Of all the ideas we’ve forgotten from math class, the most important to relearn — because it illuminates our most urgent challenges — is the difference between arithmetic and geometric progressions.
Maybe you recall. An arithmetic progression is a sequence in which the difference between numbers remains constant. For example, counting by twos: 1, 3, 5, 7 and so on. Each number adds two more.
A geometric progression is a sequence defined by a constant ratio. Doubling, for instance. Instead of adding two, you multiply by two: 1, 2, 4, 8 and so on. Each number is twice the one before.
Technology advances in a geometric sequence — since the dawn of computing, anyway. Intel co-founder Gordon Moore’s famous law predicted that processors would double in power every two years. Technologists believe we may be reaching the end of that sequence, but so far, Moore’s Law has taken us from room-sized computers back when I was in grade school to the inconceivably more powerful computer I slip into a shirt pocket today.
But the functionality of most adult human beings grows arithmetically, if it grows at all. We won’t wake up tomorrow twice as capable as we are today, and twice again the day after that. We add knowledge bit by bit and gain experience by slow increments.
For decades, this has been a manageable difference. Humans began with a huge head start over computers. And if you look back to the examples above, you’ll notice that the sequences are similar in the early stages. At step four, for example, the arithmetic sequence has reached 7 while the geometric sequence has reached 8. With time, though, the gap explodes.
Thirty steps into the sequence, adding two gets you to 59.
Doubling gets you to nearly 537 million.
With that in mind, let’s look at the striking teachers of the Denver Public Schools. At first blush, the picketing educators are seeking a simpler pay formula, which would end a years-long experiment in paying bonuses to steer teachers toward targeted results.
At a deeper level, though, the strike illustrates the incremental rate of human progress compared with the geometrical tsunami of technological change. In 2006, Denver’s then-new bonus system was state-of-the-art, an innovation embraced by teachers, administrators and taxpayers alike. Now, in the time it takes a student to go from K to 12, it has become an anachronism — a creaky contraption linking pay to measures of ever-more dubious value.
It matters little whether this experiment was noble or cracked. What matters is how long it took. During the same 13 years, technology created the smartphone, the tablet and the cloud, which, in turn, enabled countless educational apps, games and other resources that — for the first time in history — make truly individualized learning a universal possibility. The slogan of Khan Academy (founded in 2008, two years into the Denver experiment) captures this revolutionary promise: “You can learn anything. For free. For everyone. Forever.”
So while Denver was fiddling with formulas (and many other districts have been trying far less), what’s needed is a blank-slate rethinking of what teachers do, how classrooms work and what schools are for. Students from inner cities to rural hamlets can now (or soon) experience the world’s most skilled instructors delivering optimal lessons — at the student’s pace, in the student’s language, at whatever time of day the student learns best. To leverage and augment such incredible resources, on-site teachers must become life coaches, role models, facilitators, therapists, motivators, demolishers of obstacles and openers of eyes.
These are not qualities easily measured by student test scores or accumulated grad school credits. Yet they point to something essential about the future relationship of humans and our technologies. We cannot outdo the computers in terms of standardized outputs or efficiency. We can only keep pace by emphasizing those things that make us human to begin with: our capacity for connection, compassion, empathy and love.
As a kid, I was lucky enough to know a gaptoothed band director named Byron Gillette, who passed away in Colorado last month. I thought he was teaching me to play a trumpet when actually he was teaching me to live a life. My sister and I recently recalled life under his baton, and we felt the same debt piled up over nearly half a century. “I remember Mr. Gillette having me play clarinet solos at church,” Lynn texted. “As a super-shy, super-awkward teen, it was really helpful.” She’s now a life-shaping educator herself.
How do we harness technology to help more teachers awaken more students to their possibilities and resources? How do we structure schools to create mastery of both the power and the perils of future tech? These are just two questions in the long arithmetic sequence of human learning — but the answers, once we find them, could work wonders.
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