Language in mathematics

I am aware that although I am a maths specialist I haven't actually written anything about maths yet. Here is an assignment I did very early on about some misconceptions children in mathematics. Sadly the figures don't want to appear. 

Introduction

Language, and our interpretation of it, is the key to understanding and being understood in mathematics. Definitions change from person to person and can take many forms depending on the situation and how the thinker views the world. Johnston and Mason, (2005, p. 20) remind us that “Meaning depends on context. When a learner says something that seems wrong to the adult, it may be that the learner is stressing something that the adult is ignoring and vice versa.”

How we use language could mean the success or failure of a lesson and it is important to remember that children don’t always see the world as we do. If, when children are learning to define shapes, they see only one example of that shape, that image then becomes that shape. This is called the “prototype phenomenon” (Hershkowitz, 1990, cited in Fujita). Any other orientation of the shape is something else. There is a good example of this phenomenon in the introduction of Johnston and Mason’s, (2005) book, Developing Thinking in Geometry, where a young girl is show a triangle like figure 1.1. She is asked to name the shape, but can’t. Eventually she points to the girl sitting opposite her and says “It’s a triangle for her.” This example clearly shows that children will link imagers to definitions, which as Mooney, C et al (2007) points out, can aid with the meaning, but it is not necessarily a good thing in the long term.

I will look at a series of phrases said by children studying geometry. Each statement is incorrect in some way, so I will attempt to identify what is incorrect and where the problem arose from, be it from skewed definitions or misunderstood concepts. By doing this it will help me to understand some of the misconceptions about geometry that children have and reinforce my own understanding.

Children’s Misconceptions

“This cube’s got spots on all six sides!” 

The child saying this has done well to identify that shape she is observing is a cube, a three dimensional shape made from six squares, and that there are indeed 6 sides. The inaccuracy here is one of terminology. A three dimensional shape has faces instead of sides. It is quite understandable why she will have referred to the cube as having six sides instead of six faces, as in everyday English when referring to three dimensional objects, such as buildings, we would say “around the side of the building” not “around the face of the building.” Even when being taught about shapes at home or in school or nursery, it is very likely that the faces of a cube will have been called sides. This problem links  back to the idea from Mooney, C et al (2007) that definition is learnt through experiencing various examples, and not-examples, which will refine a child’s definition of a word. If a cube is never referred to as having faces then the child will never learn otherwise. As the only objects to commonly have faces instead of sides are clocks and cliffs it is little wonder that many children, and teachers, will say a cube has six sides.

“Symmetry?  Yeh, I know what it means – it’s when it’s exactly the same on both sides.”

When I look at this statement I can understand what the child means and how they came to that conclusion because until now I have never thought of what symmetry actually is. I can see that the statement is wrong but I can’t say why because I don’t know. By drawing several shapes and their lines of symmetry I have come to the conclusion that a line of symmetry splits the shape in two. Either side of the line is exactly half and a reflection of the other. So a line of symmetry can also be called a reflective line of symmetry.

After reading about symmetry in Mathematics Explained by Derek Haylock (2010) I can now see where the child and I went wrong with our understanding. There are in fact two kinds of symmetry, reflective symmetry and rotational symmetry. Reflective symmetry is a phenomenon where a shape reflected on a particular mirror line matches exactly with the other side. Rotational symmetry is where the shape rotated about a point and will fit exactly with the original shape without rotating a full 360 degrees. I think the child is only referring to reflective symmetry, which may be the only type of symmetry known to him, and should have said “it’s an exact reflection/copy on two halves/ sides of a mirror line, of the shape.” That would be a more accurate description of symmetry. It is an easy definition to forget or not understand as it does appear that the shape is “exactly the same on both sides.”

“It’s not a parallelogram, it’s a square.”

The statement is incorrect simply because a square is a parallelogram and there are many possible reasons why the child has had this misconception. According to van Hiele’s model as mentioned by Fujita (no date) this child is most likely to be at “Level 1- Visualisation: Identifying shapes according to their concrete examples.” The child most likely has what Fischbein (1993, cited in Fujita) calls a figural concept, which is a collection of imagers and definitions of a parallelogram, which does not include a square, so anything that does not conform to this image is not a parallelogram. This image comes from the way in which we teach children the names and definitions of shapes. It is highly likely that every time a parallelogram was mentioned the teacher will have pointed to something like figure 1.2 and said “this is a parallelogram.” This may be a good method initially to teach what different shapes look like, as it does follow the definition, it has two pairs of parallel sides. But unless over kinds of parallelograms are shown to also fit that rule the definition is void as a parallelogram will always take the original figural concept.

“[Referring to a pentagon drawn in her book…] you only need to check the angles to know its regular.”

All the interior angles of a regular pentagon are 108º and as the exterior angles of any simple polygon add up to 360º then each of the exterior angles must be 72º. A regular or simple polygon has all the angles equal and all the sides. So just checking angles, although a good indicator, won’t definitely mean that it is a regular pentagon. It is possible to have a pentagon with all angles 108º that is not regular, that is why you also have to check the length of the sides as well. This misconception has likely arisen from misunderstanding when talking about pentagons. If an emphasis is placed on angles of a regular pentagon and very little mention of the side lengths then the angles will be the main focus of the child when identifying regular pentagons. It is easy to overcome this problem by ensuring that children are aware of what makes a regular polygon. All angles must be equal and all sides must be equal.

"They can't be parallel lines cos they're not the same length"

Misconceptions about parallel lines seem to be common place in the classroom, Johnston and Mason, (2005) talked about a similar example where a boy said the lines weren’t parallel because they were not straight (vertical or horizontal). It is likely that this child has formed a figural concept weighted towards imagers rather than definition. When he thinks of parallel lines he will probably recall lines of the same length or specific shapes like squares and rectangles that are made up of straight lines the same length. This may be due to having more emphasis on imagers and not definition when learning about parallel lines. The definition parallel lines states that they never meet no matter how far they extend, so if more emphasis had been placed on definition and varying examples then this misconception would not have happened.