment, without truly knowing the theory behind it. As a future educator, I may potentially have to teach Chemistry and I have to challenge myself to give students deeper meaning to the subject matter of chemistry, to ensure that they are actually learning the material as opposed to simply memorizing it, as I did in high school.
Finally, as an activity, we used base ten blocks and algebra tiles to interpret multiplication and algebraic expressions. I would definitely use these in my future classroom because utilizing these types of manipulative's can intrigue hands-on and visual learners that may not get the opportunity to utilize these types of learning styles very often in a math classroom. I felt like the most challenging part of this lesson for me was the algebra tiles. It was very difficult to manipulate the tiles in such a way that I could factor expressions such as x^2+5x+6. As a teacher, I realize that a future challenge for me is to learn different ways to approach a question, so that my students have a wide range of opportunities to solve a problem using their own techniques. Something that I questioned while doing the algebra tiles in specific is that some students may just 'memorize' the steps in using the tiles, and the use of these tiles for conceptual understanding may in fact backfire and the students will not fully understand the concepts that they should know in the math classroom. To ensure that this does not happen, I must ensure that students are utilizing their higher-order thinking skills through ongoing assessment.
Overall, this lesson really made me think outside of the box and pushed me outside of my comfort zone as a teacher candidate. As someone who is so used to doing math problems their own way, it was very useful to learn about the differentiation between procedural and conceptual understanding, and using hands-on learning strategies to solve abstract algebraic problems. Although I felt challenged in this lesson, it just means that I have so much more to learn on a professional level.
Attached is a video that discusses using conceptual understanding of slopes of lines, a concept that is taught in our very own Ontario curriculum for many of the grade nine and ten courses! I have also attached a visual representation of conceptual and procedural understandings in mathematics classrooms.
I hope you guys keep coming back to my blog to learn about my experiences as a teacher candidate!
Jarrett
References
Ball, T. (2014). Conceptual Understanding of Slopes & Equations. Retrieved from https://www.youtube.com/watch?v=7e5a7n4Pv7I
Ward, J. & Williams K. (2018). Math: Cambridge Public Schools. Retrieved from https://www.cpsd.us/departments/math
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