Peter Grant had a piece about the frustrations of people who depend on computers and can’t (or won’t or don’t have time) to double-check what went into the computer and the numbers that come out. When DadRed was tutoring in the public schools, he encountered an absolute absence of the ability to estimate and round.
Now, I am not exactly innumerate, but close. Doing math in my head hurts. I do it, and do it often, but it is not easy and requires complete concentration. When that’s not possible – say, when I’m in the grocery store – I estimate.
Why this is no longer taught escapes me, but I’m sure there are modern, up-to-date pedagogical reasons. Or it’s not on the standardized tests, so no one has time to cover the material. Which is really a shame, because as my readers well know, it’s one of the most important life-skills you can have up your sleeve.
When I had a hard grocery budget of $35/week, I rounded all prices up if they were more than twenty-five cents. This allowed for sales tax. And I’d add every time I put something into the cart or basket, making certain that I was below my weekly limit. It worked very well, but it was tricky keeping a mental list of the prices of everything constantly in mind. I rarely went over budget, and those weeks usually coincided with deep-discount meat, so I could amortize the cost over two or three weeks. When stew meat hits $1/pound, you go a little wild, or at least I do when I have freezer room.
I almost always round up. It’s safer when dealing with money and budgets. Round up expenses, round down income, and you have a safe margin. Round up the stress load, round down the estimated materials strength, and your home-made scaffold is less likely to fall down with you on it. An old Boeing mechanic who had worked on the 707 and several other planes chuckled about the 707 being overbuilt. “It was the last commercial airliner designed with slide-rules, and everyone rounded up at each stage.” Made perfect sense to me. Especially for something no one has done before, allowing a little wife-and-kids-room in calculations is safer.
But, I grew up without being allowed to use a calculator until I was in college. I had to learn the multiplication tables, and one of the reasons I have trouble to this day is that I didn’t. When I changed school districts, I no longer had to learn everything cold by rote. So I didn’t, and I managed to bluff my way through until I hit algebra and bounced. It wasn’t until college and trig that I ever recovered. Trig was story problems. I can do story problems because there is a reason for them. I like my bridges to stand up, and my flight path to end up where I had intended for it to. (Ah, vectors, how I hated thee until I encountered wind correction angles!)
“An old Boeing mechanic who had worked on the 707 and several other planes chuckled about the 707 being overbuilt.”
I wonder if that’s part of the reason that 707-based airplanes are still flying today, 65 years after the design was created.
I suspect so. They are a touch overbuilt.
The elementary school had addition and multiplication charts on the walls and taped to every student’s desk. And then let them use calculators in first grade. When my wife and I went to parent-teacher night, we went to our kids’ seats and cut off the charts. And moved the charts around the room so our kids wouldn’t be able to see them from their seats. And put the calculators in the teacher’s desk, with strict instructions not to let our kids use them.
This is the reason they can do math in their heads now. Despite the best efforts of the State school board mandating New Math, err, I mean Common Core.
When I graduated High School, my graduation present was a Post Versalog slide rule. I had used a cheap one in high school for chemistry and physics, but not trig. Engineering college would require a professional slide rule. Estimation is a required skill when using a slide rule as it would tell you the raw digits, but not the order of magnitude. For example, is the answer 34.2 lbs, 342 lbs or 3,430 lbs? It makes a difference if you’re building a bridge. Also important is the ability to know what is realistic with regards to input. If you design an HVAC system to provide 50 cubic feet per minute of cool air to a room with a 100,000 Btu heat source, you missed something. Calculators provide the numbers, but not the brains.
There are some very nice mathematical tools on the computer these days. Such as Matlab, ANSYS, and R.
Are these alone sufficient for all tasks? No.
Do people try to use them by trusting them blindly, yes.
They are actually trying to teach rounding in common core– the Princess is fighting it, so about 4th grade?
It is hard to teach “approximately.” Khan does it by teaching to round and then add that. The Duchess is finding it useful for remembering zero is nothing, but still important.
Zero isn’t always “nothing” – most of the time, it’s a power of 10 (plus or minus).
*it* is always nothing, though; we just have it visible, sometimes, to let you know that something to its left is bigger than it looks.
(Ever had to explain WHY you can’t add that one  to the other one  and why you have to write down the zeros between? It is a brain bender. We got it, eventually.)
