Arts in Education · Math and Book Arts

Books, Symmetry, and Students

Pamphlets made by seventh graders
Sewn Pamphlets made by seventh graders

I’m in the busy part of my art-in-ed, itinerant artist season. The challenge is to keep what I do relevant to the students, to the curriculum, to the teachers, and to myself. Most of the work that I do in schools is done with teachers I’ve worked with in previous years. Usually I repeat a project each year with the teachers’ new classes, though there are always tweaks that are made. Then, sometimes, it’s time to retire a project that’s been working well for years.

I’ve just finished up quite a few projects in classrooms, many of which were new this year. I’m going to attempt to write a number of posts about these projects before the next set of classes that I teach start up.

This is a detail on one of my own drawings
This is a detail on one of my own drawings that starts with the graph of functions.

There’s been a shift in my approach to what I offer to the schools. Whereas I used to think of my work as a way to motivate and celebrate literacy, now I am more focused on using our bookmaking projects in a way that supporting the teachers’ math goals. I’ve been realizing that the math part of the curriculum is where many teachers most appreciate support. There is so much in the paper and book arts that can support the math that students need to learn that making this shift has been thoroughly enjoyable to me.

Symmetry is a theme that kept emerging in the projects that I presented these last few weeks. This is partly to do with the nature of making books, but I also deliberately focussed on it more than it other years. I’ve realized, just recently, that the symmetry of shapes is the visual equivalent of mathematical expressions. I probably won’t express this well, but here goes. Think about doing any sort of math problem that has an equal sign in it. 5+3 = 8. It’s balanced. If you add a 3 to one side you have to add a 3 to the other side to keep the expression true. Math calculations are all about symmetry and balance. It is, therefore, completely appropriate and desirable, to help kids develop their natural affinity to symmetry.

Starting with a pile of cards
Starting with a pile of cards

One of the projects that I did with kindergarten students had to do with these piles of square cards. Students worked in teams. The first student puts down a card, then the partner puts down a card that is a symmetrical reflection of color and shape. They take turns putting down a card and then reflecting it.

Making Symmetry
Making Symmetry

It took a bit of doing for these 6 year olds to get the hang of what we were doing, but, still, quickly, patterns emerged.

Reflection symmetry
Reflection symmetry

These cards, by the way, are an element within a larger math activity book that we made.

 Since these pieces are made from paper, I suggested to the teacher, as this becomes easy for the kids, to cut out one of the smaller squares from some of the cards so that mirroring the shape transformations becomes a bit more challenging.


Another symmetry project I tried out for the first time was with the Pre-K crowd. My friend Joan, who has worked with this age group, showed me this activity that she had developed with kids she had worked with. I’ve been excited to try it out.

Popsicle Stick symmetry
Popsicle Stick symmetry

What I did here was define a line of reflection. Then these five-year olds did the same kind of reflection symmetry that I described above, each taking turns putting down a stick, then the partner reflects it with one of their sticks.

Popsicle Stick symmetry
Popsicle Stick symmetry

Again, it was a struggle to get these students started, but it didn’t take long for them to catch on.

Four student symmetries
Four student symmetries

After a short while I combined groups so, instead of working in pairs, there were four people in a group, which led to different kinds of designs. The pattern above was made near the end of the activity. From start (first handing out the sticks) to finish, this activity took a mere twenty-two minutes, which was how long it took them to began to lose interest. At this point I suggested that they just used the sticks to make whatever arrangements that they wanted to make. Surprisingly, many started trying to use them to spell out their names. I heard their teacher remark something about the fact that they struggle to write their names but they seem to be able to construct them just fine. Which gave me an idea, which I will show in my next post. Now, though I want to jump back to the photo at the top of the post, which is the books made by seventh graders.

Folding and tearing  large paper
Folding and tearing large paper

I’ve been doing this project with the seventh grade for many years. I give them a large piece of paper (23″ x 35″), which they fold and tear to make a pamphlet.

Pamphlets in progress
Pamphlets in progress

I don’t explicitly talk about the symmetry of the folding we do, but I will talk about it in the future. The fact that the sequence of fold and tears results in a scaled down version of the original sheet is something I want them to be aware of.

Glueing out the spine piece of the pamphlet
Glueing out the spine piece of the pamphlet

In fact, every aspect of making this book is symmetrical, even the pattern of the thread that sews the pages together is totally symmetrical.
When building just about anything, even a book, symmetry rules.

