Gifted Issues Discussion homepage Forums General Discussion Semi-wonkish question about significant figures

So. I have a question about math education and significant figures.

My kids are coming home with decimal addition problems that look something like this:

  3.1416  
+2.71828

The teacher, the math books, and many internet sites all say that you should do the addition by tossing in a zero, as follows:

  3.14160  
+2.71828

I disagree. I think that "tossing in a zero" is one of those little details that seems so reasonable yet creates a blurry picture of mathematics (and science) and can lead to real problems down the road. Bottom line IMO: you don't know if that "6" in 3.1416 is 0.00055-9 being rounded up or 0.00060-4 being rounded down. So yes, you can toss in a zero, but not knowing anything else about the number, you have at best a 1/10 chance that your answer is correct in the 5th decimal place past the zero. And of course, the result in the 5th decimal place can affect the 4th decimal place and so on.

I don't think it's appropriate to be teaching the finer points of significant figures to kids learning about decimals for the first time. However, I do think it's easy to tell the kids, "You don't know if that 5 is 0.00057 rounded up or 0.00064 rounded down or just plain 0.00060. All of those different numbers will give you different answers when you add. So you should only add the numbers you're sure about, which is only as far as the ten-thousandths place. Just discard any decimal digit past the ten-thousands place in the other number." This approach would also tie in nicely with rounding.


This method, IMO, sets kids up to have a more intuitive understanding of significant figures and why they're important.

Thoughts? Am I wrong?

Yes, you're wrong, because you are assuming that such a number must be an approximation. It need not be; it might be a precise representation of a rational number, in which case appending a zero gives you a different representation of the same rational number, and is definitely correct.

I agree with you.

IF this is "necessary" as a bridging thing-- you know, to scaffold computation of differing place values-- then I'd advocate making the "0" a subscript zero.

That tallies nicely with how scientists tend to carry non-significant figures through calculations-- and we do-- in order to avoid rounding errors.

At the same time, however, it is still important to be able to evaluate-- at any point in the process, at any point in the calculation, at any point now or in the future-- where there is a question of experimental error, and how large the error might be.

Okay, might be. That's my issue with it. You don't know.

Math and physics books that deal with significant figures all tell you not to make that assumption. So my feeling here is that if it isn't an approximation, the math book should add the zero and make it clear.

Okay, this makes sense and bundles some important ideas.

Right.


2.3 =/= 2.30000

Not in scientific terms, anyway.

How many zeros would you like them to add?

More seriously, although I agree with the underlying point that approximation and sane treatment of significant figures should be treated better in school, I do think that this way lies madness.

What you're going to end up with is children thinking (correctly, in your world) that you can only add numbers if they have the same number of digits after the decimal place. Sure, you can define addition that way, and provided we have ways to coerce numbers into having the right number of decimal places, all will be well. But the children will now not be able to add 1 to 2.5 without first turning 1 into 1.0. If this were proposed in Everyday Math, imagine the outrage.

I will modify my original idea based on comments thus far: let the kids add the zeros as subscripts, but then tell them to round to the place value of the digit where there is a value in each number being added. So you would get

  3.14160  
+2.71828
  5.85988

The answer would then be rounded to 5.8599.

This method would mesh with scientific practice, and again, would create awareness of an idea that will come up in later math and science courses.

Better?

MUCH better.

Better in terms of the foundation for a later (and more intuitively correct) understanding of Sig-figs, and also better in terms of developing number sense in the here and now.

Reinforces place-value concepts, too.