Improving Students’ Mathematical Justifications

What makes a good justification anyway?

I can only directly control one aspect of the successful education system I explained in my first blog post. As I reflected on my first year of teaching and prepared for my Teacher Workshop (TW), I came to the conclusion that justifications are what tie the important aspects of mathematics together. Justifications explain your process while problem solving. They are used to make a strong argument. They help students develop critical thinking skills. They force students to think deeply about concepts. And they help students find errors in their logic.

I knew why justifications were important, but did my students? They were asked to justify their work regularly in class and on assessments. Unfortunately, quantity doesn’t imply quality. I arrived at the guiding question for my TW:

How can I improve students’ mathematical justifications?

To answer this question, I read through research and teachers’ classroom experiences. I immediately saw a need to first answer the question, “What is a mathematical justification?” Turns out, my students probably didn’t know. The short answer is a mathematical justification explains what you did and WHY you did it.

After I defined mathematical justifications, I got back to the initial question, “How do I improve these justifications?” Some key methods I discovered and hope to blog more about are:

• Model justifications
• Use questions or prompts to further understanding
• Use multiple representations at once (visual, numeric, algebraic)

To learn more about my findings or activities, please watch the video of my Teacher Workshop, read through my slides, or take a look at my handout. (Fair warning, the video is long, but it’s valuable. The second half is a discussion about my workshop with the rest of my GMWP group.)

Now keep in mind, it is currently the middle of summer vacation, so I have not tried all of these wonderful methods with students yet. As mentioned in my TW, my next steps are to:

• Explain the difference between explaining and justifying to students
• Model more justifications
• Incorporate multiple representations simultaneously
• Continue to ask clarifying/extending questions and explore the use of prompts
• Peer edit justifications in class (thank you fellow GMWP peeps for the ideas)

These are the actions I will be trying and blogging about throughout the year. I’m sure more questions and possible solutions will pop up along the way, but this is where I start. Stay tuned to see how it goes!