A few weeks ago, a factoid started bouncing around my social media feeds highlighting the lack of perspective people generally have about large numbers – specifically how much bigger a billion is than a million. The various versions explained that while 1 million seconds is about 11.5 days, 1 billion seconds is about 31.5 years. It’s a great example of how scale can slip away from us and how simple calculations can provide perspective. I started looking for other examples and wound up learning just how different other students’ introductions to back-of-the-envelope calculations were than my own.
My own introduction came from the arithmetic of paying for lunch. Looking at $5 per day for lunch may seem like an affordable amount – just $5. But what happens when you multiply that by the ~260 work days in a given year. That lunch suddenly costs $1,300 out of your annual budget. Two bucks for coffee? $520. Increase daycare costs by $10 per week – that’s another $520. When people want you to join a committee or club and suggest finding a sitter, getting a weekly sitter for three hours at the average ~12/hr will run more than $1,800 for the year. For many families, the back of the envelope is where first-order financial decisions get made.
But when I looked online for similar experiences, most of the stories I found came from college campuses instead of kitchen tables. Stanford holds these kinds of calculations in such high regard that they required all incoming physics majors to enroll in “Science on the Back of an Envelope.” Carleton College has a suite of resources for using simple calculations and estimation techniques to introduce geoscience students to time and length scales that might be hard for them to imagine. Whether calculating the cost of diapers or figuring out how to lift James and his Giant Peach with seagulls on strings (an actual assignment in the Stanford course), these calculations take an abstract idea and break it down into chunks. They provide the opportunity to make estimates about relevant details and create a sanity check.
Both science and life have a place for precise calculations, but initial guesstimates provide a place to start. Asking kids to estimate how long it would take to walk to Disneyland opens up a huge question and a series of small assumptions and answers. There are pieces to start with – starting location, a specific distance to Disneyland, hours in a day – but then they have to start breaking it into chunks. How tall is the person walking? How long is their stride? Pick some round numbers that seem reasonable. Or maybe start with an average time it takes them to walk a single mile. How many hours could they go without stopping? Will they sleep or eat? At some point they need to stop adding new details and settle on an estimate that seems good enough.
Practice at breaking down silly or low-stakes problems makes breaking down the bigger ones seem less scary. It also seems more feasible to check something that doesn’t seem quite right when you have done it before. When a kid learns to knit and wants to sell scarves, they need to know how to break things down and name a price. Fifteen dollars might sound like a nice amount of money, but a scarf that takes 4 hours to make would be $3.75 per hour without the cost of yarn. How many more bags of dirt would it take to fill a backyard garden, if you increase it by a foot on each side? How many hours per week will a person lose if their daily commute increases by 20 minutes? When people who have had practice with smaller problems read something or hear a story on the news later, they can perform their own sanity check. The huge price tag for a statewide education program may not seem so outrageous when considered in dollars per student, or per taxpayer. They can recognize that comparing millions of dollars for certain things to billions of something else is a great way to hide a huge difference in scale, and they can take a few minutes to break down the math themselves. And - as those same social media feeds later pointed out - when they hear a company is worth a trillion dollars, they can use the fact that a trillion seconds would be more than 31,709 years to gain some perspective.