For the past several weeks, our kindergartners have been playing with 2-digit numbers: they made numbers out of base-10 blocks, wrote them on the board, translated between the two representations, counted by 10’s from different starting points, and more. We also explored 2-digit addition with the base-10 blocks and the students had a great time with it.

Today I decided to spice things up a little and venture into making 3-digit numbers with the base-10 blocks. The students right away predicted that 10 rods would make a flat and 10 flats would equal a cube.

They also had no trouble counting by 10’s to 100 and by 100’s to 1000 (I was actually surprised that no one except for me said 10 hundred. They all knew it was one thousand!). Writing one hundred and one thousand on the board using digits posed no problem as well.

Then I took 2 flats, 5 rods, and 7 little cubes and asked the students what number I just made. With minimal help, they were able to figure out that it was the number two hundred fifty seven. And then came the real challenge of writing that number on the board.

One student volunteered, and wrote:

I asked the students whether they agreed that this was two hundred fifty seven and they all said yes.

So then I wrote another number under it and asked the students what that one was:

Now half the students said that the number that I wrote was two hundred fifty seven and the other half said they didn’t know what my number was, but they thought that two hundred fifty seven was 2057.

I then decided to ask the students how to write the number two hundred. Here they all agreed that it should be 200. However, when I asked them how I should write two hundred one, opinions varied once again. Half the students thought it should be 201 and they other half said 2001.

One student said that they thought that 201 is two hundred one and 2001 is two hundred ten. So of course I had to write 210 on the board and ask them what that would be. They said they didn’t know.

I decided to make one more 3-digit number with the blocks, have someone write the number on the board and see if there’s consistency to their thinking. This time we had four flats, three rods and one little cube.

I asked one student to go up to the board to write down the number and they wrote 40031. A second student raised their hand and asked, “I do not understand, why would you put two zeroes there? It should be 4031!” When I asked the student why one put the one zero there, the student replied, “That’s just the way you write it.”

So in the end we were left with the following three possibilities:

and no consensus as to which one is correct. We left it at that.

My big question now is, how should I start tomorrow’s lesson?

Do some geometry !!!!!!!!!!!!!

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More to the point the kid with 40031 is reading it as “400 and 31”.

400 and 31 is more formally written as 400 + 31, but that doesn’t get you anywhere towards 431.

Needs a bit of “place value” here.

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40031 actually makes sense to me, it’s the 4031 that I have a bit more trouble finding the reasoning for. But I totally agree, place value is the missing concept here, and they have plenty of that in their future 🙂

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The different ideas you heard remind me of alternative number systems (Greek, Roman, Egyptian, Babylonian). This, in turn, reminds me that place value is a convention, a choice about how to represent numbers. Depending on your objective, there might be mileage in exploring that idea, we can choose how we write numbers, but those choices have implications:

– do we all agree on the meaning (so that we can share ideas and learn from each other)?

– is the system clear or does it have scope for confusion?

– what is faster: for writing, for understanding?

– is it easy to extend the system to write increasingly large numbers in the system?

I did some searches to see what my (other) favorite bloggers have written. I didn’t see anything that offered specific tasks to address the point your students are at, but many talk about the same stage or come very close to this issue and might be helpful.

Here is a Marilyn Burns’s place value assessment. At the end, she has links to her books that have place value teaching material. If you have access to those books, this seems the most helpful.

Chris Danielson has a collection tagged “place value” that all seem worth reading (they are very fast reads). This particular post seems to demonstrate the same point of struggle as your students: days to xmas.

Tracy Zager has activities that attack the transition between counting by 10s vs 1s: counting circle variants.

Graham Fletcher doesn’t directly hit place value, but I wonder if there are stages related to counting that might be shaky and manifest as place value misconceptions? His latest progression video for early number and counting is at the top of this page.

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Those are some of my favorite bloggers too! Thanks for the links and for doing the research! I have seen a few of those posts before, but definitely not all. There’s also something to be gained for sure from looking at all of them together. And it gives me some reassurance that I’m in good company 🙂

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This comes at just the right time for me and the 5 and 6 year olds I teach. We are counting up to 100 quite a lot now, and I suspect they might respond in similar ways to the ways yours did.

It’s a bit like garden design, this job of ours, isn’t it? We want to arrange the space in such a way that, arriving at a certain point, between two hedges say, a certain vista will open up. We want the right amount of drama and detail, order and chaos.

