A Day in the Life of an Algebra Student: Using Concrete Examples for Algebraic Concepts

You’ve probably never thought much about how toothpaste relates to math, but Mrs. Elizabeth Williamson’s algebra classes have. Elizabeth works with students to provide concrete examples of algebraic concepts in everyday life, ensuring that the lessons Middle School math students take away from math class go beyond quantitative problem solving. For example, recently Algebra I students reviewed the Commutative Property, debating the merits of toothpaste and then water vs. water and then toothpaste. The end result — clean teeth — is the same, but the order doesn’t matter. Students also participate in multiple other ways to ensure subject matter mastery.

Below, Elizabeth outlines the teaching process for her algebra classes.

• Algebraic Concept: Commutative Property
• Attention Grabber: Toothpaste + Water = Water + Toothpaste

A real world scenario gives students context to what they are about to learn. Even if it’s a silly example about the order in which one prepares to brush one’s teeth, students begin to realize that math is all around them. They just have to learn to have eyes to see it.

“Students were actually very passionate about how the order does matter in this example; therefore, it would not be the Commutative Property. I think I grabbed their attention. Mission accomplished,” said Elizabeth.

• Whole Class Learning: We discuss as a class an example for Commutative Property and how it’s different than the Associative Property. Students call on a “friend” (aka any student in the class) to arrive at the multiple correct answers to the problem. Detailed notes are taken.
• Partner Practice: Students turn and talk to their neighbor to try the next problem. Mastery starts being displayed when students can teach it to another student. So, there are already observable qualities from partner practice that a student is mastering the material just by a simple explanation. Math is another language, so talking through the processes verbally is vital for fluency.
• Individual Mastery: It never hurts to have a little more practice because practice makes permanent. Students now navigate a problem on their own. Can they arrive at the correct answer all by themselves? That’s what homework requires of them, so let’s give them the confidence they need to be successful before class is over.
• Quick Check: After three different examples covering the algebraic concept at hand, it is time to put it all together. Do they remember what we did 20 minutes ago, 10 minutes ago, and 5 minutes ago? A three question “Quick Check” from the three different examples is given for students to know if they mastered the material before they leave class. It is not a quiz. They have beautiful notes, so they can flip back and use them just like it’s homework. Students come to check their answers.
• Self-Reflection: The more we can emphasize reflection; the better. It takes time. Based on the correctness of the Quick Check, students rate themselves on their level of understanding: novice (I don’t understand this yet.), apprentice (I can do this with some help.), master (I can do this and explain it.), and expert (I can do this and teach someone). They also categorize their mistakes: silly, computation, process, or none.

“When it’s all said and done, students have tangible evidence if they need to see me during Tutorial or not. They are confident that they need more help because time and time again they have not displayed mastery. Conversely, they are confident that they have displayed mastery of the concept at hand multiple times throughout the class period. Regardless, these mathematicians have the knowledge they need to make the next best decision to learn and grow, taking ownership of their own learning. I will always be here cheering them on in their journey as a mathematician and as a person: You are seen! You are known! You are loved! Go Team!” Elizabeth said.