My last post, Michaela’s Knowledge Deficit, is proving very popular and has led to a lot of twitter discussion around the issues of qualifications appropriate to subject specialists and desirable standards for mathematical resources. Upon looking at the incorrect diagrams discussed in that post many people, like me, had a WTF moment while others could not quite place their brains on what it was that troubled them about what they saw. I hope in exposing the incorrect nature of one of these mind-bending, eye-twisting blots that I can help readers to understand the long-term implications of not challenging such incorrect material and thus motivate them to greater concern for the standard of maths currently being delivered by some schools and personnel, perhaps even to your own children.
You may wish to reach for a pencil and a pad that, if necessary, you may doodle through the geometry as we go.
The Alternate Segment Theorem
Please look at this diagram presented on the website of Michaela Community School:
This diagram is intended to demonstrate a basic fact about circles called the Alternate Segment Theorem. It states that the angle at the point of contact between a tangent (a straight line outside a circle touching it at only one point) and a chord (a straight line segment within a circle joining any two points of its circumference) is equal to the angle in the alternate segment.
Now, this theorem can be applied twice in this diagram. First, we can see that ∠cbY, the angle between line segment bY and diameter bc, is 90°, a right angle. By this theorem its alternate angle, ∠bac, will also be 90°. So it is, by your inspection. This is easy, right? (We also know it is a right angle because of Thales Theorem, as featured earlier in these particular materials, by which any angle created at a circle’s circumference by chords anchored at either end of a diameter is a right angle).
Just take a moment to acknowledge your excellent work here. You looked at the diagram according to my description of several of its features and you saw for yourself at least the following:
- That two lines, bc and XY, are in perpendicular relation.
- That this makes the angles between these lines right angles, with 90° each.
- That there is also a right angle, ∠bac, in the triangle.
- That the right angle, ∠bac, is alternate to the right angle, ∠cbY.
- That there is a rule which proves that ∠bac = ∠cbY = 90°.
Key to your success in applying the Alternate Segment Theorem was your act of inspection, your identification in the diagram of certain mathematical features and their relationships. Inspection allowed you to recognise an angle as a right angle and acknowledge that it contains 90°. You will have learned to recognise right angles at Primary school, also learning to recognise their bisections, angles of 45°. Look at the diagram again, can you see an angle of 45°?
There are three 45° angles here but the two which might jump off the screen at you are ∠abX and its complement ∠abc (complementary angles sum to 90°). it is ∠abX to which we can apply the Theorem a second time. Unfortunately, the author of this “example” has incorrectly labelled ∠abX as 71°. To illustrate the significance of this error that you may appreciate its consequences I will show you the triangle which should stand in place of Δabc (triangle abc) if it were true that ∠abX = 71°: The pink triangle below (albeit reversed in the diameter to facilitate comparison) is drawn accurately to the angles attributed to Δabc , 19° at b’, 71° at c’ & 90° at a’.
By insisting that these triangles are identical, the diagram’s author invalidates pupils’ and your first application of the Alternate Segment Theorem, which showed ∠cby = ∠bac, by saying that none of you can be trusted to recognise an obvious angle measure such as 45° and thereby its double, 90°.
The Impossible Triangle
As if none of this were enough, this incorrect diagram also inserts into Δabc supernumerary degrees. You remember learning at Primary school that triangles have only 180°, yes? Hold onto your seat.
You have successfully used inspection to identify Δabc as right-angled but you can also use inspection to identify it as an isosceles triangle, a triangle with two sides of equal length and two angles of equal size. We have four significant cues:
- Chords ab and ac are the same length.
- Chords ab and ac cut the circle’s circumference to create two arcs of equal length.
- Chords ab and ac and these arcs together cut the area of the circle to create two segments of equal area.
- The bisection of ∠bac will give a diameter parallel to tangent segment XY.
This fourth cue may not be immediate for you but the longer you do maths the more you will be able to identify useful features like this. This is one of the reasons we emphasise to pupils the importance of looking closely at diagrams and why responsible maths teachers ensure that our diagrams are as coherent as possible.
As a gestalt these four cues cohere such that, having learned what an isosceles triangle is in Primary school, your working memory is loaded with relevant facts:
- An isosceles triangle has two sides of equal length.
