This chapter outlines the 2016 – 2017 school year. I spent one period each day teaching at Jefferson Middle School and the reminder of the day teaching at my high school. I taught a standard first-year Algebra course while following the district mandated curriculum and pacing guide. Jefferson follows a blended scheduling format with each class meeting for forty-five minutes three days a week and two block days with each class meeting one of the two days for ninety minutes. Daily and weekly classroom norms emerged and are detailed. I describe the process and justification for the selection of the analysis topics with emphasis placed on reasoning and problem solving, and the language and notation of mathematics.
Back to Middle School
In the fall 2016, I found myself in a place I had left in 2001, the middle school classroom. Five of my first six years of teaching were spent working with middle school students and the experience was invaluable. I had the opportunity to teach students of all abilities and motivations. During those years I worked with students who were preparing to compete in national mathematics competitions and others who struggled with basic computation.
This return trip to teaching eighth grade was much different. I had taught at my current high school since 2002, and I had certain responsibilities that I needed to maintain. My course load included both International Baccalaureate and post AP
Calculus mathematics courses. These courses required both training and education beyond the typical classroom teacher, making it critical that I be available to teach those courses at the high school.
There were also concerns that my presence at particular middle schools would be viewed as a recruiting effort. Because of these two issues, I pursued one of the middle schools that serves the same region of the city as my high school. There was the added benefit that this middle school also serves as a partner school to my high school with the International Baccalaureate program.
I negotiated with my building principal, district officials, and the principal of Jefferson Middle School to make this unique placement and schedule possible. Instead of moving to a different school for this research experience, I was granted permission to spend time at both schools. My day started at the middle school during the first period of their school day. I would then return to my high school to teach the remainder of the day.
When I received the official assignment, I was excited to find out that I would be sharing a classroom with another NOYCE Master Teacher Fellow, Phil Lafluer. Phil and I had worked together for the past seven years in both the Master Teacher Fellowship and as leaders of professional development for fellow math teachers in our district.
Jefferson Middle School
As described in Chapter 3, I was assigned to teach one class of first-year Algebra at Jefferson Middle School. Jefferson is a school of approximately eight hundred
students with an ethnically diverse population and more than seven in ten qualifying for free or reduced lunch. About one in three students are non-native English speakers.
The school week is a blend of traditional and block scheduling. On Monday, Tuesday, and Friday, students attend eight classes each lasting about forty-five minutes.
Students attend half of their classes on Wednesday in a block period format, each lasting about ninety minutes. They attend the other half of the classes on Thursday. Because my Algebra class met during the first period of the day, we met Monday, Tuesday,
Wednesday, and Friday during a usual week. Some changes in the schedule happened throughout the year; for example, during the first and last weeks of each semester, classes met for forty-five minutes every day instead of four out of five.
District Mandated Curriculum
The curriculum of the course was dictated by my school district along with a general pacing guide. Eleven units of study were required for the course as shown in Figure 4.1. These units aligned by title and content with the chapters of the textbook that I would be using with the students throughout the year, Algebra 1 by Glencoe (Carter et al., 2014). Additionally, district standard assessments were mandated to be given at the conclusion of each chapter. Students were allowed to review their scored work on these chapter tests, but they were not allowed to retain the scored tests.
Figure 4.1 – District pacing guide as it appears in curriculum material
Daily and Weekly Routines
Prior to the start of the school year, I planned a daily and weekly routine with Phil.
Having been away from the middle school classroom for fifteen years, Phil’s advice on topics such as classroom norms and lesson pacing proved invaluable. The classroom routine emerged and more or less remained the same after a few adjustments during the early part of the year. Our day went as follows:
1. Students arrived to class at the opening of the school day.
2. Morning announcements and the Pledge of Allegiance were read over the intercom.
3. Students spent the first five minutes of class working in small groups helping each other with problems from the previous night’s homework. Any problems no one could solve would be written on the white board at the front of the room.
4. While groups were working, I took attendance, checked in with any students who had been absent recently, and monitored the groups.
5. After going over homework, we moved on to the lesson for the day.
6. On Wednesdays we would use part of the additional time as preparation for the state math exam that would be given in the spring as well as a brain break at the half way point of the period.
Once established, this routine was maintained throughout the year and provided a daily reminder to the students that our classroom was collaborative and not individual.
