Logical Reasoning

How I spent 70 minutes defining mathematical justifications without doing any “math”

I wish you could have seen the look on my students’ faces when I told them that they wouldn’t have to do any “math” in the entire 70 minute period. Most of their faces were filled with joy, relief, and a bit of confusion. I explained that they wouldn’t be solving any equations or doing arithmetic. Instead we would spend the hour talking about logical reasoning and characteristics of a good argument.

This was my sneaky way of having students practice a key mathematical skill, building a logical argument. Logical reasoning is the foundation of mathematics. Fortunately, logic is used to create arguments in any subject, so it is easy to spend a day practicing and defining logical arguments without complicating the process with math content. This day laid the foundation for mathematical justifications in my class.

The first two examples were provided by our curriculum (College Preparatory Math, CPM). I then supplemented with some activities of my own. This is how we spent our 70 minutes:

• Students started by analyzing a list of relatively simple statements given by various characters to defend a “not guilty” plea. They needed to determine which statements were necessary and then convince me, the judge, of their verdict.
• After each argument was presented, I asked the group which statements they chose to use and why they used them. Most groups explained that they used statements with times or locations since these were key elements in their argument.
• Next we looked at an example with artifacts, such as a driver’s license and newspaper articles. They had to determine which were necessary and make logical connections between the artifacts to write a convincing argument.
• For the previous example, students approved each others arguments with an activity called “The Traveling Salesman”, a strategy found in our curriculum resources. One student from each group rotated to a new group to present their argument. The new group discussed what made it a good argument or what could have been added to make it stronger. The salesman then went back to their original group to discuss the feedback they received.
• At this point we discussed and listed the characteristics of a good argument as a class. You can see the list of student’s responses below. I made sure to briefly describe how each of these characteristics show up in mathematical justifications as they were presented.

• I then gave students a persuasive essay to analyze.
• We went back to our list from the board to determine which characteristics were also used in the essay and added any additional ones that students found. (These characteristics are in pink.)

As I mentioned in my last post, the key to students writing good mathematical justifications is to explain and model what a good one looks like. When discussing this process with the other GMWP teachers, they mentioned the need to start with non-math examples. I took this to a bit of an extreme and saved the math examples for another time. In reflecting after the lesson, I think it may have been helpful to show a mathematical justification at the end of class and compare it with the list of characteristics that we made. I did save a picture of our list and intend to do this during another class period closer to the start of proofs. I also intend to describe the difference between explaining and justifying once it shows up in the classwork relatively soon.

Overall, the students appreciated focusing on logical reasoning in a non-math context. I was impressed with the many connections to mathematical justifications the students were already making. The foundation has been laid.

Note: The 70 minute period I described above was for my geometry class, who will be doing some formal mathematical justifications in the form of flowcharts and two-column proofs later in the year. Since my Algebra 2 classes have already gone through this process, I plan on doing an abridged version of this lesson. I will spend about 15 minutes introducing justifications with the persuasive essay and characteristics list before students write justifications using multiple representations (blog post to come).