John Mighton is a mathematician, author and founder of Jump Math. His latest book is All Things Being Equal: Why Math is the Key to a Better World.
In a keynote address at the Calgary City Teachers’ Convention, I once asked 700 teachers why, when you divide seven by two-thirds, you are allowed to flip and multiply (or why seven divided by two-thirds is the same as seven times three halves).
7 ÷ 2/3 = 7 x 3/2
Someone in the audience yelled out, “Because you get the right answer.”
Over the past few years I’ve asked hundreds of people why, when dividing by a fraction, you can invert the fraction and multiply. Only a few have been able to give me a simple explanation. The vast majority will admit they learned this procedure as a rule that they never understood.
It usually takes just 20 minutes for me to teach even the most math-phobic child or adult why this mysterious procedure works. And I don’t have to explain much – I can almost always guide the learner to figure out the math for themselves by drawing a diagram and asking a few questions about the picture.
Most people believe that math is an inherently difficult subject – accessible only to people who are born with a “gift” for numbers or who display mathematical ability at an early age. They assume that topics such as the division of fractions are too complex for ordinary minds and that the majority of brains are not suited for solving problems in math. But imagine how backward our society will appear to be if one day people discover that math is actually the subject in which learners of all ages can most easily unlock their true intellectual potential.
If it turns out that everyone (or almost everyone) can learn math, we may have trouble explaining – to our children, for example – why we let so many students struggle in the subject.
We already know that for young children, success in math is the strongest predictor of success at school and that the abilities and perspectives students develop in math can be applied in all areas of life.
Not knowing how to think mathematically makes people, on average, less healthy, less financially secure, less innovative and less productive. Innumeracy damages the economy and degrades the environment. All of these facts are well established by a large body of scientific research that we have inexplicably ignored. And there are other benefits of numeracy, including an appreciation of the beauty of math, that are harder to quantify.
We may also have trouble explaining to our children why we believed that math is too hard for most people when the research clearly suggests it isn’t.
For young children, the strongest predictors of later achievement in math involve very simple skills and concepts – such as counting or associating a numeral with a quantity – that every person will almost certainly develop. And brain scans of mathematicians have revealed that expert problem solvers activate the same neural networks (and rely on the same primitive sense of space and number) that young children use when they think about math. As well, logicians have shown that even the most complex mathematical concepts can be unravelled into extremely simple conceptual threads.
All of these results suggest that math should be accessible to every brain.
Research in cognitive science also suggests there are more and less efficient ways to learn math. Lessons that cause “cognitive overload” by pushing learners too far outside of their comfort zone, or that fail to provide consistent feedback and support (called “scaffolding”) for learners, can be highly inefficient.
Teachers are sometimes blamed for poor results in math, but in my opinion they are not ultimately responsible for these problems. I believe teachers should be commended for helping their students as much as they do, especially when the resources and methods of instruction that they are required to use are not typically designed to close the gap between students.
If teachers had more authority to test approaches that have produced positive results in rigorous studies, I expect they would help even more students.
Methods of teaching that guide student learning shouldn’t be confused with rote learning. When a teacher helps her students to see connections and make discoveries using well-structured questions, activities and exercises, the students do the thinking, not the teacher. As students develop the confidence and conceptual knowledge they need to do more challenging work, the teacher can let them struggle more (and direct their own learning more).
Several years ago, Jump Math, which uses a form of guided instruction called “structured inquiry,” participated in a large randomized controlled trial in Ontario. Oddly, the U.S. Department of Education provided funding for the study. U.S. scientists believed there was enough evidence behind the methods of teaching used in Jump to justify a rigorous study – in another country.
In the second year of the study, in Grade 3, students in Jump classes made significantly more progress in solving the kinds of problems that appear on the Ontario provincial exams than students in the control group. The study, which was published in the scientific journal PLOS One, provides more evidence for the view that the best way to help students become strong problem solvers is to rigorously guide their learning.
One reason teachers should look for ways to help all students learn math is because students learn more efficiently in classrooms where there are fewer visible academic hierarchies. As early as kindergarten, children start to compare themselves with their peers and to identify some as talented or “smart” in various subjects.
Children who decide they are not talented will often stop paying attention or making an effort to do well. The cycle is vicious: The more a person fails, the more their negative view of their abilities is reinforced and the less efficiently they learn.
Psychologist Carol Dweck, who has drawn attention to the role that attitudes play in learning, once watched me teach a problem-solving lesson on perimeter. She said the lesson implicitly incorporated “growth mindset” principles and observed that “the kids are moving at an exciting pace, it feels like it should be hard but it’s not too hard for them. … They all have the feeling of progress and they all get the feeling that, ‘I can be good at this.’ ”
In the first few minutes of the lesson, I realized that 20 per cent of the Grade 6 students couldn’t find the perimeter of a simple L shape. But after 45 minutes, they were all enthusiastically solving problems at grade level and could even explain how they found their answers.
If I had been teaching a different subject, especially one that required strong reading skills or extensive background knowledge, I might have had trouble getting all of the students doing the same work, even if I had many lessons to work with them. But in math, there are usually only a small number of skills or concepts that I need to review, or misconceptions that I need to catch, in order to include everyone in the lesson.
Rather than thinking of math as a subject that is accessible only to the brilliant few, we need to recognize it as a powerful educational tool for creating a more equitable society. There is no other subject where we can mitigate differences between students so quickly or build productive mindsets so easily.
Everyone should have a right to fulfill their intellectual potential. The research suggests that math is the subject in which the vast majority of people could – if teachers were empowered to use evidence-based methods in their classrooms – enjoy that right today.