Decimals and Place Value -- How do You understand them?

[mom2bee]

So, I had a conversation with a 6th grade PS science teacher and I am more than a little confused. I hesitate to say that someone who has a masters in a Science field and several years experience teaching, is point blank wrong, but we spoke about math instruction and she refuted a comment I made and kind of shot me down when I tried to politely argue the point with her.

 

Basically, she refuted the idea that decimals are an extension of place value beneath units (or ones) and instead said that decimals are fractions.

 

I understood them to be both, place value and fractions and I think that I am right. However, it is possible that I am wrong.

 

I suppose another view is that they are exponents where the base is 10 and the exponents are the natural numbers, but that seems to be the whole point of place value and the decimal system.

 

What says the hive?

[vonfirmath]

Decimals are fractions.

 

From dictionary.com: a fraction that has a denominator of a power of ten, the power depending on or deciding the decimal place

 

So 3/10 would 0.3 but 3/100 would be 0.03

 

You need to understand place value to read it properly though, yes.

[texasmama]

I have always conceptualized decimals as fractions.

[Kuovonne]

Huh? What was her argument against place value? I get that decimals are fractions, but that doesn't prevent them from being part of place value.

 

In the number 0.123, the 1 is in the tenths place, the 2 is in the hundredths place, and the 3 is in the thousandths place. Ten of any one place is one of the next place value up. Addition, subtraction, multiplication, and division work on decimals the same as whole numbers because of place value; you just have to keep track of the decimal point.

[Paige]

Decimals are fractions, fractions are unfinished division problems, a place for everything and everything in it's place (value)... IMO, there is no conflict. Aren't whole numbers fractions with a denominator of 1? I see decimals and fractions as all part of the same system and that includes place value.

[NASDAQ]

Hold on, let me appeal to a higher authority:

 

From Parker & Baldridge:

"Decimals specify points on the number line by repeatedly subdividing intervals into tenths ("deci" means tenth). Just as a mailing address locates someone by specifying a state, a city in that state, a street in that city and a house on that street, a decimal number gives the "address" of a ppoint on the number line -- the digits give successively more accurate information which, together, precisely locate a single point.

 

This process is an extention of place value notation. Whole numbers are written as a string of digits that specify multiples of the "denominations" 1, 10, 100, 1000, etc. according to their position. Decimals are numbers written with the same logical notation, but including place value positions corresponding to the denominations 1/10, 1/100, etc. obtained by dividing 1 by powers of 10. "p. 202 Elementary Mathematics for Teachers

 

Nowhere does it say that a decimal is a fraction. A decimal and a fraction are alternate ways of expressing the same number.

 

Decimals are an extension of place value notation.

[stripe]

How does this affect her teaching style with regard to scientific notation, units conversion, and other such calculations?

 

And also, there is a difference when in comes to accuracy and significant figures.

1/10 = 10/100, but .1 does NOT equal .10, because .10 is more precise. And neither is equal to .100, which is 100/1000, which is equal to the other fractions ; .100 is more precise yet.

[kiana]

They are both, imo.

[Violet Crown]

 

And also, there is a difference when in comes to accuracy and significant figures.

1/10 = 10/100, but .1 does NOT equal .10, because .10 is more precise. And neither is equal to .100, which is 100/1000, which is equal to the other fractions ; .100 is more precise yet.

My understanding is that that's the case in scientific use, where the significant figures indicate greater accuracy; but mathematically, 0.1 and 0.1000 are the same number.

 

 

[mom2bee]

Aren't whole numbers fractions with a denominator of 1?

 

Thats what I pointed out to her at one point, but she said "But thats different, we're talking about fractions with meaningful denominators." as though I was annoying her with silly questions.

How does this affect her teaching style with regard to scientific notation, units conversion, and other such calculations?

 

I haven't the slightest idea. I didn't even think to ask her that.

Accuracy vs Precision is an excellent point.

[kiwik]

I think of them as both. Number line 1000,100,10,1,0.1,0.01,0.001 and fractions.

[stripe]

My understanding is that that's the case in scientific use, where the significant figures indicate greater accuracy; but mathematically, 0.1 and 0.1000 are the same number.

Because the OP said the person was a science teacher, not a pure mathematician, I was curious how she taught significant digits if she thought of decimals as being fractions and rejected the idea of place value. Which is a hard thing to understand, in my opinion, given that there is indeed place value at work.

