To Track Or Not to Track, That is Not the Question

My school accelerates every 8th grade student by requiring them to take the 9th grade algebra Regents in June. I have always held mixed feelings about this. On one hand, does this policy do justice to those students who would rather be taking regular 8th grade math? On the other hand, don’t all the research and trends suggest that this would benefit all students by pushing them to take higher-level math? The alternative would be a tracked system where some students were accelerated while others were not, which is precisely the situation that, based on their most recent publication, the NCTM would probably take issue against.

Thanks to Michael Pershan, I ran into an article by Rochelle Gutierrez that has helped me clarify my perspective on these issues. In it, she points out that tracking is often talked about in a vacuum, as if it were a purely bureaucratic matter rather than a policy that is “embedded in cultural and political contexts, replete with good intentions, bad intentions, and messy human decision making.” She goes on to talk about requirements for students to take high-level math classes:

“Although requiring students to begin with Algebra and take three years of mathematics has a great influence on students’ ultimate mathematics attainment during high school, it was clear from interviews that such a policy did not operate in a vacuum. That is, certain factors seemed to facilitate the successful implementation…teachers believed students could realistically achieve such goals and were committed to supporting them in this endeavour. Mathematics departments which altered their programmes to fulfill state or district requirements, but did not effect changes in course content or did not provide students with the adequate support networks for the increase in demands, were less successful in advancing their students.”

In other words, policies that require students to take higher-level math may not benefit them if there is not a corresponding set of teacher attitudes, cultural practices, and support structures committed to the belief that every student can succeed under such conditions.

I think this is where I take issue with my school’s policy to accelerate all 8th grade students, and I also believe I am complicit in the problem. For years, the teachers in our math department (me included) have regularly said statements to one another along the lines of, “[So and so] should not be taking algebra. They don’t have the foundation” and “I can’t believe [so and so] has to take algebra when they got a 1 on the 7th grade exam”. The culture in our department is decidedly not aligned with the belief that all students can succeed in algebra, but rather our discourse supposes that some students are equipped with the prerequisites while others are not. To be frank, it is a culture that, up until now, I have been participating in without even realizing it.

It’s no surprise that our school’s system of universal acceleration has not been working, and our principal has stated that he’ll be making some changes next year. There are two possibilities: remove the policy of universal acceleration or work to change teacher attitudes and the support structures coming from our math department. The principal will be doing the former, with the result that there will actually be three 8th grade tracks: regular 8th grade math, 9th grade algebra, and 10th grade geometry (reserved for our school’s gifted and talented program, which is a whole other issue).

Once again, my feelings on this are mixed. On one hand, I can’t ignore the arguments against tracking. On the other hand, what good is universal acceleration if it will be treated as an empty gesture, as a means of promoting the school’s reputation while turning our teachers into cynics and setting up some of our students for failure?

I don’t actually think the ideal situation lies within the choice to universally accelerate or not. Such thinking continues to treat tracking and acceleration as purely bureaucratic matters. Instead, I think the ideal situation is one where the formal policies, regardless of their exact specifications, reflect a departmental culture that thinks carefully about what mathematical success means, what is appropriate for each student, what each student prefers, and what can be gained for each student by the decision to accelerate them or not. Michael Pershan talks about his school’s system for tracking, which I believe reflects a positive departmental culture. Moreover, I’m not sure such a cultural shift can be made by administrators. I do know that moving forward, I need to check my beliefs about who can succeed in my algebra class and what it means, and doesn’t mean, to enter 8th grade with low 7th grade math scores.

I can’t say I’ve developed a firm opinion on whether to track or not track. What I do have is a new framework for thinking about tracking–that is, to step away from a “tracking lens” and look deeper into the surrounding values and beliefs distributed across a school’s math department and how such values and beliefs shape the learning trajectories of students.

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Mathographies

I recently gave my students a Mathography assignment. The directions were to “write about yourself and your relationship to mathematics. Start off by telling a little about yourself – your interests, hobbies, family, and any other aspects of your background that you’d like to share. Talk about your experiences with mathematics since as early as you can remember. Discuss any favorite lessons or aspects of math which you enjoy (or don’t enjoy). Talk about any math that you do or that you see outside of school. Talk about how you see yourself as a learner and doer of mathematics. Most importantly, talk about why you feel the way you do about mathematics.”

Beyond this, I did not give them any structure or constraints other than the fact that it had to be at least one page long.

The day I assigned it, I shared my own Mathography. I stressed that their Mathographies did not have to resemble my own, and that it was okay to share positive and negative experiences or views about math. It was important, however, that they try to discuss the root causes or sources of their feelings and views.

In Why (Urban) Mathematics Teachers Need Political Knowledge, Rochelle Gutierrez said, “All mathematics teachers are identity workers, regardless of whether they consider themselves as such or not.” It follows then that a Mathography may be a way to assess at least one aspect of my teaching practice this year. If mathematical learning is not just the factual repertoire and techniques students develop, and if it includes their attitudes about math, their disposition and engagement with math, and their sense of how math is defined, what doing mathematics looks like, and who can do mathematics, then a Mathography that asks students to reflect on their mathematical identities may be a way for me to know how well (or poorly) I’ve supported (or undermined) these aspects of their mathematical learning.

