When Chelsea Castilloadame left the Navy Medical Corps after five years to pursue her longtime plan of becoming a doctor, she knew her transition back to civilian life was going to be tough. But she wasn’t prepared to feel so unprepared for — of all things — math class.
“I would leave classes and exams literally in tears,” Castilloadame, 24, said. “I went to the bathroom and I cried after almost every math exam. It was very humbling to go into the professor and say, ‘I am so frustrated. I’m bawling my eyes out and I know this material.’ ”
She failed precalculus and decided to switch schools, from a university in Nevada to San Diego State University, rather than repeat and risk failing again.
Castilloadame is one of many students who experience mathematics as a roadblock to other fields — such as science, technology and engineering. In fact, about 50 percent of students don’t pass college algebra with a grade of C or above, as noted in a recent report, “Common Vision,” from the Mathematical Association of America (MAA). The report called Americans’ struggle with math “the most significant barrier” to finishing a degree in both STEM and non-STEM fields. In the worst-case scenarios, students can get stuck in remedial classes and fall so far behind that they drop out of college all together.
“There’s going to be a need for a million more STEM graduates in the next decade than we’re currently producing,” said Chris Rasmussen, professor of mathematics education at San Diego State. “I think people started looking at their courses and saying, ‘We can probably do better.’ ”
San Diego State has joined a couple of dozen math departments across the country that are rethinking the way they teach math.
In teaching assistant Natalie Nowicki’s math class at San Diego State, for example, step one is to reconfigure the classroom.
It’s just a few minutes before her Calculus II class begins. Desks stand in perfect rows, facing the wide, multitiered whiteboard. Within five minutes, students have re-formed the rows into 10 small groups — so that the desks face each other, not Nowicki or the board.
Today the students are learning about something called fractals and how to mathematically define dimension. Easy, right? A handout shows five patterned triangles with captions like n = 0, n = 1, n = 2, n = 3, n = 4 underneath. Phrases like “ad infinitum,” “Sierpinski Triangle” and “magic property” dot the page.
The students need to determine the dimension of standard objects at different scales — a point, line, square and cube — and then apply the same method to the fractal Sierpinski triangle.
“Can I hear your reasoning?” Nowicki asks the groups.
The students mumble among themselves, peeking at each other’s notes.
“Does it have to be a perfect box? Or can it just be a rectangle?” One student asks his group mate.
She answers him with her own question: “Are you allowed to do cubes or does it have to be, like, flat squares?”
Nowicki waits about a minute and says, “You can’t be wrong.”
A few years ago, administrators at San Diego State noticed high “D-F-W” (grades D and F, and withdraw) rates — 35 to 50 percent — for math courses, according to Michael O’Sullivan, chair of the math and statistics department. In 2014, the newly elected O’Sullivan, along with frustrated faculty, decided to overhaul the program. That December they formed a calculus task force. They did away with an online version of precalculus, extended class hours and established a learning center and “learning communities” to offer support for students who typically struggle in math and science classes.
And, like some of their peers at other universities, they introduced a major change in their approach: Students would continue to attend lectures, but now they would also attend smaller sessions wired for student-centered active learning, in which the focus would be on concept-based discussion — not just absorbing information. The smaller sessions would be required and would correlate to material taught in the lectures.
The changes at San Diego State and in other colleges’ math classes are similar to components of the Common Core Standards for Mathematical Practice — the movement driving change at the K-12 level in the 40-plus states that have adopted it — which also place an emphasis on asking students to communicate their reasoning and construct arguments. Educators say that colleges aren’t following Common Core, but acknowledge that current math reforms in both higher education and K-12 are based on research showing that students can thrive in “engaging” learning environments.
In Nowicki’s classroom, for example, almost every exercise starts with a prompt like “talk to your group mates” and ends with “explain yourself.”
She lays out two options on the whiteboard to answer one particularly tricky question.
“Do either of these approaches speak to you?” she asks the students.
Speak to me? This is math class?
Active versus passive
The active-learning sessions are noisy. Tables move. Teachers scuttle around. Whiteboards roll and projector screens move up and down. Students talk over each other and with each other and — exceptionally — seem less interested in their iPhones.
