Math Pedagogy and My Experience as a Teacher Candidate Learning Math

Using Manipulatives for a Robust Understanding to Mathematics

I am a conceptual artist. My medium: photo-based and mixd media. Though my work involves much aesthetic know how, my work is a response to or questions our socioeconomic surroundings. Why do I bring up the concept of art in this math reflection you ask? I have the curiosity of concepts. I am inquisitive in nature, have the innate desire to understand how things work and why the square root of 81 is nine.

Picture it: 1996. I was upgrading mathematics to an OAC level to apply to Interior Design and Architecture at Toronto Metropolitan University – formerly known as Ryerson. My tutor was a mechanical engineer student from UofT and our minds clashed. Whenever I raise an inquisitive question beginning with the letter ‘W’ or ‘H’ followed by a desperate “please explain,” his response was: “Because that’s the way it is.” Needless to say, after weeks of tutelage, I walked into the examining room and didn’t fail the exam. I never wrote it. My thinking process involves critically analyzing the ‘Who’, ‘What’, ‘Where’, ‘Why’, and ‘How’ of things and how they relate and are interconnected. And because I’m critical thinker, I realized that I wasn’t a failure at math. It was my teachers who failed me.

My anecdote relates to Dr. Small’s brief description of how math was taught in my time and how it should never (ever) be taught that way. There was no connection between math problems and real life, nor were there any logical explanation as to why a student must carry the one to the tenths column. We didn’t understand the math, we just “completed” it (Small, p. 3).

These manipulatives was an aid to ask students how many rattan balls would fill 1/3 of this tray. It demonstrates two ways to find the answer: counting in 2s or using base times height.

Now that the mathematics curriculum has taken a giant step away from rote learning, I would develop math pedagogy involving cognitive thinking. This will encourage students to connect prior knowledge, skills, and strategies to new challenges. To make my lessons more inclusive, and universal for many learners, I would include manipulatives to bring another dimension to the lesson. Hands on learning can help math become less abstract, more visual, and predictable. Since toddlers, learners are taught about the world through textbooks/literature. Picture books, for example, showcase farm animals’ colour, and shape. It’s up to children to speculate how the animals feel, sound, smell, or how large or small they are. To justify their speculations, toddlers are taken to a farm to make life connections from their books (Scholastic, 2012). It’s only then, concepts from textbooks are solidified and further understood. Like picture books, math textbooks introduce theoretical concepts into the minds of learners. And much like farm animals, manipulatives bring these concepts to life. “If students have physical evidence of how their thinking works, their understanding is more robust (Scholastic, 2012).” Before students learn abstract math concepts from textbooks, and to consider my students possible learning styles, I would begin lessons by using manipulatives. Beads, blocks, or shapes are concrete objects to allow students to explore mathematical problems hands-on. It can be especially helpful for diverse learners and ESL students and an excellent aid for teachers. For students who cannot articulate their math process for whatever reason, teachers can observe their thinking process based on how they use their manipulatives. Once most concepts are understood through the hands-on approach, textbooks and/or worksheets can be better utilized by both teachers and students. They reiterate their hands-on experience, confirm their reasoning, and provide further evidence of their findings using manipulatives.

Next, I would encourage active learning to promote social interaction and brainstorming to solve problems. Working in pairs or small groups is one example that demonstrates active learning (Small, p. 4). This environment promotes interactive learning where students gain a variety of strategies to solve math problems in multiple ways from one another. This method of learning also develops students’ problem-solving skills, a sense of autonomy, team building and collaboration. In addition, more value is added to their learning by observing multiple ways to answer a math question from peers. Interactive learning further develops their understanding to math concepts to a point when manipulatives are no longer available or necessary. Students would be able visualize and clarify values by referring to the ‘by-hand’ process in their mind to solve a problem (Small, 4). Based on my own experience in our current math class, these teaching methodologies encourage students to identify similarities and differences between each other’s solutions (as evident when students had to find multiple math equations using four 4s). Furthermore, it provokes students to make curricular connections between math strands and their own ideas and of their peers (Ontario College of Teachers, 2007).

References:

Government of Ontario. (2020). The Mathematical Process.

https://www.dcp.edu.gov.on.ca/en/curriculum/elementary-mathematics/context/the-mathematical-processes

Professionally Speaking. The Magazine of the Ontario College of Teachers. (2007).

Sketch of a Three-Part Lesson. https://professionallyspeaking.oct.ca/march_2010/features/lesson_study/three-part.aspx

Scholastic. (2012, Nov. 6). How Manipulatives Can Help Kids Learn: What are math

manipulatives and why is my child using them in school? https://www.scholastic.com/parents/school-success/homework-help/more-homework-help/math-manipulatives.html

Small, M. (2013). Making Math Meaningful to Canadian Students, K-8. Nelson

Education Ltd.