## Origami Boats and Meandering Number Line with 4-year olds

### August 24, 2016

Full disclosure: I did not try out teaching 4-year olds how to make origami boats. It’s not that I didn’t want to, or that I chose not to, it was just that there were other things I wanted to do more, and my time with these little ones was limited. Much to my delight, though, after we used the boats in our activity, the children asked me about how they were made. I did a demonstration, with hope that this may encourage an interest in paper-folding.

I chose to use these paper boats because they* stack. *Just for the record, I was curious to see if they floated. Turns out ;Yes! Until the paper absorbs too much water, these vessels are sea worthy. What was more useful for me, though, was that they can stand on their own, so that we could use them as playing pieces for a board game.

During my workshops with these children I noticed that even the most accomplished child in the group could not coordinate counting items with the movement of his hands. In other words, if there was a pile of 8 stones, these children would end up counting inaccurately *because their fingers would move out of sync with the numbers that they were reciting*. I was really interested to see this, partially because I’ve read that there is something about learning to play the piano that helps children be better at math: this now makes sense to me, as playing notes would help train a person to coordinate fingers with intention.

Wanting to try out a simple, and, yes, frugal, made-from-paper activity to encourage accurate counting skills, I worked out a sweet game that the kids seemed to like . What we did mimics classic board games where a die is thrown, and the player advances a certain number of spaces along a line. It was, however, important to me that *I didn’t want to create winners or losers. *This is how it went: the playing pieces were these paper boats, and when two boats land on the same space, they become a team, and stacked together. The point of the game is to get all the boats stacked together as a team before any boat reaches the end of the meandering number line, which, just, for no particular reason other than I ran out of space on my paper, was 42 units long.

Unfortunately, I didn’t get a photo of the kids playing this game, but, they played in groups of three and four, and they seemed to enjoy watching others play as well as playing the game themselves. Counting spaces, counting the dots on the dice, and (especially!) anticipating what throw of the die would yield the desired outcome were all challenging but doable for these kids.

Each of the five sessions that I worked with these students, one-third of my lesson plan was to focused my interpretation of relationship thinking, such as creating patterns from shaped paper, developing finger sense, estimating, discussing what was the same and different about shapes and flowers, and this unit counting game.

Other parts of my time of my time with these kids was artful numbers, which I what my next and last post about my time with these students will be about.

I had to learn how to make these origami boats for this project. I looked many different models, but this one that I’ve shown I found most enchanting. I put together a video of it, that is worth watching because there’s some pointers included that I just can’t fit onto a tutorial page.

Happy boating.

## Trying to Merge Visuals & Math with 4-year olds

### August 8, 2016

I was invited to work with 4-year olds this summer as one of five Arts-in-Ed people. My part, as I defined it, was to create an experience with math thinking that merged with visual arts. Or was it the other way around? My biggest discovery was that this math/art is a completely natural activity for four-year-olds. I could only understand this, however, after having already having unlearned then relearned some things about math. The surprise was uncovering how much math thinking has in common with the ways I think about art.

The important detail in the photo above, is that these two young fellows don’t look like they are having fun. They are not having fun, but they are not unhappy either. They are thinking. They are engaged. They are doing some independent explorations and are figuring out how to respond to a challenge I’ve posed.

The reason that I am showing this first is that it reflects how my thinking about arts-in-education has evolved through listening to math educators. There’s a teacher from Brooklyn, Michael Pershan, who is vocal about things not having to be *fun *to be valuable. Then there’s Mike Lawler, whose postings have introduced me to the way mathematicians ask questions, make predication about the answers, then go at it to see what happens. Christopher Danielson has validated the value of forming math opinions (like, “these things are the same because….”) and then forming a divergent opinion (“these things are different because….”). Malke Rosenfeld has enriched my understanding of thinking about the powerful lessons within examining the shifting relationships in the physical world. John Golden and Simon Gregg have caught my attention by the way they approach an idea, looking at it and playing with it every which way they can think of. These are things mathematicians do, and these are things that artists do. Turns out that these values resonate with 4-year-olds, too.

People who think about math as a vehicle for doing calculations might have a hard time following my thinking here. If that’s where you’re at, suspend that belief for the rest of this post (or the rest of your life!) and think of math and art both as a way to discover, examine and develop insights about the world around us. With this in mind, each week that I worked on with the students, 1/3 of my plan was to introduce an activity that aspired to develop mathematical thinking while working with artful materials.

