Posts Tagged ‘Angle’

Angle with highlighted vertex
Image via Wikipedia

The main reading for this study block (linked below) is a tricky and detailed account of current teaching relating to angles. It’s main findings are that not enough is done to develop the concept of ‘turn’ with children – that an angle can be defined at the amount of turn from one position to another, and that, if it is taught, the main focus is on right angled turns.

Beebots and roamers could be used in school to investigate the idea of turn, but why not use the school grounds? Create obstacle courses in the playground for children to be directed around – making sure that the turns aren’t always at right angles.

Mitchelmore and White state that children need to experience a wider range of angle concepts. They believe that teacher move too quickly on to the abstract idea of an angle – as shown here.

Their research looked at whether children could represent the movement of objects such as a door, or wheel in terms of diagrams and still understand what was happening – whether they could move from the physical to the abstract in one move.

For instance, the angle shown here could represent the movement of the blades of a pair of scissors, or the opening of a handheld fan, things that children could see happening, and represent in a diagram like this. However, children referred to these movements as ‘opening’, not ‘turning’.

Children were asked to represent the angles using bendy straws to demonstrate the movement and the associated angles. If children could see the angle of movement, and explain what was happening, the researchers were happy.

The researchers also looked at children’s ideas of slope, with the idea that this is an area that is overlooked in schools. I would consider this to be the case simply because of the difficulty of representing it as well as not always being able to see the angle that a hill slopes at, for instance.

…most students had some global concept of slope but that many did not quantify it by relating the sloping line to a fixed reference line. Unlike the wheel and door, where the second line may be suggested by the initial position and there is a global movement which can be copied, there is in fact very little to help a naïve student interpret a slope in terms of a standard angle.

We conjecture that many students have a global conception of slope as a single line and do not conceive it in terms of angles. Had the physical model of the hill consisted simply of a sloping plane without any supporting edges, it is likely that far fewer students would have indicated a standard angle interpretation.

Mitchelmore and White discuss how children find it easier to see the turns, angles and slopes when both elements are easily visible (the scissors, fan, etc.) and this is likely to be because it fits more readily to the idea of an angle as drawn above – something they are likely to encounter in class. They go on to say that, “the fact that the standard angle was used more frequently for the door and hill (where one arm must be constructed) than the wheel (where both arms must be constructed) supports the view that the crucial factor accounting for the rate of use of standard angles is the physical presence of the angle arms… 88% of the students used standard angle modelling when both lines were visible, 55% when only one was visible, and 36% when no line was visible.”

There are three main findings to this piece of research.

  1. That angle work can be related to the everyday concepts of corner, slope and turn.
  2. The fewer arms that are present in a particular angle context, the more that has to be constructed to bring it into relation to other angle contexts and, therefore, the more difficult it is to recognise the standard angle. It is only in exceptional cases that the relevant line has to be discovered. In most cases, it has to be invented through conscious mental activity.
  3. That many children form a standard angle concept early, but that this concept is likely to be limited to situations where both arms of the angle are visible. If the concept is to develop into a general abstract angle concept, children will need more help than is presently given to identify angles in slope, turn and other contexts where one or both arms of the angle are not visible. The slope and turn domains are particularly important for the secondary mathematics curriculum, the former because of the frequency of angles of inclination in trigonometrical applications and the latter because it provides a valuable aid in teaching angle measurement.

Again, the more hands on practice children have at experiencing  the different elements of angle, the stronger their knowledge is likely to be.

The NCETM has a series of tools for analysing how confident you feel about various areas of mathematics. Part of the MaST programme requires me to complete each area over time. I also have to complete the sections for a range of Key Stages – 1, 2 and 3 – to demonstrate a broad knowledge of the subject.

Here are my results for the Understanding Shape/Geometry sections. (1 is not confident and 4 is very confident)

Key Stage 1Understanding Shape

  1. How confident are you that you understand the relationship between angle as a measure of turn?
  2. How confident are you that you can give relevant examples to illustrate the meaning of reflection?
  3. How confident are you that you can give relevant examples to illustrate the meaning of line or reflection symmetry?
  4. How confident are you that you know common side, angle and symmetry properties of polygons?
  5. How confident are you that you know common side, angle and symmetry properties of triangles?
  6. How confident are you that you know common side, angle and symmetry properties of squares and rectangles?

