Euclid? No, but Carol, Yes

Stage: 4
Article by Lyndon Baker

Published July 2000,February 2011.


In the preface to A New Geometry for Schools, parts i and ii (1961), Durrell stated that his aim was:

..."to provide a treatment which lends itself both to class teaching and to individual use by the pupil."

Then, Durell's modus operandi was to "develop each group of geometrical facts by the following successive stages:"

  1. examples for oral discussion
  2. an exercise of (graded) numerical examples
  3. formal proofs of the corresponding theorems
  4. an exercise of readers.

It was the author's conviction that discussion not only facilitated "the learning of formal proofs of theorems" but was "the best method of strengthening the power of the pupil to tackle riders by showing... the types of construction most often required."

The material presented in the book had been faithful to the recommendations of the Second Report on the Teaching of Geometry, which in its day (c. 1955) was considered a milestone of sound educational practice.

Dipping into one "group of geometrical facts" - loci, can be found exercise 46 (Oral)

What loci are described by the following:

4. The top of your head if you slide downstairs on a tea tray?

10. The centre of a marble which rolls about inside a spherical bowl?

15. A variable point P that is due north of a fixed point A?

24. A, B are fixed points; P is a variable point on a given circle, centre A. ABPQ is a parallelogram. Find the locus of Q.

38. A circular cone rolls on a plane. What is the locus of the centre of the base of the cone?

There were plenty of questions to choose from before consideration was given to:

Theorem 29 (1): "Any point on the perpendicular bisector of the line joining two given points is equidistant from the given points."

Theorem 29 (2): "A point which was equidistant from two given points lies on the perpendicular bisector of the straight line joining the given points."

Then, more opportunity for oral discussion followed which in turn laid the foundations for:

Theorem 30: "The perpendicular bisectors of the three sides of a triangle, are concurrent." (i.e. The Circumcentre theorem), and

Theorem 31: "The altitudes of a triangle are concurrent."

(i.e. The Orthocentre theorem)

Following on was exercise 47, and dipping in again can be found:

4.Draw a triangle ABC in which angle A is obtuse. Construct a point P on BA produced such that PB - PC = AB

10. In triangle ABC, AB = AC. If the perpendicular bisector of AB cuts BC, or BC produced, at X, prove that angleAXB = angleBAC.

15. A, B, C, D are four points on the circumference of a circle. Prove that the perpendicular bisectors of AB, AC, AD, BC, BD, CD are concurrent.

24. If H is the orthocentre of triangle ABC, prove that the angles BHC, BAC are equal or supplementary. [ Triangle ABC may be acute angled or obtuse angled.]

Again there were plenty of examples to challenge one's thinking before several more theorems were considered.

Theorem 32(1): "A point which lies on the bisector of a given angle is equidistant from the arms of that angle.", and

Theorem 32(2): "A point which is equidistant from two intersecting straight lines lies on one of the lines which bisect the angles between the given lines."

There followed even more opportunity for oral discussion about bisectors of angles, internal and external and the intersection of loci, before:

Theorem 33: "The internal bisectors of the three angles of a triangle are concurrent."

This chapter was finalised with 35 examples (exercise 48) wherein you were requested to give all possible answers of the required constructions.

Dipping in yet again:

4.ABC is an equilateral triangle of side 4 cm. Construct a point P, 2cm from AC and equidistant from BA and BC

10. Draw a circle, radius 4 cm., and draw a straight line ABCD cutting the circle at B, C such that BC = 4 3 cm. Find the locus of points which lie on the line AD and are always more than 1 cm from the nearest point of the circumference of the given circle.

15. Draw a triangle ABC such that BC = 5 cm, CA = 4 cm AB = 6 cm. Construct the four points which are equidistant from the three sides of the triangle. Construct also the incircle and as much of the three escribed circles as there is room on your paper.

24. Take two points A, B 6 cm apart. A variable point P is such that PA = 2PB. Draw the curve which represents the locus of P. On AB produced take a point C such that BC = 2 cm and add to your figure the circle, centre C, radius 4 cm.

Even from these dippings it is clear what an enormous emphasis there once was on geometry teaching in our schools. But just what were the advantages/disadvantages of giving such a rigorous treatment? How applicable were they for all pupils? In what contexts did these examples flourish?

Nowadays, some pupils use LOGO and even fewer appear to use CABRI type packages, while little is mentioned (perhaps only en passant) about transformation geometry. As such it begs the question of what insight to geometry are we currently giving children. What do other countries do? Do they still need such insights?

By minimising the geometry pupils actually meet what foundation to mathematics is being laid?

Only a this afternoon a colleague lamented that in the whole of last year while visiting secondary schools across the length and breadth of England not a single geometry lesson had been seen. Her own (low level) researches had yielded anecdotes about many pupils who could not use compasses proficiently, who knew little about congruency and even less about similarity, while proof/argument was now always done 'by symmetry'. At a recent Masterclass of some 200+ pupils there was only a handful who could name Pythagoras in answer to "What mathematicians do you know?" On closer questioning no-one (apparently) had heard of Euclid, which is perhaps understandable. Just as when asked to name some female mathematicians the chorus was Carol Vordeman!

The demise of (pure) geometry in our schools is perhaps now complete.

It is believed there was once an aphorism inscribed above Plato's Academy in Athens.

"Let no one ignorant of geometry enter here"

It could have been written above many of the classrooms way back in the 1950's and 60's.

That is, in times BC -

Before Carol!