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."