Art and Math

Finger Tracing

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.


Decoration · Making Books with children · Making books with elementary students

Adirondack Birds, Books by Second Graders

Adirondack Bird Books by second graders
Adirondack Bird Books by second graders

Here’s a project I did with second graders a number of years ago, but, for a specific reason that I will divulge at the end of this post, I chose not write about. Now, having just come across this folder of picture, I liked the images so much that I decided it’s time write about these  books.

Adirondack Robin
Adirondack Robin

These second grade student chose to a local bird to research. My job was to design a project that would showcase the results of the research, display some generalized info about the life cycle of the bird, have an “About the Author” section, as well incorporate a diorama that flatten, and which included pop-ups and a paper spring.

Adirondack Birds Katelyn's Eagle 2
Adirondack Birds Katelyn’s Eagle

I can’t say for sure (though I will dig up my notes and include this info later) but I’d say that this book stand about 10″ high.  You can see that it opens from the center to reveal the habitat of the bird.

Adirondack Blue Jay
Adirondack Blue Jay

We were able to do two pop-ups; one in the sky and once on the forest floor. The Blue Jay is attached with a paper spring to give the bird some dimension and movement.

On the backside of the habitat there’s ample room for research and everything else.

Adirondack Birds, Chelsea's Hummingbird Book
Adirondack Birds, Chelsea’s Hummingbird Book

Food and Interesting facts go on one of the sides.

Blue Jay book
Blue Jay book

Facts about the bird’s appearance and their habitat are written on the far edges of the paper…

Adirondack Birds Life cycle back of book
Adirondack Birds Life cycle back of book

….with life cycle info at the center…

Adirondack Birds Chelsea's hummingbird
Adirondack Birds Chelsea’s hummingbird

…topped off by information about the author.

Adirondack Birds Chelsea's hummingbird 2. jpg
Adirondack Birds, Chelsea’s Hummingbird Book. jpg

Now here’s some details to notice. To get the front sections to stay together, the rotated center square is glued on half of its surface, the other half slides under the long strip, which is glued down just at its bottom and top.  The details of the decorative elements on the fronts of the books were created with simple, geometric symmetries. I loved the decisions that kids made with the shapes!

Adirondack Owl
Adirondack Owl

Another idea that the students worked with was the idea of using different mediums and methods to make thehabitat. The cloud is foam, there’s cut paper shapes, drawing with markers and crayons, a few shapes created with paper punches (the  butterfly and dragonflies) paper springs behind the owls, and both a one-cut and a two-cut pop-up: all with the goal of creating an interesting, texture display.

Adirondack bird nest made from Origami Pocket
Adirondack bird nest made from Origami Pocket

As you might imagine, these books are made using lots of separate pieces. For this kind of project I generally first have the students make a large origami pocket from a 15″ square paper so that we have  container in which to keep everything organized. The classroom teacher, Gail DePace, who I could always count on to enrich my projects with her own personal standards of excellence, had the idea to ask the students to decorate their origami pockets as if they were bird’s nest, complete with  appropriately colored eggs.



clay owl
clay owl

The students added another dimension to this project by creating their birds in clay and putting them on display along with their books.

At the beginning of this post I said that there was a reason that I hadn’t written about this project. As lovely as the project is, the teacher, who was a spectacular collaborator on this and all projects that we did together, didn’t love this project. She noted that this structure didn’t work well as a book, that it was awkward for the kids to open to the “pages” and read their work when it came time to do their presentation of the final project.

I’d have to agree that this project works much better as a display than as a book. Oh, and it looks great in pictures too.  Sometimes, though, the display and the documentation are the priorities, so that’s what I’d keep  in mind for this project next time.


Exploring the Language of Patterns

translation design pattern

News Flash: A Codified Language Exists to Describe Patterns. I’ve been so excited to discover the way to speak about patterns.

