This is the second part of the blog post : Visible Thinking in Math
She started out with laying a foundation from the start of the school year.
Listen to her students explain the why, how and what next of metacognition in Math class.
How could she give her students practice in articulating their mathematical thinking? We chose to use iPads and Explain Everything app.
- Students took an image of the Math problem
- Students recorded themselves solving the Math problem. Emphasis was placed on articulating their thought process, including when they thought “I really don’t know where to start”. Helping making their “fluency” of following thinking like that with strategies to continue audible.
- Once the video of them writing and talking themselves through solving the problem (correctly or incorrectly solved), the project file was saved as a video clip and exported to the camera role.
- Another student was then charged in starting a new Explain Everything project on the same iPad and importing the previously saved video clip from the Photo Gallery.
- It was the new student’s job to watch and listen to the thought process and annotate mathematical thinking and strategies observed.
- The new video (original video clip plus annotations, written and oral) was saved as a new video clip and uploaded to Google Drive to be able to be embedded into a blog post
Examples of one of the final video clips
Laurel presented at the AASSA (Association of American Schools in South America) conference this past month with an elementary school colleague, Kelli Meeker, about her findings and experience of Redefining Reflection
Laurel also developed a few questions as follow up to help her students reflect on their blogfolio on the metacognition “project”
What does metacognition, thinking about your thinking, mean to you and how has it helped you in math?
Metacognition, thinking about my thinking, ……
What does your “inner voice” say to you or what questions does it ask you as you solve a problem?
I have an inner voice that …..
How has reflecting on your thinking while solving a problem helped your mathematical thinking?
Reflecting on my thinking/listening to my inner voice while doing math ….
What have you learned about yourself as a mathematician from this project and from this whole year?
This project/This year I ….
Below are a few excerpts of student responses. Click on the students’ name to see their entire blog post and embedded video.
Thinking about my thinking is reflecting in my own words. It is thinking about how your thinking can improve and what your thinking has mastered. When I am thinking about my math thinking like when I am screen casting a video on Explain Everything, my inner voice tells me to break up the problem and then read the specific part and work on that part. Afterwards, I think about if this is a good strategy or not. I think that this Explain Everything project has helped me a lot because I solved a problem and then I listened to my thinking while solving the problem
In math, Ms. J taught us to kind of talk to our “inner voice.” I only talk to my inner voice in difficult problems, I sort of ask for help. When I’m with my inner voice, I try to think differently, and usually can get a way for my answer, but I need to concentrate a lot. While I reflect on my thinking I always think in a better way. This helps because I always question myself and see if I’m really correct. I get to a more profound way of thinking.
We have been focusing on metacognition while doing math. This means thinking about our thinking, and asking our selves, “What am I doing, and why?”Using metacognition has really helped me analyze my results in math and it has also made my work a lot more error-free. Whenever I do questions now, and I am not sure how I got my answer, or if it is right, than I always think back to what I did to find out the answer, and if I could do anything better. This is also a habit of mathematical thinking that I find that I am very good at, and I use a alot.
Metacognition, thinking about thinking. When Ms.J first introduced this to us I was like, What The heck! What does she expect us to do? But now I see that it’s a useful skill that has improved not only my math skills but my other classes as well. Very early on i realized that I loved to talk. Ever since i was little i knew this. So it’s one of the reasons why sometimes I think I get bad grades in math. I hate being alone, and in fact am afraid of being alone, so not talking is a symptom. I usually struggle in silence because I like to work through my thinking aloud. Which was why I benefited from this project so much.
I think that I can apply metacognition to lots of different things, like sports that I play, like basketball. During a game, I can ask myself: “Why isn’t this working? What can I do to improve?” The next quarter, I can work on improving in those aspects to help the team win the game.
I realized while doing the project that in my head I am thinking about more than one aspect of the problem at a time, as we call it in math class, my inner voice. It was constantly checking if what I was doing made sense and figuring out other efficient and coherent ways to solve it, so if I had any difficulties or needed to revise my work I could use them. By, also, hearing my second voice I was able to understand the problem on another level, meaning I could draw the right visuals, analyze it, and do it with a different method.
When I first came here from 5th grade, I soon realized that I was not really listening to my thinking, actually not at all. I still did not know what metacognition actually meant and could not define it in first quarter. Now I can define it, and know what it is. So then, I started to think more deeply what I am doing and why I am doing this while doing these problems in my head. This has really helped me because it can not only help you to see the reasonableness of the answer but also to read more carefully.
Metacognition helped me, because, when I make a mistake in the problem, I don’t really notice it, unless someone else shows me what the mistake was, or where it was. After hearing myself in the problem, I can tell if I made a mistake. For example, if I misread the problem and didn’t notice, then heard what my thinking was, I would’ve noticed the mistake I had made. Metacognition, to me, means understanding what works, and what doesn’t work in your head.
When I would reflect my thinking on the iPad, it helped me by looking over my homework’s, my tests and etc. It would help me now and then. My inner voice would ask me “Does this answer make sense?” “How did you get this answer?” When my father would ask me “How did you do this problem?” I would say “I don’t know?” That when I realized that I need to ask myself these things. Now metacognition helps me a lot, like when I am asking my dad for some help and when I am doing a problem by myself
I have an inner voice. I think that the whole purpose of the iPad projects, was to find my math inner voice, and use it. I think I found that inner voice. I’m pretty proud of myself for that because it was with my first projects, it was pretty hard, though now, for sure I found it. It helps me wonder, and think: Should I use this chart or this chart? Which method works best?
While doing these problems, I have sort of an “inner voice.” Not in the crazy, psychopathic way, but the thinking way. I tell myself to do this or do that, or check my work. I say hundreds of things to myself in my head. And I always ask myself how I did this. I explain to myself, and try to find mistakes. Mistakes teach you that to become great at math, you need to make mistakes. Albert Einstein once said,”A person that never made a mistake never tried anything.” I know I’ve made mistakes that that inner voice saved me from.
We are having conversations, looking at student samples, tweaking how reflection and thinking about their thinking impacts student understanding and learning as well as create peer-created resources for future students (think Alan November’s thoughts about leaving a legacy).
A million thanks go to Laurel and Adam for sharing their thoughts, questions, trials, failures and success in the process and most importantly their willingness to make it transparent for others to learn with and from their process.
Do you have student samples of making mathematical thinking visible? Please share the link for all of us to learn from and have quality examples to model after.
More examples of students “writing” in Math: