Arts in Education

Recalculating a Failed Workflow

Each year, as part of a bookmaking project with sixth graders, I bring in my impressive assortment of paper punches, and let the students decorate their handmade books with a colorful array of  items which include stars, crescents, hearts and creatures. It can be a wild free-for-all, with some students slapping on their paper-punched creations willy-nilly, and others making carefully thought out arrangements. It’s generally a messy, high-energy class period, with bits of paper and glue being  put down with excitement and delight.

Reflection, translation, glide-reflection
Reflection, translation, glide-reflection

Last year it occurred to me to rein in some of this excitement and introduce the students to different kinds of symmetry that would enhance their designs, Things seemed to go pretty well last year, so I tried it out again this year.

Paper Punch Symmetry
Paper Punch Symmetry

While some good things happened, it was clear to me that my own  ideas about teaching something new came at a cost: students didn’t do nearly as much decorating as in previous years, and much of the excitement had been sucked out of the project.

Naturally, what I want is the excitement AND to be able to teach something new and valuable.

My experience with the sixth graders was weighing on my mind this past week when I was working with second graders on cut-paper project.

Rhombi as the ready
Rhombi as the ready

What I think had been my biggest misstep with the sixth graders was that I dampened their paper-punching enthusiasm before they ever got a chance to indulge in the novel activity I was laying out for them. I depend on the  use of unusual tools and colorful materials to engage students,  but I hadn’t  given these kids a chance to experience their excitement before laying out the conditions which seemed to challenge and dampen their spirits.

In hindsight, I think things would have gone much better if I had laid out the materials, let the students create their collections of paper punched bats, balloons, dragongflies etc. and then, only after they had gathered together these personalized treasures, I could then proceed with the references to symmetries.

Cutting out Rhombi
Cutting out Rhombi


Now, this week, working with second graders, I tried to learn from what happened during my time with the sixth graders.  I had an agenda, which is to use rhombuses to construct plane shapes, such as trapezoids, hexagons, and triangles. This is supposed to be an exciting activity, full of experimentation and discovery. I didn’t want to do anything suck the joy out of playing with the rainbow of colors in search of wisdom.

The students needed to cut out rhombuses from a packet of colorful  printed strips. I decided not to tell them exactly what we’d be doing with the rhombuses. Instead, after the rhombuses were cut, I encouraged to students to slide them around on their desk, and organize them into piles or shapes that appealed to them.

Rhombuses on the run
Rhombuses on the run

This exploration time took only a few minutes, and, as different students finished their cutting at different rates, it kept everyone busy.


I traveled around the room, showing some kids how to use these shapes to make all sorts of arrangements.

Hexagon, Cube, Star
Hexagon, Cube, Star

It’s worth mentioning that they were notably impressed that three rhombuses could look like a hexagon, or like a cube in perspective, with color choice playing a significant role in creating the illusion that these same shapes were different.

playing with Rhombi
playing with Rhombi

It was only after this time of playing around that we got down to the business. I tried to keep them in discovery mode by asking them to take just a few pieces in their hands (which at this point included some rhombuses that had been cut in half to form two equilateral triangles) and to try to figure out how to make a trapezoid or a triangle or a hexagon, or a scaled up rhombus.

Plane Shapes
Plane Shapes

It all worked out.

More Plane Shapes
More Plane Shapes

At no point did I feel like my agenda had sucked the air out of the room. Whew.

My note for next time is to remember to let the kids feel the excitement and let them create their own relationships with the materials before I overlay my lessons into the moment.  Whereas I had hijacked their enthusiasm before, I think that this different approach enriches their enjoyment, and hence their learning.







Geometric Drawings

More Designs with Curves and Lines

Designs with Lines and Circles, by Paula Beardell Krieg

My workshop in making designs with lines and circles attracted a good number of people last month. I wrote about the first class already, and here’s my wrap-up post, which is a few weeks late, but, hey, it’s summertime and my children are home.

Designs with Lines and Curves class
Designs with Lines and Curves class

The most wonderful and surprising aspect of the class was not only the number but the age range of the participants! In fact, the class attracted ‘tweens through adults, and since this way of working was new to everyone, the group felt cohesive.

Michael's , Six petals
Michael’s , Six petals

At this second class there were a number of people who were coming for the first time, so with them, we started off creating intersecting circles to create the six-petal pattern that I had introduced to the group last time. People colored them in according to their own style.

Six petal pattern
Six petal pattern

I love the range of styles that I saw. Some of the work was bright and bawdy.

Joyce's six pointed star
Joyce’s six pointed star

Some of the pieces were dreamy and meditative.

Karen's Six-Petal image
Karen’s Six-Petal image

Some pieces were cheerful and carefully considered.

Kathy's 12-petal image
Kathy’s 12-petal image

For people who wanted to go just a step beyond the six-petals, I showed them how to transform the six into twelve petals.

eight-pointed-star-geometry PDF

Okay, I know that the image above is hard to see, but, actually, that’s the point. This is what I handed out to people who were ready to move on to a more complex  set of shapes. While I worked with the newcomers, I asked the rest of the group to look at this page and start finding shapes that they might like to highlight in their own pieces. That, after all, is one of the coolest things about this way of creating designs: there are an infinite number of individual responses to the same underlying architecture.

Eight Petal design

Something surprising happened. Some people were happy to use the PDF that I handed them, and to color that in. The point of the class (I thought) was to show people how to create the shapes for themselves, but not everyone wanted to do that.

Diane's work
Diane’s work

It’s fascinating to me to see what unexpected things people do in classes. The fact that there was a group that happily and prolifically just colored was fun to see.

Jeanne's Work
Jeanne’s Work

There was also a group who was very interested in learning how to make the underlying designs for themselves.

Michelle's work
Michelle’s work


I didn’t end up taking many photos during the third class that I taught. It turns out that what I brought in was way challenging, and I spent most of the class focused on helping people be successful. What happened? The first class I showed a six-pointed star (or six-petal) design, the second class I showed an eight-pointed star design, so for the last class it seemed logical to me to show a five-point star design. Making a five-pointed star design turned out to be much more difficult for people.

One of the highlights of the class was showing people a site that turned their work into hyperbolic tiles.

hperbolic tile
Hyperbolic tile

If you don’t know what a hyperbolic tile is, well, it doesn’t matter. Just upload an image into the site  and hit the “generate tiling” tab. Within days of showing this to people, I started see hyperbolic tilings show up in my FB feed.

All told, the three workshops were just wonderful to share with the people in my community.

Here are the links that I found useful for these workshops

Intersecting circles for Seed-of-Life Geometry

Eight Pointed Star Geometry PDF

Dearing Wang’s Youtube channel

Dearing Wang’s HOW 2 Draw Tesseract – Octagram Into Infinity

Golden Ratio Star Pattern

to make a Hyperbolic Tile

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.