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I came across this online quiz discussing the new Common Core education standards, and it all seemed pretty reasonable, until I came across this question:

In the number below, how many times greater is the number represented by the digit in the thousands place than the number represented by the digit in the hundreds place?

57,762

1,

10,

100,

1000

I answered "1", as the number represented by the digit in the thousands place is 7, as is the number represented by the digit in the hundreds place.

Yet, my answer was deemed incorrect.

It seems as though they interpret the "number represented by the digit in the thousands place" as 7000. I understand this interpretation, but this is not the definition of "digit" that I have used in 30+ years.

My question: Is this genuinely bad math, or did the meaning of "digit" change sometime along the way?

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The convoluted formulation "the number represented by the digit" instead of simply "the digit" should have made you frown. OK, you took the number represented by the digit if we write down the number that has exactly one digit (i.e. by the digit alone, with strict distinction between the digit $7$ and the number seven), while they apparently wanted something like the contribution of the digit to the given number - which they maybe didn't say explicit enough. Admittedly, this does not seem like a very interesting concept. Then again, someone checking your business plan and saying "you have an error in the thousands digit" may lead to more concern than someone saying "you have an error in the hundreds digit".

Additionally, I'd usually distinguish between "$n$ times as big as" and "$n$ times greater than", taking the latter to mean "$(n+1)$ times as big as". Conclusion: Language is a much more complicated issue than math.

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  • $\begingroup$ Right, I see both sides but it's not a good test question. Additionally, if the answer is (1), what does "1 times greater than..." mean? $\endgroup$ – daniel May 11 '14 at 15:07
  • $\begingroup$ @daniel I don't see there are two sides. "the number represented by the digit" can only be one of ten numbers, assuming we're using base $10$. $\endgroup$ – David Mitra May 11 '14 at 15:12
  • $\begingroup$ @DavidMitra: A digit represents one of ten numbers. But if I stipulate that it the digit 1 also represents (say) 1000 by virtue of its place in a number that might be a contract if you understand and accept it. The problem with the question as I see it is that this is not at all clear. $\endgroup$ – daniel May 11 '14 at 15:15
  • $\begingroup$ Language is a more complicated issue, but in this case it seems as though it doesn't have to be. No wonder why kids grow up to hate math. $\endgroup$ – Emily May 11 '14 at 18:09
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Three different things: place value, value, and digit.

Both digits are $7$.

The digit with a place value of thousands has a value of $7000$.

The digit with a place value of hundreds has a value of $700$.

I tutor and this is a really difficult thing to teach. That doesn't mean that it should not be taught though. Right now, books are disappearing and computerized education is taking over. It is going to take a while for a stable system to bubble up and for teachers to learn how to deal with it effectively. In the old days, I would have scolded you and told you to read the book. Now I suppose, I would suggest Kahn academy. The man could teach calculus to goats.

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