When we think about teaching students to be convincing, we sometimes hear about these three levels:

- Convince yourself
- Convince a friend
- Convince a skeptic

Additionally, I think a lot about the three types of justification from *Thinking Mathematically*:

- Appeal to authority
- Justification by example
- Generalizable arguments

To me, the first two justifications are what I see often from students that “convince yourself,” while generalizable arguments are more for convincing others.

I want to talk a little more about these ideas and levels–how my students and I develop these ideas through debate structure. Let’s take an example statement I gave to my students last year.

*True or False: The sum of two linear functions will always be a linear function*.

I gave this statement to my students as a bonus (not worth any credit) question at the end of a quiz. Some students gave an example or two and said it was true. Others jumped into writing a convincing paragraph (presumably without even trying an example). Yet others said it was false and moved on. Below are some of the highlights of our discussion about how to be convincing.

**Level 1: Convincing yourself.**

When I am first introducing debate structures to my students, a general basic level of convincing–convince yourself–is acceptable for warrants. Early on, I will accept most any warrant that is relevant. In the starting weeks of the school year, my goal is more for students to find their voice and feel included than critiquing the quality of their arguments.

That said, as time goes on, I push for stronger arguments. When I gave the question about the sum of two linear functions, I was ready to talk to students about being more convincing. After I handed back the quiz, we talked about/looked at examples that either had some sort of appeal to higher authority

“Mr. Luz said it was true.” or

“It is impossible to get a quadratic.”

or were a justification by (one) example.

“2x+3 + 4x+5 = 6x+8”

Clearly, each of these students had convinced themselves in the moment, and perhaps did not think we had to go further. Up until now, these were acceptable answers in our debates. Since we had not talked much about a quality justification yet, I shared these examples as a foundation step. It is important to first take a moment to convince yourself.

However, from now on, convincing yourself is like a rough draft; it is the work you might do on a scrap paper to start yourself on the path of justification. I compared this first step to creating a quick outline for an English or History essay. Convincing yourself is an important organizational step before jumping into writing a paragraph justification, but it is not, in itself, enough.

**Level 2: Convincing others.**

I talked to students about the next step: to imagine a friend. Yes, actually picture someone you are talking to, and write an example and some sentences that you think would convince them that you are correct. To highlight some examples of this, I shared that one student had written:

“(x+1) + (x+2) is linear because the x-value is not squared. Since you are not multiplying, the x-values are not squared, or cubed, etc. So it will be linear.”

Another student wrote:

“My claim is this is true and my warrant is because you are adding the values together. So you will still end up with a linear function. Ex: (3x+4) + (3x+4) = 6x + 8.”

One of my classes really took to this as a good way to remember this level of convincing. Picture a friend. Give them an example and an explanation (or in debate terms, a claim and a warrant) that you think would convince them. Many students had already shared arguments in our debate activities that could count as convincing a friend, but I wanted to be clear that this should be the base goal for us from now on.

I thought it was a good goal for the class to move toward this routine for Level 2 convincing, thinking about what to say to *truly* convince your friend. For some, it was going to be a good jump in growth for their mathematical thinking. For others, it would help solidify writing a basic justification, one that would get them some “credit” on future problems.

A few others were ready to talk about the next level, but I talked to them in individual feedback at this point. I wanted to save a discussion on Level 3 for the next quiz, leading us into a discussion of proof and how to convince a skeptic.

**Level 3: Convince a skeptic.**

For me, level three is where a rigorous justification or proof is needed. If I have planted the seeds for debate throughout the starting months of the school year–through Soapbox Debates, Circle Debates, using Debate Cards, and putting good debate questions (sometimes for no credit) on quizzes, students are primed to talk about a solid mathematical argument (or proof) later in the semester. What’s more, it feels natural to talk about mathematical proof. It’s not like we’re introducing this idea of argumentation for the first time as we start a unit on proof. It’s a skill we’ve been building up regularly, a skill key to being a mathematician.

I’m not going to go into proof right now, many have published great ideas and research better than I could do (and this post is already long!). The only thing I can add at this moment is when it comes to being *thoroughly* convincing, enough to convince a skeptic, I tell students to think of a counter-example a skeptic might try to come up with (like what about if 0 is involved? Or fractions?) and include how that is not a counter-argument in your explanation.

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Further reading (partly so I can find these links in the future again):

Robert Kaplinsky wrote an interesting blog on Levels of Convincing.

Jo Boaler wrote a good article Prove It To Me.

NCTM has a great article on “Promoting Mathematical Argumentation” too, but you need a membership to access it. (March 2016, Vol. 22, Issue 7)