Math Monday::Banana Dave Edition!

So, the big question here today was: What does a sea star and a banana named Dave have in common?

A: They both have pentagonal symmetry!

To explain further, here are my hooligan kids. This was originally the practice version, but then I realized it was almost 10 and I’m exhausted so we went with it. They do a good job explaining everything!

If, for some reason, we do not do a good job explaining everything, forgive us. It must be the late night and sugary dessert that got to us. Leave a question and I’ll explain it all later, after a nice long rest…

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Books and games we’ve enjoyed:
Symmetry: A Journey into the Patterns of Nature

Why Beauty Is Truth: A History of Symmetry

Studies in Symmetry with pattern blocks.

SYMMETRY SHAPES

Symmetry Letters – (26 letter shapes)

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resources on the web:

More pictures (300+) and information about echinoderms.

Symmetry in Biology.

Going Bananas for Echinoderms; a sheet that describes how you can do this same thing on your own with just a banana. You can name yours whatever you want.

Fantastic and simple article about five fold symmetry.

Images of things with Pentagonal symmetry.

Math Monday::Beach (or Pool, or Bath) Physics!

If I organized a school, kinda like what Emerson and Alcott and Thoreau did (all of them transcendentalists…I’m mad for transcendentalism and their impact on alternative education in America…) then I’d make sure to set it up on a coast where kids could play in the surf and sand.

To children, play is the fastest route to learning.

Children learn through various methods, but often their physical and social world teaches them the most.

This is true for adults, too, but I think most of us have forgotten that!

In any case, the physical aspect of playing in the water and sifting through the sand can open up so many discussions that are so mathmatically important. I know many of you don’t have a beach access, but it’s really not about the beach at all. The process of introducing math to any mind is more about opening yourself up to the discussions and details and play no matter where you are.

For us, being so close to the beach, we spend a lot of time in the waves. We do not compute math problems while there. We do not take timed tests. We do not memorize times tables. But we do discuss physics!

I, personally, never took physics, I could barely pass Math 101 in college (hello dyscalculia!) but my sister minored in it. Go figure. Anyways, I know she studied waves. Waves are a part of the science of physics, defined as a type of change that moves from place to place.

Instead of learning about waves from a textbook, we are learning about them from the real deal…the waves themselves. If you don’t have a beach close by for your kid (or yourself!) to play in, take the play to a bath or hot tub or pool or kitchen sink.

We like to throw a ball out as far into the ocean as we can, and then try to be the first person to get it back. The ball hovers over the waves, bobbing and dipping, until it gets caught in a breaking wave and comes towards shore. Often, in the midst of going to get it, one of us gets caught in a breaking wave as well, and gets tumbled on the sand. I’ll show you a video of our in depth learning:


You may be saying to yourself….but wait, you’re just playing with a ball at the ocean! And you’d be right. That’s the beauty of Math Mondays! So much of it is rooted in play, and the learning and exploring feels happy and not like learning at all. I’ll review some of the discussions we’ve had because of our beach play.

* Is the wave moving the ball, or is the ball moving over the wave?
* If a wave is approaching, where is the best place to be…the bottom (underneath the water), the middle (face) or top (crest)?
* Do big and small waves have the same force against an object?
* what makes a wave move?
* What makes a wave break?
* What is the most dangerous kind of wave?
* Can you ever throw a ball far enough out so it doesn’t come back to the same shore?

Sometimes I answer their questions, usually–because I’m no physics major, I just wonder along with them. In a break with public school, I believe the most learning happens in the questions, not the answers. I rarely introduce a subject unless my kids are questioning a lot about it, because their minds are the most receptive to studying something if they want answers.

Throw a bunch of balls in the bath, hot tub, or pool and let the play begin. Keep an ear out for the early signs of wave exploration!

Then, trips to the library can begin. There are lots of books on waves. The kids and I have poured over this wikipedia article on waves. We also liked this series of “lessons” on waves from The Physics Classroom!

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more links:

transcendental Ideas about Education

How You Can Support Children’s Learning Through Play

Math Monday::Teach Less Math in School.

I discovered last night that my favorite Math guru and total inspiration, Sue VanHattum (and her blog, Math Mama Writes was referenced in an article in Psychology Today, called When Less Is More, The Case For Teaching Less Math In School. Intrigued with the punk rock title of the article, I immediately went over to check it out.

It. rocked. my. world.

Seriously, go read it.

Right now.

Normally I don’t like people to leave my blog in the middle of my post, but I’ll make an exception…just promise to come back!

For realz, why are you still reading this?!

For all you who don’t want to hyperlink out of my little heavenly haven of bloggy goodness, I’ll sum up.

