Real-World Problems + Group Learning = A.P. Calculus Success

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Tim Jones during a math lesson.

The Urban Assembly School for Applied Math and Science, an unscreened public middle school-high school with nearly 600 sixth through 12th graders in the South Bronx, has been taking some creative steps to bridge the achievement gap, including home visits by a team of staff members for all incoming sixth graders. This year, 16 students (out of 80 seniors) took the A.P. Calculus exam in May, and 8 received a 4 or a 5 -- the highest scores -- on the exam. "Roughly 10 percent of the entire cohort (seniors) scored high enough to get college credit from an Ivy League school," the principal, Kenneth Baum, noted. Over all, 12 of the 16 students who took the exam scored a 3 or higher, an impressive feat "even for affluent districts," Mr. Baum pointed out, let alone a school in the poorest Congressional district in the country.

And how did Urban Assembly School for Applied Math and Science achieve its success? SchoolBook asked the A.P. Calculus teacher there, Tim Jones, to explain.

A function is a relationship between two sets, a domain and a range, so that each element in the domain is paired with exactly one element in the range.

Calculus is a course about functions, so it is standard to give this definition on the first day of an A.P. class. What made my class unique was that the first day of our 2012 A.P. Calculus course was four years ago, with a group of 26 ninth graders. In fact, we are a school of sixth through 12th graders, so you could argue that we started even earlier than that.

At Applied Math and Science we don't have the luxury of assessing a few hundred seniors, choosing the top 16 who happen to have both talent and work ethic, and enrolling them in an A.P. Course. We only have 80 kids in each grade, so if we don't start early, we may not have enough kids to field a class by the time they get to 12th grade.

We are a regular unscreened public school in the South Bronx, so statistically speaking, we are not drawing students from a population headed toward calculus, at least not at a rate of 20 percent. The question for us is how to deliberately groom kids who wouldn't otherwise get to calculus, beginning as early as possible if we don't know who they are going to be yet.

In order to be successful in A.P. Calculus, kids need a certain amount of natural talent, and a whole lot of work ethic. For me, the best way to foster both is to make kids enjoy what they are learning. I have taught almost every math class at our school (honors and non-honors), and if there is one thing I think I have gotten good at, it is getting kids to want to solve difficult math problems.

The other thing A.P. Calculus kids need is a pre-12th grade curriculum that sets them up to think about math in the right way. For all students, I believe this comes from using mathematics to model real-world phenomena and solve contextualized problems.

My experience with our 2012 seniors began when they were in seventh grade. It was my first year teaching, and the honors math section was what teachers describe as “full of energy.”

Learning how to manage a class, it wasn't long before I realized that kids don't want to solve 10 of the same type of problem. Instead of boring worksheets and textbook pages, I began assigning one or two interesting multi-step problems each day: "Here's a playlist of songs; choose some to fill an 80-minute CD as full as you can." "You have $100 and a couple coupons; figure out how to buy the most stuff at a store." Or: "You're throwing a party; take this brownie recipe with fractions in it and make enough batches for all your guests."

With these problems, the discussion at the end of the day was not, “What did you get for No. 3?” It was, “Why would you use the coupon on the most expensive item?”

You don't need to be a teacher to know that kids like the second approach better, or to realize it will make them smarter.

The next thing I learned was that the sooner I stopped talking, the smoother class would run. I would explain the day's problem for a couple of minutes, and then they got working.

Inevitably there would be something in the problem that needed more clarification, but not for everyone. I almost always had kids in groups, and generally only had to re-explain myself to a few of them, and when I did, I could hit five kids at a time.

I was not always adept enough to get to students who needed help, so as a matter of necessity, I told them to ask their group mates. In an attempt to coral the energy in the class, I learned to speak loudly, move quickly without stopping, and address students immediately from across the room.

The message was clear: bell rings, you start working immediately, and you don't stop. If you don't understand, ask someone.

At that time Applied Math and Science was in its third year. There was no high school yet, which meant when I eventually taught our first Algebra 2 section, the curriculum had to be designed from scratch.

Knowing that A.P. Calculus was the goal, at least for the top students, we decided that our kids would skip Geometry and begin an Algebra 2 course in ninth grade that was centered around functions.

We organized each unit around different types of functions: linear, piecewise, quadratic, radical, rational, exponential, logarithmic. The final unit on regression tied the year together.

Students graphed functions, figured out why they looked the way they did, and learned the algebra skills necessary to manipulate the formulas to solve real-world problems. One of my favorites involved the population of a town growing according to some exponential function of time. The cost of trash removal increased according to some radical function of population. "How much will trash removal cost in 2020? In what year will the cost of trash removal reach $2 million?”

Solving this problem involved compositions of functions, a topic that can be confusing. When the functions were contextualized, understanding the difference between g(f(x)) and f(g(x)) was a lot easier -- one made sense, one didn't.

I found it hard to find problems like these in textbooks, so I spent a lot of time making up the equations myself so the numbers would work out. I knew that if kids could solve real-world problems involving compositions of functions in Algebra 2, then when they got to calculus, they would be set up well for the chain rule.

Now obviously there are times when kids need to buckle down, learn a rote algebra skill, and practice it on their own. When it's time for that, I tell my kids: “You know that math is used to solve real-world problems. But this week we gotta knock out some new skills so we have the techniques to solve some harder problems.”

I generally had enough application days in the bank to get kids to play ball on the boring independent skills worksheets. No Algebra 2 students complained that complex fractions wouldn't help them in the real world because we had just spent a week creating quadratic models to maximize profits of a business (albeit without derivatives).

Designing a school curriculum centered around real-world application has two advantages. First, it makes the abstract parts of higher-level mathematics easier to grasp. Kids who might be lost in a traditional curriculum are able to participate in deep conversations about the behavior of functions.

Second, it turns kids on to math who would otherwise find it boring. I tell my kids all the time about specific careers they could have that involved that day's mathematics. How many kids want to be financial analysts when they grow up? Most don't even know what a financial analyst is.

Showing them how to use math in high school that could land them solid jobs in their 20s motivates students to work. And many of these are the kids our school needs to keep motivated to succeed in math if we have a shot at a full calculus class of seniors.

Of the 26 kids in my ninth-grade Algebra 2 class, 16 enrolled in A.P. Calculus last fall. Twelve of them passed, 8 with 4s or 5s.

Getting 15 percent of a cohort from the South Bronx to pass the A.P. Calculus exam took me six years. The two techniques that I developed Year 1, partly out of necessity and weak classroom management, stayed with me: don't give kids boring problems, and do let them work together.

The curriculum I designed for high school kids gets them thinking about math for the right reasons. Of course, not every student is capable of passing the A.P. Calculus exam, but I am confident that at Applied Math and Science, almost every kid who is capable will also end up liking math enough to do so.