I know my little twin cousins (21 now) had “estimating” in school. They both struggled in math (a lot had to do with poor teachers, and switching every day so they never got anything down) and then they get to “estimating” where if they wrote down the “correct” answer instead of the “approximate” answer, they were marked wrong. Drove them and my mother (their babysitter who got to help them with their school work) nuts.
This is where Khan academy is AWESOME.
The choices are multiple choice, and if the idea is “round to the nearest 100,” there will only be 100s.
No trick “only rounded one” or “is the actual answer.”
The switching all the time is THE fatal blow for Common Core. It simply doesn’t allow enough time for kids to be comfortable.
Common core doesn’t teach the basic math. You have to have that down before you can effectively estimate.
The standards boil down to “teach ALL the math!”
The actual implementation is more like “have the kids do a page or two of practice in a wide range of methods the teacher isn’t comfortable with and can’t actually explain, and then blame parental involvement when the kids are confused.”
Trying to be fair, I only figured out some practice stuff from school when my eldest hit it; if the kid is smart enough, they can fake understanding on accident.
This is a skill I use often – but I can’t remember when or how I learned it.
I often don’t bother with a calculator since my mental estimates are close enough.
I have said for a while that our society is sorting itself into two sectors based on knowledge and ability – it will eventually become a division of haves and have-nots, but it isn’t there yet.
Maybe the Morlochs could do math and the Eli couldn’t?
I used mental math when I worked retail: Taxes, and other percents. It got easier because I’d have to do the same calculation over and over. I understand geometry (the logic part of it) but algebra is about my speed.
Alma, how did you learn vectors if you find some types of math difficult?
Because we have mechanical slide rules that calculate the basic correction angles for us. Once in the air, the flight-path gets adjusted for what the actual conditions are, because winds-aloft forecasts are never, ever correct. Either the direction, the speed, or both are off to some extent.
My mental math is poor, yet it constantly amazes people. Being able to do a little mental algebra helps. And basic vector mechanics is fun.
I tell people that it’s not to hard multiply 16-digit numbers in your head–unless you insist on getting the right answer, and Leonhard Euler got the right answer. Someone should write a good book about all of the math he gave us. (=Dr. Euler’s Fabulous Formula= is too specialized and advanced for the general reader.)
When I went to college, I was in the first year told to get a calculator instead of a slide rule. (The SR-71 ruled the day, unless you wanted to shell out for an HP scientific.)
The greatest unsolved problem today–and for the last hundred and thirty years–is the proof or disproof of the Riemann Hypothesis. THAT problem stands proximately and squarely on Euler’s work–on a remarkable and breathtaking transformation.
…. and THAT ain’t going to fall with a calculator……
I am so old that when I was in fourth grade, every morning started out by drawing a twelve-by-twelve grid on a sheet of notebook paper. When Miss Abney said, “Go!” we began filling out the grid, from 1 X 1 = 1 to 12 X 12 = 144. Can’t remember how long we had to do it, but it was short. Nowadays this would be considered child abuse three different ways.
We had a grid-shaped toy with plastic sliders at home. You could move the 7 slider on the left side and the 5 slider on the bottom, and the number 35 on the grid would appear.
Okay, we still.had to memorize the times tables, but it was cool.
I am TOTALLY stealing that!
Always round for worst case. Fuel rounds down, weight rounds up with a reserve on top. Winds are worse than forecast, weather drops sooner, icing where they say it won’t be yet. Nobody herded the cows off the runway, the spare got traded out for a beer, the mechanic isn’t in today, and the bribe will be triple because your factor paid the loser in the struggle that happened while you were enroute. Paycheck’s going to be late, the wound always gets infected, and there’s never enough pepto bismol to dose the whole crew!
Don’t forget that Murphy’s acolytes will “upgrade” something beyond its design basis, or repurpose something for a purpose never intended by the designer. Yet somehow when it fails, it will be the design engineer’s fault.
The plot of The Pentagon Wars.
I’m dyslexic. Estimation is a necessity, because if I don’t have a good idea what the answer should be, I’ll get lost once the numbers start wandering around on the page.
Which did sometimes cause problems with teachers who hadn’t taken much math themselves and who wanted me to “show my work”.
(It also memorably got me accused of cheating off of other students a time or two. Fortunately, having a much higher test score than anyone sitting anywhere near me was a defense that was really hard for the teacher to overcome. And boy, did they get frustrated as they watched me like a hawk while never understanding how I was “getting over on them”. Even after I demonstrated the process on specific problems a number of times.)
