When was the last time your instinct answered something before your brain did? Maybe you tossed in the right pinch of spices to a recipe without looking. Maybe you instinctively knew how to spell something without looking it up. Or maybe you brought your groceries to the checkout, and your mind estimated within a few dollars how much your bill was going to be (actually…maybe we’re all under estimating a little bit these days).
Intuition can be a powerful thing. We might know people who seem to be naturally gifted to be more intuitive than others. That can be intimidating. And while to some degree intuition may be a gift, intuition is also both teachable and learnable. Can intuition be taught in a subject as laden with rules and memorization as math?
After all, how can math be intuitive? You can’t instinctively know that the diameter of a circle is equivalent to a ratio slightly greater than 3.14 without calculating it, and once you’ve calculated it, you won’t know next time without committing it to memory. It would be difficult for the first person to come up with the long division technique to simply know how to do it intuitively. It requires figuring. And following figuring, it requires memorization.
Let’s take a look at the definition of intuition. Merriam-Websters defines it as:
a. the power or faculty of attaining to direct knowledge or cognition without evident rational thought and inference
b. immediate apprehension or cognition
The first definition seems like it’s talking about knowledge pulled out of thin air. You may not be able to determine the intercepts of a parabola this way, but perhaps this type of intuition helped the great mathematicians of old bank on an intuitive belief that there was a mathematical order to the world.
But take a look at the second definition: “immediate apprehension or cognition.” This is the ability for a new truth to be easily plugged in to your existing framework of understanding. It’s the ability to pick up new knowledge quickly, because to some degree, it already fits with your expectations—even if you were previously unaware of them.
One way that we start to teach this is through problem solving strategies on basic questions that can be used for more advanced questions later. We rehearse visual modelling techniques so that when students approach a type of question they have never seen before, their brains already have the tools to start mentally mapping the scenario. Familiar techniques can also be used by instructors when explaining how to solve a question in a way that students don’t have to struggle to follow.
We also emphasize the use of reason, which is very closely related to intuition. We practice breaking down similar questions in several different ways rather than sticking exclusively to one “correct” approach. This trains students not only explore a problem from different angles, but to look for the most efficient way to solve a question based on the variable parameters of the question.
We encourage estimation when approaching a question, because making a reasonable guess sharpens a student’s eye for picking out unreasonable answers. If their mind is already calculating an approximate answer, this can act as a bit of a self-guide as they determine if they are on the right track with a question. They won’t need to wait until they get a final answer to check if they were correct; the better they are at estimating reasonableness, the more they can check in with their methods along the way.
Our strategies develop not just an understanding of a concept, but a three-dimensional understanding; rather than the memorization of problem solving techniques, we prioritize transferable application. All this can help build towards that first definition of intuition:
a. the power or faculty of attaining to direct knowledge or cognition without evident rational thought and inference
In mathematics, new knowledge is never created―only discovered. Math is numerical archaeology, digging deeper into the truths that are buried in the universe to discover the architecture of the way things are. New knowledge cannot be intuited out of thin air, but the most advanced mathematical minds can use this kind of intuition to deduce mathematical truths that were previously untaught.
While the powers of mathematical deduction may be a rare gifting of the mathematicians among us, we can steer our learners towards the same patterns of thinking that turn mathematics into more low-hanging fruit―ideas that aren’t too much of a leap from where they are currently standing. The more our students learn reasonableness, problem solving, and flexibility in their approaches, the stronger their framework of understanding becomes―the stronger their framework, the stronger their powers of cognition, intuition, and deduction.
And who knows―the more confident they become in their intuitions, the more that thinking might just spill over into other areas of their lives. They may even be looking over your shoulder one day and telling you how much spice to put in dinner.
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