With this guided discovery activity, students dissect complex equations like advanced mathematicians do, deepening their own understanding of core concepts.
Lecturer in Mathematics, University of New Hampshire
MS in Mathematics, BS in Mathematics Education, BA in Mathematics
Rebecca Hawthorne, MS, may be a math professor, but her most important teaching cue came from her experience in an English classroom. As an undergraduate at Boston University, Hawthorne was required to enroll in a writing course—an unwelcome demand for many STEM majors. Then she discovered one that fit her interests to a T: a composition course centered on science and technology.
“I had the best professor,” recalls Hawthorne, a lecturer at the University of New Hampshire. “When he taught us to write, he did it in math terms. He explained how to write each paragraph as if it were a proof statement, and it was the first time I understood how to write an English paper.”
Today, as a result of this experience, Hawthorne seeks to empower her students to succeed in her classes regardless of their comfort or familiarity with math. She explains, “I am trying to do what my writing professor did for me, but in the opposite direction. If we can see what students are good at and try to explain math in those terms, we can reach more of them.”
Challenge: Doing math problems mindlessly
When students are wading in a sea of homework problems, working through them might become muscle memory. But Hawthorne believes that to really appreciate the intricacies of the field—and to be prepared to tackle some of its more challenging questions—students need to understand the process behind the strategies they employ.
“Many students haven’t been taught to care about the why. Often, students are told the rule and then asked to go practice it 50 times. But we all need to learn how to think—to really look at a problem and determine how you can rearrange it to get what you need to get. Sometimes how we got there is even more important than the answer.”— Rebecca Hawthorne, MS
This applies specifically when Hawthorne teaches her students about derivatives. Over the years, she found that many of her students had been taught shortcuts to reaching a solution. And while they were able to give her the correct answer to a problem, they could not explain why.
“Many students haven’t been taught to care about the why,” she reflects. “Often, students are told the rule and then asked to go practice it 50 times. But we all need to learn how to think—to really look at a problem and determine how you can rearrange it to get what you need to get. Sometimes how we got there is even more important than the answer.”
Innovation: Helping students find their own shortcuts
The skills that Hawthorne’s students emerge with are not the product of a single activity but rather of “discovery learning” or “guided learning” in action—a methodically organized series of lessons that equip students with the knowledge and practice they need to understand the “guts” of equations intimately.
Although shortcuts exist to solve various types of equations, Hawthorne does not provide these shortcuts directly to her students. Instead, she challenges them to discover the shortcuts themselves during a group activity. In fact, Hawthorne guides her students through the same discovery process undergone by those who first devised the shortcuts.
“I want them to struggle with it,” she says. “I want them to really sink their teeth into the challenge, get their hands dirty, and work to manipulate the problem. If I just wrote it all on the board, they would copy it, and they would never have to prove their work again. But I want the struggle to be stuck in their minds. How did they overcome it? What did they do? It leaves them with a more intimate knowledge of how it all works.”
The method is not necessarily specific to her unit on derivatives; she sees potential for broader application in the subject. “There are a lot of functions that students can be challenged to prove the shortcut rules for,” Hawthorne says. “Sine or cosine, the quotient rule, the product rule. There are a lot of relatively basic rules that students could benefit from understanding more deeply.”
MATH 425 Calculus ISee materials
“All of my students are English-language learners. I work for a company called Navitas, which partners with universities worldwide to provide extra support for international students in their first year at a university outside their home country. Once they reach a certain level of English, they start taking academic courses, and usually the first one is math, since math symbols are universal. But they may still need help with math English—the terms we use when speaking about mathematical equations—because students don’t often get math English in an English-language course.”— Rebecca Hawthorne, MS
Course Title: MATH 425 Calculus I
Frequency: Three class meetings for a total of five hours per week, for 15 weeks
Class size: 20–35
Course description: Calculus of one variable covering limits, derivatives of algebraic, trigonometric, exponential, and logarithmic functions; applications include curve sketching, max-min problems, related rates, and volume and area problems.
Lesson: Facilitating guided discovery
Hawthorne believes in the value of hard work, and her ideology is embodied in her teaching of derivatives.
For those interested in a refresher: Derivatives help identify the rate of change at any point on a straight or curved line, also known as an instantaneous rate of change. You may vaguely remember f'(x)=2x as the derivative of f(x)=x^2.
Here are the steps she takes to guide students to discover their own shortcuts to these equations:
First, make them do it the hard way
Hawthorne’s lesson on derivatives comes on the heels of her lessons about limits. A limit describes how a function—or f(x)—behaves as it gets close to a particular value for x.