In 1929, the superintendent of schools in Ithaca, New York, sent out a challenge to his colleagues in other cities. “What,” he asked, “can we drop from the elementary school curriculum?” He complained that over the years new subjects were continuously being added and nothing was being subtracted, with the result that the school day was packed with too many subjects and there was little time to reflect seriously on anything.

Another superintendent replied back with a shocking answer….drop arithmetic. And why would he say this?! Read on:

“For some years I had noted that the effect of the early introduction of arithmetic had been to dull and almost chloroform the child’s reasoning facilities.” All that drill, he claimed, had divorced the whole realm of numbers and arithmetic, in the children’s minds, from common sense, with the result that they could do the calculations as taught to them, but didn’t understand what they were doing and couldn’t apply the calculations to real life problems. He believed that if arithmetic were not taught until later on–preferably not until seventh grade–the kids would learn it with far less effort and greater understanding.

Considering that we here at Child’s Play have intuitively done just such an outrageous thing as not seriously begin a math curriculum during early and middle elementary years, this really peaked my interest.

Benezet followed his outrageous suggestion with an outrageous experiment. He asked the principals and teachers in some of the schools located in the poorest parts of Manchester to drop the third R from the early grades. They would not teach arithmetic–no adding, subtracting, multiplying or dividing. he asked the teachers of the earlier grades to devote some of the time that they would normally spend on arithmetic to the new third R–recitation. The children would be asked to talk about topics that interested them–experiences they had had, movies they had seen, or anything that would lead to genuine, lively communication and discussion. This, he thought, would improve their abilities to reason and communicate logically.

This, also is what we’ve done just instinctively. Math is more than numbers. Numbers are a way of expressing math, but at it’s core, mathmatics is all about logic, reason, problem solving, and a certain kind of thought process. And those things can be done without numbers at all, and strengthen a mind to prepare it for dealing with numbers later. But I digress.

So, how did his experiment turn out? The article goes in to greater depth, but in short:

Benezet showed that kids who received just one year of arithmetic, in sixth grade, performed at least as well on standard calculations and much better on story problems than kids who had received several years of arithmetic training. This was all the more remarkable because of the fact that those who received just one year of training were from the poorest neighborhoods–the neighborhoods that had previously produced the poorest test results.

Isn’t that so counter culture?!

Speaking just from our experience, since we’ve unknowingly recreated that experiment ourselves, our math abilities only took off when we dropped a math curriculum altogether.

What we’ve done instead is supplement our day with lots of critical thinking skills, lots of word play, lots of measuring, counting, and questioning, lots of diagraming and organizing…

And I can’t forget all the brain Teasers that get your mind thinking.

When doing math, think outside the box…especially if you have an outside of the box thinker!

link through to past Math Monday posts here

Math Monday:: Domino Math!

Don’t faint, but I’m actually posting a Math Monday blog today!

I’ve picked up a few more readers, so I’ll explain…

I am a recovering mathphobe. I hated math growing up, panicked about it in school, felt stupid doing it for my whole life. I can’t really even remember simple addition problems, let alone memorize multiplication tables. In 4th grade, everyone got ice cream sundaes at school for remembering their times tables up to 12…and I hadn’t even cleared them past the 4’s. No ice cream for me. Not only was I humiliated (I had to go sit by myself outside while everyone ate their prize in the classroom), I really really wanted some hot fudge. I pretty much concluded I was an imbecile, end of story.

Then my daughter came along with the same issues with math, and I felt like obviously that meant everyone dealt with it but some people learned to do math better because they were smarter. And then my son was born and he started multiplying in his head by kindergarten. And I thought…wait a minute…some people actually have minds that get it! That don’t struggle like mine and Naturalists. I found out about dyscalculia and felt like I finally had some answers.

Then Naturalist and I discovered sacred geometry, which made math and numbers so much cooler. So much more right brain and visual spatial!

So we started up a Creative Math Club, and doing math our way. Using art, music, and anything else we could get our hands on to make math very experiential.

Serendipitymama (with her cool blog, here…) tweeted me (I’m childplay on twitter, come say hi!) to ask if I’ve done any math mondays while here in So. Cal. I said no, because I think of the creative math club back in Colorado as the main part of math mondays. Without them, I feel mathless. A form of worthless, but with math.

But it turns out, that’s not really true. We do math all the time! Especially because Sassy loooooves dominoes. She carries around her own bag of them everywhere we go! So, in my first math monday video ever to debut here on Child’s Play, I bring you video documentation of what math looks like with Sassy and I! It involves being by a pool, singing in an opera voice, using watercolors, and a shout out to all the ‘rapping birds’.

The great thing about dominoes is that it’s so easily transformed into math games for older kids, simply by changing the operation. Instead of adding, you can multiply the numbers, divide, or subtract. And if you have creative kids (which I’m betting you do, because you’re reading this blog and we’re all about creativity over in this part of the blogosphere!) then they can pretty much come up with their own games with nothing more than some dominoes and some pens and paper!

