Mathtubs

Give me a mathtub each morning,
Give me a mathtub at noo-oo-oon,
Give me a mathtub each evening,
But give me a mathtub soon.

One of the little perks of my job is distributing the mathtubs each Friday. {See the link below for more info about these tubs--they're boxes filled with math games, math materials, suggestions for math-related projects, and picture/storybooks with math connections; kids take them home for a few days at some point during the year.}

I walk into a classroom, around about the time pizza is delivered, carrying one or two tubs, and deliver the tubs to the children (CHOSEN AT RANDOM--NOTHING UP MY SLEEVES) who will get them for the next few days.

Children vary, of course, in how they express their excitement over getting the tub (from a small, self-satisfied smile up to an enthusiastic fist-pump and a chanted "Oh yeah, oh yeah, oh yeah"), but about 97% are very pleased to have their turn. It's quite gratifying.

The other children, meanwhile, are full of helpful comments such as "Is it my turn yet?" "When is it my turn?" "She's so LUCKY," and so on. For the moment, at least, the arrival of the Math Guy and the Math Materials outranks everything--even pizza. No small feat.



I thought I had invented the mathtub idea, or at the very least, um, repurposed it from a similar idea I'd read about somewhere in which teachers sent books home in backpacks for kids and families to enjoy. A couple of years into the mathtub project, though I was cleaning out some old papers and discovered to my surprise that in 1997, while at a conference in Rochester (NY), I had actually attended a workshop in which the presenter was describing how teachers could package up some math materials for use at home. I still take credit for the name "mathtubs"--I think that teacher used shopping bags or something similar--but as for the concept, well, I should know by now that there are few truly original ideas in education. Hey, it works, and the kids enjoy it, and that's what counts--right?

You can read more about mathtubs here, in an article I published in a teacher magazine a few years back:

http://www.highlightsteachers.com/archives/articles/the_mathtubs_are_coming_by_stephen_currie.html

People v. Tables

"I am going to have a party," read the question given to a number of our 1-2 students the other day. "I want to invite ___ people." (The blank is standard: everybody gets a different number, which a) cuts down on the Problem of Roving Eyes and b) allows us to give somewhat harder numbers to kids who are ready for a challenge while keeping the same problem frame for everyone.)

"I have ____ tables where my guests can sit," the problem continues. "Each table has room for _____ people. Do I have enough tables, or do I need to get more?"

Different kids had different ways of attacking the problem, as usual. Some sketched the tables, drew chairs around them, and counted by ones. Others dispensed with the chairs and simply wrote the number at each table, then counted by that number if they knew how. A couple didn't bother with a sketch at all. One or two made groups with checkers or other materials--7 groups of 6 checkers, for instance, to represent 7 tables with 6 people at each--and then checked the number of people to see if they'd gone over or not. The strategies were generally quite accurate, if not consistently efficient: the next step will be to move kids away from the pictures and toward more abstract skip-counting and other strategies.

At any rate, children needed to show or describe their work and then answer the question (which, if you recall, had something to do with whether there were enough tables or whether we needed to get more). Several children didn't recall--they needed a reminder to do this part--but eventually we had the answers we sought.

"I have enough tables," one child wrote confidently and accurately. (Actually, she wrote "enuff," but let that pass...)

"You have enough tables. Am I rite?" wrote another child, perhaps a little less confident than the first. (Yup, I told him, you're rite. Um, right.)

"You need to get more tables," wrote a third responder, "because seven tables is going to be a smaller number of people. You need 8 tables." She included a careful sketch with the correct number of heads jowl by jowl at each table: an arrow then pointed to the last table, with the helpful label "extra."

"I have to sell one more chair," wrote still another girl. A somewhat convoluted way of saying that she not only had enough tables--she had an extra seat. I'm not entirely clear whether the sale would be an auction for the right to attend the party, or simply an attempt to convert an unwanted and unnecessary item into cold hard cash. Either way, this is a girl who knows the value of a buck.

And perhaps my favorite: the boy who discovered that he had space for 52 when he only needed to seat 49. After showing his method, he concluded: "You need more people."

Virtual Manipulatives

We sometimes haul out the laptops during lower school math times and have kids work with Utah State University's National Library of Virtual Manipulatives website. We've made good use of this site for projects with both the 3-4 and the 1-2 classes, but my personal favorite is the subtraction.

See, you get these rods and cubes, just like base blocks only they're on the screen and exist only in pixel form, so they don't fall off the table and get lost and they can't be used as hockey sticks and pucks, or as drumsticks or grenade launchers or whatever else creative minds have in store for them.

