Diagram Deficit II: Andy’s Fallacies

mass-of-the-sun

Overview

This is written in response to @OldAndrewUK’s polemical apology for bad mathematics at Michaela Community School entitled “The Truth About Calculating Angle Questions“, which itself was written in response to me. You will not find the document on his blog, which he has cultivated for over a decade and to which most people would refer for his views, perhaps ashamed of its content. With good reason.

Inspection

The most significant of @OldAndrew’s fallacies is his bizarre degradation of inspection.

The skill of inspection and proof thereby is ancient. The Classical Greeks called this kind of a proof diknume. You can see an example of such here from The Meno, a Socratic dialogue given by Plato:

The proof shows that the diagonal of any square gives you the square root of the square double its area. You will see that this diagram consists only of lines without any additional notation but once the proof is seen it convicts you. This is a proof by inspection. Try it with your pupils & your friends.

Meanwhile, @OldAndrewUK’s polemic relies upon readers’ unfamiliarity with how mathematics works. He tells you bluntly that at Michaela “there is no skill of “inspection” being taught“, even putting the word inspection in scare quotes as though I had made up the idea off the top of my head. While it is true that what Michaela is doing here is not teaching the skill of inspection this is only so because the materials they have created for the teaching of circle theorems do not support inspection, the diagrams and numbers therein contradicting one another. In fact, inspection is a real thing in mathematics and proof by inspection is used by mathematicians every day. It is even required to be taught by Primary & Secondary teachers, as I shall show shortly.

Circle theorems, the point of this dispute, rely upon inspection. For example, to confirm that the angle at the right hand of the circle below is 90° we need only see that it is created by chords (line segments within a circle connecting any two points on its circumference) which originate at either end of the diameter (the longest chord possible within any circle), this relying upon Thales’ Theorem:

mcs-ri-cotg-ct-asum-3

This is a proof by inspection, the recognition of relationships between features to reason mathematically to conclusions & solutions, and it is vital to the teaching and learning not only of geometry but also of algebra. To give a facile example, most maths teachers will see this factored quadratic (x – 137)(x + 5238) and know its solutions at a glance not because they performed any calculation or manipulation but because they can immediately identify particular features of the expression and their meaning. We teach pupils to recognise these features, to know in advance of their work what solutions factored quadratics will yield, that they will have a means of verifying their work.

Inspection is used & taught at all Key Stages

While inspection is not named literally until KS5, it is referred to in the Aims of the Mathematics Programmes of Study for all Key Stages.

At KS1 & KS2 we read that pupils should learn to:

ks1-2-rm

Our teaching is also specified to develop inspection:

ks1-2-rm-b-2

ks1-2-rm-c

The following is found in KS3 and in KS4:

ks3-4-rm-c

KS5 requires pupils, via its first Overarching Theme, to demonstrate inspection:

ks5-ot

Indeed, examination specifications explicitly invite inspection:

alevel-inspect-1

So, inspection is a formal mathematical skill used by mathematicians ancient and modern which the Department for Education requires to be developed in pupils at all Key Stages and which is currently taught by teachers in every Primary & Secondary school. @OldAndrewUK’s defence of Michaela’s incorrect materials by degradation of the traditional & mandated skill of inspection fails by reference to accepted and required mathematical practice. We could leave this here but several other of his fallacies, which I partly anticipated in my last blog post, are worth addressing directly.

The Argument from Lemmings

That Pearson or anyone else produce bad diagrams does not excuse others teaching via bad diagrams. Pearson jumped off the cliff of bad diagrams so @AndrewOldUK thinks we are all justified in jumping off the same cliff.

Form an orderly queue, everyone.

The Argument from Notation

Diagram notation is not necessary for users to recognise their features. Inspection of the diagram below proves that the angle on the right has 90° by reference to Thales’ Theorem even without right-angle notation:

mcs-ri-cotg-ct-asum-3

Although we have seen via the Meno that notation has been unnecessary since the earliest mathematicians used diagrams, @OldAndrewUK would have pupils taught that because there is no right-angle notation in the above diagram, then they cannot trust that it contains a right angle despite theorem:

ao-little-square-b

ao-little-square-c

This is completely incorrect and a recipe for mathematical failure. Let’s take a moment to appreciate what @OldAndrewUK is saying here, that neither you nor your pupils know a square when you see one, that only the heavily annotated square on the right may be called or treated as a square:

square-ra-2square-ra-1c

Frankly, I don’t care how long a person has blogged or how many Twitter followers they have – if they think this should be taught to our pupils then they need to retake their own GCSE. While It is true that conventional notation is useful, in examination and in everyday maths it is only ever placed appropriately, where it is necessary to a solution or to a proof. Where unnecessary, as here, it is omitted.

