I will be working with about 75 second graders later on in the school year, doing what I do, which is to make books that are artful, and that relate to curriculum. The teachers haven’t settled on exactly what that curriculum link will be, so I’m trying to develop a two-prong math/design based project to pitch to them. The first prong is about numbers, which I will write about at a later date. The second part will be about how to use math to make beautiful designs.rotational-symmetry-2

I’ve been inspired by a number or educators to work with stars, including Alan Parr, who has prodded and poked the possibilities of this kind of math/design artivity, I mean activity, with a very young crowd. I won’t have nearly the amount of time with the students to go into the details that Alan explored in his star posts, but I do think I can expose these kids to a way of making interesting images with math.


I want to keep this simple, but still spectacular. I’ll be giving the students circles that are divided into 12 parts, as this is more like a unit circle than a pizza circle. What else, I will ask them, is circular or cyclical that is divided into 12 parts? Clock. Calendar. My original thought was that I would have the students use a ruler to connect the dots according to a rule, but I abandoned that strategy for one main reason…

rotational-symmetry-6…which was I didn’t like that hole in the middle of many of the stars.

I did try out some ways of filling that hole…rotational-symmetry-7

…but it just didn’t seem right. What I came up with, then, was the idea of providing the students with shapes that they could rotate around the circle, which I may or may not mention is called rotational symmetry. Students can use a square or a triangle and rotate and trace it within the circle, touching the corners to the 12 dots on the perimeter of the circle, then they can cut out a shape and line the corners of that shape up to just the black dots around the circle, always rotating around the point in the center of the circle.


The shapes above are the ones that students can use.


The designs above are some of the ones that can be made with the shapes, which…rotational-symmetry-3

…I will encourage students to color in. Notice the use of Sharpies. I think that the black Sharpies really make these designs pop. I will be buying lots of Sharpies.

rotational-symmetry-9I also hope to have some time to have students try to draw some circle on their own, divide them up just as best they can, then play around with using their freehand circles to make a different kind of drawing.

rotational-symmetry-4So that’s my plan. I hope I get to do this project with the second graders that I will be working  with!

Here are some links to what people who’ve gotten me thinking about this way of working:

Simon Gregg, Odds and Evens, 

 Malke Rosenfeld: Stars, Factoring, and Patterns,

John Golden: Stars made While I should have been sleeping, and once again

Alan Parr, Star lessons

and, finally, an interactive star by Dan Anderson (hover your cursor over the image, and move it around! )

My thanks to my son, his girlfriend and my daughter who helped me trouble-shoot this project by letting me do a math project with them on Thanksgiving Day.

Math on the Move by Malke Rosenfeld

Math on the Move by Malke Rosenfeld

I’ve been reading a book by Malke Rosenfeld about teaching children math concepts by engaging whole body movements.

I devoured every word of this Math on the Move, took copious notes, and am reading it again, even though I don’t anticipate that I will ever exactly do this thing that Malke writes about.

What I happily discovered here is what I see as a landmark book that gives a voice to the thoughtfulness and rigor within the practice of the teaching artist.

It’s funny, looking over my notes: on the right sides of my notebook I write down reference to passages that I want to remember. On the left sides I’ve sketched out flows of ideas that pop into my mind as I read what she’s written.

One thing I realize as a go through this book is that the teaching-artist in me is lonely. The dynamic, curriculum-referenced collaborations, I do with students is exciting, joyful and creative work, but I suffer from a dearth of a community of colleagues.

Malke has shared not only an innovative and accessible way of presenting complex ideas, but, in this book, she has also been generous with letting me experience of the way she brings her art into the classroom, sharing details of her process and values, details which I am assimilating with a greedy urgency.

An example: in preparation for a math/movement class, Malke writes that she has applied blue tape on the floor which looks like the silhouette of a 25 foot ladder. She then invites students to get to know the taped shaped before she begins the formal presentation. The students run around the outside of the shapes, get on the floor and slide around it, jump in and out of the delineated squares. I read this and I feel ashamed and excited. Ashamed because I know how often I hand out evocative materials and introduce them with the words don’t touch, excited because I can imagine ways of allowing for a few minutes of pre-activity exploration which I suspect will profoundly deepen the students connection to the work we do together.

Further in to the book, Malke mentions changing up a lesson the fly because of student absences.  Oh my gosh, this is something that no one ever talks about! The challenges that teaching-artist faces in providing meaning and continuity within the context of seemingly random shifts in who showing up (or who’s crying, or throwing up, or coughing incessantly) requires a calm and nimble response. We seamlessly need to be able to switch things up.

As I am writing this I’m  on-line and I’m seeing that David Butler, a PhD in finite geometry (that is to say a University prof far removed from elementary math education) is writing about Malke’s book too, also noticing her process. He’s saying to her, In the videos I noticed you say “that’s gonna take some practice” but never “that’s hard” and The only time I notice you saying “great job” is for the act of sharing their work-in-progress with others. Yes! It’s such a gift for the details in Malke’s teaching to be included with the big ideas of this book.

