How does a student know if they are truly learning the course material or just mimicking what a professor has shown them?

That is the question Faculty of Mathematics and Science post-doctoral fellow Laura Broley (BSc ’13) has aimed to answer through her research on the teaching and learning of mathematics at the post-secondary level.

Broley recently presented her PhD research in an online colloquium talk at Brock.

“My work fits in the trend of thinking about how mathematics education can focus on mathematical thinking,” she said. “We want to get students thinking like mathematicians by moving away from the robotic mimicry of just copying what the professor does and moving towards more advanced problem solving.”

Past research pointed to a trend in calculus courses where standard activities given to students, in particular the final exam tasks, tend to encourage non-mathematical practices.

“With non-mathematical practices, students are learning to identify tasks in superficial ways and following steps that ‘just kind of work’ without understanding the underlying mathematical principles,” Broley said.

Relying on non-mathematical practices can be detrimental because a small perturbation of a task or diversion from a standard question format can break a student’s normal question-solving method.

Broley seeks to identify the type of learning style students are applying to their mathematical tasks and help reshape their thinking. She aims to identify “how can we encourage students to engage with new types of tasks and to understand the underlying theory a little better.”

Not all students learn non-mathematical practices and many use a combination of styles to accomplish their goals. In her research, Broley compares the learning styles she observed in an Analysis course to understand the mechanisms and thought processes that drive a student to use a non-mathematical approach.

Students seeking to advance their learning may ask themselves how to identify if they are solving a problem sub-optimally. It may not be obvious, especially if they are getting the right answers.

“If a student is being validated by high exam scores using a non-mathematical approach, they are being told they are learning the right way and their approach to learning is appropriate,” Broley said. “It reinforces that the student should approach their next course in the same way.”

To break out of this pattern, there is an opportunity for students to recognize the difference between what scholars call “studenting” and “learning” in their approach to problem solving.

Broley describes studenting as “trying to do whatever possible to achieve success in a course.”

“We see students apply the rote memorization of tasks, copying the solution to a problem from past examples and following step-by-step methods without truly grasping the underlying mathematical processes,” she said.

In contrast, Broley said learning means a student is “operating according to the rules and standards of the mathematical discipline.”

She offers a checklist for enterprising students who wish to develop a mathematical learning practice by using these indicators:

• Can the student identify the task as being part of a general type of task that is mathematical?
• Can they choose and implement a mathematical technique for solving the problem?
• Can they explain using mathematical language how and why the technique operates?
• Can they refer to the underlying mathematical theory at work?

These learning indicators work well for mathematics and many other subjects.

“We should always critically examine our approach in learning to discover ways to improve our analytical thinking,” said Faculty of Mathematics and Science Dean Ejaz Ahmed said. “Analytical intelligence may also improve emotional intelligence, but the converse may not be true. Each new skill enhances our ability to solve and find a simple solution to the complex problems we face in our studies and in the world around us.”

Broley, who offered thanks to her supervisors for their guidance and support through her studies, will also be speaking Thursday, March 4 at noon at the MathED Seminar Series. Her presentation is titled “Computer programming in mathematics education: Some results from a literature review and an international scan.”