Better paths through maths

Understanding what children are thinking as they make a simple drawing with perspective is one of several ways that researchers at the University of Alberta are improving how K-12 students succeed at math.

Perspective drawing involves spatial reasoning, the ability to visualize and mentally manipulate objects. It is actually considered more valuable to long-term success in math and science than memorizing computational skills and basic algorithms, says math education professor Lynn McGarvey.

“Most of what we do in problem-solving, in any field, has a visual component to it,” says McGarvey, who is also vice-dean of the Faculty of Education. “So, if we can help children understand that visual side of things, they’ll improve in all fields, including math, engineering, chemistry, computer science, physics.”

Importantly, spatial reasoning isn’t an innate gift. It can be taught.

In McGarvey’s research, children in Grades 2 to 5 use an area of mathematics called projective geometry as they learn to shift their understanding between 3D objects and 2D images of those objects, while making a perspective drawing of a simple building.

The perspective drawings create a visual conflict they learn to resolve. On the one hand, the children can see that the vertical lines of the building’s corners are different lengths on paper, but they also realize that in real life those corners are all the same height.

Such skills are not typically part of elementary school math curriculum, yet “the research is showing that children in early education need a lot of experiences like these to help them understand mathematics all the way through school,” says McGarvey. Otherwise, she says, they’ll start simply memorizing the right answers and, as math problems get more complex, they’ll become frustrated and give up on math and other STEM subjects.

Assistant professor Josh Markle, one of McGarvey’s collaborators in spatial reasoning research, is further investigating exactly how students visualize when learning math. Some do get a clear mental picture they can verbally describe; others need gestures to capture what they’re seeing.

“There’s an assumption that learning is taking place only in the brain,” says Markle. “That’s how we think of math, as deskbound, cerebral, solitary work. But the interesting thing about visualization is it’s a collaborative and collective activity. You’re never stuck in your own imagination. The next step is to talk with others about what you see and refine your imagining, and maybe start sketching what you see with others.”

By discovering how visualization emerges from our embodied interactions with the world, “I hope to provide better tools and practices for teachers in assessing and practicing visualization,” he says. “We want teachers to feel confident and capable in providing meaningful feedback to support that kind of spatial reasoning.”

More than fun and games

Many commercially available board games are a terrific tool for improving not only visualizing but also deductive reasoning, says Janelle McFeetors, an associate professor in elementary mathematics education.

“I believe we have begun to challenge the idea of what ‘back to the basics’ means,” McFeetors says. “Rather than emphasizing arithmetic as a basic skill in mathematics, it really is reasoning that is basic to mathematical thinking and learning.”

When students predict moves in advance or decide on the best spot to place their game piece, they’re practicing visualizing. When they convince themselves or other players of the best move to make, they’re executing deductive reasoning. All of it builds their confidence in problem-solving, and in seeing themselves as mathematical thinkers.

McFeetors has created online resources for numerous store-bought games, so students can reflect on the math skills they learned in specific games by answering questions, using words and pictures, such as explaining why a particular move they made was a good decision. The resources also include prompts to students such as, “What did you do to not give your opponent an advantage?”

Making math relatable through data literacy

For high school students, Kwesi Yaro, an assistant professor of mathematics education, is seeking to strengthen their data literacy — the ability to put data into context by finding it, interpreting it and communicating it — by connecting it to sustainable development goals.

One of his next projects will empower mathematics teachers to use big data — for global air quality indices or food prices, for example — that’s credible and can be imported accurately. Teachers will create inquiry tasks that allow students to apply their computational thinking skills by examining complex, real-world scenarios, analyzing the data for trends and making predictions.

By thinking critically about climate change or hunger through mathematics, math becomes relatable to students’ lives and more engaging to them, says Yaro. It also lays the foundation for skills in mathematics modeling, which are critical for careers in data science and artificial intelligence.

In other research, Yaro is conducting qualitative interviews with new immigrant families in Edmonton to discover how they support their children’s mathematics learning. He says that the linguistic, cultural, navigational and aspirational capital they bring — a non-English mother tongue; different systems of counting, such as base 5 or base 10; the ability to navigate hostile environments; and the desire for a better economic future — are not barriers but assets when supporting math learning.

“It’s like having another language,” he says. “Having the ability to reframe that math concept in another language really helps (the students) understand the concept better … and It’s the duty of the school to encourage such practices and different ways of knowing.”

Interested in deepening and extending your understanding of math education? The U of A Faculty of Education offers a Graduate Certificate in Educational Studies - Mathematics Education, a four-course program for working education professionals.

Feature image: Professor of math education Kwesi Yaro is exploring how new immigrant families in Edmonton support their children’s mathematics learning (photo credit: John Ulan).