We need more young voices in science and math. I dream of children doing science alongside adults. Kids can contribute a lot to any project! They bring divergent thinking, creativity, and peacefulness to any working environment.
Joseph is a seven-year-old homeschooled boy. He loves science, engineering, and science fiction. He loves to ask questions and to tinker with things around him. He also loves climbing trees and splashing in puddles. Enjoy his poem!
Some think we are smart,
Some think we are dumb,
But our civilization
Is still very new
And in some place
In a vast Universe
There is a big old civilization
Five thousand times older than us.
We are just new,
We are just new.
Some people think
We are smart, too.
We are just new –
We are just new,
We are just new,
Like a new born baby.
With her mommy and daddy crew.
That are older than us
Old smart aliens,
Looking at us.
“Oh, how little,
Does not know how to move,
Use it hands, legs,
Can not clean after itself,
Pees all the time;
You can’t stop it –
It’s how normal baby grows.
That is the Earth,
That is the Earth”
That’s how we
Live on the Earth.
we do very bad things:
Like throwing sand
Into people’s eyes.
Of course, of course
We need to know
Stop doing that
Before we die.
Because we are doing something bad,
That if we did not, we’d be glad.
You know, you know,
That if we go
Five trillion light years
From the Earth
You will reach a place,
You will reach a place,
Where there is a new civilization
You have never seen.
But you guess what,
But you guess what -
It’s like a nanny
Looking at us.
Giving us food,
To everybody else.
Those other little civilizations
That are just born –
She needs to care about all,
About every single thing.
We are just one,
Just one of them,
That is just born,
Just newly here.
We know, we know –
The newly Earth.
All the Math Goggles challenges so far had to do with noticing math with your eyes. But for this week’s challenge, let’s try to just listen.
Let’s listen to the math in what our children talk about. I don’t mean like when we ask them what they did today in preschool, kindergarten or school. And I don’t mean like when we quiz them on how many teddy bears are in the room or what shape is the kitchen table.
Let’s listen to the math children bring up on their own. Our contributor, Malke Rosenfeld of Math in Your Feet, frequently describes such math chats on her blog. Here’s an example from her recent post:
Seven-year-old is pushing cart around the store, narrating as she goes: “Go forward, now one quarter turn to the right, now go forward, parallel park. Okay, now turn half way around, go straight, one quarter turn…”
Here’s my six-year-old who is waiting impatiently for his first baby tooth to fall out, but it seems it won’t ever happen:
Mama, I have a tiny hope, and it’s quickly approaching zero, that this tooth will fall out soon.
Or David Wees’s “Decomposing Fractions” post, in which he retells a conversation with his son:
Daddy, I’m full. I had 1 and a half…no, one and a quarter slices of pizza which is the same as five quarters of pizza,” said my son at dinner tonight…
By the way, David’s whole project, Math Thinking, is about children sharing their mathematical thoughts.
So this week, let’s just listen. You might be surprised at how your child looks at things, at math ideas she explores on her own, and at mathematical reasoning behind what she says.
You may also share your observations here on the blog.
Is 2*3 different from 3*2? My answer used to be “But of course! Don’t you know the commutative property?” Now, after following Malke Rosenfeld’s exploration of multiplication, I answer it with a lot more non-committal “it depends“. And I notice more and more examples when even though quantitatively, 3*2=2*3 and 8*1=1*8 and 3*5=5*3, qualitatively you sometimes get two distinctly different results.
Ironically, as I’m moving from quantity toward quality, my 6-year old is moving in the opposite direction. Consider these two examples:
Remember Maria’s review of Clap, Drum and Shake It by Marcia Daft? In particular, this part:
Do more multiplication. In particular, invite kids to multiply within pattern units. For example, how do you double the pattern unit “clap, clap, shake”? That is, how do you show 3×2 in the language of the book? “Clap, clap, shake; clap, clap, shake” is what the book does. You can also do “clap, clap, clap, clap, shake, shake”!
Two months ago I tried it with my son using 3 colors of PostIt notes instead of printed cards. He concentrated on the qualitative aspect (the patterns) rather than on the quantitative side (6 elements total) and insisted that the two strings had nothing in common.
Me: “But you said this one had 6 PostIts and the other one had 6 PostIts”.
