“Math is everywhere. To varying degrees, of course, but math is just something that’s everywhere,” Professor Limin Jao, assistant professor and assistant graduate program director in the Department of Integrated Studies in Education, said in an interview with The McGill Tribune.
She’s right, of course. There’s math in our kitchens, on our walks to school, and in the way the clouds move. It’s everywhere, and though many of us try our best to live in denial, it will always be everywhere.
But, math doesn’t have to be scary. Taught in the right way, math is beautiful, joyous, and important.
The way math is taught now in many primary and secondary schools across North America, however, leaves much to be desired. Most curricula foster a high-pressure learning environment, heavily focused on students’ abilities to recite formulas, perform calculations, and not ask questions. Students have numbers and equations thrown at them and are assessed on how quickly and accurately they can replicate textbook solutions. All the while, they miss out on some of the most valuable lessons that math has to offer.
Marta Kobiela is an assistant professor and graduate program director in the Department of Integrated Studies in Education. Her own experience learning math led her to research the relationship between learning material and applying the knowledge in practice.
“When I started my bachelors, I pretty soon came to realize that the math [learned] as a math major was very different from the math I had experienced for my 13 years of education before that,” Kobiela said. “Whereas before it had been very procedural, [...] in my math classes at university, I was then expected to actually engage in reasoning and proofs and form some mathematical argument. [It was a] very different kind of thinking. So I felt a little bit cheated, in a way [....] I had thought math was one thing and then now I’m realizing math was something else.”
In her research, Kobiela found that curricula often do not integrate theoretical material with real-world application.
“It’s not that they’re these separate entities, and we’re just going to fuel people with knowledge and then [...] teach them practice, but how [can] they learn both of them at the same time?” Kobiela said.
In approaching her own research, she asked herself how to bridge the gap between learning and practicing math, and how students could develop skills like mathematical argumentation and justification.
“An area I worked in was mathematical defining, so, actually constructing definitions,” Kobiela said. “We often think that definitions are created, and that’s it, and they don’t change, but [...] as new knowledge is created, new definitions are created. So part of the work I looked at was how do you engage students in that construction process, that reasoning process that’s associated with it, and [the] practices [of] constructing questions or conjectures.”
Kobiela’s move away from traditional methods of teaching mathematics and toward flexible ways of thinking has become increasingly popular among educators across North America. Jo Boaler, a professor of mathematics education at the Stanford Graduate School of Education, she has written about how students can feel constrained in math class because of a fear of judgement.
“For many people, the words ‘math’ and ‘freedom’ can’t be put in the same sentence—because they’re taught mathematics as a subject of rules, conformity and constant performance,” Boaler wrote in an article for Time Magazine.
Boaler emphasized the importance of leaving behind an outdated system in which performance is everything and replacing it with a culture that encourages free thinking. Methods as simple as leaving constructive comments instead of marks demonstrate to students the value of learning over performing and help promote a creative and open-ended learning space.
“My research on math learners suggests that when students think they’re in class to learn—to explore ideas and think freely—they understand more and achieve at higher levels than when they think the point is to get questions right,” Boaler wrote.
Kobiela also stressed the potential of a learning environment where students are empowered to explore math on their own terms and make their own discoveries. It is important to her that students get to immerse themselves in math instead of standing on the outside looking in. This shift helps them gain confidence in their problem-solving abilities and tackle the misconception that there is only one right way to find a solution.
“[Something] that teachers can do that might be less about connecting [math] to real life that is very powerful is to make students feel like they are authors of mathematics, and they are agents, and they’re not always relying on the teacher and the textbook for answers,” Kobiela mused. “If the kids feel like they’re authoring ideas, they feel like they have a voice. [If] their ideas are respected, it makes a huge difference for their lives and how they approach mathematics.”
Eugenia Cheng, a mathematician and scientist in residence at the School of the Art Institute of Chicago, has developed many outreach projects to make math a more accessible subject.
She hosts a wide range of outreach projects, including public lectures, workshops, and art, for diverse audiences. She is adamant that not only can everyone appreciate mathematics, but everyone can enjoy it too. However, she understands that everyone has a different relationship with math and believes that, since no two students are the same, the spaces in which they learn math need to reflect the variety of learners.
“The ideal learning environment is one that is sufficiently flexible to be able to respond to different students' needs and personalities,” Cheng wrote in an email to The McGill Tribune. “Trying to teach everyone the same way is not something I support [....] I support environments where we concentrate on not putting people off [of] math, rather than just trying to get them to be proficient at it.”
A central component of Jao’s research is studying the ways in which teachers engage their students and why teachers choose certain teaching practices over others. For her, connections are key, whether they are made across content, with the real world, or between teachers and their students.
