Schoolless

Mind popping math

I'm not quite sure what I've taken on.

Thirteen kids aged 7 through 13 squeezed around my dining room table and looked at me with varying degrees of anticipation.
"I don't want this to be about rules." I ventured.

Running a math club for homeschoolers has been something I've been thinking about for several months. And now, it has officially begun. I imagine that a classroom teacher stands in front of her group of kids with a fairly firm idea of what she wishes to accomplish and how that's going to happen. I don't have a teacher's training, and had no such preconceived notions. I wanted merely to present what I hoped was a fun idea, and let the kids run with it. Yet now, running was something I was a little afraid of. There were, after all, thirteen of them.

Here was the problem:

How many kernels of popped corn would fit in my van?
You may use: one bag of popped corn, an empty box, a tape measure (optional), a calculator, plus paper, pencil and a friend.

The problem immediately had to get modified due to abundant rain, and a large quantity of squishy mud around my van. I desperately wanted that mud to stay outside. After a short deliberation, we decided to use various rooms around the house instead. I divided the kids into mixed-age groups, and set them loose.

Initially, they needed a little prompting. How would you go about figuring out how much popcorn will fill your box? Could you do it without counting the kernels? Could you do it without actually filling the box?
And they were off.

At first, four groups of kids crouched of the living room floor deliberating and scribbling in their notebooks. Then, for two hours, chaos reigned. Each team picked a different room to measure, and one team picked the fireplace. There were kids inside the fireplace. Others stood on the bathtub and the toilet. Still others climbed beds and each other's shoulders before they figured out they could measure the height of a ceiling with a stiff tape measure without actually climbing to reach it. Popcorn disappeared from bowls as great quantities of it were consumed or scattered on the floor in the course of the project. Calls for help were heard from one part of the house to the other. Younger kids needed explanations of what bigger kids were doing. Bigger kids needed conversion factors. Liam entertained everyone with stories of a monthlong ski trip from which he had just returned against his will.

But, in general, everyone seemed on board with solving the problem. Peter drew 3-d cubes and penciled in numbers along the sides. In between ski stories, Liam measured several corn kernels, and calculated their average size. Zola multiplied numbers into the tens of thousands longhand, oblivious to the presence of several calculators in the room. Zeb figured out that two boxes fit in a foot, and measured the room using boxes. Nadia wrote long paragraphs documenting every step of the process. Louise and Eric meticulously converted feet into inches and then into boxes. Mattie had a little trouble with writing some numbers, but seemed to intuitively grasp the concept of volume.

In the end, it was estimated that about 149,000 kernels would fill my fireplace, just over one million would be needed for the bathroom, and about three million for each of the bedrooms. As a bonus problem, we figured out that if we wanted to play a joke on Peter and literally fill his room with popcorn, it would cost us about $5,600. There were no takers.

Here are a few things I would consider for future math club meetings:

• Briefly introducing the basic concept of what we are doing. I wanted this to be as far from a "lecture" as possible, but a short demonstration of calculating volume with a few simple cubes might have been helpful.
• Having kids work in same-age groups. The idea of dividing them into mixed-age groups was, of course, to keep everyone involved, and to balance out different levels of ability. This notion of "balance" may be somewhat flawed. Younger kids may understand something conceptually, but are frustrated by the inability to express it in mathematical terms. However, they may be able to express it through pictures or narrative. Older kids who have the facility of using mathematical notations may find it more satisfying to use these shortcuts. I dunno. Different activities may lend themselves to different approaches.
• Allowing kids to choose to what degree they want to participate in the project. Kids have different attention spans. Many of the participants seemed quite eager to get down to the business of solving the problem, while others wanted to go exploring. I think that's fine, as long as they respect the essential integrity of my house. I prefer to have kids voluntarily engage in an activity. Perhaps if those who are genuinely engaged appear to be having a good time, the more reluctant ones will come around on their own.

I consider this our warm-up project. The kids almost certainly had fun. They probably learned something from the process as well, though perhaps it was not what I would have intended. Who knows. But, as usual, I myself learned a few things, and walked away with a lot of questions.

Worksheets be damned

I don't know where math anxiety comes from in our house, but it may have something to do with worksheets. My kids, especially the older one, think of worksheets as gratuitous torture which force them to do mental contortions just for the heck of it (I think the unbridled use of Saxon worksheets in my son's third grade classroom was responsible). For my part, I have been conditioned over the years to think of worksheets as a good way to get a handle on math facts and operations.

Everyone in our house has a pretty decent math sense. Which is not the same as knowing their math facts. Though, is knowing math facts a good way to learn to love math? What if math facts could come gradually through the love of math?

As adults, we know that facility with math facts helps us at the bank, the gas station, at the store, and in the kitchen. To kids, it must seem that math facts only help you do math problems. It's hard to see how they can help you solve any other kind of problem. And I've discovered that, more than anything, my kids, and many other kids I've observed, want to solve problems.

So, here's a problem for you (straight from Marilyn Burns' I Hate Mathematics):
How many popcorn kernels can fit in a mailbox?

This problem captures the kids' attention immediately for two reasons: it is completely silly ("Are we really going to stuff popcorn into a mailbox?!"), and it requires making popcorn ("Can I, can I?").

After the popcorn was popped, I found a smallish lightbulb box and asked the kids if they had any ideas about how to guess the number of pieces of popcorn that would fit in it without counting every one. Nadia's plan was to figure out how many were in one layer at the bottom of the box, then count how many layers the whole box could accommodate, and add them all up. Peter liked the idea of measuring the average kernel, then measuring the sides of the box, and mathematically (!) figuring out how many kernels the box could accommodate. Both perfectly good solutions, both arrived at independently, and not a worksheet in sight.

