#### Math did not come easily to this professor. By providing a path from Novice to Journeyman to Mastery, she helps students work through their struggles.

#### Educator

### Amanda Harsy, PhD

Assistant Professor of Mathematics, Lewis University, Romeoville, IL

PhD, MS, MA, and BA in Mathematics

Amanda Harsy knows firsthand that an instructor’s passion for a subject can go a long way toward keeping students motivated—particularly in a discipline that does not come easily to them.

“In high school, math wasn’t my best subject,” she says. “In fact, I got my worst grade in high school in calculus. But I knew I had to persevere.” The attitude of the adults in her life certainly helped: Her parents encouraged her not to focus on the grade but to work hard, and her instructors fostered her love of the subject, even though she was not the star student in the class.

“My math teachers’ joy in the subject was easy for me to latch onto as a way to be as passionate about the subject as they were,” she says. “I responded to their love of math theory.”

In time, Harsy’s perseverance earned her better grades and eventually a PhD in the world of mathematics, and today she is an assistant professor of mathematics at Lewis University in Romeoville, Illinois. Her overarching goal is to provide the necessary support to inspire the same kind of passion in her students, including those who struggle with the subject.

Interestingly, she soon saw that displaying her own love of math theory was only part of the equation. But, of course, she persevered.

## Challenge: Traditional “lecture and test” does not promote mastery

When Harsy started teaching, she noticed that many students had preconceived notions about how the class would be structured—basically, a series of lectures followed by a test (rinse, repeat)—and what they would need to do to eke out a decent grade.

“In the traditional [math] world, if you get enough problems partially right, you can pass,” she says. “[Students today] are so worried about landing good grades and getting as much partial credit as they can [that] they’re missing the bigger picture of why they are [in my class] in the first place.”

In addition to their hyper focus on points, students also seemed to be giving up too easily on understanding math problems and the theories behind the solutions. In other words, they were not truly learning the subject. And often the lack of comprehensive understanding early in the semester would lead to greater issues as the course progressed.

“The stuff you use early in the semester you continue to use later,” she says. “So it’s really important to learn the earlier material. Otherwise, you’ll continue to be hampered by what you missed, and that can be demotivating.”

## Innovation: A mastery-based approach to mathematics

An encounter at a MathFest conference led Harsy to apply a new technique—mastery-based testing—to her class. “I was meeting with other mathematicians, and one of them [told me they] used mastery-based testing,” she says. “It sparked something in me, and I thought this could really work for my students.”

With linear algebra, Harsy says, students are typically expected to learn a concept by this or that date, then show what they know on a midterm and a final. However, she feels that what really matters is not this week-by-week timeline, but the *eventual* mastery of each section. What happens between the first and last day of class can be achieved with steady progress or in fits and starts, so long as the students can to do the work by the end of the semester.

Using mastery-based testing, she notes, enables her to divide the lessons into smaller sections that students can use as building blocks throughout the course. This approach can take more time, she admits, but it ensures that students develop the foundation of knowledge necessary to preserve motivation and foster future success. Plus, the grading is easier than you might expect.

## Context

“If you’re grading on an all-or-nothing scale, the only way to do that successfully is to offer students multiple chances to demonstrate mastery.”

— Amanda Harsy, PhD

**Course: **MATH-30700 Applied Linear Algebra

**Frequency: **Three 50-minute meetings per week for 16 weeks

**Class size: **25–35 students

**Course description: **The study of matrices and matrix algebra, systems of linear equations, determinants, and vector spaces with a focus on applications. Topics include LU-decomposition, inner products, orthogonality, the Gram-Schmidt process, and eigenvalue problems. Applications include differential equations, Markov processes, and problems from computer science.

## See resources shared by Amanda Harsy, PhD

See materials## Lesson: Applying mastery-based testing to math—or any subject

Harsy admits that mastery-based testing may seem daunting to instructors, since it does require a greater volume of work—after all, there are more tests to give. But it is not always as tricky as it sounds.

“There’s *more* grading,” she says, “but it’s *easier* grading. It’s an all-or-nothing scale: Students either understand the concept or they don’t.”

Here is how she puts this approach into practice so that it adds up to success for her students:

##### Set the stage

Harsy usually starts the class by asking students to give her some information about themselves. “One of the questions I ask them is, ‘What is one of your hobbies, and how did you become good at it?’” she says. “Usually students say they became good at their hobby through practice, spending a lot of time doing it, and learning from mistakes. That gives me a good transition to [discuss] the work that is required for this course and how we often are not good at things until we practice and spend time working at them.”

##### Offer (and explain) the multiple checkpoints

A mastery-based approach is new for many math students, who tend to stress about grading. So, at multiple points in the class, Harsy offers a detailed explanation of how she tests.

