Why do this problem?
The overall goal of this
is for students to leave with an enhanced
understanding of clear mathematical communication which they can
apply in all other areas of mathematics. It encourages this clarity
of mathematical communication by means of trying to negate
statements. This provides a good way for students to become aware
of the ambiguities present in everyday communication and naturally
reinforces the need for clarity in more advanced mathematics. It is
also conversational, quick and good fun.
This problem works well if approached through group
conversation: there is nothing like asking for a verbal explanation
to make students realise that they do not understand the concepts
or are speaking unclearly. Start with the first statement and
simply ask: Which do students think is the correct negation?
Various points will arise: what, EXACTLY, does negation mean? How
might we tell if a statement has been properly negated? How can we
CLEARLY argue this? Does it matter if the statement is actually
true or false in reality? It is important to follow the resulting
discussion so that students realise the need for clarity and then
attempt to resolve this need for themselves.
The goals of this first part would be:
1) Understand clearly the point that the negation of a
statement P is true if and only if the statement P is false.
2) Understand that we are not asking directly as to the truth
or falsehood of P or any of its constituent parts.
3) Understand that clarity of communication is very important,
and that a
is perhaps the clearest way forward.
Once these points have been raised, try to work on the second
statements and then, finally, the last compound statement.
Throughout, encourage precise, clear mathematical communication:
with each explanation ask the rest of the group: Was that
As a collective follow up activity, students could try to
invent their own sentences with four suggested negations. This is a
really powerful part of the activity, as constructing the questions
will encourage full engagement with the underlying concepts.
At each stage, are you absolutely sure that your arguments are
clear and unambiguous?
Did you think that that was a clear explanation?
The activity involving constructing their own sentences for
negation can be taken to complicated levels. Students with a clear
grasp of logic might try to flip through a text book, find a
theorem and construct the negation of that.
Students who initially struggle with constructing the
negations are likely to be able to spot flaws in the explanations
of others. Initially, such students could be given the role of
'judge' to determine whether a particular argument in favour of a
negation is valid.