In the list below, I’ve shared snippets from my students’ Mathographies. A couple of comments. First, by way of background, I teach at a public midle school in eastern Queens. Technically, it’s part of the NYC DOE, but it’s also in one of the wealthiest districts in NYC. The school is racially diverse and there are some students from lower socioeconomic backgrounds, but the school mostly draws on students from the surrounding middle class and upper middle class communities. The snippets come from 6th graders in my school’s gifted and talented program. This year, they took Prealgebra, and next year they’ll be taking the 9th grade algebra Regents exam.

I chose passages that stood out in some way. Maybe I thought it was funny. Maybe what a student said cohered with my philosophies on teaching math. Maybe it didn’t. I realize my particular selection likely reflects my biases about math and math education and maybe also some desire to “look good”. But I did try to choose a wide range of thoughts and experiences, some reflecting well on my teaching and some perhaps reflecting poorly. I also acknowledge that there’s something fundamentally lacking by leaving out the context surrounding each passage. Then again, I was careful to include only brief passages so as not to include anything that could be revealing or identifying about my students.

Mathography Snippets

“I see myself as a limited mathematician who can do greater things. I see myself as a person who wants to do more; wants to learn more…People who think math is hard; they quit and give up easily. But to others, math can be a hobby, a passion, a love for a subject. And that’s me: a person who doesn’t give up, a person who challenges himself to do his best.”
“My classmates seemed like everything was easy, like they’ve done this a million times so I felt alone…Knowing that I was not smart when I came to math it put me down not that people were better than me but it was just that I felt…dumb. Then we had our first test which I didn’t do really well on which made me even more sad and everyone was doing really well and again I felt alone. Not only that but it put a lot of stress on me.”
“I remember in third grade, when I first was taught to do long division, I just sat for over an hour just dividing any numbers that popped into my head.”
“I can’t wait to learn the math skills that ninth graders learn. I can impress my friends with my exceeded knowledge in the field of math, and even use it to help overcome the challenges of daily life.”
“I wish that I tried harder in class. I used to like math because it was easy so I’m guessing that the reason that I don’t like math anymore is because I’m bad at it. But I’ll slowly try and improve in math and maybe one day I’ll like math again.”
“I thought that math was (excuse me) dumb, because I didn’t understand why math was so important in life and why a person should be learning math. Everything came back down to counting, I said to myself, and I knew how to count…It was my fatal flaw that led me to like math. I think it was because that I wanted to be just like my sister, with her walls covered with paintings and pastel drawings, plus all the awards, medals, and certificates that cluttered her bookcases.”
“Math is not only about the patterns and numbers and symbols and signs, but about the way you interpret the numbers. Math is like a ladder. If you are missing one step, you can’t go higher. So, if you miss something, you can’t go on.”
“Ms. […], one of them, was so interested in both math and art, and brought math to life with her art. Art was my talent, and I was exhilarated to beĀ  applying art to math.”
“What I don’t like to do are word problems and decimals. I don’t hate math, but I don’t love math either, so I just like math. Math is a part of my life because my family is conflicted in it…If there wasn’t math in my life, then who would I be?”
“Mr. Peralta, our new math teacher not only taught, but encouraged us to use creative thinking and create our own formulas…Instead of sticking to the curriculum, he encourages us to make our own. There isn’t just one way to do math, but multiple ways, and none of them are incorrect. In math class, mistakes don’t matter, but but rather benefit you, for they allow you to learn and grow.”
“I remember the summer before second grade my grandfather would give my sister and I about ten problems everyday to solve. I never really understood what I was doing…I recall crying because he had to explain the same thing over and over, but I still couldn’t understand.”
“My mom was an after school teacher and wanted to teach me about numbers. However, I threw a big tantrum that I didn’t want to study and ended up hurting myself from all the running away and kicking. That was just the beginning though.”
“Math can be really helpful. It could get you the job of your dreams. Being good in math looks good on your resume.”
“Most people have a strong opinion about math but I don’t. Either they like it or not. For me, it depends on the teacher.”
“At my Sunday school for math, the teachers don’t speak English well and have a Russian accent so it’s hard to understand and they teach the ‘Russian way’.”
“I remember that when I was testing to get into the G&T program, the tester asked me which shape was the circle and I chose the triangle. A TRIANGLE…Another time in pre-K, the teacher mad me spell mop and I spelled it wrong. Then she made everyone spell it slowly and loudly RIGHT IN MY FACE! After that experience, I never felt very confident about my spelling skills.”
“One of my favorite lessons of math is when you get to learn to think critically and solve certain problems outside the box…Also, it is very satisfying when you solve a hard problem on your own. I like math because it is sort of an open free world that you can use any method or idea you want, as long as you can solve the problem and get the right answer…Why I feel the way I do about mathematics, is probably because of my mom, who influences me a lot.”
“A math topic that I really enjoy is our current topic – multiplying binomials (forwards and backwards problems)…The only topics that I don’t like are State Test Prep topics. I don’t find it fun because we have already learned everything and I find it boring to do the same thing over and over.”