One of our projects was based on research I had read about http://indy100.independent.co.uk/article/the-simple-dot-test-that-can-massively-improve-a-childs-maths-skills–WJ4LCbVq8EZ which I wrote about at the end of a post a few weeks ago. The idea is to have two different piles of items (we used mancala beads) then estimate which pile has more. No counting allowed, which is great because even the students who can count do not count accurately! I extended the ideas that the researched suggested by then telling each child that I thought that they were wrong, that their other pile had more. I then asked them to* prove *their estimation. That’s when this got really delightful!

The first configuration that emerged for proving which pile of beads was bigger was to line up the beads. The longer line had the most beads. But not everyone connected to that system. One of the students lined up the beads with the short edge of the folded paper, then when he ran out of room he continued his line by forming a sideways “L,” finally comparing the shape of the L’s made by each pile.

One young lady made circles with her beads, explaining that the larger circle had more beads.

Here’s someone who worked on making arrays out of beads in order to compare them!

Then there was the student who just didn’t participate. I was able to figure out pretty quickly that she didn’t know what the word “more” meant. What was hard for me was figuring out how to explain to her what more meant. Go on, try to explain that word. Finally, after many failed attempts to convey the meaning I made two lines of beads, each of which were made up of six beads. She knew word “same.” Yes, she agreed, the lines were the same. Then I took one bead away from one of the lines

.Are they still the same?

No.

Okay.

Not the same. This line has more.

She got it! Very exciting.

This is about all I can write for today. I will write about the other activities that we did in a later post.

## The Great Big Number Line

### August 4, 2016

I haven’t posted in a while. I haven’t been able to figure out how to organize my thoughts around this work I’ve been doing with 4-year olds. This is an age group that is mostly mysterious to me, and although I had a plan going in, my intent was to be responsive to what I was able to learn about and from them each week. The fact is that there is too much for me to write about. So, rather than fret about it any more, I’m going to try to just get to it and see what happens. I am thinking that I will do this in small bits, maybe, hopefully, posting frequently, until I write about most of the things that we did during our weekly meetings over the past five weeks.

One of the bits of info that I was given at the beginning of these journey was that the teachers at the local school wanted the students to be able to recognize numbers. Within the small group of students that I worked with, most seemed to already be familiar with the numbers.

The part that surprised me was that it became clear to me that counting and recognizing numbers are two completely different skill sets. For instance, there was one child who could count to four, could count four things, do four actions, but could not seem to fathom a connection between the concept of four and the symbol for four.

Among the projects that we did together was this great big number line. My idea had been that I would mount the numbers on to an accordion-supported structures that could stand on itheir own so that students could pick the numbers up and carry them around.

Coloring in these big numbers was part of what we did each week. Although these photos are what I am showing first, I want to say that we would always do a number-sense activity before we settled into the more relaxing activity of coloring in numbers.

The second week that we colored in numbers I brought in protractors, circles, and other shape templates to see if the kids would like using them. They *loved *tracing around the shapes that I brought in.

We used a combination of crayons and markers. Markers are tricky with this age group, as the markers are fairly readily destroyed when the kids press them really hard, but I have too many markers anyhow….

When a number was done, I would cut it out, smear some white glue on the back of it and mount it on the accordion-supported structure.

Then, each week, the numbers that were mounted were placed in a place where the students could grab them, and I would ask them to put the numbers in order.

Since there is a front and back to these numbers, there was a surface to add shapes that could relate to the numbers. This is what I mean:

I brought in my strips from rhombus project that Malke Rosenfeld and I had developed together (see http://mathinyourfeet.blogspot.com/2015/04/some-thoughts-on-hands-on-math-learning.html and https://bookzoompa.wordpress.com/2015/03/31/piecing-together-a-project-over-land-and-sea/) The idea was that on the back of the 1 there would be one shape, on the back of the 2 we”d make two shapes, and on and on. I’m not really sure if the students made the connection, or cared about it, but they did do lovely work.

