May 22, 2016
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.
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.
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.
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.
Food and Interesting facts go on one of the sides.
Facts about the bird’s appearance and their habitat are written on the far edges of the paper…
….with life cycle info at the center…
…topped off by information about the author.
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!
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.
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.
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.
June 2, 2015
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 lynda.com, 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.
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.
A Translation takes a motif and repeats it exactly.
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.
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.
Horizontal Reflection mirrors the design across an imaginary horizontal line.
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:
So that’s it:
- Reflection (horizontal or vertical)
- 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.
April 17, 2015
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.
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).
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.
Some students choose to work alone and spread their papers out all out in front of them
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.
These 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.
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.
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.
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.
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.
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.
March 31, 2015
This story begins in a teachers’ lunchroom, a couple of years ago, in Upstate NY. I was sitting with some teachers when another member of the staff started talking to a first grade teacher, Mrs. K, about a new math mandate. It was something about using manipulatives to create a variety of shapes. I’m a bit foggy on this part but it seems to me that they were required to use rhombuses (or rhombi, both are correct) for their shape building.
Upon being told that she would have to incorporate these manipulatives into her math unit Mrs. K asked if there was any money in the budget for manipulatives. The answer was no.
After school I sought out Mrs. K and showed her some paper-folding and shape transformations that referenced rhombuses. This teacher seemed delighted with what I was showing her. I volunteered to send her something that I thought she might find useful, then went home and created these images for her, which are equilateral triangles that become a rhombus.
I never asked Mrs.K if she used what I sent her. I recognize that what I sent was, unfortunately, not a project. Instead, it was just the bones, the beginning of a project that needed to be developed. Every so often I’ve revisited these images, wondering what I could do with them. Then a few nights ago Malke, from Indiana, asked me about projects for a family night.
It was late, and we decided to resume the conversation the next day. The next morning, before Malke and I reconnected, I saw this post from Simon Gregg, in France:
I had an Aha! moment. It suddenly came together. I sent off this note to Simon:
Malke, who I included in the conversation, responded with a reference to a beautiful manipulative that I wasn’t familiar with, but which also showed that she immediately recognized what I was getting at with my DIY paper version of manipulatives.
Since Malke seemed to know exactly what I was thinking about I got to work creating the pieces for this activity. I’m pretty happy with how this has developed. It requires triangle paper, and matching paper shapes that can be printed on colorful papers. My thought is that simple, bold shapes can be created in sort of a free form way…
…or more challenging shapes can be drawn on to the paper…
…and filled in, while trying to make as few cuts as possible and being mindful about cutting along the lines defined by the triangles.
So, where can you get these papers to do a do-it-yourself shape building set? Right here. I’ve created a couple of PDF’s to get you started:
Make beautiful shapes. Send photos. Thank you.
Addendum: Take a look at Malke’s post on hands-on math: she collected and organized many interesting perspectives. It’s a fabulous piece of writing. http://mathinyourfeet.blogspot.com/2015/04/some-thoughts-on-hands-on-math-learning.html
Addendum #2 (April 2016) Malke liked working with smaller rhombuses so I made her this PDF rhombi with spaces So far she is planning on using them without the triangle grid paper. Here’s a link to some images she’s created as samples for an upcoming project https://www.facebook.com/MathInYourFeet/ rhombusphotos