“[The goal is] developing good thinkers [who...] have an ability to do mathematics in flexible ways, [...make] connections [...] between the different [...] content areas of math or [...] between math as a field and baking, or whatever else is going on in their life, so they don’t think of math as just this own little pocket thing that has nothing to do with language arts or history or riding a bike to school or anything else,” Jao said.
Jao is a strong advocate for developing teacher-student relationships that help engage individuals in the classroom. Students who develop personal relationships with their teachers are much more likely to attend classes and actually learn.
“It’s important for teachers to understand [their students’ interests, backgrounds, and experiences] and get to know those parts first [because] that’ll inform [their] teaching practices,” Jao said. “We’re teaching a subject area, but we’re also developing people, and we’re working with people. So, at the end of the day, I still want to get to know my students, and I care about them as people, too. If I can connect to what they’re interested in or if I can just help them grow as people, [...] that’s still a benefit to them, too. If [...] they learn some math content at the same time, that’s great, too, but that [does not] necessarily [have] to be a math teacher’s only focus.”
As math curricula progress, they are shifting toward making clearer connections, between math’s different subjects, math and the real world, and the application of skills developed in math to other disciplines. For many students, math is frustrating and pointless because it is difficult for them to see these connections: Why are there letters in math? How did we get from shapes to integrals? When will I ever need to use trigonometry?
The connections are all there, but, too often, they are hidden by seemingly-meaningless formulas and endless calculations. Math is broad and continuous, but for practical reasons, math curricula cannot replicate the fluidity of mathematics. As the curricula gets broken down into separate bits and pieces, student interest can run up against the barriers of textbook chapter divisions.
“In a textbook, you have chapter one, chapter two, chapter three, chapter four, so it would be up to the student, or the teacher, or the curriculum guidelines to say, ‘By the way, these might be in nice units because the publisher needs to have some way to organize it, but can you create thread between them,’” Jao said.
Combating the disconnect between mathematical subjects does not have to be a complex process. Kobiela used the equals sign as an example of a potentially-simple adjustment in elementary instruction that can help make a clearer connection between arithmetic and algebra.
“In elementary school, sometimes [...] students are taught to think of the equals sign as ‘do it’, […] and what the equals sign is supposed to represent is ‘the same’ and that’s a really core idea for algebra.” Kobiela said. “Solving something like this [6+5=x+3], relies on you understanding what the equals sign means [....] At an early age, if we change what kids see, [as] they’re very flexible thinkers, they will start to understand [that the equals sign] means that the two sides are the same.’”
Another way to strengthen the connection between different areas in math is through problems that require students to integrate those different areas into one solution. In Quebec, for instance, math standards now include situational problems: A set of more in-depth exercises that combine several topics in mathematics in a real-world problem-solving context. These problems are not designed to have a single or immediately-obvious solution, thus promoting a multi-disciplinary approach to problem solving. Students have all the tools that they need to come up with a solution, ranging from skills they have learned in math class to their ideas and insights gained from lived experiences.
“These are really complex problems that require [students] to use a lot of areas of mathematics and, in some cases, [...] areas outside of math,” Kobiela said. “So, there is that motivation from standards to encourage [teachers] to foster this kind of thinking, but it does take time [....] It’s not just a problem on a piece of paper [....] They have to think about ‘what do we need to find out, what resources do we need to mobilize,’ and go through [those...] problem-solving processes as they would in real life.”
Jao commends this effort by the Ministry of Education in Quebec to give students hands-on practice in making connections.
“[The situational problem is] one of the ways that the Ministry has tried to support the idea that math isn’t these siloed segments, [and] that math topics are related to one another,” Jao said. “It’s not a straightforward calculation, but you have to reason with it and problem-solve and then also use these math topics to solve the problem.”
Math is so much more than the equals sign and algebra and trigonometry, but, in grade school, students’ exposure to the subject is limiting and off-putting, so, it’s easy to lose interest. Cheng encourages students to take a step back and reflect before they give up.
“Remember that the subject is beautiful, open-ended, ambiguous and flexible,” Cheng wrote. “And, then, hope that you can make it through the less interesting stuff and get to what you really want to do.”
Many of Kobiela’s students are future elementary school teachers. When they come to her claiming that they are not ‘math people’ or ‘bad at math,’ she is quick to contradict them.
“It’s not that you’re not good at math, it’s that maybe you just didn’t have opportunities that would have allowed you to excel at math,” Kobiela said. “You are good at math, you have all the potential to excel at math so there’s nothing innate about you, but it’s really about the opportunities that you were given. You just have to work on developing your relationship with mathematics.”