“It is the one subject where there is a strict right and wrong. No ‘good’ answers, only right and wrong answers.”
“I started to gain momentum and by the time I made it to 100 by the end of 5 minutes, my mom hugged me and said, ‘Oh my gosh! You are correct!’ and that’s when I knew math was my thing.”
“How I see myself as a learner and doer of math is not just memorizing formulas and facts, but as in finding patterns, solving puzzles, and doing something that will benefit me in the future.”
“Math, so far, was just something that I didn’t want to be beaten at. Not because I was amazingly good at it, but because of my pride…Because of my realization of my identity as a math student, the burning flame inside me has (temporarily) died down, and I can appreciate mathematics for what it really is, not as a field of study where I am compelled to compete with others concerning my skill at it.”
I began middle school at […]. I’d never before in my life been taught such material in such a way. I actually began enjoying math. I began to see and be awed by the complexity and simplicity that occurred at the same time in mathematics.”
“I got really bored at the idea of ‘learning’ math, and I say ‘learning’ because I didn’t actually learn much. I didn’t get the idea or the concept of doing and ‘learning’ something I had learned by myself. I’d also had some more common sense and kind of figured out everything on my own. I didn’t understand why I had to do something I already knew and just keep doing examples. According to my previous teachers, it was to make sure we understood wheat we were learning, to me we weren’t learning anything.”
“The way I view math is…complicated to say the least of it. It’s kind of a love hate relationship. I can really love the challenge, but I also partially have anger issues so I get frustrated kind of…A LOT.”
When I was in third grade a thing I had trouble with was the times tables. Whenever we were waiting on line, the teacher would always ask us times tables. She expected us to answer in a snap. However, I was always too slow.”
“I actually have to work now, and think about whether or not the answer I got is right or makes sense. Basically, the reason I don’t love math is because I’m a very lazy person and I don’t like to work.”
“One other detail that I do not like about math is the amount of test prep that is created by the NYS exam.”
“If I learned one thing from mathematics in my entire math life, it is probably the fact that mathematics is easier to people who enjoy it, not to the people who are forced to do it.”
“I’ve always disliked ‘school math.” My earliest memories in math were about my mom who taught me a simple, but challenging view of numbers. Those were my happiest memories of math. I was challenged, but happy to spend time with my family. I felt that that was when I progressed the fastest.”
“I feel the way I do about mathematics because doing math required you to think. If you do not think, there is no point of working in the first place.”
“I learn Korean math from a Korean teacher by Skype. Korean math is basically math but just a bit more challenging.”
“Math has always been my favorite subject. I don’t really know why, but it was. My sister excelled in it, so I guess I wanted to be like her…My parents always compared me to her…Well, when she was my age, nothing was this competitive, the work wasn’t as hard, nor did she do 13 extracurricular activities.”
“This was probably why I loved the cubes so much. I would fiddle and fidget with them when I was done with the classwork involving them. It made me think about how that one tiny thing the size of the tip of a thumb could turn into something the size of my head and then even bigger by putting multiple of them together.”
“The thing I like about math is how it’s so flexible and that it can be used to solve some of your everyday problems. What I don’t like about math is the way math is presented, because it takes away the creativity and fun of math, and turns math into a thing that most kids dislike.”
“I think I hate it because I’m bad at it, but my math average went from an 87% to a 93%, so I say I’m good.”
“Growing up as an Asian-American in an Asian family and household, I was always involved with math and learning outside of schools such as academies, my whole life.”
“As an (very bad) analogy, math would be a daily skills, like getting ready for school in the morning, while “hands-on” learning would be like taking a field trip outside of school.”
“Sometimes I wonder how people know how to solve questions so quickly while I’m struggling with them. It sometimes makes me feel annoyed and jealous that I’m so slow…I have realized that to solve my problems involving math, I should just keep trying and take my time if it takes me a lot of time to figure things out.”
“To this day, I still picture the Abacus in my head when doing mental math like multiplying two-digit numbers by one-digit numbers or dividing two or 3-digit numbers by one-digit numbers.”
“In certain topics of math, I am often fascinated and amazed at how everything fits together and makes sense. I usually learn well in math when I see someone else solve a problem and use it as a model that I can use to solve the same sorts of problems.”
“Math is sort of fun outside of the curriculum, which sometimes a student must do in order to get the most out of his or her educational life. Math is not the same when you view it from a different perspective, which is why math is sort of enjoyable to certain students and not to others, who see it as set-in-stone numbers and formulas, and nothing more.”
“Some subjects require you to memorize things, then forget about them after a test–but, math is different. If you understand a topic, you can remember it for the rest of your life.”
“I didn’t really understand most strategies used for visualization. Of course, it’s helpful for students to visualize problems. But, when forced onto students, it can confuse more than contribute.”
“Extremely long word problems that are used to trick you don’t help comprehension. By conning a student into not understanding a word problem doesn’t make them better at the subject. I don’t find it difficult, just tedious.”