This is not to say that there I wasn’t interested in her big ideas, which are about connecting students with math in a completely unexpected but totally reasonable way. Anyone who knows my work has seen how I’ve been interested in the connections between math and artful projects. Malke combines  dance movement with math. This is such an utterly brand new idea for me that I had no idea what she would be talking about. However, step by step, from chapters entitled The Body as an “Object to Think With” and How is this Math?  to Assess, Extend, and Connect I totally understood what she was saying, so much so that it’s hard for me to understand why I hadn’t thought about this before.

Malke references Seymour Papert, who wrote “…the teaching of mathematics, as it is traditionally done in our schools, is a process by which we ask the child to forget the natural experience of mathematics in order to learn a new set of rules.”

The deep truth in this quote stops me in my tracks every time. It reminds me that the most profound teaching that I can do is mostly about guiding students to make sense of and give new context and language to that which they already know. For instance, Malke points out that “…developing a useful understanding of angle has been shown…to be be extremely challenging for learners but critical to students’ success…” Angles, hmm. Think about this in terms of the body – any child who can bring two fingers on the same hand together then spread them apart has the tools to explore the properties of angles. Once I got the hang of thinking about math concepts in terms of the body, the possibilities seem endless.

This is where the power of the Malke’s ideas are in full bloom. Why restrict the learning of important concepts to the dimensions of a desk when there is a whole body there that already knows so much of what we teach and is ready to learn more?  Malke shows how to figuring out how to tap that inner knowledge and build on it.

My teaching toolkit just got infused with a whole new way of working!












Birthday Blog Girl

Birthday Blog Girl


Once again acknowledging the happiness that this blog gives me, for seven years now.

Blog map

Blog map from 10/22

It used to be that the USA and the United Kingdom were always where my blog was most viewed. While it still stands that the US is by far where most visitors originate from, I ‘ve noticed a changing trend in the #2 spot of most-views-from. Yesterday, Israel provided the second most views. In the past two weeks, after the US, top number of views have come from the Czech Republic, Spain, Italy, Russia, Germany, the Netherland and India. Occasionally I see views from unexpected places like Qatar, Lebanon, and Guam (which I think isn’t even shown on the wordpress map?). Yes, it’s satisfying to know that people are interested in what I’ve been publishing on this site.

The two most clicked images this year, and last, were these:

How to make an origami Pocket by Paula Krieg

Click to enlarge for reading or printing


How to make 2 basic Pop-ups

How to make 2 basic Pop-ups

Two of my favorite posts from this past year were the Ta-Dah! post which featured the collaboration of brave souls to create two beautifully diverse stars.

First Star


Second star


Steve Morris wrote a timely, prescient blog post  in which he used one of the images above. You can view it on his blog at

I have lots more favorite posts of this past year, but here’s another image from just one more:

I’ve had lots of fun this year responding to geometric and numeric sequence equations that Dan Anderson and Martin Holman have posted on twitter.

Also, this past year, Mike Lawler responded so enthusiastically to a post I wrote about a pentagon box  that I got inspired to make lots more boxes. Maybe I will write more about these at a later date.

Template for Puff Box

Template for Puff Box

Finally, I’ve been recently most enthralled by the Zhen Xian Bao, a structure that Sue Cole had introduced me to. To date, I’ve written seven posts on these, and most recently have been working on deciphering the secrets of creating my own version of this traditional piece of folk art. It turns out that all the attention I’ve given to geometry and math has been invaluable in sorting out ways to work the details of this Chinese folded structure! What a surprise…img_0513

Actually, as I look back on the year, I am seeing many more posts I want to highlight as a highlight….but I will stop here. It’s just that, OMG, I am just so grateful to have to place to share all of this with you.


Chinese Thread Book, Zhen Xian Bao, from the collection of Ed Hutchins

Detail of a painted Twist Box from a Chinese Thread Book, Zhen Xian Bao, from the collection of Ed Hutchins

I can’t believe my good luck.

I’ve been reading about the Chinese Thread Book, devouring anything in print that I could find about it, scouring the internet then just thinking about and trying to make sense of this structure. It didn’t even occur to me that I might actually get my hands on an authentic Zhen Xian Bao.

Front of  Zhen Xian Bao, from the Collection of Ed Hutchins

Front of Zhen Xian Bao, from the Collection of Ed Hutchins

This is how it happened. I wanted to share what I’ve been studying with book artist Ed Hutchins. When he told me that he wasn’t familiar with what I was talking about I drove to his house and dropped off Ruth Smith’s book on the subject. then received this mysterious message a few days later. Ed wrote: ” LOVED THE BOOK. I devoured it cover to cover. I’m going to try to find my zhen xian bao before you get back. keep your fingers crossed…” then a day or two later “You won’t believe this: I found the sewing kit book–AND you are going to love it!”

