Quick…one sentence…

What is the meaning of the decimal point?

Here is an illustration of the most common answer in my class (a math content course for prospective elementary and special ed teachers):

It’s not wrong, but it’s incomplete.

Thinking of the decimal point as the border between whole numbers and decimals limits our ability to think about relationships between these two domains.

In particular, look at that picture again. The essence of whole-number place value is right-to-left; the essence of fraction (decimal) place value is left-to-right if we’re thinking about the decimal point as the border.

But the same relationships hold on either side: If we move one place to the left, the value is 10 times greater and if we move one space to the right, the value is 1/10 as great.

No, the decimal point isn’t really the border between two different settlements.

And it’s not a the marker of symmetry either. Thinking about the decimal point this way leads us to expect that there should be a *oneths* place to the right of the decimal point.

No, the decimal point marks the most important location in a place value system: the ones place. Once we know where the ones place is, we know the value of every other place.

It is just an unfortunate accident of history that the decimal point lies to the right of the ones place, when really it should sit underneath it.

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Wow, interesting thoughts. The nice thing about putting the point to the right of the ones place, though, is that it then becomes quite natural to conveniently omit the point when dealing with whole numbers.

(Side note: This kind of discussion — about the aesthetic choices that mathematics is built upon — should be happening for just about every topic in grade school through college. It could be so enlightening, and build such an appreciation for the humanity and accessibility of the subject.)

Interesting, er, point about aesthetics,

Roy. I recently went back and forth with Chris Lusto about some similar ideas. You may find his high school perspective interesting.But I’m not ready to concede on aesthetic grounds. Imagine an underscore under the unit digit. We would write 357.6 as 35

76 This would round to 358You could just as easily leave out the underscore and write 358Of course I’m aware that we’re not going to be changing decimal notation anytime soon. And I agree wholeheartedly that there are aesthetic choices that get made in mathematics (and especially in mathematics teaching). But there are also unfortunate accidents of history. I am not at all convinced that the decimal point is the most elegant of several choices. When I get some time, I’ll look into it. My hunch is that it really was an accident involving convenience and the separation between wholenumberville and fractionland.

I love these discussions of the “why’s” in mathematics. Would love to write “what is the role of a decinal point” on my board and see what kids come up with. I had a student from Belgium a few years back in my AP Statistics class, and he made my brain spin whenever I read his papers and commas were used in place of decimal points. It all comes down to developing a system of symbols which clearly communicates the value.

I have similar discussions with 9th graders about fractions. By 9th grade, they certainly have had plenty of experience with fractions, but it takes some arm twisting to have them stop writing “slanty” fractions. As we get into rational expressions, they begin to see why fractions written with a slant line can potentially mis-communicate values.

How neat of a poject would it be to have kids develop a new system for expressing fraction or decimal values, and then defend the improvement made by their system.

Every time I teach decimals (6th grade) I am reminded that I grew up using a comma instead of a point for “decimal point.” So it seems our world is split pretty much in half as to what countries use what notation: http://en.wikipedia.org/wiki/Decimal_point. I expect kids to know place values before they get to me, but no such miracle. I have 8th graders who don’t even SEE the blessed decimal point, and that’s when I say to them, “Oh, I was going to give you $100 today, but apparently that’s the same as a penny to you, so here you go…”

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