The Argument from Scaling

@OldAndrewUK knows, or should know, that the only scaling taught in KS3 & KS4 preserves angles and would not transform only two sides of a triangle leaving the third, here the hypotenuse along the circle’s diameter, untouched:

mcs-ri-cotg-ct-asum-3

The Argument from Inaccuracy

A variation on the last argument, by which @OldAndrewUK wishes you to believe an angle drawn with a discrepancy of 26° is no more significant an error than an angle drawn with one of 2° or 3° degrees. In the Michaela diagram below I have drawn ΔPinky to the angles attributed to Δabc:

asum4-comparison-b

By all mathematical convention, to which @OldAndrewUK appeals when it suits him, his affirmation that 45° = 71° produces an impossible triangle of 232°  yet he believes with all his heart that Δabc is a perfectly acceptable representation of ΔPinky. Michaela’s head of maths justifies this deliberate error pragmatically as a time-saver:

mcs-dq-save-time-2

Earlier they expressed disquiet at what they believed was their pupils’ inability to generalise:

mcs-dq-uikb-tmmtmm

It should be clear to anyone that teaching pupils all triangles may be represented as the same triangle is taking generalisation absurdly far, to the distant shores of misconception in fact. Also, it is a drastic swerve to the antipode of Michaela’s previous position, that pupils should be presented with as many variations as possible:

mcs-dq-uikb-tmmtmm-question

This is also problematic, as I explained in a previous post. Now, note above the reference to inspection of diagram features. This contradicts OA’s position, given first in his title, that the purpose of these diagrams is simple calculation. These materials, as reference to their original source shows, were created primarily to support the application of theorems by inspection of diagrams:

mcs-ri-cotg-intention-b

The Argument from Mathematical Sadism

Many people reading this, or perhaps their children, will have sat in a mathematics lesson and felt entirely lost. A bad maths teacher can turn this from a temporary confusion to a lifelong memory of humiliation at the hands of a capricious bully. This kind of teacher, rare though they may be, are so disproportionately horrid that they have become almost the archetype of the maths teacher. Given this, why would any maths teacher think it good practice to “mislead” and “catch out” children who are giving answers in good faith and to the best of their knowledge?

ao-catch-out-b

It is one thing to highlight common misconceptions, it is quite another to lead pupils into the trap of deliberate misconception when they are simply doing as they have been taught. Such ill-advised practice only confuses pupils already insecure in their understanding. Misconceptions should be taught by direct instruction, not by deceit – children are not toys for the amusement of their teacher.

The Argument Ad Hominem

While he does not use my name in his peculiar document many people reading this will know that @OldAndrewUK is traducing me in particular when he speaks of “trolls“. He has blocked me on Twitter but merrily & frequently impugns my sanity to his followers. You may not think this definitively identifies him as someone lacking confidence in his ability to argue with me directly but it is an excellently reliable proxy. His misbehaviour says nothing about me.

Thanks for reading.

If you would like further advice on any of the mathematics here then please don’t hesitate to comment. If you do not believe what I have said here because it is me who said it then please refer to the links under the section “Inspection is used & taught at all Key Stages” and decide for yourself whether or not your pupils will benefit from being taught mathematics “The Michaela Way” or the way we all traditionally teach it and examine it in schools in every village, town and city in the UK. The first will certainly save time planning, as will the abrogation of any necessary duty, but it is only the second which will ensure your pupils learn mathematics.

Peace.

diagrams-hilbert

ALL TEACHING MATERIALS USED IN THIS ARTICLE ARE PUBLICLY AVAILABLE AND COMMENT REPRESENTS LEGITIMATE ANALYSIS & REVIEW. IF ANY INFORMATION HERE IS FACTUALLY INCORRECT THEN PLEASE COMMENT AND I SHALL ATTEND TO ANY ERRORS. THANK YOU.
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