Art and Math

Designing for Considering Boundaries around Infinity

Thinking about squares and infinity

This was bound to happen, that I would put up a post on my book and paper arts blog that appears to just be about math.

Initial impressions sometimes need refinement.

Anyone who follows me has probably noticed my attention to math ideas emerging as a theme. I’ve been paying attention to math and it is shaping how my creative work is evolving.

What I am here to say right now is that I think that math needs more designers and paper engineers.

In calculus there is this thing called Integration-by-Parts. It requires either a fluency with many rules or access to tables that contain these rules. The two problems with this is that it is not typical for the student to be fluent with the rules and the tables are not at all friendly looking and are embedded in humongous textbooks

When I was doing integration by parts problems it occurred to me to make a foldable that organized the information I needed to do these problems into a handy reference page.

I got lots of input and help from folks in the math community. I now have a greater appreciation for people who write whole textbooks, as just this one foldable was a big deal to do.

four page booklet
four page booklet

Here are the PDFs for

I’m going to be making a series of videos integration-by-parts.  As they are done, I will be editing them into this post so that I don’t flood my book arts followers with math videos. Still, I hope some of my non-math friends will take a look at this up to the 6:05 minute mark and tell me if it makes sense to them at all.  I am really enjoying being an artist who thinks about math instruction as a design issue.

Much of my thinking about math, as in my thinking about book arts instruction, centers around the weak links, meaning I search for places where misunderstanding sabotages learning.

This next video tries to address the disorientation of no longer solving for x, after solving for x for so many years.


Here’s the third video. At this point there’s not much here of general interest as it’s getting more specific to this specific method.


Finally, some worked examples. So far I’m showing two problems, but hope to add two more in the near future. Then these will be done!


The one below has a bit of bonus material about near the beginning.



Art and Math · Arts in Education

100-Book for Kindergarten, Trying it Out

Rubber Bands books to use for Hundred Book
Rubber Band book for 100-Book

About a month ago Simon Gregg posted a string of images about 100-books that were assembled by his class of 5 and 6 year olds.  On each of page there was a group of items that added up to 10.  That’s 10 pages of with 10 items, so it’s a 100-book. I want to do this project with 5 and 6 years olds! I ‘ve been in kindergarten classrooms all this week, doing a literacy based book project, but I noticed that there was one 40 minute block of unscheduled time in my schedule so I asked one of the teachers if I could try out this project with her class.

cover of 100-Book
cover of 100-Book

40 minutes was just enough time to get this book started, to get a feel for it. I assembled books for the whole class, just to expedite getting to the content. Although I didn’t realize about how this project would be received, the bottom line is that the kids enjoyed it, were enthusiastic to continue working on it, and seemed to be making some new connections. I debriefed with the teacher the next day and we agreed it’s a project worth developing. She is continuing the book without me: her kids are demanding to finish!


five plus five equals ten
five plus five equals ten

I wanted this to be a tactile book, and one that combined objects and finger counting. I brought in lots of foam, sticky-backed items. Students put a number of items on the page, hopefully in an arrangement, then traced the number of fingers they would need to make the total of items and fingers equal ten.

Eight plus two equals 10
Eight plus two equals 10

When there were less than five fingers to trace, the finger image looked rather odd, but the kids didn’t seem to mind.

Our first page, which I don’t have a photo of, was two tracings of hands, so that was five plus five equals ten. I noticed, when looking through the books afterwards, that a few of the kids wrote  5 + 5 = 10 on every page: these kids were not connecting the items on the page to the number sentences.  When I talked to their teacher about this she said that having them assemble the items, make them equal ten, and writing the number sentence was probably too much to do so quickly.

If I get a chance to properly do this project, these are the things I’d change or keep in mind:

  • I’d say we should do groupings of items on all of the pages as being one step (rather than finishing each page completely before moving on)
  • Next step would be going back to figure out how many fingers it would take to make ten and then creating the tracing of the right number of fingers.  Even though the fingers looked funky, I really like including them in this book.
  • Last step would be to revisit each page for a third time, this time to write the number sentence that describes the page.
  • I’d slow down and make sure kids were putting their items in groupings that could be recognizable to them.
Six and four make 10
Six and four make 10

Given enough time, I think that I would have the students make each page on single sheets, then bind them together with a simple pipe-cleaner binding, kind of like this one :

I’d probably have the kids make a big origami pocket to store all their pages in before they are bound together.

The three traced fingers
The three traced fingers

I like the idea of using items in the book that are slightly 3D, like these foam sticky-back pieces, but I’d think it would also be great to have other things, like cotton balls or popsicle sticks, that the kids glue in. I like having them work with glue.

The kids that Simon Gregg worked with included some playing cards in their books. This is something I’d like to do. My local thrift store sometimes has used sets of playing cards that I can pick up cheaply.

Hopefully I’ll be writing more about this project as I get more of a chance to do it with more kids.

Do take a look at this twitter thread that got me thinking of this project. The videos towards the end of the thread are precious.



Art and Math · Arts in Education

Making more Beautiful Letters with Pre K

The Letter T
The Letter T

Each time I do this project with 4 year olds I understand better how to approach it and how it is a valuable activity. The images that these children create are so gorgeous that it’s easy to feel they don’t need any justification. Beauty for beauty’s sake is great, but I’m happier if there is more going on.

