Feb / 17
21 Feb / 17
This post is co-authored by:
Meg Escudé, Jake Montano and Natalie Freed
In our XTech program for Middle and High School youth, we started the year off by revisiting a book binding project. Students explored graph theory concepts and computational algorithms all while stitching bundles of paper together with needle and thread. The project engages youth in the centuries-old practices of Chinese and Japanese stab bookbinding with the assistance of a web-based designing app created by Natalie Freed. Natalie is a long-time friend and collaborator of ours and we invited her back to share her project with our special yearly girls-only XTech session. Since then we’ve done the project with the integrated group of intermediate XTech students and again with the youth at the Visitacion Valley and Don Fisher Boys & Girls Clubs.
The process for designing and making a notebook can be a little long, and the spatial and mathematical concepts embedded are a little abstract. In order for students to have some foundational experience with both the graphing concepts and practices of sewing, we developed a warm-up activity. Actually, we designed four warm-up activities! In the four times we’ve introduced this to a new group of students or facilitators, we’ve made changes to the warm-up activity, adjusting for things we noticed in the previous workshops. In all iterations, the prompt was based on the principles that are present in the traditions of bookbinding, and which incidentally make them a “Euler path.” They are that the design should be one, continuous graph* (a combination of points connected by lines) that can be stitched with one piece of string without doubling any lines and so that the design is identical on the front and the back of the paper.
Natalie: These groups of points connected by lines are called graphs as in “graph theory.” Oddly this use of the word graph is, as far as I can tell, completely unrelated to graphs of functions (eg. plotting X and Y on a grid). Graphs show up in all kinds of places, but a few examples are that you can use them in a more abstract way to model social networks (people connected by relationships) or transit networks (such as train stations connected by routes).
To explain a bit about what you’re doing when you figure out the stitch order for the books, take the train example. Let’s say you want to see the scenery on every single train route, but you only want to see it once. You’re allowed to stop as many times as you want at the same stations, but you just don’t want to travel the same route twice. This is the exact same problem as trying to stitch each connection between holes only once: you can go through the same hole as many times as you want (as long as you’ve punched the holes big enough!), but you can’t go over the same stitch more than once. The path through the railway network or stab book that accomplishes this goal is the Euler path.
Once you start experimenting with a particular train route, you might figure out that some train systems are set up in a way that makes this possible, and some aren’t. You might also figure out that as you plan your trip, sometimes the order you decide to take the routes works, and sometimes you get stuck at a station with no way to return other than paths you’ve already taken. Eventually, you might figure out that there’s a general rule to find out a) is there such a path? and b) in what order should I visit the stations to find this path? Finding that general rule is finding an algorithm (ie. repeatable procedure) to solve your graph.
This activity is perhaps not the most direct way to explain graphs or algorithms, but one thing I like about it is that as you stitch books (or patches or paper), you begin to develop an intuitive, tactile sense of the algorithm.
Warm-up Iteration #1
We came up with a design for a small, 3” square stitching pattern that follows the book-binding requirements above. I tested it to be sure there were several different interesting designs possible when connecting the same dots:
We then made multiple copies of this configuration of dots and handed them out to students. They were asked to draw a design that they liked that involved making a closed circuit of lines connecting all the dots. We purposely didn’t include as extensive an explanation of graph theory or algorithms as Natalie provides above at this point- so that it would be more open for discovery or question-generation. (Natalie did share these analogies and explanations as she introduced the app for the notebook designing after we were done with the warm-up activity.)
They then punched holes on the dots with these great long-reach, tiny hole hole-punches and started stitching the designs with colorful crochet thread and large, leather stitching needles. Many students learned to thread a needle using a needle threader, and the basics of back to front stitching during this time. Once they got going, it sometimes took a couple tries to figure out the order of stitching that would give them an identical design on both sides of the paper, without doubling any lines. This ordering of stitches is an algorithm. The designs turned out really beautiful but Jake noticed at the end of the day that none of the students seemed interested in taking them home. We decided to try and find a way to make the warm-up activity a little more meaningful- both by trying to pull out the math a bit more, or in having the final product be something more useful than a piece of paper with string on it.
