Drawing Kids into Mathematics (Part 3)
In parts 1 and 2 of Drawing Kids into Mathematics, you learned the basics of turtle geometry and explored the constructivist and constructionist underpinnings from which it was developed. In this last part of the series, we will discuss how this might influence deep student learning both within and beyond turtle graphics.
Effects ‘With’ vs Effects ‘Of’
If you ever wonder at the end of a day just what your kids learned while working at the computers and you are dissatisfied with your thoughts, consider the following simple model. Gavriel Salomon has posed an analysis of the difference between effects with and effects of computers.
“Effects with are the changes that take place while one is engaged in intellectual partnership with peers or with a computer tool, as, for example, is the case with the changed quality of problem solving that takes place when individuals work together in a team. On the other hand, effects of are those more lasting changes that take place as a consequence of the intellectual partnership, as when computer-enhanced collaboration teaches students to ask more exact and explicit questions even when not using that system.”
In other words, the effects with are the enhanced ability one gets from the use of technology. Salomon elaborates: “The combined product of human-plus-machine yields a higher level of performance.” The effects of are the lasting individual changes resulting from the computer-supported collaboration, the cognitive residue, one might say, the transferable knowledge or skills.
Jeffrey (a Grade 2 rascal!) made a most interesting leap from Logo to a completely different domain one day.
We were having a discussion inspired by the flight of the space shuttle piggybacked on a jumbo jet. Our Grades 2/3 class had the opportunity to watch the flight. When we returned to the classroom, a discussion of space naturally arose. One child asked if Earth was in space, and in asking the question, she determined it must be, because it wasn’t sitting on anything. The discussion continued until Jeffrey piped up.
“You know . . . it’s sort of like Logo (Turtle Art).”
We stopped and looked at him curiously.
“What do you mean?” I asked him studiously.
He replied, “Well, Earth is like a procedure. It’s like a subprocedure inside the solar system. The solar system is the superprocedure. And the solar system is like a subprocedure inside the universe. The universe is like the superprocedure.”
“Fascinating,” I said, then asked, “What’s the biggest superprocedure?”
After a moment he replied, “I don’t know. I guess the universe.”
Well, I was truly amazed at the generalization across domains that Jeffrey had made. He clearly demonstrated significant transfer of a concept from his experiences with Logo to an authentic event. Although Jeffrey’s illumination happened spontaneously, I learned that I could play an important role in helping students to acquire Salomon’s effects of by providing opportunities for them to look for these comparisons across subject areas.
I started playing a game with students that I called Metaphoria. I gave them sentence starters such as:
“Programming in Logo is like …” or,
“Finding a bug in a program is like …”.
Students have answered: “Programming in Logo is like playing tennis. First, I take a turn, then the computer takes a turn.” “Finding a bug in a program is like looking for a needle in a haystack.”
Here is another example that provides students with a mental model—a model that is durable and independent of coding—a residual effect.
“…provide students with mental models with which to think.”
One example of this residual effect became evident after the students’ experience with bug collecting during their time with Logo (Turtle Art). They had learned that identifying problems in their Logo code meant that they had mistakes, or bugs, in their thinking.
Of course, actively seeking bugs was a necessary component to getting the program to do what they wanted. Bug seeking naturally evolved into bug collecting. Every time a bug was solved, the kids squished it—metaphorically, of course!
The class had built a large papier maché turtle, and one of the students suggested that perhaps when a bug was solved it could be fed to the turtle instead. This was delightful and useful in and of itself, but the transfer of this model became clear as I overheard two students working on a traditional paper math task. They knew their answer wasn’t right.
One student said to the other:
“There’s a bug in here somewhere. We’d better find it!”
I believe it is important to maximize the opportunity for the acquisition of these skills as your students are coding. Do not leave it solely to the use of the computer. Be explicit in building the bridge—in making the connection of this skill to other domains. Discuss them in class. Have students describe other situations where these skills might be used.
In fact, take it one step further. Ask your students to think of particular aspects of that give them generalizable skills. In this way, you are empowering them to take more responsibility for their own learning.
What other ways might you think of that encourage transfer or effects of?
Online Resources for Getting Started
Be sure to go to the Getting Started pages on the Turtle Art website for some excellent beginning tutorials.
Ok. Now for the Big Leap!
One of the best ways for students to learn Turtle Art is to look at some samples—both of the final artwork but also the code. Have them deconstruct the code. Check out the Samples, Snippets, and art in all the ‘books’ on the Turtle Art website.
Come to understand them and take lessons from the way they are constructed.
Give your students much time to tinker. Give them opportunities for guided and free inquiry. This combination of tinkering and inquiry, which Brenda Sherry and I call Tinkquiry, is essential.
When you click on the little yellow block at the bottom left of the picture (on the website), it will show you the code.
What Mathematical Processes are in Play in an Effective Implementation of Turtle Art?
The mathematical processes that support effective learning in mathematics as stated by the Ontario Mathematics curriculum are as follows:
- problem solving
- reasoning and proving
- selecting tools and computational strategies
I would add another such as:
- wondering, tinkering, playing (Tinkquiry)
My Total Turtle Trip
I shall finish this post as I started it, thus doing a total turtle trip:
Let your students amaze you with their mathematical minds as they create art with turtle geometry!
- Mindstorms: Children, Computers and Powerful Ideas, Papert, S., Basic Books, 1980
- The Children’s Machine: Rethinking School In The Age Of The Computer, Papert, S., 1994
- Harel, I., & Papert, S. (1991). Constructionism. New York: Ablex Publishing.
This page is a compilation, and rewriting, of some materials already posted in this site…and is here for the purposes of The Quest 2016.