Seems to me that this activity supports spatial reasoning skills, the value of which you can read about at

These 4 year olds are practicing precise finger control, which  is discussed in this long article

Students sort items, which is discussed at

Preparing Materials to fill in letters
Preparing Materials to fill in the letter H

Basically, what we do is use materials to fill in the letters of the alphabet, which this age group is still learning to identify. Although there are no hard and fast rules about what I use, basically the three categories of materials we used this year for the assemblages are:

  1. Fresh natural materials, such as flower petals (rose and others), carrots, fresh greenery, red potatoes, but also used pine cones, and small Indian corn (for letters A- H)
  2. Dried materials, such as dried yarrow, Japanese lanterns, artemisia, globe amaranth, nigella,and statice, along with some cedar ( letter I, J and S – X) and
  3.  Items students picked out from the classroom, including lego pieces, rubber bands, jigsaw pieces, small building blocks, crayons and dominoes (the rest of the letters).
Dried Materials
Dried Materials

We absolutely spent time talking about the materials, naming them, smelling them, familiarizing the kids with their properties. Oh, and we avoided using white and black items, as both of these colors are problematic in photographs.


The Letter M
The Letter M

The letters are outlined on papers that I provide. After the children fill in the letters,  I take a picture, then the students re-sort the items back into bins before startomg the next letter. Children worked in groups of two. We had some good chunks of time to work on these, so no one was rushed. We encouraged the students to take their time, work precisely, be inventive, and keep adding materials. Occasionally these kids would make a beautiful letter then immediately deconstruct it before it was photographed, so we had to be attentive!

The Letter E
The Letter

I brought the digital versions of these letters home with me, put them into Photoshop, got rid of the background with cropping tools and color range selections, and tinkered with the highlights, saturation and shadow.

The letter J
The letter J

When all the letters were done I assembled the letters into a PDF that became little booklets, so each student has their own alphabet book to keep.  Finally, on my last day with these kids, we were able to project the pages of the PDF on a large screen, and did the magnificent “Which One Doesn’t Belong” activity, aka #WODB which is discussed at 

Which One Doesn't Belong
Which One Doesn’t Belong

There are no right or wrong answers in the Which One Doesn’t Belong activity. To get the kids started, we went through the letters one by one, stating reasons why is could be said that each letter might not belong in the grouping. Because there is such an emphasis at this school on inclusion, I think it made the kids a bit uncomfortable to think about exclusion. However, seeing that a case could be made to exclude every letter is, in fact, a lesson about inclusion.

So, that’s it, this year’s alphabet book.





Art and Math

Bubbles & Connections

Sometimes it feels like the universe is conspiring to remind me of connections.

Years ago I played around with decorating papers with colored bubbles.

“Bubble marbling” is a simple technique that can create some really fun images. I’ve hardly ever done this with kids because it can get really messy. I did teach it in an adult workshop at Dieu Donné Papermill, NYC many years ago. It caught the eye of Helen Heibert, and, in 2001, she included an image I made with a brief description of the technique in her book, Paper Illumninated, which is a gorgeous collection of instructions about making paper lanterns.

hey, look, I’m in this picture!

Yesterday I showed this bubble technique to groups of kindergartners. By delightful coincidence, I also heard from Helen Hiebert yesterday.

Helen is still making paper lanterns. With the proliferation of LED light strips and other safer options for illumination, paper lanterns make so much more sense now then they did it 2001. Helen was telling me about the on-line course she is teaching, which sounds fabulous, so I am sharing this info with you, too before going on about bubbles any further.

Helen Hiebert's Shadow Lantern Screen
Helen Hiebert’s Shadow Lantern Screen

While it doesn’t appear that she showing any bubble marbling this time around, she is teaching an impressive array of projects that include paper cutting, tessellations, and pop-ups. I am happy to spread the word about this. You can allow yourself to be inspired by looking a video she made at

You can learn about the bubble marbling from me. Now.


Dressing up to do something special
Dressing up to do something special

While it’s not required, it’s not a bad idea to dress up before making a mess. This sets a mood, but it also protects clothing.

Bubble Marbling in action
Bubble Marbling in action

Here’s what to do: put about a tablespoon of paint (tempra, acrylic, any strong pigment but not ink because you would need too much) in a fairly shallow container, preferably round. Add bubble mixture. I buy this ready made, or make it with Ultra Dawn, water, and a touch of glycerin. Now mix the paint REALLY REALLY well with the bubble solution. Place a straw in the bubble solution, blow gently, like blowing bubbles into milk. Make the bubbles just high enough to be above the rim of your container. Then GENTLY lay a piece of paper onto the bubbles and remove.

Bubble Prints
Bubble Prints

That’s it. They dry fast. You can overlay colors on top of each other. So much fun. But there’s more. There’s something to notice.

Three fold symmetry
Three fold symmetry

Yesterday, doing this with these kids, I reminded them of the three-fold symmetry projects that we did a couple of weeks ago…then

Looking at the symmetry of how the bubbles meet each other

I showed them that the shape we used to make the our three-fold symmetries is the same as the shape that the bubbles make where they meet. And everywhere they meet they make this same shape.

Now that’s a connections worth noticing.

They loved seeing this. Then they said, It looks like a soccer ball!


Symnetry like a soccer ball
Symmetry like a soccer ball

I’m learning that anything that looks like hexagons reminds kids of soccer balls. I can live with that.