It Stands to Reason
An important aim in teaching mathematics is to help pupils develop some specific strategies to use when they are challenged to solve problems that are ‘new’ to them (when they can’t just follow a sequence of steps that they’ve used time-and-time-again to ‘solve’ previously-frequently-met, exactly-similar, problems). To provide ‘space’ in pupils’ minds so that they can concentrate on problem-solving strategies, rather than on mathematical techniques, you can present interesting tasks that involve relatively simple techniques that most pupils will have mastered. Tasks in which pupils discover simple number patterns are ideal for this purpose.
Here we present one example task. It is demanding enough for teachers to enjoy working-on-it-together prior to challenging pupils with a wide variety of different number-pattern tasks from a source that we link to below.
In-shuffles: explore and explain
A possible introduction
This single in-shuffle of EIGHT items …
… changes the order from 1-2-3-4-5-6-7-8 to 1-5-2-6-3-7-4-8. It is called an in-shuffle because the items that are first and last do not change.
This is what happens if we continue to in-shuffle only SIX items in the same way (if we keep on repeating the same procedure) …
The original order is restored after four in-shuffles!
If we continued to in-shuffle EIGHT items would we also eventually restore the original order, and if so, after how many in-shuffles?
There are many patterns and regularities to notice and to try to explain. For example, with 14 items …
… the first six in-shuffles reverse the order of all the items except the first and last. Does this suggest a conjecture about the number of in-shuffles that will restore the original order?
Also, the sum of the two central items is always 15! Why are the two central items always different colours?
Is the following conjecture generally true?
When the number of items is doubled, the number of in-shuffles to restore the original order increases by 1
Ways of working
Your role should be one in which you place less emphasis on detailed explanation and on knowing answers, and more emphasis on encouragement and transferable strategic guidance. So you might ask “How can you make it simpler?” rather than “Have you looked at 1, then 2, then 3?”.
Pupils will benefit from plenty of opportunities to discuss, in pairs, in groups and as-a-whole-class, what they notice, and strategies that work for them. Encourage them to reflect on, and explain-through-talk-and-in-writing, their approaches to problems and their discoveries.
It is easy to underestimate how difficult it is for many pupils to express patterns directly using conventional algebra. It helps them to …
- first SAY their ‘rule’ in ordinary language
- then WRITE it in ordinary language
- then EXPRESS it using their own choice of symbols and shorthand
- then, and only then, try to express it conventionally and algebraically.
If pupils refer to lists of ‘strategic hints’, as is suggested in the material described below, it is extremely important that such hints are NOT regarded, or interpreted by anyone, as lists of questions for pupils to answer! Pupils should understand that they are there to help ONLY IF pupils get stuck! Some pupils may be able to solve problems without ever needing to look at any hints-from-a-given-list!
Encourage pupils to think of explanations and proofs as efforts to help others ‘to see what you are seeing in the way that you are seeing it’ (John Mason, QCA Conference, 2001).
This article constitutes a brief introduction to the considerable and valuable advice (together with a very large bank of exciting classroom tasks) that you can draw on in the free-to-download resource, Problems with Patterns and Numbers, available at the Shell Centre for Mathematical Education Publications Ltd.
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