I apologise to any of my previous students who I messed up by saying that the perimeter is "... the distance around the outside of a shape." I meant well, I really did.
But I lied. The perimeter is most certainly not the distance around the outside of a shape.
I realised this error by looking at students' responses to a seemingly straightforward length question:
This is the type of response we were expecting:
Notice that in the left hand path, the student has got 15 in two different ways (the 'outside' path and the 'inside' path) but that this is by luck: the student counted squares that are not involved in the length. Other students made similar mistakes:
There is a definite theme here: counting squares as opposed to line segments.
This response is very telling:
The student has got the right answers (15 and 17) but for the wrong reasons. And if they hadn't have shown how they got the answer (by writing the numbers in the squares) I would have been none the wiser for it. I probably would have thought that they 'got' it. Again, mea culpa.
So how did this misconception arise?
Probably as a result of the 'count the distance around the outside' rule that has been impressed upon the students. Or maybe students' initial concrete experiences with measuring length involved using, say, square tiles to measure a pencil and so they now think 'squares' are used to measure length. As a result of this, I would certainly get students to use popsicle sticks, string, etc to measure length (i.e. objects that are more blatantly 'length' rather than 'area'). I also now refer to the perimeter of a shape as the sum of all its sides.
So if you have a student who thinks that the perimeter of this shape is 18...
... then show them how highlighting the sides forces us to find the sum of the sides of the rectangle: