Pulling on Loose
Threads
Oliver Thomson
MA Conference, Oxford, April 2016
This session
• I will talk for a bit
• Provoke thinking
• Your chance to discuss
• Share ideas
Pulling the thread
• Simplify:
𝑥 2 −𝑦 2
𝑥−𝑦
=𝑥−𝑦
• “A careless mistake with signs”
• Year 12 revision for C1/2
• Pulling the thread…
• Explain what you did
• What do you mean by cancelling?
• When can you cancel, what can you cancel?
• Give an example of how this works for numbers?
• Why does it work?
Method: factorise, then cancel
• Factorise
• What does it mean? How do we define it?
• “Opposite of expanding brackets”
• Do we call these factorising?
• 12 = 6 × 2
• 12 = 22 × 3
• Why do we have to factorise before we cancel?
Method: factorise, then cancel
• Cancelling
• How do we explain this?
• What does the fraction line “mean”?
• What does cancelling tell us about the relationship between
multiplication and division?
• Do/should we make this explicit? With which students?
• What else could we cancel?
• 𝑥+3−3
2
•
5
• sin−1 sin 𝑥
2 methods
• I will show two methods, and compare them
• To illustrate my points
• You may agree or disagree!
• Only 2 methods compared
• Both relate to the symbolic manipulation of fractions
• Clearly there are other methods and representations that you
might use as a teacher
Simplifying fractions: method 1
÷4
4
1
=
8
2
÷4
• (Nearly) ubiquitous
• A limiting method
• Tougher examples (which arrive quickly) require you first to find a
factor (or, better, hcf)
• Arbitrary: why can you multiply/divide, but not add/subtract?
• Does not (necessarily) lead to deeper understanding
Simplifying fractions: method 2
4 1×4 1 4 1
1
=
= × = ×1=
8 2×4 2 4 2
2
15
3×5
3
=
=
70 2 × 5 × 7 14
• Conceptual understanding
• Highlights the need to factorise first
• Rules of arithmetic are reinforced
• Rearranging (commutative, associative)
• Identity and inverse under multiplication
• Link between multiplication and division
• Embedding of principles of what is a fraction
Simplifying fractions: method 2
4 1×4 1 4 1
1
=
= × = ×1=
8 2×4 2 4 2
2
15
3×5
3
=
=
70 2 × 5 × 7 14
• Enabling method
• Move to algebraic fractions and more complicated terms
• Links to different areas
• Functions/operations and their inverses
• Multiplication of negative numbers (through factorisation and
rearranging)
Would you ever use method 1?
• “Simpler” (for which examples?)
• “They will never need to simplify anything more complicated”
(never?)
• “They will get it” (do you mean be able to use it?)
• “Similar to method for proportion/ratios” (do we want this?)
• “Their understanding of factors is not really strong enough for
method 2 yet” (!!)
• “Time constraints” (all the more reason to use method 2?)
What leads up to this?
• Fluency with integers: lots of practice
•
•
•
•
•
•
12 × 25
13 × 7 − 3 × 7
7×8 + 6×4
6 × 25 = 3 × ___
14 × 15
13 × 10 ÷ 5
• This is also for extension and interest
• N.B. examples above from Extension Mathematics: Book 𝛼, by Tony
Gardiner
• He suggests these books for “top 25%” of the cohort, but I think many of the
exercises are useful for many more students!
What am I saying?
• Perception of students: maths is a list of arbitrary rules and methods to
learn and apply
• It is our job to show them the simplicity and beauty!
• Teaching approaches matter; methods matter
• Progression to the next stage
• Enabling vs. limiting
• Generate deeper understanding
• Highlighting fundamental concepts, or underlying structure
• Inspiring students
• Which method to favour
• The best one! Most appropriate?
• How many methods should students see?
• As many as possible? If they are understood (and limitations of each)
• Methods vs. representations
• Risk of confusion
• Scaffolding has to be removed (by definition)
Over to you
• Have you ever pulled on a thread and unravelled a lot?
• What examples can you think of that underpin further
development?
• What are the methods/approaches?
• What are their relative benefits?
• Which are enabling/limiting?
• Which are useful scaffolding?
• At what stage will it have to be removed?
• Which is “best”?
• Then we will share some ideas