The negative reaction I’ve seen to the notion that it’s teaching kids to count change leaves me scratching my head. Through almost the entire 90s, I worked retail, and one of the most common complaints from the older folks was, “What’s the matter, can’t you just count out the change in your head?”
Well, no, lady, it’s that I could lose my job for doing that, if I screw up in a rush. I could do it in my head just fine. I’ve run into plenty of cashiers who fail the Dilbert test, though.
It’s considerably less mental overhead, though YMMV. I’ve fat fingered a cash register more than once when punching in the cash I’ve been given. Suddenly, the twenty dollars I was supposed to punch in is $265.65. Generally it’s easier to count in sets of “complete tens” or “complete fives” and working out the differences in the tail of the process. It never takes more than a few seconds, but when I first started teaching myself how to do it seemed more complicated than it was. When I took calculus my university didn’t allow calculators and this method served me very well, even with paper and pencil it was faster for a lot of calculations.
Carrying and borrowing doesn’t work out well for me mentally because I always end up mucking up my memory of what numbers got carried, added to, etc. I try to hold a mental image in mind but I get confused quickly.
Coming back from a casino I made a $2.50 purchase. i handed the cashier a Sacajawea dollar, a Susan B Anthony dollar, and a JFK fifty cent piece. The look on their face was priceless.
Suppose you buy coffee and it costs $4.30 but all you have is a $20 bill. How much change should the barista give you back? You sure as hell aren’t going to get out a sheet of paper …
I can do this in my head, because I learned the Old Math.
Okay, snark aside, the new method seems useful as a double check, or if something is too hard early on (14.73 - 5.92 = 0.08 + 8 + 0.73 = 8.81). In fact it seems much more useful in this case than the example given. Though you are probably going to need to get out a sheet of paper and write down those intermediate sums. It does seem useful as a learning and visualization tool (what are we actually doing?).
Finally, this is a long way from what was briefly inflicted on me as New Math. That was all set theory.
The core of this system is a very strong intuitive grasp of working within five. My kindergartener is only being taught to manipulate numbers within five: addition, subtraction, basically algebra, but only in a system from one to five. Some of the parents in his class were upset about this, but the way the teacher explained it, to me, it made sense. Please start my kid off with being extremely fluent in operations and manipulating numbers, even if it means starting with one hand’s worth of numbers. When I saw this post, I thought, ohhhh… this is where this is leading.
There’s been many times I’ve handed people $2.14 for a bill of $1.89 precisely because I want a quarter back (for the laundry machine). In my mind, having the change be a quarter is easier for the cashier. Most look at you like you’ve just made their life really difficult though. They expect to get $2 and that the change will be 11 cents. They’re utterly perplexed as to why you added this “random” 14 cents into the problem. Most can’t easily see how this works out to be a quarter. Once they finally figure that out, they act like it’s magic.
The beauty of the method comes with practice. You’ll find that remembering the intermediate sums is easy because the pattern is uniform. You don’t use the memory space to remember the history of the process, any more than you use up memory counting up by fives. You remember 3, then 5, then 10, then 2. The other stuff you just dump as you process it. With carrying and borrowing you have to remember a more complete process history, which tens you borrowed from and which zeros have become nines etc.
I’m sorry, but there’s something wrong if you can’t subtract $4.30 from $20. By the time you’ve rounded up and all of the other things you’re doing with the “new” math, I would already be with my next customer.
I do hope that the math illustrated in this article is a departure from reliance on calculators. I’m struggling to recall a single math class, other than differential equations, that allowed them.
I wish Scott Adams had chosen better numbers for his comic, because Dilbert comes off looking dumb when the cashier just hands him his $5 back. A better example would have been a $9.87 purchase that he gave the cashier $10.12 for.
Converting “lesser” change into quarters is always a win for me. I call it the change game. If you come out of a transaction with fewer coins that you started with, it’s a win. Upgrading smaller coins to quarters is also a win. Not jangling around with a golf ball sized lump of coins in your pocket is always a win.
The only time it was difficult was Europe, where cashiers hated dealing with their overcomplicated coinage and gave only scornful glances when I handed them exact change instead of a nice easy bill.
That’s rather the point, though. The example is contrived to make the algorithm look foolish, precisely beause it’s a set of inputs where a different algorithm would be far better. The counterexample from Reddit skewers it precisely on this point, by contriving an example that’s just as horrible for the alternative.
No, you’d probably do it in two steps: “$4.30 is 70 cents short of five bucks, which is 15 short of 20. So the answer is $15.70.” I don’t see what the “all the other things” you’re talking about are. They don’t seem to be there.
Tim, my one step is still less than two steps.
There is no way in which the thing you describe is different from the “arithmetic jiggery pokery” that you seem to think it isn’t.
Unless your “one step” is having memorized 20-4.3, it’s not really one step, by definition.
Wouldn’t it be so much nicer if there was a decent value coin in regular circulation? Like, say, a dollar, or two, or five?
A quarter is worth jack shit. There need to be much larger value coins.
We’ve been here before: Quote from New Math, wikipedia:
In 1965, physicist Richard Feynman wrote in the essay New Textbooks for the “New” mathematics:
“If we would like to, we can and do say, ‘The answer is a whole number less than 9 and bigger than 6,’ but we do not have to say, ‘The answer is a member of the set which is the intersection of the set of those numbers which is larger than 6 and the set of numbers which are smaller than 9’ … In the ‘new’ mathematics, then, first there must be freedom of thought; second, we do not want to teach just words; and third, subjects should not be introduced without explaining the purpose or reason, or without giving any way in which the material could be really used to discover something interesting. I don’t think it is worth while teaching such material.”
"4622 - 1873
7 + 20 + 100 + 2000 + 600 + 20 + 2 or 2749."
See, when I do that mentally:
I bring 1873 up to 2000 by adding 127,
then take the 2000 off of 4622 to get 2622,
then add the 127 back in to get 2749
Which seems to me to be a compressed version of the math illustrated in the article. And since I am pretty good at math, this must be the right way. 
"It’s also not much different from the math you learned back when you were learning how to count change. "
I’m sorry but there is no math involved when you count back change. It is counting. Counting does not count as math in my book. Total bill $7.36. Get a $10 bill. Count out 4 pennies. Count out 1 dime. Count out 2 quarters. Count out 2 dollar bills. There’s your change.
It is how I have always done math for myself, and watching my oldest work on it I am just fine with him struggling. What I see is him working on and refining principles that once he has them will be automatic. However, it would be simpler and more straightforward to start teaching algebra as algebra has all the logic and algorithms one might need. No reason that can’t start in kindergarten, we do it verbally and concretely at home and progress to symbolic math as it is brought home from school.