Him: “But these are not the same 6s, Mom.”
Yesterday we finished listening to “Chitty Chitty Bang Bang” audio book. After giving the story some thought, my son asked me: “Do you know what it’d be if there were two Chitty Chittys?” He then explained that it’d be “Chitty Chitty Chitty Chitty Bang Bang Bang Bang”. He then went on to tell me what three Chitty Chittys would be like. You guessed it: “Chitty Chitty Chitty Chitty Chitty Chitty Bang Bang Bang Bang Bang Bang”.
Me: Can two also be “Chitty Chitty Bang Bang Chitty Chitty Bang Bang?”
Him:”Sure because you know, Mom, it’s the same thing”.
Which reminds me of a story we recently re-read. It was a chapter from the Karlson on the Roof by Astrid Lindgren. In it, a little boy says that he will have one birthday cake with eight candles on it (1*8); to which his friend adds that it would be a whole lot better to have eight cakes with one candle (8*1).
It also reminds me of a video in which Malke’s daughter is sharing her perspective on the quality vs. quantity issue.
What do you think? Share your examples of when you’d rather have a*b than b*a.
My name is Marie, and I am ten years old. Since I was six years old, I have attended a math circle. Last year I started to help out with the class for little kids. This year, I decided that I could start teaching a math circle on my own. Now, I am teaching a Pre-K math circle for little kids that are about four or five years old. The kids are used to me now, and I am really enjoying the teaching experience.
This is what I observed during one of the classes; it’s very funny. When I told the kids that the problem they were solving was a game, even if it was a just an ordinary problem, the kids started getting much more involved in it, because they thought that they weren’t actually solving a problem, but that they were playing a game! An example of when this happened was when we were using the board and pieces of the game, “Othello” (we weren’t actually playing the game though). The kids were reluctantly solving geometric problems using Othello pieces, until I told them that we were playing a game. “A game? Let’s play!” yelled the kids, excitedly going back to the exact same problems they were solving before.
Kids are much more involved in the class when:
- The problems involve them
- They get to choose what the problems are about, or at least change the details of the problems
- The problems involve real-life situations, especially if they have to do with their life
- The problem has a fairy tale, or some other kind of story woven into it.
For example, when the class was doing Venn diagrams, I made the diagrams about who in the class had sisters, brothers, or both. The kids were delighted with the very idea that they would be inside a mathematical problem!
I observed that warm-ups and easy problems that kids can solve on their own, correctly, help the kids relax and build more confidence in themselves, when encountering harder problems later in the lesson. Discussions and introductory examples to the topic are a good way to start a lesson. DO NOT start a lesson with a game! Kids will lose their attention, become over-excited, and be unable to return to the topic.
Teaching a math circle turned out being much more fun and interesting than I expected. I enjoy listening to the kids’ ideas and thoughts, and observing how they react. The kids consider me adult enough that they listen to me, but they see that I am still a child, so they are not afraid to share their ideas, and make mistakes.
If you would like to see the lessons of the math circle, and more details, please visit my blog at:
Check out my newest home decor item, a hundred chart. The amount of work I put into it, I consider getting it framed to be proudly displayed in the living room. The thing is monumental in several ways:
1. It is monumentally different from my usual approach to choosing math aids. My rule is if it takes me more than 5 minutes to prepare a math manipulative, I skip it and find another way.
2. It is monumentally time-consuming to create from scratch all by yourself.
3. It is monumentally fun to show to a child.
My son, like many other kids, is fascinated with big, huge, stupendous numbers – a million, a billion, a googol. He is also comfortable with very small numbers, all the way up to 7 or 8. But the space in between, particularly the first hundred, is confusing and thus boring to him.
Some of the math tools we’ve tried so far – fingers, counting sticks, pebbles, marbles, counting bears, abacus, Cuisinaire rods, the number line, even (gasp) rote memorization… No matter what we tried, number facts for anything greater than 7 remained incomprehensible.
Getting slightly desperate, I spent several evenings making this chart. It doesn’t look exactly like the usual hundred charts. Instead, each number has bars underneath. There are bars for units and there are bars for tens and 100 has a bar for hundreds. Each bar is made up of 2 rows of 5 cells. So, the number 35 has 3 bars in the tens space and one bar in the units space. All three of the tens bars are colored in, but only 5 cells of the unit bar are colored.