In case you need some help, here are some links:

Domino Math Book (grades 1-4) that we enjoy!

Domino Math Book (grades 2-6) that we also enjoy!

Fun (free) domino games with domino math mats.

Math Monday::Memory Games = Better Math Skills

I noticed with Naturalist and myself that a lot of our problems with math actually have very little to do with the numbers and lots to do with our poor short term memory. (well, actually, we also have big problems with numbers, too, but one thing at a time!)

If you have a mathphobic kid that you are trying to help out, I suggest doing fun memory exercises rather than keep focusing on the rote memorization stuff or straight computation. Because the foundation for doing any of that well is a strong memory. How can you memorize math facts if your memory is full of swiss cheese holes?!

This is a great thing to do if your child has panic attacks surrounding math, or who says they hate it with a passion. If you can make math fun, then they have a harder time hating it, and when they stop hating it, they relax, and when they relax their brain works better. So spend your math time doing fun games instead! And if you can combine fun games with candy, then it’s all that much better.

Here’s a few that we like to do…without pictures because I’m in over my head with life a little bit this week. But I’ll explain it really well for you.

Take a handful of M&M’s. Put up a barrier of some sort, so you can see them on the table but your kid can’t, then arrange 4 of them in a pattern. Lift up the barrier for 2 seconds, then lower it, then have them recreate the pattern in front of them. If they can do that without a problem, increase it to 5 M&M’s. Then 6, then 7…the point is to build up to as many M&M’s as they can. You may need to adjust the time they can see them from 2 seconds to something longer. Then switch and have them come up with the sequence and you try to reconstruct it.

If you don’t want to use candy (but really why wouldn’t you?!), this can be recreated with just about any colorful toy. Legos work really well! Just sequence the colors in a specific pattern, let your child see it, then cover it up again, then have them try to reconstruct it with their legos. Keep building up to a greater and greater number of legos. You can also vary the time they reconstruct it…have them wait 30 seconds after you cover yours back up, or a minute, or even 5 minutes after they see it.

This has been a fun way for Naturalist and I to work on our memory skills, which in turn has helped our math recall. And, we’ve gotten to eat candy in the process! It’s a win win!

Other Resources:
My favorite fun math skills gamebook…125 activities to build skills for better math, and not necessarily through math computation drills! Mega-Fun Math Games and Puzzles for the Elementary Grades: Over 125 Activities that Teach Math Facts, Concepts, and Thinking Skills (Jossey-Bass Teacher)

our family favorite memory game, this one is so much fun, and no math at all!!! But totally develops memory skills that will help with retaining math facts/ideas. Tell your kid it’s time for math, then play this game, they’ll think they’re in heaven! STARE! Game

(STARE! JUNIOR if your kids are younger!)

Math Monday::Pentominoes, part 2.

When I last left off our pentominoes investigation, the kids had made the pentomino shapes out of square pieces of paper and laminated them. The sheets of pentominoes looked like this, and then we cut them out:

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So now that they each had 12 pentomino shapes as their game pieces (review of what a pentomino shape is linked through here) it was time to make the game board. The board was for a solitaire game of pentominoes–the goal being to get every one of the 12 pieces to fit inside the game board without any piece overlapping or going outside the board. So, we brainstormed. The question: how can we determine the size of the game board using the pentominoes as a reference.

No one really had any ideas, so I rephrased. ‘What we want to know is, how many pentominoes will fit exactly into a game board. How much room does each pentomino take up? (5 squares, because they’re each 5 squares.) So how much room total will you need if you want to fit them all in a game board?’ This kicked some activity off…counting started…some kids counted each square individually, some kids picked up their pieces and counted by 5’s, and some kids just did the mental math 12 pentominoes times 5. They all came up with 60 squares.

I wrote 60 down. So our game board needed to fit 60 squares, no more or less. To use mathmatical speak, the area of our board shape needed to be 60. Now that we knew that, we could find the dimensions of the board and then make one out of contruction paper.

So the next question was, what size board would give us an area of 60.

Not a lot of ideas.

Rephrase: “to find the area of something, you multiply the length times the width. We want to find the length and the width…what two numbers multiplied together equal 60?”

At this point, they could use calculators, multiplication cards with the math facts, or their own mind. Numbers started trickling in.

“1 and 60!”
“4 and 15!”
“6 and 10!”
“2 and 30!”
“3 and 20!”
“5 and 12!”

With those factors, we could start to narrow down our game board shape. Some of the dimensions wouldn’t fit any pentaminoes in them, like the 1 by 60 board, so we crossed that out. We kept crossing out number combinations until we arrived at the 4 shapes that were possible.

