Posted on February 16th, 2015
Hello from the Institute for Learning, this is Steve Miller.
What do you think of when you think of the word argument?
Do you think of a heated exchange between two individuals? Do you see people agreeing? Do you see professionals taking different positions to uncover the results that might occur?
In mathematics, proficient students are expected to organize a sequence of facts using definitions and postulates and things that mathematicians agree to be truth that lead the listener to an inevitable conclusion. Just like in the court of law, a case is built and presented that is intended to lead those listening to arrive at an intended conclusion. Students should sequence a set of evidence that logically leads to their conclusion.
While I typically think of an argument as oral communication, in mathematics as in a court of law, presenting an argument includes submitting evidence such as a picture, table, graph, or equation, and talking or writing about how the evidence supports my conclusion. The tangible artifacts allow the listener to follow my explanations. This evidence conveys my meaning.
Picture any convex polygon with more sides than a square. Maybe you visualize a pentagon or an octagon. It may help you to draw out your polygon on paper. I'm sure you know the number of sides that your polygon possesses. I want to make an argument that will convince you that I can determine the sum of the measures of the interior angles of your, or anyone else's, polygon.
On the drawing of your polygon, start at one vertex and label it point A. Continue counter-clockwise around your polygon, labeling the points (B, C, D, and so on) until every vertex has a label. Now choose a point anywhere inside your polygon, let's label it M, as long as you haven't already used M. Create a line segment from M to each vertex of your polygon. How many triangles have you created? You should have created the same number of triangles as there are numbers of sides in your polygon. Lets call that number s, for sides. We know by definition that the sum of the interior angles of a triangle is 180 degrees. You have a number of triangles that we have called s. So the sum of the angles of the triangles in your picture is 180 times s or 180s. But wait, take a look at all those angles around point M. They are not included in the interior angles of your polygon. We need to subtract the measure of all the angles around point M. Since those angles create a complete revolution around M, we know by definition the total measure will be 360 degrees. I will therefore conclude that the sum of the interior angles of your polygon, or any polygon, will always equal 180 times the number of sides minus 360.
So what makes up a good argument? What would you expect to hear? I would want to hear students using accurate and appropriate vocabulary. I would want to see students use labels to add precision whether they are referring to a specific angle or giving a value that needs the context of a unit. I would also want to hear students include appropriate understandings from their prior knowledge using definitions and properties that have already been accepted in the mathematics community. Their steps and language in the argument need to be clear enough that the listener can follow and understand what is being described.
As students progress in their ability to build strong arguments, they begin looking for generalizations in their arguments by examining multiple examples and looking for counter-examples. For example, in the case of the sum of the interior angles, we can examine multiple polygons because you the listener can pick any number of sides. When a counter-example surfaces, students would be alerted to a possible error in reasoning or in calculation. A double check will allow them to correct their calculations or change their conclusions.
In the presentation of an argument, students respond to other people questioning their evidence. If students are truly convinced by their own collection of evidence, they should not feel defensive in the course of being questioned. When students listen and respond to the questions of others, they build capacity to understand what constitutes a good argument.
That's what the Common Core State Standards for Mathematics' Mathematical Practice #3 (Creating a viable argument and critiquing the reasoning of others) is looking for in the creation of a mathematical argument. I hope this makes you think about what you see and hear in your classroom.
Once again, from the Institute for Learning, this has been Steve Miller.
Steve Miller is a Fellow of the Institute for Learning's mathematics Disciplinary Literacy Team.
More information on Steve Miller.