realistic constructionism

I came to understand that making Shape 31 with variables was a nice challenge in real maths and understanding of the application of variables, measurement, ratio, proportion and fractions



Students in schools are invariably brought up on textbook maths and so to apply real maths to a challenge like Shape 31 is daunting for most. Hence, the need arose for me to help them scaffold this challenge, to make it more accessible to them.

I would see this as realistic constructionism, avoiding the twin errors of:
a) Just throw them off the deep end and hope that they swim. A few will but most won't
b) Not setting a real challenge through fear that they will find it too hard, just stick to the safe exercises in the textbook

The way I scaffolded it was to first demonstrate an easier shape that nevertheless required the thoughtful and not just mechanical use of variables.

In looking at the Barry Newell (BN) shapes from this perspective it becomes clear that some shapes have sides that vary (eg. rectangle) while others have sides that are all the same (eg. regular polygons such as square, triangle, pentagon etc.).

So I decided to demonstrate how to do the rectangle because the variation in the sides (2 are longer, 2 are shorter) is relatively easy to follow. So, here is is:


I then set as challenges shape 3, shape 4, shape 6 and shape 31 (all shapes here). They had to be done as follows:

"Shapes with variable sizes using the box. The variation has to work for small and large sizes"

Nevertheless, I still found that students found this hard. The following problems arose:

• using subtraction instead of or as well as multiplication and/or division. ie. not understanding that proportions or ratios alter with subtraction and do not alter when using multiplication and/or division
• trial and error instead of using measurement and knowledge of fractions or proportion

I gave direct advice about these things. Sink and swim has it's place in highly controlled or 1:1 or self directed learning environments but in a larger class is mainly irresponsible teaching. But in retrospect I could have done better and in fairly obvious ways. For example, I should have supplied rulers and more encouragement / demand for students to measure shape lengths and achieve more accuracy in proportions.

Nevertheless, one thing I discovered was that to quickly check whether the variable shapes scaled correctly was to quickly type in a very large value (eg. 1000) into the box and then see if any gaps in the shape resulted.

I asked the students to write this up in a blog. The best write up so far has been from namelessurl, especially this remark, which provides an insight into how student's often operate in ways not intended by the teacher but that a well constructed task might alter that line of least resistance:

"There were two ways to work out what values were needed in order to create a shape which could change in size and still keep it's correct dimensions. First was to use trial and error and we had to simply guess each value until we got it correct. The other way was to use mathmetics and actually calculate the values. I mostly used trial and error because i was too lazy to do the maths but in the end i found that using maths i got a much more accurate shape."

turtle art: using heading to change colours

Some of the samples in turtle art such as colors.ta and candyvortex.ta are quite spectacular visually

I want my students to explain how they work, then make some modifications and explain how the modifications worked. Here are some questions I could ask them about candyvortex:

• How does colour work? What numbers represent which colours?
• How does shade work?
• How does heading work? Which numbers represent which directions?
• What do xcor and ycor represent? Describe the co-ordinate system.
• Why does the pen thickness vary in this example?
• What happens if you rotate right rather than left?
• What happens if you vary the forward and back values?
• What happens if you take out shade?

Then I realised this would be too much all at once for some students. So I thought up an introductory exercise rather than trying to do all of the above at once. My introduction just explains one thing clearly, how can the heading can be used to set the colour. Here are the screenshots

Update 14th September:
Tony Forster has pointed out how to display the colour number without the complication of storing variables in boxes. We can do this because the fill screen tile also erases any lines drawn on the screen, whether by the turtle or the show command. The screenshot below illustrates the better method:

Also Tony has demonstrated a clever method of displaying both the colour and the shade effects in the one script. To follow this one you need to know that colours vary between 0-90 in ROYGBIV fashion and that shades vary between 0-100 with 0 darkest, 100 lightest and 50 median. Once again see the screenshot below:

turtle tricks

Armed with my Barry Newell worksheet (40 shapes challenge) I introduced Turtle Art to my students today

Some interesting things from my perspective

Shape 13: the circle
The students used arc to do that one! Shock! That defeats the purpose of the original Seymour Papert logo idea of drawing a circle with only straight lines (such as forward something) are available. I had to laugh. Still, I can set that task as a separate challenge.

Shape 17: six triangle rotated around a point
One student did that using only forwards and turns. I showed him and a couple of others how it could be done using repeats, in fact a repeat inside another repeat. His solution was about 25 tiles, mine was less than 10. So the issue of elegance is one to point out here.

Shape 38: a bunch of squares rotated around the centre of a square
Every time I use this sheet I get some students saying that have done shape 38. Invariably they haven't. Instead they have done lots of repeats with sharp (reflex) angles and it ends up looking something like shape 38. What I eventually did was show them the secret of shapes 6, 7 and 38. Namely, to first draw a square using pen up and pen down so that the turtle finishes in the middle of the square. Once you have done this for one square the more complex shapes are not too hard

Here's an example showing how to do shape 7: