I make mistakes all the time. We all do. As you’ve probably heard, mistakes are an important part of our learning. However, I don’t think it is always as easy to learn from our mistakes as any catchy statement might make it sound. So, I thought I’d share three quick stories and then reflect on them.
Story 1
The other day I was leading a string mini-lesson (similar to a number talk) and was modelling on a number line what the students were saying. The string involved the question 78-29 and I took answers from a few students.
Student 1 explained that they started at 78, went back 30, then forward 1. However, I modelled it on the numberline incorrectly. I took what they mentioned, started at 78 and landed at 38 and 39 instead of 48 and 49.
Story 2
A few weeks ago I was co-planning / co-teaching with two teachers starting a new Contexts for Learning unit (Cathy Fosnot’s The T-Shirt Factory). In our planning we looked at her landscapes of learning, the progression of lessons, did some of the questions together, anticipated student responses, and started gathering needed materials. I joined the classrooms on day 4 of the unit and came in to hear some issues they had with some group members not doing much of the work. When asked, they had told me they split the class into groups of 4 students because the resource told them to. They put each group in charge of figuring out 1 size of T-shirts (as is the context in the unit). They placed struggling students together to figure out the easier sizes, and “stronger” students together to work on the larger numbers.
After the first few lessons both teachers noted that none of the groups had all students working. I suggested that it would be far easier for us to work with pairs instead of groups of 4. I co-taught the same lesson with each teacher that day having students in pairs, but something was still not right. We read through the unit again and realized that we had grouped students incorrectly and had been assigning problems incorrectly! We should have placed 4 random students together and given EACH student their own size T-Shirt. That way each student could work as part of the group, been given their own problem, and we could assign numbers that were appropriate for each student.
We ended up having to redo the previous 2 lessons over again so each student would have proper groups and their own responsibilities!
Story 3
In my third year of teaching I went to a workshop where a teacher shared their practices with the group. The workshop was about having students create their own math dictionaries where definitions (with examples and pictures) were kept, and lessons and worked examples from each school day would be stored. I quickly started working on making this a reality. Over the summer I created the template for these books, I wrote out the notes I wanted every student to copy for each day of the year leaving blanks where their samples would go, and put together a package for each student that would be continually added upon throughout the year.
I was excited because these resources would house all of the definitions, all of the examples, all of the thinking from the entire year, and they would be able to use this resource for studying, for homework, and keep it for future years as a reference!
June quickly came and every student had their very own personal math dictionaries we created from countless hours of work. Each was neat and complete, full of all of our thinking from the year! On the last day of school, I watched as students packed up their belongings (forgotten sweaters, old shoes, pencil cases…), handed in their textbooks, and threw out their their unwanted worksheets and duo-tangs. However, I wasn’t counting on just how happy many of my students were to throw out their math dictionaries. I tried to pull out of the garbage and recycling bins each book to save anyway.
Some thoughts about the kinds of mistakes we make
We tend to make mistakes at different levels. Story 1 showed one of the many simple mistakes I make. They are quick little things that are often due to carelessness, forgetting things, accidents… When we make mistakes like this, what matters is how we show our students how WE handle mistakes. If we make a calculation error, do we get embarrassed? Do we pretend it didn’t happen? Do we turn it into a teachable moment? This really matters! If we show our students that we aren’t comfortable with us making mistakes, what unintended messages will this send?
In my second story I shared a mistake I made by not reading carefully and not being fully prepared for a lesson. Mistakes that cause us to redo something because we noticed that something isn’t working are great mistakes to make. When we realize that we have taken a wrong path, we come to see just how important the bigger things are. We learn to be more intentional because of our mistakes! In this situation, we realized that we could differentiate the learning in the classroom without ability grouping, and that if we have larger groups (4 students) we still need to give everyone something that they are responsible for.
The third story is a story I don’t share much. Probably because I had spent an entire year getting my students to memorize, follow procedures, copy out worked examples… My students eagerly throwing out their books was great feedback for me (I didn’t think it was great at the time though). Over time, I came to see that I was teaching my students instrumentally! Not only did my mistake help me to notice just how unhappy my students were in my math class, it helped me realize why my students had done so poorly on their provincial testing (they were the lowest results I’ve ever had). I was also able to reflect on what it means to learn mathematics and what it means to be engaged in thinking mathematically. Recognizing we are taking a path that isn’t beneficial for students requires us to see other approaches, understand the research, experience learning mathematics ourselves in ways that help us understand the concepts deeper. Mistakes like this are not only difficult to recognize, they are difficult to change. It has been a long road for me to continue to develop my own math knowledge for teaching, but I know that it started when I started realizing math is more than memorizing, more than rules and procedures, more than a collection of unrelated topics.
Some things to reflect on:
- When you make little errors in front of your students, how do you react? What do your students think about making mistakes in front of others? Is there a relationship?
- Can you think of a time you made a mistake with a lesson, or in teaching a unit? These types of mistakes are easy to learn from, but we need to take advantage of these opportunities. How have you learned from your mistakes?
- Recognizing and reflecting on our practices that reflect our beliefs is probably the most difficult for us to do. So, how can we find out which teaching practices are helping our students learn? How can we find opportunities for our students to give US feedback? What experiences help us reflect on our own beliefs about what is important for our students to do? What experiences help us reflect on our own beliefs about how students learn mathematics?
- How do the beliefs I have shared here in this post relate to yours?
As always, I encourage you to continue the conversation here or on Twitter (@MarkChubb3)
Thinking Mathematically 