folding · How-to

The Paper Spring in the Classroom

Teaching kids how to make a paper spring is always thrilling. Children ooh and ahh, and practically jump out of their seats when I show them what we’ll be making.

The only problem has been is that it takes up a big chunk of my teaching time, as only about 55% of the students (who are usually 6-8 years old) in the classes I teach are able to make paper springs without extra help.

I’ve been teaching kids how to make paper springs for probably 20 years. Have shown it to thousands of students. We usually glue something to the top of it it, like a cut-out of their hand, to give the books we are making another dimensional element.

About a year ago, driving to another of my itinerant teaching-artist jobs, I was stressing over the fact that, due to time constraints I needed to cut something from my agenda . Realized the paper spring was going to have to be eliminated…unless…unless I could figure out how to get all of the kids to do make it without any extra help.

A caterpillar of paper springs
A caterpillar of paper springs

The way I’ve been teaching it is to glue two paper strips together to form a right angle, then alternate folding the strips on top on each until the papers fold down into a square. It’s easy to teach this method to adults, but kids keep folding in front then wrapping behind, which sabotages their springs.

 

 

Notice the corner is like a square, Draw a happy face on the square.

What if I ask students to fold the other way, to fold it below the glued corner, rather than above it? And to keep them from folding forward, draw a happy face which they are told should not be covered up?

Really, no one wants to cover up a happy face.

So I tried it out. Asked the students to alternate colors folding behind the happy face, said what we wanted to end up with is a little square.

Almost done

Couldn’t believe how well this went when I first tried it out. There is still a bit a confusion that happens when they see these flaps at the end.  I probably should say to cut off these pieces, but…

Last step

…these flaps can be  folded back too, then secured with a bit of glue.

This method of teaching has worked out for me unbelievably well. Unbelievable, even to me. Students have been nearly 100% successful in class after class.  So exciting to have discovered this way of teaching the paper spring.

Here’s a video:

 

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 Desmos.com near the beginning.

 

 

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.

Stars
Stars

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.

 

 

 

 

 

 

folding · Paper Toy

Six-fold, flat-fold, Paper-fold

Paper Folding the Ferozkah Jaali
Paper Folding the Ferozkah Jaali

I found a fold.

If paperfolding graps your attention, prepare to be overwhelmed.  There’s three things to unpack here: the fold, the pattern on the fold, and how they interact.

I had been wondering if I could fold a tetrahedron out of a rectangle.

Tetrahedrons and other shapes
tetrahedrons and other shapes

Turns out, yes. I can make a tetrahedron with a square base or a triangular base out of the same piece of paper using the same folds in different ways.

Looks like a fish
Looks like a fish

Then I started seeing that I could make other shapes out of the same pieces of paper using the same folds differently.

Some shapes are flat, others are dimensional.

I’ve been playing with these all week, and I am still finding different shapes that these folds create.

 

I’ve also been drawing this six-fold pattern from Islamic Geometry called the Ferozkoh Jaali. It occurred to me that it would go perfectly with the folds I was making.

detail of Ferozkah Jaali
detail of Ferozkah Jaali

 

This is just a small portion of the pattern. I’ve been coloringing copies of these in all week, trying to get to know the shapes.

Here’s the fold that I’m using:

 

Mountain and Valley folds
Mountain and Valley folds

It’s four mountain folds (diagonals) and two valley folds (horizontal and vertical) that are created around equilateral triangles. Oh, and there’s a slice in the middle. One horizontal slice.

Now here’s the first wonderful thing about using this image with my folds:

No matter how you use the creases (which are around the equilateral triangles) , the pattern lines up. In the photo above, a corner is peeking through that slice in the paper, and, look, the pattern lines up.

Equilateral triangle(s)
Equilateral triangle(s)

I printed the design on the fronts and backs of my papers, and look, when the paper wraps around itself, the pattern lines up.

Now there is one more thing to mention. Hold on to your seats. This is wonderful. But, first, here’s the foundation of the image I created, first by hand, then on the computer, because I needed the precision of the computer image.

Six-fold-geometry
Six-fold-geometry

Okay, so as I’ve been folding and refolding and refolding again, and finding different shapes all the time, the last final amazing thing that I noticed (and this makes so much sense) ….

Some heart shapes?
Some heart shapes?

…is that every shape I make with these folds can is echoed somewhere in the lines of the  geometric drawing that is printed on the paper.

This makes me so happy, well, I can’t even describe it.

Another heart shape
Another heart shape

Well, there you have it. Hope you love it as much as I do.

covered with NOT geometry
covered with NOT geometry

Oh, and just in case you’re wondering, I think this fold looks good with just about anything on it.

Am seriously considering making a bunch of these and offering them for sale, probably through Etsy. Stay tuned….