When I went back to school to get my math endorsement, my teachers in several classes asked us to use our 'mathematical intuition'. The term caused me to furrow my brow. What is mathematical intuition?
I knew of intuition. Merrill-Webster defines it as a natural ability or power that makes it possible to know something without proof or evidence, a feeling that guides a person to act a certain way without fully understanding why or something that is known or understood without proof or evidence. Going with your gut. Your first instinct. You have a feeling. A sixth sense. Back when I was in the pre-married world and dating, people would tell me, "When you know, you know."* People trust their intuition.
The part of definition that says "without proof or evidence" seems like a contradiction to mathematics. However, after taking a few courses, I understood better what mathematical intuition is. It's building a trust in your own mathematical abilities. In the past we've used words like 'reason' or 'common sense,' but those do not encompass the feeling of confidence that people should have in math and their ability to do math. "That answer feels wrong." "Something doesn't add up." "Something has gone awry." Having good mathematical intuition helps you see when the math isn't right. And with math all around us - life is a story problem- being able to trust your mathematical intuition is important.
So here are 3.14 ways to improve your mathematical intuition:
1. Recognize Patterns: A key part of mathematics and the applied mathematics all around you is recognizing patterns. I'm one of those people who looks at how gas prices move and wonders what was going on in the geopolitical world that caused the movement. I look at big patterns. When are milk prices the lowest during the year? When are certain vegetables in season? How much is packaging and product shrinking? There are also small patterns. How fast do my girlios grow through shoes? How many times does my dog have to bark before someone let's him out? How many calories am I eating daily? We recognize patterns in people's and animal's behavior. Start recognizing patterns in numerical behavior. There is no need to obsess in the patterns. Just recognize them and their anomalies.
2. Estimate: In order to guess better guess often. On the days that we go grocery shopping together, my husband and I over/under the bill. Guess on how much gasoline will fill the tank. Guess on how many steps to the mailbox. Guess on how long it will take you to get to work or school. Guess how much that new phone will cost with its down payment and installments. Make those numbers real. When I was in my science education classes in college I read about the Nobel Prize physicist Richard Feynman.** After Richard and his father read about T-Rex's height in the encyclopedia Richard's father explained to Richard that the dinosaur's head would fit through their second story window. I loved that. Richard's father was making the numbers real. Estimating not only can be used for guessing answers. It can also be used to show ratios, to equate values, to make numbers tangible and to build mathematical intuition.
3. Try and Err: To build some trust in your mathematical intuition, you have to test your mathematical intuition. Just like your regular intuition builds off of what you have already experienced, your mathematical intuition needs some experience. Positive experience. I do a lot of math tutoring on white boards. Why? Because white boards are forgiving. Easy to erase. People have a hard time making mistakes and getting past them. In math, you may have to make a lot of mistakes before getting it right. And when you get it right, you probably won't trust yourself until you have done it enough to feel comfortable with it. This may take some time. Just like breaking bad habits and replacing them with good habits takes repetition so does building your mathematical intuition.
0.14...Don't Fear Fractions: Fractions may not become your friends, but they are necessary. We use them in money without a second thought. A quarter really is a quarter of a dollar, 1/4. Cents are one hundredths. Decimals, which seem to make people much more comfortable, are really fractions in disguise. Decimals are fractions hiding in a base ten system. You often buy things in fractions without knowing it. Milk is sold by the gallon. Meat and vegetables are sold by the pound. Gasoline is sold by the gallon. Those are all fractions. Yes, the word 'denominator' seems like it has the word 'demon' in it, but it doesn't.
Good luck improving your mathematical intuition. Have your pi and eat it too.
*That always caused me to furrow my brow too. 50% of marriages end in divorce. Should I trust my 'know'? Statistically, intuition didn't seem very trustworthy, and yet people often trust it wholeheartedly. There are times when my normal intuition and mathematical intuition collide.
** Richard Feynman's books are fantastic. He's writes well and is very down to earth.