I answered 4 for each of these, giving me an outcome of very confident. I chose 4 for each of the answers as, reading through the examples given, I use the techniques described and go deeper too, being a Key Stage 2 teacher.

Key Stage 2Understanding Shape

  1. How confident are you that you understand through practical activity and the use of ICT the meaning of congruence?
  2. How confident are you that you understand through practical activity and the use of ICT the meaning of translation?
  3. How confident are you that you understand through practical activity and the use of ICT the meaning of reflection?
  4. How confident are you that you understand through practical activity and the use of ICT the meaning of rotation?
  5. How confident are you that you understand through practical activity and the use of ICT the meaning of Reflective (or line) symmetry?
  6. How confident are you that you understand through practical activity and the use of ICT the meaning of rotational symmetry?
  7. How confident are you that you can establish through practical activities the side, angle and symmetry properties of regular polygons?
  8. How confident are you that you can establish through practical activities the side, angle and symmetry properties of equilateral, isosceles, scalene and right-angled triangles?
  9. How confident are you that you can establish through practical activities the side, angle and symmetry properties of squares, oblongs, parallelograms, rhombuses, kites and trapeziums?
  10. How confident are you that you can establish through practical activities the nets of common 3D solids?
  11. How confident are you that you understand the terms right angle, acute angle, obtuse angle, reflex angle?
  12. How confident are you that you can show that the sum of the angles in a triangle is 180° in two different ways?

Again, I answered 4 for each of these questions, giving me an outcome of very confident. I chose 4 because of the ways I have used to teach shape over the years in Years 5 & 6. I use pull up nets to show how the 5 Platonic solids are made, regularly discuss the properties of shapes – especially the range of triangles – with my class. One minor concern was the use of ICT in the first 6 questions, but I consider my SMART Notebook slides to be using ICT and I rarely teach a maths lesson without one.

Key Stage 3Geometry

  1. How confident are you that you are aware of a range of visualisation activities to help pupils to appreciate properties and transformations of shapes? (3)
  2. How confident are you that you understand through practical activity and the use of ICT the meaning of translation? (4)
  3. How confident are you that you understand through practical activity and the use of ICT the meaning of reflection? (4)
  4. How confident are you that you understand through practical activity and the use of ICT the meaning of rotation? (4)
  5. How confident are you that you understand through practical activity and the use of ICT the meaning of enlargement? (4)
  6. How confident are you that you know the meanings of alternate angles, corresponding angles, supplementary angles, complementary angles? (3)
  7. How confident are you that you can prove that the exterior angle of a triangle is equal to the sum of the two interior opposite angles, the sum of the angles in a triangle is 180° and the sum of the exterior angles of any polygon is 360°? (4)
  8. How confident are you that you can prove that the opposite angles of a parallelogram are equal? (3)
  9. How confident are you that you know the conditions for congruent triangles and can prove that the base angles of an isosceles triangle are equal? (3)
  10. How confident are you that you know how to establish through geometrical reasoning the side, angle and diagonal properties of quadrilaterals? (3)
  11. How confident are you that you know how to execute and prove the standard straight−edge and compass constructions? (3)
  12. How confident are you that you know how to describe simple loci? (2)
  13. How confident are you that you know how to explain and prove some circle theorems? (3)
  14. How confident are you that you understand Pythagoras’ theorem and its application to solving mathematical problems? (4)
  15. How confident are you that you can explain the conditions for similar triangles? (4)

This held some trepidation for me, as I haven’t ever really considered the Key Stage 3 curriculum before now for geometry while teaching. My answers are in brackets above and a mainly a mix of 3s and 4s with one 2. This gave me and outcome of confident. The 2 is for the question about simple loci – a choice made because I can’t remember having done any locus work in years! (The locus of a point is its path when it moves according to given rules or conditions. The plural is loci.) I think, having read the examples on the NCETM site, that I could certainly do the work for myself, but would probably struggle to teach it.

Where I chose 3, it is often because I felt I fully understood most of the content but there were areas where I may not have been able to give examples. In question 6, for instance, I would be fine with alternate angles, corresponding angles and complementary angles but may confuse supplementary angles.

Clearly from this, I need to develop my knowledge of some of the Key Stage 3 geometry material.

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