I think question f above is very revealing. If the kid understood that the individual digits have value then they would’ve said “0”, not “00,000.” I think what it reveals is that the kid just understands the above exercise on a procedural level: “Can you keep that underlined number and put zeroes past it?”

I’d have a conversation about that “00,000” first. I’d ask, “Which number is bigger: 0 or 00,000?”

Here’s an idea for a question that might help: “Here’s a game: You start with 100 points. I give you a number, and you can swap it with any of the digits in 100. The number is 1. Which would you swap? Why?” Hopefully we can get at the idea that some places in a number are worth more than others. Which place is worth the most? How much is it worth?

Well, I can see why the student is confused. I don’t really like the wording of the question. Maybe they’ve done similar problems and should know what this means, I don’t know. Maybe this version is too lengthy, but it would help me understand what it’s asking for, “Based on position within the number, which digit represents the greatest value?”

To move forward with the student I would make the same distinction. I’d begin with asking, “Why is 18 different than 81? What does the 1 represent in each number?” If it makes sense to the kids (depending on their level), I’d take the number in the problem and ask them to read it if they could (“four hundred fifty six thousand eight hundred two”) then break it into: 456802 = 400000 + 50000 + 6000 + 800 + 00 + 2. (The double zero might confuse for a minute, but it fits a pattern they could always use until they get used to the idea.)

mpershan

Interesting. My comment, above, assumed that “00,000” would work against a student forming an understanding of place value. You disagree.

Anyone have thoughts on this disagreement?

Dan Allen

The strategy this student used was only good for questions a – h. The understanding was not extended into future questions and therefore was not a true understanding but a cleverly disguised coping strategy.

Dan Allen

Another question that could be asked as a follow up is “Why didn’t you use 090,000 for your answer for part a?”

Ooh. And now I want to ask, “In the number 456,802 what digit is in the millions place?”

Dan Allen

I had the discusion “Is 0456 a four digit number?” with a group of seniod students yesterday and there was some general disagreement with regards to this question. The student in question has learned a strategy for solving the problems but I’d be a little bit leary to say they understand place value because of their answer to parts i & j. That being said, the student may only require a small amount of remediation to correct their mistakes.

We can all agree that (i) is potential trouble. I concur with Dave on the idea that the wording of the question is sort of awkward. I find that often to be the case with these kinds of items. We ask “how many tens in 268?” when we really mean to ask “what digit is in the tens place?” (I’m looking squarely at you, Everyday Math).

So I submit that we need a moratorium on place-based questions.

But (f) caught my eye right away as interesting. You could make an argument either way-that it may be evidence of rote non-comprehension, or that it may be evidence of important understanding (namely that 0 isn’t so special at all; it’s just a digit like the others).

We would really have to talk with this kid to know.

But as an example for discussion with learners of place value? This is great. I would want to ask a group of second graders whether this one is correct; and see if we can draw out both sides-that it’s correct but needless.

And it makes me think of other things I’d want to ask these kids. Like, “Which is greater? 1000 or 0100?” or-more problematic-“Which is greater? 1000 or 00100?” In the latter one, the “more digits implies greater number” rule gets challenged. And, either way, we get to talk about another important piece of place value understanding, which is that zero is different from the other digits.

I hadn’t thought to ask these questions before seeing this particular piece of work. So thanks for that.

mpershan

One thing: if the student’s operating model is “greatest value means just the highest digit” then the student’s response to “least value” is a little bit puzzling. Shouldn’t this student have answered 0 for the second part of this question, instead of 2?

louise

That was the first comment that came to my mind. I would start from there.
Pretty interesting that we would probably all fail this test…