Abuse of the implication symbol

Posted by Kevin Houston on Mar 11, 2019 in How To Think Like A Mathematician, Mathematical thinking, Mathematics education | 0 comments

By the time students come to me many have picked up a bad habit. They use the implication symbol, , incorrectly. This is not thought of as a symbol to use for (i.e., statement is true implies that statement is true) but more as “this is what we do next”. This is understandable as most proofs students have seen will be one deduction after another. But it does mean that I see lots of examples where is used at the start of a sentence.

Steps need to be taken to overcome this. The one I use at the moment is to just tell them at the start of the introductory course that they need to be careful and shouldn’t use the symbol until we’ve studied it in the course. (This does violate my “Show, don’t tell” rule but it seems to work quite well.) There is an air of mystery about why and I think that makes them pay attention. After a few lectures we start studying it and the period of abstinence in using it gives a fresh start.

I begin by observing that the symbol is rarely used in written mathematics. Almost any textbook will do to show this. For me the important thing is to get across the idea that we are connecting two statements. To do this I use everyday examples: “If I am Winston Churchill, then I am English” can be written as “I am Winston Churchill” “I am English”. With that sort of example students can see that using “ I am English” would not make sense and hence we can’t use $\latex implies $ at the start of a sentence.

My usual approach to implications is: statement, converse statement and contrapositive statement. I shall discuss the latter tomorrow.

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