What Does it Mean for Students to Construct a Viable Argument and Critique the Reasoning of Others?

To create an environment for this third standard of mathematical practice, a teacher must create a safe environment where students’ do not fear risks and where it is okay for them to be wrong.  In our current school culture most students do not feel comfortable being wrong for fear of being laughed at or belittled by their peers or worse–maybe the teacher.  This culture must be replaced by the idea that everyone’s thinking is worth examining either for why it is correct or incorrect.  For students, gone is the day when they simply agree or disagree with someone’s answer–might I add usually following the tone of the teacher’s voice to know if the answer is correct.  Students must be able to explain why they agree or disagree.

We must value each student’s response.  Instead of getting an answer and moving on, teachers should collect the thinking of at least several students and ask if the answers are reasonable or unreasonable.  For example, for the problem 14 x 3, a teacher might go around the room and gather the answers 17, 34,  32, 42, and 44.  The teacher can ask students which answers are unreasonable and why.

A student who is critiquing the reasoning of other with a viable argument might say something like:

“Well, I know that 17 can’t be correct because if you add 10 twice that makes 20 and fourteen is more than 10 and 20 is more than 17, so 17 couldn’t be correct.  Also, I think they accidentally added because 14 plus 3 is seventeen.”

In the above case the student who gave the incorrect answer is not named because no one remembers who gave the response 17.  Since 17 is written on the board as a response and no name is written by the response the answer is viewed merely as something to discuss.  The student, however, who gave the response 17 is most likely intently listening because he gave the response.  Further, the student answering 17 may have even recognized his mistake when the teacher wrote all of the responses on the board and saw how his was far from other students’ answers.

A practical approach of a student constructing a viable argument for his correct answer might sound something like this after being asked what is 14 x 3?

“I know the answer is 42 because I used the distributive property to multiply my tens first and then my ones.  I broke the fourteen into one ten and four ones.  Then I multiplied the ten times three and I got thirty.  Next, I multiplied the three ones and the four ones and got twelve.  Then I added thirty and twelve and got 42.

I know what you are thinking, “My students wouldn’t say all of that!”  BUT, if you praise and expect this behavior, you will be surprised at what your students will be able to explain by the end of the year.  Eventually you will no longer prompt students to go beyond the answer “42”, they will be explaining their reasoning without being asked.