I’ve worked with thousands of students. Most of the names and many of the faces have faded from memory. But there are a few who are memorable because they challenged my identity as a math teacher and my purpose as an educator. All of them were at the heart of a dilemma, at a crossroad between my mission and my mandates. They are memorable students because they taught me a thing or two about myself, about my integrity, about the potential impact of my choices and the limitations of my influence.
Hector is one of those students, and I would like to tell you about him. But first,
There are few things I detest teaching more than quadratic equations. The amount of content is vast. The standards ask students to create tables and graph parabolas by hand, find x-intercepts by factoring and the vertex by completing the square, apply algebraic understanding to
real-world pseudo-context applications, derive the quadratic formula, and fluently navigate tables, graphs, equations, and verbal descriptions of quadratic relationships.
This content often requires 8-10 weeks (or longer) of instruction and assessment in Algebra 1 and nearly an equivalent amount of time in Algebra 2. I don’t think it’s time well spent. There’s a long list of topics that would be more worthy of the time (statistics and media literacy, voting systems and democracy, proportional reasoning and wealth inequality, conceptual calculus), but that’s another blog post.
I’m here to talk about why teaching quadratic equations (specifically) and Algebra 2 (generally) represent one of the most frustrating professional obligations as a math teacher. But back to Hector for a moment.
Hector (fall term)
Hector was affable and humorous and equipped with a charming smile. He was also a spirited person with a kind heart, a charismatic entertainer who would lift spirits when he walked in our classrooms. He was a kid you wanted to root for.
Hector was not, however, a good math student. He was a senior in my class, taking Algebra 2 for the second time. This was before the adoption of the CCSS, and Algebra 2 comprised of California State Standards that were a hodgepodge of content cobbled together without much attention to coherence. Algebra 2 credit was not only a requirement to get in to college, it was also a requirement for Hector to earn his high school diploma. (The school was an independent, college-prep, public charter school that had higher graduation requirements. All students had to be accepted to college in order to graduate.)
For him, math was a series of tricks, a disconnected jumble of isolated skills, and he was not able to build any enduring understanding on his weak math foundation. He crammed for tests, made use of extra help, and would retake exams for passing grades. (My policy was unlimited extra help and an open time window to retake exams as many times as necessary for credit.) While he would pass tests (eventually), I sure as hell knew he didn’t really know the content. But he managed to jump through the hoops I had set up during his fall term and did enough to earn credit. If anything, I had helped him learn how to navigate bureaucratic procedures in the education system. That’s something I guess.
The spring term was a different story for Hector.
My Algebra 2 Class
When I taught Algebra 2, I modified much of the curriculum. I skipped all content regarding conic sections because they were not heavily assessed on the California State Test or the Entry Level Mathematics (ELM) exam (or an essential component for Pre-Calculus and AP Calculus). I nixed the statistics and probability content for the same reasons. I focused a bulk of the work on linear, exponential, rational, quadratic, and higher-order polynomials functions. I focused on factoring and solving algebraic equations (and, yes, I focused heavily on solving quadratic equations). After state testing, I carved out 4 weeks for a financial literacy unit that leveraged exponential functions as a gateway to understanding compound interest. Students investigated how credit cards work, how to establish a good credit history, how payday lending businesses exploit low-income earners, and how to make financing decisions when it comes to buying depreciating assets like new and used cars. Students completed the unit with an individual project that asked them: How much money do you need to invest each year if you want to retire as a millionaire?
This course structure was my grand compromise. It was my attempt to set students up for access to college level math (more depth and fluency in fewer topics that were heavily assessed) while also providing an opportunity for them to learn something useful and meaningful (and for me to teach something I was passionate to teach).
But the course still felt like a turd sandwich.
There was so much boring content to slog through. But it was a graduation requirement, and it was on the ELM, so we had to march onward. I made it the best I could.
(A Quick Note about the ELM: The ELM was a placement exam used by community colleges and public universities to determine math proficiency and to place incoming freshman in appropriate math classes. Failing the ELM meant being placed in a remedial (high school level) math classroom. Sometimes incoming students had to take 3 math classes (essentially repeating high school math) before gaining access to courses that led to college credit and progress toward a diploma. The wasted time, money, and human resources was dispiriting, not to mention that most remedial classes are disproportionally comprised of students from disadvantaged backgrounds. Fortunately, things seem to be changing and less focus is being placed on the ELM.)
Hector (spring term)
Hector fell behind in math during the spring term. He had narrowly averted failure in the fall, and he was banking on a similar outcome in the spring. With a few weeks left in the term, I started pressuring him and other failing students to come to extra help and work to retake their exams, reminding them the consequences of failure (no high school diploma and no college in the fall).