“There are two ways to take the derivative,” Hawthorne explains. “There is the limit definition and the shortcut.” At first, all of the derivatives she works on with students are done using limits. You might say this is working through the problems the “long” way—without taking any shortcuts. “Limits can be very tedious,” she adds. “They take a long time, and they aren’t always easy. After we spend some time with them, though, students start to get good at them.”
While she knows she could simply hand out a list of shortcuts, she feels that would be doing students a disservice. “There are an infinite number of functions in the world,” she explains. “I’m not going to be able to teach them a shortcut for every single one of them.” Which is why she wants them to begin with the “long way” and then work backward to figure out the shortcuts for themselves.
Next, create generalized functions
Once the class becomes comfortable with the concept of derivatives, Hawthorne shares with her students a set of what she calls 5 Questions. These questions appear similar to the derivatives problems they have become familiar with. However, Hawthorne has “generalized” them by removing the numbers. This allows students to focus on simplifying them rather than solving them. The simplifications they are left with can be used as shortcuts for future derivatives problems, simply by plugging in the numbers where appropriate.
Hawthorne explains that, by working with generalized functions (as the original inventers of the shortcuts once did), they are in fact inventing the shortcuts on their own. This is a far cry from simply memorizing shortcuts and applying them, without knowing where they originated.
“These shortcuts did not just appear out of thin air,” she adds. “Somebody had to do all of the work to discover them. By going through that same process, the students can really see the guts of the problem and, ultimately, understand it better.”
Leverage collaboration to differentiate instruction
Hawthorne assigns students to groups of four or five so that they can discuss and work together on the 5 Questions. These questions are organized in increasing difficulty, so students who struggle with the topic can contribute at the beginning, and those who thrive can help teach the group toward the end. This arrangement—of students teaching and learning from each other—has the added benefit of giving learners at all levels the opportunity to succeed.
“For my students, this is their first semester at a university in America,” Hawthorne explains. “The chance to debate—to have more than one person to bounce an idea off of—that is really important.”
Be ready with helpful hints
As the groups work, Hawthorne moves around the room to assist, making herself available when needed to offer guidance, answer queries, and offer encouragement.
Having learned from past iterations of the activity, she has introduced another in-class time-saver: supplying “hints,” which she makes available to groups that get stuck on common challenges.
She explains, “The hints are a relatively recent addition. I realized there were a few sticking points that I had to keep writing on the board, so to save myself time and make it easier to help groups, I decided to write them on a piece of paper and provide them at the outset.”
Check for understanding while building soft skills
Each group’s work culminates in a class presentation on one of the 5 Questions. These are doled out by Hawthorne based on the degree of understanding she observed during the activity. For this, she uses what she calls an “old school” method: overhead transparencies.
“These questions involve a lot of work,” Hawthorne says. “If everybody has done the problems already, it’s inefficient to spend students’ time having them watch just a few students writing the work on the board. So as groups are finishing, I’m handing out transparencies and assigning each group the question they will present.” This way, all of the students are writing at the same time, instead of going one by one up to the board and making the whole class wait as they rewrite their notes.
Using the transparency as a visual, each group explains its work, engaging presentation skills in the process, which is a valuable exercise in itself for the English-language learners who make up Hawthorne’s class. She asks each group some targeted questions to ensure that they touch on the most important aspects of the process, and then she asks for questions from the class.
She has notices that, sometimes, students feel uncomfortable quizzing their peers, so she concludes by inviting the class to ask questions of her. “I want to make sure I can address anything they are confused about,” she says.
Record and apply the learnings
After the groups present, Hawthorne provides each student with a chart on which they can record each shortcut. On the chart, students write the name of each shortcut, the rule, and an example that applies to the rule.
Last, they use their shortcut sheet to apply their newfound learnings by working on some sample problems, both individually and in pairs. Once class is over, students walk away with an artifact of their learning (the shortcuts sheet) and the pride of having “discovered” them with their peers.
Hawthorne has found that her approach has successfully influenced student engagement.
“There is always someone in the class who rolls their eyes because they would have wanted the answers sooner,” Hawthorne says, “but most of them are actually pretty excited. They enjoy the challenge, and they want to [figure out the shortcuts so they will] be able to do their problems faster.”
She has found, too, that for the students who go on to take higher levels of math, the results are noticeable.
“I’ve found that my engineering students, when they get to those harder math classes, are prepared. They know how to think through a challenge.”