If the customer doesn’t, how do they know when they’ve been stiffed?
I also do it this way. Around freshman year in college I just started doing it, not sure why, I guess the stress of certain exams caused me to start doing it the fastest possible way.
The problem with the by hand technique is you make errors. If you can just mentally visualize the numbers adding, subtracting, or multiplying/dividing in your head, by using low resolution representations of the number, you can check your work for errors. You can also use it to eliminate choices on multiple choice exams.
We have calculators. There is no longer any use whatsoever for an exact on paper technique that works for any fine decimal number. You need a rough technique to save time and check for mistakes, and a fine technique that involves pushing buttons on the calculator.
One of my favorite short stories:
[QUOTE=Isaac Asimov]
“Aub! How much is nine times seven?”
Aub hesitated a moment. His pale eyes glimmered with a feeble anxiety. “Sixty-three,” he said.
Congressman Brant lifted his eyebrows. “Is that right?”
“Check it for yourself, Congressman.”
The congressman took out his pocket computer, nudged the milled edges twice, looked at its face as it lay there in the palm of his hand, and put it back. He said, “Is this the gift you brought us here to demonstrate. An illusionist?”
http://www.themathlab.com/writings/short%20stories/feeling.htm
[/QUOTE]
I am about to make perhaps the most pedantic edit I have ever made:
The bolded words should be replaced with “with low frequency of error”.
Any reason you passed up the chance to make this point in your reply #s 70, 73 and 76? I guess we should just drop the math requirment altogether, huh? That’s that that Lockhart guy thinks, so some might consider you to be in good company.
As a practical matter it does not benefit the customer to audit for penny-level errors. As a matter of principle (and law!) the business is supposed to deliver penny-level accuracy in all its accounts.
I didn’t write post #70, but as for why I said “The business does so with cash registers and computers”, you brought up the business’s need to keep track of to-the-penny totals. How should I have replied to that other than to point out the salient empirical fact as to how businesses do so? Was I supposed to refrain from pointing out how your comment did not support your position?
I consider Lockhart excellent company. There are some things worth hammering into children universally. Some of these are even in math. The basic conceptual understanding of arithmetic, for example; that is, what addition and multiplication and so on mean and how they behave, how to interpret decimal notation, etc. But there aren’t a lot of such things, I dispute the contention that drilling execution of standardized decimal notation arithmetic algorithms is among them, and I am in general perfectly happy to drop, at every level, requirements that force people to sit through a bunch of classes and homework on things they don’t care about and have little substantive need for.
Sure. This was the attitude I espoused in the comment of mine you originally took issue with…
My school was racially integrated in the 70’s and all of us could make change in our heads using the traditional method and that was in grade school. It was considered an easy task back then. Not sure why we didn’t break math down by political affiliation and skin color. Would have seemed racist and politically antagonizing back then.
But thanks for sharing.
Yet I’ve had a clerk hand me back $19.06 for an $18 purchase paid for with a $20. Didn’t blink an eye. It took a great deal of patience to walk her through the error.
There’s nothing new about math. In fact, you can use math to calculate which teaching method is most effective. You can also use math to calculate how much time is saved by only using one method versus 8 and which version works best with various learning disabilities.
Teaching ever person every possible method is both time consuming and confusing.
Do you suppose the problem was that this person was not exposed to traditional subtraction algorithms in their childhood math classes? It seems the problem was instead that they had not developed “number sense”, as it was called previously in the thread. Perhaps even the emphasis on “Shut up and run the algorithm!” had interfered with their development of such “number sense”; a manually implemented black box would still be a black box.
If the point is instead merely that it is possible to erroneously use a calculator, it is trivial to note that people also make errors when performing manual arithmetic. If the response to this is “They should be taught to do manual arithmetic carefully!”, my response to that is “We could just as well try to train them to use calculators carefully…”.
Math a 3rd grader could easily do in my era of learning. I doubt any of us use the word “algorithms” or “number sense” much in a sentence. But then we could make change without a calculator. So there’s that.
but carry on with your explanations and theories.
I will!
It seems most likely that what happened is that the clerk (correctly) determined the final price with tax included, then conflated this with the amount of change to return, forgetting to move on to the next step of calculation incorporating the amount paid. How would doing subtraction by hand rather than by machine have helped here, when they didn’t even do a subtraction in the first place?
(Also, I don’t much care how easily people could make change without a calculator back in the day, on its own; I don’t even care if the number of people who could make change with access to calculators denied has decrease over time. Show me that the number of people who can make change with the tools around them now is less than the number of people who could make change back in the day, and then I’ll be interested…)
Missing letter restored in bold:
I really don’t think change making has much to do with your second grade math education. My mom (in her 60s) reckons the difference you see in change making skills has to do more with store training. My mom said plenty of people (including her) were bad at arithmetic when they got their first jobs in retail in high school or early college, but the stores themselves trained people how to correctly bag items and make change efficiently.
When she got a second job at Target a few years ago for some extra money, one of the things she said is that she doesn’t think the newer hires were any better or worse than she was when she took those jobs; it’s that nowadays they get absolutely no real training in anything.
This made me laugh out loud! Thank you!
<blink>. <blink>. Can you explain to us how she got her answer? I can’t see it.
what happened is that she saw a number on a piece of paper and just decided that was the change.
Got it. can’t come up with anything to say so you’re moving on to dismissing math skills.
It’s Friday night. We’re done.
Twice now, people have conflated math with execution of standardized decimal notation arithmetic algorithms, while at the same time acting like it is a strong burn on me to note that I continue to voice a position I have made clear that I hold.
Artihmetic skills are not math skills. They are a subclass of math skills, but it’s nowhere near a universal thing. Equating being able to quickly add numbers with “math skills” is a very Hollywood way of looking at it. In some ways, having good arithmetic skills strikes me as being the regional spelling bee champ. Cool, and it shows some degree of persistence, intelligence, and dedication. But mostly a neat parlour trick. I do research-level mathematics and there is a wide, wide range of arithmetic skills among people I interact with. If specific numbers are actually important you get a computer to do them, and probably also have a second person verify your results for safety. Especially if the number seems like it should be wrong for a given domain.
Knowing when an answer is “reasonable” is by far the more common skill among mathematicians. In that way, the cashier really should have known that $19 in change is probably not correct when handed a $20 bill for a charge of over a buck, regardless of whether she could actually calculate the exact change value in her head (which most POS terminals do for you nowadays).
I had a friend with a physics instructor who would take off points if you didn’t explain why your answer seemed reasonable on a test. They would also give a small amount of partial credit if you got the exact answer wrong but could explain why it seemed wrong and unreasonable. (“It makes no sense for the ocean to freeze solid in 15 years” and such). I think that’s a pretty reasonable assessment of knowledge and skills in an area. Adding arbitrary scalar numbers doesn’t often come up, even in hardcore number theory.
It actually goes something like this in your head…
225 + 300 = 525, 35, 45, 55, 556
You don’t have to carry numbers or rely on remembering the result of a previous computation (like your 25 + 31 = 56}.