I’ve been teaching decorative techniques for a long time now. I’ve started trying to use more precise terminology in my teaching, and I suspected there was more to know. I started out looking at artistic and graphic design sites, really I did. I  looked on, I looked on youtube,  and poked around the internet in general. Then Maria Droujkova  pointed me in the direction of something called Wallpaper Groups, and guess what, I landed on sites that described pattern making with precision, using the language of mathematics.

The more I learn the more I understand that what math does is enhance the way that people can describe what’s in the world. It appears that hundreds of years ago mathematicians figured out how to understand and talk about patterns.

Example of Pattern Rotation

This summer I’ll be teaching a week’s worth of classes to young children at our community center. I enjoy showing students decorative techniques, so my immediate interest has been to develop a modest curriculum that focuses on making books that are embellished with style. Even though many of the students will be at an age where they are still struggling with concepts such as “next to” and “underneath”  I hope to introduce them to ways of thinking about concepts of transformation.

Strip Symmetry  is where I landed when I was surfing for a way to find words to describe the kind decorations I’ve been thinking about.  In other words, the patterns I am looking to teach will have a linear quality in the way that they occupy a space, as opposed to being like a central starburst, or an all-over wallpaper pattern. It turns out that there are only a handful of words that are used to describe every single repeating linear pattern ever made.

Translation repeat pattern

 A Translation takes a motif and repeats it exactly.

Example of Vertical Reflection
Vertical Reflection

Vertical Refection mirrors a motif across an imaginary vertical line. The name of this particular transformation confused me at first, as the design itself extends in a horizontal direction, but once I prioritized the idea of the vertical mirror, it made more sense.

Example of Glide Pattern

Glide Reflection can be described as sliding then flipping the motif,, but that description sounds confusing to me. Instead, understand glide reflection by looking at the pattern we make with our feet when we walk; Our feet are mirror images of each other, and they land in an alternating pattern on the ground. Imagine footsteps on top of  each of the paper turtles you might better be able to isolate the glide refection symmetry.

Example of Horizontal Reflection
Horizontal Reflection

Horizontal Reflection mirrors the design across an imaginary horizontal line.

Example of Translation Pattern

Here’s a translation that shifts horizontally, but there’s no such thing as a strip symmetry that translates top to bottom. Instead, convention dictates that the viewer turns the pattern so that it moves from left to right.


Rotation rotates a design around an equator.  The pattern above, as well as the first image of this post, I had considered these both to be rotatation( ( I imagined the equator drawn across the middle of the page), especially if it’s 7 year-olds that I am talking to, but close inspection reveals more. To highlight that I am presenting these concepts with  broad strokes, here is what Professor Darrah Chavey wrote about the image above (the one with the leaves) when I asked for his input:

“As to this particular pattern, there’s a slight problem in viewing these leaves as a strip pattern. The leaves you show are made from a common template, but that template isn’t quite symmetric, and the way the leaves are repeated across the top isn’t quite regular. For example, the stem of the maple leaf in the top row, #1, leans a little to the left, and has a bigger bulge on the left. If we view this as a significant variation, then the maple leaves on the top row go: Left, Right, Left, Left, Right, Right, which isn’t a regular pattern, i.e. it doesn’t have a translation. On the other hand, if we view those differences as being too small to worry about, then the leaves themselves have a vertical reflection, successive pairs of leaves have vertical reflections between them, and the strip pattern on the top is of type pm11. The bottom strip is a rotation of the top strip, but if we view those differences as significant, then it still isn’t a strip pattern (it would be a central symmetry of type D1), and if we view those differences as insignificant, then it would be a pattern of type pmm2, since it would have both vertical reflections, and rotations (and consequently also have horizontal reflections).”
I was excited to get this response to the leaves image, as it reminded me that my newly acquired understanding of symmetries, though useful, is simply just emerging.