The idea is not mine, but taken from an old Russian book by Nikolay Zaitsev. In it, Zaitsev explains that with a chart like this, a child gets to see exactly what each number is made of and develops an idea of place value without lengthy and confusing explanations.
We’ve just started working with this hundred chart. By this I mean we finished putting it together, looked at it, counted to 55 or so, then skip counted by 10 to 100. Oh, and my son put stickers on the most important numbers (the ages of all family members, our cat and Preston Stormer, his fave toy of the moment). Along the way, my son asked some terrific questions that never came up before:
1. Why are the numbers from the bottom repeat in all over numbers?
2. What are these bars and why not all of them are colored in? Can I help you color them in? How many do I color in for this number? Why?
3. Look, if I go this way (moving from right to left), numbers are getting smaller! And this way too! (top to bottom)
4. (Halfway through chart-making) I think you will run out of space because the cards are getting bigger. How many cards can fit on the board?
5. Why are there more bars for this number (78) than here (18)?
I gave my son a couple of left-over cards since I printed a couple of pages twice. He was busy drawing on them and coloring in the cells. Then he created his own card for the number 5 (this is how many weapons of a certain type Preston Stormer has on him):
He now stops by this hundred chart a few times each day. And the most frequently asked question nowadays seems to be when are we going to expand this chart.
This is a story inspired by Vi Hart’s “Wind and Mr Ug” video. I so wanted to tell it to my son myself, but my drawing skills fall way short. So instead we talked about ants.
Once upon a time two ants lived on a strip of paper. A strip of paper has… how many sides? how many edges? [I drew two ants on a strip of paper] Each ant lived on his own side of the strip. They never saw each other, but since ants have an excellent sense of smell, they smelled each other. And they really, really wanted to have a playdate or maybe a tea party one of these days. They tried visiting one another, but each time they reached an edge of their little flat worlds and would get scared.
Then one day something happened. There was much shaking and twisting and the ants got scared and closed their eyes and covered their heads and tucked their antennas. When the twisting and shaking stopped, they opened their eyes and saw something strange. Their flat world was no longer flat. Instead, it became cylindrical [At this point I glued the edges of the strip together to create a cylinder]
Hooray! said the ants. Maybe now we can visit each other! One of the ants, who was a bit braver and more adventurous than the other, immediately set out on a round-the-world trip in hopes of meeting his friend. He crawled and crawled along, leaving tiny prints behind him [I'm drawing ant's path with a marker]. Will he ever see his friend?
Soon the ant came to a set of tiny footprints. At first he got excited. Was that the marks left by his never-before-seen friend? Is he getting closer? But soon the ant realized that those were his own prints and he’s been crawling round and round his little world.
But what about the other ant? He too sat out on a journey, crawling along. Will he have better luck? [I'm drawing second ant's path with a different colored marker]. Nope, he too finds no one, just his own footprints. Their world had… how many sides? how many edges?
Poor tired ants needed a rest. But just as they were going to take a nap, their cylindrical world shook and twisted again. Again they got scared and closed their eyes and covered their heads and tucked in their antennas. [Here I cut the cylinder to turn it back into a strip; then I twist the strip and glue to form a Moebius strip]. When the twisting and the shaking stopped, they opened their eyes and looked at their strange new world. Maybe now that it changed they will be able to meet each other.
The first ant, the braver one, set out on his round-the-world trip once again. He walked up the hill and down the hill and across the valley [I'm tracing the ant's path with a marker] until… he saw the other ant! Hooray, the two cried and hugged each other. And then they walked back to the first ant’s home [a child is tracing the ants' path with a different color marker]. Their world was no longer flat. Was it a cylinder? Nope, it became something called a Moebius strip. How many edges does it have? How many sides?
And that was the story. But then we experimented some more. We made another cylinder and another Moebius strip, each with its own pair of ants. This time my son traced ants’ paths all by himself. Then I brought out the scissors and both worlds underwent another cataclysm, this time it was a continental drift (thanks, Ice Age 3, for the idea) as I cut the cylinder and the strip in half. Want to know what happened to the ants? Try it for yourself. It’s really very fun!