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In a companion article Logic, we state the definition of logic as the science of reasoning, proof, thinking or inference (according to the Oxford Compact English Dictionary). It is the ability to reason that is central to logical thinking. For many of us, these reasoning skills are often put to the test during arguments. Being able to reason is clearly a valuable skill! But is it something that we should be "teaching"? Do children learn how to form logical arguments within the home?

Clotilde Pontecorvo and Laura Sterponi conducted research to investigate "how young Italian children are socialized to argumentative discourse" which they summarise in the book "Learning for Life in the 21st Century". They approached argumentative discussion as being ways of reasoning used during different speech activities within a range of contexts. The two contexts they chose were a narrative activity in a pre-school (the children were all between 3 and 5 years old) and family dinner conversations.

In the school setting, the narrative was always jointly constructed so that the children did not accept each other's opposing views but used them to reformulate ideas. The discussion followed complex patterns involving the use of hypothetical statements with possible negative or counter-factual consequences. For example, in the story read by the children, a girl had run away. A discussion ensued about her age. One boy suggested that she could not be too young (hypothetical statement ) because then she would not have been clever enough to run away (counter-factual consequence ).

At the dinner table, the child again works collaboratively to produce narrative. However in this situation, the roles of the participants change and this requires the use of more complex cognitive processes. This is as a result of the nature of relationships within a family. The more familiar the people around you, the more risks you are prepared to take in expressing opinion. Often in the home, argumentative discourse is linked with violation of rules. This produces a pattern of conditional statements with negative consequences with which children become familiar. For example, in the study the mother warns her 3 year old daughter that she should not fall asleep late (conditional statement ) because when she had done so on a previous occasion it had made her feel sick (negative consequence ).

Pontecorvo and Sterponi suggest that these two structures of discussion (one taking place at home, one at school), are in fact very similar. Hence, before they attend school, children will have already experienced complicated reasoning patterns.

So, how does this affect us as teachers? When children engage in narrative as part of a group, their contrasting views lead to a high level of revision and improvement, and through this process they become more aware of "thinking". Providing opportunities for this kind of narrative in our classrooms is vital but just as important is the way we handle them. The teacher should try to be responsive to the children's contributions, perhaps by mirroring them, while facilitating the "multi-voicedness". If we can build relationships with the children that foster familiarity and ease, at the same time as encouraging this kind of interaction amongst the pupils themselves, then the quality of reasoning will be enhanced.

Logical thinking within the Maths lesson

There is no doubt that being able to think logically is a cornerstone of mathematics. Is there anything that we can do to encourage and develop this skill in a mathematical context?

Anne Watson and John Mason describe their view of mathematics as one which is based on structures of pure mathematics and mathematical thinking. Within any topic in maths there are many different types of statement that can be made. The statements relating to a particular topic could be termed its structure.

Table 1: Mathematical Statements

Definitions Facts Properties Theorems
Examples Counter-examples Techniques Instructions
Conjectures Problems Reprentation Notations
Symbolisation Explanations Justifications Proofs
Reasoning Links Relationships Connections


Watson and Mason propose that there are also many different branches of mathematical thinking which they grouped for convenience:

Table 2: Mathematical Thinking
 
 

exemplifying completing comparing changing generalising explaining
specialising deleting sorting varying conjecturing justifying
  correcting organising reversing   verifying
      altering   convincing
          refuting

Their philosophy is that pupils can achieve higher order mathematical thinking if they are focused by the teacher's appropriate use of questions and prompts. They suggest that questions to promote these six areas of mathematical thinking could be asked in relation to all the mathematical statements in the first table. Looking at Table 2 above, we can see that the modes of thinking most closely related to logic and reasoning are contained in the last column. Mason and Watson give many examples of the sort of question a teacher could use to develop these particular thinking processes. In the general examples below, each question relates to a different statement in Table 1:
  • Is there anything else which is not an X which is described by this?
  • Why is ... an example of ...?
  • How can we be sure that ...?
I cannot possibly go into all the detail of Mason and Watson's research here, but suffice to say it is a fantastic book which is of immense practical help in the classroom. They propose three different ways of using the questions to stimulate mathematical thinking:
  1. Take a topic and use some specific questions within certain statements and certain groups of mathematical thinking.
  2. Take a mathematical process from Table 2 e.g. "explaining" and try to find similar examples in different topics to help you make links between topics.
  3. Take a certain mathematical statement from Table 1 and look for similar questions in different topics, again helping you to make connections but also to understand how that statement differs within topics.
The book refers to several examples of the above to help you get a feel for what you may be aiming for.

Of course logic is needed in many mathematical problems and the whole of NRICH Prime this month contains such puzzles. There are also several famous conundrums of this type which you may like to tackle with your class. Here are a select few:

The Tower of Hanoi

A legend states that there was once a monastery in Hanoi that had 3 needles. One held 64 different sized discs which were arranged in order of size, with the largest at the bottom. God ordered the monks to move all the discs to another needle so that they ended up in the same order. To do this they were allowed to use all the needles but a larger disc could not be put on top of a smaller one. The legend had it that when they moved the last disc the world would end.

How would this be done? Here is a simpler version which has just 3 discs:
Three towers, the first with three rings on (largest lowest, smallest highest) and the other two towers empty

Try to move all three discs from this starting position obeying the rules above.

For an interactive version of this puzzle, visit
www.cut-the-knot.com/recurrence/hanoi.shtml

The Journey across the Stream

A fox, a hen and a bag of grain need to get to the other side of a stream. The fox and the hen can't swim. A man with a boat can take them but the boat can only hold one thing as well as him. However, the fox will eat the hen if they are left on the bank together, or if they travel in the boat at the same time. Similarly, the hen will eat the grain if she is left with it or if it travels with her in the boat. How can the man carry all three safely to the other side of the stream?

Mazes

Logical thinking can help find the quickest way through mazes, which have fascinated people for centuries. This article contains more information and there are many maze problems in the archive. Try July 01 and Sep 01.

And if that's not enough

The following website has a whole page that is devoted to maths logic puzzles.
Go to www.fi.edu/sin/school/tfi/spring96/puzzles/index.html .

Don't forget Lewis Carroll's stories of Alice and his wonderful use of logic. Read this month's sister article for more details.

We hope to have given you some guidance and inspiration on the development of logical thinking in your classroom. This time of year would be perfect for trying out the above activities and practising your questioning techniques. Good luck.

References

Watson, A. and Mason, J. (1998). Questions and Prompts for Mathematical Thinking. Derby: Association of Teachers of Mathematics.
Wells, G. and Claxton, G (eds) (2002). Learning for Life in the 21st Century. Oxford: Blackwell Publishing.