Going around in circles

I’ve often puzzled over why students seem to struggle so much with angles - whether it be comparing the sizes of angles in diagrams drawn to scale, measuring angles or deducing their value. As I develop the angle waypoints for the Framework I’ve been forced into rethinking my own conception of what an angle is and what it can represent. 

I’ve just finished a wonderful paper: Young Students' Concepts of Turning and Angle by Michael Mitchelmore (1998). The paper highlights quite how confusing angles are for our young learners and the variety of situations that we expect them to join together concepts of angle without explicit explanation. We work with an angle as a frame that we could model from wire, an angle as something more solid related to 2D space such as the corner of a shape, the angle view of a camera or spread of water from a sprinkler, and in movement, whether that be rotating an object or turning a corner. Pupils meet angles when discussing symmetry, properties of shapes, loci and constructions and carrying out transformations, to list just a few areas. Angles are sometimes constructed accurately and sometimes not drawn to scale. Even the language is confusing…why don’t left angles exist? The word LEFT even has a 90˚ angle in it – right at the beginning! 

Mitchelmore suggests classifying angles in six ways: 

Unlimited rotations: such as wheels and revolving doors 

Limited rotations: such as the volume dial on your car stereo or a key in a lock 

I-hinges: a linear object which is hinged about one end and can rotate between well -defined limits such as a door 

V-hinges: two linear objects hinged at a common end point, such as the cover of a book 

X-hinges: two linear objects hinged about a midpoint, such as a pair of scissors 

Bends: two line segments with a common end point, where the turn is the result of moving around the corner formed. This one is a bit more difficult to imagine but if might help to imagine a turn in the road or an external angle of a polygon. 

In addition to these categories angles can have a dynamic or a static nature (Magina, S., 1994). Are you looking at the final picture and measuring an angle – or watching the angle being formed? 

In several of the categories above the angle isn’t explicit and additional line segments are needed in order to measure any angle; for example measuring the angle a door handle has rotated through. To measure the angle of turn you would need to identify the centre, a starting position and therefore define both the leg of the angle and the finishing leg (it’s difficult even to find simple vocabulary to explain these concepts clearly). 

Other considerations include: the true representation of these angles; how to label them accurately and communicate precisely what is to be measured; how the degree unit is developed – and why 360˚? Moreover, when do angles remain invariant? For example, lengthening the arms of an angle doesn’t affect its size, and similar ideas can be explored in reflection, rotation, translation and enlargement. 

These are the kinds of consideration that inform this very important area of the Framework. Mitchelmore suggests that the successful study of this area relies on the abstraction theory of conceptual development in the way that Piaget, Dienes, Skemp and others explain. Having spent 13 years in the classroom, it’s something I recognise in my learners. Abstraction begins when students can draw similarities between a class of experiences, and this similarity itself then becomes a new mental object. In other words, by experiencing each of the situations above and identifying similarities students then develop the concept of an angle. Furthermore, Skemp (1986) would say that this new mental object is at a higher level of complexity than each of the initial abstractions. We study the pieces, recognise the pattern and stand back and see the beautiful picture.

^References: 

^{Magina, S.M.P. (1994) Investigating the factors which influence the child’s conception of angle, Institute of Education, University of London.}

^{Mitchelmore, M.C. (1998) 'Young Students’ Concepts of Turning and Angle', Cognition and Instruction 16, 265–284.} 

^{Skemp, R. (1986) The psychology of learning mathematics (2nd ed.), Harmondsworth, England: Penguin}