Framing mistakes as a learning opportunity

In a mathematics class mistakes are really tricky. Many students have a very simple view of mistakes; they are bad and should be avoided. Unfortunately being human beings, we all make mistakes, even in a math class. Even me, at the front of the classroom, I will make mistakes. Hopefully we use these mistakes as learning opportunities, which we can only really do if we stop and reflect on what happened, and how to prevent it from happening again.

I made a mistake last week during one of my class's online quizzes, I made a very simple mistake in the setup of the question. Whenever this happens I am usually alerted by an industrious student, and fix the mistake.

During a previous class I shared a list of questions I had to answer when writing a student recommendation for an internship. One of the questions we discussed was "Does the student ask for help when needed?" In most situations it is preferable for someone to ask questions in order to know what they should do, than it is to have someone perform work they do not know how to complete. The time in detecting these errors, fixing them, and mediating any negative consequences is considerable, not counting money or other resources. No wonder an employer would want an intern (or employee) to ask questions when confused.

So this error cropped up in an online quiz, and we just talked about asking for help when needed. So I naturally asked "How many students will send me a message?", if they really listened to our discussion of this question they would ask for help when needed. Out of 29, 3 students sent me a message asking about this question. Ok, so they didn't hear me, not a big surprise. Whenever I talk about topics that are not math (study skills, organization, soft skills, group work, etc.) students usually tune out.

In class the next day I shared the story of the Mars Climate Orbiter. Long story short, a $350 million probe burned up in the Martian atmosphere because someone did not check the units of a calculation. This definitely go their attention, as it usually does; no one expects literal rocket scientists to make mistakes, let alone really simple ones. After talking about the orbiter I had a few questions for students:

• Was this a preventable mistake?
• Was this a significant mistake?
• Do you think NASA learned from the mistake?
• How many people here have made mistakes in their classwork during the last week?
• How many people learned from those mistakes?
• Are your mistakes preventable?
• Are your mistakes significant?
• Did any of your mistakes cost $350 million?
• Do you have space in this class to make mistakes through unlimited attempts completing homework questions, Quiz corrections, and working with others in class?

We had some real-talk on making mistakes in math, and how they are part of learning the course material. By making it clear that I expect students to make mistakes ("Make them early and often.") and to learn from them I hope to lessen student fear of making mistakes, the stigma associated with them, and to reframe student's expectations of me. I regularly ask students to reflect on making study plans, if they were able to follow their initial study plan, what factors prevented them from following their plan, and what they are going to do in the future to prevent these factors from getting in the way. 

In the future I would like to do a few things with this theme of making mistakes:

• A POGIL-like activity where students read through a description of the Mars Climate Orbiter, and come to some of these conclusions themselves.
• Demonstrate the role reflection plays in learning from mistakes. 
• Tie this into the growth mindset, a very natural place to discuss these issues. 

How do you address mistakes in your classroom? Do you do any specific activities with students to frame mistakes as learning opportunities?

Course Observation - Definitions

Life has been pretty hectic lately, I moved apartments (in the same building), interviewed for a full-time position at the college I am currently teaching at, and have generally been a mess teaching 20 credits, two new classes (to me), developing interesting lectures and activities, and grading (oh the grading). I enjoy it and love the challenge all of this offers me, but makes it hard to post regular updates. Hopefully you've enjoyed (or been amused by) the worksheets, activities, and quizzes I am posting every Wednesday. Feel free to drop me a note or post a comment below if you find anything on this blog particularly engaging or interesting.

This week a seasoned math instructor asked if he could observe my class and I gladly said yes. Sure, it can be stressful having someone in your class that isn't one of your students, but I am always looking to improve my teaching, and if that means a bit of discomfort, so be it. My desire to become a better teacher is greater than my desire to avoid criticism, constructive or otherwise. Lucky for me the feedback was very constructive, and I plan on discussing it in a few posts. Here I will only talk about one part of that feedback, the balance between core skills and more abstract ideas.

They observed a day when we went over graphing functions based on their derivatives. I have had previous conversations with this instructor about this lesson, and it was by chance the they observed this class. The previous conversations were about an esoteric point about the definition of a phrase that impacts how we answer certain questions. I mentioned this in class and how the book and I differ on this one definition, and how it could change our answer.

The observer commented that this fine distinction might not be the best use of time and energy. They were worried that few students would understand what the impact of this difference is, and most students would get nothing from it. Students have a difficult enough time understanding the book's definition, mentioning my definition may confuse them.

I understand this concern, and appreciate the fact that this instructor has spent much more time with this student population. I will likely not mention this difference in definitions in future classes, but I do want to spend time explaining why I included this difference. This lesson is at the end of a section that is very computation focused where they apply different rules. With this small diversion into how the book and I differ, I was hoping to demonstrate that mathematics isn't just a matter of following rules and computing things. This aspect of having and using precise definitions is also important and one they are going to be exposed to in future classes. I still like this idea and would like to incorporate it into another lesson. If you have any ideas about how to do this, feel free to share below.

I'll be discussing the other feedback they provided in future posts. I'm doing so partially to help myself think through the feedback and to reflect on how I can use it to become a better teacher. I also want to be able to talk about these ideas without sounding defensive. I think about my teaching pretty carefully, so sometimes I get wedded to an idea without talking to others about it. When this happens and I get feedback to the contrary I can feel myself respond defensively, and I don't find that response very useful, or frankly, professional. Part of it is this math culture of being right that I've internalized, but it also comes from my own self-doubt, both things I need to shed.

On that happy note, feel free to share your comments or email me directly.