Well if it's not about what we are using now but what will teach a kid to know roughly how big some numbers are, we could always ban calculators in favor of slide rules.

What I mean, I guess, is that calculators are by their nature supposed to eschew estimation for the sake of simplicity. Porting estimation back into them seems like a mistake.

> if you turn off estimate mode, a flashing light comes on... so an instructor or test proctor would be able to see it.

And somebody will find a way to disable the LED so that students can "cheat".

Also, the students for whom this thing is intended (those who would rather just get the answer) will probably study the calculator, learning how it works and probably learning ways to beat it. Then we can return to business as usual: students who "get it" and reason about the problem will continue to do so; students who just grab for answers will continue to grab for answers; and teachers will be complacent--everyone's getting their results by typing an accurate-enough estimate into the calculator, right?--until people realize what's happening, and this will become one more failed educational experiment.

Here's the first approach that occurred to me. How quickly can you type in estimates, and how accurate do they have to be? (The first thing I would do with one of these things is find that out, simply for curiosity and control over my tools.) If, say, it's "same order of magnitude of the correct answer", then you just have to guess 1, 10, 100, and so on (maybe .1, .01, etc. if the numbers are small). If it's "within a factor of 2", then you could guess 1, 4, 16, 64, and so on (or something that's easier to type; maybe guess 1 10 100 1000, 4 40 400 4000).

It may take a while to do, but if the problem is difficult and frightening, it is easier and safer to apply some brute force strategy like this--even if you had to type "1 2 3 4 5 6 7 8 9 10 20 30 40 50 ...". I wouldn't be surprised if the answer-grabbers got really fast at it (1e5 2e5 3e5 ... even handles many digits in more or less constant time). It would still take longer than a normal calculator, but that would be the case for people who estimated it "the right way" too.

Maybe you could put in some kind of penalty for repeated wrong guesses, but I'm skeptical. First, even the students who understand what they're doing will make some wrong guesses, and you don't want to punish them too hard; second, I suspect it would actually be hard to do that: the student can turn the calculator off and on at any time, and I think giving it some non-volatile storage of timestamps of wrong estimates would make the calculator somewhat more complex and expensive than it looks like it's supposed to be.

(Actually, I think the "problem" of "misusing" the calculator is not limited to pure answer-grabbers. I could see myself respecting the challenge and doing my estimates, then being off by a bit and making a correct estimate that was right next to it, then realizing I could just be lazy about my estimates--truncate everything to [single digit] * 10^n, or whatever--and just make two or three guesses if necessary to cover the relatively wide range. I might possibly still enjoy the experience or the challenge of making estimates and not considering myself done until I can justify my estimate--just like I enjoy doing this stuff http://setgame.com/set/puzzle_frame.htm even though I can write a program to solve those puzzles, or not solve them at all--but if I was bored with the assignment and just wanted to finish it, or annoyed with the people who thought such a calculator was a good idea, then there would be nothing stopping me from using perverse strategies.)

Edit: Turns out it is somewhat more complicated. The degree of tolerance depends on the perceived difficulty of the mental calculation (in particular, transcendental functions give a large room for error). http://qamacalculator.com/qama/complicated.jsp

That is kind of an impressive thing, actually. The way error bounds are determined might be somewhat complicated to work out, and maybe even a determined answer-grabber would have to do a bit of estimating work... I dunno. Maybe you could figure out that "as long as there's a sine in the expression, then the error bound is at least this much".

OH MY GOD you would simply take any expression and put "times sin(89°)" or something in it, something close to 1. Get your answer, then use that as an estimate for the original thing. Maybe put in a ton of little transcendental-but-you-know-it's-about-1 expressions in there; maybe you can get your error bounds so wide that you need only guess once. Now maybe the programming will notice things like that, will give small error bounds for something very close to a known thing like sin 90. But then put in A * (your expression) * B, where A and B are transcendental expressions whose ratio is very close to 1, but which individually are definitely not 1, and which don't obviously cancel out. Like e and 2^-1.44.

I am, let's say, 95% confident that kids would figure out something that would defeat this.

Do you have any idea how much mathematical knowledge you've deployed in coming up with ways to defeat this thing? Anyone can come up with all of that (or even part of that!) is either not part of the target audience or has learned enough that I'd be comfortable calling the experiment a success.

