### Back in the day

My maternal grandfather’s family fled Poland in 1934 on the last boat to Canada before the war. He, his brother, and his parents spent a week on the ship, and then another two weeks quarantined for chicken pox, before they headed off to Montreal with ten dollars in their pockets. Around this time, according to family legend, my paternal grandmother’s brothers were spending their weekends playing hockey with frozen cakes of horse manure. My great uncle, who told this story, has been known to tell tales, but this one rings true, because I can’t think of a more Canadian way to get through the Great Depression: *We might not be have been able to afford rubber, but we had horse shit and damned if we were going to give up hockey.*

My parents had it better, but apparently life in the fifties still sucked. For instance, my father and his siblings, for reasons I’m still not sure I understand, had to brush their teeth with salt. *(Correction: Mom tells me it’s my grandmother who did that.)* And the six members of my mother’s immediate family lived in a small house with one bathroom *and when your grandfather went in, God bless him…*

Not me. I grew up in an era of fluoridated water, small families, and multiple household bathrooms. For a time I resented my parents somewhat for depriving me of stories to tell my own children and grandchildren when they started bitching about how tough their lives were. I mean, what was I going to say? That our surround sound system had only *six* speakers? That we had to watch movies on lower-quality *videotapes*, without extras?

I needn’t have worried.

“I can’t believe you guys had to do all this stuff without graphing calculators,” said an incredulous high schooler this morning. “What did you *do* in your math classes?”

“Math,” replied Grandma Moebius Stripper, Luddite.

“No, seriously” (”seriously!”) “how did you guys do questions like this one?” - pointing to the *Solve for x: tan x sin x - tan x = 0* example, which our tech-savvy student, following the instructions given by MATHPOWER 12, had just solved by first plotting the functions *y = tan x sin x - tan x* and *y=0* on his TI-83+ and then invoking the INTERSECT command.

[*I’d originally accidentally written tan x sin x - tan x = 1. Sorry! Thanks to Dog of Justice and Ronald for pointing that out.*]

And from there, a five-minute digression into how *when I was his age* we knew the trigonometric ratios of the special angles by heart, and if we didn’t, at least we knew how to derive them from the triangles. (”You mean the ones you showed me yesterday?” “Yeah, those.” “Oh, so like you had to *learn* those?”) We had to actually *factor* expressions such as the one he was working on. Sometimes, we would sketch the graphs *by hand*, which we got to be very fast at, and then we’d apply our knowledge of periods and phase shifts and suchlike in order to find general solutions.

Replied the student, clearly a product of west coast ethos, “That musta taken lot of paper.”

Damned straight, it did. And this was before pulp and paper mills. And we had to cut down the trees by ourselves. *With our teeth*.

Solve for x: tan x sin x - tan x = 1Well, I got this down to the cubic y^3 - y^2 + y + 1 = 0 (y = sin x) fairly quickly, and was prepared for a clean solution like x = -36 or x = -40 degrees… damn you for leading me on! The boy was essentially right about this particular equation!

(perhaps you made a typo in that equation? I assume you weren’t expected to know the general solution to a cubic? — and even if you did, you could only express the final answer as an inverse trig function.)

wait… how do you solve that?

When I have to invoke the “old ways,” I bend over, put a hand on the small of my back and pretend to be walking with a cane in the other hand, and say in my best aged grandma voice, “When I was your age, we used to browse the internet with a TEXT-BASED browser!!” What’s funniest is that it’s totally true. I spent many an evening with Lynx in the basement of the math building.

I find that the “old ways” make some topics much, much more enlightening to students. Logarithms in particular. Most textbooks just present logarithms as some magic numbers that come out of a calculator. If you take 5 minutes to explain to students why logarithms were invented, they understand them a whole lot better.

D’oh! should be =0, not =1 on the right hand side. Sorry! So it’s actually one of those super-contrived trig questions for which you can find the exact solutions.