Here where the system is a convention, but a logical convention, we want the students to appreciate the order, the system too it, but know we also have to lead them a bit, as the system is one of a number of possible systems. It would be great if we could maximise the Aha! and Ooh! moments as the satisfying and economical logic of the place value system becomes clear. I can’t say I know how to do that – so thanks for opening up more possibilities!

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This is my first year teaching this age group with any sort of regularity and I must say that it’s been a truly wonderful experience. Their excitement and openness to new ideas can’t be paralleled by any other age group that I’ve had experience with (older or younger). I would love to hear more about the sort of things you do with your students of this age!

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I don’t know that didn’t say who I am – Simon Gregg! I guess you see lots that I do on twitter!

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I knew it was you :-). I’m a big fan of everything I see from you on twitter, but I’m sure that I miss a lot because I don’t always go there regularly. Should go back and look at things more systematically. How many different age groups are you currently teaching?

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Just my K3s, 5 & 6 year olds. (Grade 3s last year.) So, when I go back after this 2-week break, it’ll soon be time to hit 3-digit numbers. I’m trying it out lots of different ways in my mind.

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Pingback: 3-digit numbers are tricky! Part II | Musings of a Mathematical Mom

I enjoyed reading your blog and many ideas popped into my head as I read:

You wrote: “They also had no trouble counting by 10s to 100 and by 100s to 1000.” That prompted several thoughts:

1. It reminded me of when a child counted, “. . . seventy, eighty, ninety, tenty.” We talked about how to write that number, following the pattern of 7 with a zero, 8 with a zero, 9 with a zero, then 10 with a zero. It was then that the students noticed that the numeral for “tenty” looked just like “a hundred.” In a way, “hundred” is the correct term that makes less sense that “tenty” but is one of the social constructions of math.

2. When I’ve had students count by 10s, they typically stop after they say “one hundred.” I always ask them what would come next if they continued to count by 10s. I get different responses. Some know it’s 110, others think 200, others think 101. I’ve found this quick assessment to be revealing.

3. I’ve found that children typically can read and write 100 and 1000, but that’s not necessarily an indication that they understand the place value structure of those numbers.

I’ve had experiences where I give children a heap of base ten flats, rods, and little cubes and the problem: If we cut all of the flats and rods apart into little cubes so we had a pile of cubes instead of these blocks, how many little cubes would there be? You posed the problem differently, asking them to write the number you made with the blocks. I’m now curious if the children who were able to figure out to write 257 would also understand that this means that the blocks cut apart would result in 257 cubes.

About the notation of incorrectly writing 201 as 2001, this is an error I see often. I like the idea of writing both on the board and seeing, first, if they agree that the two numbers look different. And then if they agree that there’s only one way to write a number; that is, that 201 and 2001 can’t both represent two hundred one, and then to talk about which number is “right.”

Then I thought more about place value and how hard this seems to be. I know that I’m two months late responding to your question about what to do the next day, which is long gone, but I’d be curious about assessing their understanding of place value, even with tens and ones. I wrote about my favorite protocol for doing this in a blog post: http://www.marilynburnsmathblog.com/place-value-how-to-assess-students-understanding/ (Joshua, who left a comment on your blog, included a link to this as well.)

Again, I apologize for such a late response to your blog. But thanks for sparking my thinking.

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Thank you for so many wonderful thoughts!

1) I like “tenty” and I don’t think I’ve ever used that one before. I do, however, find myself often saying things like “forty-ten” and “sixty-ten” when counting by 1’s, either to emphasize that we have another ten or when students forget the usual words for these quantities.

2) I know that we’ve often counted beyond 100 by 1’s, at which point the students have no trouble saying one hundred and one, but I don’t remember whether we’ve done it when counting by 10’s (at least not recently). I should try this before the school year is over and see what they say.

3) I completely agree that being able to write 100 and 1000 correctly does not at all mean that they understand the place-value structure (in fact, I see that in my students). Next time we play with base-10 blocks with these students (which will either be next week or next year 🙂 I will try posing the question the way you did and report back on what they say!

I also really like the idea of interviewing the students one by one to evaluate their understanding. I actually did it earlier in the year with my 2nd graders, and it was very telling. I haven’t done them with these kindergartners yet, but it’s in the plans for early next year.

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