--Oh, and then when you model regrouping (which I prefer not to call "borrowing," as I've said before, because you don't ever give it back--I prefer to use the phrase "stealing") you actually grab one of the virtual tens rods and bring it over to the ones column and let go and watch as it separates itself into ten little ones cubes.

Then you hear the kids saying WHOA! and COOL! and NEATO TORPEDO! (well, not that one, maybe) and the like.

Then you get to separate hundreds into tens the same way and thousands into hundreds and the whole thing is utterly charming and truly awesome and the best thing next to...

[Down, boy.]

[The picture below isn't actually from the virtual manipulatives website--it's from a powerpoint presentation I made dramatizing the process. What you see here is the ones stealing a ten, in the dead of the night of course, dragging it back to Ones Street, and breaking it into ten little ones cubes so there'll be enough ones to carry out the subtraction.]









Anyhow, children sometimes ask how they can get to the site at home. Unfortunately, the address isn't straightforward. If you google "virtual manipulatives," it's the first site that comes up (as of today, anyway).

The whole site's URL is http://nlvm.usu.edu/en/nav/vLibrary.html. If you're interested, take a spin around the site with your child(ren). It may not be the equivalent of a medieval European cathedral, but as the Michelin guide would put it, it's quite definitely worth a visit.

Corn

How many kernels on an ear of corn? we asked the third and fourth graders the other day. They've been studying the Mayan people, who called themselves "People of the Corn," so it was a worthwhile question.

We started by having students find approximations; as you should know by now if you've been reading this blog, us Math Guys consider this a very important step. We asked students to choose a round number (a number that is a multiple of 10); the point, after all, wasn't to guess the exact number, but to use a number that makes some sense and is relatively easy to work with. You can always revise your estimate later, we assured them.



What is the estimate based on? Well, we gave them each an ear of dried corn to eyeball. Some did some quick-n-dirty calculations, fourth graders in particular. (Yes, we asked them to justify their reasoning. Some of them HATE this, but it's oh-so-good for them.)

"About 20 in each row," wrote one student. "Maybe 10 rows. 10 x 20 = 200. I estimate 200 kernels in all."

"I think there are 20 rows and 30 in each row," reported someone else, "but that might not be enough so I added a few more. I say 640."

"I think 260," wrote a third grader, who would have been happy to leave it at that, but who added, under duress from a teacher, "because it looks right. And because it's a good number." We might call this strategy "Pick-a-large-number, any-large-number, and-assign-it-great-virtue-so-critics-will-be-cowed."



The next step: Count the kernels! The classroom teachers had prepared egg cartons with ten cuplets (better them than me). Kids used their fingernails to push the kernels off the cob (great fun). Then they distributed the kernels 5 or 10 at a time into the cups, making groups of 50 or 100. Record the number, dump out the kernels, lather, rinse, repeat.



At some point along the way several students noticed that their estimates weren't looking as accurate as they had back before counting had begun. This was especially true for those whose initial strategy had been "Pick-a-large-number, any-large-number &c," but other more careful estimators ran into this difficulty too. No problem! we said. Just revise your estimate, record it--oh, and explain why you wanted to change your original prediction. (My favorite: "Because I passed my first estimate a long time ago.") You will no doubt be shocked to learn that the second set of estimates were considerably closer than the first.

Eventually, all corn kernels were off the cobs and had traveled through the eggcups and into plastic bowls or paper bags (except for a few strays which had found their way onto the floor), and everyone had an exact answer. Some were surprised to see how many there were. Others found the results unsurprising in the extreme, or claimed they did: "I knew it," crowed one boy whose answer was not, perhaps, as close as he thought.

As students finished, they compared their totals with friends and thought about questions such as Why aren't all the totals the same?, What could you do to get a better estimate next time? ("Nothing," said the young man quoted above), and About how many kernels do you think there might be in the whole class?

So, three-digit numbers, ordering, estimating, grouping by tens, fives, 50s, and 100s, and explaining reasoning. Plus, a fun project (there's something truly satisfying about flicking those kernels off the cob, and something even more satisfying about running your fingers through a nice big tub full of everyone's kernels), and one that relates to science and social studies. A worthwhile math period indeed. Next up: data analysis with these results. On Thursday we'll be in the Chapman Room calculating the median and range of the data and forming a Living Histogram. Pictures to follow, assuming my camera behaves itself...

p-a-t-t-e-r-n-s

What do you call it
When things repeat?

We call it...
A pattern.



Head, shoulders, knees, and feet,
Head, shoulders, knees, and feet,
That
Is a pattern.



A, B, C, A, B, C, A, B, C, A, B, C
That
Is a pattern,

1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3
That
Is a pattern.



What do you call it
When things repeat?

We call it...
A pattern.