I think the students were quite taken by the stars that they made for the backs of their big numbers. In the background of this photo is one of the kindergarten teachers from the local school. Like me, her first name is Paula. She was awesome to work with, and we traded ideas back and forth. She came up with the brilliant idea of suggesting that the students *wear* the numbers, which is exactly what we did, although the first time I slipped one of the numbers over the head of one of the girls, she burst into tears. I don’t know if it scared her, or if she didn’t like the particular number, or what it was, but it shook me up! However, after that first outburst, it became just another fun thing.

Turns out that wearing numbers makes it super easy to move them around.

This was one of two number lines that we made: this number line starts with 1, but a second number line project started at zero By our last class these kids were asking why we didn’t start this number line at zero. I liked being able to introduce them to the fact that there was no one right place to start the number line. Yee-ha!

Just as I am posting this I’ve come across this article about math and young children. http://blogs.edweek.org/edweek/early_years/2016/08/ask_a_scientist_whats_the_best_type_of_math_to_teach_in_kindergarten.html?cmp=eml-enl-eu-news1 I guess I’ll write my next post about some of the number-sense work that we did, which was all very hands on with materials….it was fascinating to watch these kids really figure things out!

## Zhen Xian Bao, Post #3 Twist Box for top layer

### June 27, 2016

Zhen Xian Bao, Chinese Thread Book, Post #1 intro

Zhen Xian Bao, Chines Thread Book, Post #2, starting with the finished size

Of all the posts that I plan to write about the Zhen Xian Bao, this will likely be my favorite post. Figuring out both how to make this twist box and explain it have kept me happiliy distracted. I’ve looked closely at photos of people making them, watched clips of the twist box being opened and closed, examined templates, and studied videos (all available for you to see on my Zhen Xian Bao Pinterest board). Then I just kept making these boxes until I was happy with the results.

These boxes are the top layer of the Chinese Thread Book, a structure that is made up of layers of collapsible boxes. Not all of the Zhen Xian Bao have twist boxes on the top, but it’s the style of box that I like the best, as it’s so very different from any of the other boxes in the rest of the structure.

There are numerous versions of the twist box.

The version of the twist box that I originally fell in love with was designed and demonstrated by Chrissy Paperkawaii. While I still like much about her no-cut, no-glue version of the twist box, it’s just a bit too bulky for my liking; also, it was really difficult to twist, and if I could barely make it her way I knew I wouldn’t be able to teach it.

A twist box that I *thought* I liked the least, but which I now appreciate, appears to be made from a template created by Lori Sauer, which Rachel Marsden wrote about. Rachel’s post is one of the most beautiful pieces I’ve seen on the Zhen Xian Bao. You really must take a look at it. Here’ s the photo of the not-yet-folded twist box that Rachel Marsden made from the template.

You might notice that this shape is almost exactly like the *nineteenth *drawing on my tutorial page, meaning Rachel probably got her twist box done in few less steps than me. The only reason that it’s not my favorite is that it’s made from a template. So what do I have against templates?

The first answer that comes to mind is that it can be tricky to scale a template. As part of a collection of different styles of origami boxes, I want a reasonable way to scale all the different elements so that I can make whatever size thread book that I want.

The other reason that I prefer not to rely on a template is that being able to figure out the system of folds from a rectangle gives me the chance to fully understand and appreciate the foundational symmetry of the structure that I am folding.

Not long ago I wrote about a fairly tricky folding structure. I included in my post the hard-to-read-and-decipher tutorial page that I had followed as well as a video. I then aimed to entice my highly accomplish weblog friend Candy Wooding to make the piece (she did!) and then asked her if she preferred the video or the written directions. She said that the video was good to start with, but then the written directions provided her with reminders. With this feedback in mind, here’s a video of making the twist box.

Disclaimer? I know I’m not that good at making these videos. Sometimes things fall off the edge of the frame, and my hands get in the way, but I’m hoping that the more I make these the better I will get at it. Still, I think this imperfect video is plenty helpful.

One last note:

Here’s the black and white version of the Zhen Xian Bao tutorial page. You can color in this one in yourself. It’s a small PDF file, unlike the huge color file at the top of the page. I haven’t figured out how to make a *small*, colored downloadable file to post on my blog.If someone wants to offer me some pointers, I’m interested!

My next Zhen Xian Bao post will either be about the box that is the next level down, or about ways to decorate this box. Haven’t decided yet. Though I know I won’t be able to get to it right away, still, I’m looking forward to post #4! Make boxes! Thank you.