A powerful principle for developing “math sense” in physics and chemistry
BY ROBERT OÑOZ
What do we teach as physics and chemistry teachers? Some say it’s the study of matter and energy. Others say it’s the study of space and time. Personally, I believe what we’re teaching is just common sense.
The trick, however, is that common sense is not always obvious! More times than not, common sense actually takes a great deal of studying, practicing with examples and deriving complex equations from the ground up. Between tough concepts, long problems and abstract mathematics, the physical sciences can make a student’s brain spin (with great torque!) if you’re not clever in how you present the material.
Some teachers try to rectify this by reducing the amount of math in their courses. Others double down on math and devote whole lessons to introducing calculus to high school students.
I propose a third way: using mathematics as an extension of physical intuition.
“Math sense” is one of the greatest superpowers you can give your students. When I write a number like 1023 on the board, I pay careful attention to how my students react. If they stare at the number like dry paint on a wall, I know I have failed at conveying how numbers should feel in their mind.
To put this number in perspective, the average student has 10 fingers—let’s call that 101. A small classroom of 10 students would, by common sense, have 10 times this number of fingers, 100 fingers in all—let’s call that 102. Ten small classrooms would have 1,000 fingers—103.
A pattern has obviously emerged and, if you are a careful reader, your sense of magnitude for the original number I wrote on the board has shifted tremendously.
If we return to our original number, 1023, we now recognize this number as one hundred sextillion or a one followed by twenty-three zeroes (100,000,000,000,000,000,000,000,000,000,000). To have these many fingers, we would need ten sextillion people (100,000,000,000,000,000,000,000,000,000,000 or 1022 people) to raise their hands. With only about eight billion people on our planet (~8×109 people), we need about one trillion of our Earths (~1012 Earths) to reach that number of fingers.
Why overwhelm students with such unfathomably large numbers? Am I not creating more cognitive demand for my students? I stop and let my students feel the utter magnitude of this number. Then, I give them the kicker: “Although it would take a trillion Earths worth of people to hold up 1023 fingers, you can hold 1023 atoms in the palm of your hand.”
The overwhelming dread of numbers now transforms into shock and awe. The enormously large has now transformed into the conceptually small. You haven’t told students how small atoms are, you’ve made them feel how small atoms are. You’ve built an intuition for the physical world. More importantly, you’ve replaced a fear of numbers for an intuition with numbers—the “math sense” that students need to be successful in physical science.
Students can now progress through their physical science courses with more confidence in their sense of numbers. If their calculators say a car weighs 170 newtons, they can say, “Wait a minute. If Mr. Oñoz says one newton is about the weight of an apple, there’s no way a car can weigh 170 newtons. That’s just not enough apples!” A quick math check can confirm their suspicions that the weight of a car should really be 17,000 newtons, not 170 newtons.
Numbers can easily be used to extend students’ physical intuition about the universe, especially if those numbers tap into students’ previous real-life experiences.
Robert Oñoz is a student at Montclair State University and an NJEA Preservice ambassador. To learn more about NJEA Preservice, visit njea.org/preservice.