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Secondary Magazine - Issue 129: It Stands to Reason


Created on 17 December 2015 by ncetm_administrator
Updated on 02 February 2016 by ncetm_administrator

 

Secondary Magazine Issue 129'Close up of The Thinker' by Brian Hillegas
 (adapted), some rights reserved
 

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 ready-made intermediate stepping-stones 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 one-step action

Ask pupils to give examples of one-step 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   un-tie 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   multiply-out

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 one-step 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 upside-down
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. ‘self-inverse’ 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 two-step action

With two-step 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 ‘A-followed-by-B’ is ‘the-inverse-of-A’ (which, for speed, we’ll now write as A−1) followed by ‘the-inverse-of-B’ (or, more succinctly, B−1). They are likely to make this incorrect generalisation if they start by investigating two-step actions in which the direct action ‘A-followed-by-B’ has the same effect as the direct action ‘B-followed-by-A’; this is because when, and only when, the combination of direct steps (of a two-step 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 two-step 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 (G-followed-by-H) −1 is G−1-followed-by-H−1 because it has the same final effect as the actual inverse, which is H−1-followed-by-G−1. However, when all four diagrams are looked at together they show that…

the inverse of the whole action, that is (G-followed-by-H) −1
     is not the sequence of actions G−1-followed-by-H−1
     but is the sequence of actions H−1-followed-by-G−1

and that...

the inverse of the whole action, that is (H-followed-by-G)−1
     is not the sequence of actions H−1-followed-by-G−1
     but is the sequence of actions G−1-followed-by-H−1

Therefore, if pupils are going to generalise correctly for themselves, they need to explore right from the start two-step 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 two-step action (ensuring that the two steps are not commutative under combination), and then offer two alternative sequences of images representing the undoing of that two-step 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 two-step process?

Image credit
Page header by Brian Hillegas (adapted), some rights reserved

 
 

 

 
 
 
 
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Comments

 


08 February 2016 09:48
Being someone who values problem solving which, of course, involves reasoning I was heartened to read the above: ""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 ready-made intermediate stepping-stones that reduce every inverse problem to a sequence of direct steps”." I find there is a disjoint between the use of textbooks and the effective, and affective, learning of mathematics. Similarly between exams and the impact this can have upon effective learning of mathematics. Thus causing learners to reason why, for example, 17 - 9 = 8, or why the roots of y = 5 - x^2 are +root 5 and -root 5 are more valuable learning processes than merely doing lots of similar questions which require answers only but do not require the reasoning to accompany the 'answers'.
By mikeollerton
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01 February 2016 13:29
Really helpful article. I have always 'followed' Tony's idea of the importance of Doing and Undoing ... I like example 3 as this is the type of thing we can do with Cuisenaire Rods! Also transformations is also a rich topic to explore. Thank you for these ideas!
By HeatherNorth
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