In the Common Core, the Standards for Mathematical Practice push for student discourse in mathematics. Here's the standard as written in the Common Core:
I've been pushing students to do this for quite some time in my teaching experiences. But whether I was teaching 1st graders or gifted 5th graders, I found it a bumpy road due to 2 main difficulties:
Discourse Difficulty #1: Students need a lot of practice in order to articulate their mathematical reasoning in a way that makes sense to others. As a teacher, I did a lot of inferring when I heard the students explain their thinking. I had to fill in the blanks often when students trailed off or couldn't completely describe the steps in their problem solving process.
Discourse Difficulty #2: Students have a hard time listening to each other during their problem-solving explanations. When you have a whole class of students with different abilities and diverse prior experiences in math, you lose about 1/3 of the class whenever you have one kid explaining their thought process to everyone. Part of this is due to the attention span of the class; Part of this is due to Discourse Difficulty #1; And part of this is due to some students just not being ready to hear and understand the reasoning of the student speaking.
Difficulty #1 is best solved by offering many opportunities for math discourse throughout the school year. There are some other strategies for overcoming these problems, but I'll save those for another post.
In this post, I want to show you how Google Drawings can solve Difficulty #2 and keep all students engaged and participating while still "listening" to the arguments of their peers.
For a recent 2nd grade lesson in Measurement and Data, Mrs. Ahearn and I were interested in getting the students to use mathematical arguments to explain how to measure using standard and non-standard units. And we wanted to use Google Tools to do this.
First, I started taking photos of different things being measured with linking cubes, paper clips, and rulers. I took about 18 photos and loaded each to its own Google Drawing. The image at the top of this post is an example of one of the "trickier" photos I took.
Then, we shared those Drawings with the class in one Measurement folder on Google Drive.
In the next class, we explained to students that they needed to write a comment for an assigned Drawing. The comment needed to measure the item using the structure, "I think _____ because ____" Mrs. Ahearn emphasized the idea of backing up your thinking with evidence.
After students made their first comment, they chose another Google Drawing in the same shared Measurement Folder. They had to reply to their classmate's answer and reasoning using the structure, "I agree/disagree with ____ because ____." Again, the students were encouraged to back up their thinking with evidence. If they finished that comment, they were told to find a new photo in the folder and make another comment using the same structure.
On our 3rd day of discourse within the Google Drawings, we took a look at some of the arguments and thought about what good qualities they had and what can be done to make it even better. I think the reflection on these arguments is key towards improving the articulation of the students.
Because students always had something to do, they didn't have a chance to lose interest in the discourse process. Most students actively participated in at least 3 arguments. Some students participated in more.
I was happy with how the students stayed engaged in the process. I do think that, as 2nd graders, they still have growing to do in terms of articulating their logic and truly understanding the logic of their peers, but that will come with practice and plenty of teacher scaffolding.
I'm interested in hearing how other teachers have helped their students become skilled at making mathematical arguments. Any ideas?