Q&A #315

Value of two-column proofs

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From: Mary Lou (for Teacher2Teacher Service)
Date: Mar 17, 1998 at 20:56:18
Subject: Re: Value of two-column proofs

When I speak to adults, they tell me that they learned how to think doing 
two-column proofs. I think what they really mean is that it was in geometry 
that they learned how to put together an argument to establish something.

There are all types of approaches to two-column proofs. Some teachers expect 
the student to do the proof the same way that the teacher would do the proof.  
It is rare that two people think exactly alike, especially when there is more 
than one fact to be established. Some teachers expect the student to do the 
proof in the most efficient way possible. Students do not always think of the 
most efficient way the first time they approach a problem. Other teachers 
believe that a two-column proof is the only way to prove something.

Now that I have told you what some teachers will tell you, I will tell you 
that I think that two-column proofs are a good way for some students to 
organize their work. However, when I am the teacher I teach students to do 
proofs in a flow chart manner so that they can see what is a consequence of 
knowing something. Once they establish two or more facts they can see the 
consequence of their thinking. In this way they will see that different 
"flows" can be rearranged but that the final consequence has to come at the 

This methodology establishes good thinking as well. Accompanying a flow chart 
is a listing of reasons that for a two-column proof would be the righthand

When doing proofs, work hard on them, do them every day, and collect the work 
from the students. If their work is not correct, have them do it over again 
until it is correct. Do not expect that the average student can do many of 
them on a daily basis. Pick and choose. Once a student shows you that he or 
she can do a proof, assign fewer but more difficult proofs. However, do not 
spend more than four weeks doing them. By that time the students who will 
learn from them will have learned, and the other students who have not 
learned will probably need more work one-on-one with you, the teacher.

What material were you considering covering when you thought of cutting out 

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