Chinese Thread Book, Zhen Xian Bao, from the collection of Ed Hutchins

First Opening, Chinese Thread Book, Zhen Xian Bao, from the collection of Ed Hutchins

Turns out that even though Ed had no idea of what I was talking about, once he saw the Ruth Smith book those memory gears kicked in, and he suspected that something he had in storage, might be of interest to me. Turns out he had, many years ago, bought this item on Ebay, without knowing what it was. When he asked the seller about it, well, the seller didn’t know anything much about it either. Ed suspects that this thread book was part of an estate that was being sold off.

Chinese Thread Book, Zhen Xian Bao, from the collection of Ed Hutchins

Two kinds of origami on the top layer of the Zhen Xian Bao (grey strip of paper has been placed in the open box on the right, to hold it open for the photo)

Here’s a variation of boxes on the top layer that I hadn’t seen in all of my perusing: this Zhen Xian Bao features both twist box, and a masu-type box on the top layer, in an alternating pattern.


Looking under the flap of the square boxes with the star on top readily reveals that this box is an embellished masu box.

Edge of Masu-type box

Edge of Masu-type box

There are a few things about this masu box that I’ve deemed particularly noteworthy. The first is that the green and red backgrounds of the star shapes are not hand colored, rather, these are colored papers that are adhered to the masu-box paper. The star motif is also decorated with collaged bits of colored papers. The other detail that I thought was interesting is that the masu boxes were made from a lighter weight paper than all the rest of the boxes in this thread book.

Second Layer, rectangular domino proportions, under each set of twist box and masu box Chinese Thread Book, Zhen Xian Bao, from the collection of Ed Hutchins

Second Layer, which are boxes (or trays?) that have rectangular domino proportions; Gray paper strips placed in the boxes to hold them open for photo-op

Okay, so there’s 16 square boxes, each two of which reveal a box underneath, so, between just the first and second layers there’s 24 boxes.

not origami

not origami

There’s some precise folding going on with these rectangular trays, but it’s also clear that it’s not what we think of as origami. Its been my impression that the rectangular trays traditionally are more like simple folded templates, but I will continue to make mine with origami methods for the reasons I’ve discussed in earlier posts. The decision mostly has to do with the paper. Oh, and it’s the paper in this book that makes it most convincing to me that this Zhen Xian Bao was made in China. The paper is thin, strong, and has an uneven texture. It’s certainly handmade paper, and it’s not like paper I’ve seen. Actually, this paper’s closest counterpart in my paper stash is the common grocery bag (though I am sure that this similarity is purely cosmetic!).

Next layer down, from Chinese Thread Book, Zhen Xian Bao, from the collection of Ed Hutchins

Next layer down, from Chinese Thread Book, Zhen Xian Bao, from the collection of Ed Hutchins

Next layer down! Here, each set of four top boxes pull away from each other. Now the count is up to 28 boxes.

Next layer, tray under eight boxesChinese Thread Book, Zhen Xian Bao, from the collection of Ed Hutchins

Next layer, tray under eight boxes Chinese Thread Book, Zhen Xian Bao, from the collection of Ed Hutchins

Please excuse the purple straw holding the next layer open. Now we’re up to 30 boxes. If you are confused by the count, remember that each set of boxes has a symmetrically placed counterpart, so this open box on the right side is mirrored, but currently hidden, on the left.

The Big Box of the Zhen Xian Bao

The Big Box of the Zhen Xian Bao

Finally, here’s the Big Box layer. There are some major tears in the part of this box that articulate the spine. With the big box, there;s a total of 31 individual compartments in this book.

A secret is revealed on the big box layer that I loved seeing…. one thing that bothered me about this structure was the cover. Although there are no rock-solid rules for the cover of the Chinese Thread Book, I found the cover of this one to be somewhat out of place. But at the edges of the material that covers the big box there’s a hint of something different.


Look, at the head of the box there’s an indigo pattern on material that is underneath the red cover paper.

img_0535There it is again, at the tail edge of the box. The red cover was somehow added on, over the original indigo cover, which is a color that makes more sense for this book. Maybe the original cover was damaged and a seller thought to recover the book to make it more sale-able?

Decoration on a twist box of  the Zhen Xian Bao,

Decoration on a twist box of the Zhen Xian Bao,

I kind of plan on kind of replicating this book using my own methods. Using the measurement methods I’ve been writing about, the only measurement I will need to replicate this book is the diagonal measurement of the square that is made with the 2 x 2 square of the top-tier of boxes.

Chinese Thread Book, Zhen Xian Bao, from the collection of Ed Hutchins

Back cover

That’s it. Now I better get this book of boxes back to Ed before I get too used to having it here.

%d bloggers like this: