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Check for Math Comprehension with Small-Group Mini Quizzes

Math professor Lawrence C. Udeigwe, PhD, draws on lessons from his other passion—music—to reframe math quizzes as collaborative opportunities for practice.

Educator

Lawrence C. Udeigwe, PhD

Assistant Professor of Mathematics, Manhattan College, Riverdale, New York

PhD in Mathematical Information Science, MA in Mathematics, MS in Applied Mathematics, BS in Mathematics, BA in Computer Science

While the rigidity of mathematics and the fluidity of West African jazz fusion may seem to inhabit different worlds, Lawrence Udeigwe, PhD, has found that both can be approached with the same dynamic passion and creativity.

Working under the name Lorens Chuno, this Nigerian-born singer, composer, and pianist has produced several studio albums and collaborated with well-known international acts and with independent jazz musicians seeking to introduce their work on his radio show and podcast. Working as Dr. Udeigwe, assistant professor of mathematics at Manhattan College in Riverdale, New York, he champions the fusion of solo and collaborative work in his approach to his research interests, which include differential equations and dynamical systems, oscillations and pattern formation in biology, theoretical neuroscience, and applications of neural computations in machine learning.

And in the classroom, where the traditional concepts of “quizzes” and “collaboration” also seem to inhabit different worlds, Udeigwe has found that their fusion, too, creates harmony—among mathematics students.

Many teachers would evaluate student learning simply by handing out a test. Udeigwe tried that, but he was not satisfied with the results. He found that “the students would go home and, when I graded the quizzes, I would realize that some of them didn’t quite get what I’d taught.”

He knew he would have to rethink the way he supported student understanding.

Challenge

Infrequent solo quiz work does not reveal where practice is needed

It is not always easy to tell whether a carefully crafted lecture has landed and the students truly grasp the material. Udeigwe has found that traditional end-of-unit exams or quizzes allow too much time to pass before revealing any gaps in knowledge or understanding. They also can leave students feeling frustrated, unsupported, and insecure about their work.

Innovation

Quiz students in groups to check understanding

Udeigwe wanted a way to determine what the students had learned that day, reinforce what was learned, and offer opportunities for them to ask questions to increase their understanding, so he decided to improvise a bit. Today, he delivers at least one quiz a week on the same day as its accompanying lesson, with students working in small groups to come up with and justify their answers.

Context

“Through group quizzes, students empathize with each other and work together and discover new ways to learn. The good thing is this is happening in the classroom.”

— Lawrence Udeigwe, PhD

Course: MATH 286 Differential Equations

Course description: This course focuses on techniques of solving first-order, second-order, and systems of first-order ordinary differential equations. Methods include separation of variables, variation of parameters, and the Laplace transform. Applications include linear and nonlinear models.

See resources shared by Lawrence C. Udeigwe, PhD

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Udeigwe’s approach to group quizzes

In at least one class a week, Udeigwe ends the session with a group quiz. (This approach, he notes, can be applied to any subject matter—not unlike creating a jazz arrangement of a classic rock song.) Here is how he does it:

Let students select the groups

Udeigwe allows the students to choose two or three classmates with whom they feel comfortable collaborating. If he sees that someone cannot work effectively with his or her friends, however, Udeigwe will move that student to another group. “I don’t tell them why, but they respect it,” he says.

Keep the quiz very short

The number of questions on each quiz varies depending on the material, but four is the average for Udeigwe. If he presents too many more questions than that, he finds that students run out of time or suffer burnout.

Allow open book and notes

Students are invited to use their textbook and notes to help out. However, Udeigwe reminds them that the point of working in groups is that they can ask questions of their peers and work out the solutions together.

Not for Beginners Only

Udeigwe finds that encouraging group collaboration on mini quizzes works both for his introductory classes and his upper-level courses. He uses this approach in all the mathematics classes he teaches, which also include Math 185 Calculus 1, Math 186 Calculus 2, Math 285 Calculus 3, and Math 386 Partial Differential Equations.

“In advanced mathematics courses, I use group quizzes to not only measure how well the students understand the concept of the day but also how well they can apply it,” he explains. “So, unlike in lower-level courses where I assign mostly drill problems in group quizzes, in upper-level courses I also throw in challenging application problems to inspire vibrant discussions.”

Limit questioning of the professor

Students cannot discuss the problems with people in other groups, but they can ask Udeigwe up to two questions without penalty. After that, he tells them he will deduct points for each additional question. That is because, as he says, “I want to encourage students to help each other rather than get help from me.”

Know when to step in

Sometimes Udeigwe finds that a group will ask him, “Which answer is correct?” He will not answer that, but he instead turns the tables and asks the students questions to spark their thinking. Other times he notices that a group is going in the wrong direction—then he joins the group to redirect it.

Have them show their work

Udeigwe wants his students to understand that a quiz is not just about going for the answer; it is about showing they understand the work. By taking group quizzes, they will first justify their work verbally to their peers, and then they will put their steps in writing on the paper they hand in.

No copying, no makeups

Ultimately, each student must complete and submit his or her own quiz showing individual work. Udeigwe monitors the groups to ensure that they are all doing the calculations together and that no student is merely copying the work and resultant answers of others.

There is no makeup opportunity for students who missed class, because an instructor cannot recreate the collaborative environment for just one student. “However,” Udeigwe adds, “I usually give one or two more quizzes than I need, so that I can drop the student’s lowest or zero-grade quiz.”

Offer extra challenge, as needed

The groups usually have about 20 minutes to complete a quiz. If one group finishes long before the others, Udeigwe assigns them extra-credit problems to keep them working until the end of class.

The benefits of group quizzes

Udeigwe finds that students indeed gain a better sense of the material and ultimately receive better quiz grades. Furthermore, they no longer just write down their answers, he says. He sees them “begin to understand the importance of details. And they learn from others how to justify their work.”

The classroom experiment has yielded unexpected benefits, too: Natural leaders emerge, he says, and individual students learn that they are not the only ones who are confused by certain concepts. They realize they can join forces to help each other—and they take turns lifting one another up and assuring each other that they have the right idea.

Many of his students also form study groups outside of class, Udeigwe says. As with collaborative jamming in music, these collaborations in mathematics can create beautiful harmony.

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