Should we still teach long division?
By Linden Faculty: Beth Alexander, with contributions from Soteira Hortop, Nasrin Matini, Martha Muraira, Niki Popper, and Ruthie Szamosi
Traditional vs. Constructivist Model
This question, currently debated by the provincial government and outlined in this recent CBC News article, is an interesting one. There has long been a battle in math education. On one hand, the proponents of a "back to basics" approach call for repetition, teacher-directed instruction, and value speed and accuracy when performing calculations like long division or working with fractions. On the other, the constructivists argue that conceptual understanding is the best foundation for mathematical achievement, and focus on concrete materials and a student-directed discovery approach. Both camps value problem-solving, although the meaning of this varies. Some believe it means an ability to interpret word problems. Others believe that true problem-solving is shown when applying math skills to authentic situations, such as writing and balancing a budget, or figuring out how much paint is needed to redecorate a room. Others still believe it refers to using logic and abstract reasoning to figure out how to use math skills in an unfamiliar setting, like coming up with a proof.
The Ontario Curriculum, which is re-written every ten years or so, has swung back and forth on its approach, but now leans toward a constructivist model. The media, sounding warning cries whenever Canada's math students slip in worldwide rankings, often argue that our children are too busy moving counters and journaling about their math discoveries to compete globally at "proper" math. It is true that when it comes to some math tasks—including performing calculations quickly—Canadian children do not rank as highly as their peers in one of the “top 10” countries, including Singapore and Finland (we ranked 13th at last measure). On the other hand, we do very well in the sorts of critical thinking tasks that require you to synthesize information and make decisions. In 2014, the OECD ranked Canadian teenagers first among their English-speaking peers when it came to these sorts of problem-solving tasks.
Linden’s Approach to Teaching Girls Math
So which is the best approach? There are strengths in both approaches, and this is why most math programs—and the Ministry of Education—favour a model that calls for some memorization of core facts, and some use of inquiry to consolidate important concepts. But even the best model, backed by research and proven by test scores, is doomed to fail some kids.
There simply isn't one "best" way to teach or learn math. This is why, time and again, two variables stand out when studying the most successful math programs: small classes and responsive teachers. Linden bases its math instruction on some core principles, including:
- Posing real, interesting questions and using math to solve genuine (not textbook) problems
- Presenting more than one way to perform basic calculations, and letting students choose which work best for them or come up with their own method
- Giving girls the opportunity to talk through their thinking, with the teacher, and among each other
- Solving a combination of concrete and abstract problems so that students can learn using a method that they are comfortable with, but also challenge themselves to try something unfamiliar
- Valuing girls' individual ways of thinking and solving mathematical problems
- Using a mindset approach to tackling challenges, which encourages students to view mistakes as growth opportunities
But, in the end, we are responsive to our individual students. There are some students who respond very well to a traditional "drill and kill" model, some who grow by leaps and bounds when given objects to manipulate, some who need to hear metaphors that relate new concepts to their own lives, and some who thrive in the face of competition.
Most kids need all of these, at different times in their years of learning math. So, as math teachers at Linden, we plan our instruction with the understanding that our plans will adapt to the evolving needs of and input from our students. And, since our classroom instruction is based on a foundation of community and trust, our students are encouraged to be honest with us about what they need. Research—but most importantly, our own experiences as teachers and the feedback we get from graduates—tell us that this approach is successful.
And yes, we still teach long division.
Our Math Teachers Say…
“We nourish our girls to become confident math learners and help them overcome any anxiety they might have about tackling difficult math questions. Our small class settings allow me to work one-on-one with students and address individual difficulties and set goals.” —Martha Muraira, Intermediate School Math Teacher
“Although the pendulum of opinions in math education is always swinging, we stay grounded in our practice of being responsive to girls’ individual needs.” —Niki Popper, Junior School Math Teacher
“Rather than being pitted against each other based on things like speed, our students are encouraged to collaborate on problem-solving, to learn from each other, to make mistakes, and to build creative solutions together.” —Soteira Hortop, Junior School Math Teacher
Our Students Say…
"The material taught is ahead of our grade, but it is taught in a way that we understand. Teachers talk with us, not at us." —Lia, Grade 8
"Over the past 11 years that I have attended Linden, I have found that math has become easier for me. When I started out doing complicated word and arithmetic problems, I would always memorize the instructions without understanding what I was doing or what purpose they served. I have worked closely with two amazing math teachers, Nasrin Matini and Martha Muraira, and both have spent countless hours with me working on helping me understand the concepts of solving various mathematical questions. I'm very appreciative of both teachers and I fully believe that I will be able to expand my knowledge over the next year so long as I keep working with them. I hope to come back and continue to work with them in my free time after I have graduated Linden. Overall, I am more confident of my mathematic skills and it's all thanks to ambition and these two women who helped me achieve my goals each and every day." —Maud, Grade 11
"At Linden, the teachers make sure you understand the concept and teach you multiple ways of solving problems." —Sara, Grade 8
"The fun way to do math is by making mistakes. First, when you look at the page you think it will be so hard and then you do it and keep working on it. Sometimes we have fun when we work in small groups, with our partner trying to figure it out." —Ellie, Grade 3
"Students gain an understanding of what must be done to solve a problem and why they do that, as opposed to memorizing formulas." —Mieko, Grade 8