By the time Hector was ready to learn and retake, the pile of content was too high come June. Hector had completed most of the content work and his financial literacy project, but he couldn’t solve quadratic equations or sketch graphs of parabolas. The deadline for senior grades came and he couldn’t do it. I told him I had to submit an “F.” His face fell. He knew what that meant. He muttered a “Thank you,” shook my hand, and left.
I felt like shit for failing him. It didn’t seem right. It never did when students failed Algebra 2. It was a bullshit course, an arbitrary gatekeeper in the machinery of math education that seems to serve the function of making mathematics a miserable subject for so many people under the guise of “college preparedness” (whatever the hell that means).
I resented the system for entrusting me the power to determine if a student graduates or not while also making me feel so powerless in the face of Hector’s reality (and the reality of so many students before and after him). Furthermore, Hector was a product of our teaching, the result of four years of our efforts (and his) to learn how to reason mathematically. What burden of blame did we (his teachers) bear in Hector’s failure?
“But Chase, teacher’s don’t give Fs; students earn them.” Bullshit. Grades are a terrible way to measure anything. They are a shallow language that can’t describe the depth and quality of the growth we value. But they are a useful device. How else would we calculate GPA?
Regardless, I hid behind my syllabus and grading policy under the guise of fairness to all. I made a choice and the case was closed.
Until it wasn’t.
The Assistant Principal wasn’t going to be able to finalize senior grades until the following afternoon and offered Hector an extension. Hector showed up at my classroom doorstep again and pleaded for more help. If it seems more than slightly unprofessional for the AP to undercut my enforcement of school policy without a conversation with me first, you have a fair point. I was more dismayed that I had waded through an unpleasant and dispiriting quagmire the day before, and here I was staring at the prospect of wading into the same quagmire again. But screw it. Like I said, Hector was a kid you wanted to root for.
Short story long, we spent a few hours breaking down quadratic equations and parabolas into isolated DOK1 skills and slowly checked them off to create an illusion of progress. He answered released CST questions. He’d get them about half right, I’d tell him to check his answers in the graphing calculator, he’d identify mistakes and explain them. To call it a watered down version was an understatement. If Algebra 2 was meant to be a flavorful fruit salad, I was teaching a version that was more akin to flavored soda water.
But this time, I passed him. He knew just as little Algebra 2 as he did the day before. He was still going to have to take the ELM and remedial math classes at the local community college. I just felt like passing Hector was the best thing for Hector. He brought to the surface all the frustrations I had about teaching math, but I wanted to use what little power I had to defend Hector’s right to have access to opportunities than defend our practice of making fluency in Algebra 2 a college requirement.
I wonder if what I did was fair to the other handful of seniors who had failed Algebra 2 that year. Was it their fault they never came to extra help? Or was it family, work, or some other legitimate reason? Did I not root for them hard enough? What biases was I blind to?
To this day, I still wonder if what I did was right. And I taught Hector nine years ago.
Why This Still Matters
I currently support the math teachers at a pilot school in LAUSD. Part of my work is to help the 9th grade teachers navigate CPM curriculum in a way that their students can experience success and be challenged. A few of the students are at grade level, but most never matriculated from middle school and their math ability shows it. Many are impacted by trauma in ways that have delayed their emotional growth. Several don’t have a home. Even when they act their worst, I pause and remind myself that they are all Hectors in some way. They are all worth rooting for.
We are about to start covering quadratic equations and graphing parabolas. It’s still the same boring crap that it was before CCSS. And I find myself crafting lessons using CPM problems and other engaging lesson resources like Desmos activities and 3-Act Math tasks. But designing lessons on this content still seems so contrived, so unnecessary when we are step back and look at what the students need to learn and how they need to grow as people.
So, my disgust for teaching quadratic equations (and other unnecessary math topics) continues to this day. Yet, I feel powerless to change anything in that reality. It’s the Hector situation all over again, but this time as a coach. And I’m still not sure what’s the most just and fair course of action for the students, the teachers, and the school.
What do we do when the needs of our students conflict with the mandates of our profession?
I share this dilemma because I think it’s important that we do so as educators. Too often, we privatize our experiences in isolated silos, unwilling to expose our sense of conflict and turmoil as we navigate the messy dilemmas inherent in our work. Failure seems safer when no one is watching. We need to have the courage to make failure cheap and make it public so that learn from each other.
I invite your thoughts and perspectives. Who are your Hectors? And what did they teach you about the professional dilemmas you face in your work? What dilemmas are currently pulling at you?
One More Thing
I followed up with Hector a few months ago on Facebook when I knew I wanted to write this piece. He remembers the moment as vividly as I do. He says the experience taught him to make the most of his opportunities. He works for a major airline and is happy and well.