“We’re getting into the ‘why’ factors. Why do we do this? Why do we want to know? Why do we care?” Castilloadame said. Such questions are not always easy if you’re just used to “solving for x.”
According to Rasmussen, “There’s deep engagement in mathematics by students and lots of peer-to-peer interaction. … Students are building up ideas for themselves rather than passively sitting back and listening to someone else do the thinking for them.”
Other schools are making similar changes. In 2012, Judy Walker’s department of mathematics, at the University of Nebraska-Lincoln, extended class time, trained teaching assistants with active-learning techniques and, importantly, got tables conducive to students working together.
Initially, Walker said, the student pass rate dropped from 62 to 59 percent. But then, after some tweaks and polishing, it went up to 80 percent in fall 2013 and has not dipped since.
Educators at San Diego State University mostly based their initiative on a nationwide 2010 survey (results published in 2015) by Rasmussen and his co-investigator, David Bressoud, professor of mathematics at Macalester College.
In 2015, they wrote, “Our survey revealed that Calculus I, as taught in our colleges and universities, is extremely efficient at lowering student confidence, enjoyment of mathematics, and desire to continue in a field that requires further mathematics.”
Yet, Bressoud wrote, in 2010, almost two-thirds of the 700 calculus instructors surveyed felt that lectures — the status quo — were the best way to teach students.
“If we were talking about medicine it would be illegal to continue doing that kind of method of instruction. It’s unethical to continue with straight lecture,” said Walker. “Now, interactive lecture, that’s a completely different ball game.”
In the largest study of its kind, Scott Freeman, a biology instructor at the University of Washington, set out to investigate whether active learning or lecture “maximizes learning.” He and co-investigators found that, on average, 66 percent of students pass a lecture-based undergraduate STEM course, while almost 80 percent pass an active-learning-based course. If they had been studying a medical issue, the authors explained, “the control condition” — in this case lectures — “might be discontinued because the treatment being tested was clearly more beneficial.”
“When you teach, you are trying to create an environment where all students can be successful — you make a high bar, but give students all the tools they need to rise and get over it. When you select (or ‘weed’), you don’t worry about underprepared or struggling students, because you see your job as differentiating the star students from the others. You throw students into the deep end, and expect them to learn how to swim,” Freeman said in an email. “We’re teaching, not just selecting.”
In 2015, after Rasmussen had visited five universities (of varying sizes and types) that run successful calculus teaching models, he and Bressoud produced what amounts to a seven-step how-to-succeed-at-teaching-calculus regimen.
As of this year, San Diego State has implemented all seven pieces, said Michael O’Sullivan. It is too soon to know long-term results, of course, but for now the professors are happy that this semester’s Calculus II midterm grades increased by five to eight percent compared to previous years, according to Ricardo Carretero, professor of applied mathematics.
Rasmussen and Bressoud’s ideas seem to be gaining ground. In a recent follow-up survey, their research revealed that 44 percent of institutions (that offer advanced mathematics degrees) surveyed consider active-learning techniques “very important” in precalculus through calculus courses.
However, similar teaching changes — related to the Common Core — in K-12 schools have been met with criticism and concern. Some experts, like Wilfried Schmid, professor of mathematics at Harvard University, caution against thinking that any one technique is the answer.
“Yes, we should teach calculus not totally by old-fashioned lecturing,” he said. “On the other hand, I also think that — at least in many cases — the switch to student-centered practices has gone too far.”
Rather, he added, solid teaching and understanding that student needs are unique at every institution contribute to successful outcomes.
A history of reform
This is not higher education following the Common Core, Judy Walker, of the University of Nebraska-Lincoln explained, but it’s also not a coincidence.
The Common Core and college math reform movements are connected by their advocates and by shared concerns that math education and performance in the U.S. have been stagnant for decades and need to be refreshed if the country is going to stay competitive.
William McCallum, professor of mathematics at the University of Arizona, for example, was one of the three lead writers of the Common Core and a founding member of the Harvard Calculus Consortium.
“In the ’90s [we] had the calculus reform movement that was all about teaching calculus in a way that kids would actually understand the concepts [and] be able to use the mathematics,” McCallum said.
Calculus reform spurred significant progress at some universities — notably at the University of Michigan, which several experts credit with leading the way.
