Saturday, September 19, 2009

shape 31

I asked some of my more able students to create the above shape in Turtle Art (from the Barry Newell shape sheet) using variables, so that by varying the input value they could create the same shape in different sizes, as shown below:

Well, this did set the cat amongst the pigeons even amongst my better students who had been happily in the groove making complex shapes from the BN sheet. One problem was that in their procedure they were using subtraction to alter side length and of course this will not work for variable size shapes.

Another issue is that despite doing algebra out of a textbook they didn't really grasp the box model of a variable, that you need for programming. Imagine a box which has a name, which doesn't vary, but you can put numbers (or other things) in the box which do vary.

So, I realised this was a nice challenge in real maths and understanding of the application of variables, measurement, ratio, proportion and fractions. I see this as an excellent example of constructionist maths in contrast to textbook maths.

Clue: When I measure the lengths of the "curly rectangle" (not sure what the correct name is) on a larger diagram in BN's book they are 46mm, 32mm and 39mm


Mark Miller said...

It seems that we all get mixed up at some point about the meaning of "variable" when we go between math and programming. In fact we can get mixed up about the concept of a "function" as well.

While variables can be known or unknown at the outset in arithmetic, in mathematics they're representations of concepts or parts of concepts. They can just be representative of a set of characteristics, with no expectation that they will be resolved down to some concrete value.

In programming there are assumptions or assertions about variables, even if they're implied. Some of their characteristics are either known or expected. This is somewhat compatible with the sense of variables in mathematics, though with rare exception every programming model I've used expects a variable to always resolve to a concrete value. It's just a place holder for a value that is expected to exist, though it may not always be the same value or a value with the same set of characteristics. That's the distinction.

MariaD said...

Question: should we be doing an art/math/paper activity on spirolaterals sometimes BEFORE this programming challenge? Mentioning the word "spirolateral" in the vicinity of kids and internet-enabled computers will probably do the trick, too.

But the question remains - SHOULD we?

Bill Kerr said...

hi maria,

Playing turtle can be a useful introduction. ie. clear a space and one student is the robot and you have to tell him / her what to do to make the required shape. Write up the instructions as we do it.

This enables some formalisation of concepts which can be helpful:
- to make a regular polygon the turtle has to turn through the external angle, not the internal angle
- you can find out the external angle by 360 / N

I've been using the Barry Newell sheet for years without knowing about spirolaterals. My instinct is that if there is an opportunity for exploration before formalisation then it's best to take the exploration path first, without forgetting that formalisation along the way might be very helpful

Bruner: "Doing with images makes symbols" is one way of expressing the idea that playful exploration can lead to the gradual development of symbolic or abstract knowledge