Daily lessons included multiple points where the students worked together to analyze situations, solve problems, explain reasoning, and inquire into the mathematics that they were studying. By starting class off with the students working in small groups on
homework, conversations about mathematics were the first event of the period making further discussions easier and more natural for the small groups.
Three Topics for Analysis
The district mandated curriculum lists eleven units of study. Within these units are a myriad of potential topics for research. Any unit of first-year Algebra contains topics and sub-topics worthy of academic scrutiny. For instance, Unit 2 is the study of solving linear equations where the final topic is solving equations symbolically. I could have examined how students connected their mathematical understanding of solving one-variable equations to solving symbolic ones.
From the planned eleven units of study in the curriculum, I chose three for the analysis portion in the subsequent chapters. The three units are Unit 5: Linear
Inequalities, Unit 7: Exponents, and Unit 9: Quadratic Functions and Equations. To choose these, I examined the data I had collected throughout the school year. After reviewing my notes, student work, and a sample of the video recorded lessons, these three units emerged because they connected to both my problem of practice and research questions. They were also convenient to study because the volume of data I had collected in these units allowed for a robust analysis of my learning and student learning.
While there were many topics worthy of study, these three units presented a great opportunity given the development of the students during the course as mathematical thinkers and learners. These were chosen because they contained material that was not covered in prior course, allowing me to study student learning and my teaching without students entirely relying on prior learning.
Teaching these three units was interesting to me both mathematically and pedagogically. Mathematically, studying inequalities requires the use of nearly every concept and procedure developed prior to the unit, e.g. solving linear inequalities uses the same procedures as solving linear equations. Exponents form a distinct algebraic
structure that interacts with the algebra of linear equations, e.g. simplifying expressions with both variables and coefficients. Parabolas presented the opportunity to build a complex set of mathematical relationships that serve as a bridge between first-year Algebra and upper-level mathematics, e.g. the vertex of a parabola is a maximum or minimum value.
There were characteristics unique to each unit that made them compelling to teach. Inequalities offered me the opportunity to connect common ideas to mathematical representations when designing lessons. The result was the opportunity for students to make sense of mathematical concepts through authentic situations. Teaching properties of exponents was an opportunity for students to discover algebraic properties in a
mathematically authentic way. The complexity and volume of information learned about parabolas was challenging to teach in a way that was still approachable for students.
The eighteen students in the class had all successfully completed a Pre-Algebra course in the prior year. Three of the eighteen students began in a first-year Algebra course at the beginning of seventh grade; two were moved to Pre-Algebra within the first six weeks of the first semester, and the third was required to repeat the course. The Pre-Algebra curriculum in my district covered similar material contained within units 0 – 4 of the first-year Algebra course. Phil had warned me that some of the students would rely on their knowledge from Pre-Algebra during those chapters and not pay attention in class.
This meant that the first topic that would be new to all but one of the students was Unit 5 and made it and any subsequent chapters ideal for study of my teaching because it would not be directly influenced by what was learned in the previous year.
Reasoning and Problem Solving
Each of these units contained distinct concepts and processes of abstract
reasoning and problem solving. In Unit 5: Linear Inequalities, the students worked with equations that presented families of solutions instead of particular values solved for, e.g.
𝑥 > 5 instead of 𝑥 = 5. Unit 7: Exponents exposed students to an algebraic structure that was different from the previous ones studied, e.g. product of powers compared to
multiplicative property of equality. Unit 9: Quadratic Functions and Equations engaged students in the algebra of parabolas, which utilizes the algebra of linear equations as a part of its analysis, e.g. completing the square.
Language and Notation of Mathematics
These three units also presented the opportunity to study how students interacted with the language and notation of mathematics critical to future success in further
mathematical study. These interactions were foreign to any of their previous experiences in school or their lives outside of the classroom. Additionally, the structure of the algebra studied in these units is distinct. The algebra of linear inequalities and quadratic
functions both make use of the algebra of linear equations in their development but have different goals. The algebra of exponents is distinct from the algebra of linear equations.
Also, it does not lend itself easily to the modeling of ideas that are relevant to eighth grade students.
The three chapters that follow are an analysis of my teaching of Inequalities, Exponents, and Parabolas. My focus within these chapters are the problems and struggles that I discovered while examining the data I collected over the course of the school year. Some of these issues reflect the problems that current literature has
identified in learning algebra, while others are specific to my practice. They represent the story of my 2016-2017 school year and act as a guide to my year-long learning
experience returning to the middle school classroom after over a decade away from it.
CHAPTER 5