Harsy offers three exams *and* three retesting opportunities (one per exam). Students are expected to review their graded exams so that they can learn from their mistakes and see what they need to spend more time on. In each case, the retest will not offer the exact same problems, but it will present all of the same concepts, so they can see if their understanding improved.

For Harsy, the objective when creating the tests is to create rich problem sets that she can easily modify in a way that makes each test and problem a little bit different, while still assessing the concepts students are expected to master for each exam.

##### Demonstrate progress by designating levels of mastery

Harsy uses a three-point grading scale that makes it easy for students to understand their current standing: Novice, Journeyman, and Mastery. Mastery, she says, means that the student **“**understands the concept and has a (mostly) correct solution (with maybe a small technical error).” The Journeyman designation means the student **“**understands some of the concept, but his/her solution is missing key components or details. The student still needs to review the concept in order to reach full understanding.” The Novice, says Harsy, **“**does not understand much, if any, of the concept. His/her solution is incomplete or missing, and the student needs to review the concept in order to reach full understanding.”

Students only succeed on a given concept within an exam if they receive the designation of Mastery.

The three exam scores are each based on the number of concepts students master; Harsy also uses those scores when determining the student’s final grade in the course.

##### Resources for Mastery-Based Testing

Harsy recommends these resources for math instructors interested in mastery-based testing. Some are explicit discussions of the concept, while others provide active learning materials to enhance the class. “This helps with providing buy-in and trust with alternate methods of teaching and assessment in the classroom,” Harsy points out.

- Mastery-Based Testing in Undergraduate Mathematics. This blog is written by Harsy and colleagues from universities around the country.
- Mastery-Based Testing in Undergraduate Mathematics Courses. Harsy coauthored this research paper into the mastery-based grading method.
- IMAGEmath Labs. These applications can help students learn advanced math concepts.
- Clicker questions from MathQuest. Information on the use of classroom clickers for students doing in-class quizzes.
- IOLA project. The Inquiry-Oriented Linear Algebra project develops student materials for use in teaching. Harsy has used this resource as the basis for many of her lessons.

##### Clarify what concepts will be tested

Harsy breaks down each exam into several core concepts, which students are told about in advance. For example, Harsy provides students with this list of what to study for Exam One:

- Mastery Concept 1: Gaussian elimination
- Mastery Concept 2: Understanding solutions to systems of equations
- Mastery Concept 3: Vector spaces and subspaces
- Mastery Concept 4: Span
- Mastery Concept 5: Basis/linear independence
- Mastery Concept 6: Linear transformations

##### Promote mastery above all else

When students are preparing for their first exam on these six concepts, Harsy reinforces the notion that she would rather have them master a few concepts than to earn Journeyman grades on all six.

“‘If you score a 75% on a traditional exam, what does that even mean?’” she asks students. “‘Is it relevant? It means you could understand some aspects fully but not understand the others.’ Once I explain it like that, students realize that mastery-based testing will help them spotlight the areas where they have the concepts down and where they need to practice more.”

Harsy reminds students that she wants to see full understanding of all of the concepts; it does the student no good to have partial understanding.

She also says students usually are not able to fully use the strategies she teaches them until the second exam. “It is a new way of thinking about preparing, and I think the first exam is the toughest just because it is new,” she says.

##### Offer retests—on a delay

Another innovative aspect to Harsy’s approach? To make sure students have sufficient time to learn more about a concept before retesting, she has instituted a rule: A retest can occur no sooner than two weeks after the first attempt.

Students may want to retest immediately, but her “delayed retest rule” prevents them from jumping the gun and ensures that they will work through enough problems to embed the learnings in their memory.

“I want to encourage a growth mindset for students,” she says. “I want them to know that mistakes are learning opportunities, and it’s OK to not get something right away.”

## Outcomes

Harsy says that mastery-based testing can be a double-edged sword. It will likely help the hardworking B student get an A, but it also means that students who could have eked out a C minus may well get an F: There is no middle ground of partial credit to allow them to “skate by.”

However, Harsy adds that, in general, she has seen students’ grades improve with mastery-based testing. “I rarely have students who work hard and retest and get help [who then] get Fs,” she says. “I try to personally reach out to struggling students and have them meet with me in my office to talk about the course and strategies for how to succeed.”

## Student feedback

Students give Harsy high marks for her teaching style and her use of mastery-based testing. Some recent verbatims include:

“Dr. Harsy made it easy for students to do well in the course, given that the student is willing to put in the work.”

“I had Dr. Harsy for Calc 1, 2, 3, and now linear algebra. She is my favorite teacher and she always makes me work hard in her class to achieve great success.”

“Mastery-based testing took a lot of stress off me, and if I fail a concept and still want to master it. I am forced to take it until I learn it, so I definitely learn the material.”

“This unorthodox style of testing was really helpful, and I would really love to see it implemented across more classes at Lewis.”