This video by the National Council of Teachers of Mathematics (NCTM) explains the importance of conceptual understanding. They explain that many students are taught to just memorize algorithms but when they are faced with a problem where they need to use the math they have learned, they have no idea how to use the algorithm. It is easy to give students specific questions that look similar so they can practice the algorithm and develop procedural fluency, but once an open ended, real-life problem is to be solved they are not able to pick out a solution. The video explains that this is because the students don’t have a conceptual understanding, they are never taught the “why?” behind a mathematical procedure.
Category: Research
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Jo Boaler (1998) examined a math class in two different schools in England. One school, Amber Hill, was very traditional, textbook heavy and procedural while the other, Phoenix Park, was more open ended in the math classroom. Boaler conducted an ethnographic study of these two classes over the course of 3 years and throughout her time conducted various assessments to see where they were at with their learning. Boaler explained that the demographic in both schools was basically the same, but their teaching styles were different. Phoenix Park had a philosophy in the math classroom that believed students should work through real-world problems that require the math they are learning and in doing so, they will understand the concept well. This led to Phoenix Park presenting open-ended real life math problems to its students and Boaler found that they scored better on her assessments than the students at Amber Hill. This article again, shows that working through real-life examples promotes mastery of the concepts over memorizing steps.
Reference:
Boaler, J. (1998). Open and Closed Mathematics: Student Experiences and Understandings. Journal for Research in Mathematics Education, 29(1), 41–62. https://doi.org/10.2307/749717
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Juan Garzon and Julian Bautista did a case study with 40 first year engineering students in Columbia who were taking an algebra course. They presented virtual algebra tiles for a random set of 20 of them to use and recorded their understanding of the concepts and their long-term retention compared to the other 20 students who were given a traditional lecture. They found that the students who used the virtual algebra tiles learned the concept better and long-term retention increased.
Reference:
Garzón J, Bautista J. Virtual Algebra Tiles: A pedagogical tool to teach and learn algebra through geometry. J Comput Assist Learn. 2018; 34: 876–883. https://doi.org/10.1111/jcal.12296
My last conversation for the project was with a faculty member at the University of Victoria. It was a lecture that this faculty member gave on Conceptual vs Procedural understanding that prompted this project in the first place. In my conversation with the faculty member, she explained that there is much literature on the theory that conceptual understanding (i.e. concept mastery) is better retained than procedural understanding and she directed me to Deborah Ball and Jo Boaler who are leaders in the field. She also shared a story with me that a student teacher she was supervising was trying to promote conceptual understanding before teaching the students the procedural steps but there was backlash from the students. She explained that the students just wanted to the steps to memorize to answer the questions because it was quick and easy. The issue with quick and easy though is that it is quickly and easily forgotten.
Another conversation I had was with another math teacher at my LINK2PRACTICE school about concept mastery in a Foundations of Math class. This teacher uses many real-world examples that the students can relate to, which she says is one of the best ways to help them understand what is happening. In the class I observed she was teaching the concept of a “scaling factor” and used examples of candy ring bracelets, pizzas and cone shaped popcorn bags. Each example demonstrated the scaling factor in each dimension (1-D, 2-D,3-D) which was the main objective of the lesson. After the lesson, the teacher explained that she also uses online manipulatives, simulations and other hands-on activities often because that also helps concretize concepts and helps with understanding. With a Foundations of Math class, many students have a hard time thinking abstractly and need to visualize what is going on, so she explained examples and activities are critical. The also shared many resources that she uses to find activities and manipulatives, some of which can be found in the resources section of this site.
I had a conversation with a teacher at my LINK2PRACTICE school about teaching for understanding instead of memorizing, or in other words, conceptual teaching instead of procedural. The teacher was happy to have this conversation because he focusses much of his teaching on understanding concepts over memorizing steps. The teacher does this through a couple different techniques he has developed over the years including, explaining to his students the “why?” behind mathematical procedures instead of just showing the steps. He also encourages his students to solve problems in different ways which helps them to understand a problem more deeply, leading to more retention. For example, if a student is asked to multiply binomials, he has three different ways they can do that: algebra tiles, drawing shapes similar to algebra tiles and using the FOIL method. He also explained that the three options works well for assessment because if a student can’t complete all three solutions they usually can figure out one, so they can still solve the problem. This conversation led me to investigate Algebra Tiles and other manipulatives.