# Approaching Mathematics in a Less Than Mathematical Way

I recently had the opportunity to teach a lesson to a group of 6th grade students on writing algebraic expression involving addition and subtraction. The group consisted of 18 students, of which 10 were classified as English language learners two were classified as students with special needs. I started the lesson by introducing myself and the objective of the lesson that we would be addressing throughout the lesson. Then, I moved on to the pre-assessment by posting the five different algebraic expressions using the document camera and LCD projector and directing the students to translate each expression into as many different phrases as they could. The students wrote all five of the expressions on their papers and began translating. I circulated around the room and noted a few observations. Some students worked through each expression sequentially, while others provided at least one phrase for each expression before adding more phrases. Interestingly, the first phrase that every student wrote for addition expressions included the word “plus” and for subtraction expressions included the word “minus.”

After administering the pre-assessment, I modeled how to represent algebraic expressions using visual models. One of the models that I wanted to focus on was the bar diagram. I used the bar diagram to differentiate between the operations of addition and subtraction and to illustrate the commutative property of addition. Since there were 18 students, I divided them into six groups of three students each and gave each group a set of algebraic expressions to represent using a bar diagram. I circulated around the groups and engaged them with questions to further their thinking. After five minutes passed, I brought the class together and had the groups share their solutions with the rest of the class.  This provided an excellent opportunity to record their responses using the document camera and LCD projector. I had the students help me organize the responses to see the order of complexity of each response. For example, for the expression “n + 5,” we categorized “n plus 5” as a simple statement and “n increased by 5” as a complex statement. Then, we looked at the phrase “5 more than n” and contrasted it with the first two based on the order of the factors. I had each group brainstorm other ways of translating the expressions into verbal phrases. We did the same for “10 – m” and highlighted the impact of commutativity on writing verbal phrases. This led into our discussion on multi-step algebraic expressions and the syntax necessary to translate these expressions into verbal phrases.

I administered the post-assessment by posting a set of five different algebraic expressions using the document camera and LCD projector and directing the students to translate each expression into as many different phrases as they could. I changed the factors from the expressions used during the pre-assessment and gave the students several more minutes to complete this task before concluding the lesson. Compared to the pre-assessment, the students took almost double the amount of time to consider verbal phrases to represent each algebraic expression with during the post-assessment, signaling the students increased level of understanding. I also observed an increase in the number of grammatically written phrases used to represent the expressions given. Unlike the syntactic errors that the students made on the pre-assessment, they rearranged the factors in the correct syntactic order.

After reflecting on the pre- and post-assessments that I administered, I would consider changing the post-assessment. For the post-assessment, I would have the students organize a list of verbal phrases and explain their reasoning for organizing the phrases the way they did. Instead of repeating the same style of assessment as the pre-assessment, this type of assessment would require the students to apply their critical thinking skills as they consider the application, the syntax, and the logical implication of each verbal phrase. This could lead into a nice introduction for the following lesson by having the students work with a partner, reflecting on each other’s responses and critiquing their partner’s reasoning.

Throughout this whole experience, I really appreciated the students’ level of engagement. The students were eager to participate in class discussions and provide responses that facilitated the critical thinking process. They were also open to the concept of analyzing mathematics through language. As I circulated around the classroom, I heard groups engaged in collaborative discussions, using their communication skills to create solutions that made sense to everyone in the group. They shared ideas and built consensus as they brainstormed different verbal phrases to represent the algebraic expressions given. It was great to see so many students receptive to approaching mathematics in a less than mathematical way.