It’s hard not to enjoy sessions when you’re already drinking the kool-aid. That said, Amy Ellis did a fantastic job of balancing research and practice around laying a foundation for proof well before Geometry class (on a side note, I still hope to lead a session called “Proof Doesn’t Start in Geomety” at some point in the future). She gave a convincing argument for the importance of introducing the idea of proof in early elementary school, and more importantly discussed structures and cultures that promote proof at any age.
Ellis started with a simple, but robust problem asking students to explore a growing rectangle in Sketchpad. She showed some student data
She then talked about student conjectures and their explorations, including some nice proofs in pictures.
Ellis then swung over to the research side, making a number of claims, including something we teachers can always use reminding of: “Students over-rely on examples as proof (Chazan, 1993; Harel & Sowder, 1998; Healy & Hoyles, 2000; Porteous, 1990).” “Students have no problem generalizing, these generalizations just aren’t always right or productive.” “Students tend to generalize in recursive ways before explicit ways.”
In middle school parlance, we’re talking about the difficulty in getting our students to shift from “I do 3 examples. They all work. Done.” to proof.
We talked with partners about how you know when something is correct in mathematics. My personal answer: when you can convince your skeptical peers that your reasoning is valid.
Convince myself, convince a friend, convince a skeptic.
She then shared some student responses to this same question, many of which were pure gold.
“You experiment until you find five examples to see if the answer is right.”
“When you get the test back.”
“You don’t really ever know if anything is right, you just have to hope and pray that you are right.”
Ellis then introduced a new problem, something she called Eric’s Property.
Eric came up with a new mathematical property. He thinks this property is true for every whole number.
First, pick any whole number.
Second, add this number to the number before it and the number after it.
Your answer will always equal 3 times the number you started with.
Do you think this is true for every whole number?
She shared a few examples of “proof by three examples”, and then moved on to the following abstraction.
“So here are random objects. Okay. This is a number. 3 Triangles. This one has more triangles. It has 4 triangles. And this one has 2. So if you take one of the triangles and you put it over here, it becomes 3 of the same thing. So 3 of these would be the same thing as this plus this plus this.”
She also shared a traditional algebraic proof of this. I noticed she referred to the former as a proof and the latter as a formal proof. An audience member asked the same question I was thinking, asking why the former was not a formal proof. I was unconvinced by her answer (while I do think this student is using the triangle symbol to represent both “some number” and “one”, I don’t otherwise see how this is any different from x-1, x, and x+1).
Ellis ended the talk discussing classroom practices and habits that promote proof. It’s fine (and important) to allow incorrect conjectures to play out. Generalizing doesn’t have to be algebraic. In earlier grades this can be done visually or verbally. Proofs too often start as exercises to verify versus proofs to explain (and in my opinion, students have a much harder time proving things that are “obvious” to them). Give kids the opportunity to conjecture and prove. Extend tasks to encourage more general reasoning. Make observations, conjectures, and argument/proofs. Don’t get so caught up in the structure of proof. Provide problems for which examples fail.
Use peer critiques. Have students explain why their statement is true to a student in an earlier grade. Ask students: will it always be true, when will it be true, are there times when it won’t be true. Make a conjecture wall. Share (and I would add create as a class) criteria for a valid proof.
And to add my own final recommendation, don’t fall into the trap that proof must require algebra, two columns, and/or a less eloquent rehash of Euclid.