So that’s it:

  • Translation
  • Reflection (horizontal or vertical)
  • Rotation
  • Glide Reflection

Darrah Chavey, who is a  professor at Beloit college, turned out to be the hero in this journey of mine, for having made and posted videos on youtube. Here’s a link to one of his many lectures on patterns: Ethnomathematics Lecture 3: Strip Symmetries

Now here’s some nuts-&-bolts of what I’ve learned from making the samples that I’ve posted here:

  • the book I made was too small (only 5.5″ high) because the cut papers then had to be too small to handle easily.  I’m thinking that any book I make with students needs each page to be at least 8.5″ tall.
  • It was easier to create harmonious looking patterns when I started out with domino rectangles (rectangles that have a 2:1 height to width ratio), then cut them in half and half again to make squares, tilted squares,triangles and rectangles.
  • I like the look of alternating plain paper and cubed paper. Folding paper that has cubes printed on just one side accomplishes this.

I am going to enjoy teaching these college level concepts to young elementary children.

Rubber Band Books · simple book binding

The Humongous Rubber Band Book

A bounty of books
Eight Humongous Rubber Band Books, showing off their colorful spines

The sixth grade English teacher in this school likes the idea of each of her students making a book that they can use as (her words) a memory catcher. Writing, pictures, and ephemera will go into these books. The design challenge is that I can’t count on having more than 40 minutes to work with the students. I want them to end up with something large, sturdy, and I want them to enjoy making it.

Setting out the Papers
Setting out the Papers

On my day with these sixth graders, they walked in the library, saw the colorful papers and were immediately delighted. “Do we get to do this today?!” They were all so happy! My papers here are tabloid size, 11″ x 17″ 67lb papers (which, by the way, are getting more expensive and harder to source every time I look).

Choosing papers
Choosing papers

Each student chooses eight papers. We have plenty of space to work. It’s interesting to notice how each student chooses to arrange their stash.

setting out papers
Getting ready to work

Some students choose to work alone and spread their papers out all out in front of them

Getting ready to work
Getting ready to work

Other students work two, three or four to a table and have to stack their papers.

Next step is to fold the papers then nest them together in groups of two.


I’ve worked with these students many times before, and they are all have expert paper-folding skills.


The trick to accurate paper-folding is to hold the paper with one hand, then slide the other hand towards the curl.

foldingThese students have been using my bone folders just about every year they’ve been in school. If I forget to hand them out they will ask for them. In schools I refer to them “folding tools” to avoid  vegetarian discussions. If the fact that they are made of bone comes up, I advise vegetarians not to eat them.

These papers will  get nested together in groups of two.
These papers will get nested together in groups of two.

The students end up with four groups of two folded papers. This grouping is completely non-intuitive: students want to nest them all together, one inside of the other, and wrap one rubber band around the spine and be done. In fact, the book would work just fine that way, but I’m here to show them something different, and, arguably, better. By asking them to make four groups of paper they will end up with a thicker, and much cooler looking book spine, one which shows off some of the colors in the book.

Snips off the top and bottom of the spine edge of the paper
Snips off the top and bottom of the spine edge of the paper

Once the pages are grouped together, there’s one more step before the assembly starts. The corners of the tops and bottoms of the folds are snipped off. These snips create valleys that the rubber bands will settle into.

Attaching nested groups of paper
Attaching nested groups of paper

Two groupings of papers are set next to each other side-by-side, opened in the middle. The rubber band slides over the four adjacent pages, binding the page groupings together.  I use  Quill Brand Rubber Bands, 7Lx1/8″W which are humongous in just the right way. Smaller rubber bands will actually work for this, but the tighter the rubber band stretches, the sooner it will rot and break. I want these books to stay together for a good long time.

Attaching signatures together
Binding together two groupings of papers, with the rubber band

On goes the rubber band! This is done until all four sections are linked, in sequence, one group right next to each other. This book can be made to be just about any number of pages long.

Book made with humongous rubber bands
Book made with humongous rubber bands

It’s a good idea to decorate the cover of this book right away, as the flexible nature of the spine can make it tricky to figure out which page is the front once it’s been opened and looked through. Students make pockets to go on the front and back covers, to store items that will be eventually attached into the books. I’ve been making these books with this school’s sixth graders for a number of years, but I don’t get to see them finished. Students, however, will joyfully tell me about them, and they will also tell me, oh I remember when my brother made these! From what I understand, they hold a plethora of memories.