Does turtle graphics have the wrong associations?

[Vincent Manis]

On 2009-11-12, at 19:13, Peter Nilsson wrote:
> My recollection is that many children struggled with Turtle
> graphics because they had very little concept of trigonometry.
> [Why would they? Many wouldn't learn for another 2-10 years.]
> Adults tend to have even less concept since they almost never
> use trig (or much else from school ;-) in the real world.
> 
This paragraph is based upon a complete misunderstanding of turtle geometry. You do NOT use trigonometry to teach it, because the goal isn't to replicate cartesian geometry. The whole point about turtle geometry is that the student viscerally imagines him/herself BEING the turtle, and moving around the room according to the succession of FORWARD and TURNRIGHT commands. This is easier to visualize when one has an actual robot that draws pictures on butcher paper, as the second iteration of the MIT/BBN turtle work did (and they worked in middle schools, Grades 4-8, so there was no issue of trigonometry). 

> They can see the patterns and understand there's a logic to
> it, but they struggle replicating it. Get an angle wrong
> and you end up with a mess where it's not clear whether it's
> your algorithm or the maths that's at fault.
Kindly explain to me the difference between `algorithm' and `maths' here. I believe you just said that if there's a bug in the design, the program won't work. Hmmm.

This reminds me of a well-known anecdote about the original LOGO study done at Muzzey High in Lexington, MA, in 1968. A group of NSF funding people was doing a tour of the school, and they came across a Grade 5 student who was doing a family tree program. The NSF people were impressed by the complexity of the program. One of them said in a patronizing tone, `I guess this stuff really helps you learn math'. She got quite angry, and responded, `This stuff has NOTHING to do with math!'

> The visual aspect might pique interest, but may put just as
> many people off. In any case, it won't relieve the difficulty
> of having to teach what is fundamentally an abstraction that
> doesn't have very good parallels with how people approach
> problems in the real world. Humans simply don't think like
> mathematicians^W computers. :-)
Having taught grade 8 math, I can tell you that cartesian geometry is much LESS intuitive to people that are learning it than the relative polar coordinates of turtle geometry are. (`Oh, you want to get to the mall food court? Turn left, and walk past about 10 stores. The food court is right after the Gap.')

> I've met a lot of mathematics and comp sci teachers who
> honestly believe that you can't teach these subjects without
> a mathematical perspective. That stands in contrast to the
> number of people I see using spreadsheets with a very high
> proficiency who would never dream of saying they were good
> at mathematics or programming.
It is true that you can't teach computer science to children without having a strong understanding of the mathematical foundations. It is also true that when you teach it to children that you very carefully hide the mathematical formalism. 

I might point out that the people who had most to do with the invention of turtle geometry were Wally Feurzeig (who was my boss when I worked at BBN in the 1970s) and Seymour Papert. Papert had spent a lot of time working with Jean Piaget in Switzerland. If you read the original LOGO memos, you will see his clear explanations on how this material ought to be taught to people with no math background, including children who are too young to do symbolic thinking (that kicks in in the early teens). That's why the visceral `I am a turtle' approach works well with middle-school kids. 

-- v
> 
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> Peter
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