It Stands to Reason
Solving a mathematical problem sometimes involves using inverse processes intelligently, fluently and flexibly. In Teaching Mathematics At Secondary Level, Tony Gardiner argues (page 13) that in many textbooks and exams such processes “have been neglected, or have been distorted by providing readymade intermediate steppingstones that reduce every inverse problem to a sequence of direct steps”. He gives examples of situations in which pupils fail to recognise that they need to use the inverse of a direct procedure that they know, or what the inverse of that procedure is. For example, he writes (page 61) “A pupil may know how to ‘find 75% of £120’ yet fail to relate this direct operation to inverse variations, such as ‘A price of £90 is raised to £120. What percentage increase is this?’.”
Here we look at some aspects of, and classroom approaches to, developing pupils’ reasoning about undoing mathematical transformations, in the widest sense.
Doing and undoing a onestep action
Ask pupils to give examples of onestep actions or ‘doings’, and to describe what action would be the ‘undoing’ of each one. Examples might include…
doing 

undoing 
open a door 

close the door 
put a cake in a tin 

take the cake out of the tin 
tie a knot 

untie the knot 
Now ask them to think of mathematical examples…
mathematical doing 

mathematical undoing 
add 3 

subtract 3 
rotate 90˚ clockwise 

rotate 90˚ anticlockwise 
factorise 

multiplyout 
Ask for another example, and another, and another… and establish the conventional vocabulary:
‘direct’ (action or process) means ‘the doing’ 
‘inverse’ (of the action or process) means ‘the undoing’ 
Now ask them to think of onestep actions where the ‘undoing’ is the same as the ‘doing’. With some prompting (rotating a book through 180˚ usually generates a collective “oh yes…”) they should come up with, or at least agree with, examples such as…
turn upsidedown 
rotate through 180˚ 
divide into 12 (but not “divide by 12”…the distinction is worth making clear) 
subtract from 10 (but not “subtract 10”) 
reciprocate (a fraction) 
negate 
….i.e. ‘selfinverse’ actions.
Now can they think of ‘doings’ (or actions) that share another special property – ‘doings’ that do not change whatever they act on? Pupils’ mathematical examples should include…
add 0 
multiply by 1 
rotate through 360˚ 
...ie 'identity' actions.
Doing and undoing a twostep action
With twostep actions the question for pupils is:
How do you undo the overall, combined, effect of doing one thing after another?
Pupils will realise that each separate step has to be undone. That is, they will probably be able to say that they need to do the inverse of each step. The challenge is to get them to see for themselves that the inverse of each step must be done in the order that is the reverse of the order of the direct steps – making sure they can distinguish between the meanings of the words ‘reverse’ and ‘inverse’.
Pupils often think that the inverse of ‘AfollowedbyB’ is ‘theinverseofA’ (which, for speed, we’ll now write as A^{−1}) followed by ‘theinverseofB’ (or, more succinctly, B^{−1}). They are likely to make this incorrect generalisation if they start by investigating twostep actions in which the direct action ‘AfollowedbyB’ has the same effect as the direct action ‘BfollowedbyA’; this is because when, and only when, the combination of direct steps (of a twostep action) is commutative, then so is the combination of the inverses of those steps. For example, since the two separate steps, G : enlarge ×2 with centre P and H : rotate 90˚ clockwise about P, of a twostep transformation have the same combined effect whichever single step is done first…
…the inverses of those two particular single steps are also commutative under combination…
Consequently, pupils may conclude incorrectly that (GfollowedbyH) ^{−1} is G^{−1}followedbyH^{−1} because it has the same final effect as the actual inverse, which is H^{−1}followedbyG^{−1}. However, when all four diagrams are looked at together they show that…
the inverse of the whole action, that is (GfollowedbyH) ^{−1} 
is not the sequence of actions G^{−1}followedbyH^{−1} 
but is the sequence of actions H^{−1}followedbyG^{−1} 
and that...
the inverse of the whole action, that is (HfollowedbyG)^{−1} 
is not the sequence of actions H^{−1}followedbyG^{−1} 
but is the sequence of actions G^{−1}followedbyH^{−1} 
Therefore, if pupils are going to generalise correctly for themselves, they need to explore right from the start twostep actions in which the two steps are not commutative when combined. The wider the variety of sequences of visual images representing ‘real life’, geometrical and arithmetical processes they see, the more secure their grasp, and recall, is likely to be.
One strategy is to show them a sequence of images depicting the doing of a twostep action (ensuring that the two steps are not commutative under combination), and then offer two alternative sequences of images representing the undoing of that twostep action. Pupils have to decide which of the two sequences truly represents the undoing, and explain why. Here are three examples:
Example 1
Example 2
Example 3
Next you could give the sequence of images showing the doing, and challenge pupils to create for themselves a sequence of images showing the undoing. For example, what sequence of images would show the undoing of this twostep process?
Image credit
Page header by Brian Hillegas (adapted), some rights reserved