Yet, two decades later, in 2012, the United States President’s Council of Advisors on Science and Technology (PCAST) released a widely influential report saying that students — especially those underrepresented in STEM fields, such as women and minorities — are still not getting what they need to succeed.
“It did create a lot of alarm within the mathematics community,” O’Sullivan said.
In 2014, prompted by the PCAST report, Karen Saxe, professor of mathematics at Macalester College, and Linda Braddy, deputy executive director of the Mathematical Association of America, set out to investigate the state of undergraduate math education by looking at guides released by five of the big professional mathematical societies. Their investigations led to the MAA’s “Common Vision” report, released in January. In it they highlight specific ways to improve undergraduate mathematics. They also note that the majority of students entering community colleges need remedial math class — and because most don’t pass the class, they never complete a degree.
There are deeply rooted issues at play that need to be addressed. “From second grade on, math is used as a sort of benchmark for all students, no matter what they’re going to do. By the time people get to college there’s these huge disparities in their backgrounds,” Saxe said. “Other subjects don’t really have that in the same way. You get to college French class and there’s not this decade of built-up failure and animosity and anxiety. It’s just not there.”
Back to the basics
It’s 9 a.m. on a frigid winter morning but the students at this City University of New York (CUNY) Start class are nonetheless already engaged by their math lesson — they don’t really have a choice. Their teacher, Gregory Fein, bounces around the room in a bright yellow collared shirt, questioning every answer with “why?”
Today the problem is about field crickets and whether they chirp faster or slower based on the temperature. A function of c = 4t-150 is meant to measure the number of chirps (c) per minute, depending on the temperature (t).
“Could you say something about the crickets and the temperature in a sentence?” Fein asks.
A young woman in boxy glasses mumbles something about the amount of chirps and simultaneously giggles a little.
“See if you can explain it in more detail,” Fein prompts before calling on another student for more input.
Looking on, Kevin Winkler, a curriculum and professional developer for mathematics at CUNY Start, noted, “It’s hard to get kids to write in sentences this early on. This is math class.”
CUNY Start is a program for students entering one of CUNY’s seven community colleges (and two other CUNY colleges that offer associates degrees) who need to raise their scores on math, reading or writing placement exams in order to take for-credit classes in those areas.
But getting students to engage, in this way, is tricky. They’re accustomed to emphasis being placed on answers. So how do you get students onboard?
“Poker face,” Winkler says.
The instructor must present a poker face, Winkler explains, because the goal here isn’t necessarily to get the right answer out of students — it’s to get them thinking and articulating reasons. If Fein hesitates before pivoting back to the board, a student might feel the hint to quickly change an answer.
“The teacher creates confidence in us,” said Monica Garcia after the three-hour session wrapped up. She initially failed the math placement exam by only 1 point. She plans to major in math and hopes to become a math teacher herself.
Administrators established the program in 2009. They’ve seen over 9,000 students take the intensive 15- to 18-week program. Of those, 76 percent of the students who needed math help passed out of the course, according to CUNY. While the program did not draw its curriculum from Common Core, administrators said, “philosophically it’s basically the same idea.”
Back at San Diego State University, Janet Bowers, director of the Mathematics and Statistics Learning Center, is sprinting across campus to supervise a precalculus break-out session.
Students in shorts and flip-flops are modeling the spread of infectious diseases. The session leaders instruct them to pair up, flip pennies and jot down heads or tails. If a partner lands on tails that means they’ve got “the disease.” One by one, “infected” students get up, and soon enough they’re graphing the results.
Students watch carefully and interrupt with questions.
It turns out, one of the session leaders is an undergraduate student and the other just recently graduated. They’re training to possibly become math teachers themselves. Bowers explained that using active-learning techniques also helps teachers-in-training prepare to move on to full-time jobs in K-12 schools.
Unlike many of her peers, Bowers is both aware of the Common Core connection and embraces it. “We took our cues for developing the activities from the eight standards for math practice that are in the Common Core standards,” Bowers said back at her office. “As a math educator, you know, we’ve been doing research supporting those ideas for years.”
It may be a bumpy road, but Bowers, for her part, is glad that math is changing from kindergarten on up.
As for Chelsea Castilloadame, she’s loving math again. Next semester, she’s going to start tutoring.