On February 28th, 2019, I visited the Environmental Sciences Magnet School at Mary Hooker (ESM) to guest-teach a sixth grade math lesson on substituting to evaluate addition and subtraction problems. Prior to my visit, the students had been learning about variables and expressions for about two weeks. A day before I came to the class, the regular class teacher, Mr. Smith, notified me that he had introduced the students to substitution in addition and subtraction expressions, as well as multiplication and division expressions, and furthermore had already completed the workbook pages associated with my lesson topic. Thus, I had to adapt my initial plan so my lesson would serve primarily as a review and practice of skills rather than an initial introduction to it. The students were already very familiar with the topic which allowed me to facilitate more advanced activities than I had originally planned for with them.
The learning objectives of my lesson were as follows:
- “Write, read, and evaluate expressions in which letters stand for numbers”
- “Use variables to represent numbers and write expressions when solving a real world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specific set
These objectives are outlined in the Eureka Math Instructional Guide, which is the math curriculum used by ESM.
Most of what I had originally planned for the students to be doing was pattern finding from tables. This was to help them see the basis of the expressions they were writing and to help them understand the concept of the variable as represented by the values in the table. As I previously said, this was shifted slightly due to the students having more comfort with pattern finding in tables and expressions. Therefore, I loaded more on solving real world mathematical problems, to put this skill they had already practiced in context.
In my original lesson plan I had outlined my lesson time as follows:
This was my plan for the introduction to my lesson. It included:
- A review of the concept of a variable
- Modeling a problem similar to those in the workbook:
- Completing a table comparing student ages to my age as a class
- Facilitating a discussion to look for the patterns in the table
- Interactively writing the expressions represented by the table, modeling what each variable represents
Click here to view the table and questions that I prepared to go over during the launch.
Collaborative workbook practice
The students sit in desk clusters of 2-3, and during my observation I saw that they frequently complete group work within their seating groups. I planned to instruct the students to solve their workbook section on substituting for addition and subtraction expressions following the steps in the model problem we had completed in the launch. I also planned to circulate the classroom guiding students and answering their questions.
Below is a photo of the workbook problems the students had in their workbook.
Challenge Activity and Exit Ticket
For any students who completed the workbook effectively and quickly, I had a challenge activity planned in which students were to write their own word problem and corresponding addition and subtraction expressions. Then, I planned to select the most thoughtful of these word problems and present it to the class, asking them to create the table for the given expression and word problem as their exit ticket.
Click here to view original design for this activity.
Note that this was my initial plan for my lesson, which in a short time before my teaching had to be altered due to the students having more experience with the topic than I had anticipated. The lesson summary that follows reflects the changes I made to this plan to adapt.
To begin the lesson, I briefly reviewed the original lesson which the students had begun two days prior as a refresher. Additionally, Mr. Smith had provided me with a powerpoint of one of the problems from the student Eureka problem set corresponding to this lesson, so I called on students to model this problem on the board as well. The students were very interactive and generally attentive, reflective of their prior experience substituting to evaluate addition and subtraction expressions and with variables. It seemed to come easily to the students!
See Mr. Smith’s Problem Set PowerPoint from the Eureka Math workbook.
Below is the updated edition of my challenge activity. I had shared my original design with Mr. Smith, and he suggested most of the edits which were made to create the final version of it which I used in my lesson.
My introduction of the activity:
Nearly all of the students actively collaborated moving through the worksheet as myself and Mr. Smith circulated the room checking in with them as they went. I periodically called the class’s collective attention for a moment to review the instructions for a section at a time when it seems that several students were ready to move on, however I always emphasized that students should move at their own pace and make sure their work was complete and high quality before moving to the next step.
Here are some clips of the groups working on the challenge activity, including my “check-ins” with the group seated closest to the camera:
Rather than selecting one problem for the entire class to complete as an exit ticket, since every student had completed the challenge problem I used the remaining time to allow the students to present their situations on posters and display them for their peers. As a form of formative assessment, I collected the students’ completed challenge activities. Below are some examples of student work on this activity:
My lesson addressed equity to the very best of my abilities. While I did not ask enough questions during my launch for every student in the classroom to answer one, I deliberately never called on the same student twice, aiming to give as many students opportunities to speak as possible. Additionally, in the bulk of my time in the classroom during which I was circulating throughout the room to provide support, I paid close attention to who was asking me questions and who wasn’t. When I sensed students might be behind in understanding any parts of the instructions, I made sure to go over that section again with their group. For this I looked at the worksheets themselves as students progressed. Some students seemed to actually just be copying their peer’s work rather that understanding it for themselves, so I made a point to ask individuals questions about their group’s work to guide them to understand and be engaged in it. This helped to give less outgoing students who might be hesitant to express their misunderstanding the opportunity for scaffolding just as much as students who either understand or are more willing to ask direct questions.
The aspects of checking in on group work as well as looking at individuals within the group’s own efforts, and the collection of completed worksheets serve as formative assessments of the student’s understanding.
Credit to Adam Smith for edits to worksheets
My greatest overarching critique of my own teaching was my lack of preparedness in regards to adapting the lesson to the student’s level of knowledge on this topic. I will take into account in the future that Mr. Smith prefers to introduce any topic I’ll be teaching to his class so that their first exposure to it doesn’t come from me, as he expects me to serve rather as a review for his class to reinforce and advance what they’ve already learned.
With this taken into account, I feel my launch, which I left relatively unchanged, was likely too simple. I think it could have been more effective to additionally model a more advanced step of the basic skill being taught, as I only focused on the types of problems which the class had already been working with rather than doing a sort of activity which I would subsequently be asking the students to complete as an activity, in this case meaning practice coming up with real life situations in which there are patterns from which expressions can be written. Additionally, having reviewed the video of myself teaching, I think it would have been helpful to slow down my speaking, which was difficult as I was admittedly nervous to be teaching. My fast pace of talking led me to misspeak several times and also potentially move too fast for some students to understand.
The biggest change from my original lesson plan was the elimination of the student workbook practice problems as group work, as the students had already completed these before my workbook. My challenge activity thus became the primary focus of my lesson, leaving me with fewer activities planned. From this I learned that I should absolutely check in with Mr. Smith about a week prior to my scheduled visit, as finding out a day before did not allow sufficient time for me to effectively design new activities catered to what the students had already done.
I think the challenge activity, though probably less challenging considering this was the class’s second exposure to this lesson topic, was an effective learning activity which deepened their understanding and allowed them to apply the mathematical practice in a more advanced way. The student’s completed worksheets which I collected all had accurate pattern relationships written as real life situations and expressions representing them, indicating their successful understanding. While students certainly benefited from my answering questions and guidance whilst checking in with their groups, I didn’t sense that any students were entirely at a loss, as most of their questions for me were relatively specific about the activity.
I regret here not having further activities planned, as I don’t feel that asking the class to rewrite their problems as a poster presentation necessarily added to their learning in any form, as it was just copying what they had already written on their challenge activity sheets. However, I unfortunately had run out of activities at that point and had to fill the remainder of the lesson time somehow. In the future I will ensure that the entire duration of my lesson is going towards enhancing learning.