A conversation was taking place over at Dan Meyer’s Blog (http://blog.mrmeyer.com/?p=17964) about proofs, which is a topic that I find myself faced with about this time every year.
This isn’t a new conundrum for me. I’ve been working for while now trying to make this idea of proof, which, when compared to the typical form of textbook Algebra I should be an easier sell. But it just isn’t.
Here are some discussions of my previous attempts to sell it. Posts from Nov 2, 2012, Nov 16, 2012, Dec 7. 2012 are a few examples of my thoughts from around a year ago when geometry hit this place last year.
The problem I have is that the academic norms seem to prefer deductive reasoning to inductive and use of the theorem names. Those two things seem important to decide on before starting the journey of proof. If you are going to prefer deductive measurements, it rules out using measurements in proofs and it requires a lot more formal geometric language.
The problem that I see is that to rule out measurements (at least from the very beginning) and to strongly increase the formal geometric language in a way that makes deductive proofs possible from the very introduction of proofs creates… well… what Christopher Danielson is quoted as saying in Meyer’s post… “one of the most lifeless topics in all of mathematics.”
In order to breathe life into the topic, from the experience I’ve had, you need to let students engage in ways that make sense to them at first. The target to start the process is simply to get them comfortable with the idea of designing a functional persuasive argument about a mathematical situation. This requires recognizing that they need to start with a clearly stated claim (preferably something that is provable) and then start supporting it.
I find it helpful to let them pull measurements from pictures first and use those in the proof. The idea of comparing two things by length and NOT measuring them to get the length seems to a lot of kids like we are making the math difficult simply because we want it to be difficult. If they sense there is an easier way to solve a problem, then the explanation for why that method is against the rules had better be very strong, or else buy-in is going to suffer some pretty heavy causalities.
Once they get the hang of making an argument, then we can start by having discussions about what kinds of evidence are more compelling than others. This is usually where the students can figure out for themselves that each piece of information needs its own bit of mathematical support.
Next we can start deliberately exposing the students to different ways of proving similar situations. Triangle congruence seems to be a popular choice. We can have conversations about proving a rigid motion or proving pairs of sides and angles. Eventually certain kinds of explanations become more and more cumbersome. For example, using definition of congruent triangles to prove that two triangles are congruent as shown here:
Do we really need to keep going to find the three pairs of congruent angles?
Then, we can start pushing into shortcut methods. Mostly because those angles are going to be somewhat tricky to find (and why do more work than you need to… the students DEFINITELY identify with that.)
By using this method, I am trying to create what I’ve heard Meyer call “an intellectual need” for additional methods to prove this claim. (Keyword: trying… not sure how successful it is, but I’m trying.)
Then, that transitions fairly smoothly into stuff like this:
… where we standardize the side lengths of two different triangles and see how many different triangles we can make and in what ways they are different.
Now, the tougher question is whether or not you allow the class consensus following the “Straw Triangle Activity” (which was a gem that came out of Holt Geometry, Chapter 3) to count as proof of the SSS theorem. In an academic sense, now we should “formally prove” SSS theorem. To most of the students, it’s settled. Three sides paired up means the triangles are congruent. What are we risking by avoiding the formal SSS proof? Do we risk giving the impression that straws and string are formal mathematical tools? But wait… aren’t they? What do we risk by doing the formal SSS proof? Do we risk our precious classroom energy by running them through an exercise there isn’t a lot of authentic need for right now?
Am I able to say that this is the definite recipe for breathing life into geometry proofs? Not even close. I am sure there are students who are completely uninspired by this. I can say using anecdotal evidence that engagement seems significantly and satisfyingly higher then when we used to run deductive two-column proofs at students from the very beginning.
But, we’ll have to see what the consequences are as we keep going.
Mr. Shauver - Learner Educator 
We know that, at least in the US, geometry proofs remain a fleeting topic usually visited some time during the course of a single school year, typically a year 10 geometry course, yet when less than 1/2 of 1% of American seniors can complete this rudimentary proof (from the NAEP):
https://drive.google.com/file/d/0B6lw97EHbvfHYkVfRGF5bU14YzA/edit
we know the current arrangement isn’t successful. Common Core, to its detriment, maintains the status quo, so we’ll pipe up again and submit that geometry proofs can crafted around primary school topics, such as “the area of a triangle” and “subtraction”, and therefore can (and should) make their entry into the curriculum beginning in primary school:
http://fivetriangles.blogspot.com/2013/09/93-first-proof.html
We’ve been asked (by whom, we won’t say) how to “do” this proof, so we know that what makes understanding even a “first” proof like this a challenge is that it requires visualisation and extended thinking, which have no surrogates in graphing software, apps, games, and other learning gimmicks.
The idea behind what proving something means needs time to sink in; proofs shouldn’t be given short shrift.
I would agree that proving mathematical statements requires a certain mindset and that the mindset doesn’t just show up overnight. Sadly the students often aren’t given much of a chance to develop that mindset before they make it to me. So, to a lot of them, the class I describe in the blog post doesn’t even really seem like a math class… at least not the way they’re used to.
Problem is, I don’t see any kind of movement toward changing that (at least not that I can tell from reading blogs and following twitter feeds). In fact, at our school, all the emphasis is on Algebra, which seems to combat the idea of proof, judging by the way it gets taught. In most algebra classes that I’ve seen, students are asked to show their work, but there isn’t a lot of generalization done by the students. All the general function forms are given to them. All the equations are given to them.
I think of a fairly simple statement like “the product of two odd numbers is odd.” You would probably see a lot of students showing example after example, but I suspect many of them wouldn’t know how to model abstractly in a way that allows that statement to be proven.
This year after introducing the idea of congruent polygons I used this lesson over the course of three days: http://map.mathshell.org/materials/download.php?fileid=1302
I have not yet read over the final assessment (they just completed it yesterday) but it seemed to go pretty well. Our motivation for the usual congruence postulates is now not just “it would be nice to have shortcuts” but “I really can’t think of a counterexample, can I explain why the specified properties _must_ produce congruent triangles?” At least this is the motivation for my students who are ready for it. I definitely have a good chunk of kids who are at those lower vanHeile levels and struggling just to make sense of the whole activity. For those who are ready for it, this was a really nice set-up. For the weaker students, the entry into the problem by drawing was something they all could latch onto.
I’m still struggling on the larger question of the role of proof in my classroom. I’m using Harold Jacob’s text and I really like the way he sets up students to understand the structure of deductive arguments before asking them to write deductive arguments about geometric situations early on. It makes this feel like a truly transferable skill (especially because my colleagues in the English & social science departments are building on this work in their classes). I feel like asking students to write the short textbook-y geometric proofs helps them understand the structure of argument and apply it to math. However, these are the very proofs that feel “lifeless” to me. I would rather my students prove more interesting, surprising, or important things (like the Pythagorean Theorem, the various area formulas, or the midpoint quadrilaterals are always parallelograms). Do these “lifeless” proofs actually help students understand the structure of mathematical argument in a way that can be transferred to these bigger theorems? I see them as a means to an end, but is there a better way to get to the same end? And then the next question is: what do we do with these theorems? Simply collecting/memorizing them is ridiculous. Either these results need to be played with in the real world (not sure why that door isn’t closing? Measure the diagonals of the door & of the frame to see if they are both rectangular…), or built upon in the mathematical world (used as tools to explore and discover other interesting results).
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