There was a while when Tabitha (five and six years old at the time) would try to get away without wearing underpants when she dressed herself. Those days are pretty much over, but I still like to make sure she has done the complete job, so I ask her from time to time.
Tabitha (7 years old): I’m dressed!
Me: Are you wearing underpants?
T: Yup—Dora the Explorer.
Don’t worry. The child is not wearing these in the picture.
Me: Nice. How do you feel about your Dora the Explorer underpants?
T: I don’t really like Dora that much, but I have a thousand of them.
Me: That’s a lot.
T: I counted them once.
Me: All one thousand?
T: No. I don’t really have a thousand. I don’t even know how to count to a thousand. Just to ten hundred.
I pause for a moment. Does she mean one-hundred-ten? Can’t be. She must know that one-hundred-eleven comes next.
Me: Ten hundred. You mean like after nine hundred is ten hundred?
T: Yeah. That’s as high as I know how to count. I don’t even know how many a thousand is.
Me: A thousand is ten hundred.
T: Oh. Cool.
A few minutes later, I get an idea. I wonder how she would write ten hundred. She needs to get out the door for school so I make it quick. I ask her to read some numbers out loud as I write them.
- 900
- 832
- 110
- 1000.
For that last one, she says one thousand.
I ask how she would write ten hundred.
She writes, “1000”.
T: It’s the same.
Me: Because I just told you that. Right. How would you have written ten hundred before I told you it was the same as one thousand?
She shrugs her shoulders. Drat. Moment lost. We talk about hundreds for a moment. One hundred, two hundred, etc. up to ten hundred.
Then I have one more.
Me: OK. Last one, then off to school. How would you read this one?
I write 10,000.
She looks for a moment. And thinks.
T: Ten….
More thinking.
T: Ten thousand?
High five!
I zip up her sweatshirt and send her out the door to catch her bus.
So What Do We Learn?
A recent research article argued that children learn a lot about place value through everyday conversation, and that kindergarteners know a lot more about the structure of the number system than parents and kindergarten teachers (on average) think they do.
Here you can see that knowledge in action. Tabitha knows that 1000 is a big and important number. She knows the pattern that allows you to keep counting by hundreds. She has not put these two pieces together. A short conversation helped her put those two pieces together, and then to extend the pattern.
Starting the conversation
This didn’t start out as a math talk. It began as a clothing inspection. But the opportunity presented itself. Listen for those times your children use numbers, and ask follow up questions about them. You won’t get this much learning out of every such conversation, but if even 10% of those opportunities turn into a little bit of learning, the interest compounds.
“and that kindergarteners know a lot more about the structure of the number system than parents and kindergarten teachers (on average) think they do.”
This is so true. We forget that the number names themselves are based on the base ten system (except to some extent in France). We also tend to ignore prior exposure to ideas about fractions, and rush to overwrite them with excessive formalism.
That is simply fabulous. I also appreciated the link to the research. It has seemed to me for a while that kids “get” place value before they were expected to, but I’m not sure I have done the best job in advancing their learning in that area.
Something I wanted to ask you: Do you think that kids MUST have a lot of exposure to math manipulatives in order to develop number sense or other math skills? When I teach kids with poor attention spans, they tend to get lost in the manipulatives, building towers or train tracks.
All the evidence I have seen suggests manipulatives are really valuable tools for learners at all levels. As a slightly more expanded framework, think about taking your back and forth between Concrete (physical manipulatives)/Pictorial (drawings and diagrams)/ Abstract (purely symbolic). Also, trust me that there isn’t anything inherently more advanced about abstract or inherently more childish about concrete. Check out the amazing stuff Erik Demaine and his father do if you need reassurance about the last point.
For your question about manipulatives being distracting, I saw a tip elsewhere (sorry, don’t recall the source): plan to allow some free play when you introduce a new manipulative. The source suggested the free play could be before your target activity or promised for afterward. However, my (limited) experience is that it works best if the free play is done well in advance, say a prior day.
Here are the reasons why:
(1) It gives them a chance to explore the materials without a strict time pressure, so that they get a real chance to investigate. I believe this phase is actually important for the later math learning as they are building their intuition for the physical properties of the manipulatives that you will then invoke as a model for the mathematical concepts.
(2) You separate the free play from the instruction time so that moving from one to the other doesn’t feel like a punishment (“No, you can’t play anymore, you have to do math”) or a release (“enough of that stuffy math, now time to play!”)
(3) You can practice/teach responsible habits for using the manipulatives through multiple sessions. For example, where are they stored, who gets them out, how are they distributed, how do you collect them at the end, etc.
Frankly, of all these reasons, I think the first is most important. In many cases, I would offer that a thorough play, investigation, exploration of the manipulatives is even more valuable than the specific math lesson.
Thank you so much for taking the time to respond to my question. Great advice! You could post a whole section on the use of manipulatives! I am excited about moving forward with this approach.
My 6 yr old and I are in the middle of underwear inspections these days, hoping I can find more math out of it than “how many days since you last changed those?” Great post!