Book, peeking out of its box by Paula Beardell Krieg

Book, peeking out of its box

When Miriam Schaer was assembling her teaching collection to send to Telavi University in the Republic of Georgia, I very much wanted to contribute, but nothing I had on hand seemed right. In the nick of time, some thoughts came colliding together. Polygon Fractal book by Paula Beardell Krieg

This structure started out with an exploration of a shape which I wrote about a few weeks ago after watching family math video made by the Lawlers.

Inside Outside Book by Paula Beardell KriegThe book  opens in an accordion-like fashion, but front and back are structurally different.

Polygon Fractal book by Paula Beardell KriegThe colorful pages rotate open to create these double layered corners. The polygon fractals on the pages here are harvested from Dan Anderson’s openprocessing page then toyed with in Photoshop.

To see the fractals in their full radiant radial symmetry one must rotate the book. There are six completely different images to be seen. But it gets more interesting, because there is a whole other side to see.

The folds of those double layered corners completely reverse to form a cube!

You can’t imagine how excited I was when I saw this cube emerge from the folds!

This folded structure totally suggested that, whatever I use on it, that it be about the dual nature of….something….a suitcase (no, too obvious), a politician’s statements (ugh, too boring)…actually wanted to use images that didn’t imply any hierarchy, hiding, agendas, or judgement about contrasting inner and outer manifestations.

It was this thinking, about duality but equality of visuals, which led me to using Dan’s code along with the polygon fractals that it creates. So perfect. Code and images are perfectly linked, simply completely different ways of seeing the same thing. You know, like Blonde Brunette Redhead 

Now, I do have a lingering unresolved issue with this book. I’m not thrilled with the paper that I’ve used. It’s 32lb Finch Fine Color Copy paper. It takes color beautifully, folds well, but I’m thinking that the folds might be more prone to tearing than is comfortable. Not sure what else to use…am open for suggestions. Miriam’s copy has been shipped, but I’m still happy to check out different papers to use.

I can’t help but wonder if people will be able to figure the transformation of these  pages without seeing this post or reading the brief explanation I’ve provided on the back page of the book? Dunno.

Oh, and here’s my favorite variation:

Hanging a tea light from a pencil so I can see the inside and outside at the same time.


Paper folding in the morning

Paper folding in the morning

Yesterday I watched a video that showed the Lawler family looking at shapes.

One of Mike’s sons said he liked the top shape in the image above. You can’t see from the photo, but it’s a full sphere. The image above is only half of the sphere, the other half looks no different than the half that is showing. I’ve played around with constructing foldable versions of shapes that look something like the one above, and I thought I’d be able to make a foldable version of what was on Mike’s screen.

Template for folding and cutting

Template for folding and cutting

I’m not showing all the steps that led to this map of folding and cutting because what I’m most interested in showing here are the wonderful visuals I got to experience along the way of creating the final structure.

Folding and Cuttin

Folding and Cutting

Silly as it may seem, one of my first realizations was that the indoor, nighttime lighting in my workspace was just all wrong for photographing what I was about to fold. Morning light would be best. So I went to bed.

Assembling in the morning light

Assembling in the morning light

Of course I forgot to recharge the battery of my phone camera before I went to bed, so I didn’t get the earliest light.


Hmm, I don’t really want to say much more about this process. I’m just going to post pictures now. (Haven’t had my coffee yet.)






Ok. Time for coffee. Am heading to Rochester today to bring my daughter back to college. Will be thinking about all shapes that this structure made. (Which reminds me of a question someone once asked me, “What, do you just sit around thinking about folding paper?” Well, yeah.)

Finger Tracing

June 16, 2016

Hand of a four year old

Hand tracing by a four-year-old

This post is for people who have the good fortune to be working with four- and five-year olds.

For quite some time I’ve been exploring ways of drawing the attention of young students to their fingers. In my post Counting to Ten, March 2015  I wrote”My thinking here is that I want these students to create a visual that connects the numbers that they are learning to the fingers that they count on.” I see the fingers as the original number line.

