Last year, I figured out about a week or two after the first introduction to proofs in Geometry last year that I should have started with a more clear connection to the ideas we had been working on in the classes before. We did a progression of logical statements, conditional statements, working on biconditionals as definitions, and then the laws of detachment and syllogism. I realized then that I never made strong references to these concepts and how they all fit together – I just hoped that the students would see how the proofs were built out of these ideas without formally telling them as such.
This year I was much more explicit in how the ideas fit together, particularly by showing a paragraph proof as a series of conditional statements with true hypotheses. I was really happy with the results. I created two videos to use as part of the instruction:
Students watched the video and then worked on identifying the properties of equality and congruence being applied in a few different situations, and completing statements given that a particular property is being applied. This led to some great conversations about subtle differences between the transitive property of equality and the substitution property of equality. (‘If a = 5 and b = 5, then a = b’ is the latter, not the former. This assumes only one property is being applied at a time, of course.)
Once I was satisfied with their progress, I sent them on to watch this video:
Some students immediately took the equation, solved it, and said they were done. This led to more good discussions about the purpose of this lesson. We already knew how to solve equations – what was new this time was justifying each step using a property. It was an opportunity to push these students to then focus on what was happening in each step and not so much on the algebra that they did quickly. Once I was convinced that they did understand what was going on in each step of the video, I had them move to either some problems with the steps written out, missing only the justifications (for weaker students), and for others, full algebraic problems that they do from start to finish.
The thing I did differently here (and which was made easier by the magic of video) is emphasizing that the different steps in a proof are really all either conditional statements, statements of fact (the given information), and possibly steps of arithmetic simplification. Each line should be connected to the previous one in the form of a conditional statement. I have said it in the past, but never explicitly written it out each time so that the students think of it this way.
The whole point of doing things this way is so that students are not introduced to two concepts simultaneously: writing steps in a proof, and proving a statement deductively from scratch. Having a good sense of algebra, this lesson focused on introducing students to the process first. Next time we will move on to actually finding and proving theorems about line segments, with the idea that they already have a basic sense for how different thoughts can be linked together logically in a proof. I am hoping that being this deliberate will pay off – this was definitely the smoothest this lesson has ever gone for me.