Argument Writing in Math: Getting Started

Are you an English teacher prepping a training on argument writing for the teachers in your building/district (and you want to be applicable to those STEM folks)? Are you a teacher that just went through an argument writing training that left you wanting more? Are you a math teacher that wants to integrate more logic, reasoning, and writing into your course? (Let's be friends!) Whoever you are, I wanted to share a few thoughts and resources to get you started on your argument writing (and thinking) in a math class. If you'd like more of a primer on argument writing in general, check out this post I wrote after I practiced some argument "writing" with my 3 year old daughter last year. (Teaching Argument Writing for Preschoolers...and anyone else!)

The most natural application of argument writing is in proofs, which most often come up in Geometry, and should also usually be used in Algebra (but rarely are in my experience). To make a common core connection, standard for mathematical practice #3 requires students to "Construct viable arguments and critique the reasoning of others."

Most students experience with proofs (and perhaps you remember your own) is with column proofs like this, for proving things about angles, line segments, or shapes.

Have I induced any terror sweats yet?

But proofs can also be written in paragraph form, which is where the English training and Hillcock can be applied. Reasons are the warrants, the statements are evidence, and claims will be what must be proved. "Prove that angle A is a supplementary angle," or "prove that the lines defined by y=2x+3 and y=2x-25 are parallel"

But...I don't even know how to explain what a proof is! 

Watch this adorable TED-Ed video introducing and explaining the basis and application of mathematical proof.

Want some more? Here's a resource from Berkeley on mathematical logic
"First, a proof is an explanation which convinces other mathematicians that a statement is true. A good proof also helps them understand why it is true. The dialogue also illustrates several of the basic techniques for proving that statements are true.

Table 1 summarizes just about everything you need to know about logic. It lists the basic ways to prove, use, and negate every type of statement. In boxes with multiple items, the first item listed is the one most commonly used. Don’t worry if some of the entries in the table appear cryptic at first; they will make sense after you have seen some examples.

In our first example, we will illustrate how to prove ‘for every’ statements and ‘if. . . then’ statements, and how to use ‘there exists’ statements. These ideas have already been introduced in the dialogue." - from Introduction to Mathematical Arguments (http://math.berkeley.edu/~hutching/teach/proofs.pdf)

A couple more resources for your classroom:

1. "Making arguments with equations, figures, and images" - (http://wacillinois.wordpress.com/2014/04/22/making-arguments-with-equations-figures-and-images-writing-in-stem/)

As soon as you have students start writing more in math class, some of them will start trying to write out EVERYTHING. The point of this post is that sometimes mathematical symbols are still most appropriate

2. "Developing argument writing in math using crime scene investigations" -(http://teacherleaders.wordpress.com/2012/12/15/developing-argument-writing-in-math-using-crime-scene-investigations/)

This blog post from a teacher directly integrates an argument writing text by George Hillocks, Jr., Teaching Argument Writing, Grades 6-12 (with a bonus handout!) as a strategy for students to attack math word problems

Face-Off: Which Bono Venn Diagram Is Correct?

Have you seen the top Venn diagram on Facebook or Twitter based on the U2 song "With or Without You?"

I'm always pleased with math-type subjects creep their way into everyday life, and this diagram is no exception. (Little known fact to the general public: English and history teachers stole Venn Diagrams from mathematicians)

The top diagram is cute, and a good conversation-starter, but I don't know if it's correct.


Here's the lyrics in the chorus we need to dissect:

With or without you

With or without you

I can't live

With or without you

Quick background on Venn Diagrams:

• The Venn Diagram was published by John Venn in 1880. The intentional was to show relationship between elements (things) in any given comparison of sets (groups). (Source)
• The overlapping area of two or more sets is called the intersection. An intersection would be the subset of both overlapping sets.
• A set that describes everything that is in two or more sets is called the union
• In common language, an intersection is usually appropriate when "and" is the conjunction between two sets, and a union is appropriate when "or" is the conjunction.

It seems to me that the correct placement of Bono depends on where you're choosing to stress your interpretation. Is it in the double-negative spiral of "can't" and "without," or is it as cut and dry as "or and "and"?

Which diagram is correct? 

Further study on this topic:

"With or Without You Venn Diagram Meme"

"You, Bono, and a Venn Diagram"

"Bono Residency"

Logic with Minesweeper

Have you ever played minesweeper? You know, that game with the smiley face on any Windows PC? Most people know about it, as in, they know it exists, but far fewer actually know how to play it. Besides randomly clicking on boxes (which can sometimes lead you to victory by chance on beginner), there is reason and logic built into the game.

Which of these do you identify with?

I've chosen to focus on logic this fall in my Saturday morning SMART Academy classes for a few reasons:

• reasoning and logic make us better, more creative problem solvers (math and real world)
• logic intellectual building blocks are probably done forming in the brain long before a student gets to me, but I can at least build on what a student does have, and nurture/encourage refinement
• students often get lost in the middle of solving problems because they don't know what to do next, or they run out of gas. Practicing that perseverance in minesweeper or a logic puzzle lowers the risk of failure.

Coming next, we'll see how my kids did.