Each game board has it’s own possible solutions, so I used the wikipedia article here to explain which board would have the most possible solutions and which board would have the least. They decided on what their game board would look like, then started measuring and cutting it out.

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and then, of course, we laminated them. Any excuse to pull out the laminator!

So then they played around with trying to fit their pentominoes onto the game board.

There is a section on pentominoes in the book Shape Patterns (Let’s Investigate) which I notice is for sale on Amazon, used, for .88! (In fact, Amazon has most of the ‘Let’s Investigate’ book series for great used prices…I’ve found these books to be the best math related investigative books anywhere!)

In the shape pattern book there are a handful of two player pentomino games. A few of our favorites were:

*Two people each play with all their 12 pentominoes. They take turns laying down their pentominoes (without a board). Points are scored for each square of another pentomino that you touch with your own pentomino. You cannot overlap, but you can leave gaps and flip the pentomino around. Highest score wins.

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*Two people take turns choosing pentominoes from one (12 pentomino) set. Each player has one minute to arrange all their (6) pieces into a 5 by 6 rectangle. No overlapping, but they can be flipped and turned. Every square outside the rectangle counts as one point…the winner is the one with the lowest score.

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And we just played with the pentominoes for the rest of the time.

Yay for math!

Math Monday::Pentominoes!

{as an aside, let’s just all pretend that I actually write and post these on Monday, it would help keep the whole ‘math monday’ alliteration theme going…and help me keep my dignity…}

A few weeks ago, our Creative Math club met together and did even more things with the square inch pieces of paper that Sonja and I had cut out for the Square Numbers Math activities. I went to town with my paper cutter one night and ended up with about a bazillion squares before I realized it.

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When the kids arrived, they just sat around the table and played with the squares for a little while. Then I focused it a little more and asked them to choose 5 squares, and see what kind of pictures or shapes they could make. Some were really pretty!

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And finally, after a few minutes of that, I told them we were going to play a game, but that first we needed game pieces made from 5 squares. Each piece needed to have 5 squares connected–no more, no less. There would be a total of 12 game peices per person. Their job was to put the 5 squares together to see if it was a game peice…each piece had a specific rule to follow in order to use it to play the game with, but, I wasn’t going to tell them what the pieces looked like, I would only tell them yes or no if it was a piece as they showed me their ideas.

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This is a technique I discovered works really well with visual spatial, right brain thinkers. Rather than tell them the rule, or explain how something works, I just give them the free reign to experience it for themselves and then tell me the rule or how it works. The rule for the game pieces is simply the definition for pentominoes…shapes that are formed by joining a series of 5 squares attached edge to edge (not edge to corner!). Or, another way of putting it: A pentomino is a polyomino composed of five (Greek πέντε / pente) congruent squares, connected orthogonally.

So, the kids would arrange 5 squares, then ask me if it was a game piece. Out of these 3 examples, only the two on the ends are pentominoes, because the middle one joins the squares at the corners.

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If you’ve ever played Tetris, then you’ll recognize the pentomino shapes…even though Tetris only uses four block shapes.

Slowly they honed their skills, and it didn’t take long for them to figure out the characteristics of the pentomino shapes.

Picnik collage

Once they all had their 12 pieces glued to a sheet of construction paper, they cut them out and then I brought out my most favorite thing in the world….the laminator. It’s hard to express how much love I have for this thing, I’m totally addicted to laminating things. The kids were also totally addicted to watching their pentominoes go in with a thin sheet of plastic over them, and come out laminated and sturdy.

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Now, they had their game pieces ready…but we’d run out of time. So, we’ll make the game boards next week and use the pentominoes in a very mathmatical and fun way. And then next math monday I’ll explain what games we played with them. In the meantime, here’s an Online Pentomino Gameand another online Pentomino Game.

To finish up our Math Club, we played Blokus, which is a family favorite in our household, and one that uses pentominoes and other shapes to create a simple and addicting game. This has been in heavy rotation around here for the past year, solid.

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So, here’s a quick note about pentominoes, and why I love them. They are totally mathmatical, but totally non computational. Meaning, you don’t have to sit around with a calculator or worksheet. Pentominoes are meant to be played with, flipped around, discovered, explored, and used in a very hands on way. If you have a mathphobic kid, or are mathphobic yourself, I highly recommend getting a hold of some Pentominoes games and puzzles for a while. Math is fun! Just play with them! If you can’t resist a more ‘schooly’ way of dealing with them, there are lots of school standard type Fiction and Non Fiction books about Pentominoes.

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More about Pentominoes here

Wikipedia article about pentominoes here.

Blokus game if you only have 2 players.

and then I just now saw this Blokus 3D Game, which looks pretty cool!