Why do this problem?
The overall goal of this activity
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 truth table 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 explanation clear?
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.