Making Meaning of Angles and Degrees: Art and Programming the LogoTurtle

Exploring geometry by making art with Logo programming using a physical floor turtle allows for deeper comprehension of mathematics concepts like angles and degrees. A computer screen is too precise to capture the nuances that a floor turtle immediately exposes when students program geometric shapes to create art. The physical construction of a floor turtle is also an ideal STEAM project, combining 3D printing, electronics, microcontroller programming, and debugging. Students who use this analog tool to create art easily develop personally meaningful mathematics challenges that provide for the construction of authentic mathematical knowledge.

- The pedagogy of Logo programming: learning to think about thinking 

• When we teach Logo programming, we teach people to think about thinking
• The goal of learning to program in Logo is to think algorithmically: you program the turtle to express a sequence of designs that express over time
• Solomon, Papert, and others emphasized the importance of the student's body in space and relating one's personal position and movement to that of the turtle
• Much of the earliest work with Logo programming occurred with a physical turtle complete with stepper motors, sensors, and a pen
• Educators lost track of the importance of the physical turtle with the availability of screens, as Walter Bender explained it to me
• Scalability
• Cost: the stepper motor was $30 and required a huge power supply
• Logo programming has always been about debugging
• Counter to today's template-based approach to computer science where you learn to program in a certain way, do it right the first time, and move on to the next cookie cutting
• Logo programming is particularly well suited to the exploration of geometric art
• As educators we can use Logo as a microworld to explore students' hypotheses about angles, degrees, arcs, rays, and combinations thereof

- The screen is too precise

• The possibility of serendipitous discovery made through the ease of block-based programming is a great way to get students interested in programming
• I used Turtle Blocks since October with middle school students in Bridgeport, CT in an after school club that meets for an hour each week
• Blocks come with numbers attached: for example, the "right" movement block comes with "90" already attached
• Some student might already know that 90 refers to 90 degrees
• Ease of use at risk of shallow exploration
• Turtle Blocks offers engagement with the possibility of deep exploration provided a couple of prerequisites:
• Scaffolding: demonstrate how to make the turtle go forward, how to go right, how to connect blocks; encourage them to get out of their seats
• Time: allow adequate time for the free exploration of the software and the individual creation of knowledge
• Collaboration: allow for the sharing of ideas, techniques, code

- LogoTurtle

• STEAM project
• 3D Printed parts: recycled from another project that my friends and I found on the internet
• Microcontroller: simple inexpensive computer that is not without some limitations
• Hand-built electronics: I developed an illustrated set of directions so anyone can assemble their own LogoTurtle using about $60 of electronics parts
• Text-based programming
• Interesting designs can emerge from fewer than 10 lines of code
• Aspiring towards "elegant" code: define a square procedure and use it in a master procedre
• A tool that lets you create work that you otherwise would be incapable of producing
• Relative precision
• Repetition
• An "imprecise" tool
• Alpha release and unreturned email to Brian about a noticeable drift that occurs
• Realization that programming an "ideal" design is about mucking around with rays, angles, and degrees
• Mathematics
• No floating point means no decimals means tricky math: microworld approach 
• Science
• Friction of pen and wheels
• Electronics
• Stepper motors
• Batteries 

- Making Sense of Angles and Degrees: An Example

• Transfer from block-based design to text-based syntax
• The syntax matches, but you need to learn the abbreviations
• There is also formatting to consider, especially the use of brackets
• Spelling becomes important
• Imprecision apparent
• Angles and degrees appear to not have worked as they did on the screen
• Checked calibration: debugging, troubleshooting, scientific method
• Student chose to iterate
• Six tries to "get it right"
• Constructing mathematical knowledge while creating art
• Played with angles and degrees to work towards the idealized design
• How many students choose to do over a math problem they got “wrong” six times until they get it “correct”?
• Combining art with a mathematic and science challenge encouraged iteration and construction of knowledge

- Obstacles and Work Arounds

• What is easy and what is right
• Not an out of the box solution
• End up with a thorough understanding of how a robot works
• Hard fun
• mouth up versus mouth down fun
• Debugging is at its core, and mucking around rewards the user
• Require expensive technologies
• outsource your 3D printed parts
• workshop the electronics
• work with the community of users
• Child programs the computer, rather than vice versa
• Papert's Mindstorms
• Behaviorist encroachment in the classroom in the name of efficiency and standards