Which it doesn't, and then I thought "meh". It's truly easy to be spoilt by this clicky-shiny web thing.
An electronic calculator having means for enabling a user to input a calculation, characterized in that the calculator has a mode of operation in which the user is required to input additionally at least one estimate of the result of the calculation before the correct result of the calculation is displayed.
That's nice, but you haven't any understanding of what a patent actually is. It is not an idea, any more than the title of a book is a story. It is an implementation. In the case of Qama, the clever bit is scaling the estimation tolerance with the difficulty of the calculation.
It seems that maths isn't the only area that needs education.
Patents are intended to be for inventions, and this feels like it qualifies.
This wouldn't need to give away any details - if need be, just have 10 example calcs with the tolerance hard-coded.
But then they lose me a bit when they say (on the scope/content page) that they don't accept rational-number estimates because there's no point. I'll tentatively agree (with reservations) for numbers greater than one, but decimal numbers are just a subset of fractions, and there are a lot of problems where an answer like "1/20" is a lot more natural than "0.05" or "5E-2".
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.
And somebody will find a way to disable the LED so that students can "cheat".
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.
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.
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're talking about general decimal numbers, this is provably false. Pi, for example, cannot be written as a fraction.
If you're talking about floating-point numbers on a piece of finite hardware, they can all be written as fractions, and, depending on whether you use multiple-precision math, you might be able to represent all the fractions your computer can work with as decimal numbers.
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.
The tolerance - how far out one may be for the estimate to be accepted and the answer shown - must at all times appear reasonable ... The student should never feel that an estimate was unreasonably rejected, and on the other hand should not find an odd guess being accepted
The tolerance - how far out one may be for the estimate
to be accepted and the answer shown - must at all times
appear reasonable... The student should never feel that
an estimate was unreasonably rejected, and on the other
hand should not find an odd guess being accepted.
The first problem to be solved is that most introductory math education is still about calculating and not about applying mathematical principles to real problems. I think that being able to estimate a solution is a valuable skill, but for many problems with a high degree of precision (lots of decimal places) or large numbers, this just becomes problematic.
I'll definitely admit that at the end of a string of transformations of complex numbers, integrations, and other things, the last thing on my mind is guessing whether my solution is close. Understanding all the steps that lead up to that point is more crucial (and worth more marks!). The only thing I worry about is whether I typed the equation into the calculator correctly to get my final result.
That being said, when students are first learning about a new 'operator', like when learning division, multiplication, logarithms, and exponents, this could be a useful tool for enhancing a student's intuition about how that operator works. But after that, I'll stick with a normal calculator.
I think this is a great idea, though - too many times I've caught an in put error simply because the answer was an order of magnitude or two off - or I had forgotten to convert units.
That said, this calculator does allow a no-estimation-necessary mode, but entering that mode causes some LEDs along the top to blink. If the student thinks he or she can get away with it while the teacher is looking away, those LEDs actually keep flashing for a while after switching back to estimate-first mode.
I think it's a good idea! The calculator's ease of use is stopping people from thinking about what a correct answer should look like.
Suppose you are studying math, and come upon a question to which you don't know the answer. Which better helps you to understand, starting from "I need to figure out how to do this" or "I need to figure out why the answer is 0.0001."?
Further, this has little to no application in the real world classroom. For tests in which calculators would simply give the student the answer, students are simply not allowed to use calculators at all. I would argue that, in an exam environment, there is effectively no margin between "a normal calculator would be cheating so you have to use paper" and "this has multiple steps such that a simple calculator will not give you the right answer without understanding." In light of the existence of pen and paper, and simple 8-digit calculators, this product addresses a problem that simply doesn't exist.
We all know that skills unpractised are skills lost. You need to work continually and constantly on skills to keep them honed, and you need to work hard to gain new skills.
Clearly this calculator isn't suitable for every context, but neither is Python, C++, Haskell or Lisp. (Well, except for Lisp. We all know that's useful in every context. 8-)
The problem is that this is a tool that an author designed for another people to use. There is an essay from pg that says that the good languages are designed by the authors for themselves, but I can't find the link. The same idea applies to the other tools.
I'm teaching elementary calculus and linear algebra in the university, and I think that it is very important that the students get a general idea of what they are doing. (For example: The integral from 0 to 1 of e^-x is less than 1, because the area is inside a 1x1 square. If a linear equation system includes x+y+z=1000 and each variable is positive, then x=2117 is not a good answer) But an annoying tool is not the answer to this problem.
But I also deal with students, and constantly, constantly struggle against their willingness to accept just any old number simply because it came from a calculator, a program, a newspaper, or wikipedia. Recently a colleague recounted how a student had been doing some work and had come up with a result. When asked "How accurate is this?" the student clearly just didn't understand the question, let alone have a clue how to answer it.
There is no single solution, there is no single way to make the lazy work harder, there is no single tool that will solve all the problems that exist in education as a whole.
But having a collection of tools, a collection of techniques, and a collection of approaches has to be a good thing.
We also know that skills unused are skills unpractised. I don't know how useful keeping up with my handwriting is, for example, I haven't done it in about ten years.
I also see this being really useful at getting better at stuff like dimensional analysis.
Maybe I should just build one and wait for the knock.
"Hey, what do you need me for! That's the answer."
"Glad I'm here. You're off by more than 10%"
"This is not even the right order of magnitude"
"You don't even have the sign right!"
"um, what? This is...not even wrong..."