We considered both solutions, decided on Nadia's for the time being, and established that the box would hold about 412 kernels of corn. Now the problem was to translate the contents of the box into the contents of the  mailbox. Peter suggested that one way to tackle it was to see how many layers of boxes would fit in a mailbox, how many boxes were in each layer, and multiply it out. Not a bad plan. We went to check it out. Just in case, I brought along a tape measure.

First, we measured the mailbox using our small box. We omitted the curvy top of the mailbox, because it was non-conforming, and also, if we really stuffed popcorn in there, then it wouldn't open. We discussed the fact that so far we have been using non-standard units of measure: kernels of corn and lightbulb boxes.

For comparison, I suggested we should also take standard measurements, and that's where the tape measure came into play.

Back at home, we tabulated our findings, and came up with the following results:

• Using the non-standard units, the mailbox could hold about 101.25 lightbulb boxes.
• If each lightbulb box held 412 kernels, that the mailbox could hold 41,725.
• Using the standard units, the mailbox had the volume of 135,675 cm3, while the lightbulb box had the volume of 1,200 cm3.
• The mailbox volume was 113 times that of the smaller box.
• Thus, by this method, the mailbox would hold 46,556 pieces of popcorn.

So, if you really want to turn your mailman's day into a nightmare, you need about 50,000 corn kernels, which translates to roughly 75 bags of corn, at maybe 50 cents each, or between $35-40. Is it worth it?

Pure reasoning

The budding mathematician writing in her journal (spelling dictated by dad).

This always happens after a visit with someone who has accomplished children. You know, a six year old who reads at fifth grade level, a kid who is writing computer programs in second grade. This time, it was the cousins' twin children who, at age seven, were doing division and multiplication with ease. This was early was last year. Nadia was barely five. So, of course, on the way to preschool I decided to quiz her own mathematical ability.

I started off real easy: Nadia, if you have four cookies and there are four children, how many cookies does each child get? One.
If there are two children, and each one has two cookies, how many cookies do they have together? Four.
How about if there are three children, and each has two apples, how many apples all together? Six.
Wow, I was beaming with maternal pride. I decided to go for something harder. I had meant to ask how many apples would each of two children get if there were four apples. But what came out of my mouth was:
What if there were two apples and four children, how many apples would each child get? Nadia was unfazed. A half, she promptly replied.
At this point, it was clear to me she was ready for a challenge. So I asked:
What if there was only one apple, and four children. How many would each child get? They would get little triangles, mama.
I was laughing so hard, I had to pull over.
Nadia is now six and a half, and has a full year of homeschooling under her belt. During her "kindergarten" year, she has enjoyed some of the math books I purchased, as long as I let her color and decorate the pictures. This was OK with me. So some days, we would spend and hour on one page, because she was busy drawing bows and polka dots on the little girls in the illustrations, and on other days, when she was feeling less artistic, we would breeze through fifteen pages in as many minutes.
She seems to have an intuitive understanding of arithmetic, and absolutely no patience for mathematical symbols. She has stopped using her fingers to add and subtract, and will get the right answer nearly every time, provided I remind her what the little cross symbol or the little dash mean. She's not sure if thirteen or thirty come after 29, has not mastered counting by fives or tens, and can only tell me what time it is by reciting the symbols on the digital clock in the exact order. But she does know, without anyone having consciously "taught her" how many half-hours there are in two hours, and that if you try to divide five apples among two children, you will end up with halves, unless mom gets one. What if mom is not around? Well, then who will cut the apple in half for the children?
Go figure!
She has been.

Legomath

This was back in May, when I thought we really should get back into math.
We used Legos to figure out which numbers were prime numbers. Nadia took groups of like-colored legos and tried to group them into even groups. The numbers that were prime could only be grouped into one large group, or groups of singles (divisible by itself and by 1). The numbers that could be evenly divided into smaller groups were not prime.

While Nadia threw herself wholeheartedly into this project, Peter got fed up pretty fast. "I HATE math," he chanted like a mantra. Then he added unexpectedly: "but I like doing math the fun way." I wasn't quite sure what he meant. He had never previously put the words "math" and "fun" in the same paragraph, let alone the same sentence. As he spoke, he was putting together a lego pyramid with a square base. Chris commissioned him to build a pyramid for a king, the base of which would be a square with a side of ten bricks (measured in little square lego's with four dots). As the royal engineer, his job was to figure out how many bricks would be needed to construct such a pyramid (solid, not hollow). First, he established how many bricks were needed for each level. This provided a thorough review of the perfect square numbers up to 100. Then, he determined that you would have to add the total of bricks on each level to figure out the total number of bricks needed.

Since he understood how to do the calculation, I decided to introduce him to the magic of spreadsheets to do the actual math. Peter learned how to enter values into cells, how to use autofill, and how to enter a simple formula into a cell. He concluded that we would need 385 bricks. Once he knew how to do that, he figured out how many bricks would be needed to build a pyramid with a square base whose side was 200 bricks. For that, we would need 2,686,700 bricks.

Since we were on the subject of pyramids, we spent a little time talking about equilateral and isosceles triangles. Here is what we covered, according to Peter: the sum of angles in any triangle equals 180; an equilateral triangle has at least two sides of the same length, at least two angles have to be the same, the third angle equals 180 minus the sum of the other two angles; an isosceles triangle must have three sides of equal length, and three equal angles. He reminded himself that each of its angles is 60, because 180 degrees divided by three is 60.
OK it took longer than I planned, but we were not bored. And, I was reassured to see that some of the math actually stuck.