Slide rules were great – we’d multiply 2 by 3, read off “looks like about 5.99, let’s call it 6…”. But we did have to worry about those pesky decimal points: like when it was 2000 by .003…….
One time I was calculating how much power an electric motor needed to move a big door. I did the calculations, according to the tables, came up with a result. I talked to a Professional Engineer friend, he check the work, then said “OK, now multiply that by 4…”
There is (or maybe, was) a standard book of math for engineers – by Steinmetz – where the first two chapters are on estimating the result.
If you’re going to be building things that carry people around, 5 miles up, or carry cars over long bridges, you want to put a lot of safety factor in your design. Computer calculations will tell you that this part will probably fail at 234.567 pounds of force – you don’t want to allow 235 pounds on it when it goes out for use.
I believe schools should make students memorize the multiplication table up to 12×12 (the “tens” are easy….), mainly because it’ll save a lot of time later.
Heinlein pointed out that, in his day, you learned the multiplication table to twenty. Hmm. Why not learn it for the prime numbers from thirteen to 53?
There is still a Steinmetz Guide for that. They’ll never quite go out of print, because a lot of their examples also wind up as test problems.
I can still estimate flow, forces, and a lot of the trig in my head, and get a good approximation. Scares the kids who are button pushing. If you memorize the multiplication tables, 4-5 constants, and tring properties for roughly 4 angles, most people will think you’re a genius instead of prepared.
Another important feature is setting up problems with the correct units, and using dimensional analysis and estimates to get to the right place. Rhen use a correction of 2.5 to be on the safe side. Lexcorconstans is absolutely correct about respecting Marge N. OfError, who is mean and unforgiving. In combination with Low Bidder or MOUS (Material of Uncertain Strength), always be careful.
Yep, same was true for the early space capsules… Estimation is always better than nothing, and rounding will save your butt! And there is always the EA-6B or variants thereof! Old enough that we memorized math tables, and FINALLY got to use slide rules. Didn’t see my first calculator until I was in the Navy.
I’m guessing you mean E6B “flight computer”, rather than an EA-6B Prowler. 😉
Calculators became somewhat affordable between my first and last year of HW, but I had already taught myself to use a slide rule before then (thanks, RAH!).
In my first-year college chemistry class, the professor and I were the only ones who used slide rules. He was a stickler for understanding calculations, not just plugging them in blindly – for instance, he would mark lab results and calculations down by a letter grade if they tried to provide more than 3 digits in the results (you had to get the units correct – but he pointed out that three digits was about the maximum accuracy possible with the freshman chem lab equipment, and giving 8 just because your calculator displayed it showed them proved you didn’t understand the work).
In my case, I had a calculator, but liked to save the battery – and the calculations we were doing were so simple I could do them nearly as fast with the slipstick
But I was *really* glad I had a programmable calculator when I got as far as circuit analysis – a phasor matrix is MUCH faster when when you could concentrate on the why and how of the problem rather than getting bogged down in calculations.
And it’s been so many years since I used the skill I’d need to start over from the very beginning to understand the work I was doing then.
I never learned to use a slipstick. Lucky(?) me.
Miscellaneous notes on the theme:
There is a PBS show that is occasionally annoyingly PC, but still quite good at explaining basic math concepts like estimating at a grade-school-kid level: “Cyberchase.”
I recall an incident where I was buying a single item in a state that had sales tax (mine doesn’t) and the cash register was down, so the clerk was unsure how to calculate the sales tax amount. 5.5% of $22.95 is a nontrivial problem without pencil and paper. I showed her a simple method I learned somewhere: take the list price of the item and multiply by 1+(sales tax/100), so a 22.95 item with 5.5% tax came to $24.21.
I once looked at some sample lesson plans for Common Core math courses. They were ridiculous, taking whole pages to describe goals that I could sum up in a sentence. They’d clearly been written by a committee… or a bureaucrat, which amounts to the same thing.
Anyone who works in software engineering learns to estimate FAST. And to get it right the first time, because “adding manpower to a late software project makes it later.” Lesson Number One in The Mythical Man-Month…
High School physics course,. early 1980’s. Instructor loved to turn the lights off to thwart “solar’ (NOT ‘dual power’ yet!) calculators. My battery-eating LED TI-30 kept chugging along. One day a “NO CALCULATOR” quiz was announced. “What about slide rules?” some wag (not me, really!) asked. “Slide rules? Yeah, OK.” And the next day (after some Very Rapid Self-Teaching) 1/4 to 1/3 of the class showed up with slipsticks – and knew how to use them.