A device for learning mathematics by the physical manipulation of numbers
It is widely accepted that creativity in learning mathematics fosters improved know-how and prepares students better for real-world applications of mathematics concepts. However, due to the burden of testing, curricular constraints, as well teacher misconception, creativity in mathematics is often neglected in the classroom.
Base Converter is a creative task focused device that aims to facilitate mathematics teaching and learning using the real-world application of textile weaving. The device introduces primary school aged children (ages 7 to 12) to the concepts of modular arithmetic and alternative numeral systems. It enables them to do basic arithmetic and base conversion operations by physically modelling and manipulating numbers. The device is unique in how it promotes physical number exploration and presents the opportunity to children to formulate their own mathematical problems. A cross between an abacus and a loom, it is a device envisaged to provide creativity enhancing opportunities in mathematics classrooms in a low-cost manner.
Compared to the traditional method which involves finding powers and multiples of the base number, Base Converter only requires one type of division and everything else is calculated through the physical weaving process.
Modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value (the modulus). The system is best explained on a circle and one common use of this is the 12-hour clock. On a clock, the circle is divided into 12 sections. If the clock hand is pointing to section 8 now, then 5 hours later it will be pointing to 1 because clock time "wraps around" every 12 hours with the hour number starting over after reaching 12. This is arithmetic modulo 12. The Base Converter was modeled after the circular loom as circles lend themselves effectively to basic modular arithmetic operations, and basic modular arithmetic is an effective foundation for eventually teaching base conversion to young children.
The Base Converter kit includes a square base, a set of round center pieces, 12 warp sticks, a roll of cotton yarn and 10 instruction/sample exercise cards.
Base Converter kit
Base Converter cards
Initial Ideation & Research Through Making
The context of this project is the intersection of three domains: ‘the long now’, ‘human-machine symbiosis’ and, a domain I will attempt to define through this project, ‘augmented craft’.
Inspired by the work of The Long Now Foundation - specifically their 10,000 Year Clock project - specific elements of 'the long now' domain with particular importance to this project are the ideas of slow technology, long-term responsibility and mindfulness.
'Human-machine symbiosis' is defined as the collaboration and/or mutually beneficial relationship between humans and machines, especially when working towards a previously defined optimum output to a task.
‘Augmented craft’ takes human-machine symbiosis one step more specific, and alludes to craft-specific concepts such as craftsmanship, craft skills and creativity. For this project, the focus was weaving as weaving is commonly considered as a time consuming, meditative process that requires a certain level of craftsmanship.
Mindmap: What is craft-augmented technology and what is augmented craft?
User Testing & Iterations
Preliminary qualitative testing of prototype #1 with four adults revealed that the Base Converter’s circular design is intuitive and users could easily associate warp sticks to a number. However, there also emerged an idea about the possibility of removing the woven structure from the base while keeping the woven design intact. It was speculated that our real users, children aged from 7 to 12, may want to keep their woven structure as a keepsake to take home if the device is introduced to them at school. Although this point was dismissed as not been specifically related to the learning of mathematics, it informed the modular design of the next iteration.
Prototype #2 featured a removable center piece in the base that accommodated for the universality of the device. It was envisaged that each Base Converter would come with a set of interchangeable center pieces that have different number of holes in them. A warp stick would sit in each hole and the user would decide on which center piece to use based on which modulo, or base, they need to work in. As well, warp sticks used in prototype #2 were made of thinner, 11.5 gauge aluminum wire and were covered in soft plastic tubing to make the design safer for children to work with. The base of the prototype was changed to a circular shape instead of a square to get feedback from users on their preferences. Both users who tested this prototype preferred the previous square design. Additionally, it was revealed during user testing that the holes in the center piece need to be smaller and the warp sticks should fit more snugly in them.
Prototype #1 in use
Prototype #2 round base
Prototype #3 was designed with silicone center pieces so that the warp holes can be made slightly smaller while the warp sticks still remain easily attachable and detachable. The base of the device was changed back to a square and a user test was done with a ten-year-old boy named Mitchell from the Long Island based Nova School. Mitchell was taught how to use the Base Converter to do basic modular arithmetic and basic base conversions. He was then tasked to convert a number from one base to another and, after 2 repeats, he started to get the answers right.
