Read aloud the following number:
182,356
Now mentally answer this question: What is the value of the 8 in this number?
I see two correct ways of stating this:
Eighty thousand, and
Eight ten-thousands
And I’m trying to decide whether I care about the difference between these. I’m not sure that I do.
So now I do what I always do to test ideas. I ask, What if? Specifically, What if we looked to the right of the decimal point? What would this question look like there?
So consider the number 0.0008.
If you said Zero-point-zero-zero-zero-eight, then I’ve got a lot more work to do with you.
Place value language matters in math classrooms.
And the sentence in the picture below is confusing.
No, I’m guessing we would all agree that this is eight ten-thousandths, which is not at all the same as eighty-thousandths (although it is the same as eighty hundred-thousandths).
So now I see that my What if? question has muddied the waters, rather than clarified them.
What can we conclude?
I suppose that the major conclusion is this:
We need to stop pretending that decimal place value (i.e. to the right of the decimal point) behaves exactly like whole-number place value (i.e. to the left of the decimal point).
In the abstract, this is certainly true. But composing units is not conceptually (or linguistically) equivalent to partitioning them.
Like I was saying here:
Overthinking my teaching 
I’m not sure I understand your point. Who is assuming that place value behaves exactly the same on both sides of the decimal point? I think one of the main things we want students to learn with regards to place value is that regardless of the decimal point, the place immediately to the left of a given place has a value 10 times greater while the place immediately to the right of a given place has a value 10 times less. The extension of this is that looking to the left of a given place, you are increasing by powers of ten. When looking to the right of a given place, you are decreasing by powers of ten. This is a huge understanding for students to grasp. In a number like 3.33, the students see the digit 3 three times, but to really understand place value is to appreciate that the 3 in the leftmost place (ones) is 100 times greater in value than the 3 in the rightmost place (hundredths).
182,356
The 8 in 182,356 can be interpreted as:
…
.008 ten millions
.08 millions
.8 hundred thousands
8 ten thousands,
80 thousands
800 hundreds
8000 tens
80,000 ones
800,000 tenths
8,000,000 hundredths
…
ad nauseum
The 8 in 0.0008 can be interpreted as:
…
.000008 hundreds
.00008 tens
.0008 ones
.008 tenths
.08 hundredths
.8 thousandths
8 ten thousandths
80 hundred thousandths
800 millionths
…
One reason for confusion is that we generally don’t verbally mention the specific unit when our unit is ONE…as in 80,000 ones and .0008 ones above. We just express the naked numbers as “eighty thousand”, not “eighty thousand ONEs” and “eight ten thousandths”, not “eight ten-thousandths ONEs.” We assume that the unit is “one” unless we call it something else. Another confusion is that we often use the words “unit” and “one” interchangeably. Why does ONE keep changing? You did a great job in the video showing the nature of how “ONE” can be interpreted in many ways. The CCSSM Practice Standard “Attend to precision” should serve as a reminder for us not to be so cavalier with our treatment of the “understood” one and to be more explicit about the meaning of unit.
Elaine, you are totally correct. A quick reflective observation: I think of eighty thousand as missing a unit label (ones). I do not think of eighty thousandths as missing a unit label. Not sure what this says about me.
bstockus, I agree that the following is a learning goal:
But that sameness is at a purely formal level. I don’t think we can start (or really even explore very much) with the purely formal. I think we need to explore the messy related ideas that the formal definitions clean up.
For example, we think of 100 as a composed unit. We get 100 by putting ones into groups of 10, and then putting those groups of 10 into a group of 10. 100 is the result of moving from right to left. In order for things to be conceptually the same, we would need to be just as comfortable thinking of 1 as being the result of partitioning 100; i.e. that we start with 100 and cut into groups of 10, then cut those groups into groups of 10, resulting in 1. But I don’t think most of us are. We go the other way.
Conversely, 0.01 is the result of partitioning. We get it by cutting 1 into 10 pieces, then cutting each of those into 10 pieces. We don’t tend to think of 0.01 as the original unit, from which we compose 1. It is in this sense that I claim 100 and 0.01 are very different beasts-very conceptually unalike.
It has sort of become one of my life’s missions to challenge the mistaken idea that place value is easy. It isn’t, and the better we understand the ways that it’s hard, the better able we are to understand our students’ struggles (at all grade levels).
For instance, consider the following question: Why are there repeating decimals (0.333…) but not repeating whole numbers (…333.0)? Or this one: Why do so many people write 0.05¢ when they mean $0.05? Or this one: Why is the symmetry around the decimal point imperfect (alternately, why is there no oneths place)?
I argue that answering these questions requires more conceptual firepower than “the place to the left is 10 times greater, while the place to the right is 1/10 as great”.
Thanks for the thoughtful response. I definitely got more out of that than the original post. I agree that place value is deceptively “easy” to teach because as long as students can learn specific patterns, they can read and write numbers and answer trivial questions on tests. That doesn’t mean they actually *get* the structure of place value and the power of what we can do with it. It also means they can end up having a lot of troubles as you pointed out with regards to interpreting numbers.
The reason I called out the pattern of 10 times greater, 10 times less is because I have seen this with students who make errors in calculations. Forgetting to regroup a 1 while adding or transposing digits within a number can seem insignificant to a child. If a digit is two places away from where it should be, what does it really matter? In fact, it matters a whole lot! Students need to appreciate that movements from place to place are significant. I like how you suggest that we don’t always start with 1 either. Students should understand that you can start with a larger number like 100 and decompose it to tens and then down to one and even down to tenths or hundredths if you want. Flexibility with regards to numbers and place value is essential.
Thanks again for the response. I appreciate it!
And thank you, bstockus for sticking with us. I dig your observation about relative sizes of numbers. As an example of how tremendously hard this is, I submit our dear friends Sal Khan and Vi Hart.
As for your comment about decimal categories in number naming, one alternative is to consistently refer to the “ones” category as “units”, making it more similar to the rest of the places. Also helpful is to establish the convention, at least early on, of naming “zero tens” or “zero hundred(s)” when a decimal category has a zero.
Of course, that still won’t solve the problem with decimal categories relative to whole-number categories. I suspect that that problem may be due, at least in part, to language-specific issues of number naming and fraction terminology, but I don’t have any additional cross-linguistic evidence at hand.