Drawing Kids into Mathematics (Part 2)
In Drawing Kids into Mathematics (Part 1), we described turtle geometry and how to get started with Turtle Art. Now we will play with some pedagogies underlying successful implementation.
Constructivism vs Constructionism
I wish to take a moment to describe the difference between constructivism and constructionism as these philosophies are central to optimal outcomes for students when engaged in creating with Turtle Art.
These understandings arise through experiencing events and then reflecting on those experiences.
If we encounter something new, we either assimilate it into our previous ideas and knowledge—our schema—or we must accommodate it by changing what we believe, therefore modifying our schema. Or maybe we just discard the new information as irrelevant.
In neuroscience terms, we are building neural pathways, cognitive routines, habits of mind, models with which to think, webs of ganglia, mental models.
This is when he coined the term constructionism.
“Constructionists believe that deep, substantive learning and ‘enduring understandings’ occur when people are actively creating artifacts in the real world.” Papert & Harel, 1991
Constructionism holds that children learn best when they are in the active role of the designer and constructor. This building of artefacts mediates conversation among kids.
“Constructionism relies on visible and discussable thinking.”
In a constructionist approach, it is not merely the act of constructing that is essential. Constructionism relies on visible and discussable thinking. Powerful things happen when that act of constructing mediates deep conversation with others. The very act of articulating ideas, sharing thoughts, confusions, ahas, questions, potential solutions makes knowledge building explicit. Sometimes words are spoken. Oftentimes facial expressions and body language communicate. We might draw diagrams or build prototypes. All these serve to make the thinking visible and, therefore, discussable—not only with others but for oneself. We learn our subject matter well as we think hard about it and are very intentional about constructing not only the artifact at hand but also our knowledge and success.
OK…back to the turtle!!
Total Turtle Trip Theorem
This is a lovely piece of mathematics fun and students love it.
The Total Turtle Trip Theorem suggests that “If a turtle takes a trip around the boundary of any area and ends up in the state in which it started, then the sum of all turns will be 360 degrees.” Mindstorms, 1980
So, as an advance organizer or a minds on as we might say today, I asked my grade twos, to write a story about a total turtle trip before we explored this powerful idea on the computer.
Leeanne, aged 8 wrote…
Once there lived a turtle. He was very curious. He wanted to take a trip. So he said to his mother, I’m going on a trip.”
“Oh, but what if you fall and land upon your back?” his mother asked.
“I won’t do that.”
And off he went. When he was walking along he met Father Bunny.
“Where are you off to on this fine spring morning,” Father Bunny asked.
“I’m going on a trip around the world.”
“But what if you…”
Turtle didn’t hear. He was halfway down the road. He started down a big hill but he tripped and tumbled down, down, down the hill and landed on his back. And there he stayed. Meanwhile Turtle’s mother got worried and went to find him. She found him on his back. She helped him to his feet.
And he said, “I guess I took a total turtle trip!”
Total Turtle Trip Theorem in Practice in Turtle Art
In this example of a square, you can see that the turtle is asked to repeat forward 100 right 90 4 times. So the turtle turns 90 degrees 4 times. 90 X 4 = 360 degrees. A Total Turtle Trip!
Here we see a pentagon, made by 5 turns of 72 degrees each. 72 X 5 = 360! You can imagine that the student has to think— “5 times what number equals 360?” Or, “360 divided by 5 equals what number?”
Leanne, of course, had to do it differently. Now, remember, this is an 8 year old! She decided to make an Onzagon – an eleven-sided figure. (We were a French Immersion school!) But, she decided to make the computer do the math! So she had the turtle turn 360/11. I thought that was pretty clever!
Think About Making an Equilateral Triangle
I would like you, as a teacher, to think about how you would make an equilateral triangle. It is very often a great lesson in the difference between Turtle Art and Cartesian geometry. What angle do you have to turn those three times? Turtle geometry is relative to my heading and to the amount I, as a turtle, need to turn.
Procedures and Subprocedures
One of the most powerful aspects of Logo is that students can teach the turtle new vocabulary. This ‘naming’ can be extremely powerful as students are building and structuring their own language.
So, for example, the turtle does not have the command Square in its list of commands. But, you can teach it. Pull the yellow diamond out from the My Blocks tab, and type the word square on it and add your commands. A new procedure block called square will appear in the My Blocks panel for your use in the future. You have now taught the turtle a new word—a new procedure. Smart wee thing.
You can now use the new procedure, square, inside another procedure. Square becomes a subprocedure and, in this following case, flower becomes a superprocedure. You can see that, once again, the Total Turtle Trip Theorem comes into play as flower is made by making a square and then turning 10 degrees and making another square 36 times. 36 X 10 = 360!
Now you will be getting comfortable having some hard fun with the mathematics of turtle art. But, how might this impact the generalizable learnings a child might have? This will be the topic of Drawing Kids into Mathematics (Part 3).