After a great three weeks in Utah, I flew over to Boston for a one week conference run by the Boston Debate League on using debates and debate techniques in all subjects. Formerly called Debate Across Curriculum, the program is now called Evidence-Based Argumentation (EBA). The week I went to was particularly focused on math and science teachers. More information can be found on their website here. I have a TON to say about this conference: thoughts to express, ideas to share, projects to try out, etc etc. So the next few posts will slowly unravel all that is bouncing around in my head about this. Many people may wonder how debate works in math or why try to blend the two. I have tons to say on that and maybe my next post will explore the why. For now, I just want to get some of the basics of debate (at least as the Boston program taught it) to give a starting point.

The program has a 5-step process to developing EBA, and the first step is making a basic argument. The formula for an argument is:

Argument = Claim + Warrant

where

Claim = a controversial statement

Warrant = reason why your controversial statement is true

I know many people instantly think arguments fit well in an English or History classroom but believe that “controversial statements” are rare in math. This is not the case, but I will expound upon this in a future post. Right now, I want to focus on the basics. Suppose I said: “Exponents make a number bigger.” All the math teachers may instantly snap to a counterexample. However, from the point of view of a student (especially one who has not seen rational exponents), this may appear controversial. What I would want from students at the beginning of the year, when they are first learning to mathdebate, is a response such as

“I claim the statement is true, and my warrant is that an exponent makes a number multiply by itself and thus get bigger.”

Other students can agree or disagree and add their own comments. When developing basic arguments, a warrant can be a bit general or could be an example. The focus here is on the structure. A lesson or two later things will get more serious…

I begin my classes (Geometry and PreCalc) with a bit of basic logic (conjunctions, disjunctions, conditionals and negations), and I think it’d be great to define argument on the first day, as we define statement and open sentence. This will blend debate into the basic structure of the class and help as develop explanations and proofs. I think students will get so accustomed to the gimmick of “claim + warrant” that explanations will become second nature to them in a more natural way.

One last note: controversial statements can come in a wide variety. From less mathy opinion questions (math or non-math related) to more mathy equations and applications. Using words like best or worst easily make many statements controversial. Some quick examples off the top of my head are:

- Math is the most important subject in preparing for a future career.
- A calculator is the best tool for solving a math problem.
- Every number has a square root.
- The best way to solve this quadratic is by completing the square.
- Verizon has the best cell phone plan for someone who uses a lot of daytime minutes.

I particularly like the last one as it requires students to research on their own. Much of the research these days encourages teachers to give less direct scaffolding and promote student “struggle.” Controversial statements and open ended application problems provide these opportunities for students struggle.

I will get into more interesting (and complex) uses of arguments and debate in math in the upcoming posts. I hope this post summarizes the basics. I could easily get students engaged in this first lesson by making several of these controversial statements (ones that don’t require research) and having students respond in one of two ways. First, I could set up the room for a “Four Corners” activity, where each corner of the room has a label (agree, strongly agree, disagree, strongly disagree) and have the students react to each statement by moving to the appropriate corner of the room. Students would then be called on to explain their stance with a basic argument. Second, I could use a “Soapbox” activity, where I call on students one at a time to stand and give a basic argument. Again, this is all very simple, but my focus is on developing the structure (and build excitement around debate). We will get to the really fun stuff quickly.

If anyone has good ideas for some simple but controversial math statements, please comment below!

Lastly, as I said in the last post, I want to end each post with a topic for debate. However, I ask that you please reply in the basic argument format, explicitly using the words “I claim…and my warrant is…”

CLAIM: High school math’s most important aim is to prepare students for Calculus.

You might recall that Kate Nowak’s 5 minute short at PCMI provided a wealth of excellent questions (Good Questions at Cornell), and seems to aim at similar goals as yours.

I’ve done similar things , but often much more “closed.” However, the closed nature of the questions (they often have one correct mathematical answer) does not prevent them from getting students into the important mathematical connections I want them to achieve. I often ask students the good old “Always, Sometimes, Never” statements. These require a lot of logical thinking, technical detail in verifying, and clear communication.

“Always true?” A deductive proof is expected.

“Sometimes true?” One example and one counter-example.

“Never True?” Following the remises until a contradiction must exist, and clearly juxtaposing two contradictory conclusions.

Not-so-deep example: “If a polynomial function of the form p(x) = x^3 + Bx^2+Cx + 4 (B and C are real numbers) has real zeroes, then all of those zeroes will be at x = +/- 1, +/- 2 or +/- 4. ”

Answer? Sometimes true. This would be always true if the statement mentioned “rational zeroes.” But it doesn’t, and the process of actually constructing counter-examples and verifying/ checking hit the communicating, technical fluency, and thinking I want my students to do.

Another example: If the mean and the median of a set of data are equal, then the distribution is symmetric.

Answer? Sometimes again: {7, 8, 8, 8, 9} vs {5, 7, 8, 10, 10}

I like the idea of defining an argument. I find myself asking students to “defend your answer” regularly and this makes that idea more precise. I have done “True, False, Fix” with similar statements to the ones you are looking for. I will share once I find them.

Oh, and: I claim that high school’s aim is not to prepare students for calculus and my warrant is I am more interested in creating logical thinkers and problem solvers than math majors.

I play a related definition game. The group tries to define something. When a definition is made, the step in the game is to claim it’s too wide, or too narrow, and to provide a counter-example showing that.

It’s impossible to finish that game with almost any object outside of mathematics! So we usually play with something like “chair” first (exercise in frustration), and then something like “triangle” (which kids define in a finite number of steps, and then appreciate math for it).

I want to integrate the game with what you described here…