How do you know if a child in grades 3-5 has a strong understanding of
place value?  What are some good indicators?

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Request for Question Clarification by czh-ga on 27 Sep 2005 00:29 PDT

Hello darwin29-ga,

Please explain what you mean by "place value" as I'm not familiar with
this term. Where are you located? What country or school system?
Thanks.

~ czh ~

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Clarification of Question by darwin29-ga on 27 Sep 2005 18:13 PDT

Hello,
Thank you for your response.  Place value is a mathematical term that
is used in our number system in the United States.  The number 239 can
be broken down into two hundreds, three tens, and 9 ones.  So
depending on which column(place) the number is in, determines its
value.  I hope this helps.
Thanks

I'd say a series of questions of two types would assess someone's understanding:

What place is the number 4 in?
40 (tens)
40,000 (ten thousands)
0.004 (thousandths)

What digit is in the tens place? What digit is in the thousands place?
641 (4, none/0)
51,128 (2, 1)
0.528 (none/0, none/0)

I think someone with good understanding would answer all the questions
correctly without any trouble.

I agree with the previous suggestions, and would add these.  (I am
educational researcher who focuses on middle and high school
mathematics.)

There are two prime candidates as far as ways a student can
misunderstand place value.  The commentor above gives you a nice way
to test the first way, how well the student can work with the
vocabulary of place value.  Sometimes what seems like a confusion
about the concept of place value is really a confusion about the
terminology.  "Hundreds" and "hundredths" can be difficult to
separate. :)

The second major misunderstanding is related to the concept of how the
quantity a number is meant to represent: how "big" a number is.  It's
hard to understand what the "thousands" place is if you don't have a
concept of "thousand."

First, establish a basic understanding of size.  Ask the student which
is bigger, 9 or 12?  98 or 201?  897 or 1001?  I know these sound very
basic, and he/she will most certainly get at least the first one
right, and likely all of them.  That's great.  The key is to talk
about HOW the student knows which one is bigger.  If you can get the
student to think about what relative size really means, you're on your
way.

Use some pictures to talk about how MUCH bigger one number is than
another, e.g. the difference among 10, 100, and 1000.  For example,
draw a small circle on a piece of paper, and say that one "garumph"
(or other nonsense word) can fit into that circle.  Draw a bigger
circle say that about 10 garumphs can fit into it.  Ask/talk about how
big the cirle would have to be for 100, 1000, or even 1,000,000
garumphs to fit.  That kind of exercise can really start the student
thinking about what relative size means, which is the essence of place
value.

In my opinion, it's important to establish an understanding of whole
numbers before moving on to decimals.  To start that area, you can do
the same type of comparison questions (which is bigger) with number
pairs like 0.9 and 1.1, 0.09 and 0.01, 0.009 and 0.01, etc.  Using
food examples here is often helpful- would you be more full if you ate
0.5 of a pizza or 0.0005 of a pizza?  0.10 or 0.099?  A discussion of
that last pair leads to the realization that those numbers are very
close in value, a concept that's also very important in understanding
place value and often overlooked.

Best of luck!
valerie