It is 4:22pm on Wednesday, April 4th. An eccentric billionaire has gathered $5 billion in 500-dollar bills and proceeds to hand them out, one each second, without stopping. What is the exact date and time when the last bill is given away? -A puzzle I got from Mo Page

I’m flabbergasted. I have a number of students—maybe 10? 20?—who determine by division how many bills there are, then figure out by multiplying 60x60x24 how many bills are given away in a day. Fine. But then they start subtracting … after day 1 there are 9,913,600 bills left. After 2 days there are 9,827,200. Almost immediately many students lose interest, but there are a few arithmetic ox that start chugging through it (with calculators, to be sure). 9,740,800. 9,654,400. I watch in disbelief as the markerboards are filled in, line by line. 8,617,600. 8,533,000. After a while I can’t help myself. I casually mention that people sometimes use division to do repeated subtraction, and I countdown from 10 by 2’s and compare to 10/2. They are a little chagrined at not having thought of that, but they try it. Then they face confusion about handling the remainder.

I don’t believe math must be learned in a particular order. My seniors don’t need to model repeated subtraction with division in order to learn the basics of trigonometry. But, really? Should I really keep teaching precalc instead of throwing it out and teaching pre-algebra? I don’t even know how to teach pre-algebra! But I’ll learn … if that’s what’s needed.

What should be the primary goal for instruction of students who are placed at one level but who are missing huge chunks of what they were “supposed to have learned” years ago? I feel like I keep asking this same question in different ways on this blog. Instead of elegantly reasoning my way to a solution, I just keep doing what I know how to do, day after day, hopefully making progress, gobsmacked at the glacial slowness.

Brute force. Them and me both.