Prototype making process
Mitchell was also introduced to a mini game whereby one user thinks of a number and starts to model (weave) it on the device, and a second user finishes the number. The only rule that applies to the second user is that the final modeled number has to be a real number that can be interpreted by the first user. The task of the first user is then to interpret the resulting woven structure into a real number. Mitchell was able to interpret a number in this way on first go and expressed that he enjoyed the game.
Mitchell modeling on the Base Converter
WHAT MEDITATIVE EXPERIENCES (for humans) CAN WE BUILD AROUND HUMAN INTERACTION WITH TECHNOLOGY?
HOW CAN TECHNOLOGY BE DESIGNED TO AUGMENT CRAFT SKILLS WITHOUT MAKING CRAFT REDUNDANT?
WHAT IS THE FEELING OF BEING A CRAFTSMAN?
WHAT VALUE/S DOES CRAFT HAVE? MONETARY, CULTURAL, SOCIAL?
WHEN DOES A THING BECOME OBSOLETE?
In trying to answer some of these questions, I felt the need to really understand the kind of tasks a weaver performs and the experience they go through while operating the loom - from spinning the yarn to setting up the loom, from seemingly endless repetitive motions to the precise control of each yarn's tension. This naturally presented many more 'implementation' focused questions such as:
How does one set up the loom?
How long does it take?
How is the weaving pattern represented, and then coded into the loom? and so on...
A loom consists of many different parts, and when setting up the loom, the weaver handles all the different parts to successfully feed the yarn through the heddles, adjust individual yarn tension, as well make sure all yarns are on the same tension. The different parts are highlighted on the left.
This was an implementation-focused paper prototype for understanding how the heddles and treadles of a floor loom work. In short, each treadle (or pedal) is connected to a shaft (aka harness) that carries an X number of heddles. Pressing on a specific treadle lifts up the shaft connected to it.The heddles are long strings (sometimes rods), usually made of metal, that has an eyelet in them. The yarn goes through this eyelet and depending on which eyelets have been threaded and which shaft is lifted by pressing on the treadle, the woven pattern changes.
The purpose of this prototype was to visualise how threading a specific heddle, and which combinations of treadles, will produce what type of pattern. Each cut strip of tracing paper (the horizontal ones) represents a shaft, and there are 4 different types of shafts. By selecting two shafts at each step (overlaying them), I get the pattern at the bottom right corner. But I could also select only one shaft at one step, then select 3 shafts at the next, then select 2 different shafts etc. to create a much more complex pattern.
While cutting many strips of paper by hand late in the evening, I got reminded of how I used to cut lots and lots of paper patterns when I used to do fashion. Each assignment required us to make patterns both in paper and in cardboard and we'd stay up late into the night cutting patterns by hand. After a while, our hands would start to get sore and tiny blisters would start forming where the scissors were held too tightly. The cardboard patterns required more strength... and lots of patience.
This got me thinking about the sound of cutting paper. I know that cutting patternmaking paper produces different sound from cutting cardboard. And it is the same with weaving. An expert weaver has better rhythm and in fact makes less noise, while an amateur weaver's rhythm is inconsistent and there is generally more noise, from the beater slipping out of the hand or the shuttle dropping to the floor.
Prototype 2 attempted to investigate this by using the sound produced while cutting four different types of paper.
The last prototype became more of a multi-material prototype and incorporated a cardboard box, wooden doll pins and cotton yarn. The purpose of this prototype was to visualise and also understand how a card loom would work. From preparing the yarn strands to setting up the card loom, there are many small variations that can be tweaked that change the final outcome. Threading the yarn through the holes in the yarn from the back to the front would create a different result than if it was threaded from the front to the back. I could also have 2 yarns threaded from the back, 2 from the front. Despite being one of the simplest loom designs, the card loom is still able to produce unexpected complexities.