 

I think it's really quite a horrid idea to try to distinguish between 10/100 and 1/10, at least if one is using a calculator and/or performing any calculations whatsoever. I was wondering how she does it in practice.

 

What is the definition of a "meaningful" denominator, anyway? Why say 5/10 in that case when it should properly be reduced to 1/2, but then why should 500/1000s be kept that way, or should it be reduced to 100/200 or some such nonsense.

 

Putting aside my comments about the idea of this, I don't really understand how it works in practice.

[Paige]

Thats what I pointed out to her at one point, but she said "But thats different, we're talking about fractions with meaningful denominators." as though I was annoying her with silly questions.

 

 

 

 

All denominators are meaningful! Wow. I would be concerned if she were teaching my child math or science.

[Violet Crown]

 

Because the OP said the person was a science teacher, not a pure mathematician,

Oh, I missed that.

[Rebel Yell]

I don't think I ever in my life thought if decimals as fractions until I started teaching it to my kids... actaually, until my kids started teaching math to themselves and we ditched Saxon for college textbooks.

 

I always learned decimals as havning a place value.

[merry gardens]

So, I had a conversation with a 6th grade PS science teacher and I am more than a little confused. I hesitate to say that someone who has a masters in a Science field and several years experience teaching, is point blank wrong, but we spoke about math instruction and she refuted a comment I made and kind of shot me down when I tried to politely argue the point with her.

 

Basically, she refuted the idea that decimals are an extension of place value beneath units (or ones) and instead said that decimals are fractions.

 

I understood them to be both, place value and fractions and I think that I am right. However, it is possible that I am wrong.

 

I suppose another view is that they are exponents where the base is 10 and the exponents are the natural numbers, but that seems to be the whole point of place value and the decimal system.

 

What says the hive?

I say I wouldn't want this person teaching my children if she's going to "shoot down" someone for trying to politely make a point that she disagrees with, especially when she's wrong. I suppose that's the way a 6th grade teacher keeps the class focussed on whatever it is she wants to teach, but seriously, the idea that decimals is an extention of place value beneath units is a great way of understanding decimals, as is fractions, If she "corrects" people who understand concepts more fully then she can do a lot of damage as a teacher. You're right. She's only about half right--or approximately .5 right and .5 wrong. :tongue_smilie:

 

Singapore math sells these http://www.singapore..._p/mnpvdecs.htm Place Value Decimal Strips in their section on manipulatives for schools. But I suppse she might "shut down" the people at Singapore Math who disagree with her too. I pity her students.

[stripe]

I say I wouldn't want this person teaching my children if she's going to "shoot down" someone for trying to politely make a point that she disagrees with, especially when she's wrong.

 

Or if one can think if something in two ways, get hot and bothered by someone using a different method.

 

Personally, I am delighted any time my kids solve a problem differently than I would.

[nmoira]

I understood them to be both, place value and fractions and I think that I am right. However, it is possible that I am wrong.

 

I suppose another view is that they are exponents where the base is 10 and the exponents are the natural numbers, but that seems to be the whole point of place value and the decimal system.

 

In the context of place value, integers.

 

As for the question, it's two sides of the same coin:

 

Th, H, T, U, t, h, th,

x1000, x100, x10, x1, x(1/10), x(1/100), x(1/1000)

x10^3, x10^2, x10^1, x10^0, x10^(-1), x10^(-2), x10^(-3)

 

Musing:

 

There's no difference between saying 532 is 5H+3T+2U and 0.532 is 5t+3h+2th ... it's just more difficult (not intuitive?) to think about values to the right of the decimal point this way due to the lack of a common denominator when looking at each "place."

 

Sometimes people get hung up on saying 0.532 is 532/1000, but forget or never knew it is built up the same way using place value as is 532.

 

Also people are used to thinking about number lines and zero being the "mirror line" (stolen from DD the Younger) between positive and negative, but with place value its analogue is 1 (i.e. to the zero exponent). Maybe that's tougher for people to wrap their heads around?

[stripe]

I wonder too if it would be confusing for children dealing with fractions that are infinitely long (either repeating or not) to conceive of a decimal exp ression for those, and being told it's a fraction. (In this case, each digit is its own fraction, but really.) Imagine each digit of pi as a fraction while pi itself cannot be expressed as a fraction....shudder....