I don't think anyone's suggesting that this thing is a panacea. It's a tool, one of many in a well-stocked pedagogical toolkit. I think a decent teacher could make good use of it, and more importantly, I refuse to dismiss it just because the experiment might fail.

Heh heh heh. I'm flattered, but really, all it takes is for one kid to come up with one strategy that works; after that, others could find out about it through word of mouth or the internet. I could imagine a kid noticing by accident that... not having access to the calculator, I can't be sure whether any specific example will work, other than the ones described on the website... but I can imagine a kid finding by accident that, while "A * B * C" fits a certain estimate, if you ask the calculator what A * B is (about 20) and put in "20 * C" and give the same estimate, it will reject it. It's likely that either he will figure some things out (trying things like putting (A * B / 20) * C, and trying that with other values of C), or he will be confused and announce it to the class, in which case someone else will probably figure things out. And this is to say nothing of a clever kid who knows math and who deliberately looks in the first place for ways to defeat the device (perhaps after the device offends him by rejecting an honest estimate).

I would warn in general against underestimating the cleverness of children, even those who appear not to understand the material of the class. From John Holt's "How Children Fail" (letter from May 10, 1958), describing some elementary school classes:

Children are often quite frank about the strategies they use to get answers out of a teacher. I once observed a class in which the teacher was testing her students on parts of speech. On the blackboard she had three columns, headed Noun, Adjective, and Verb. As she gave each word, she called on a child and asked in which column the word belonged.

Like most teachers, she hadn't thought enough about what she was doing to realize, first, that many of the words given could fit into more than one column and, second, that it is often the way a word is used that determines what part of speech it is.

There was a good deal of the tried-and-true strategy of guess-and-look, in which you start to say a word, all the while scrutinizing the teacher's face to see whether you are on the right track or not. With most teachers, no further strategies are needed.

This one was more poker-faced than most, so guess-and-look wasn't working very well. Still, the percentage of hits was remarkably high, especially since it was clear to me from the way the children were talking and acting that they hadn't a notion of what nouns, adjectives, and verbs were. Finally one child said, "Miss —, you shouldn't point to the answer each time." The teacher was surprised, and asked what she meant. The child said, "Well, you don't exactly point, but you kind of stand next to the answer." This was no clearer, since the teacher had been standing still. But after a while, as the class went on, I thought I saw what the girl meant. Since the teacher wrote each word down in its proper column, she was, in a way, getting herself ready to write, pointing herself at the place where she would soon be writing. From the angle of her body to the blackboard the children picked up a subtle clue to the correct answer.

This was not all. At the end of every third word, her three columns came out even, that is, there were an equal number of nouns, adjectives, and verbs. This meant that when she started off a new row, you had one chance in three of getting the right answer by a blind guess; but for the next word, you had one chance in two, and the last word was a dead giveaway to the lucky student who was asked it. Hardly any missed this opportunity, in fact, they answered so quickly that the teacher (brighter than most) caught on to their system and began keeping her columns uneven, making the strategist's job a bit harder.

He adds later:

Not long after the book came out I found myself being driven to a meeting by a professor of electrical engineering in the graduate school of MIT. He said that after reading the book he realized that his graduate students were using on him, and had used for the ten years and more he had been teaching there, all the evasive strategies I described in the book—mumble, guess-and-look, take a wild guess and see what happens, get the teacher to answer his own questions, etc.

But as I later realized, these are the games that all humans play when others are sitting in judgment on them.

if you do this, e.g. sin89 x 2^-1.44 x sin89 it will know what you are up to and firstly blot out your sin89 and require an estimate for the 2^-1.44 ...! (and if it accepts your estimate for this (e.g. 0.3 will be fine) it will then let you do you x sin...

If we can get the estimation of answer, then we can answer it. Calculator are in math when i was on highschool.

> Pi, for example, cannot be written as a fraction.

Pi cannot be written as a decimal either.

Decimals _are_ a subset of the rationals. Any rational whose denominator has a prime factor other than 2 or 5 cannot be written using decimals.

I don't think we're agreeing on the basic definitions, and until we do it's pointless to continue.

Under what definition of "decimal" is Pi a decimal number?

When you use it as a very informal synonym for 'real', or as a short form for 'infinite decimal'. (This is even more understandable when you realize every decimal is an infinite decimal, and our convention of truncating an infinite string of zeroes has little mathematical reality.)