The instinct and intuition of mathematics
There were some people on a train.
Nineteen people get off the train at the first stop.
Seventeen people get on the train.
Now there are 63 people on the train.
How many people were on the train to begin with?
Periodically, as I’m scrolling through social media, I see problems like this one. In my opinion, these problems well, actually, the way they’re presented, is indicative of one of the greatest misconceptions about math. This particular example was in an article titled, “This Math Problem for 7-Year-Olds is Going Viral Because No One Can Figure It Out.” I understand the concept of clickbait. They want you to read their article, and they’ve got to make math, of all things, sound exciting. However, the comments generally only confirm my suspicions.
To quote the opening of the TV show “Numb3rs”: “Math is more than formulas and equations. It’s logic; it’s rationality. It’s using your mind to solve the biggest mysteries we know.” Throughout the past three and a half years of study towards a major in math, I have come to see the absolute truth in this statement. More than anything, math is a way of thinking. Certainly, there are formulas, equations, and theorems to memorize, but truly practicing math requires much more than just plugging in numbers to calculate an answer. These facts are not merely static, separate entities to be called upon in specific narrowly defined situations. Rather, they are a collection of tools to be used in tandem to solve a wide variety of problems.
Learning to think mathematically is developing an intuition that connects mathematical concepts and summons them almost instinctively. Undoubtedly there will still be difficult problems to solve, but the more one develops this mathematical instinct, the easier it becomes to find a foothold, or a place to start, even if this initial insight ends up being incorrect. Furthermore, the more one develops a mathematical intuition, the more its logical foundation will begin to enrich thoughts outside of the realm of mathematics. Again, “Numb3rs” reminds us that, thinking mathematically is about, “using your mind to solve the biggest mysteries we know,” and many of life’s greatest mysteries aren’t really mathematical at all. Nonetheless, a strong rational instinct can go a long way in contemplating them.
Let’s return now to that impossible train problem, and see how with a little logical intuition it’s really not impossible at all. The first step in approaching almost any problem is determining what you know and what you’re looking for:
There were some people on a train. Since we don’t know how many, let’s say for now there were ‘x’ of them.
Nineteen people got off the train at the first stop. Thus, after the people get off, there are x minus19 people left on the train.
Seventeen people get on the train. Once these people are on the train, there are a total of x – 19 + 17 people on the train.
Now there are 63 people on the train. This means that 63 = x – 19 + 17.
By working through the problem step by step, we’ve reduced it to a simple algebraic expression and can determine that the train originally had 65 people on it. See, not so difficult after all!
Although it can be discouraging wading through the comments on these “viral math problem” posts, the problems themselves give me a sense of hope. It seems to me that for one reason or another many people in today’s generation were taught math in the vein of rigid formulas and inflexible equations, and it seems that learning math this way often leads to a whole host of misunderstandings about math centered around impressions like boring, useless, and impossible. What I hope is that today’s kids are being taught that math is dynamic and flexible, and that it will lead a need generation of thinkers to view math as intuitive, valuable, and dare I say, fun.
