Why are justifications important?
This is the question that I focused on for the past year through the help of the Greater Madison Writing Project. When I started, I knew the answer, but I wasn’t sure if my students felt the same way. After some research on this topic this summer, I developed some plans for the following year to help me build a culture for justifications in my classroom and if you’ve followed along with my posts you’ve seen some of the results. (If you’d like to go back and read some, I’d recommend starting here.) Now it’s May, and I decided to ask my students the same question that got me started on this journey.
Why are justifications important?
With no other prompting, I asked nearly 100 of my students for their honest responses.
The dreaded, though least common, response:
“Justifications are important to show your work so teachers don’t think you are cheating and to get full points on tests and quizzes.”
This is the response I feared most. In fact I was so nervous about getting responses like this that I put off reading any of them until it was time to write this blog post. Unfortunately, this response is a common feeling shared by math students around the country, which is why many of us are trying to instigate some changes. Thankfully, there were only two or three responses similar to this out of all my classes (and hopefully students were as honest with me as I asked). Despite my enthusiasm for writing justifications, students often complained about all the writing and they struggled with my endless requests for them to explain their reasoning. My goal this year was to help students feel the importance of justifications, and this feeling doesn’t come without struggle.
I was relieved to see how many students had responses similar to my own when I started trying to explain the importance of justifications.
The most common response:
“Justifications are important because they show how you got your answer, and allow people to see the reasoning behind your answer and if it is correct.”
Hallelujah! Although there’s no way to know if students truly feel this or if they are just trying to butter me up before final grades, they at least know the intent of justifications. There are two main parts to this response that I saw repeatedly: showing your work and explaining your reasoning. These two parts go hand in hand to create the ultimate justification. Many of my geometry students (who studied proofs this year) referenced the facts and reasoning that work together to make a convincing point. I also hope that this response comes as a result of the communication in my classroom. I am constantly asking students to explain their work to their team, and they appear to have learned that their efforts will be most useful if they write out their work and explain it thoroughly.
In class, I mention the importance of problem solving and often stress the process over the solution to my students. I have been working on general process explanations instead of formulas, and many times we create these ourselves. We also discussed the importance of thorough justifications and how to convince others to listen to you. These were a few other responses that caught my eye and tugged at my heartstrings a bit:
“Justifications are important because if you don’t know how something works, you might not be able to figure out a certain problem in the future. For example, in class, we have cheat sheets with equations on them but in 20 years, we won’t have those sheets right in front of us. So if we know how to justify equations and answers, we will be able to solve and answer any question.”
“Justification is important because if you do not justify something, there is honestly little point in saying it. Anyone can say anything without justification. You need proof to make your claim legitimate.”
“Without justification your answer will have no credibility.”
“They explain your process to people who might now know how to do what you are justifying.”
These are profound ideas for students in high school to be considering. The ability to persuade or help others learn something new will help them in many other aspects of life, particularly when it comes to making a name for yourself in the workforce or community.
I also had students write about how writing justifications have helped them learn. Many students mentioned that it helped them understand the problem better and one student said, “It helps us find mistakes through our process.” These are the types of responses that are most rewarding to me. These show that students are finding personal growth and value through doing the justifications.
I could continue gushing over my students’ responses, but I wanted to end with a short story. One day in class, one of my students said that her grandpa loved math because it was black and white. This cliché saying has often been said by students and math lovers (including teachers) and I understand why. It is comforting to know that there is a right and wrong answer to a problem, which is often the case in math. Then the student asked me if that’s why I liked math. I paused for a second to filter my response because in all honestly, I completely disagree with the cliché and my love for math is almost the complete opposite. I then explained that at one point I did like this aspect of math, but then I went to college and started teaching. I learned that when you dig deeper into mathematics you can see so many connections entwining together. You see multiple perspectives on the same problem and you see how that problem now connects to other concepts you’ve learned and even other subjects like science and music. That’s when I fell in love with math.
Math is not black and white or right and wrong to me. Contrary to the cliché, it is full of many colors and perspectives coming together. My passion for all of math’s colors shines through justifications. They are where the connections and perspectives emerge. This is the message I hope to share with my students.