Recently I watched a TedX  Stanford video of Jo Boaler, an educator who is involved in research in how to support learning and growth. I’m already of fan of Jo Boaler, but particularly liked this talk of hers and particular like what she says about counting on fingers. Starting at about the point 7:22 on the video, she tells the audience that when we calculate, the brain area that sees fingers lights up. She then goes on to tell us that the amount of finger perception grade 1 students have is a better predictor of math achievement in grade 2 than test scores.

Hand tracing with a cut-out tunnel to fit over the student's wrist.

Hand tracing with a cut-out tunnel to fit over the student’s wrist.

After hearing Jo Boaler’s talk, I couldn’t help but wonder if it is possible to modify my finger tracing project in a way that could possibly help children strengthen their finger perception? This isn’t something that I’ve ever thought about before, but I guess that some people are more challenged than others in connecting the sensation of having a finger touched to the finger that is being touched. More simply said, when I touch your finger, do you know, without peeking, which finger I am touching?

First we did some counting together, up to five. Then the children traced their hands. The adults wrote down the numbers on the drawn fingers. We cut out a tunnel so the child’s hand could rest under the drawing. Then a partner would touch a finger, and the child would reference the drawing and say which finger had been touched.

Here’s a few short video clips of how it went:

Mostly it was children doing this with each other, but it was easier for me to use these clips that show the adults working with the students. When the students were working with each other I put little stickers on the student’s hands, labeling the fingers 1-5 so that the student who was doing the finger touching would know what number was matched to the finger that he or she was touching.

It was interesting to see that there was a huge range between how hard and how easy it was for children to identify which finger was touched. One child simply could not make the connection at all. It finally occurred to me to let her see her hand and the map of her hand at the same time, to see if she could develop the connection. This seemed to work out for her. Here’s what it looked like towards the end:

This was my first attempt at this kind of…um….hands-on drawing project/ finger game with young students. It was a really quick, let’s-see-what-happens-if-we-do- this kind of thing, but there seems to be something interesting going on here. and I hope to do more of this, and hope other people will try it out too! Thanks Jo Boaler!

Addendum July 21, 2016

I’m seeing a trend… my addendums are getting more frequent and could become posts in themselves. Seems like once I post something I stumble across something that’s totally relevant, or someone tags me with content that applies. This addendum has both.

Part 1

First, here’s an absolutely accessible, research based activity to do with young children, to help their brains develop its mathematical regions. It’s based on the idea that there’s a correlation between preschooler’s sense of approximation and their general ability to do mathematics. It looks like this:

from a report on the work of Jinjing Wang of John Hopkins Universitys

from a report on the work of Jinjing Wang of John Hopkins University

Looks to me that there’s lots of possibilities for here for books that I could make with kids for them to  bring home. Even though the task here is to estimate which side of the page has more dots, I can see all sort of other kinds of vocabulary that can be come up with 4 year olds that this kind of graphic can facilitate. Here’s a link the the article about this, that Dave Radcliffe @daveinstpaul pointed out to the twitter community:–WJ4LCbVq8EZ

Part 2

Ted Lewis saw my post and alerted me to his, which, discussed the  longer and more satisfying exploration and examination of ” number sense and how we create it.” This is the graphic that accompanied Ted’s post:


 Okay, this is a perfect reminder that I don’t need a computer screen and Big Bird and Elmo to do this kind of work with students. Ted is using the inequality signs for this exercise. (Silly Ted, inequality signs are the downfall of many, but let’s talk about that some other time….)  The point though, is, if you are interested in visuals and the brain and the want some insights and food for thought, read this post for yourself .

I’ll be working, weekly, with four-year-olds starting the first week of July. Can’t wait to see if I can corral them into doing any of the activities that are inspired by these dots and other ways of thinking.