- An isosceles triangle has two angles of equal size.
This happens so quickly and so effectively that it is impossible to see it as any other kind of triangle. Knowing that Δabc is isosceles, compare it again with the pink, make-believe triangle. Are they the same triangle?
No. ΔPinky is right-angled but it is scalene, with sides of different lengths and so has three different angles. It is absurd, then, to be asked to treat Δabc as though it were ΔPinky. If anyone still believes that ∠abX = 71° then I invite them to take their belief literally by applying the Alternate Segment Theorem (the raison d’être for this diagram) to ∠abX = 71°. This gives ∠acb = 71°. Having confirmed by inspection that Δabc is isosceles so it is necessary that ∠abc = ∠acb = ∠abX = 71°. This gives us an angle sum for Δabc of 71° + 71° + 90° = 232°. This is impossible. As we all learned in Primary school, all plane triangles have an angle sum of 180°. The diagram and the label for ∠abX = 71° contradict one another. We must conclude either that this diagram is drawn incorrectly or that it is labelled incorrectly. Either way it is a useless, educationally damaging diagram.
Inspection is of prime importance to the learning of geometry. To expand on an earlier point, the more you enable pupils to see with their eyes, the more they will be able to deduce with their mathematical reason so whereas full diagrammatic accuracy is not compelled, it is vital that diagrams have internal coherence of inspection, arithmetic & algebra and that these accord with learned mathematical facts and topical propositions. Otherwise, why call a thing mathematics?
This principle is especially important to GCSE success where 25-30% of the marks depend explicitly on pupils’ ability to reason mathematically from observed and learned facts. Deliberately thwarting that ability with contradictory information and instructions, with doublethink, not only prevents pupils from solving the problems in front of them by adding unnecessarily to their immediate cognitive load and leading to a lack of learning but also, in the long term, causes them to doubt their ability to reason mathematically and lose confidence. The result is disaffected, disengaged pupils who who are at real risk of failing at GCSE mathematics and thereafter in multiple ways. No amount of discipline, “character education” or assemblies about pop Stoicism can recover this.
Mathematics teachers, subject-qualified or not, for the sake of your pupils please ensure that your diagrams and all of your materials are coherent and correct.
There is one last thing to be addressed here. Several Twitterers have attempted to wave away this and all the other false “examples“ by arguing they are “not drawn to scale” deliberately, to prevent measurement via protractors or guessing. This apology fails on several counts:
- If a teacher does not wish pupils to measure angles with protractors then they can tell pupils that they are not allowed to use protractors. Teachers can’t make 45° = 71° but they can do this.
- If a teacher does not wish pupils to guess then they should tell pupils not to guess. A good teacher knows or will check when and which pupils are guessing.
- Isotropic scaling, by which lengths vary to a constant factor while angles are preserved, is practised in Primary & Secondary maths classrooms across the world but it scales entire objects rather than selected features. A scaling would not affect only two of a triangle’s three sides. Anyone who teaches pupils to scale in this way needs to retake their own GCSE.
No appropriately qualified mathematics teacher is going to put Δabc in front of a class and tell them it has the same angles as ΔPinky, nor can any who has watched such a lesson laud it and these materials without necessarily looking foolish.
Thanks For Reading
If you cannot see the necessity of each geometrical argument outlined here then it’s possible that I might have put them to you badly. You may find that a second reading resolves any confusion but the message I chiefly wish to convey to you does not depend upon your understanding the geometry here: There is as much mathematical power in diagrams as there is in arithmetic and algebra, and their coherence is just as important to teaching and learning mathematics as are correctly composed algebra and efficiently executed arithmetic. Schools do not have pedagogical license to ignore this nor do Secondary maths teachers, irrespective of their qualifications, have license to require their pupils to forget any mathematical facts they learned from their Primary teachers. That is the message.
In my last post I looked at the “knowledge grids“ used in the maths department of Michaela Community School and described their inconsistency with Michaela’s psychological totem, cognitive load theory. For another useful discussion of the disjoint between that theory and these WTF maths diagrams I refer you to a good post by Sue Gerrard, Going Round In Circles.