Some finished books
Some finished books

These kids are so proud of their books.

Ok, enough for now. More tomorrow…..

8 1/2" x 11" Book Making · Accordion Books · Art and Math · geometry and paperfolding · Making Books with children · Making books with elementary students

Five Days of Summer Workshops with 3rd and 4th Graders

Pop Ups with more Pop ups
Pop Ups with more Pop ups

In the summertime, when school is not in session, I’m on my own in terms of deciding on what kinds of projects that I want to teach in workshops. Last week I taught for five days  at the local community center.  My sessions with the kids were 40 minutes long, and although I prepared for 30 rising third and fourth graders, there was no telling how many students would attend each day. I had originally thought I would make a plan for the week, but quickly realized that it was more satisfying to create projects each day based on what I found interesting in the children’s work from the day before.

Making Pop-ups in an Accordion Structure
Making Pop-ups in an Accordion Structure

My own goal for the week was to do explorations with shapes and symmetry.  On Day 1 we made a four-page accordion book and did some cut-&-fold to make pop-ups. The students were amazing paper engineers;  With impressive ease, they created inventive structures.

Pop-up Worksop
Pop-up Worksop

There were plenty of counselors in the room, and from this very first project, these counselors joined right in with creating their own projects.

Overlapping Rotated Squares
Overlapping Rotated Squares

I was so impressed with the students’ folding skills that the next day I helped them create an origami pamphlet that contained more pop-ups, as well as some interesting other cut-outs. What turned out to be the most interesting work on Day 2 was how much the kids liked the little bit of rotational symmetry that I encouraged them to do: I gave them each a square of paper, asked them to trace it on to the cover of their book, then rotate it and trace again.

summer squares 3

These students like the shapes created by shapes, so the next day I brought in a collections of shapes and asked them to arrange tracings of these shapes on a piece of heavy weight paper, which was folded in half.

Tracing Shapes to Create  More Shapes
Tracing Shapes to Create
More Shapes

Students seemed to enjoy creating these images.

summer shapes 2

After they created the outlines they added color.

Colored Shapes
Coloured Shapes

When the coloring was done we folded the paper, and attached some pagesto the fold so that the students had a nice book to take home. The kids seemed to like this project and made some lovely books, but I ended up  feeling like there wasn’t anything particularly interesting going on with this project in terms of explorations of building with shapes. So …

Building Stars and Hexagons with Regular Rhombuses
Building Stars and Hexagons with Regular Rhombuses

…the next day I brought in colored papers that were printed with rhombuses, as well as some white paper printed with a hexagon shape. Each student filled in their own hexagon with 12 rhombuses.

Making a Hexagon with a Star in the Middle
Making a Hexagon with a Star in the Middle

My plan for this project was to have each student make their own individual hexagon then put them all together on a wall so that it would be reminiscent of a quilt.

Paper Hexagon Quilt
Paper Hexagon Quilt

Here’s our paper quilt made from 22 hexagons!

The next day, Day 5, was my last day at this program. I liked the engagement with and results of how the students worked with shapes when they were given structure. There’s a balance that I try to honor of providing structure while allowing individual choices. For my last day, then, I decided to give the students a page that I created that is based on the geometry that uses intersecting circles and lines to create patterns.

A work in progress by one of the couselors
A work in progress by one of the counselors

If you look closely at the photo above you’ll see many different lines and curves overlapping and crisscrossing.

summer geometryI asked students to look for shapes that they liked, to use the lines that they wanted to use, and to ignore the lines that they did not want. It was interesting to watch how the students worked; I was particularly interested in seeing how some children chose to start looking at designs starting in the center, while other children gravitated to the outside edges first.

summer geometry 8

Some students filled areas with color, while others were happy to make colorful outlines of shapes.

summer geometry 2Some drawings were big and bold.

summer geometry 5

Some drawings were delicate and detailed.

summer geometry 4I think that every one of the teenage counselors sat and made their own designs, right alongside of the students. Actually, I think that my favorite unexpected outcome of the week was how involved the teenagers got with the projects.

summer geometry3

This last project of the week was my own personal favorite (though the quilt project runs a really close second). I had never done anything quite like this before with students, and was really surprised to see how much they enjoyed this work, and how differently they each interacted with the lines and curves. This kind of surprise is what’s so great about summertime projects.