Does anyone have a sense that Trice’s teacher may have been part of the Benghazi cover-up? Because I think it’s demonstrably true.
Now let’s get into a deeper meaning. Is “9” as a digit different than “9” as a number? If so then I think 999 is correct since 9 is being used as a digit it that.
A few nitpicks and more.
One of Mr. Trice’s underlying complaints is that young students are shot down for knowing something more than they ought to know; something that All-Knowing All-Wise Teacher hasn’t taught them yet. Whatever Mr. Trice’s fable may be, that does happen, I think. Haven’t we all read or heard (or experienced!) umpty-ump stories to that effect over the years? It seems a popular complaint that schools sometimes extinguish rather than encourage the student’s curiosity and desire to learn. How much does this really happen in real life?
Taking Mr. Trice’s fable at face value (or is that Facebook Value?), I’d have gone along with him up to but not including the point of demanding that all the other children who answered 999 should be marked wrong. The question is ambiguous, unless it’s somehow understood that only digits and no operators may be used (and for that matter, only base-10 digits). (ETA: And similarly that digits are being used as Saint Cad discusses just above.)
A bit of vocabulary: In any expression like 3 + 5 - 7 / 9 ^ 12 (where ^ means exponent), the numbers being acted upon are the operands and the symbols indicating the operations are the operators. As Chronos points out, the expression 9[sup]9[/sup] still has an operator in there, it’s just implicit, but it’s still there. Same with multiplication: In the expression 3ab we have two implicit multiplication operators.
If we were to allow Knuth’s up-arrow notation, the sky’s the limit (which is to say, there is no limit), since there can be any number of up-arrows. I will use the symbol ↑ hoping you can all see that in your browsers. You can write: 9↑9↑9 (but I don’t know if that means (9↑9)↑9 or 9↑(9↑9) let alone which is bigger). You can write 9↑↑9 or 9↑↑↑9 or 9↑↑↑↑9, each extra arrow indicating an additional layering of exponent stacks (and you still get another 9 since the problem calls for three digits). Heck, you can simply pile on factorials too: 9!!! ad infinitum; combine that with arbitrarily many of Knuth’s up-arrows.
I’m with Mr. Trice for making a fuss about the teacher – and the principal! – being jerks and shitting on his little student for being smart (or maybe even smart-ass). I’m with him for taking the trouble to fight this up to whatever level it would take to force that teacher and principal to eat their own shit (apparently that meeting at the state capitol in this story). Beyond that is probably going to far. Again, commenting on the story at face value.
Context is important, and recognising appropriate context is just as important. And kids do need to learn that.
So when a teacher introduces a concept (like the interior angles of a triangle adding up to 180[sup]o[/sup]), the context is plane (euclidean) geometry. This should be explained and emphasised, but most students at that point will have no conception of non-euclidean geometries, so the exceptions are not usually presented. Mentioning that these rules may not apply if your geometry is not flat may be a good idea for some students, but may also confuse weaker students.
Now if a student is pointing out the exceptions (like the triangle on a sphere) a good teacher should respond with something like - I agree that this rule does not apply when we look at a triangle on the surface of a sphere, because that is a curved (non-euclidean) geometry. However, in the context of plane geometry, the triangle interior angle rule does apply, and this is what we are focussing on right now.
In the same way, the question about the largest number represented by three digits has a context of number theory. Unless the concept of exponentiation has been introduced, or operators explicitly included, the context seems to indicate the largest three digit number is required.
If the story is true, then the young Trice lass has to learn about identifying context to go with her advanced math knowledge.
Also, (as far as I can recall, and wikipedia backs me up) 9^9^9 is not ambiguous - it is 9^(9^9) - higher order exponentiation is evaluated first before lower order exponentiation.
If Knuth can invent his own notation for representing ridiculously large numbers, then what’s to stop anyone taking a standardized test from coming up with their own notation for representing ridiculously larger ones?
When I first saw the problem, I thought the answer (to the question of what’s the largest number that can be represented with three digits) was:
FFF[sub]HEX[/sub] = 4095[sub]DEC[/sub]
When I saw the answer postulated, I figured I’d leap frog over it:
F^F^F[sub]HEX[/sub] ≈ 4.17 x 10[sup]264[/sup][sub]DEC[/sub] >> 9^9^9[sub]DEC[/sub]
Nothing except the fact that you need a heck of a lot of imagination to come up with any larger numbers to begin with. Most things you can think of will be hugely smaller than anything expressible in arrow notation.
And while I nitpicked about operators before, I should counter-nitpick myself that numbers are themselves operators.
On the contrary; coming up with an operator that produces numbers larger than those produced with the most prolific known operator is trivial. Let ⋄ be some binary operator such that x ⋄ y produces some ridiculously large number. I therefore define my own operator ⋄′ such that x ⋄′ y = (x ⋄ y) + 1. No matter what existing or invented operator ⋄ you start with (including Knuth’s up-arrow notation), I can always define one that produces a larger result for the same arguments.
He specified that it was a national test. The only national math test given to grade schoolers is the National Assessment of Educational Progress, which is administered by the National Assessment Governing Board. I suppose he might have assumed the NAGB was part of the federal Department of Education.
Still, there’s a problem with the story even then: I’m pretty sure NAEP scores are not reported (they’re just to see how students are doing overall). So they could not possibly have affected the daughter’s grade.
i just find it funny that people who still gave that guy the benefit of the doubt and believed his transparently false story still thought he was an a-hole. 
What a colossal failure.
This is an interpretation of “operators” (as opposed to operands) that I’m not familiar with. In what sense are numbers themselves operators?
The daughter is now a biology major [rolls eyes].
On the general point; does anyone have first hand experiences of getting into trouble for correcting a teacher?
My experience was the opposite. After the mid-term, my physics teacher put the scores breakdown on the board to show how he arrived at our grades. I pointed out that he started the top of the curve at 110 points - “That’s right, someone got them all and the 10 point bonus question!”
I pointed out that if the bonus question was counted as part of the curve, then it’s just one more question on the test and to call it a bonus question was deceptive.
He thought about it and begrudgingly redrew the curve and several people got better grades.
I had many many such experiences but I must admit that the teachers nearly always backed down and conceded that I was right. I don’t recall ever being sent to the principal’s office for it.
Yes, if you already know of Knuth’s arrow notation. But if you don’t, you’re likely to come up with something that seems HUGE! to you, but which pales in comparison to it.
I can’t show you one, or five. I can show you one apple or five apples, I can show you one car or five cars, I can show you one person or five people. The number itself has no meaning except when connected to something else. A number is an operator which can act upon apples, or cars, or people.
Let us define the digit ◊ as the last digit in a base-infinity system. Then just the one digit would be an amount infinitely close to infinite.
This is absurd. There is no “last digit,” in a base infinity system.
Let us define the digit ◊ as one more than the largest number that will be proposed as a counterexample.
Better …
Then what is the value of 0.◊◊◊… ?