So, the following Saturday when our XTech Intermediate group met, we started the day with a meeting of our teen staff and tried out this second iteration with them:
Warm-up Iteration #2
We printed out three different historical bookbinding designs, sized for small, pocket sized notebooks and pre-cut the paper for this size. We asked everyone to choose a design and went over the steps for punching the holes, and trying to figure out the algorithm for stitching the book following those same requirements discussed above. We finished our small notebooks and I asked the group if they thought this was good way to introduce the activity. One of our facilitators who was present in the previous girl’s day workshop said she thought the first iteration was better. She said she liked that students got to choose their own design and that she noticed a lot of them brought elements of those initial designs into their final book design when using the app. This was a very convincing argument! So, for that day, we quickly switched back to iteration #1 and scrapped #2.
Warm-up Iteration #3
In preparing for the start of our semester of tinkering after-school programming at the Boys & Girls Club in Visitacion Valley, Jake expressed his continued disappointment that nobody took their squares home with them. With the age of the younger, elementary aged youth at the clubhouse in mind, we decided to focus the first day entirely on the warm-up activity and continue into book making the following week. That meant that this warm-up better be something worth investing their time in! I had the idea of preparing colorful felt patches with the same hole configuration instead of zeroxed paper squares. Jake and I laser-cut 4” round patches with the holes in place.We also felt like we could have done a better job naming the mathematical concepts present in the process of stitching these designs. In my introduction, I did some live stitching, asking students to do some group-thinking about which stitch I should make next. After we did a few stitches, I let them know that those decisions we were making were the elements of an algorithm and that there were many successful algorithms possible for most designs.The kids drew their designs on their felt with markers and stitched them in the same way as the papers. When they were finished, we gave them safety pins so they could be attached to clothes and backpacks. This turned out great and several students walked out of the workshop proudly wearing their creations:
Warm-up Iteration #4:
The session at Visitacion Valley (VV) went rather well, so we decided the following week at Don Fisher’s Boys & Girls Club to do largely the same. We brought the pre-laser-cut circular badges for students to draw out their own designs, but also brought along square-shaped pieces that possessed no holes at all. We had noticed only a small smattering of students at VV that wanted to create designs entirely from scratch, and didn’t expect there to be much interaction with the square pieces at Don Fisher because of that. Surprisingly, and gleefully, we were very wrong!
Before we began constructing, I had the group participate in helping me draw a familiar “puzzle” which resembles a house with an x inside to convey the challenge of creating a design without doubling back, then illustrated the multiplicity of designs available on the pre-cut circular badges. Meg passed around the circular badges, thread, and needles and the kids started designing and stitching.
Many of the students had recently taken a sewing class elsewhere, and once they’d finished a first badge - completed in about fifteen minutes - they decided to add their own holes to the square pieces. Generally, their designs were representational - a cat or the insignia for a favorite superhero - and all possessed well more than the eight holes we cut. Other students went straight to the square patches, as some seemed either intimidated or bored by the challenge of drawing lines from pre-made holes.
One aspect I think we could play around with some more is how computational and algorithmic thinking can be intuitive, meaning the students developed ways of troubleshooting the stitching of their patterns as they progressed and regressed (and the process most certainly included a balance of both progress and retracing steps with the needle). Through stitching between a finite number of holes, we’re utilizing mathematics to reach completion, which sometimes means going backwards to a step in order to move forward altogether. To help demonstrate the challenge of creating a pattern in which the thread passes through to create a line only once on each side we started referring to a widely-known children’s “puzzle” in which a house with an x must be drawn without lifting the pen or doubling back with it. With this in mind, figuring out a way to have students describe the stitching of their patches or books using only words might be a great way to further explore Euler circuits and trails and to elicit the steps students took to complete their stitched items.
It has also been mightily impressive to see the wide variety of directions the students have taken to personalize their designs. Some have used multiple strings (sometimes of two or more colors) to stitch with, while others have “stacked” strings and patterns on top of each other to produce an almost three-dimensional effect on their patches. The designs themselves have also spanned a spectrum of styles, from the minimal and abstract to homages to 8-bit videogames like Pac-Man. Other designs have stretched what is possible for binding books, and it’s of great curiosity to see how book-making is pushed into new territories based on their bindings.
Natalie: This is so cool! As a follow up, I’d be interested to ask the students how they decided what step to stitch next, whether they felt like their technique improved over time (in terms of not getting stuck and having to undo stitches), and whether they could put into words some of the strategies they intuitively developed.