Wacky Hermit, re logarithms - god, YES. I’ve seen three or four high school/intro college math texts that present the laws of logs as though they were handed down by God Himself (log ab = log a + log b…just…

because). The authors don’t even seem to see the point in justifying those formulas. And then, in the next chapter, we see that log_a(b)=ln (b)/ln (a), so you don’t even have to know how to evaluate something like, say, the base 3 log of 81 without your handy TI-83+.BTW if it was tan x sin x - tan x = 0, it’d be a helluva lot easier to solve.

Ah.

P’shaw. That hardly takes any paper at all!

Oh, thank god that is equal to 0. I was feeling very, very dumb.

when i teach something that (at first, at least) seems relevant only to pre-calculator times, i like to do a little Wayne’s Worldesque move…the one where you make vertical squiggles with your fingers and say “doodleloooloooo, doodlelooloo”. then we imagine it’s the time before calculators. maybe it’s just because they’re kids, but that takes some of the edge off.

i remember going with my dad to an electronics store during my junior year in high school and getting my first scientific calculator (which i still own and use). ah, nostalgia.

When I was in High School, my grandmother gave me my uncle’s old slide-rule. It was an interesting enough gadget that I actually went to the trouble to figure out how to use it. When I went off to college, I took it with me, mostly because it was a low-mass, low-volume, yet interesting, toy.

While in college I took to carrying it around with me, almost purely for its geek-chic factor. Since it got me a few “Are you insane?” looks, it must be accounted a success.

Then came the Chem test when my calculator battery died (a custom rechargeable battery, so not replaceable at the time). That slide-rule saved my life, although I actually had to use the whole testing period to finish the test.

So there, you whippersnappers.

8-)

Polymath — similarly, I point out how we could use our calculators to do this, and in general we probably would use our calculators to do this. But, I say, the war is coming, and the machines will no longer do our bidding, so we will need to remember how to do it ourselves.

Ha! I can beat you Doug:

I used my Dad’s sliderule throughout my trig & algebraic geometry class in high school in 1988-1989. My sisters kept stealing my calculators, and I hated using the tables and linear interpolation (I’m a pro at that).

Even better, on one test I was the first to finish.

I still have that slide rule, with its orange leather case (my dad went to Clemson) and the instruction book.

What kills me about the ubiquity of graphing calculators is that I have a whole arsenal of answers to the oft-asked question “Why do I have to learn this?” -

all of which are rendered null and void by the calculator-based curriculum. Being able to prove mathematical statements (or, to use less scary language - to “justify [their] answers”) requires students to be able to form logical arguments - a useful transferable skill. But when “We know that the minimum value of the function y=x^2-6x+8 is -1 and occurs at x=3″ occurs in the section on completing the square, it becomes alotharder to explain the difference between legitimate mathematics and technological shortcuts.I think one of the best things about slide rules is that they reinforce the concept of significant digits. There really aren’t many things in life that need more than four digit approximation. Frankly, much of life has measurement errors in the first digit.

Calculators are brilliant at false precision, and far too many people completely fail to understand that limitation. “2+2=5 for sufficiently large values of 2″ is a concept foreign to most, but it can be critical.

Sometimes the best you can do is, “We don’t know and you don’t either”, but it’s a hard sell.

The argument I made was “You’ve got to do it by hand before using the tool, so you can know when it gives you a wrong answer.”

In my work, I use spreadsheets alot, and oftentimes I’ve screwed up something somewhere. Sometimes, it’s been my bosses who caught it, because they know where the answer should lie and I don’t have enough experience. I’ve been catching my mistakes more often, but they still get through.

Any of y’all can tell your students that even the best in math screw up when they’ve got calculator in hand. Knowing more than one way of doing things, and thinking on a higher level about what kind of answer you expect, makes one’s final error rates drop quite a bit.

Doug Sundseth:

If I had a dollar for every time a student checked their work by plugging in the n-decimal-place approximation into the original equation and then said, “Oh, I guess I must be wrong - the answer should be 0 but the calculator says 1.3235902590E-7″…I’d be rich. You’d think that when calculators form the

backboneof the math curriculum, students would actually learn how to interpret their output. And understand their limitations.Meep - ah, but see, the difference is that in your work, you have to think about how you’re going to use the spreadsheet before you use it. Whereas my students explicitly “learn”

exactly what they need to plug into their calculatorsin order to solve this question or that one. So to find, say, the maximum of y=x^2-6x+8, they graph that function, and then select the MAXIMUM command. They’re a lot less likely to screw that up than they are to complete the square incorrectly. So the “you need to learn multilple ways of solving a problem” explanation falls flat: technology may not be infallible, but it’s a hell of a lot more reliable than the shoddy math skills they bring to the table - particularly when the technology method doesn’t require them to do any actual thinking.Something else that calculators (and spreadsheets) don’t do:

unit analysis. This is bending me all out of shape on a rewrite of (what should be) a simple blog posting, because I’ve mixed units in too many places and my tools won’t help me untangle them.The solution to the calculator-dependent mentality might be problems based in physics. If the student has to work it out so that the answer is in kilometers per hour by dividing kilometers by hours, they’re not nearly as likely to treat the process as meaningless.

Something else that calculators (and spreadsheets) don’t do: unit analysis.For better or worse, it doesn’t seem hard to design tools that will automatically do this for you in many contexts.

MS: Yes. which is why I never allowed students to use technology to solve the problem until they knew how to do it by hand.

Cart before the horse, and all that.

Of course, you can’t do much about a crappy text and even less about a crappy primary education.

The funny thing is I spent an entire dry lecture sitting and talking about Sig. Figs and unit analysis and all that jazz. All the non-mathy people in lecture sitting there and nodding their heads.

We should just create a society that if you can’t do math by age X, where X is a small number (not too small, greater than 10 I’d say) you must be sent to a distant planet of math nazis or something and trained to love mathematics. Maybe they can learn to fetch water too. Don’t tell them to fetch H2O though, they’ll freak out and explode.

I really don’t understand where our education system is going. Too soon we’re going to have a bunch of people that don’t even know how to develop the series that make calculators work and people will be upset that they can’t use them anymore.

In response to Tarid’s comment about machines boycotting us, has anyone seen the Terminator movies? I had never seen them and saw one of the last ones quite recently. In between that and Stealth, our society will very soon think that machines are against us. That’s when they will drop the calculators and we can all be farmers and not worry about this kind of stuff. We can work with Complex and Real analysis by hand and they can learn it in high school. Then they will understand what it’s all about. ;).

So, I understand in this case why you can’t tell the kid to leave the calculator in his backpack, but is there a reason you can’t do that in your classes? My eleventh and twelfth grade math teachers did not allow us to use our calculators on quizzes and tests until a couple of months before the AP exam, when they showed us all the shortcuts. Although I know how to use my calculator just fine, I’m much faster and much prefer to work calculatorless until I get to a point where it’s not solvable by hand.

Vanes63 - yeah, there’s so much really cool math out there, and none of

itis taught in high school, either. (And there’s plenty of fun-but-elementary math that could easily be taught at the high school level, or even before. Personally, I wouldn’t even consider teaching infinite geometric series without spending a day or two on Escher.) Somehow, though, the standard math curriculum has evolved in such a way as to contain neither interesting mathematics nor useful mathematics. Argh.Caddie - oh, I can and I have forbidden (fancy) calculators in my intro college classes. I don’t forbid calculators altogether, largely because I am usually one of several instructors teaching the courses, so we have to synchronize our subject matter somewhat. However, the problem is that I, as a college teacher, am basically forcing calculator-dependent students to quit the habit cold turkey. So I ever have to spend a LOT of time teaching the basic math skills that they should have learned ten years earlier…or I have to brace myself for a lot of failures. (Usually, I get lucky and do both!)

How many of your students factored the expression but forgot that tan x doesn’t always exist?

I wonder if perhaps your relative was actually using

baking sodato brush her teeth and mistook it in her memory for salt. Baking soda is not a salt but a weak base however it does have a slightly salty taste, and it has been used since the begining of time to cleanse teeth.Speaking of old calculators, I still have the HP-35 I bought when the price dropped from $400 to $300 in 1972.