The Number Five in Beans

April 26, 2016


The number five in beans, herbs, pasta and violets

The number five in beans, herbs, pasta and violets

I still have some in-school arts-in-ed projects to show up for before my season ends, but it’s not too early to think about the summer. I’ll be working with pre-K children this summer, in the local Lunch, Learn & Play camp. I’ll be there three hours once a week for five weeks. It’s just the sort of situation that I’m best at: completely unpredictable.  This is a free, show up when you want program so I won’t ever know the number of children who will be in attendance and I won’t be able to predict continuity of the participants.

The goal of the program is to support preparing children for kindergarten. Two teachers from the local school will be on premises. Mostly the mandate for these kids is to work on literacy, and that’s what most of the involved adults will be doing. I will be the only one of the group that will be focusing on numbers and relationship thinking. The teachers have said the goal for these 4- and 5-year olds is simply number recognition: assigning value or even writing the numbers is not part of what I will be doing. Given all these conditions,  I’ve worked out a curriculum that I’m very excited about.

finger counting

Happy Hands

I’ll likely start out with some form of this finger counting drawing activity that I’ve previously done with kindergartens. I like planting this happy connection between fingers and counting into the minds of these young children. I am happy to see current research  supporting this kind of thinking. Addendum:  There’s a wondeful 12-minute talk by Jo Boaler, in which she speaks about finger counting between 7:22 and 8:57. 

I won’t be asking children to draw numbers but I will be asking them to interact with them. The number 5 at the top of the page is the filled in outline of the number five.. Here’s a video, in 8x, showing how I built up the form:

There’s no gluing done here. I will be taking photos and hopefully even videos of the children making these numbers to help instill lasting impressions.


We’ll also be working cooperatively to fill in big numbers. I will have these number drawn on heavy weight paper…

Filling in a number

Filling in a number

— and ask children to paint, draw and collage items on the number. Staying in the lines won’t be an issue because…


Stand alone number

Stand alone number

…the numbers will be cut out and mounted on these accordion supported structures. This number two is 25 inches tall.

Hiding behind numbers

Hiding behind numbers

I’m considering adding in some sort of door so that children can actually get into the space behind the numbers. My thought is to scatter these numbers around the hallway  outside our meeting room each week, and asking children to gather them into our room and line them up in order.

Same and Different

Same and Different

In addition to the numbers I will be packing other kinds of activities that will, hopefully, support relationship thinking.  Included in these other activities, I will be bringing along a variation of a project that  Christopher Danielson has developed around the concept of “which one doesn’t belong?”  What I will be creating are  cards to accompany the questions: How are these the same? How are they different?

Ways to consider Bigger Smaller Higher Lower etc

Ways to consider Bigger Smaller Higher Lower etc

My thought is that I will be trying to coax out or introduce words that have to do with scale, position, shape and color and whatever else those active minds come up with.

We’ll so come coloring, too.

Circles Coloring Page

Circles Coloring Page


Likely I will make coloring pages like this one but I will wait until after I start to create these as I want to get a feel for the things first.

Also, I will bring books to read.

One Was Johnny by Maurice Sendak

One Was Johnny by Maurice Sendak

In the Night Kitchen Farm donated this gem One Was Johnny by Maurice Sendak to LL&P, so this is the beginning of my travelling library. I would be thrilled to hear recommendations for other books, so please help me out here!

For the grand finale, I want the kids to make really humongous numbers, which, hopefully, will be filmed and/or photographed as the numbers are made. I was flailing about, trying to figure out what to use to make the numbers….

A Number of Children

A Number of Children


… and finally it seemed to me that having the children become the number, in other words, choreographing them into the shape of the numbers, like the number four in the drawing above, might be a fun thing to try out. Hmm. Wish me luck and look for updates on how this goes during July and August!

Addendum: When Malke Rosenfeld saw this post, she tweeted me a link to this post of hers, which referenced her math work as a five-year old. It knocked my socks off, so I’m sharing it here 1970’s Kindergarten Worksheets

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