## Two Beautiful

### August 28, 2016

The four-year-old children in my summer workshops made numbers so lovely that I had to make an accordion book out of photographs of the numbers. These images are so exquisite that I don’t expect anyone to believe that they were assembled by such young children, so at the end of this post I’ve embedded video links that show, in fast motion, a couple of the numbers being made. You’ll notice that the children created the assemblages without adult interference.

The materials that students used were mostly from my husband’s garden. One particular harvest of beans had been buried in the back of our pantry for too long and dearest was going to throw it into the compost. These, therefore, are rescued beans. I also picked a selection of flowers (marigolds are the best) Some of the students went outside and brought in leaves from the community garden. There was also pasta in the activity boxes in the room, so we used pasta too.

We worked as a group to make the first number. I pondered over whether to start with zero or one, and took to twitter, asking the math community that shows up there what they thought. There was no definitive consensus but seemed to me that there was more said in favor of starting with zero.

I’ve come to a way of thinking about what number to start with. In the course of my five sessions with these students we made or used three different number lines. Each one was different. Our Great Big Number Line went from one to ten. The meandering number line went from zero to 42. This sequence went from zero to 10. Did the children notice the differences? Turns out, yes, they did!… which gave them the opportunity to see that the number line is not a fixed item.

Students worked mostly in pairs of two. It took about 8 to 10 minutes for each number to be made. We did not glue anything down. These assemblages were created to be photographed.

I was a bit worried that these kids would be unhappy about the fact that, as soon as a the picture was taken, the number was undone.

As usual, these kiddos totally surprised me. Dumping the contents of the numbers back into the big bowl was one of their favorite moments!

.My daughter Angela did a great job of photographing this process.

The first thing I did with the photos was, with Photoshop, isolate the numbers from the background, vectorize, then print them up. The next day that I saw these children I showed them the prints and we did a number line clothes line.

I’ve been inspired by Joe Schwartz and Tracy Zager, who have written about facilitating the building of number lines with clothes lines and Post-its, So the first thing we did with our numbers was to hand them out in random order, and have the student estimate where the numbers should be located….

Next stop was Kinko’s to make copies of the numbers on standard sized copy paper, that could be folded into accordion books. I had one problem. I didn’t want to *just* have a line of numbers. I wanted there to be some corresponding items that could be counted, you know, like five *things *associated with the number five. After agonizing over what these things should be I realized that we had already created designs on the backs of the Great Big Number Line, so I recreated, with acceptable accuracy, the students’ designs and made them part of the book.

Now, here’s the little accordion number line book:

What’s great about accordion books is that they have fronts *and* backs. Flipping the book upside down reveals the designs that correspond to the numbers.

Completely opening up the book reveals numbers and images!

I’ll be making a few copies of this book to give to the kindergarten teachers who will have these students in their classes.

Now if you haven’t seen enough images on this project, here are two clips of the children working, in fast-action mode.

Yup, love this project.

## 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!